a stress approach model of moderately thick, homogeneous...

Research ArticleA Stress Approach Model of Moderately ThickHomogeneous Shells

Axel Fernando Dom-nguez Alvarado and Alberto D-az D-az

Centro de Investigacion en Materiales Avanzados SC Miguel de Cervantes 120 Complejo Industrial Chihuahua31136 Chihuahua Chihuahua Mexico

Correspondence should be addressed to Alberto Dıaz Dıaz albertodiazcimavedumx

Received 12 October 2018 Accepted 6 December 2018 Published 31 December 2018

Academic Editor Francesco Tornabene

Copyright copy 2018 Axel Fernando Domınguez Alvarado and Alberto Dıaz Dıaz This is an open access article distributed under theCreative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium providedthe original work is properly cited

This paper presents the theoretical development of a new model of shells called SAM-H (Stress Approach Model of Homogeneousshells) and adapted for linear elastic shells from thin to moderately thick onesThemodel starts from an original stress polynomialapproximation which involves the generalized forces and verifies the 3D equilibrium equations and the stress boundary conditionsat the faces of the shell Hellinger-Reissner functional and Reissnerrsquos variational method are applied to determine the generalizedfields and equations The generalized forces and displacements are the same as those obtained in a classical moderately thickshell model (CS model) The equilibrium and constitutive equations have some differences from those of a CS model mainly inconsideration of applied stress vectors at the upper and lower faces of the shell and the stiffness matrices Another feature of theSAM-H model is the inclusion of the Poissonrsquos effect of out-of-plane normal stresses on in-plane strains As a first applicationexample to test the accuracy of the model the case of a pressurized hollow sphere is considered The analytical results of stressesand displacements of the SAM-H and CS models are compared to those of an exact 3D resolution In this example SAM-Hmodelproves to bemuchmore accurate than the CSmodel and its approximation of the normal out-of-plane stress is very precise Finallyan implementation of the SAM-H model equations in a finite element software is performed and a case study is analyzed to showthe advantages of using the SAM-H model

1 Introduction

In structural engineering models of plates and shells arenecessary for carrying out reliable predictions of displace-ments and stresses without the excessive computationalcost of three-dimensional solid finite element calculationsDisplacement and stress approaches are commonly used asstarting point for the construction of these models In a dis-placement approach the displacement field is approximatedby linear combinations of friendly functions of the thicknesscoordinate such as polynomials In a stress approach ananalogue approximation is made with the components of the3D stress field Additionally in this last approach it is idealthat the approximate stress field verifies the 3D equilibriumequation and stress boundary conditions [1]

In the family of linear elastic plate models the Kirchhoff-Love theory [2] is the simplest one In this theory the plate

thickness does not change during deformation and the in-plane displacements are approximated by first degree polyno-mials of the thickness coordinate while the out-of-plane dis-placement is uniform across the thicknessThe coefficients ofthese polynomials depend only on the three displacements ofthe middle surface and their derivatives (these displacementscan also be called generalized displacements) This modelneglects out-of-plane shear forces and is usually applied tothin plates A more complete plate model is Mindlinrsquos one[3] which considers the same polynomial degrees in thedisplacement approximation but includes two rotations inthe coefficients of the in-plane displacement polynomialsFive generalized displacements are then involved inMindlinrsquosmodel (three displacements and two rotations) and the linearout-of-plane strain remains zero as in Kirchhoff-Love theoryAnothermodel of plates is Reissnerrsquosmodel [4]which is basedon a stress approach the in-plane stresses are approximated

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 6141683 16 pageshttpsdoiorg10115520186141683

2 Mathematical Problems in Engineering

by first degree polynomials while the out-of-plane shear andnormal stresses are approximated by second degree and thirddegree polynomials respectively The stress approximationverifies the 3D equilibrium equation and the stress boundaryconditions at the upper and lower faces of the plate Thecoefficients of the stress polynomials are the stress resultantsalso called generalized forces Owing to the shear correctionfactor introduced by Mindlin the models of Reissner andMindlin have similar plate equations this explains the com-mon denomination ldquoReissner-Mindlinrdquo plate theory Thesemodels provide accurate stiffness predictions of moderatelythick plates and good but less precise results for thick shellsThese models have been the starting point for the analysisof composite laminated plates and the development of othermodels such as higher-order models [5] zig-zag models[6] extensions of Reissner-Mindlin-like models [7 8] andlayerwise models [9] In this last family of models Paganorsquosmodel [1] which proposes a polynomial stress approximationthat verifies the 3D equilibrium in each layer is worthciting Pagano used Reissnerrsquos variational mixed formulation[10] to build the model No displacement approximation ismade despite the use of a mixed formulation The Hellinger-Reissner functional and the stress approximation helps toidentify 2D generalized displacements More recent develop-ments of Paganorsquos approach can be found in [11ndash13] wheresome simplifications are adopted to obtain more operationallayerwise models In [13] Alvarez-Lima et al enhanceda layerwise model called M4-5N (originally developed byChabot in [11]) in order to include the Poissonrsquos effect of out-of-plane normal stresses on in-plane strains which may havean important impact on plasticity and failure mechanismsMost models of plates do not take into account this effect

Shell models can be classified depending on how the ratio120578 between the thickness to the smallest principal radius isconsidered in themodel equations thin andmoderately thickshell models neglect any term multiplied by 120578119896 for 119896 ge 1and 119896 ge 2 respectively thick shell models consider higher-degree terms The simplest thin shell models are Loversquos firstapproximation theories [2 14ndash17] which neglect transverseshear deformation and are based on a displacement approachsimilar to that in Kirchhoff-Love plate theory Other thin andmoderately thick shell models apply a displacement approx-imation (in the curvilinear coordinate system) analogue toMindlinrsquos plate model [18 19] for example in [20] Tooraniand Lakis proposed a generalization of the classical Reissner-Mindlin plate theory in order to include the curvature andtransverse shear strains of the shell These authors improvedtheir model in [21] by enhancing the kinematic fields Inthese displacement approach models the 3D constitutiveequations yield linear in-plane stresses and constant out-of-plane shear stresses across the thickness of the shellFor these models the stress field does not verify the 3Dequilibrium equations and the stress boundary conditions atthe faces of the shell [22] Moreover the normal out-of-planestresses and the linear out-of-plane strains are neglected andthis may lead to important errors in thicker shells Thesedrawbacks have motivated the development of higher-degreedisplacement approximations the use of better techniques

to recover the 3D stress and the construction of modelsbased on stress ormixed stressdisplacement approximationsIn [23 24] Tornabene and Viola used an approximationof the 3D displacement field to obtain all the generalizedequations and generalized fields for shells of revolutionAfter solving the static problem they applied the generalizeddifferential quadrature method in order to approximate thespatial derivatives of the stress field This approximationand Hookersquos law allowed recovering and correcting the 3Dstress field which took into account the boundary conditionsat the faces of the shell One of the first stress approachmodels for shells is the one developed by Trefftz [25] inthis model an approximation of in-plane stresses in termsof generalized forces is made but transverse stresses areneglected This implies that both 3D equilibrium equationsand boundary conditions at the faces of the shell are notverified In [26 27] Synge and Chien proposed a mixedstressdisplacement approach model for thin shells wherestresses and displacements are approximated by polynomialsof the thickness coordinate In their theory the boundaryconditions at the faces of the shell are not fulfilled In [28]Fang et al developed a thick shell model using a mixedstressdisplacement approach by applying Hellinger-Reissnerfunctional [10] In this mixed model the stress field verifiesthe boundary conditions at the faces of the shell but not the3D equilibrium equations Shellmodels based on a pure stressapproach are less usual than those based on a displacementapproach two noteworthy stress approach based modelsfor cylindrical and spherical thick shells can be found in[29 30] respectively A summary and comparative studyof several shell and plate theories was made by Leissa in[31 32] In Leissarsquos work all the shell models start withan approximation of the displacement field There are fewmodels based on a stress approach although these types ofmodels have advantages over those based on displacementapproximation

Many commercial finite element software packages offersurface elements to analyze shell structures These elementsare based on classical shell models A family of these surfaceelements is the one introduced in the MITC (Mixed Interpo-lation of Tensorial Components) shell concept this family isreferred to in the literature as MITC119899 where 119899 is related tothe number of nodes in the element [33] Lee and Bathe [34]state that these elements development is based on the use ofdisplacement interpolations and the selection of appropriatestrain field interpolation Finally the element rigidity matrixis built using a stress-strain constitutive equation of classicalshell models Obviously with this family of elements thestress field does not verify the 3D equilibrium equations andthe stress boundary conditions at the faces of the shell

In this paper an original model of linear elastic moder-ately thick shells (120578 le 03) called SAM-H (Stress ApproachModel of Homogeneous shells) is developed by making useof a method similar to that applied by Pagano in [1] andbased on a stress approach which in the limit case of ahomogeneous plate involves less generalized forces thanPaganorsquosmodel Reissnerrsquos variationalmixed formulation [10]is applied to obtain the generalized fields and equations Themain contribution of the paper resides on the quality of the

Mathematical Problems in Engineering 3

stress approximation and the simplicity of the model the 3Dequilibrium equations and the stress boundary conditionsat the shell faces are verified with the same number ofgeneralized forces moments strains and displacements asthe linear elastic version of the classical model of generalmoderately thick shells proposed by Reddy [19] Anotheroriginality of the model resides on taking into account anddistinguishing the effect of stress vectors applied at the innerand outer faces of the shell Finally the inclusion of thePoissonrsquos effect of out-of-plane normal stresses on in-planestrains is another feature of themodel that has not been foundon any thin or moderately thick shell model

This paper is organized in six sections In the first sectionthe adopted notations and assumptions are detailedThe sec-ond section recalls the Hellinger-Reissner variational princi-ple of elastostaticsThen in section three the proposed stressapproximation is shown The generalized displacementsstrains equilibrium equations and force edge conditionsare obtained in section four The generalized constitutiveequations and displacement edge conditions are determinedin section five In the last section the main differencesbetween the SAM-H model and a classical shell model arediscussed a comparison of analytical and numerical resultsof these two models is also shown Finally an appendixcontaining further details of equations is included

2 Notations Geometry Definitionand Assumptions

A doubly curved shell is considered Its thickness ℎ isuniform (1205851 1205852 1205853) denote the orthogonal curvilinear coor-dinateswhere 1205851 and 1205852 are the lines of curvature at themiddlesurface (1205853 = 0) The principal radii of curvature are 1198771

and 1198772 the principal curvatures are then 1205811 = 11198771 and1205812 = 11198772 The Lame coefficients are defined by1198601 = 1198861 (1 + 12058531198771

) 1198602 = 1198862 (1 + 12058531198772

) 1198603 = 1(1)

where 1198861 = radic119892211 1198862 = radic1198922

22 119892120572120573 (120572 120573 = 1 2) are thesurface metric tensor components In what follows the nextnotations are adopted

(i) Greek subscripts take the values 1 and 2(ii) Latin subscripts take the values 1 2 and 3(iii) the shell lies in the volumeΩ its middle surface is 120596(iv) bold face characters denote vectors matrices and

tensors(v) unit vectors in the directions of the lines of curvature

are e1 and e2 the unit vector normal to the middlesurface is e3 = e1 times e2 where times is the cross product

(vi) the dot product of vectors and simple contraction oftensors is noted ldquosdotrdquo while the double contraction of

tensors and the tensor product are noted ldquordquo and ldquootimesrdquorespectively

(vii) an underline and a double underline below a bold facecharacter (for exampleU andM) denote respectivelyvectors and second-order tensors of the 2D spacedefined by the basis (e1 e2)

(viii) 120581 = 1205811e1 otimes e1 +1205812e2 otimes e2 and 1205811015840 = 1205812e1 otimes e1 +1205811e2 otimes e2are curvature tensors

(ix) the stress vectors applied on the outer (1205853 = ℎ2) andinner (1205853 = minusℎ2) faces are s+ and sminus respectivelythese vectors are defined by

s+ (1205851 1205852) = 120591+ + 120590+e3and sminus (1205851 1205852) = 120591minus + 120590minuse3

where 120591+ = 120591+1 e1 + 120591+2 e2 and 120591minus = 120591minus1 e1 + 120591minus2 e2 (2)

in the above expression 120590+ and 120590minus are the appliednormal stresses while 120591+120572 and 120591minus120572 are the applied shearstresses

(x) f is the 3D body load vector and its components in thecurvilinear coordinate system are 119891119894

(xi) n0 = 11989901e1 + 11989902e2 is the outward-pointing normalvector to the edge of the middle surface 120596

(xii) 120590 and u denote the 3D stress tensor and displacementfield

(xiii) dik 120590 denotes the 3D divergence operator applied tothe second-order tensor 120590

dik 120590 = 1119860111986021198603

(k1 + k2) (3)

where vectors k1 and k2 take the array form

k1 = ((

12059711986021198603120590111205971205851 + 12059711986011198603120590121205971205852 + 1205971198601119860212059013120597120585312059711986021198603120590211205971205851 + 12059711986011198603120590221205971205852 + 1205971198601119860212059023120597120585312059711986021198603120590311205971205851 + 12059711986011198603120590321205971205852 + 12059711986011198602120590331205971205853))

(4)

and

k2 = (11986021198862 12059711988611205971205852 12059021 minus 11986011198861 12059711988621205971205851 12059022 + 1198602

11988611198771

1205903111986011198861 12059711988621205971205851 12059012 minus 11986021198862 12059711988611205971205852 12059011 + 1198601

11988621198772

12059032minus1198602

11988611198771

12059011 minus 1198601

11988621198772

12059022 ) (5)

(xiv) grad g is the 2D gradient of the 2D vector g = 1198921e1 +1198922e2 its components take the following array form

grad g

= ( 11198861 12059711989211205971205851 + 119892211988611198862 12059711988611205971205852 11198862 12059711989211205971205852 minus 119892211988611198862 1205971198862120597120585111198861 12059711989221205971205851 minus 119892111988611198862 12059711988611205971205852 11198862 12059711989221205971205852 + 119892111988611198862 12059711988621205971205851) (6)


(xv) grad 119901 = (11198861)(1205971199011205971205851)e1 + (11198862)(1205971199011205971205852)e2 is the2D gradient of the 2D scalar function 119901

(xvi) dik M is the divergence of the 2D second-order tensorM = 119872120572120573e120572 otimes e120573 (sum on 120572 and 120573) its componentshave the following array form

dik M = 111988611198862sdot (1205971198862119872111205971205851 + 1205971198861119872121205971205852 +11987221

12059711988611205971205852 minus11987222

120597119886212059712058511205971198862119872211205971205851 + 1205971198861119872221205971205852 +11987212

12059711988621205971205851 minus11987211

12059711988611205971205852) (7)

(xvii) the divergence of the 2D vector field k = V1e1 + V2e2is defined by

div k = 111988611198862 (1205971198862V11205971205851 + 1205971198861V21205971205852 ) (8)

The following assumptions are adopted

(1) small displacements and strains are assumed(2) the material is homogeneous and orthotropic and

one of its orthotropy directions is direction 3 S is the3D 4th-order compliance tensor and its componentsare 119878119894119895119896119897 while handling plates and shells it is usefulto define the following compliance tensors and con-stants

S = 2sum120572120573120574120575=1

119878120572120573120574120575e120572 otimes e120573 otimes e120574 otimes e120575Sc = 2 2sum

120572120573=1

11987812057212057333e120572 otimes e120573SQ = 4 2sum

120572120573=1

11987812057231205733e120572 otimes e120573

and 119878120590 = 1198783333(9)

(3) in all calculations the terms 1205853119877120572 and ℎ119877120572 are con-sidered but higher-order terms (1205853119877120572)119899 and (ℎ119877120572)119899(119899 ge 2) are neglected

(4) the curvatures vary smoothly along the lines ofcurvature and their derivatives with respect to 120585120572 areneglected

(5) in order to ease the generalized compliance calcu-lations the divergence of the shear stress vectorsapplied on the faces of the shell are assumed to bezero

div 120591+ = div 120591minus = 0 (10)

(6) the body load vector f is uniform through the thick-ness direction

(7) no 3D displacement constraints are applied on theinner and outer faces of the shell

Ωs

ΩsΩu

Figure 1 Example of a shell with different regions of boundaryconditions

3 Hellinger-Reissner Functional

On the boundaries 120597Ωu and 120597Ωs of Ω the body is subjectedto the displacement vector ug and stress vector sg respec-tively In Figure 1 an example of the boundaries 120597Ωu and120597Ωs is shown The Hellinger-Reissner functional for elasticproblems [10] applied to the shell is119867119877 (ulowast120590lowast) = int

Ω[120590lowast 120576 (ulowast) minus f sdot ulowast minus 119908lowast

119890 ] 119889Ωminus int120597Ωu

(120590lowast sdot n) sdot (ulowast minus ug) 119889119878minus int120597Ωs

sg sdot ulowast119889119878 (11)

where n is the 3D outward-pointing normal vector to theboundary ofΩ ulowast is a piecewise C1 first order tensor field 120590lowastis a piecewise C1 second-order symmetric tensor field 120576 is theCauchyrsquos infinitesimal strain tensor and 119908lowast

119890 = (12)120590lowast S 120590lowast is the elastic energy density In what follows a field with anuppercase lowast denotes a field that is not necessarily the solutionof the problem and varies in its corresponding space If a field(stress generalized displacement force or moment) appearswithout uppercase lowast this field is solution of the problem

In this paper as in [35] an integration by parts isperformed for the term containing 120590lowast 120576(ulowast)119867119877 (ulowast120590lowast) = minusint

Ω[dik 120590lowast sdot ulowast + f sdot ulowast + 119908119890] 119889Ω+ int

120597Ωu

(120590lowast sdot n) sdot ug119889119878+ int120597Ωs

(120590lowast sdot n minus sg) sdot ulowast119889119878 (12)

Reissnerrsquos theorem [10] states that

(i) the solution to the mechanical problem is the 3Ddisplacement fieldstress tensor field couple (u120590)that makes the HR functional stationary (notice thatno uppercase lowast appears for the solution)

(ii) the stationarity of HR with respect to ulowast yields theequilibrium equations and the boundary conditionson the stress vector while its stationarity with respectto 120590lowast yields the constitutive equations and the bound-ary conditions on the displacement field


In this paper a stress approach is proposed The intro-duction of the stress approximation in the HR functionalin (12) allows identifying the generalized displacements inthe thickness integral of the term dik 120590lowast sdot ulowast The stressapproximation will lead to 5 generalized displacements (3displacements and 2 rotations)

4 Stress Approximation

41 Polynomial Expressions Let us define the followingpolynomial basis of 5-th degree 1205853-polynomials1198750 (1205853) = 11198751 (1205853) = 1205853ℎ 1198752 (1205853) = minus6(1205853ℎ )2 + 12 1198753 (1205853) = minus2(1205853ℎ )3 + 310 (1205853ℎ ) 1198754 (1205853) = minus143 (1205853ℎ )4 + (1205853ℎ )2 minus 140

and 1198755 (1205853) = (1205853ℎ )5 minus 518 (1205853ℎ )3 + 5336 (1205853ℎ ) (13)

This polynomial basis is orthogonalintℎ2

minusℎ2119875119898 (1205853) 119875119899 (1205853) 1198891205853 = 0 if 119898 = 119899 (14)

The following stress approximations are selected120590lowast120572120573 (1205851 1205852 1205853) = 1sum119899=0

120590119899lowast120572120573 (1205851 1205852) 119875119899 (1205853) (15)

120590lowast1205723 (1205851 1205852 1205853) = 3sum119899=0

120590119899lowast1205723 (1205851 1205852) 119875119899 (1205853) (16)

120590lowast33 (1205851 1205852 1205853) = 4sum119899=0

120590119899lowast33 (1205851 1205852) 119875119899 (1205853) (17)

where 120590119899lowast119894119895 are stress coefficients which will be expressed aslinear combinations of the generalized forcesThepolynomialdegree of each stress componentwas chosen so as to verify the3D equilibrium equation as will be shown in Section 43

42 Generalized Forces Let us define the components of themembrane forces Nlowast shear forces Qlowast and moments Mlowast asfollows 119873lowast

120572120573 = intℎ2

minusℎ2120590lowast120572120573 (1 + 12058531198773minus120573

)1198891205853 (18)119876lowast120572 = intℎ2


)1198891205853 (19)119872lowast120572120573 = intℎ2


)1198891205853 (20)

where the subscript 3minus120573 is 2 if 120573 = 1 and 1 if 120573 = 2The samecan be said for the subscript 3minus120572 Let us point out thatNlowast andMlowast may be nonsymmetric while the stress tensor is alwayssymmetric Owing to assumption (3) and to the symmetry ofthe 3D stress tensor11987321 and11987221 are related to11987312 and11987212

as follows 11987321 = 11987312 + 12057512058111987212

and 11987221 = ℎ21212057512058111987312 +11987212 (21)

Here 120575120581 = 1205812 minus 1205811 By making use of assumption (3) inSection 2 and introducing the stress approximation definedin (15) and (16) one obtains the following119873lowast

120572120573 = ℎ1205900lowast120572120573 + ℎ2121198773minus120573

1205901lowast120572120573 (22)119876lowast120572 = ℎ1205900lowast1205723 + ℎ2121198773minus120572

1205901lowast1205723 (23)119872lowast120572120573 = ℎ3121198773minus120573

1205900lowast120572120573 + ℎ2121205901lowast120572120573 (24)

In the limit case of a plateNlowast andMlowast are symmetric andthus the model contains 8 generalized forces and moments 2less than Paganorsquos model [1] applied to a homogeneous plateThe generalized forces and moments of SAM-H are the sameas those in the Reissner-Mindlin plate theory

43 Determination of Coefficients in the Stress ApproximationOwing to (3) the 3D equilibrium equation yields the follow-ing

k1 + k2 + 11986011198602f = 0 (25)

Introducing (4)-(5) and (15)ndash(17) in (25) yields

(((((

2sum119902=0

1205781199021 (1205851 1205852) 119875119902 (1205853) + 4sum119902=0

1205821199021 (1205851 1205852) 119875119902 (1205853) + 1199031198911 (1205851 1205852)2sum

119902=0

1205781199022 (1205851 1205852) 119875119902 (1205853) + 4sum119902=0

1205821199022 (1205851 1205852) 119875119902 (1205853) + 1199031198912 (1205851 1205852)3sum

119902=0

1205781199023 (1205851 1205852) 119875119902 (1205853) + 5sum119902=0

1205821199023 (1205851 1205852) 119875119902 (1205853) + 1199031198913 (1205851 1205852))))))

= (000) (26)


where 119903 = 11988611198862(1 + 1205853(1205811 + 1205812)) (the terms in (1205811205721205853)2 areneglected owing to assumption (3) in Section 2) and 120578119902119894 arenonnegligible linear combinations of the stress coefficients120590119899lowast119894119895 and their derivatives with respect to 1205851 and 1205852 (none ofthe terms in 120578119902119894 are multiples of (120581120572ℎ)2) The terms 120582119902119894 arealso a linear combination of the stress coefficients but theyare neglected because they aremultiples of (120581120572ℎ)2 Bymakinguse of the linear independence of polynomials 119875119902 the valuesof the stress coefficients are selected so as to obtain

(i) 1205782120572 = 0 this is ensured by selecting 1205903lowast1205723 judiciouslyin this manner as mentioned at the end of Section 3the terms 1205780120572 and 1205781120572 will provide two generalized equi-librium equations and two generalized displacementsfor each 120572 value

(ii) 12057813 = minus11988611198862ℎ(1205811 + 1205812)1198913 (in order to obtain 120578131198751 +1199031198913 = 0) 12057823 = 0 and 12057833 = 0 this is ensuredby a suitable selection of the values of 1205902lowast33 1205903lowast33 and1205904lowast33 in this manner the term in 12057803 will yield onemore generalized equilibrium equation and anothergeneralized displacement

Owing to (22)ndash(24) assumptions (3) and (4) in Section 2and the definition of the stress vectors applied at the innerand outer faces the obtained stress coefficients 120590119899lowast

119894119895 areexpressed in terms of the generalized forces and applied shearand normal stresses at the faces of the shell In (A1)ndash(A3)in Appendix A B and C the expression of these stresscoefficients is shown Let us point out that with these stresscoefficients the 3D boundary conditions at the faces of theshell are verified forall (1205851 1205852) isin 120596 120591+120572 (1205851 1205852) = 1205901205723 (1205851 1205852 ℎ2) 120591minus120572 (1205851 1205852) = minus1205901205723 (1205851 1205852 minusℎ2) 120590+ (1205851 1205852) = 12059033 (1205851 1205852 ℎ2)

and 120590minus (1205851 1205852) = minus12059033 (1205851 1205852 minusℎ2) (27)

The obtained stress field in a classical model of moderatelythick shells (such as that based on a displacement approachand presented by Reddy in [19]) does not verify theseconditions

5 Generalized Fields and Equations

51 Generalized Displacements Equilibrium EquationsStrains and Stress Boundary Conditions The application ofthe stress approximation in the Hellinger-Reissner functionalin (12) makes the term int

Ω(dik 120590lowast sdot ulowast + f sdot ulowast)119889Ω becomeint

120596(dik Nlowast + 120581 sdotQlowast + ℎ2 (1205811 + 1205812) (120591+ minus 120591minus) + 120591+

+ 120591minus) sdot Ulowast119889120596 + int120596(divQlowast + 120590+ minus 120590minus2 ℎ (1205811 + 1205812)+ 120590+ + 120590minus minus 119873lowast

111205811 minus 119873lowast221205812)119880lowast

3 119889120596 + int120596(dik Mlowast

minusQlowast + ℎ24 (1205811 + 1205812) (120591+ + 120591minus) + ℎ2 (120591+ minus 120591minus))sdotΦlowast119889120596 + int

120596(ℎf sdot Ulowast + ℎ1198913119880lowast

3 + ℎ3 (1205811 + 1205812)12 f

sdotΦlowast)119889120596(28)

where f = 1198911e1 + 1198912e2119880lowast3 = intℎ2

minusℎ2

1198750ℎ 119906lowast31198891205853 (29)

Ulowast andΦlowast are 2D vectors whose components are defined onthe basis of lines of curvature as follows

Ulowast = (119880lowast1119880lowast2

) 119880lowast1 = intℎ2

minusℎ2

1198750ℎ 119906lowast11198891205853119880lowast2 = intℎ2

minusℎ2

1198750ℎ 119906lowast21198891205853Φ

lowast = (120601lowast1120601lowast2) 120601lowast1 = intℎ2

minusℎ2

121198751ℎ2 119906lowast11198891205853120601lowast2 = intℎ2

minusℎ2

121198751ℎ2 119906lowast21198891205853

(30)

Expression (28) allows identifying the following 5 gener-alized displacements 119880lowast

1 119880lowast2 119880lowast

3 120601lowast1 and 120601lowast2 (the two lattercan be regarded as rotations) Let us point out that in the stressapproach the approximation of the 3D displacement field u isnot required Nevertheless one can state that its componentstake the following array formu

= (1198801 (1205851 1205852) 1198750 (1205853) + ℎ1206011 (1205851 1205852) 1198751 (1205853) + 1205751199061 (1205851 1205852 1205853)1198802 (1205851 1205852) 1198750 (1205853) + ℎ1206012 (1205851 1205852) 1198751 (1205853) + 1205751199062 (1205851 1205852 1205853)1198803 (1205851 1205852) 1198750 (1205853) + 1205751199063 (1205851 1205852 1205853) ) (31)

where 1205751199061 1205751199062 and 1205751199063 verifyintℎ2

minusℎ21198751198941205751199061205721198891205853 = 0 for 119894 ge 2

and intℎ2

minusℎ211987511989512057511990631198891205853 = 0 for 119895 ge 1 (32)


The stationarity of the HR functional (in (12)) withrespect to the generalized displacements leads to the follow-ing generalized equilibrium equations

dik N + 120581 sdotQ + ℎ2 (1205811 + 1205812) (120591+ minus 120591minus) + 120591+ + 120591minus+ ℎf = 0 (33)

divQ + 120590+ minus 120590minus2 ℎ (1205811 + 1205812) + 120590+ + 120590minus minus 119873111205811minus 119873221205812 + ℎ1198913 = 0 (34)

dik M minusQ + ℎ24 (1205811 + 1205812) (120591+ + 120591minus) + ℎ2 (120591+ minus 120591minus)+ ℎ3 (1205811 + 1205812)12 f = 0 (35)

Let us point out that in other shell models such as thatin [19] a supplementary algebraic equilibrium equation isconsidered 119872121205812 minus119872211205811 = 11987321 minus 11987312 (36)

In SAM-H model this equation is not considered as anequilibrium equation Equations (21)will be considered in theconstitutive equations (C4) of SAM-H model and with thisconsideration it can be easily verified that condition (36) isautomatically satisfied by making use of assumption (3)

The aforementioned stationarity for the boundary inte-gral int

120597Ωs


in the HR functional provides the boundary conditions onthe generalized forces Let us point out that the term 120590lowast sdotnminussgis zero on the inner and outer faces of the shell due to (27)Thus the boundary integral reduces to an integration on thelateral boundaries (not the faces) of the structure At theselateral boundaries the following assumptions are adoptedfor the stress vector sg the first and second components ofthis vector (1199041198921 and 1199041198922 ) are first degree 1205853-polynomials whilethe third component (1199041198923 ) is uniform through the thicknessof the shell Owing to these assumptions and to assumption(3) in Section 2 the aforementioned stationarity yields theboundary conditions on the generalized forces

N sdot n0 = F119892Q sdot n0 = 119865119892

3 M sdot n0 = C119892 (38)

Here F119892 1198651198923 and C119892 are the given membrane force vector

shear force and bending moment vector at the shell edgesrespectively Their components are119865119892

120572 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 1199041198921205721198891205853

1198651198923 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 11990411989231198891205853

and 119862119892120572 = intℎ2

minusℎ2

ℎ212 ((11989901)2 1205811 + (11989902)2 1205812)1198750 (1205853) 1199041198921205721198891205853+ intℎ2

minusℎ2ℎ1198751 (1205853) [1 minus ℎ6 ((11989901)2 1205811 + (11989902)2 1205812)]sdot 1199041198921205721198891205853

(39)

Once the generalized displacements are identified bymeans of an integration by parts the term minusint

Ω(dik 120590lowast sdot ulowast +

f sdot ulowast)119889Ω in the HR functional becomesint120596(Nlowast 120576lowast119905 +Qlowast sdot dlowast +Mlowast 120594lowast119905)119889120596 + 119879 (40)

where the superscript 119905 denotes the transposition of a tensor119879 is a term involving the generalized forces anddisplacementsevaluated at the edges of the middle surface 120596 120576lowast 120594lowast and dlowastare the generalized strain tensors defined as follows

120576lowast = gradUlowast + 119880lowast

3 120581120594lowast = gradΦlowast

dlowast = grad119880lowast3 +Φlowast minus 120581 sdot Ulowast (41)

52 Generalized Constitutive Equations and DisplacementBoundary Conditions Let us now introduce the stressapproximation in the expression of the volume elastic energydensity 119908lowast

119890 in (12) The integral of this energy can be writtenas followsint

Ω119908lowast119890 119889Ω = int

120596(119908119904lowast

119890 + 119908119899lowast119890 + 119908119876lowast

119890 + 119908119888lowast119890 ) 119889120596 (42)

where 119908119904lowast119890 119908119899lowast

119890 119908119876lowast119890 and 119908119888

119890 are the surface densities ofelastic energy due to in-plane stresses 120590120572120573 elastic energydue to normal out-of-plane stresses 12059033 elastic energy dueto out-of-plane shear stresses 1205901205723 and elastic energy due tocoupling between in-plane stresses and normal out-of-planestresses respectively Let us point out that inmoderately thickplate models and shell models 119908119888

119890 is neglected this energyaccounts for the 3D Poissonrsquos effect of normal stresses 12059033on in-plane strains Owing to the definitions of compliancetensors and constants in (9) these surface densities of energyare defined by

119908119904lowast119890 = 12 intℎ2

minusℎ2120590lowast S 120590lowast (1 + 12058531205811) (1 + 12058531205812) 1198891205853

119908119899lowast119890 = 12 intℎ2

minusℎ2120590lowast33119878120590120590lowast33 (1 + 12058531205811) (1 + 12058531205812) 1198891205853

8 Mathematical Problems in Engineering119908119876lowast119890= 12 2sum

120572=1

2sum120573=1

intℎ2


and 119908119888lowast119890= 12 intℎ2

minusℎ2(120590lowast Sc) 120590lowast33 (1 + 12058531205811) (1 + 12058531205812) 1198891205853

(43)

Let us express the surface densities of energy as algebraicfunctions of the generalized forces the third component ofthe body load (1198913) and the components of the stress vectorsapplied at the shell faces (s+ and sminus) Let us first apply inthe equations above the stress approximation detailed in(15)ndash(17) and its corresponding stress coefficients determinedin (A1)ndash(A3) in Appendix A B and C Some terms in thesestress coefficients yield directly the desired algebraic combi-nations and the others involve the following divergences

(i) divQlowast ℎ div (1205811015840 sdotQlowast) ℎ div (120581 sdotQlowast)(ii) dik (Mlowast sdot 1205811015840) ℎ div [dik (Mlowast sdot 1205811015840)](iii) div 120591+ div 120591minus div (1205811015840 sdot120591+) div (120581 sdot120591+) div (1205811015840 sdot120591minus) and

div (120581 sdot 120591minus)These divergences can be expressed as linear algebraic func-tions of the generalized forces and 1198913 In this manner oneobtains that119908lowast

119890 is a quadratic function of the algebraic valuesof the components of f s+ sminus and the generalized forces

By making use of the stationarity of the HR functionalwith respect to the generalized forces one obtains thegeneralized constitutive equations in a tensor form It is easierto write the generalized constitutive equations in terms ofthe following engineering vectors of generalized forces q andstrains 120598

q = (N

M)

120598 = ( ) (44)

where

N = (11987311119873221198731211987321

)M = (11987211119872221198721211987221

)

= (12057611120576221205761212057621)and = (12059411120594221205941212059421)

(45)

In this manner the constitutive equations can be groupedin two vector equations

120598 = Cq + 120590+c+ + 120590minuscminus + 1198913c3and d = DQ +D+

120591+ +Dminus120591minus (46)

where

(i) C is an 8 times 8 compliance matrix(ii) c+ cminus and c3 are 8-component compliance vector(iii) DD+ andDminus are 2 times 2 compliance matrices

The components of these matrices and vectors are providedin Appendix B

Appendix C proves that there is an alternative form ofthe constitutive equations which expresses the generalizedforces as linear combinations of the generalized strains andthe applied loads

The constitutive equations can be grouped in the follow-ing two vector equations

q = K120598 + 120590+k+ + 120590minuskminus + 1198913k3Q = Ld + L+120591+ + Lminus120591minus (47)

The expressions of the above matrices (K L L+ and Lminus)and vectors (k+ kminus and k3) are shown in Appendix CThe resulting 11987321 and 11987221 generalized forces obtained bymeans of (47) verify (21) and the supplementary algebraicldquoequilibriumrdquo (36) included in Sanders model [14]

The stationarity of the HR functional (in (11)) withrespect to the generalized forces also provides in theminusint

120597Ωu(120590lowast sdotn) sdot (ulowast minusug)119889119878 term the generalized displacement

boundary conditions on the edge of the shell

U = intℎ2

minusℎ2

1198750ℎ (1199061198921e1 + 1199061198922e2) 11988912058531198803 = intℎ2

minusℎ2

1198750ℎ 11990611989231198891205853Φ = intℎ2

minusℎ2

121198751ℎ2 (1199061198921e1 + 1199061198922e2) 1198891205853(48)


6 Results and Discussion

61 Main Differences with a Classical Model of Shells In [19]Reddy builds a model of homogeneous general shells (fromthin tomoderately thick) which generalizes Sanders shell the-ory [14]The generalization consists of including vonKarmannonlinear strains In order to compare the linear SAM-Hmodel to Reddyrsquos model the nonlinear terms are not con-sidered herein Hereafter the linear version of Reddyrsquos modelwill be referred to as the classical shell model (CSmodel)TheCS model is based on an approximation of the 3D displace-ment field components in the curvilinear coordinate system119906120572 (1205851 1205852 1205853) = 119880119888

120572 (1205851 1205852) + ℎ1206011198881205721198751 (1205853)1199063 (1205851 1205852 1205853) = 1198801198883 (1205851 1205852) (49)

which involves 5 generalized displacements (1198801198881 119880119888

2 1198801198883 12060111988811206011198882)The first 3 generalized displacements are the 3D displace-

ments at the middle surfaceThe 5 generalized displacementscan be expressed as thickness integrals of the 3D displace-ments in a similar manner to that in (29) and (30) In thissense the generalized displacements of the SAM-Hmodel areequivalent to those of the classical shell model (CS model)SAM-H model does not propose an approximation of the3D displacements but it has the same number of generalizeddisplacements as the CS model From SAM-H generalizeddisplacements one may obtain some characteristics of the3D displacement field By applying the 3D gradient to the 3Ddisplacement field in (49) one obtains the 3D strains of theCS model Let us point out that the 3D linear normal straincomponent 12057633 is zero These 3D strains can be expressed aslinear algebraic combinations of the generalized strains ofthe CS model The generalized strains are related to the 2Dgradients of the generalized displacements of the CS modelin a similar manner as in the SAM-H model (see (41))

120576119888 = grad Uc + 119880119888

3120581120594119888 = grad Φ119888

d119888 = grad 1198801198883 +Φ119888 minus 120581 sdot Uc (50)

Let us point out that the SAM-H model does not proposeany approximation of the 3D strain tensor By applyingHamiltonrsquos variational principle to the shell Reddy obtainsthe generalized equilibrium equations of the CS model [19]

dik Nc + 120581 sdotQc + feq = 0

divQc minus 119873119888111205811 minus 119873119888

221205812 + 1198911198901199023 = 0

dik Mc minusQc = 0

(51)

where Nc Qc and Mc are the generalized forces of the CSmodel and have a definition similar to those of the SAM-Hmodel in (18) to (20) feq and 119891119890119902

3 are the equivalent middlesurface in-plane and out-of-plane loads respectively Theadditional equilibrium equation (36) in Sanders model [14]is also considered

The equivalent middle surface loads can model the effectof the body loads and the loads applied at the faces of theshell Let us point out that in SAM-H model the body loadsand the applied stresses on the faces appear explicitly in itsequations Besides in SAM-H model in some terms of theequilibrium equations these loads and stresses aremultipliedby the principal curvatures and the loads applied on the innerand outer faces of the shell are distinguished

In the CS model after obtaining the approximate 3Dstrains the 3D constitutive equations for a homogeneous shellyield the stress field approximation The CS model neglectsout-of-plane normal stresses 12059033 An integration through thethickness yields the generalized constitutive equations

(Nc

Mc) = (Ac Bc

Bc Dc) sdot (120576119888120594119888)

and Qc = Kc sdot dc (52)

where Nc Mc 120576119888 and 120594119888 have definitions analogue to thosein (45) Ac BcDc and Kc are stiffness matrices Let us pointout that in the CS model an ldquoABBDrdquo stiffness matrix appearswhile an ldquoABCDrdquo stiffness matrix is expected in the SAM-H model (see the expression of the generalized compliancematrices in (B1)) Another difference is the fact that in theCS model the constitutive equations do not involve the bodyloads and the loads applied at the faces of the shell in SAM-H model these loads appear explicitly in the constitutiveequations The Poissonrsquos effects of the out-of-plane normalstresses on in-plane strains are considered in the SAM-Hmodel but not in the CS model

62 Analytical Results Let us now compare the analyticalsolutions of the CS and SAM-Hmodels in the case of a hollowsphere subjected to an internal pressure 119901 The thickness ofthe sphere is ℎ the radius of the middle surface is 119877 Thematerial is isotropic and its Youngrsquos modulus and Poissonrsquosratio are respectively 119864 and ] Vector e3 points from insideto outside the sphere The analytical resolution of the 3Dproblem with spherical symmetry leads to the followingradial displacement (1199063) and normal stress 12059033

11990631198633 (1205853) = 119901119877119864 [[[(1 minus 1205782)3 (2 (1 minus 2]) (1 + 1205781205853)3 + (1 + ]) (1 + 1205782)3)2 ((1 + 1205782)3 minus (1 minus 1205782)3) (1 + 1205781205853)2 ]]]


and 120590311986333 (1205853) = 119901 (1 minus 1205782)3(1 + 1205782)3 minus (1 minus 1205782)3 (1 minus (1 + 12057821 + 1205781205853)3) (53)

where 120578 = ℎ119877 is the thickness to radius ratio and 1205853 =1205853ℎ is the normalized position through the thickness Theanalytical resolution of the equations of the CS model yieldsthe generalized displacement and the normal stress119880119862119878

3 = 119901119877119864 [1 minus ]2120578 ]and 12059011986211987833 = 0 (54)

The generalized displacement and normal stress of the SAM-H model are119880119878119860119872

3 = 1198801198621198783 minus 119901119877119864 (1 minus 2]2 + 3140120578 + 13]30 120578 + ]101205782)

and 12059011987811986011987233 (1205853)= 119901(minus1 minus 1205782 + 1205853 minus 212058523120578 + 12057828 minus 1205782120585232 ) (55)

Let us first compare the displacement solutions of the CSand SAM-H models with the mean 3D displacement1198803119863

3 = intℎ2

minusℎ2

1198750ℎ 11990631198633 (1205853ℎ )1198891205853= 119901119877119864 [[2 (1 minus 1205782)3 (1 minus 2]) + (1 minus 1205782)2 (1 + 1205782)2 (1 + ])2 ((1 + 1205782)3 minus (1 minus 1205782)3) ]] (56)

Let us define the relative errors1205751198801198781198601198723 = 119880119878119860119872

3 minus 119880311986331198803119863

3

and 1205751198801198621198783 = 119880119862119878

3 minus 119880311986331198803119863

3

(57)

in the evaluation of the mean displacement for the CS andSAM-H models These errors only depend on 120578 and ] InFigures 2(a) and 2(b) the relative errors 120575119880119878119860119872

3 and 1205751198801198621198783 are

plotted against the thickness to radius ratio 120578 for five values ofthe Poissonrsquos ratio (-04 -02 0 02 and 04) respectivelyThemaximum value of the thickness to radius ratio considered inthis example is 05 formoderately thick shells this ratio is lessthan 03 (this limit will be justified below)The absolute valueof the relative error 120575119880119878119860119872

3 of the SAM-H model increasesas 120578 and ] increase but is less than 4 for moderately thickshells and less than 14 for thicker shells The relative error120575119880119862119878

3 of the CSmodel increase as 120578 increases and ] decreases

itmay bemore than 50 formoderately thick shells andmorethan 100 for thicker shells The SAM-H model provides afar better approximation than the CS model for the meannormal displacement Let us point out that SAM-H accuracyin the prediction of 1198803 is mainly due to the consideration ofthe elastic energy119908119888

119890 (see (43)) corresponding to the couplingbetween in-plane stresses and normal out-of-plane stressesIn Figure 3 the relative error 120575119880119878119860119872

3 in the calculation of1198803 without taking into account 119908119888119890 is plotted against 120578 for

five values of the Poissonrsquos ratio (-04 -02 0 02 and 04)Neglecting 119908119888

119890 can make this error greater than 20 for thethickest moderately thick shell (120578 = 03) this value is 5 timesgreater than that considering 119908119888

119890 This confirms the utilityof taking into account 119908119888

119890 (the Poissonrsquos effect of the out-of-plane normal stresses on in-plane strains) In what followsthis energy is taken into account in the SAM-Hmodel results

Let us now analyze the normal stresses 12059033 In the CSmodel these stresses are zero The SAM-H model predictsnonzero 12059033 stresses and in order to compare the SAM-Hmodel results to those of the 3D case the normalized stresses12059033119901 are analyzed In Figure 4 the normalized stressescalculated by the SAM-Hmodel and the 3Dmodel are plottedagainst the dimensionless position 1205853 for three values of thethickness to radius ratio 120578 = 01 03 and 05 One canobserve that the accuracy of the SAM-H results decreasesas 120578 increases In spite of this the SAM-H results are veryaccurate even for the thickest case (120578 = 05) and the boundaryconditions at the inner (1205853 = minus05) and outer (1205853 = 05) facesof the shell are verified

63 Numerical Results In this subsection SAM-H modelequations are solved by means of the finite element methodimplemented in COMSOL Multiphysics software (53 ver-sion) To test the accuracy of SAM-H model its results arecompared to those obtained with

(i) solid finite elements (SFE) available in COMSOLMultiphysics

(ii) the classical shell model (CS model) its equations areimplemented in COMSOL Multiphysics in a similarmanner to how SAM-H equations were treated

(iii) MITC6 shell finite elements [33] (triangular ele-ments) available in COMSOL Multiphysics

Let us consider a 2m tall hollow cylinder The radiusof the midsurface and the wall thickness are 1m and 03mrespectively A 1MPa pressure is applied on a 025m tallcircumferential section at the outer face of the cylinder Theresolution of the problem using solid finite elements (SFE)can be simplified due to symmetries as shown in Figure 5(a)The shell-type computations (SAM-H CS and MITC6) use


0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus0160 01 02 03 04 05

U

SAM

3

] = 0

] = minus04

] = minus02

] = 02

] = 04

(a)

] = 0

] = minus04

] = minus02

] = 02

] = 04

0 01 02 03 04 05

12

1

08

06

04

02

0

UCS 3

(b)

Figure 2 Relative displacement errors vs thickness to radius ratio for the SAM-H (a) and CS (b) models

01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

minus04

U

SAM

3

0 01 02 03 04 05

] = 0

] = minus04

] = minus02

] = 02

] = 04

Figure 3 Relative displacement error vs thickness to radius ratio for the SAM-H without taking into account 119908119888119890


= 01

= 05

3D

= 03

SAM-H

0

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08

minus09

minus1

33p

minus06 minus04 minus02 0 02 04 063

Figure 4 Normalized 3D and SAM-H normal stresses 12059033119901 vsdimensionless position 1205853the mesh shown in Figure 5(b) The material is isotropic andits Youngrsquos modulus and Poissonrsquos ratio are 200GPa and 03

The resolution of the generalized equations of SAM-Hand CS models allows building the approximate 3D stressfield and plotting the stress components on the cross sectionof the cylinder drawn in Figure 5 COMSOL Multiphysicsoffers the possibility of obtaining from theMITC6 results the3D stress field in a given position through the thickness of thesolid In Figure 6 the normal stresses 120590119903119903 calculated by solidfinite elements (SFE) the shell models (SAM-H and CS) andMITC6 shell elements are plotted on the cross section The120590119903119903 stresses calculated by the CS model and MITC6 methodare neglectedwhereas that calculated by SAM-Hvaries acrossthe thickness in a similar manner to the SFE stress Thepressure application on a delimited portion of the outer faceprovokes the existence of a 120590119903119911 shear stress In Figure 7 thecolor maps of the calculated 120590119903119911 stresses by SFE SAM-HCS and MITC6 methods are plotted on the cross sectionof the cylinder The CS model predicts a constant throughthickness shear stresswhereas SAM-Hmodel yields a variableshear stress that verifies the boundary conditions at the innerand outer faces in a similar manner to the MITC6 methodThe values of the 120590119903119911 stress calculated with SAM-H modelseem more to the SFE results than those calculated withthe CS model In Figure 8 the von Mises stresses obtainedby SFE SAM-H CS and MITC6 are plotted on the crosssection of the cylinder Once again one can note that SAM-H results are more accurate than those of the CS modeland MITC6 method The maximum von Mises stress at theinner face of the cylinder calculated by SFE SAM-H CSand MITC6 methods is 215MPa 210MPa 191 MPa and163MPa respectively In Figure 8 it can be seen that thelocation of the maximum value of the von Mises stress is atthe inner face of the shell for SFE and SAM-H models while

that for CS model is at the outer face of the shell SAM-H provides a better stress approximation than the CS andMITC6 methods for this mechanical problem

7 Conclusions

To conclude a new model of linear homogeneous shells(from thin tomoderately thick) called SAM-Hwas developedin this paper Hellinger-Reissnerrsquos functional was applied tobuild the model equations and fields The model is basedon a polynomial approximation of the stress field across theshell thickness The stress field verifies the 3D equilibriumequation and the coefficients of the stress polynomials are thegeneralized forces (membrane and shear forces) and bendingmoments of the model The 3D stress boundary conditionsat the faces of the shell are also verified Hellinger-Reissnerrsquosfunctional allowed identifying the generalized displacements(3 displacements and 2 rotations) and strainsThe stationarityof this functional with respect to the generalized displace-ments yielded 5 generalized equilibrium equations and thegeneralized boundary conditions on the forces at the edgesof the shell The stationarity with respect to the generalizedforces provided the generalized constitutive equations anddisplacement boundary conditions at the shellrsquos edges Ascompared to the classical linear model (CS model) of generalmoderately thick shells adapted from Reddyrsquos model in [19]SAM-H model involves the same number of generalizeddisplacements strains and forces The stress approximationof the CS model does not verify the 3D equilibrium equationand the 3D stress boundary conditions at the faces of theshell SAM-H model involves explicitly in its equilibriumand constitutive equations the body loads and the appliedstress vector at the shell faces while the CS model involvesonly an equivalent middle surface load in the equilibriumequations The constitutive equations of the SAM-H modelare more accurate since they involve Poissonrsquos effects of out-of-plane loads on in-plane strains To prove the accuracy ofthe SAM-H model the case of a pressurized hollow spherewas considered and the analytical results of the SAM-H andCS models for displacements and 12059033 stress were comparedto a 3D solid model solution The SAM-H proved to be veryaccurate for moderately thick shells having a thickness toradius ratio less than 03 Its results are by far more accuratethan theCSmodel SAM-Hmodel equationswere also imple-mented in a finite element commercial software and the caseof the constriction of a cylinder was analyzedThe numericalresults were compared to those of solid finite elements (SFE)the classical shell model (CS model) and MITC6 shell finiteelements SAM-H proved to be very accurate it gives a betterapproximation of SFE results than the other shell models

From a resolution point of view the SAM-H modelproposed in this paper is not more complex than a CS model(both have the same number of generalized displacements)but is more complete since it is able to calculate the normalout-of-plane stresses which should not be neglected in thefailure prediction of shells The higher polynomial degreeof out-of plane stresses does not complicate the resolutionof the SAM-H model equations the calculation of thesestresses is performed in a postprocessing stage Another


012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)

Figure 5 Constriction of a thick hollow cylinder modelling using solid finite elements (a) and mesh for shell models (b)

SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1

Figure 6 Normal stress 120590119903119903 (in MPa) in the cross section of aconstricted cylinder (SFE SAM-H CS and MITC6 results)

r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005

Figure 7 Shear stress 120590119903119911 (in MPa) in the cross section of aconstricted cylinder (SFE SAM-H CS and MITC6 results)

important advantage of the SAM-Hmodel is that while beingapplied to plates it is equivalent to the extended Reissnerrsquosplate model in [13] (adapted to a homogeneous plate) thatincludes the above-mentioned Poissonrsquos effects For thesereasons the SAM-H model can be applied by engineers andimplemented in commercial finite element software to obtainmore accurate predictions of the structural behavior and

r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402

Figure 8 Von Mises stress (in MPa) in the cross section of aconstricted cylinder (SFE SAM-H CS and MITC6 results)

strength of linear homogeneous shells having a moderatethickness (120578 le 03)

In subsequent papers the static model developed hereinwill be extended so as to include dynamic effects and calculatevibration modes and eigenfrequencies of shells Also thesame stress approachwill be applied to develop newmodels oflaminated composite shells The consideration of geometricnonlinearities is an important topic that will also be treatedin a future work

Appendix

A Coefficients of the PolynomialApproximation of Stresses

In this subsection the stress coefficients in the polyno-mial approximations in (15)ndash(17) are detailed The followingexpressions are obtained

(i) for 120590119899120572120573 (0 le 119899 le 1)1205900120572120573 = 1ℎ [(N minusM sdot 1205811015840) sdot e120573] sdot e120572and 1205901120572120573 = [(12ℎ2M minusN sdot 1205811015840) sdot e120573] sdot e120572 (A1)


(ii) for 1205901198991205723 (0 le 119899 le 3)12059001205723 = [1ℎQ minus ℎ121205811015840 sdot (120591+ + 120591minus)] sdot e12057212059011205723 = (120591+ + 120591minus) sdot e120572 + 151205903120572312059021205723 = [1ℎQ minus 12 (120591+ minus 120591minus) minus ℎ121205811015840 sdot (120591+ + 120591minus)] sdot e120572and 12059031205723 = [2dik (M sdot 1205811015840) minus (4120581 + 31205811015840)sdot (Q minus ℎ2 (120591+ minus 120591minus))] sdot e120572

(A2)

(iii) for 12059011989933 (0 le 119899 le 4)120590033 = 120590+ minus 120590minus2 + 120590233 + 115120590433120590133 = 120590+ + 120590minus + 15120590333120590233= ℎ30div [dik (M sdot 1205811015840) minus 2120581 sdotQ] minus 1ℎM 120581+ ℎ12div [120591+ + 120591minus + ℎ10 (4120581 + 51205811015840) sdot (120591+ minus 120591minus)]+ ℎ12 [(1205811 + 1205812) (2120590133 minus 35120590333 + ℎ1198913) minus 45ℎ120590433] 120590333 = minusdivQ + ℎ4div [2 (120591+ minus 120591minus) minus ℎ120581 sdot (120591+ + 120591minus)]and 120590433= minus3ℎ14div [dik (M sdot 1205811015840)] + 3ℎ7 div [(2120581 + 1205811015840) sdotQ]minus 3ℎ214 div [(2120581 + 1205811015840) sdot (120591+ minus 120591minus)]

(A3)

B Generalized Compliances

In this section the components of the compliance matricesand vectors appearing in (46) are determined Matrix C isdefined by

C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)

CB1 = CC1 = 1205811 minus 1205812ℎ (1198781111 0 0 11987811210 minus1198782222 minus1198782212 00 minus1198781222 minus1198781212 01198782111 0 0 1198782121)CB2 = minus 110ℎ

sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)

Vectors c+ and cminus are defined by

c+ = minus 119878120590140 (3ℎca70ca)+ 160 ( CS (15cb + 8ℎcc + 5ℎcd)1ℎCS (36cb + 18ℎcc + 3ℎcd)) (B3)

and

cminus = 119878120590140 (minus3ℎca70ca )+ 160 ( CS (minus15cb + 8ℎcc + 5ℎcd)1ℎCS (36cb minus 18ℎcc minus 3ℎcd)) (B4)

where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)

The compliance vector c3 is

c3 = minusℎ2119878120590210 ( ca0 sdot ca) + 1120 (ℎ2CS (6cc + 5cd)12CScb) (B7)

The components of the shearing compliance matrices DD+ andDminus are given by

D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)

D+ = minus 160sdot ( 11987811987611 (6 + 5ℎ1205812) 11987811987612 (6 + 6ℎ1205811 minus ℎ1205812)11987811987621 (6 minus ℎ1205811 + 6ℎ1205812) 11987811987622 (6 + 5ℎ1205811) ) (B9)

and

Dminus = 160sdot ( 11987811987611 (6 minus 5ℎ1205812) 11987811987612 (6 minus 6ℎ1205811 + ℎ1205812)11987811987621 (6 + ℎ1205811 minus 6ℎ1205812) 11987811987622 (6 minus 5ℎ1205811) ) (B10)

C Alternative Form of the GeneralizedConstitutive Equations

In this section a method to obtain the alternative form (47)of the two generalized constitutive equations (46) is shown

The constitutive equation

d = DQ +D++ +Dminusminus (C1)

is transformed directly by multiplying it by the shearingstiffness matrix L (it is the inverse of the shearing compliancematrixD) and one obtains

Q = Ld + L++ + Lminusminus (C2)

where L+ = minusLD+ and Lminus = minusLDminusFor the other constitutive equation

120598 = Cq + 120590+c+ + 120590minuscminus + 1198913c3 (C3)

the task is more complicated because the compliance matrixC is not invertible Its fourth and eighth columns are linear

combinations of the third and seventh columns This is dueto the fact that 11987321 and 11987221 are linearly dependent of 11987312

and11987212 as shown in (21) Matrix operations allow obtainingthe alternative form of the generalized constitutive equation

q = K120598 + 120590+k+ + 120590minuskminus + 1198913k3 (C4)

where the stiffness matrix K and vectors k+ kminus and k3 aredefined by

K = H2 (H1CH2)minus1 H1

k+ = minusKc+kminus = minusKcminus

and k3 = minusKc3(C5)

In the previous definition the components of matrices H1

andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)

One can verify that with these matrices (21) and the sup-plementary ldquoequilibriumrdquo equation (36) included in Sandersmodel [14] are satisfied

Data Availability

The equations used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] N J Pagano ldquoStress fields in composite laminatesrdquo InternationalJournal of Solids and Structures vol 14 no 5 pp 385ndash400 1978


[2] A E H Love ldquoPhilosophical transactions of the royal societyof londonrdquo Mathematical Physical amp Engineering Sciences vol179 Article ID 491546 pp 491ndash546 1888

[3] R D Mindlin ldquoInfluence of rotatory inertia and shear onflexural motions of isotropic elastic platesrdquo Journal of AppliedMechanics vol 18 pp 31ndash36 1951

[4] E Reissner ldquoThe effect of transverse shear deformation on thebending of elastic platesrdquo Journal of Applied Mechanics pp 69ndash77 1945

[5] J N Reddy ldquoA simple higher-order theory for laminatedcomposite platesrdquo Journal of Applied Mechanics vol 51 no 4pp 745ndash752 1984

[6] E Carrera ldquoHistorical review of Zig-Zag theories for multilay-ered plates and shellsrdquo Applied Mechanics Reviews vol 56 no3 pp 287ndash308 2003

[7] A Lebee and K Sab ldquoA Bending-Gradient model for thickplates Part I Theoryrdquo International Journal of Solids andStructures vol 48 no 20 pp 2878ndash2888 2011

[8] A Lebee and K Sab ldquoOn the generalization of Reissner platetheory to laminated plates Part I Theoryrdquo Journal of ElasticityThe Physical and Mathematical Science of Solids vol 126 no 1pp 39ndash66 2017

[9] E Carrera ldquoTheories and finite elements for multilayeredanisotropic composite plates and shellsrdquo Archives of Computa-tional Methods in Engineering State-of-the-Art Reviews vol 9no 2 pp 87ndash140 2002

[10] E Reissner ldquoFree boundaries and jets in the theory of cavita-tionrdquo Journal of Mathematics and Physics vol 29 pp 90ndash951950

[11] A Chabot Analyse des efforts a linterface entre les couches desmateriaux composites a laide de modeles multiparticulaires demateriaux multicouches (M4) [PhD thesis] Ecole Nationaledes Ponts et Chaussees 1997

[12] R P Carreira J F Caron and A Diaz Diaz ldquoModel of multilay-eredmaterials for interface stresses estimation and validation byfinite element calculationsrdquoMechanics of Materials vol 34 no4 pp 217ndash230 2002

[13] R Alvarez-Lima A Diaz-Diaz J-F Caron and S ChataignerldquoEnhanced layerwise model for laminates with imperfect inter-faces - Part 1 Equations and theoretical validationrdquo CompositeStructures vol 94 no 5 pp 1694ndash1702 2012

[14] J L Sanders Jr ldquoAn improved first-approximation theory forthin shellsrdquo Tech Rep NASA TR-R24 United States Govern-ment Publishing Office Washington DC USA 1959

[15] W Flugge Stresses in Shells Springler-Verlag 1966[16] E Ventsel and T Krauthammer Thin Plates and Shells Theory

Analysis and Applications Marcel Dekker New York NY USA2001

[17] V Z Vlasov ldquoNASA-TT-Frdquo Tech Rep National Aeronauticsand Space Administration 1964

[18] P M Naghdi In LinearTheories of Elasticity andThermoelastic-ity Springer 1973

[19] J N ReddyTheory andAnalysis of Elastic Plates and Shells CRCPress 2nd edition 2006

[20] M H Toorani and A A Lakis ldquoGeneral equations ofanisotropic plates and shells including transverse shear defor-mations rotary inertia and initial curvature effectsrdquo Journal ofSound and Vibration vol 237 no 4 pp 561ndash615 2000

[21] M H Toorani and A A Lakis ldquoFree vibrations of non-uniformcomposite cylindrical shellsrdquo Nuclear Engineering and Designvol 236 no 17 pp 1748ndash1758 2006

[22] D D Fox ldquoTransverse shear and normal stresses in nonlinearshell theoryrdquo Computers amp Structures vol 75 no 3 pp 313ndash3192000

[23] F Tornabene and E Viola ldquoStatic analysis of functionallygraded doubly-curved shells and panels of revolutionrdquo Mecca-nica vol 48 no 4 pp 901ndash930 2013

[24] E Viola F Tornabene and N Fantuzzi ldquoStatic analysis ofcompletely doubly-curved laminated shells and panels usinggeneral higher-order shear deformation theoriesrdquo CompositeStructures vol 101 pp 59ndash93 2013

[25] E Trefftz ldquoAbleitung der Schalenbiegungsgleichungen mit demCastiglianoschen Prinziprdquo ZAMM - Journal of Applied Math-ematics and MechanicsZeitschrift fur Angewandte Mathematikund Mechanik vol 15 no 1-2 pp 101ndash108 1935

[26] J L Synge and W Z Chien ldquoThe intrinsic theory of elasticshells and platesrdquo Von KAirmAin Anniversary Volume ArticleID 103120 pp 103ndash120 1941

[27] W-Z Chien ldquoThe intrinsic theory of thin shells and plates IGeneral theoryrdquo Quarterly of Applied Mathematics vol 1 pp297ndash327 1944

[28] Y-G Fang J-H Pan and W-X Chen ldquoTheory of thick-walled shells and its application in cylindrical shellrdquo AppliedMathematics and Mechanics-English Edition vol 13 no 11 pp1055ndash1065 1992

[29] G Z Voyiadjis and G Shi ldquoA refined two-dimensional theoryfor thick cylindrical shellsrdquo International Journal of Solids andStructures vol 27 no 3 pp 261ndash282 1991

[30] G Z Voyiadjis and P Woelke ldquoA refined theory for thickspherical shellsrdquo International Journal of Solids and Structuresvol 41 no 14 pp 3747ndash3769 2004

[31] A W Leissa ldquoScientific and Technical Information OfficerdquoTech Rep NASA SP-160 National Aeronautics and SpaceAdministration Washington DC USA 1969


[33] M L Bucalem and K J Bathe ldquoHigh-order MITC generalshell elementsrdquo International Journal for Numerical Methods inEngineering vol 36 no 21 pp 3729ndash3754 1993

[34] P-S Lee and K-J Bathe ldquoThe quadratic MITC plate andMITC shell elements in plate bendingrdquoAdvances in EngineeringSoftware vol 41 no 5 pp 712ndash728 2010

[35] A Diaz Diaz J-F Caron and R P Carreira ldquoSoftware applica-tion for evaluating interfacial stresses in inelastic symmetricallaminates with free edgesrdquo Composite Structures vol 58 no 2pp 195ndash208 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


by first degree polynomials while the out-of-plane shear andnormal stresses are approximated by second degree and thirddegree polynomials respectively The stress approximationverifies the 3D equilibrium equation and the stress boundaryconditions at the upper and lower faces of the plate Thecoefficients of the stress polynomials are the stress resultantsalso called generalized forces Owing to the shear correctionfactor introduced by Mindlin the models of Reissner andMindlin have similar plate equations this explains the com-mon denomination ldquoReissner-Mindlinrdquo plate theory Thesemodels provide accurate stiffness predictions of moderatelythick plates and good but less precise results for thick shellsThese models have been the starting point for the analysisof composite laminated plates and the development of othermodels such as higher-order models [5] zig-zag models[6] extensions of Reissner-Mindlin-like models [7 8] andlayerwise models [9] In this last family of models Paganorsquosmodel [1] which proposes a polynomial stress approximationthat verifies the 3D equilibrium in each layer is worthciting Pagano used Reissnerrsquos variational mixed formulation[10] to build the model No displacement approximation ismade despite the use of a mixed formulation The Hellinger-Reissner functional and the stress approximation helps toidentify 2D generalized displacements More recent develop-ments of Paganorsquos approach can be found in [11ndash13] wheresome simplifications are adopted to obtain more operationallayerwise models In [13] Alvarez-Lima et al enhanceda layerwise model called M4-5N (originally developed byChabot in [11]) in order to include the Poissonrsquos effect of out-of-plane normal stresses on in-plane strains which may havean important impact on plasticity and failure mechanismsMost models of plates do not take into account this effect

Shell models can be classified depending on how the ratio120578 between the thickness to the smallest principal radius isconsidered in themodel equations thin andmoderately thickshell models neglect any term multiplied by 120578119896 for 119896 ge 1and 119896 ge 2 respectively thick shell models consider higher-degree terms The simplest thin shell models are Loversquos firstapproximation theories [2 14ndash17] which neglect transverseshear deformation and are based on a displacement approachsimilar to that in Kirchhoff-Love plate theory Other thin andmoderately thick shell models apply a displacement approx-imation (in the curvilinear coordinate system) analogue toMindlinrsquos plate model [18 19] for example in [20] Tooraniand Lakis proposed a generalization of the classical Reissner-Mindlin plate theory in order to include the curvature andtransverse shear strains of the shell These authors improvedtheir model in [21] by enhancing the kinematic fields Inthese displacement approach models the 3D constitutiveequations yield linear in-plane stresses and constant out-of-plane shear stresses across the thickness of the shellFor these models the stress field does not verify the 3Dequilibrium equations and the stress boundary conditions atthe faces of the shell [22] Moreover the normal out-of-planestresses and the linear out-of-plane strains are neglected andthis may lead to important errors in thicker shells Thesedrawbacks have motivated the development of higher-degreedisplacement approximations the use of better techniques

to recover the 3D stress and the construction of modelsbased on stress ormixed stressdisplacement approximationsIn [23 24] Tornabene and Viola used an approximationof the 3D displacement field to obtain all the generalizedequations and generalized fields for shells of revolutionAfter solving the static problem they applied the generalizeddifferential quadrature method in order to approximate thespatial derivatives of the stress field This approximationand Hookersquos law allowed recovering and correcting the 3Dstress field which took into account the boundary conditionsat the faces of the shell One of the first stress approachmodels for shells is the one developed by Trefftz [25] inthis model an approximation of in-plane stresses in termsof generalized forces is made but transverse stresses areneglected This implies that both 3D equilibrium equationsand boundary conditions at the faces of the shell are notverified In [26 27] Synge and Chien proposed a mixedstressdisplacement approach model for thin shells wherestresses and displacements are approximated by polynomialsof the thickness coordinate In their theory the boundaryconditions at the faces of the shell are not fulfilled In [28]Fang et al developed a thick shell model using a mixedstressdisplacement approach by applying Hellinger-Reissnerfunctional [10] In this mixed model the stress field verifiesthe boundary conditions at the faces of the shell but not the3D equilibrium equations Shellmodels based on a pure stressapproach are less usual than those based on a displacementapproach two noteworthy stress approach based modelsfor cylindrical and spherical thick shells can be found in[29 30] respectively A summary and comparative studyof several shell and plate theories was made by Leissa in[31 32] In Leissarsquos work all the shell models start withan approximation of the displacement field There are fewmodels based on a stress approach although these types ofmodels have advantages over those based on displacementapproximation

Many commercial finite element software packages offersurface elements to analyze shell structures These elementsare based on classical shell models A family of these surfaceelements is the one introduced in the MITC (Mixed Interpo-lation of Tensorial Components) shell concept this family isreferred to in the literature as MITC119899 where 119899 is related tothe number of nodes in the element [33] Lee and Bathe [34]state that these elements development is based on the use ofdisplacement interpolations and the selection of appropriatestrain field interpolation Finally the element rigidity matrixis built using a stress-strain constitutive equation of classicalshell models Obviously with this family of elements thestress field does not verify the 3D equilibrium equations andthe stress boundary conditions at the faces of the shell

In this paper an original model of linear elastic moder-ately thick shells (120578 le 03) called SAM-H (Stress ApproachModel of Homogeneous shells) is developed by making useof a method similar to that applied by Pagano in [1] andbased on a stress approach which in the limit case of ahomogeneous plate involves less generalized forces thanPaganorsquosmodel Reissnerrsquos variationalmixed formulation [10]is applied to obtain the generalized fields and equations Themain contribution of the paper resides on the quality of the







) 1198602 = 1198862 (1 + 12058531198772

) 1198603 = 1(1)


















dik 120590 = 1119860111986021198603

(k1 + k2) (3)


k1 = ((

12059711986021198603120590111205971205851 + 12059711986011198603120590121205971205852 + 1205971198601119860212059013120597120585312059711986021198603120590211205971205851 + 12059711986011198603120590221205971205852 + 1205971198601119860212059023120597120585312059711986021198603120590311205971205851 + 12059711986011198603120590321205971205852 + 12059711986011198602120590331205971205853))

(4)

and

k2 = (11986021198862 12059711988611205971205852 12059021 minus 11986011198861 12059711988621205971205851 12059022 + 1198602

11988611198771

1205903111986011198861 12059711988621205971205851 12059012 minus 11986021198862 12059711988611205971205852 12059011 + 1198601

11988621198772

12059032minus1198602

11988611198771

12059011 minus 1198601

11988621198772

12059022 ) (5)


grad g

= ( 11198861 12059711989211205971205851 + 119892211988611198862 12059711988611205971205852 11198862 12059711989211205971205852 minus 119892211988611198862 1205971198862120597120585111198861 12059711989221205971205851 minus 119892111988611198862 12059711988611205971205852 11198862 12059711989221205971205852 + 119892111988611198862 12059711988621205971205851) (6)




dik M = 111988611198862sdot (1205971198862119872111205971205851 + 1205971198861119872121205971205852 +11987221

12059711988611205971205852 minus11987222

120597119886212059712058511205971198862119872211205971205851 + 1205971198861119872221205971205852 +11987212

12059711988621205971205851 minus11987211

12059711988611205971205852) (7)


div k = 111988611198862 (1205971198862V11205971205851 + 1205971198861V21205971205852 ) (8)




S = 2sum120572120573120574120575=1


120572120573=1

11987812057212057333e120572 otimes e120573SQ = 4 2sum

120572120573=1

11987812057231205733e120572 otimes e120573

and 119878120590 = 1198783333(9)




div 120591+ = div 120591minus = 0 (10)



Ωs

ΩsΩu





119890 ] 119889Ωminus int120597Ωu







120597Ωu










and 1198755 (1205853) = (1205853ℎ )5 minus 518 (1205853ℎ )3 + 5336 (1205853ℎ ) (13)


minusℎ2119875119898 (1205853) 119875119899 (1205853) 1198891205853 = 0 if 119898 = 119899 (14)


120590119899lowast120572120573 (1205851 1205852) 119875119899 (1205853) (15)

120590lowast1205723 (1205851 1205852 1205853) = 3sum119899=0

120590119899lowast1205723 (1205851 1205852) 119875119899 (1205853) (16)

120590lowast33 (1205851 1205852 1205853) = 4sum119899=0

120590119899lowast33 (1205851 1205852) 119875119899 (1205853) (17)



120572120573 = intℎ2


)1198891205853 (18)119876lowast120572 = intℎ2


)1198891205853 (19)119872lowast120572120573 = intℎ2


)1198891205853 (20)


as follows 11987321 = 11987312 + 12057512058111987212

and 11987221 = ℎ21212057512058111987312 +11987212 (21)


120572120573 = ℎ1205900lowast120572120573 + ℎ2121198773minus120573



1205900lowast120572120573 + ℎ2121205901lowast120572120573 (24)



k1 + k2 + 11986011198602f = 0 (25)


(((((

2sum119902=0

1205781199021 (1205851 1205852) 119875119902 (1205853) + 4sum119902=0

1205821199021 (1205851 1205852) 119875119902 (1205853) + 1199031198911 (1205851 1205852)2sum

119902=0

1205781199022 (1205851 1205852) 119875119902 (1205853) + 4sum119902=0

1205821199022 (1205851 1205852) 119875119902 (1205853) + 1199031198912 (1205851 1205852)3sum

119902=0

1205781199023 (1205851 1205852) 119875119902 (1205853) + 5sum119902=0

1205821199023 (1205851 1205852) 119875119902 (1205853) + 1199031198913 (1205851 1205852))))))

= (000) (26)















3 119889120596 + int120596(dik Mlowast



3 + ℎ3 (1205811 + 1205812)12 f

sdotΦlowast)119889120596(28)


minusℎ2

1198750ℎ 119906lowast31198891205853 (29)




minusℎ2


minusℎ2

1198750ℎ 119906lowast21198891205853Φ


minusℎ2


minusℎ2

121198751ℎ2 119906lowast21198891205853

(30)




= (1198801 (1205851 1205852) 1198750 (1205853) + ℎ1206011 (1205851 1205852) 1198751 (1205853) + 1205751199061 (1205851 1205852 1205853)1198802 (1205851 1205852) 1198750 (1205853) + ℎ1206012 (1205851 1205852) 1198751 (1205853) + 1205751199062 (1205851 1205852 1205853)1198803 (1205851 1205852) 1198750 (1205853) + 1205751199063 (1205851 1205852 1205853) ) (31)


minusℎ21198751198941205751199061205721198891205853 = 0 for 119894 ge 2

and intℎ2

minusℎ211987511989512057511990631198891205853 = 0 for 119895 ge 1 (32)









120597Ωs



N sdot n0 = F119892Q sdot n0 = 119865119892

3 M sdot n0 = C119892 (38)



120572 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 1199041198921205721198891205853

1198651198923 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 11990411989231198891205853

and 119862119892120572 = intℎ2

minusℎ2

ℎ212 ((11989901)2 1205811 + (11989902)2 1205812)1198750 (1205853) 1199041198921205721198891205853+ intℎ2


(39)










Ω119908lowast119890 119889Ω = int

120596(119908119904lowast

119890 + 119908119899lowast119890 + 119908119876lowast

119890 + 119908119888lowast119890 ) 119889120596 (42)


119890 119908119876lowast119890 and 119908119888



119908119904lowast119890 = 12 intℎ2


119908119899lowast119890 = 12 intℎ2



120572=1

2sum120573=1

intℎ2


and 119908119888lowast119890= 12 intℎ2


(43)






q = (N

M)

120598 = ( ) (44)

where

N = (11987311119873221198731211987321

)M = (11987211119872221198721211987221

)

= (12057611120576221205761212057621)and = (12059411120594221205941212059421)

(45)



120591+ +Dminus120591minus (46)

where










U = intℎ2

minusℎ2

1198750ℎ (1199061198921e1 + 1199061198922e2) 11988912058531198803 = intℎ2

minusℎ2

1198750ℎ 11990611989231198891205853Φ = intℎ2

minusℎ2

121198751ℎ2 (1199061198921e1 + 1199061198922e2) 1198891205853(48)




120572 (1205851 1205852) + ℎ1206011198881205721198751 (1205853)1199063 (1205851 1205852 1205853) = 1198801198883 (1205851 1205852) (49)




120576119888 = grad Uc + 119880119888

3120581120594119888 = grad Φ119888





221205812 + 1198911198901199023 = 0

dik Mc minusQc = 0

(51)





(Nc

Mc) = (Ac Bc

Bc Dc) sdot (120576119888120594119888)








3 = 119901119877119864 [1 minus ]2120578 ]and 12059011986211987833 = 0 (54)


3 = 1198801198621198783 minus 119901119877119864 (1 minus 2]2 + 3140120578 + 13]30 120578 + ]101205782)



3 = intℎ2

minusℎ2



3 minus 119880311986331198803119863

3

and 1205751198801198621198783 = 119880119862119878

3 minus 119880311986331198803119863

3

(57)


3 and 1205751198801198621198783 are


















0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus0160 01 02 03 04 05

U

SAM

3

] = 0

] = minus04

] = minus02

] = 02

] = 04

(a)

] = 0

] = minus04

] = minus02

] = 02

] = 04

0 01 02 03 04 05

12

1

08

06

04

02

0

UCS 3

(b)


01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

minus04

U

SAM

3

0 01 02 03 04 05

] = 0

] = minus04

] = minus02

] = 02

] = 04



= 01

= 05

3D

= 03

SAM-H

0

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08

minus09

minus1

33p





7 Conclusions




012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)


SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1


r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005



r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402




Appendix






(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018














) 1198602 = 1198862 (1 + 12058531198772

) 1198603 = 1(1)


















dik 120590 = 1119860111986021198603

(k1 + k2) (3)


k1 = ((

12059711986021198603120590111205971205851 + 12059711986011198603120590121205971205852 + 1205971198601119860212059013120597120585312059711986021198603120590211205971205851 + 12059711986011198603120590221205971205852 + 1205971198601119860212059023120597120585312059711986021198603120590311205971205851 + 12059711986011198603120590321205971205852 + 12059711986011198602120590331205971205853))

(4)

and

k2 = (11986021198862 12059711988611205971205852 12059021 minus 11986011198861 12059711988621205971205851 12059022 + 1198602

11988611198771

1205903111986011198861 12059711988621205971205851 12059012 minus 11986021198862 12059711988611205971205852 12059011 + 1198601

11988621198772

12059032minus1198602

11988611198771

12059011 minus 1198601

11988621198772

12059022 ) (5)


grad g

= ( 11198861 12059711989211205971205851 + 119892211988611198862 12059711988611205971205852 11198862 12059711989211205971205852 minus 119892211988611198862 1205971198862120597120585111198861 12059711989221205971205851 minus 119892111988611198862 12059711988611205971205852 11198862 12059711989221205971205852 + 119892111988611198862 12059711988621205971205851) (6)




dik M = 111988611198862sdot (1205971198862119872111205971205851 + 1205971198861119872121205971205852 +11987221

12059711988611205971205852 minus11987222

120597119886212059712058511205971198862119872211205971205851 + 1205971198861119872221205971205852 +11987212

12059711988621205971205851 minus11987211

12059711988611205971205852) (7)


div k = 111988611198862 (1205971198862V11205971205851 + 1205971198861V21205971205852 ) (8)




S = 2sum120572120573120574120575=1


120572120573=1

11987812057212057333e120572 otimes e120573SQ = 4 2sum

120572120573=1

11987812057231205733e120572 otimes e120573

and 119878120590 = 1198783333(9)




div 120591+ = div 120591minus = 0 (10)



Ωs

ΩsΩu





119890 ] 119889Ωminus int120597Ωu







120597Ωu










and 1198755 (1205853) = (1205853ℎ )5 minus 518 (1205853ℎ )3 + 5336 (1205853ℎ ) (13)


minusℎ2119875119898 (1205853) 119875119899 (1205853) 1198891205853 = 0 if 119898 = 119899 (14)


120590119899lowast120572120573 (1205851 1205852) 119875119899 (1205853) (15)

120590lowast1205723 (1205851 1205852 1205853) = 3sum119899=0

120590119899lowast1205723 (1205851 1205852) 119875119899 (1205853) (16)

120590lowast33 (1205851 1205852 1205853) = 4sum119899=0

120590119899lowast33 (1205851 1205852) 119875119899 (1205853) (17)



120572120573 = intℎ2


)1198891205853 (18)119876lowast120572 = intℎ2


)1198891205853 (19)119872lowast120572120573 = intℎ2


)1198891205853 (20)


as follows 11987321 = 11987312 + 12057512058111987212

and 11987221 = ℎ21212057512058111987312 +11987212 (21)


120572120573 = ℎ1205900lowast120572120573 + ℎ2121198773minus120573



1205900lowast120572120573 + ℎ2121205901lowast120572120573 (24)



k1 + k2 + 11986011198602f = 0 (25)


(((((

2sum119902=0

1205781199021 (1205851 1205852) 119875119902 (1205853) + 4sum119902=0

1205821199021 (1205851 1205852) 119875119902 (1205853) + 1199031198911 (1205851 1205852)2sum

119902=0

1205781199022 (1205851 1205852) 119875119902 (1205853) + 4sum119902=0

1205821199022 (1205851 1205852) 119875119902 (1205853) + 1199031198912 (1205851 1205852)3sum

119902=0

1205781199023 (1205851 1205852) 119875119902 (1205853) + 5sum119902=0

1205821199023 (1205851 1205852) 119875119902 (1205853) + 1199031198913 (1205851 1205852))))))

= (000) (26)















3 119889120596 + int120596(dik Mlowast



3 + ℎ3 (1205811 + 1205812)12 f

sdotΦlowast)119889120596(28)


minusℎ2

1198750ℎ 119906lowast31198891205853 (29)




minusℎ2


minusℎ2

1198750ℎ 119906lowast21198891205853Φ


minusℎ2


minusℎ2

121198751ℎ2 119906lowast21198891205853

(30)




= (1198801 (1205851 1205852) 1198750 (1205853) + ℎ1206011 (1205851 1205852) 1198751 (1205853) + 1205751199061 (1205851 1205852 1205853)1198802 (1205851 1205852) 1198750 (1205853) + ℎ1206012 (1205851 1205852) 1198751 (1205853) + 1205751199062 (1205851 1205852 1205853)1198803 (1205851 1205852) 1198750 (1205853) + 1205751199063 (1205851 1205852 1205853) ) (31)


minusℎ21198751198941205751199061205721198891205853 = 0 for 119894 ge 2

and intℎ2

minusℎ211987511989512057511990631198891205853 = 0 for 119895 ge 1 (32)









120597Ωs



N sdot n0 = F119892Q sdot n0 = 119865119892

3 M sdot n0 = C119892 (38)



120572 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 1199041198921205721198891205853

1198651198923 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 11990411989231198891205853

and 119862119892120572 = intℎ2

minusℎ2

ℎ212 ((11989901)2 1205811 + (11989902)2 1205812)1198750 (1205853) 1199041198921205721198891205853+ intℎ2


(39)










Ω119908lowast119890 119889Ω = int

120596(119908119904lowast

119890 + 119908119899lowast119890 + 119908119876lowast

119890 + 119908119888lowast119890 ) 119889120596 (42)


119890 119908119876lowast119890 and 119908119888



119908119904lowast119890 = 12 intℎ2


119908119899lowast119890 = 12 intℎ2



120572=1

2sum120573=1

intℎ2


and 119908119888lowast119890= 12 intℎ2


(43)






q = (N

M)

120598 = ( ) (44)

where

N = (11987311119873221198731211987321

)M = (11987211119872221198721211987221

)

= (12057611120576221205761212057621)and = (12059411120594221205941212059421)

(45)



120591+ +Dminus120591minus (46)

where










U = intℎ2

minusℎ2

1198750ℎ (1199061198921e1 + 1199061198922e2) 11988912058531198803 = intℎ2

minusℎ2

1198750ℎ 11990611989231198891205853Φ = intℎ2

minusℎ2

121198751ℎ2 (1199061198921e1 + 1199061198922e2) 1198891205853(48)




120572 (1205851 1205852) + ℎ1206011198881205721198751 (1205853)1199063 (1205851 1205852 1205853) = 1198801198883 (1205851 1205852) (49)




120576119888 = grad Uc + 119880119888

3120581120594119888 = grad Φ119888





221205812 + 1198911198901199023 = 0

dik Mc minusQc = 0

(51)





(Nc

Mc) = (Ac Bc

Bc Dc) sdot (120576119888120594119888)








3 = 119901119877119864 [1 minus ]2120578 ]and 12059011986211987833 = 0 (54)


3 = 1198801198621198783 minus 119901119877119864 (1 minus 2]2 + 3140120578 + 13]30 120578 + ]101205782)



3 = intℎ2

minusℎ2



3 minus 119880311986331198803119863

3

and 1205751198801198621198783 = 119880119862119878

3 minus 119880311986331198803119863

3

(57)


3 and 1205751198801198621198783 are


















0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus0160 01 02 03 04 05

U

SAM

3

] = 0

] = minus04

] = minus02

] = 02

] = 04

(a)

] = 0

] = minus04

] = minus02

] = 02

] = 04

0 01 02 03 04 05

12

1

08

06

04

02

0

UCS 3

(b)


01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

minus04

U

SAM

3

0 01 02 03 04 05

] = 0

] = minus04

] = minus02

] = 02

] = 04



= 01

= 05

3D

= 03

SAM-H

0

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08

minus09

minus1

33p





7 Conclusions




012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)


SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1


r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005



r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402




Appendix






(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018











dik M = 111988611198862sdot (1205971198862119872111205971205851 + 1205971198861119872121205971205852 +11987221

12059711988611205971205852 minus11987222

120597119886212059712058511205971198862119872211205971205851 + 1205971198861119872221205971205852 +11987212

12059711988621205971205851 minus11987211

12059711988611205971205852) (7)


div k = 111988611198862 (1205971198862V11205971205851 + 1205971198861V21205971205852 ) (8)




S = 2sum120572120573120574120575=1


120572120573=1

11987812057212057333e120572 otimes e120573SQ = 4 2sum

120572120573=1

11987812057231205733e120572 otimes e120573

and 119878120590 = 1198783333(9)




div 120591+ = div 120591minus = 0 (10)



Ωs

ΩsΩu





119890 ] 119889Ωminus int120597Ωu







120597Ωu










and 1198755 (1205853) = (1205853ℎ )5 minus 518 (1205853ℎ )3 + 5336 (1205853ℎ ) (13)


minusℎ2119875119898 (1205853) 119875119899 (1205853) 1198891205853 = 0 if 119898 = 119899 (14)


120590119899lowast120572120573 (1205851 1205852) 119875119899 (1205853) (15)

120590lowast1205723 (1205851 1205852 1205853) = 3sum119899=0

120590119899lowast1205723 (1205851 1205852) 119875119899 (1205853) (16)

120590lowast33 (1205851 1205852 1205853) = 4sum119899=0

120590119899lowast33 (1205851 1205852) 119875119899 (1205853) (17)



120572120573 = intℎ2


)1198891205853 (18)119876lowast120572 = intℎ2


)1198891205853 (19)119872lowast120572120573 = intℎ2


)1198891205853 (20)


as follows 11987321 = 11987312 + 12057512058111987212

and 11987221 = ℎ21212057512058111987312 +11987212 (21)


120572120573 = ℎ1205900lowast120572120573 + ℎ2121198773minus120573



1205900lowast120572120573 + ℎ2121205901lowast120572120573 (24)



k1 + k2 + 11986011198602f = 0 (25)


(((((

2sum119902=0

1205781199021 (1205851 1205852) 119875119902 (1205853) + 4sum119902=0

1205821199021 (1205851 1205852) 119875119902 (1205853) + 1199031198911 (1205851 1205852)2sum

119902=0

1205781199022 (1205851 1205852) 119875119902 (1205853) + 4sum119902=0

1205821199022 (1205851 1205852) 119875119902 (1205853) + 1199031198912 (1205851 1205852)3sum

119902=0

1205781199023 (1205851 1205852) 119875119902 (1205853) + 5sum119902=0

1205821199023 (1205851 1205852) 119875119902 (1205853) + 1199031198913 (1205851 1205852))))))

= (000) (26)















3 119889120596 + int120596(dik Mlowast



3 + ℎ3 (1205811 + 1205812)12 f

sdotΦlowast)119889120596(28)


minusℎ2

1198750ℎ 119906lowast31198891205853 (29)




minusℎ2


minusℎ2

1198750ℎ 119906lowast21198891205853Φ


minusℎ2


minusℎ2

121198751ℎ2 119906lowast21198891205853

(30)




= (1198801 (1205851 1205852) 1198750 (1205853) + ℎ1206011 (1205851 1205852) 1198751 (1205853) + 1205751199061 (1205851 1205852 1205853)1198802 (1205851 1205852) 1198750 (1205853) + ℎ1206012 (1205851 1205852) 1198751 (1205853) + 1205751199062 (1205851 1205852 1205853)1198803 (1205851 1205852) 1198750 (1205853) + 1205751199063 (1205851 1205852 1205853) ) (31)


minusℎ21198751198941205751199061205721198891205853 = 0 for 119894 ge 2

and intℎ2

minusℎ211987511989512057511990631198891205853 = 0 for 119895 ge 1 (32)









120597Ωs



N sdot n0 = F119892Q sdot n0 = 119865119892

3 M sdot n0 = C119892 (38)



120572 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 1199041198921205721198891205853

1198651198923 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 11990411989231198891205853

and 119862119892120572 = intℎ2

minusℎ2

ℎ212 ((11989901)2 1205811 + (11989902)2 1205812)1198750 (1205853) 1199041198921205721198891205853+ intℎ2


(39)










Ω119908lowast119890 119889Ω = int

120596(119908119904lowast

119890 + 119908119899lowast119890 + 119908119876lowast

119890 + 119908119888lowast119890 ) 119889120596 (42)


119890 119908119876lowast119890 and 119908119888



119908119904lowast119890 = 12 intℎ2


119908119899lowast119890 = 12 intℎ2



120572=1

2sum120573=1

intℎ2


and 119908119888lowast119890= 12 intℎ2


(43)






q = (N

M)

120598 = ( ) (44)

where

N = (11987311119873221198731211987321

)M = (11987211119872221198721211987221

)

= (12057611120576221205761212057621)and = (12059411120594221205941212059421)

(45)



120591+ +Dminus120591minus (46)

where










U = intℎ2

minusℎ2

1198750ℎ (1199061198921e1 + 1199061198922e2) 11988912058531198803 = intℎ2

minusℎ2

1198750ℎ 11990611989231198891205853Φ = intℎ2

minusℎ2

121198751ℎ2 (1199061198921e1 + 1199061198922e2) 1198891205853(48)




120572 (1205851 1205852) + ℎ1206011198881205721198751 (1205853)1199063 (1205851 1205852 1205853) = 1198801198883 (1205851 1205852) (49)




120576119888 = grad Uc + 119880119888

3120581120594119888 = grad Φ119888





221205812 + 1198911198901199023 = 0

dik Mc minusQc = 0

(51)





(Nc

Mc) = (Ac Bc

Bc Dc) sdot (120576119888120594119888)








3 = 119901119877119864 [1 minus ]2120578 ]and 12059011986211987833 = 0 (54)


3 = 1198801198621198783 minus 119901119877119864 (1 minus 2]2 + 3140120578 + 13]30 120578 + ]101205782)



3 = intℎ2

minusℎ2



3 minus 119880311986331198803119863

3

and 1205751198801198621198783 = 119880119862119878

3 minus 119880311986331198803119863

3

(57)


3 and 1205751198801198621198783 are


















0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus0160 01 02 03 04 05

U

SAM

3

] = 0

] = minus04

] = minus02

] = 02

] = 04

(a)

] = 0

] = minus04

] = minus02

] = 02

] = 04

0 01 02 03 04 05

12

1

08

06

04

02

0

UCS 3

(b)


01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

minus04

U

SAM

3

0 01 02 03 04 05

] = 0

] = minus04

] = minus02

] = 02

] = 04



= 01

= 05

3D

= 03

SAM-H

0

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08

minus09

minus1

33p





7 Conclusions




012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)


SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1


r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005



r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402




Appendix






(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018












and 1198755 (1205853) = (1205853ℎ )5 minus 518 (1205853ℎ )3 + 5336 (1205853ℎ ) (13)


minusℎ2119875119898 (1205853) 119875119899 (1205853) 1198891205853 = 0 if 119898 = 119899 (14)


120590119899lowast120572120573 (1205851 1205852) 119875119899 (1205853) (15)

120590lowast1205723 (1205851 1205852 1205853) = 3sum119899=0

120590119899lowast1205723 (1205851 1205852) 119875119899 (1205853) (16)

120590lowast33 (1205851 1205852 1205853) = 4sum119899=0

120590119899lowast33 (1205851 1205852) 119875119899 (1205853) (17)



120572120573 = intℎ2


)1198891205853 (18)119876lowast120572 = intℎ2


)1198891205853 (19)119872lowast120572120573 = intℎ2


)1198891205853 (20)


as follows 11987321 = 11987312 + 12057512058111987212

and 11987221 = ℎ21212057512058111987312 +11987212 (21)


120572120573 = ℎ1205900lowast120572120573 + ℎ2121198773minus120573



1205900lowast120572120573 + ℎ2121205901lowast120572120573 (24)



k1 + k2 + 11986011198602f = 0 (25)


(((((

2sum119902=0

1205781199021 (1205851 1205852) 119875119902 (1205853) + 4sum119902=0

1205821199021 (1205851 1205852) 119875119902 (1205853) + 1199031198911 (1205851 1205852)2sum

119902=0

1205781199022 (1205851 1205852) 119875119902 (1205853) + 4sum119902=0

1205821199022 (1205851 1205852) 119875119902 (1205853) + 1199031198912 (1205851 1205852)3sum

119902=0

1205781199023 (1205851 1205852) 119875119902 (1205853) + 5sum119902=0

1205821199023 (1205851 1205852) 119875119902 (1205853) + 1199031198913 (1205851 1205852))))))

= (000) (26)















3 119889120596 + int120596(dik Mlowast



3 + ℎ3 (1205811 + 1205812)12 f

sdotΦlowast)119889120596(28)


minusℎ2

1198750ℎ 119906lowast31198891205853 (29)




minusℎ2


minusℎ2

1198750ℎ 119906lowast21198891205853Φ


minusℎ2


minusℎ2

121198751ℎ2 119906lowast21198891205853

(30)




= (1198801 (1205851 1205852) 1198750 (1205853) + ℎ1206011 (1205851 1205852) 1198751 (1205853) + 1205751199061 (1205851 1205852 1205853)1198802 (1205851 1205852) 1198750 (1205853) + ℎ1206012 (1205851 1205852) 1198751 (1205853) + 1205751199062 (1205851 1205852 1205853)1198803 (1205851 1205852) 1198750 (1205853) + 1205751199063 (1205851 1205852 1205853) ) (31)


minusℎ21198751198941205751199061205721198891205853 = 0 for 119894 ge 2

and intℎ2

minusℎ211987511989512057511990631198891205853 = 0 for 119895 ge 1 (32)









120597Ωs



N sdot n0 = F119892Q sdot n0 = 119865119892

3 M sdot n0 = C119892 (38)



120572 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 1199041198921205721198891205853

1198651198923 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 11990411989231198891205853

and 119862119892120572 = intℎ2

minusℎ2

ℎ212 ((11989901)2 1205811 + (11989902)2 1205812)1198750 (1205853) 1199041198921205721198891205853+ intℎ2


(39)










Ω119908lowast119890 119889Ω = int

120596(119908119904lowast

119890 + 119908119899lowast119890 + 119908119876lowast

119890 + 119908119888lowast119890 ) 119889120596 (42)


119890 119908119876lowast119890 and 119908119888



119908119904lowast119890 = 12 intℎ2


119908119899lowast119890 = 12 intℎ2



120572=1

2sum120573=1

intℎ2


and 119908119888lowast119890= 12 intℎ2


(43)






q = (N

M)

120598 = ( ) (44)

where

N = (11987311119873221198731211987321

)M = (11987211119872221198721211987221

)

= (12057611120576221205761212057621)and = (12059411120594221205941212059421)

(45)



120591+ +Dminus120591minus (46)

where










U = intℎ2

minusℎ2

1198750ℎ (1199061198921e1 + 1199061198922e2) 11988912058531198803 = intℎ2

minusℎ2

1198750ℎ 11990611989231198891205853Φ = intℎ2

minusℎ2

121198751ℎ2 (1199061198921e1 + 1199061198922e2) 1198891205853(48)




120572 (1205851 1205852) + ℎ1206011198881205721198751 (1205853)1199063 (1205851 1205852 1205853) = 1198801198883 (1205851 1205852) (49)




120576119888 = grad Uc + 119880119888

3120581120594119888 = grad Φ119888





221205812 + 1198911198901199023 = 0

dik Mc minusQc = 0

(51)





(Nc

Mc) = (Ac Bc

Bc Dc) sdot (120576119888120594119888)








3 = 119901119877119864 [1 minus ]2120578 ]and 12059011986211987833 = 0 (54)


3 = 1198801198621198783 minus 119901119877119864 (1 minus 2]2 + 3140120578 + 13]30 120578 + ]101205782)



3 = intℎ2

minusℎ2



3 minus 119880311986331198803119863

3

and 1205751198801198621198783 = 119880119862119878

3 minus 119880311986331198803119863

3

(57)


3 and 1205751198801198621198783 are


















0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus0160 01 02 03 04 05

U

SAM

3

] = 0

] = minus04

] = minus02

] = 02

] = 04

(a)

] = 0

] = minus04

] = minus02

] = 02

] = 04

0 01 02 03 04 05

12

1

08

06

04

02

0

UCS 3

(b)


01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

minus04

U

SAM

3

0 01 02 03 04 05

] = 0

] = minus04

] = minus02

] = 02

] = 04



= 01

= 05

3D

= 03

SAM-H

0

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08

minus09

minus1

33p





7 Conclusions




012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)


SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1


r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005



r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402




Appendix






(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018






















3 119889120596 + int120596(dik Mlowast



3 + ℎ3 (1205811 + 1205812)12 f

sdotΦlowast)119889120596(28)


minusℎ2

1198750ℎ 119906lowast31198891205853 (29)




minusℎ2


minusℎ2

1198750ℎ 119906lowast21198891205853Φ


minusℎ2


minusℎ2

121198751ℎ2 119906lowast21198891205853

(30)




= (1198801 (1205851 1205852) 1198750 (1205853) + ℎ1206011 (1205851 1205852) 1198751 (1205853) + 1205751199061 (1205851 1205852 1205853)1198802 (1205851 1205852) 1198750 (1205853) + ℎ1206012 (1205851 1205852) 1198751 (1205853) + 1205751199062 (1205851 1205852 1205853)1198803 (1205851 1205852) 1198750 (1205853) + 1205751199063 (1205851 1205852 1205853) ) (31)


minusℎ21198751198941205751199061205721198891205853 = 0 for 119894 ge 2

and intℎ2

minusℎ211987511989512057511990631198891205853 = 0 for 119895 ge 1 (32)









120597Ωs



N sdot n0 = F119892Q sdot n0 = 119865119892

3 M sdot n0 = C119892 (38)



120572 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 1199041198921205721198891205853

1198651198923 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 11990411989231198891205853

and 119862119892120572 = intℎ2

minusℎ2

ℎ212 ((11989901)2 1205811 + (11989902)2 1205812)1198750 (1205853) 1199041198921205721198891205853+ intℎ2


(39)










Ω119908lowast119890 119889Ω = int

120596(119908119904lowast

119890 + 119908119899lowast119890 + 119908119876lowast

119890 + 119908119888lowast119890 ) 119889120596 (42)


119890 119908119876lowast119890 and 119908119888



119908119904lowast119890 = 12 intℎ2


119908119899lowast119890 = 12 intℎ2



120572=1

2sum120573=1

intℎ2


and 119908119888lowast119890= 12 intℎ2


(43)






q = (N

M)

120598 = ( ) (44)

where

N = (11987311119873221198731211987321

)M = (11987211119872221198721211987221

)

= (12057611120576221205761212057621)and = (12059411120594221205941212059421)

(45)



120591+ +Dminus120591minus (46)

where










U = intℎ2

minusℎ2

1198750ℎ (1199061198921e1 + 1199061198922e2) 11988912058531198803 = intℎ2

minusℎ2

1198750ℎ 11990611989231198891205853Φ = intℎ2

minusℎ2

121198751ℎ2 (1199061198921e1 + 1199061198922e2) 1198891205853(48)




120572 (1205851 1205852) + ℎ1206011198881205721198751 (1205853)1199063 (1205851 1205852 1205853) = 1198801198883 (1205851 1205852) (49)




120576119888 = grad Uc + 119880119888

3120581120594119888 = grad Φ119888





221205812 + 1198911198901199023 = 0

dik Mc minusQc = 0

(51)





(Nc

Mc) = (Ac Bc

Bc Dc) sdot (120576119888120594119888)








3 = 119901119877119864 [1 minus ]2120578 ]and 12059011986211987833 = 0 (54)


3 = 1198801198621198783 minus 119901119877119864 (1 minus 2]2 + 3140120578 + 13]30 120578 + ]101205782)



3 = intℎ2

minusℎ2



3 minus 119880311986331198803119863

3

and 1205751198801198621198783 = 119880119862119878

3 minus 119880311986331198803119863

3

(57)


3 and 1205751198801198621198783 are


















0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus0160 01 02 03 04 05

U

SAM

3

] = 0

] = minus04

] = minus02

] = 02

] = 04

(a)

] = 0

] = minus04

] = minus02

] = 02

] = 04

0 01 02 03 04 05

12

1

08

06

04

02

0

UCS 3

(b)


01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

minus04

U

SAM

3

0 01 02 03 04 05

] = 0

] = minus04

] = minus02

] = 02

] = 04



= 01

= 05

3D

= 03

SAM-H

0

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08

minus09

minus1

33p





7 Conclusions




012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)


SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1


r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005



r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402




Appendix






(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018
















120597Ωs



N sdot n0 = F119892Q sdot n0 = 119865119892

3 M sdot n0 = C119892 (38)



120572 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 1199041198921205721198891205853

1198651198923 = intℎ2

minusℎ2[1198750 + ℎ ((11989901)2 1205811 + (11989902)2 1205812)1198751] 11990411989231198891205853

and 119862119892120572 = intℎ2

minusℎ2

ℎ212 ((11989901)2 1205811 + (11989902)2 1205812)1198750 (1205853) 1199041198921205721198891205853+ intℎ2


(39)










Ω119908lowast119890 119889Ω = int

120596(119908119904lowast

119890 + 119908119899lowast119890 + 119908119876lowast

119890 + 119908119888lowast119890 ) 119889120596 (42)


119890 119908119876lowast119890 and 119908119888



119908119904lowast119890 = 12 intℎ2


119908119899lowast119890 = 12 intℎ2



120572=1

2sum120573=1

intℎ2


and 119908119888lowast119890= 12 intℎ2


(43)






q = (N

M)

120598 = ( ) (44)

where

N = (11987311119873221198731211987321

)M = (11987211119872221198721211987221

)

= (12057611120576221205761212057621)and = (12059411120594221205941212059421)

(45)



120591+ +Dminus120591minus (46)

where










U = intℎ2

minusℎ2

1198750ℎ (1199061198921e1 + 1199061198922e2) 11988912058531198803 = intℎ2

minusℎ2

1198750ℎ 11990611989231198891205853Φ = intℎ2

minusℎ2

121198751ℎ2 (1199061198921e1 + 1199061198922e2) 1198891205853(48)




120572 (1205851 1205852) + ℎ1206011198881205721198751 (1205853)1199063 (1205851 1205852 1205853) = 1198801198883 (1205851 1205852) (49)




120576119888 = grad Uc + 119880119888

3120581120594119888 = grad Φ119888





221205812 + 1198911198901199023 = 0

dik Mc minusQc = 0

(51)





(Nc

Mc) = (Ac Bc

Bc Dc) sdot (120576119888120594119888)








3 = 119901119877119864 [1 minus ]2120578 ]and 12059011986211987833 = 0 (54)


3 = 1198801198621198783 minus 119901119877119864 (1 minus 2]2 + 3140120578 + 13]30 120578 + ]101205782)



3 = intℎ2

minusℎ2



3 minus 119880311986331198803119863

3

and 1205751198801198621198783 = 119880119862119878

3 minus 119880311986331198803119863

3

(57)


3 and 1205751198801198621198783 are


















0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus0160 01 02 03 04 05

U

SAM

3

] = 0

] = minus04

] = minus02

] = 02

] = 04

(a)

] = 0

] = minus04

] = minus02

] = 02

] = 04

0 01 02 03 04 05

12

1

08

06

04

02

0

UCS 3

(b)


01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

minus04

U

SAM

3

0 01 02 03 04 05

] = 0

] = minus04

] = minus02

] = 02

] = 04



= 01

= 05

3D

= 03

SAM-H

0

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08

minus09

minus1

33p





7 Conclusions




012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)


SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1


r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005



r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402




Appendix






(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018









120572=1

2sum120573=1

intℎ2


and 119908119888lowast119890= 12 intℎ2


(43)






q = (N

M)

120598 = ( ) (44)

where

N = (11987311119873221198731211987321

)M = (11987211119872221198721211987221

)

= (12057611120576221205761212057621)and = (12059411120594221205941212059421)

(45)



120591+ +Dminus120591minus (46)

where










U = intℎ2

minusℎ2

1198750ℎ (1199061198921e1 + 1199061198922e2) 11988912058531198803 = intℎ2

minusℎ2

1198750ℎ 11990611989231198891205853Φ = intℎ2

minusℎ2

121198751ℎ2 (1199061198921e1 + 1199061198922e2) 1198891205853(48)




120572 (1205851 1205852) + ℎ1206011198881205721198751 (1205853)1199063 (1205851 1205852 1205853) = 1198801198883 (1205851 1205852) (49)




120576119888 = grad Uc + 119880119888

3120581120594119888 = grad Φ119888





221205812 + 1198911198901199023 = 0

dik Mc minusQc = 0

(51)





(Nc

Mc) = (Ac Bc

Bc Dc) sdot (120576119888120594119888)








3 = 119901119877119864 [1 minus ]2120578 ]and 12059011986211987833 = 0 (54)


3 = 1198801198621198783 minus 119901119877119864 (1 minus 2]2 + 3140120578 + 13]30 120578 + ]101205782)



3 = intℎ2

minusℎ2



3 minus 119880311986331198803119863

3

and 1205751198801198621198783 = 119880119862119878

3 minus 119880311986331198803119863

3

(57)


3 and 1205751198801198621198783 are


















0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus0160 01 02 03 04 05

U

SAM

3

] = 0

] = minus04

] = minus02

] = 02

] = 04

(a)

] = 0

] = minus04

] = minus02

] = 02

] = 04

0 01 02 03 04 05

12

1

08

06

04

02

0

UCS 3

(b)


01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

minus04

U

SAM

3

0 01 02 03 04 05

] = 0

] = minus04

] = minus02

] = 02

] = 04



= 01

= 05

3D

= 03

SAM-H

0

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08

minus09

minus1

33p





7 Conclusions




012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)


SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1


r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005



r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402




Appendix






(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018











120572 (1205851 1205852) + ℎ1206011198881205721198751 (1205853)1199063 (1205851 1205852 1205853) = 1198801198883 (1205851 1205852) (49)




120576119888 = grad Uc + 119880119888

3120581120594119888 = grad Φ119888





221205812 + 1198911198901199023 = 0

dik Mc minusQc = 0

(51)





(Nc

Mc) = (Ac Bc

Bc Dc) sdot (120576119888120594119888)








3 = 119901119877119864 [1 minus ]2120578 ]and 12059011986211987833 = 0 (54)


3 = 1198801198621198783 minus 119901119877119864 (1 minus 2]2 + 3140120578 + 13]30 120578 + ]101205782)



3 = intℎ2

minusℎ2



3 minus 119880311986331198803119863

3

and 1205751198801198621198783 = 119880119862119878

3 minus 119880311986331198803119863

3

(57)


3 and 1205751198801198621198783 are


















0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus0160 01 02 03 04 05

U

SAM

3

] = 0

] = minus04

] = minus02

] = 02

] = 04

(a)

] = 0

] = minus04

] = minus02

] = 02

] = 04

0 01 02 03 04 05

12

1

08

06

04

02

0

UCS 3

(b)


01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

minus04

U

SAM

3

0 01 02 03 04 05

] = 0

] = minus04

] = minus02

] = 02

] = 04



= 01

= 05

3D

= 03

SAM-H

0

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08

minus09

minus1

33p





7 Conclusions




012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)


SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1


r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005



r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402




Appendix






(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018











3 = 119901119877119864 [1 minus ]2120578 ]and 12059011986211987833 = 0 (54)


3 = 1198801198621198783 minus 119901119877119864 (1 minus 2]2 + 3140120578 + 13]30 120578 + ]101205782)



3 = intℎ2

minusℎ2



3 minus 119880311986331198803119863

3

and 1205751198801198621198783 = 119880119862119878

3 minus 119880311986331198803119863

3

(57)


3 and 1205751198801198621198783 are


















0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus0160 01 02 03 04 05

U

SAM

3

] = 0

] = minus04

] = minus02

] = 02

] = 04

(a)

] = 0

] = minus04

] = minus02

] = 02

] = 04

0 01 02 03 04 05

12

1

08

06

04

02

0

UCS 3

(b)


01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

minus04

U

SAM

3

0 01 02 03 04 05

] = 0

] = minus04

] = minus02

] = 02

] = 04



= 01

= 05

3D

= 03

SAM-H

0

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08

minus09

minus1

33p





7 Conclusions




012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)


SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1


r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005



r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402




Appendix






(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018









0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

minus0160 01 02 03 04 05

U

SAM

3

] = 0

] = minus04

] = minus02

] = 02

] = 04

(a)

] = 0

] = minus04

] = minus02

] = 02

] = 04

0 01 02 03 04 05

12

1

08

06

04

02

0

UCS 3

(b)


01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

minus04

U

SAM

3

0 01 02 03 04 05

] = 0

] = minus04

] = minus02

] = 02

] = 04



= 01

= 05

3D

= 03

SAM-H

0

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08

minus09

minus1

33p





7 Conclusions




012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)


SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1


r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005



r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402




Appendix






(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018









= 01

= 05

3D

= 03

SAM-H

0

minus01

minus02

minus03

minus04

minus05

minus06

minus07

minus08

minus09

minus1

33p





7 Conclusions




012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)


SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1


r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005



r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402




Appendix






(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018









012

5m

03 m

1m

z

r

085m

axial symmetry

symmetry

1MPa

(a)

e3

e1

e2

z

symmetry

(b)


SFE MITC6CSSAM-H

r

z

r

z

r

z z

r

0

minus02

minus04

minus06

minus08

minus1


r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

045040350302502015010050minus005



r

z

r

z

r

z z

r

SFE MITC6CSSAM-H

218161412108060402




Appendix






(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018










(A2)


(A3)



C = ( CA CB1 + CB2

CC1 + CC2 12ℎ2CA ) (B1)

where

CA = 1ℎ (1198781111 1198781122 1198781112 11987811211198782211 1198782222 1198782212 11987822211198781211 1198781222 1198781212 11987812211198782111 1198782122 1198782112 1198782121)


sdot( 6119878C111205811 119878119862221205811 + 5119878119862111205812 119878119862121205811 119878119862211205811119878119862111205812 + 5119878119862221205811 6119878119862221205812 119878119862121205812 1198781198622112058125119878119862121205811 5119878119862121205812 0 05119878119862211205811 5119878119862211205812 0 0 )CC2 = minus 110ℎ

sdot( 6119878119862111205811 5119878119862221205811 + 119878119862111205812 5119878119862121205811 51198781198622112058115119878119862111205812 + 119878119862221205811 6119878119862221205812 5119878119862121205812 5119878119862211205812119878119862121205811 119878119862121205812 0 0119878119862211205811 119878119862211205812 0 0 )(B2)



and


where

ca (1205811120581200)CS = (11987811986211 0 0 00 11987811986222 0 00 0 11987811986212 00 0 0 11987811986221)

cb = (1111)(B5)


cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018









cc = (1205811120581212058121205811)and cd = (1205812120581112058111205812)

(B6)




D = 65ℎ (11987811987611 1198781198761211987811987621 11987811987622) (B8)


and



















andH2 are

H1 = ((((((

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 1))))))

(C6)

and

H2 =(((((((((((((

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 1 0 0 1205751205810 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 ℎ212120575120581 0 0 1

)))))))))))))

(C7)


Data Availability




References












































Journal of












Journal of







Volume 2018






Volume 2018


















































Journal of












Journal of







Volume 2018






Volume 2018








a stress approach model of moderately thick, homogeneous...

Documents