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2009 14: 747 originally published online 11 March 2009Mathematics and Mechanics of SolidsX.-L. Gao, S.K. Park and H.M. Ma

Simplified Strain Gradient Elasticity TheoryAnalytical Solution for a Pressurized Thick-Walled Spherical Shell Based on a

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Analytical Solution for a Pressurized Thick-Walled SphericalShell Based on a Simpli�ed Strain Gradient Elasticity Theory

X.-L. GAO

S. K. PARK�

H. M. MADepartment of Mechanical Engineering, Texas A&M University, College Station,TX 77843-3123, USA� Currently at JP Kenny Inc., Houston, TX

(Received 27 June 2008� accepted 15 August 2008)

Abstract: The problem of a pressurized thick-walled spherical shell is analytically solved using a simplifiedstrain gradient elasticity theory. The closed-form solution derived contains a material length scale parame-ter and can account for microstructural effects, which qualitatively differs from Lamé’s solution in classicalelasticity. When the strain gradient effect (a measure of the underlying material microstructure) is not con-sidered, the newly derived strain gradient elasticity solution reduces to Lamé’s classical elasticity solution.To illustrate the new solution, a sample problem with specified geometrical parameters, pressure values andmaterial properties is solved. The numerical results reveal that the magnitudes of both the radial and tangen-tial stress components in the shell wall given by the current strain gradient solution are smaller than thosegiven by Lamé’s solution. Also, it is quantitatively shown that microstructural effects can be large and Lamé’ssolution may not be accurate for materials exhibiting significant microstructure dependence.

Key Words: Strain gradient theory, elasticity, Lamé’s solution, thick-walled spherical shell, microstructural effect, lengthscale, pressure vessel

1. INTRODUCTION

For the problem of a pressurized thick-walled spherical shell, Lamé’s solution is well known(e.g., [1]) and has been playing an important role in spherical pressure vessel design (e.g., [2,3]). However, this solution is based on classical elasticity and cannot capture microstructure-dependent size effects due to the lack of a material length scale parameter.

Higher-order strain gradient elasticity theories are capable of explaining the size effectsobserved at the micron scale, since they contain additional elastic constants that are relatedto the underlying material microstructure and can serve as length scale parameters. For ex-ample, Mindlin [4] developed a general strain gradient elasticity theory that contains 18independent material constants for an isotropic elastic material, with two being the classicalLamé constants. The strain gradient theory proposed by Casal [5, 6] includes four elasticconstants, with two being Lamé’s constants and the other two being the material length scale

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748 X.-L. GAO ET AL.

parameters. In view of the difficulties in determining microstructure-dependent length scaleparameters (e.g., [7, 8]) and in applying higher-order theories (e.g., [9]), it is very desirableto have strain gradient elasticity theories involving only one additional elastic constant.

One such strain gradient model was proposed in Altan and Aifantis [10] (without deriva-tions) by simplifying a strain gradient elasticity theory of Mindlin and Eshel [11] that in-volves five additional material parameters. This simplified model contains three elastic con-stants, with two being Lamé constants and the third one being a strain gradient coefficient.A variationally consistent formulation for this model has recently been provided by Gao andPark [12] by using the principle of minimum total potential energy, which leads to the simul-taneous determination of the governing (equilibrium) equations and all boundary conditions,thereby completing the development of the theory. By directly applying this strain gradienttheory, the solution for the problem of a pressurized thick-walled cylinder is derived in Gaoand Park [12].

In the current paper, the pressurized thick-walled spherical shell problem is solved byusing the above-mentioned simplified strain gradient elasticity theory. This problem is threedimensional, and the higher-order tensors and boundary conditions involved are more com-plex than those in the pressurized cylinder problem. The rest of the paper is organized asfollows. In Section 2, the simplified strain gradient elasticity theory is reviewed, with themajor equations in the theory recorded to facilitate the formulation in Section 3. The bound-ary value problem of the pressurized spherical shell is solved in Section 3 by applying thestrain gradient elasticity theory, where the newly derived solution is also compared withLamé’s solution in classical elasticity to illustrate the differences between the two solutions.The paper concludes in Section 4 with a summary.

2. REVIEW OF THE SIMPLIFIED STRAIN GRADIENT THEORY

The simplified strain gradient elasticity theory [10, 12] is based on the following strain en-ergy density function, w, for an isotropic, linearly elastic material:

w � w��i j � �i j�k� � 1

2��ii� j j � ��i j�i j � c

�1

2��ii�k� j j�k � ��i j�k�i j�k

�� (1)

where � and � are the Lamé constants, c is a strain gradient coefficient having the dimensionof length squared, and �i j are the components of the infinitesimal strain, �� = �i j ei�e j , givenby

�i j � 1

2�ui � j �u j �i �� (2)

with ui being the displacement components.It follows from Equation (1) that the constitutive equations are

� i j � w

�i j� ��lli j � 2��i j � � j i � (3)



ANALYTICAL SOLUTION FOR A PRESSURIZED THICK-WALLED 749

�i jk � w

� i jk� c

��lli j � 2��i j

��k � c� i j �k � � j ik� (4)

where � i j are the components of the Cauchy stress, �� i j ei�e j , �i jk are the componentsof the double stress,�� =�i jkei�e j�ek , and � i jk are the components of the strain gradient, ��= ��, given by

� i jk � �i j �k � 1

2�ui � jk �u j �ik �� (5)

Equations (3) and (4) can be combined to obtain the modified constitutive relations:

i j � � i j �i jk�k � � i j c� i j �kk � ��lli j � 2��i j c��lli j � 2��i j��kk � (6)

where i j are the components of the total stress, = i j ei�e j (e.g., [13]). The expression of i j as a combination of � i j and �i jk given in Equation (6) arises naturally from the variationof the strain energy [12].

As shown in [12], the Navier-like basic governing equation reads

�1 c�2�[�� 2�� u� �curl�curlu�]� f � 0 in �� (7)

and the boundary conditions have the form:

n� ��n�� : n� n�� n�� : n� n�n

�� : �n� [��n�n]� � t or u � u�� : �n� n� � q or ��u�n � u

n

��

on �� (8a,b)

where u (=ui ei ) is the displacement vector, f is the body force vector, and t and q are, respec-tively, the Cauchy traction vector and double stress traction vector. In terms of u, the totalstress is obtained from Equations (2) and (6) as

� �1 c�2�� u�I� �[�u� ��u�T]

�� (9)

and the double stress�� is determined from Equations (2) and (4) to be

�� c�� u�I� �[�u� ��u�T]�� (10)

In Equations (7)–(10), I is the second-order identity tensor, �, � and �2 are, respectively,the gradient, divergence and Laplacian operators, � is the region occupied by the elasticallydeformed material, and � is the smooth bounding surface of �. For the general case inwhich � is not entirely smooth but contains edges, additional boundary conditions arise foreach edge formed by the intersection of two smooth sub-surfaces [12]. In Equations (8a,b)and in the sequel, the overbar represents the prescribed value.




Figure 1. Pressurized thick-walled sphere.

Equations (7) and (8a,b) define the displacement form of the simplified strain gradientelasticity theory, which, as tensorial expressions, are coordinate-invariant. Clearly, the newparameter c is explicitly involved in Equation (7) in addition to the two Lamé constants � and�. When the strain gradient effect is absent (i.e., c = 0),�� = 0 (see Equation (10)) and = ��(see Equations (9) and (3)), and Equations (7) and (8a,b) reduce to the governing equationsand the boundary conditions in terms of displacement in classical elasticity (e.g., [14]).

3. BOUNDARY VALUE PROBLEM

Consider a spherical shell having the inner radius ri and outer radius ro and is subjected tothe internal pressure pi and external pressure po, as shown in Figure 1. The usual sphericalcoordinate system (r, � , �) is used in the formulation together with the associated base vec-tors {er , e� , e�}. This boundary value problem is solved here by using the simplified straingradient theory reviewed in the preceding section.

Due to the geometrical and loading symmetry, the displacement field u has the form:

u � u�r�er � (11)

where u is the radial (only non-vanishing) displacement component, depending on the radialcoordinate r. It then follows from Equation (11) that

� u � du

dr� 2u

r� curlu � 0. (12a,b)

Substituting Equations (12a,b) into Equation (7) results in, with f � 0,




�d2u

dr2� 2

r

du

dr 2u

r2

� c

�d4u

dr4� 4

r

d3u

dr3 4

r2

d2u

dr2

�� 0 (13)

as the equilibrium equation in the absence of body force, which is a fourth-order ordinarydifferential equation for c �� 0. Equation (13) can be rewritten as

L�1 cL�u � 0� (14)

where L is a linear differential operator defined by

L � d2

dr2� 2

r

d

dr 2

r2� (15)

When c = 0, Equation (14) reduces to, after using Equation (15),

Lu � d2u

dr2� 2

r

du

dr 2u

r2� 0� (16)

which is a homogeneous second-order ordinary differential equation of the Euler type. Thesolution of Equation (16), u = u0(r), is given by

u0 � Ar � B

r 2� (17)

where A and B are two constants yet to be determined.When c �� 0, a comparison of Equations (14) and (16) indicates that the solution of

Equation (14) can be obtained through solving

�1 cL� u � u0� (18)

where u0, as the solution of Equation (16), is given in Equation (17). This is based on thefactorization of differential operators (e.g., [15, 16]). Substituting Equations (15) and (17)into Equation (18) yields

d2u

dr2� 2

r

du

dr�

2

r2� 1

c

�u � 1

c

�Ar � B

r2

�� (19)

The solution of the homogeneous part of Equation (19), uh(r), is found to be

uh�r� � C�r �c

�e

r�c

r 2� D

�r ��c

�e

r�c

r2� (20)

where C and D are two additional constants. A particular solution of Equation (19) has theform:




u p�r� � Ar � B

r2� (21)

which happens to be the same as that given in Equation (17). This agrees with what was statedin [15]. Combining Equations (20) and (21) then gives the general solution of Equation (19),and thus of Equation (18) or Equation (13), as

u�r� � Ar � B

r 2� C

�r �c

�e

r�c

r2� D

�r ��c

�e

r�c

r2� (22)

Clearly, it is seen from Equation (22) that the radial displacement explicitly depends onthe strain gradient coefficient c (and thus the underlying material microstructure). Note thatthis closed-form solution for the spherical shell problem is quite different from the solutionfor the cylinder problem obtained by Gao and Park [12] in terms of two modified Besselfunctions.

With the displacement field known from Equations (11) and (22), the strain field in theshell wall can then be determined from Equations (2), (11) and (22) as

��r� � 1

2

��u� ��u�T

� du

drer � er � u

re� � e� � u

re� � e�� (23)

From Equations (23) and (3) the Cauchy stress is found to be

��r� � � �� u� I� � ��u� ��u�T � � rr er � er � � ��e� � e� � ��e� � e�� (24)

where

� rr � �� 2��du

dr� 2�

u

r� � �� du

dr� 2�� u

r�

�� du

dr� 2�� u

r� � �� (25)

The double stress,��, is determined from Equations (4) and (24) as

�� c[� �rr er � er � er � 1

r�� rr � �� er � e� � e� � er � e� � e� � e� � er � e�

� e� � er � e�� e� � e� � er � e� � e� � er�]� (26)

where use has been made of the gradient of the Cauchy stress, �� (see Appendix for thederivation), and � rr = � rr (r), � �� = � �� (r) are given in Equations (22) and (25). Here and inthe sequel, the single prime and double prime denote, respectively, the first- and second-orderderivatives with respect to r.

The total stress, , is obtained from Equations (6) and (24) as




�� rr c

�� rr �

2� �rr

r 4� rr

r2� 4� ��

r 2

��er � er

�� c

��

2� ��r 2� ��

r2� 2� rr

r2

��e� � e�

�� c

��

2� ��r 2� ��

r2� 2� rr

r2

��e� � e�� (27)

where use has been made of the Laplacian of the Cauchy stress, �2�� (see Appendix for thederivation), and � rr = � rr (r), � �� = � �� (r) are given in Equations (22) and (25).

The boundary conditions of the current spherical shell problem (see Figure 1) are

�� rr c

�� rr �

2

r

�� rr � ��

� 4

r2�� rr � ��

��r�ri

er � pi er � (28a)

�� rr c

�� rr �

2

r

�� rr � ��

� 4

r 2�� rr � ��

��r�ro

er � poer � (28b)

c � �rr

��r�ri

er � 0� (28c)

c � �rr

��r�ro

er � 0� (28d)

Note that Equations (28a) and (28b) follow from the traction boundary conditions given inEquation (8a), with the prescribed traction being t � pi er on the inner surface (r � ri ) and t � poer on the outer surface (r � ro). Equations (28c) and (28d) are obtained from thedouble stress traction boundary conditions specified in Equation (8b), with the prescribeddouble stress traction being 0 (i.e., q � 0) on both the inner surface (r � ri ) and the outersurface (r � ro). Also, Equations (26) and (27) have been used in reaching Equations (28a–d). When c = 0 (i.e., in the absence of the strain gradient effect), Equations (28a–d) reduceto the two traction boundary conditions used in classical elasticity to derive Lamé’s solution(e.g., [1]).

Substituting Equations (22) and (25) into Equations (28a–d) then yields

A �3� 2�� B

�4�

r 3i

� 12c�

r 5i

�

� C

�2�

r2i

� 2��

c

r3i

4�

c�

r 3i

� 12c�

r4i

12c3�2�

r5i

�e

ri�c

� D

�2�

r2i

2��

c

r3i

� 4�

c�

r3i

� 12c�

r 4i

� 12c3�2�

r5i

�e

ri�c � pi � (29a)




A �3� 2�� B

�4�

r 3o

� 12c�

r 5o

�

� C

�2�

r2o

� 2��

c

r3o

4�

c�

r 3o

� 12c�

r4o

12c3�2�

r5o

�e

ro�c

� D

�2�

r2o

2��

c

r3o

� 4�

c�

r3o

� 12c�

r 4o

� 12c3�2�

r5o

�e

ro�c � po� (29b)

B

�12�

r4i

�� C

1

cr 4i

�12�c3�2 � 12�ri c �� 6��

cr2i � �� 2��r 3

i

e

ri�c

� D1

cr4i

�12�c3�2 � 12�ri c � �� 6��

�cr2

i � �� 2��r 3i

e

ri�c � 0� (29c)

B

�12�

r4o

�� C

1

cr 4o

�12�c3�2 � 12�roc �� 6��

cr2o � �� 2��r3

o

e

ro�c

� D1

cr4o

�12�c3�2 � 12�roc� �� 6��

�cr 2

o � �� 2��r3o

e

ro�c � 0� (29d)

Equations (29a–d) form a system of four linear algebraic equations for determining the fourconstants A, B, C and D, which depend on the material properties �, � and c, the geometricalparameters ri and ro, and the applied pressures pi and po. For given values of �, �, c, ri , ro,pi and po, Equations (29a–d) can be readily solved by using a computer program to obtainthe four constants A, B, C and D. The displacement field (u) will then be determined fromEquations (11) and (22), and the Cauchy stress (�� ), double stress (��) and total stress ( )from Equations (24), (26) and (27), respectively.

Clearly, the involvement of c (the strain gradient coefficient) explicitly in Equations(29a–d) and (22) and subsequently in Equations (24)–(27) shows that the current solution canaccount for microstructure-dependent effects, which is qualitatively different from Lamé’sclassical elasticity solution that contains � and � only. Note that Lamé’s constants, � and �,are related to Young’s modulus, E, and Poisson’s ratio, �, by (e.g., [1])

� � E�

�1� ��1 2�� E

2�1� �� (30)

To illustrate the newly derived solution, a sample problem with specified geometricalparameters, pressure values and material properties has been solved, with the numerical re-sults presented here. The material properties used are E = 135 GPa, � = 0.3, and c = 0.05�m2, which are the same as those used in [12, 17]. For comparison, two other values of c arealso considered, as shown in Figure 2. The geometrical parameters used are ri = 1 �m andro = 5 �m, and the pressures are taken to be pi = 10 MPa, po = 0 MPa in the calculations.

Figure 2 shows the stress distributions along the shell wall given by both the currentstrain gradient solution (with c �� 0) and Lamé’s classical elasticity solution (with c = 0),




Figure 2. Stress distributions along the shell wall.

which are determined using Equation (27) together with Equations (22), (25) and (29a–d).As displayed in Figure 2, the magnitudes of both the tangential stress, �� (= ��), andthe radial stress, rr , given by the strain gradient solution are smaller than those given byLamé’s classical elasticity solution (with � �� from Equation (27) when c = 0) in all casesconsidered. Also, Figure 2 shows that the differences between the current strain gradientsolution and Lamé’s solution are negligibly small when c is very small (e.g., c = 0.01 �m2

here), but are significant when c becomes larger (e.g., c = 0.25 �m2 here). The former agreeswith the fact that Lamé’s solution is the special case of the current strain gradient solutionwith c = 0, while the latter indicates that the effect of material microstructure (measured byc) can be large. As a result, Lamé’s solution may not be accurate for materials that exhibitsignificant microstructural effects.

4. SUMMARY

An analytical solution based on a simplified strain gradient elasticity theory is derived for athick-walled spherical shell subjected to internal and external pressures. The newly obtainedsolution contains a material length scale parameter in addition to two classical elastic con-stants and can capture microstructural effects, which is qualitatively different from Lamé’ssolution. The current strain gradient solution recovers Lamé’s solution in classical elasticitywhen the strain gradient effect is not considered.

To illustrate the new solution, a sample problem is solved. The numerical results showthat the magnitudes of the stress components in the shell wall given by the strain gradient




solution are smaller than those given by Lamé’s solution. Also, it is revealed that microstruc-tural effects can be large and Lamé’s solution may lead to inaccurate results when the shellmaterial exhibits significant microstructure dependence.

Acknowledgement. The work reported here is funded by a grant from the U.S. National Science Foundation, with Dr.Clark Cooper as the program manager. This support is gratefully acknowledged.

APPENDIX

It is shown here that the double stress has the expression (see Equation (26)):

�� c[� �rr er � er � er � 1

r�� rr � �� er � e� � e� � er � e� � e� � e� � er � e�

� e� � er � e�� e� � e� � er � e� � e� � er�]� (A.1)

and the total stress is given by (see Equation (27))

�� rr c

�� rr �

2� �rr

r 4� rr

r2� 4� ��

r2

��er � er

�� c

��

2� ��r 2� ��

r2� 2� rr

r2

��e� � e�

�� c

��

2� ��r 2� ��

r2� 2� rr

r2

��e� � e�� (A.2)

Proof. From Equation (24) it follows that

��r� � �er � er�� rr � � rr��er � er�� e� � e� �� e� � e� �

� �e� � e�� e� � e�� (A.3)

where use has been made of the identity:

��A� � A� �� A� (A.4)

with � being a scalar field and A being a second-order tensor field. Note that in the sphericalcoordinate system (r, � , �),

��

rer � 1

r

�

�e� � 1

r sin �

�

�e�� (A.5)

It can be shown, after lengthy derivations involving the coordinate transformations be-tween the spherical coordinate system with the base vectors {er , e� , e�} and the Cartesiancoordinate system with the base vectors {e1, e2, e3}, that




��er � er� � 1

r

�er � e� � e� � er � e� � e� � e� � er � e� � e� � er � e�

��

��e� � e� � � 1

r[e� � er � e� � �cot ��e� � e� � e� er � e� � e�

� �cot ��e� � e� � e�]�

��e� � e�� 1

r[e� � er � e� � �cot ��e� � e� � e� � er � e� � e�

� �cot ��e� � e� � e�]� (A.6)

Using Equations (A.5) and (A.6) in Equation (A.3) then results in

��r� � � �rr er � er � er � � rr � ��r

�er � e� � e� � er � e� � e�

� e� � er � e� � e� � er � e�� e� � e� � er � e� � e� � er

�� (A.7)

Equation (A.7) gives the gradient of the Cauchy stress in the spherical coordinate system,which is a third-order tensor. It is seen from Equation (A.7) that �� has 7 non-zero com-ponents for the current spherical shell problem, while it has 5 non-zero components for thecylinder problem [12]. Using Equation (A.7) in Equation (4) will immediately give the ex-pression for�� listed in Equation (A.1).

Next, the gradient of the third-order tensor �� , which is a fourth-order tensor, can be de-termined from Equation (A.7), after very tedious derivations involving the coordinate trans-formations mentioned above, as

��r� � � ��rr er � er � er � er �� rr

r 2� rr

r2� 2� ��

r2

��er � er � e� � e�

� er � er � e� � e�� rr

r �

��

r � rr

r2� � ��

r 2

��er � e� � er � e�

� er � e� � er � e� � er � e� � e� � er � er � e� � e� � er

� e� � er � er � e� � e� � er � e� � er � e� � er � er � e�

� e� � er � e� � er�� e� � e� � er � er � e� � e� � er � er�

�� r� 2� rr

r 2 2� ��

r2

��e� � e� � e� � e� � e� � e� � e� � e�

�

� � ��r

�e� � e� � e� � e� � e� � e� � e� � e�

�

�� rr

r2 � ��

r2

��e� � e� � e� � e� � e� � e� � e� � e�

� e� � e� � e� � e� � e� � e� � e� � e� �� (A.8)




where use has been made of Equation (A.5) and Equation (A.4), which also holds whenA is a third-order tensor field. It is seen from Equation (A.8) that �� has 21 non-zerocomponents for the spherical shell problem here, while it has only 10 non-zero componentsfor the cylinder problem [12].

It follows from Equation (A.8) that

�2��r� � � ��r� �� rr �

2� �rr

r 4� rr

r2� 4� ��

r2

�er � er

��

2� ��r 2� ��

r 2� 2� rr

r2

��e� � e� � e� � e�

�� (A.9)

Substituting Equations (24) and (A.9) into Equation (6) will immediately yield Equation (A.2).

REFERENCES

[1] Timoshenko, S. P. and Goodier, J. N. Theory of Elasticity, 3rd edition, McGraw-Hill, New York, 1970.

[2] Gao, X.-L. An exact elasto-plastic solution for a thick-walled spherical shell of elastic linear-hardening material withfinite deformations. International Journal of Pressure Vessels and Piping, 57, 45–56 (1994).

[3] Gao, X.-L. Strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic-plastic material. Mechanics Research Communications, 30, 411–420 (2003).

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[5] Casal, P. La capillarite interne. Cahier du Groupe Francais d’Etudes de Rheologie C. N. R. S., VI, 31–37 (1961).

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[10] Altan, B. S. and Aifantis, E. C. On some aspects in the special theory of gradient elasticity. Journal of the Mechan-ical Behavior of Materials, 8(3), 231–282 (1997).

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[12] Gao, X.-L. and Park, S. K. Variational formulation of a simplified strain gradient elasticity theory and its applicationto a pressurized thick-walled cylinder problem. International Journal of Solids and Structures, 44, 7486–7499(2007).

[13] Vardoulakis, I. and Sulem, J. Bifurcation Analysis in Geomechanics, Blackie/Chapman & Hall, London, 1995.

[14] Gao, X.-L. and Rowlands, R. E. Hybrid method for stress analysis of finite three-dimensional elastic components.International Journal of Solids and Structures, 37, 2727–2751 (2000).

[15] Zettl, A. Factorization of differential operators. Proceedings of the American Mathematical Society, 27, 425–426(1971).

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