355_1- coulamb yield surface and the dissipation function

7/29/2019 355_1- Coulamb Yield Surface and the Dissipation Function

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Soil Mechanics and Theories of Plasticity

ON THE COULOMB YIELDSURFACE AND RATE OFDISSIPATION OF ENERGY

bWAI-FAH CHE

January 196

Fritz Engineering Laboratory Report No.355


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SOIL MECHANICS AND THEORIES OF PlASTICITY

On The Coulomb Yie ld Sur fa ce andRate of Dissipation of Energy

byWA r't'FiH CHEN'

This work has been carried out as partof an invest igat ion sponsored by The Inst i tuteo f Resea rch, Lehigh University

Fr i tz Engineering LaboratoryD e p a r t m e n ~ of Civil Enginering

Lehigh UniversityBethlehem, Pennsylvania

January 1968

Fr i tz Engineering Laboratory Report No. 355.1


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TABLE OF CONTENTS

Page1. THE FLOW RULE AND DISSIPATION FUNCTION 12. APPENDIX - THE COULOMB YIELD SURFACE IN 6

PRINCIPAL STRESS SPACE

3. REFERENCES 94. SYMBOLS 105. FIGURES 116. ACKNOWLEDGMENTS 15

i i


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1 . THE FLOW RULE AND DISSIPATION FUNCTION

The purpose of this repor t is to show tha t the rate of diss ipat ionof energy which is uniquely determined by th e p la sti c s tr ai n rate isgiven by

D = 2c t a n ( ~ - L: E: t (1 )

for the idealized pe r f ec t l y p la s t i c so i l s obeying Coulomb's yie ld*r i te r ion and i t s associated flow rule [ lJ where c is the cohesion, 0

is the angle of internal f r ic t ion of the so i l and t denotes a tensi lepr inc ipa l component of th e plas t ic s t ra in rate tensor.

According to Coulomb's c r it er io n , p la s ti c flow can occur undera constant s tate of s t ress which is represented by a point on th e r igh thexagonal pyramid eq ually in clin ed to th e a I ' a Z ' 03 axes [2J (Fig. 1),or for example, by a point on th e hexagon in Fig. 2 which is the in te r -sect ion of the pyramid with a plane perpendicular to t he GZ-axis and a t

... ..

"'''Numbers in b ra ck ets designate referencesa t end of repor t .

-1-


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a distance crz from the origin. Since th e so i l is i sot ropic , th e princ ipa l"axes of the p la st ic s tr ai n ra te must coincide with the princ ipa l axes ofs t ress and the principal components of the strain rate in th e 0'1' 02' cr3direc t ions wil l be denoted by Z' e3 in the (cr l , GZ' 03 ) diagram.The rate of dissipat ion o f energy is given by

(2 )

which is seen most eas i ly i f one takes the end of s t ress vector a t thevertex of th e pyramid.

Consider a point on th e pyramid in Fig. 1 which does not coincidewith an edge. The conc ept o f perfec t plast ic i ty [3J requires that theassoc ia ted plast ic s t r a i n rate vector must be normal to the face of th epyramid. For example, on th e face V-C-D, th e associated p la st ic s tr ai n

rate vector is normal to O'Z-axis so that. 2 = O. Hencel -3 tan2 (TI/2 - 0/2) from the geometrical relation of l ine C' n'in Fig. 2. Substituting the strain ra tes into (2) the resul ts of (1 )is then fol lowed. Here t = 83 > [e l l

For a s t ress point which coincides with an edge, however, th e.plastic strain rate vector must l ie between th e direc t ions of the normalsto th e tw o faces of the pyramid which meet a t th e edge. For example)th e p la st ic s tr ai n ra te vectors drawn in th e f ac e bounded by the normals

Ito th e sides which meet a t th e corner of the hexagon, e .g . , point Cin Fig. 2, are th e projections on the plane of Fig. 2 of possible plast ic


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s t ra in rates for stress points lying on th e edge V - C of the yieldpyramid in Fig. 1.

Fo r the stress point (crI , aZ' G3) on the edge V - C, one can2choose a neighboring point [cr1 + p, crz + p, 03 + p tan (n/4 - 0/2J on

the same edge and since th e two points should r e su l t in a samediss ipa t ion o f e ne rg y, it follows that

~ )2 (3)subst i tu t ing e l + 82 of (3 ) into (2), and again the resu l t of (1) isfollowed because th e components of the plas t i c strain ra te on edge V - Care 8 1 ~ 0, 82 ~ 0, 83 > and also 3 has th e numerically larges t valueamong those three components.

As for the specia l edges V - B, V - D, V - F, where two non-zero tens i le components of the p la st ic s tr ain rate ex i s t , one can p ro ceedin a similar way as for edge V - C and obtain the corresponding expressionof (3). When, for example, edge V - B is se lec ted, the expression is

2 (4 )

Again, th e resu l t of (1 ) is followed on account of (4).

In the same way, it can be s ~ o w n tha t Equation (1 ) holds a t everystress point of the pyramid since the components of the plas t i c s t ra inr a tes sat isfy the condition

2 TTtan (4 (5 )


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. where denotes th e principal compressive component of p la st ic s tr ai ncrate tensor.

Tresca yield cr i te r ion may be considered as a so i l for whicho= 0 and c = k where k is th e shear yield s tress , i t follows from th eexpression (1 ) that th e rate of energy dissipat ion reduces to

D 2k max; (6 )

where max \ \ denote s the absolute value of the numer ica l ly l a rges tpJ?incipal component of the p la st ic s tr ai n ra te . Expression (5)" then.becomes the familiar incompressibi l i ty condition

o (7 )Expression (6 ) for the r a t ~ of dissipat ion of energy was f i r s t obtainedby Hodge and Prager [4J for the special case of plane s t ress , 02 = 0and la te r extended by Shield and Drucker [5J to th e gener al cas e.

For the par t icular case of plane s t ra in , one of th e princ ipa lcomponents of th e plast ic s tra in rate is . always zero, and th e other twocomponents have the relat ion

so that th e maximum rate o f en gin ee rin g she ar strain is given byY max = t - c = 1 + 8in0

(8)

(9 )


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thus for plane s t ra in

D = c cos0 ymax

(10)

-5

agreeing with the expression obtained by Drucker and Prager [ l J .


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-6355.12. APPENDIX

THE COULOMB YIELD SURFACE IN PRINCIPAL STRESS SPACE

Shield [2J , following upon related work by Drucker [6J, extendedCoulomb's Law of Failure in two-dimensional problems to a unique yieldsurface appropriate for th e gener al t re atment of three-dimensionalproblems. The purpose here i s to outline a geometrical method ofconstruct ing such a surface in principal s t ress space showing that thisyield surface is a r igh t hexagonal pyramid equa ll y i nc li ned to thecr1 , GZ' G3 axes (see Fig. 1). I ts intersect ion with the plane bf Fig.2, perpendicular to the GZ-axis and a t a distance GZ from th e origin,is a hexagon. In t he f ol lowing the method of constructing th e yieldcurve in th e plane of Fig . 2 wil l be chosen. The value of a2 is fixedfor th e curve in th e (cr1 ' coordinate plane.

According to the Mohr's graphical representation of st ressesin (cr, T) coordinates, it i s well-known that a l l po in ts r ep resen tingpossible pairs of (cr, T) values a t a point of th e so i l are within th eshaded cu rv i li nea r t ri ang l e bounded by the three principal s tr es s c ir cl esas shown in Fig. 3(a) . Failure of th e so i l can occur only when thelargest of the circles touches the two straight l ines while the inter-mediate principal stress can have any value between the largest andsmallest principal stresses . The det ermina ti on o f th e cr i t i ca l circlerequires th e considerat ion of the relat ive magnitudes of th e threeprincipal s t resses . There are s ix possible orderings of th e relativemagnitudes of the st resses cr l , a2 , cr3 which determine th e s ix yieldl ines A 'B ', B 'C ', C'D', D'E' , EtF ' and F'A' as shown in Fig. 3. For


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a r ight hexagonal pyramid e qu ally in cline d to the a2 , IT3 axes andwith i t s vertex V a t the point crl = crz = G 3 = C cot0 (see Fig. 1).Clearly, such a pyramid is fully defined by th e hexagon A'-B'-C'-Df-E'-F'-A'in the (cr l , 03 ) coordinate plane. Figure 1 shows t he hexagonal pyramidwith the l ine VO as i t s center l ine and every two faces of th e pyramidopposite to each other i s para l le l to a corresponding axis . The stresspoint V in the f igure corresponds to a s tate of s t ressescr l = aZ = a3 c cot0 tha t i s , th e point Mo of figure 3(a) .


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3 . REFERENCES

-9

1 . "So i l Mechanics and P l a s t i c Analys i s or L im i t Des ig n" ,by D. C. Drucker and W. Prager , Quarterly ofApplied Mathematics , Vol. 10 , pp . 157-165, 19522. "On Coulomb's Law of F ailu re in Soils",

by R. T. Shield, Journal of th e Mechanics ofPhysics of Solids, Vol. 4, pp. 10-16, 19553 . "A More Fundamental Approach to Stress-Stra in Relations",

by D. C. Drucker, Proceed ings of th e Firs t U. s.National Congress of Applied Mechanics, pp. 487-491,June 1951

4. "Limit Design of Reinforcements of Cut-Outs in Slabs" ,by P. G. Hodge, Jr. and W. Prager, Brown UniversityReport Bll-2 to Office of Naval Research, August 1951

5. "The Application of Limit Analysis to Punch-IndentationProblems", by R. T. Shield and D. C. Drucker,Journal of Applied Mechanics, A.S.M.E., Vol. 75,pp. 453-460, 1953

6. "Limit Analys is of Two an d Three Dimensional Soil MechanicsProblems", by D. C. Drucker, Journal of the Mechanicsand Physics of Solids, Vol. 1, pP. 217-226, 1953


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4. SYMBOLSc0DDA81 , 2' 3

t

ecYmax'T

CY

CY l , 0"2' c:r3

CohesionAngle of internal f r ic t ionRate of diss ipat ion of energy per un i t volumeRate of diss ipat ion of energy per uni t areaPrinc ipa l component of th e plast ic s t ra in

rate tensorTensile pr incipal component of th e plas t ic

s t ra in rate tensorCompressive pr inc ipa l component of th e plas t i c

s t ra in ra te tensorMaximum rate of engineering shear strainShearing s t ressNormal s t ressPrinc ipa l component of th e s t ress tensor


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5. FIGURES


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o

E......... --JII'F

OA = OC = OE= 2c tan ( . .1L_i)4 2

Fig. 1 The Coulomb.Yield Surface

/

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0-1= 0-4 =2 c to"n (...1!:.. - i )4 20-2 =0 - 3= 2 e ta n ( ~ + -!L )4 2

0""1/0//

/

3

EI------....... - - ~ F1I/I

//

2

Fig. 2 Section of the Coulomb 'Yield Surface in F ig . 1.The Yield Curve in a Plane Perpendicular to th eGZ-axis and a t a D i s t a n ~ e Gz from the Origin


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......... .,........

0 b - e + ~ - e - I ~I +

~ ~ I ~ ~ I ~ .........., -....,.....AI c cc cb .... ...o 01\1 C\J C\Jb' II IIq rr>

........,. 0- I..c I"'--" 0 0

.. II- (\ JI I0 0-LL

....................... '"tr .......... rtl

u b0 0 b:E w

(\ J

bAI0- tr1\1 AI~ If\1

", ,-...b 0---- '..........

~ C~ ..........

Fig . 3 Mohr's Representa t ion of a St res s and theCoulomb Yield Cr i te r ion

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6 ACKNOWLEDGMENTS

This-report is par t of a research project on Soil Mechanicsand Theories of Plas t ic i ty carried out a t F ri tz EngineeringLaboratory, Lehigh University, Bethlehem, Pennsylvania. Dr. L. S.

tBeedle is Director of the Laboratory. The project sponsor isthe Inst i tute o f Resea rch, Lehigh University. Professor G. R.Jenkins is Director of the Ins t i tu te . The author is t hank fu l f orhis support.

The author is appreciative of th e enc-ouragement of Dr.H. Y. Fang and also indebted to ' Mrs. F ie ld in g fo r typing themanuscript.

355_1- coulamb yield surface and the dissipation function

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