324 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS

The arguments of CTMCThe arguments here are those of SUMC (exept for the array DGAM) plus DMATX and EPFLAGwhich carry, respectively, the tangent operator and the elastoplastic ag that indicates whetherthe elastic or elastoplastic operator is to be computed in CTMC. The output values of thearguments of SUMC are taken as input values by CTMC.

Function calls from CTMCFunction calls here are the same as from subroutine CTTR (for the Tresca model implementa-tion). Refer to page 295.

8.3. The DruckerPrager modelWe now move on to the last model implementation discussed in this chapter. Here we focus ona DruckerPrager model with piecewise linear isotropic strain hardening. The most importantresults of this section can be found as indicated in the table below.

integrationalgorithm

owchart

pseudo-code

FORTRAN code

Figure 8.15

Boxes 8.88.10

subroutine SUDP(Section 8.3.2)

iso-error map Figure 8.16

consistent tangent FORTRAN code

subroutine CTDP(Section 8.3.5)

Following the format of the previous sections, a brief summary of the constitutiveequations adopted in the present implementation is presented below.

The implemented DruckerPrager constitutive equations

The DruckerPrager yield surface is dened by means of the yield function

(, c) =

J2(s()) + p() c, (8.101)

whereJ2 = 12s : s; s= p() I, (8.102)

p = 1/3 tr[] is the hydrostatic pressure and c is the cohesion. The constants and are chosen according to the required approximation to the MohrCoulomb criterion.Their formulae for outer edge, inner edge, plane strain, uniaxial tension/compression andbiaxial tension/compression matching are given, respectively, by expressions (6.122), (6.123)and (6.124), (6.126) and (6.127).

COMPUTATIONS WITH OTHER BASIC PLASTICITY MODELS 325

The generally non-associative ow rule is adopted in the present implementation of theDruckerPrager model. The corresponding ow potential is

(, c) =

J2(s()) + p(), (8.103)where the constant depends on the dilatancy angle, , and according to the adoptedDruckerPrager approximation to the MohrCoulomb law may be given by (6.163), (6.164)or (6.165) respectively for the outer, inner edge and plane strain match. The ow rule reads

p = N, (8.104)where, on the smooth portion of the yield surface, the ow vector is given by

N=

=1

2

J2(s)s+

3I. (8.105)

At the cone apex, where the potential is singular, the ow vector is a subgradient of ; thatis, N is a vector contained in the complementary cone (refer to the schematic illustration ofFigure 6.20(b), page 176).

As for the MohrCoulomb implementation of the previous section, associative isotropicstrain hardening is adopted here. The hardening law is dened by (8.61), together with theassociative evolution equation for the accumulated plastic strain derived for DruckerPragerplasticity in Section 6.6.3:

p = . (8.106)Analogously to the MohrCoulomb implementation, the DruckerPrager counterpart ofrelation (8.64):

p =

pv, (8.107)

will be needed in the derivation of the return-mapping equation for the apex of the DruckerPrager cone. This relation is obtained by combining the non-associative version of (6.161)with the evolution law (8.106).

8.3.1. INTEGRATION ALGORITHM FOR THE DRUCKERPRAGER MODEL

The integration algorithm for the DruckerPrager model is simpler than its counterpartspresented earlier in this chapter for the Tresca and MohrCoulomb models. This relativesimplicity stems from two main features of the present material model. Firstly, only onesingularity exists in the DruckerPrager yield surface its apex. This is in contrast withthe two and three possible singularities identied, respectively, in the implementation of theTresca and MohrCoulomb models. Secondly, the DruckerPrager yield surface, as well asthe ow vector eld resulting from the generally non-associative ow potential, are fullysymmetric about the hydrostatic axis.

Consider the general return-mapping update formula for the stress tensor for materialswith linear elastic law:

n+1 = trialn+1 De :Nn+1, (8.108)where De :Nn+1 is the return vector. As a consequence of the symmetry about thepressure axis (see Figure 6.20, page 176), whenever the above formula is applied (regardless

326 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS

of whether n+1 lies on the smooth portion of the cone or on its apex), the return vectoris always parallel to the plane that contains trialn+1 and the hydrostatic axis (see Figure 8.14,page 329). Thus, without loss of generality, the return-mapping algorithm can be completelyformulated in such a plane of the stress space. The corresponding algorithms are derived inwhat follows. Note that, since the denition of the ow vector Nn+1 in the smooth portionof the cone differs from that at the apex singularity, two possible explicit forms exist for thereturn-mapping algorithm. These are treated separately below.

Return to the smooth portion of the coneOn the smooth portion of the cone, the ow vector is dened by (8.105). The correspondingincrement of plastic strain then reads

p = Nn+1 = (

12

J2(s)sn+1 +

3I

). (8.109)

The corresponding stress update formula is

n+1 = trialn+1 [2G (Nd)n+1 + K (Nv)n+1]

= trialn+1 (

GJ2(s)

sn+1 +K

3I

). (8.110)

Simplication of the above expression can be obtained by noting that, due to the denition ofJ2, the following identity holds

sn+1J2(sn+1)

=strialn+1

J2(strialn+1). (8.111)

Its substitution into the stress update formula gives

n+1 = trialn+1 (

GJ2(strial)

strialn+1 +K

3I

), (8.112)

which, equivalently, in terms of deviatoric and hydrostatic components reads

sn+1 =(1 G

J2(strialn+1)

)strialn+1

pn+1 = ptrialn+1 K .(8.113)

It should be noted that the deviatoric updated stress, sn+1, is obtained by simply scaling downthe trial deviatoric stress, strialn+1, by a factor that depends on . Recall that in the von Misesimplicit return mapping derived in Chapter 7 the updated deviatoric stress is also obtained byscaling its elastic trial counterpart.

The consistency condition in the present case is given by

n+1

J2(sn+1) + pn+1 c(pn+1) = 0, (8.114)

COMPUTATIONS WITH OTHER BASIC PLASTICITY MODELS 327

where the update accumulated plastic strain is obtained from the discrete version of (8.106):pn+1 =

pn +

p (8.115)with

p = . (8.116)The substitution of the above expression and (8.113) into the consistency condition results

in the following (generally nonlinear) equation for :

()

J2(strialn+1)G + (ptrialn+1 K ) c(pn + ) = 0. (8.117)

After solution of the above equation, the stress is updated by (8.113).

Return to the apex

At the apex, the return vector must be contained in the complementary cone schematicallyillustrated in Figure 6.20(b) of page 176. The derivation of the return-mapping procedurein this case is completely analogous to that of the MohrCoulomb apex return described initem 4 of page 299. The consistency condition (8.114) is reduced to

c(pn + p)

ptrialn+1 + K pv = 0. (8.118)

Further, with the introduction of the discretised form of (8.107) in the above equation, weobtain the nal return-mapping equation for the DruckerPrager apex:

r(pv) c(pn + pv) ptrialn+1 + K pv = 0, (8.119)where

,

. (8.120)

This equation is analogous to the return-mapping equation (8.85) of the MohrCoulombimplementation. Its geometrical representation is shown in Figure 8.13. After the solutionof (8.119) for pv, we update

pn+1 := pn +

pv,

n+1 := (ptrialn+1 K pv) I.(8.121)

Note that for non-dilatant ow ( = 0), the return to apex does not make sense in the presentcontext and the comments made immediately after expression (8.86) for the MohrCoulombimplementation apply.

Remark 8.4. For perfectly plastic materials, c is constant and, for linearly hardening models,the hardening function reads c(p) = c0 + H p, where H denotes the hardening modulus.In such cases, the return-mapping equations (8.117) and (8.119) respectively for the smoothportion and apex of the DruckerPrager cone are linear and is computed in closed form.

328 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS

p

J2( )s

-K I

updatedDruckerPrager

cone

complementarycone

pv

n+1trial

n+1= p In+1 pn+1trial

Figure 8.13. DruckerPrager model. Return mapping to apex.

Selection of the appropriate return mappingHaving derived the two possible forms of return mapping for the DruckerPrager model, astrategy for selection of the appropriate one is necessary for the complete denition of theintegration algorithm. Recall that the purpose of the selection procedure is to ensure that thevalidated return mapping in the present case, either the return to the smooth wall or thereturn to the apex rigorously satises the general implicit return-mapping equations (7.25),derived in Chapter 7.

The selection strategy for the DruckerPrager model is simple. There are only two possiblereturn mappings. Thus, if the application of one of them generates a contradiction, then theother one must be valid. A possible selection strategy is summarised in the following steps(refer to Figure 8.14 for a graphical representation):

Firstly, apply the return algorithm to the smooth part of the DruckerPrager cone. Fromthe corresponding deviatoric stress update formula (8.113)1, it follows that

J2(sn+1) =

J2(strialn+1)G . (8.122)

If, after determination of from the related consistency condition, the following issatised

acone

J2(strialn+1)G 0, (8.123)then the returned stress indeed lies on the DruckerPrager cone and the return mappingis validated.

Otherwise, the returned stress lies outside the updated elastic domain and is notadmissible. In this case, the return mapping to the apex must be applied. The resultsobtained by the apex return are then necessarily valid.

The above selection procedure ensures the consistency of the selected return algorithmregardless of any prescribed hardening/softening law.