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26 Micro-Mechanical Finite Element Models for Crystal

Plasticity

Surya R. Kalidindi

26.1 Introduction

“Crystal Plasticity” refers to study of plastic deformation in single crystal and polycrystalline

materials while taking into account explicitly the details of physics and geometry of deforma-

tion at the crystal (also called grain) level. At low homologous temperatures, the dominant

mode of plastic deformation in crystalline materials is slip on specific crystallographic planes

in specific directions. Crystal plasticity aims to incorporate these and other such details at the

grain scale in the description and formulation of elastic-plastic constitutive models for single

crystals and for polycrystalline materials.

Detailed accounts of the development of the crystal plasticity theory and its many suc-

cesses (especially in predicting anisotropic stress-strain response of high stacking fault energy

polycrystalline cubic metals together with the concurrent evolution of the underlying averaged

texture) have been presented in other chapters. In many of these models, simplifying assump-

tions are routinely made in making the transition from the response of a constituent single

crystal to the response of the representative polycrystalline aggregate. A typical consequence

of such assumptions is that the solutions for the polycrystals often violate either equilibrium

or compatibility or both. Consequently, although these simplistic models provide reasonably

accurate predictions in the averaged sense at the polycrystal level, their predictions at the

constituent single crystal level can be quite erroneous.

With the advances made in finite element techniques, it is now possible to build micro-

mechanical polycrystalline finite element models that can provide accurate solutions of stress

and strain fields in a given polycrystalline aggregate that automatically satisfy both equilib-

rium and compatibility (in a weak numerical sense). Such models therefore raise the exciting

possibility of predicting accurately the underlying physical phenomena occurring at the grain

scale. This chapter expounds on the development and use of micro-mechanical finite elementmodels for elastic-plastic response of polycrystalline aggregates.

26.2 Theoretical Background

The necessary theoretical background is briefly reviewed next. For further details, the reader

is referred to Chapter 5 and to the references listed therein.

Continuum Scale Simulation of Engineering Materials: Fundamentals – Microstructures – Process Applications.

Edited by Dierk Raabe, Franz Roters, Frederic Barlat, Long-Qing Chen

Copyright c 2004 Wiley-VCH Verlag GmbH & Co. KGaA

ISBN: 3-527-30760-5


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26.2 Theoretical Background 517

Figure 26.1: Schematic description of updated and total Lagrangian schemes for integration of

constitutive relations.

why the theoretical framework presented above is formulated somewhat different from that

presented in the other chapters. Figure 26.1 illustrates schematically the main features of both

the total Lagrangian and the updated Lagrangian methods. The updated Lagrangian scheme is


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518 26 Micro-Mechanical Finite Element Models for Crystal Plasticity

centered around updating the end configuration of a given time step to become the initial con-

figuration for the next time step. For example, in Figure 26.1(a), configuration corresponding

to time t serves as the initial configuration for the time step (t, τ ). As one marches forward inan updated Lagrangian scheme, there is no need to carry forward the information related to the

initial configuration corresponding to the beginning of the last time step. On the other hand,

a total Lagrangian scheme continues to use always the initial configuration corresponding to

time zero as the reference configuration, and carries forward only the necessary state variables

from the end of one time step to the next. The state variables contain enough information to

define uniquely the relaxed configuration at the end of each time step. There are several advan-

tages of using a total Lagrangian scheme. For example, by using total Lagrangian schemes in

crystal plasticity models, one can avoid the need to compute the current lattice orientation of

the crystal at the end of each time step; instead the orientation of the crystal can be computedonly when it is required. Also, the total Lagrangian scheme for crystal plasticity model de-

scribed in Section 2.1 allows easy formulation of a fully implicit time integration procedures

(Kalidindi et al. 1992). Finally, although deformation twinning as a mode of plastic deforma-

tion is not addressed in this chapter, it has been demonstrated that the total Lagrangian scheme

offers significant advantages when deformation twinning needs to be incorporated into crystal

plasticity models (Kalidindi 1998; see also Chapter 27).

26.2.3 Fully Implicit Time Integration Procedures

The computational problem at hand is posed as one with fully prescribed deformation his-

tory. However, it can be easily extended to situations where only certain components of the

deformation gradient history are prescribed along with certain other components of the stress

history. It is assumed that at any time t, the deformation gradient F(t) and the list of variables{F p(t), sα(t)} are known at a given material point in a single crystal region. Also given isthe deformation gradient, F(τ ), after a small increment of time ∆t (i.e. τ = t + ∆t). Theobjective of the computational scheme is to find the Cauchy stress T(τ ) and update the list of variables {F p(τ ), sα(τ )} at the given material point.

Fully implicit time integration of Equation (26.3a) leads to (Kalidindi et al. 1992)

Fp(τ ) ∼= {1+ ∆t L p(τ )}Fp(t), (26.6)

and

F p−1(τ ) ∼= F p−1(t) {1− ∆t Lp(τ )} . (26.7)

For brevity of notation, Equation (26.3b) may be conveniently expressed as

∆t L p(τ ) =α

∆γ αSαo (26.8)

In Equation (26.8), ∆γ α is a function of the resolved shear stresses and the slip resistancesdefined by equations (26.4) and (26.5). Therefore, it is reasonable to express ∆γ α as

∆γ α = ∆γ̂ α (T∗(τ ), sα(τ )) (26.9)


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26.2 Theoretical Background 519

Substituting Equations (26.7), (26.8), (26.1) and (26.2c) in Equation (26.2a), we can derive

(Kalidindi et al. 1992) the following equation for T∗(τ ),

T∗(τ ) ∼= T∗ tr −α

∆γ αCα (26.10)

where

T∗ tr = C

1

2{A− 1}

(26.11)

Cα = C

1

2Bα

(26.12)

A = F p−T

(t) FT (τ ) F(τ ) F p−1

(t) (26.13)

Bα = A Sαo + SαT

o A (26.14)

For the case where the slip systems are assumed to be non-hardening, Equation (26.10) rep-

resents a set of six non-linear equations with six unknowns. The non-linearity is due to the

dependence of ∆γ α on T∗(τ ). Equation (26.10) can be solved for the stress componentsusing a modified Newton-type algorithm (Kalidindi et al. 1992).

For situations involving hardening of the slip systems, additional equations have to be

formulated by integrating Equation (26.5). However, it is suggested that Equation (26.10)

be solved first using the best guesses of the various slip resistances and then to solve for the

new values of the slip resistances using a two-level iterative procedure. Such a two level

iterative procedure was found to work very well for the type of models described here. After

solving for T∗

(τ ), and updating the slip resistances {sα

(τ )}, ∆γ α

can be computed usingEquations (26.4) and (26.5),F p(τ ) using Equation (26.6),F∗(τ ) using Equation (26.1),E∗(τ )from Equation (26.2c), and finally T(τ ) from Equation (26.2b). These calculations mark theend of the time-step, and by repeating these calculations successively, one can march forward

in time and simulate a large deformation process. Note that there is no need to update the

crystal orientations at the end of each time step, since the initial configuration is kept fixed in

these calculations (corresponding to time zero).

At any point during the simulation of a given deformation process, the lattice orienta-

tions in the current configuration can be computed using the rotation component R∗, defined

through polar decomposition of F∗. Let [g] represent the coordinate transformation matrixfrom a frame aligned with the initial crystal lattice to the fixed global reference frame, i.e.

{m}g

= [g] {m}c

(26.15)

where {m}c represents a specific crystal direction in the crystal coordinate frame and {m}g

describes the same vector in the global reference frame. The rotation R∗, whose components

are denoted as [R∗] in the fixed global reference frame, causes the same vector to take the newpositionmt given by

{mt}g

= [R∗] {m}g = [R∗] [g] {m}c (26.16)

Therefore the product [R∗][g] then provides us the new coordinate transformation matrix be-tween the new lattice orientation of the crystal and the fixed global reference frame.


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26.2.4 Polycrystal Homogenization Theories

A number of polycrystal homogenization theories are used in practice to scale up to the re-

sponse of polycrystals from the response of the constituent single crystals. Taylor (Taylor

1938) proposed a simple isostrain model for predicting the macroscopic yield strength of

polycrystals, and this model with relatively minor improvements continues to be the most

widely used theory even today. Major improvements to the basic Taylor model have been pro-

posed in literature. These include the self-consistent models (Molinari et al. 1987; Toth et al.

1994) and the LAMEL model (Van Houtte et al. 1999; Van Houtte et al. 2002) that have been

discussed in Chapter 22. A common feature of all these analytical approaches to the polycrys-

tal homogenization theory is that they violate either equilibrium or compatibility or both in

the given polycrystalline aggregate. As an alternative to these analytical approaches, micro-mechanical finite element models have been proposed (Bronkhorst et al. 1992; Kalidindi et al.

1992; Beaudoin et al. 1993; Beaudoin et al. 1995; Sarma and Dawson 1996) that satisfy both

equilibrium and compatibility in a weak numerical sense in the given polycrystal. These are

described in the next section.

26.3 Micro-Mechanical Finite Element Models

In a typical non-linear finite element stress analyses problem, the weak form of the principle

of virtual work (including equilibrium and the boundary conditions) is satisfied in a given do-main, discretized into a finite element mesh, by evaluating the constitutive response at specific

locations in each element called integration points. In the present case, the given polycrys-

talline aggregate is discretized into a finite element mesh, such that each grain is modeled

by one or more finite elements to allow for non-uniform deformations between the grains

and within the grains. The single crystal constitutive behavior, together with the fully im-

plicit time integration procedure described above, can be incorporated into a finite element

code. In particular, it has been successfully incorporated into the commercial finite element

code ABAQUS, through use of a user-defined subroutine for material behavior called UMAT

(ABAQUS 2003). The code has been written such that it keeps track of the necessary solution

dependent internal state variables at each of the integration points in the mesh, and updates

them during an imposed deformation increment using the approach described earlier. In ad-

dition, the implicit finite element codes also require a computation of the Jacobian matrix

at each integration point in the mesh in order to compute a better guess of the overall non-

uniform deformation (and strain) field in its iterative attempts to satisfy the principle of virtual

work during an imposed increment of deformation. This Jacobian matrix can be computed

numerically (Kalidindi et al. 1992) or analytically for the crystal plasticity model described

here.

The algorithms used for coding the user subroutine in ABAQUS are beyond the scope of

this chapter. Interested readers can find the details in (Kalidindi et al. 1992). Instead, a number

of different applications using these codes are described in the next section.


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26.4 Examples 521

26.4 Examples

26.4.1 Predictions of Deformation Textures

Figure 26.2: A simple micro-mechanical

finite element model of a polycrystalline ag-gregate.

One of the most elementary finite element models that was used in modeling a polycrys-

talline aggregate (Kalidindi et al. 1992; Bachu and Kalidindi 1998) is shown in Figure 26.2.

In these models, the polycrystalline aggregate is given a rectangular parallelepiped geometry.

Each element in the finite element mesh (shaded region in Figure 26.2) is cube-shaped, and

represents an individual crystal or grain in the aggregate. The different finite elements in the

mesh are given different initial crystal orientations such that the average initial texture in the

model is representative of the initial texture in the sample. Figure 26.4.1 shows the measured

and predicted textures for a representative high stacking fault energy cubic polycrystalline

metal (the measurement is for copper) from both a Taylor-type model as well as a simple

micro-mechanical finite element model of the type shown in Figure 26.2. Although the grossly

simplified Taylor-type model captures quite well all of the major features of the measured tex-

ture in the deformed sample, there are definitely certain subtle features that are better captured

by the micro-mechanical finite element model. Quantitative comparisons based on the posi-

tion and intensities of the skeletal lines in the orientation distribution functions corresponding

to the textures shown in Figure 26.4.1, along with the corresponding results from other poly-

crystal homogenization theories have been presented in literature (Delannay et al. 2002; Van

Houtte et al. 2002; see also Chapter 22). These studies concluded that the micro-mechanical

finite element models provide significantly improved predictions of the deformation textures,

compared to the highly simplified Taylor-type model.

The coarse finite element models of the type shown in Figure 26.2 have also been used

successfully to predict shapes of anisotropic yield surfaces for cubic polycrystals (Kalidindi

and Schoenfeld 2000).

26.4.2 Predictions of Micro-Texture

In spite of the remarkable success of the modern crystal plasticity theory in predicting rea-

sonably accurately the anisotropic stress-strain response and the averaged texture evolution

during large plastic strains in polycrystalline metals, there does not yet exist a comprehensive

theory that is currently capable of predicting accurately the crystal-scale evolution of the mi-

crostructure, sometimes referred to as micro-texture. Micro-texture is extremely important in

developing recrystallization models. It is also intuitive that understanding the accommodation


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(b) Taylor Model Prediction (c) FE Model Prediction

(a) Measured Texture

Figure 26.3: Measured and predicted

textures in deformed samples of OFHC

copper in plain strain of −1.

0.

of plastic deformation at the local scale has important bearing on fracture and other failure

related properties of a number of brittle solids. The original experiments of (Barrett and

Levinson 1940), and the more recent repetitions of these experiments by (Panchanadeeswaran

et al. 1996) and by (Bhattacharyya et al. 2001), have cast serious doubts on the veracity of the

currently used crystal plasticity theory. Below, we summarize the results from a direct com-

parison between measurements in an Aluminum polycrystal and the corresponding predictions

from both the Taylor-type model and the micro-mechanical finite element model.

A sample was cut from a high purity (99.9%) Aluminum slab having a columnar grain

structure. Figure 26.4 shows the initial sample dimensions, the orientation of the columnar

crystals with respect to the sample geometry, and the surfaces that were scanned by the OIM

(Orientation Image Microscopy) technique. In this study, the plastic deformation was imposedusing a channel-die set-up, shown schematically in Figure 26.4. Although an overall reduction

of about 40% was obtained, the deformation was imposed in about four stages. At each stage

of deformation, the sample microstructure was characterized using the OIM technique.

The evolution of lattice orientations in the polycrystalline sample was modeled using both

a grossly simplified Taylor-type model and the micro-mechanical finite element model. The

finite element simulation was carried out assuming that the grains were perfectly columnar.

ABAQUS/CAE was used for generating grain boundaries in the sample. Figure 26.5 shows

the finite element mesh used in this study. It comprised of 615 C3D8 (three-dimensional 8-


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26.4 Examples 523

Figure 26.4: A schematic description of the channel-die set-up used for imposing plastic defor-mation on the quasi-columnar grained sample of polycrystalline aluminum. Also shown are the

sample initial dimensions and the final dimensions and the orientation of the columnar axis of

the constituent crystals with respect to the sample geometry.

noded solid) elements (ABAQUS 2003). The three dimensional mesh was compressed along

Side 2 (see Figure 26.5) by 40% (keeping Side 1 fixed). Sides 3 and 4 were left unconstrained.

Figure 26.6(a) shows the deformed mesh. It was found that the shape of the deformed mesh

from the finite element model matched well with the experimentally observed shape of the de-

formed sample. In particular, it was observed both in the experiment and in the finite element

simulation that the deformed sample had undergone a shear of 0.16 along positive RD on a

plane normal to TD.

In spite of the grossly simplifying assumptions, the Taylor-type model performed remark-ably well in predicting the averaged texture for the polycrystal as shown in Figure 26.7. The

finite element model predictions for the averaged final texture are better than the Taylor-type

model predictions, in that they are more diffuse as in the experiment. It should be noted

that the Taylor model, by its inherent assumption, could not predict the development of mis-

orientations in the individual crystals. More importantly, the Taylor-type model also failed

completely in predicting the rotations of several individual crystals, even in the average sense

(see Figure 26.8). On the other hand, the finite element model provided significantly better

predictions of both the rotations of individual grains as well as the development of average

misorientations (Figure 26.8). However, the finite element model failed to predict the forma-

tion of deformation bands in any of the grains, as can be seen from Figure 26.6(b), which

shows the contours of the magnitude of the neo-Euler angle between the initial measured and

the FE predicted orientations. The regions with similar orientation will show similar shadingin the plot. Experimentally, grains 5 and 10 showed development of banded orientation fields,

but the corresponding grains in the FE model showed no signs of similar deformation banding

within the grain.

However, in this experiment, it was possible to dig a little deeper by successfully esti-

mating the deviations in local deformation history compared to the macroscopically imposed

deformation. This was possible in our experiment because of the channel-die configuration,

which allowed only a smaller number of additional shears both at the sample scale as well

as the crystal scale. Using the crystal plasticity models mentioned earlier and examining all


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Figure 26.5: The finite

element mesh used to

model the columnar-

grained Aluminum

sample.

possible trial local strain rate tensors allowed by the sample geometry and imposed boundary

conditions, we were able to identify for each crystal of interest the local strain rate tensor that

resulted in the lattice rotations that matched with the OIM measurements during all stages of

the deformation history. In this analysis, we also carefully tracked the changes in the local

energy dissipation rate (due to plasticity).

This combined experimental-modeling approach resulted in establishing new insights into

the underlying physics behind the deformation of polycrystals. Certain grains in the deformed

sample exhibited more or less uniform lattice orientation (with minimal misorientations) and

were labeled as homogeneously oriented crystals. Our analyses indicated that all these grains

were relatively hard grains compared to their neighbors. In all the homogeneously oriented

crystals, there were additional shears locally in the crystal (in addition to the macroscopicallyimposed plane strain deformation history) that resulted in a decrease of the plastic energy dis-

sipation rate. Consequently, the Taylor factors of all these grains in the final deformed state

decreased significantly compared to those corresponding to their initial orientations. There-

fore, the local deformation history of the homogeneously oriented crystals is governed largely

by the need to minimize its plastic energy dissipation rate.

Certain other grains in the deformed sample were found to fragment into regions with re-

peating alternating orientation fields (Bhattacharyya et al. 2004). These grains were found to

exhibit inherent instability in crystal lattice rotations. That is, it was found for these grains that


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26.4 Examples 525

Figure 26.6: (a) The deformed FE mesh on side A after 40% compression and (b) the corre-

sponding contours of neo-Euler angles between the initial orientation and the final orientation

of the grains.

with a slight perturbation of the orientation before fragmentation (from different regions of the

crystal), different slip system activities and consequently different crystal rotations were pre-ferred in different regions of the grain. Finally, a certain number of crystals in the deformed

sample showed grain fragmentation, but the lattice orientations in the deformed grains did

not exhibit repeating orientation fields, i.e. the deformation bands in these crystals showed

non-repeating orientation fields. These grains were found to be relatively soft crystals (lower

Taylor factors than neighbors) with larger grain-sized and harder neighbors. There is a strong

positive correlation between the additional shears experienced in these bands and the addi-

tional shears experienced by their adjacent harder neighbors. Furthermore, in these crystals,

it was often noted that the Taylor factor corresponding to the final orientation was higher than


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Figure 26.7: (100) pole figures of the columnar-grained sample after 40% deformation in

channel-die compression. (a) Experimental measurement, (b) Taylor-type model prediction,

and (c) Finite element model prediction.

that corresponding to the initial orientation of these grains. We therefore conclude that differ-

ent regions of these crystals are responding to different influences from different neighbors.

Further details of this study can be found in (Bhattacharyya et al. 2004).

In summary, it is noted that using a combination of sophisticated experimental and mod-

eling techniques, it is now possible to follow the local evolution of the microstructure at theindividual crystal scale in polycrystalline metals and extract the underlying physics behind

how individual grains respond to a specific macroscopically imposed deformation history.

The reader would also benefit from the recent work of (Raabe et al. 2001) where a novel

method to experimentally measure the local deformation history is described. These develop-

ments open tremendous new possibilities for undertaking similar studies in a large number of

other material systems with more complicated micromechanisms of deformation and failure.

It should also be recognized that it would also be important in future to further expand the

methods described here to extend the investigations to real microstructures with more realistic

grain sizes and morphologies.


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26.4 Examples 527

Figure 26.8: Comparison of evolution of lattice rotations in individual grains depicted by (100)

pole figures. Shown are the initial orientations as well as the final measurements after 40% re-

duction in plane strain compression along with the predictions from both the Taylor-type model

and the finite element model prediction.

Acknowledgements

The author is grateful for his collaborations and many discussions with his colleague, Pro-

fessor Roger D. Doherty, and his ex-PhD student, Dr. Abhishek Bhattacharya, and for their


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528 References

many contributions to this work. This work was supported by NSF-DMR9612343 and NSF-

DMR0303395.

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Barrett, C. S. and L. H. Levinson (1940). “The structure of aluminum after compression.” Trans. Metall. Soc. AIME

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polycrystal plasticity with consideration of macrostructural and microstructural linking.” International Journal of

Plasticity 11: 501-521.

Beaudoin, A. J., K. K. Mathur, P. R. Dawson and G. C. Johnson (1993). “Three-Dimensional Deformation Process

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ments and numerical simulations.” International Journal of Plasticity 17: 861-883.

Bhattacharyya, A., S. R. Kalidindi and R. D. Doherty (2004). “Detailed Analyses of Grain-Scale Plastic Deforma-

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6. Implementation of plasticitymodels into finite element code

6.1 Introduction

Several commercial finite element software packages (e.g. ABAQUS, LSDyna, and

MARC) provide the facility for users to specify their own material models. In

ABAQUS, the user is required to provide a Fortran subroutine called a ‘UMAT’. This

chapter is concerned with the implementation of plasticity models into finite element

code, which is carried out, by example, in using ABAQUS. In particular, we start by

developing an ABAQUS UMAT for elasticity, and discuss the tests necessary to verify

the model implementation into ABAQUS. We go on to consider isotropic hardening

plasticity with explicit and implicit integration, with continuum and consistent tan-

gent stiffnesses, large deformation formulations using rotated variables provided by

ABAQUS, and from first principles using the deformation gradient. We use the prob-

lem of simple shear with elasticity to verify the large deformation implementations.

We then present an implicit implementation for elasto-viscoplasticity (and creep).

All the Fortran coding, together with the necessary ABAQUS input files, are available

through the OUP website.

Finite element code is often modular in structure, whether it be commercial code orwritten in-house. An important module is that relating to material behaviour; in other

words, the constitutive stress response of the material given prescribed conditions of

deformation. In ABAQUS, but in a similar way for all codes, a range of information

is passed into the material module relating to both the beginning and end of a time

increment. In particular, stress, strain, and deformation gradient are provided at the

beginning of the time increment. Strain and the deformation gradient are also provided

at the end of the increment. Within the module, it is then necessary to execute three

tasks. First, the stresses at the end of the time increment must be determined and,second, for the case of an implicit analysis (i.e. the finite element momentum balance

or equilibrium equations are solved implicitly) using ABAQUS standard, for example,

the material Jacobian, or tangent stiffness, must also be provided. Third, any state

variables (such as the isotropic hardening variable or effective plastic strain) must


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170 Implementation of plasticity models

be updated to the end of the time increment. In fact, in coding an ABAQUS UMAT,

a large range of information is provided, some of which will be referred to later.

However, the job of the UMAT is clear: to update the stresses and state variables to

the end of the time increment and to provide the Jacobian. We start by addressing

elasticity.

6.2 Elasticity implementation

We discuss elasticity and its implementation into a UMAT for two main reasons.

First, it provides a good introduction to writing and testing a UMAT subroutine

for those who are new to the process. Second, it provides a basis from which elasto-

plasticity models may be developed. For the latter reason, while is no way essential

for elasticity, we use an incremental approach in implementing the linear elastic

equations.

In Chapter 5, Hooke’s law was given in an incremental form in Equation (5.21)

together with the material Jacobian, for the case in which there are no out of plane

shears (e.g.planestrainandaxisymmetricproblems). Withknowledgeof theincrement

in strains (provided to the UMAT), together with the specification of elastic constants

(we shall take E = 210 GPa and ν = 0.3 throughout), the stress increment is obtained

either from Equation (5.21), or its equivalent written in Voigt notation,

σ = Cεe =

2G + λ λ λ 0

λ 2G + λ λ 0

λ λ 2G + λ 0

0 0 0 G

ε11

ε22

ε33

γ 12

. (6.1)

Because ABAQUS provides most tensor quantities in vector (Voigt) form, it may be

more convenient to use Equation (6.1) than the tensor form given in (5.21). However,this is not always the case, and later we shall use the tensor form in preference,

particularly where it is necessary to work from the deformation gradient. Note that the

shear strain quantities provided by ABAQUS are always engineering shears, that is,

twice the tensorial shear strains. It is also important to check the ordering of shear

quantities, which can vary depending on element type used. Equation (6.1) is suitable

for plane strain and axisymmetric problems, but not for problems of plane stress

or three dimensions, for which different stiffness matrices are required. For linear

elasticity, the material Jacobian is just the elastic stiffness matrix so that specificationof the Jacobian in the UMAT is easy. The coding required for the UMAT for linear

elasticity for plane strain, axisymmetric, and three-dimensional problems, is available

through the OUP website. A complete list of all the UMAT coding, together with

ABAQUS input files, is given in Appendix B.


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Verification of implementations 171

6.3 Verification of implementations

The verification of the model implementation is vital. For complex material models,

this requires the development of an independent solver (which will often be numerical)for uniaxial and pure shearproblemsso that direct comparisonwith the resultsobtained

using the UMAT can be made. In addition, it is necessary to test the UMAT using

a single, and where appropriate, multiple elements, for conditions of strain control

and load control under uniaxial and pure shear conditions. If possible, perhaps by

‘switching off’ parts of the model implemented into the UMAT, make comparisons

with a multiaxial problem with non-uniform strain and stress distributions for which

the solution is known (either an independent solution or one obtained using an internal

ABAQUS model). In this chapter, we shall not carry out all of the verification testsdescribed for all the model implementations, but it is certainly advisable to do so for

new model implementations into UMATs. Because of their importance, we go through

some of the possible verification tests step by step.

1. Single and multiple element uniaxial tests. Figure 6.1 shows an axisymmetric

single- and four-element unit square which is subjected to uniaxial displacement or

force control in the z-direction producing uniform, uniaxial stress, σ zz, and strain,

εzz, with εrz = σ rr = σ θ θ = σ rz = 0, which can be compared with independent

closed-form solutions. The four-element problem is important since it introduces

a ‘free’ node, which does not exist for the single four-noded axisymmetric element.

The force controlled test is important for checking errors in the Jacobian.

2. Single element simple shear test. The uniaxial tests in (1) do not involve the shear

terms at all, so it is important to include a problem which tests these terms, particularly

because of the potential for errors with the use of engineering shear strains rather than

their tensorial counterparts. Figure 6.2 shows a plane strain single-element unit square

under simple shear loading. For small deformations, εxx

= εyy

= σ xx

= σ yy

= 0,

r

u z =0.0

Displacement

or force

0.0

0.0

1.0

1.0

z

ur =0.0

Displacement

or force

u z =0.0

0.0

0.0

1.0

1.0

z

r

ur =0.0

Fig. 6.1 Schematic diagram showing an axisymmetric single- and four-element unit square underuniaxial displacement or force controlled loading.


18/122


g

u x = u y =0.0

x 0.0

0.0

1.0

1.0

y

Fig. 6.2 Schematic diagram showing a plane strain single-element unit square under simple shearloading.

and uniform shear strain, γ xy , and stress, σ xy , are produced which can be comparedwith independent closed-form solutions.

3. Non-uniform strain and stress field. Often, a comparison test which generates

non-uniform strain and stress fields will not be possible because a comparison may

well not exist, and a closed-form solution is now no longer possible. However, it is

sometimes possible to simplify the implemented plasticity model (e.g. by turning

off the porosity in a Gurson-type porous plasticity model) such that a comparison

with another solution (e.g. produced using one of the many built-in models contained

within ABAQUS) is then possible. While this will not test all the features of themodel implemented into the UMAT, it nonetheless may test a good number of them,

and may therefore be worthwhile. Later in the chapter, both an implicit and explicit

implementation of isotropic hardening plasticity are tested in this way by comparing

the results obtained with those produced using the built-in ABAQUS model.

6.4 Isotropic hardening plasticity implementation

In Chapter 5, we presented both explicit and implicit integration of the equations

for linear strain hardening isotropic plasticity, together with the consistent tangent

stiffness for the implicit scheme. We will implement both integration schemes into

ABAQUS UMATs and discuss the advantages and disadvantages of each. We start

with the explicit scheme which was described at the beginning of Section 5.2, and

introduce the continuum Jacobian.

6.4.1 Explicit integration for isotropic hardening plasticity withcontinuum Jacobian

We may summarize the implementation as follows. All quantities are assumed to

be given at time, t , that is, at the start of the time increment, unless indicated


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Isotropic hardening plasticity implementation 173

otherwise:

(i) Determine the yield function

f = σ e − r − σ y =

3

2σ

: σ

1/2

− r − σ y. (6.2)

(ii) Determine if actively yielding

Is f > 0?

(iii) Determine the plastic multiplier

f > 0, dλ =n · C dε

n · Cn + h,

f


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We may write the Jacobian (here symmetric) as

J = ∂ dσ∂ dε

=

J 11 J 12 J 13 J 14

J 22 J 23 J 24J 33 J 34

sym J 44

(6.8)

so that, for example, the first term is

J 11 =∂ dσ 1

∂ dε1= C11 −

∂ dλ

∂ dε1(C11n1 + C12n2 + C13n3 + C14n4).

With some algebra we may show, therefore, that

J = C − Cn ⊗∂ dλ

∂ dε, (6.9)

where ⊗ is the dyadic product of two vectors, details of which may be found in

Appendix A. In a similar way, by considering the numerator of the plastic multiplier

given in Equation (6.3), we may show that

∂ dλ

∂ dε=

Cn

n · Cn + h

so that the continuum Jacobian is

J = C −Cn ⊗ Cn

n · Cn + h. (6.10)

The explicit integration and provision of Jacobian is then complete. An ABAQUS

UMAT containing this formulation, together with various input files for uniaxial dis-

placement and load control, together with a four-point beam bending problem, are

available via the OUP website (full details are given in Appendix B). In addition, the

very same problems are analysed using the built-in ABAQUS linear strain hardeningplasticity model (chosen to represent a material with E = 210,000 GPa, ν = 0.3,

σ y = 240 MPa, and h = 1206 MPa). In all the analyses using the explicit UMAT,

the maximum time increment allowed in the analysis is carefully chosen to ensure

stability and accuracy. Despite this, at the elastic–plastic transition, the stress at first

yield is overestimated because the stress at the start of the increment was determined

on the basis of elastic behaviour in the previous increment. With an explicit scheme,

this is never corrected so that the stresses remain slightly overestimated throughout the

analysis. The error in stress can be reduced by decreasing the time increment size, butat the cost of the computer CPU time and error accumulation. In any case, many more

time increments are required using the explicit UMAT than are required using the

built-in ABAQUS implicit plasticity integration. This is a further serious shortcoming

of the explicit integration method, which needs to be weighed against the advantage


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Isotropic hardening plasticity implementation 175

of simplicity, particularly for complex constitutive equations. However, despite con-

siderably longer computer CPU times, the results obtained for the four-point bend

simulation from the explicit UMAT and the built-in ABAQUS plasticity model are

found to be near-identical. We consider an implicit implementation in Section 6.4.2.

6.4.2 Implicit integration for isotropic hardening plasticity withconsistent Jacobian

We may summarize the implementation as follows. As opposed to the explicit case,

all quantities are now assumed to be given at the end of the time increment, that is,

at time t + t , unless otherwise indicated.

(i) Determine the elastic trial stress

σ tr

= σ t + 2Gε + λI ε : I . (6.11)

(ii) Determine the trial yield function

f = σ tre − r − σ y =

3

2σ

tr: σ

tr1/2

− r − σ y. (6.12)

(iii) Determine if actively yielding

Is f > 0?

(iv) If yes, use Newton iteration to determine the effective plastic strain increment

r (k) = rt + hp,

dp =σ tre − 3Gp

(k) − r (k) − σ y

3G + h, (6.13)

p(k+1) = p(k) + dp.

Otherwise,

p = 0.

(v) Determine plastic and elastic strain and stress increments

εp =3

2p

σ tr

σ tre,

εe = ε − εp, (6.14)

σ = 2Gεe

+ λI εe

: I .(vi) Update all quantities to the end of the time increment

σ = σ t + σ ,

p = pt + p.(6.15)


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(vii) Determine consistent Jacobian

δσ = 2GQσ tr

σ tre

σ tr

σ tre: δε + 2GRδε + K − 2

3GR II : δε. (6.16)

(viii) End.

An ABAQUS UMAT containing this formulation, together with various input files

for uniaxial displacement and load control, together with a four-point beam bending

problem, are available through the OUP website (full details are given in Appendix B).

In addition, asbefore, the very sameproblemsare analysedusing the built-in ABAQUS

linear strain hardening plasticity model. Using implicit integration with the consistent

tangent stiffness eliminates theproblem which occurred at theelastic–plastic transitionwhen using explicit integration. In addition, the use of implicit integration enables

much larger time increments to be used, therefore significantly reducing CPU times.

The disadvantage of the implicit formulation is simply the difficulty in obtaining the

consistent tangent stiffness for complex constitutive equations. Often, analytical forms

are simply not obtainable, in which case a numerical procedure may be possible. The

results obtained for the four-point bend simulation from the implicit UMAT and the

built-in ABAQUS plasticity model are found to be identical.

6.5 Large deformation implementations

We have not yet differentiated between small and large deformation implementations

in this chapter. In fact, the UMATs discussed above for both explicit and implicit

schemes are suitable for both small and large deformation problems using ABAQUS.

This is because the necessary rigid body rotations for the strains and stresses have

already been carried out by ABAQUS before they are provided to the UMAT routine.

That is, referring back to Section 5.6, the large deformation stress update necessary is

σ = ∇ σ + (Wσ t − σ t W )t =

∇ σ + Rσ t R

T (6.17)

in which ∇ σ is the co-rotational stress increment. The stress, σ t , at the start of the

time increment (and the strain) has already been rotated by ABAQUS (i.e. the stress

provided at the start of the time increment is effectively Rσ t RT) so that all we need

to do within the UMAT is to carry out the stress update. Sometimes, depending on

the plasticity model employed, it is necessary to use internal variables which are alsotensor quantities. An example is the back stress in a kinematic hardening model. The

components of the back stress will need to be updated within the UMAT ( just like

the scalar isotropic hardening variable in the previous sections) and stored in what

are called state variable arrays in ABAQUS. Because state variables are not modified


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Large deformation implementations 177

by ABAQUS, it is necessary for the user to carry out the rigid body rotations on

tensorial state variables. In fact, ABAQUS provides a utility subroutine to simplify

this process. If, for example, the tensorial variable recovered from the state variable

array at the start of the increment is, say, xt , then the user must carry out the rigid body

rotationRx t RT before updating x to the end of the time increment within the UMAT.

The rotation is carried out by a single call of the utility subroutine rotsig detailed in

the ABAQUS manuals.

Sometimes, constitutive equations for elasticity and plasticity are formulated in

terms of the deformation gradient, F . Examples include hyperelasticity—large strain

non-linear elasticity and crystal plasticity. It may be, however, that the user simply

wishes to work from the deformation gradient rather than use the strain and stress

quantities provided by ABAQUS to the UMAT subroutine. In Section 6.5.1, we present

a large deformation implementation based on an explicit scheme.

6.5.1 Implementation using the deformation gradient

We may summarize the implementation as follows. In this explicit approach, all

quantities are assumed to be given at time, t , that is, at the start of the time increment,

unless indicated otherwise.

(i) Determine the velocity gradient

L = ḞF −1. (6.18)

(ii) Determine the rate of deformation and spin

D =1

2

(L + LT), W =1

2

(L − LT). (6.19)

(iii) Determine the yield function

f = σ e − r − σ y =

3

2σ

: σ

1/2− r − σ y. (6.20)

(iv) Determine if actively yielding

Is f > 0? (6.21)

(v) Determine rate of plastic deformation

f > 0, Dp = as specified by constitutive equation,

f


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(vi) Determine rate of elastic deformation and Jaumann stress rate and isotropic

hardening rate

De = D − Dp,

∇ σ = 2GDe + λIDe : I , (6.23)

ṙ = hṗ.

(vii) Determine stresses with respect to material reference

σ̇ = ∇ σ +Wσ − σW . (6.24)

(viii) Update all quantities to the end of the time increment, using explicit integration

σ t +t = σ + σ̇ t,

rt +t = r + ṙt.(6.25)

(ix) Determine Jacobian.

(x) End.

We will next address the verification of a large deformation implementation, which

is valid for both the implicit and explicit implementations in Sections 6.4.1 and 6.4.2.

Note that uniaxial displacement or force controlled loading will not test whether

the rigid body rotations are being calculated correctly, since for these cases, the

continuum spin, W , is zero. A good test, however, is provided in the form of

simple shear if we allow the strains to become quite large. To simplify matters,

we will ‘switch off’ the plasticity and allow elasticity only, since the test is being

carried out to check the rigid body rotation calculation rather than the constitutive

response.

We obtained the deformation gradient for simple shear in Section 3.5.1 as

F =

1 δ

0 1

.

Considering a constant rate of shearing, δ̇, the velocity gradient is

L =

0 δ̇

0 0

so that the rate of deformation and spin are

D =1

2

0 δ̇

δ̇ 0

, W =

1

2

0 δ̇

−δ̇ 0

.


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Large deformation implementations 179

Note that the spin is non-zero so that rigid body rotation, together with stretch, is

occurring.

The Jaumann stress rate is given by

∇ σ = 2GD + λID : I = G

0 δ̇

δ̇ 0

(6.26)

and the stress rate with respect to the material (undeformed) reference is

σ̇ =

σ̇ xx σ̇ xy

σ̇ xy σ̇ yy

, (6.27)

which is given in terms of the Jaumann stress rate by

σ̇ = ∇ σ +Wσ − σW

so that substituting for the spin and (6.26) and (6.27) gives

σ̇ xx σ̇ xy

σ̇ xy σ̇ yy

= Gδ̇

0 1

1 0

+

1

2δ̇

σ xy σ yy

−σ xx −σ xy

−

1

2δ

−σ xy σ xx

−σ yy σ xy

so that

σ̇ xy = Gδ̇ +1

2δ̇(σ yy − σ xx ), (6.28)

σ̇ xx = δ̇σ xy , (6.29)

σ̇ yy = −δ̇σ xy . (6.30)

Equations (6.29) and (6.30) give, with the initial condition that all stresses are zero,

σ yy = −σ xx .

Differentiating (6.28) and substituting for (6.29) and (6.30) gives

σ̈ xy + δ̇2σ xy = 0,

which has solution

σ xy = A sin δ̇t + B cos δ̇t .

The initial conditions require that B = 0. Solving for σ xx and σ yy using

Equations (6.29) and (6.30) and imposing the initial conditions gives the solution

σ xy = G sin δ̇t, σ xx = −σ yy = G(1 − cos δ̇t). (6.31)


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–200

–150

–100

–50

050

100

150

0.00 0.20 0.40 0.60 0.80 1.00

s yy

s

xy

x

y

Time (s)

S t r e s s ( G P

a )

s xx

Fig. 6.3 Unit square under simple shear at a rate of 5.0 s−1 and the corresponding stresses.

For small strain, δ, at constant strain rate, these reduce to

σ xy = Gδ and σ xx = −σ yy = 0. (6.32)

The harmonic variation in (6.31) results from the large deformation, and in particu-

lar, the rigid body rotation taking place. This non-physical result arises because we

are using a small strain, linear elasticity model under conditions of large deformation.

Despite its non-physicality, it provides a good test of the calculation of the rigid body

rotations in the large deformation UMAT. In order to do this, a single plane strain

element, as shown in Fig. 6.2, has been subjected to simple shear to large strain.

This has been carried out using the UMAT with elasticity described in Section 6.2,which uses ABAQUS-provided stresses and strains, a further UMAT based on the

deformation gradient, described in this section, and using the built-in elasticity model

in ABAQUS. The results obtained are identical and are shown in Fig. 6.3. The various

UMATs, together with the ABAQUS input files, are detailed in Appendix B and are

available via the OUP website.

6.6 Elasto-viscoplasticity implementationViscoplasticity, meaning rate-dependent plasticity in which the plastic multiplier is

determined through the use of a viscoplastic constitutive equation as opposed to the

use of the consistency condition, was introduced in Chapter 4. The radial return,

implicit backward Euler integration for viscoplasticity was discussed in Chapter 5.

Here, we present an implicit implementation for linear isotropic strain hardening

elasto-viscoplasticity. Such an implementation can readily be simplified for the

implicit analysis of creep. We employ a sinh-type viscoplastic constitutive equation

and for simplicity, use the initial tangent stiffness (i.e. the elastic stiffness) for the

material Jacobian.

The viscoplastic constitutive equation is taken to be

ṗ = φ (σ e, r) = α sinh β(σ e − r − σ y)


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Elasto-viscoplasticity implementation 181

and the multiaxial plastic strain increments are given by

εp

= pn =

3

2 p

σ

σ e .

We determine the increment in effective plastic strain as described in Chapter 5 as

follows. Note that all quantities are now assumed to be given at the end of the time

increment, that is, at time t + t , unless otherwise indicated.

(i) Determine the elastic trial stress

σ tr

= σ t +

2Gε +

λI

ε : I

. (6.33)

(ii) Determine the trial yield function

f = σ tre − r − σ y =

3

2σ

tr: σ

tr1/2

− r − σ y. (6.34)

(iii) Determine if actively yielding

Is f > 0?

(iv) If yes, use Newton iteration to determine the effective plastic strain increment

φ (σ e, r) = α sinh β(σ tre − 3Gp − r − σ y),

φp = −3Gαβ cosh β(σ tre − 3Gp − r − σ y),

φr = −αβ cosh β(σ tre − 3Gp − r − σ y),

r = rt + hp,

dp =φ(p, r) − (p/t)

(1/t) − φp − hφr,

p(k+1) = p(k) + dp.

(6.35)

(v) Determine plastic and elastic strain and stress increments

εp = 32

pσ tr

σ tre,

εe = ε − εp, (6.36)

σ = 2Gεe + λI εe : I .


28/122


(vi) Update all quantities to the end of the time increment

σ = σ t + σ ,

p = pt + p. (6.37)

(vii) Determine Jacobian.

(viii) End.

An ABAQUS UMAT containing this formulation, together with various input files

for uniaxial displacement and load control, together with a four-point beam bending

problem, are available via the OUP website (full details are given in Appendix B).

In addition, uniaxial, closedform implicit andexplicit solutionsareprovided in Fortran

programs for verification of the implementation.


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CHAPTER SEVEN

DISCRETIZATION AND SOLUTION

7.1 INTRODUCTION

The equilibrium equations and their corresponding linearizations have been

established in terms of a material or a spatial description. Either of these

descriptions can be used to derive the discretized equilibrium equations and

their corresponding tangent matrix. Irrespective of which configuration is

used, the resulting quantities will be identical. It is, however, generally

simpler to establish the discretized quantities in the spatial configuration.Establishing the discretized equilibrium equations is relatively standard,

with the only additional complication being the calculation of the stresses,

which obviously depend upon nonlinear kinematic terms that are a function

of the deformation gradient. Deriving the coefficients of the tangent ma-

trix is somewhat more involved, requiring separate evaluation of constitu-

tive, initial stress, and external force components. The latter deformation-

dependant external force component is restricted to the case of enclosed

normal pressure. In order to deal with near incompressibility the mean

dilatation method is employed.

Having discretized the governing equations, the Newton–Raphson so-

lution technique is reintroduced together with line search and arc lengthmethod enhancements.

7.2 DISCRETIZED KINEMATICS

The discretization is established in the initial configuration using isopara-

metric elements to interpolate the initial geometry in terms of the parti-

1


30/122

2 DISCRETIZATION AND SOLUTION

vδ aub · »

(e)

(e)

ab

a b

X 1 x1,

X 3 x3,

X 2 x2,

time = 0

time = t

`

FIGURE 7.1 Discretization.

cles X a defining the initial position of the element nodes as,

X =na=1

N a(ξ 1, ξ 2, ξ 3)X a (7.1)

where N a(ξ 1, ξ 2, ξ 3) are the standard shape functions and n denotes the

number of nodes. It should be emphasized that during the motion, nodes

and elements are permanently attached to the material particles with which

they were initially associated. Consequently, the subsequent motion is fully

described in terms of the current position xa(t) of the nodal particles as (see

Figure 7.1),

x =

n

a=1 N a

xa(t) (7.2)

Differentiating Equation (7.2) with respect to time gives the real or

virtual velocity interpolation as,

v =na=1

N ava; δ v =na=1

N aδ va (7.3)

Similarly, restricting the motion brought about by an arbitrary increment u

to be consistent with Equation (7.2) implies that the displacement u is also


31/122

7.2 DISCRETIZED KINEMATICS 3

interpolated as,

u =na=1

N aua (7.4)

The fundamental deformation gradient tensor F is interpolated over an

element by differentiating Equation (7.2) with respect to the initial coordi-

nates to give, after using Equation (2.135a),

F =na=1

xa⊗∇0N a (7.5)

where ∇0N a = ∂N a/∂ X can be related to ∇ξN a = ∂N a/∂ ξ in the standardmanner by using the chain rule and Equation (7.1) to give,

∂N a∂ X

=

∂ X

∂ ξ

−T ∂N a

∂ ξ ;

∂ X

∂ ξ =na=1

X a⊗∇ξN a (7.6a,b)

Equations (7.5) and (7.6b) are sufficiently fundamental to justify expansion

in detail in order to facilitate their eventual programming. To this effect,

these equations are written in an explicit matrix form as,

F =

F 11 F 12 F 13

F 21 F 22 F 23

F 31 F 32 F 33

; F iJ =n

a=1

xa,i∂N a

∂X J (7.7)

and,

∂ X

∂ ξ =

∂X 1/∂ξ 1 ∂X 1/∂ξ 2 ∂X 1/∂ξ 3

∂X 2/∂ξ 1 ∂X 2/∂ξ 2 ∂X 2/∂ξ 3

∂X 3/∂ξ 1 ∂X 3/∂ξ 2 ∂X 3/∂ξ 3

; ∂X I

∂ξ α=na=1

X a,I ∂N a∂ξ α

From Equation (7.5) further strain magnitudes such as the right and left

Cauchy–Green tensors C and b can be obtained as,

C = F T

F =a,b

(xa ·xb)∇0N a⊗∇0N b; C IJ =

3

k=1

F kI F kJ (7.9a,b)

b = F F T =a,b

(∇0N a ·∇0N b)xa⊗xb; bij =3K =1

F iK F jK (7.9c,d)

The discretization of the real (or virtual rate of deformation) tensor and the

linear strain tensor can be obtained by introducing Equation (7.3) into the

definition of d given in Equation (3.101) and Equation (7.4) into Equation


32/122


(6.11a) respectively to give,

d = 1

2

na=1

(va⊗∇N a + ∇N a⊗ va) (7.10a)

δ d = 1

2

na=1

(δ va⊗∇N a + ∇N a⊗ δ va) (7.10b)

ε = 1

2

na=1

(ua⊗∇N a + ∇N a⊗ua) (7.10c)

where, as in Equation (7.6), ∇N a = ∂N a/∂ x can be obtained from the

derivatives of the shape functions with respect to the isoparametric coordi-

nates as,

∂N a∂ x

=

∂ x

∂ ξ

−T ∂N a

∂ ξ ;

∂ x

∂ ξ =na=1

xa⊗∇ξN a; ∂xi

∂ξ α=na=1

xa,i∂N a∂ξ α

(7.11a,b)

Although Equations (7.10a–c) will eventually be expressed in a standard

matrix form, if necessary the component tensor products can be expanded

in a manner entirely analogous to Equations (7.6–7).

EXAMPLE 7.1: Discretization

This simple example illustrates the discretization and subsequent calcu-

lation of key shape function derivatives. Because the initial and current

geometries comprise right-angled triangles, these are easily checked.

x2 2

, X

,

1 x

X

1(4,0)(0,0)

(0,3)(2,3)

(10,3)

time = t

time = 0

1 2

3

3

1 2

»

»

»

2

1

1

(10,9)

»2

(continued)


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7.2 DISCRETIZED KINEMATICS 5

EXAMPLE 7.1 (cont.)

The initial X and current x nodal coordinates are,

X 1,1 = 0; X 2,1 = 4; X 3,1 = 0;

X 1,2 = 0; X 2,2 = 0; X 3,2 = 3;

x1,1 = 2; x2,1 = 10; x3,1 = 10

x1,2 = 3; x2,2 = 3; x3,2 = 9

The shape functions and related derivatives are,

N 1 = 1 − ξ 1 − ξ 2

N 2 = ξ 1

N 3 = ξ 2

; ∂N 1∂ ξ

=−1−1

; ∂N 2∂ ξ

=

10

; ∂N 3∂ ξ

=

01

Equations (7.1) and (7.6b) yield the initial position derivatives with

respect to the nondimensional coordinates as,

X 1 = 4ξ 1

X 2 = 3ξ 2;

∂ X

∂ ξ =

4 0

0 3

;

∂ X

∂ ξ

−T

= 1

12

3 0

0 4

from which the derivatives of the shape functions with respect to the

material coordinate system are found as,

∂N 1∂ X

= 112

3 00 4

−1−1

= − 1

12

34

;

∂N 2∂ X

= 1

12

3

0

;

∂N 3∂ X

= 1

12

0

4

A similar set of manipulations using Equations (7.2) and (7.11) yields

the derivatives of the shape functions with respect to the spatial coor-

dinate system as,

∂N 1∂ x

= 1

24

3 0

−4 4

−1

−1

= −

1

24

3

0

;

∂N 2∂ x

= 1

24

3

−4

;

∂N 3∂ x

= 1

24

0

4


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EXAMPLE 7.2: Discretized kinematics

Following Example 7.1 the scene is set for the calculation of the defor-

mation gradient F using Equation (7.7) to give,

F iJ = x1,i∂N 1∂X J

+ x2,i∂N 2∂X J

+ x3,i∂N 3∂X J

; i, J = 1, 2; F = 1

3

6 8

0 6

Assuming plane strain deformation, the right and left Cauchy–Green

tensors can be obtained from Equation (7.9) as,

C = F T F = 1

936 48 0

48 100 00 0 9

; b = F F T =

1

9100 48 0

48 36 00 0 9

Finally, the Jacobian J is found as,

J = detF = det

1

3

6 8 00 6 0

0 0 3

= 4

7.3 DISCRETIZED EQUILIBRIUM EQUATIONS

7.3.1 GENERAL DERIVATION

In order to obtain the discretized spatial equilibrium equations, recall the

spatial virtual work Equation (4.27) given as the total virtual work done by

the residual force r as,

δW (φ, δ v) =

v

σ : δ d dv −

v

f · δ v dv −

∂v

t · δ v da (7.12)

At this stage, it is easier to consider the contribution to δW (φ, δ v) caused

by a single virtual nodal velocity δ va occurring at a typical node a of ele-

ment (e). Introducing the interpolation for δ v and δ d given by Equations(7.3) and (7.10) gives,

δW (e)(φ, N aδ va) =

v(e)σ : (δ va⊗∇N a)dv

−

v(e)

f · (N aδ va) dv −

∂v(e)

t · (N aδ va) da (7.13)

where the symmetry of σ has been used to concatenate the internal energy

term. Observing that the virtual nodal velocities are independent of the


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EXAMPLE 7.3 (cont.)

From Equation (7.15b) the equivalent nodal internal forces are,

T a,i =

v(e)

σi1

∂N a∂x1

+ σi2∂N a∂x2

dv;

a = 1, 2, 3

i = 1, 2;

T 1,1 = −24t

T 1,2 = −12t;

T 2,1 = 8t

T 2,2 = 0;

T 3,1 = 16t

T 3,2 = 12t

where t is the element thickness. Clearly these forces are in equilibrium.

The contribution to δW (φ, N aδ va) from all elements e (1 to ma) con-

taining node a (e a) is,

δW (φ, N aδ va) =mae=1ea

δW (e)(φ, N aδ va) = δ va · (T a − F a) (7.16a)

where the assembled equivalent nodal forces are,

T a =mae=1ea

T (e)a ; F a =mae=1ea

F (e)a (7.16b,c)

Finally the contribution to δW (φ, δ v) from all nodes N in the finite elementmesh is,

δW (φ, δ v) =N a=1

δW (φ, N aδ va) =N a=1

δ va · (T a − F a) = 0 (7.17)

Because the virtual work equation must be satisfied for any arbitrary virtual

nodal velocities, the discretized equilibrium equations, in terms of the nodal

residual force R a, emerge as,

R a = T a − F a = 0 (7.18)

Consequently the equivalent internal nodal forces are in equilibrium with

the equivalent external forces at each node a = 1, 2, . . . , N .For convenience, all the nodal equivalent forces are assembled into single

arrays to define the complete internal and external forces T and F respec-

tively, as well as the complete residual R, as,

T =

T 1

T 2...

T n

; F =

F 1

F 2...

F n

; R =

R 1

R 2...

R n

(7.19)


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7.3 DISCRETIZED EQUILIBRIUM EQUATIONS 9

These definitions enable the discretized virtual work Equation (7.17) to berewritten as,

δW (φ, δ v) = δ vT R = δ vT (T− F) = 0 (7.20)

where the complete virtual velocity vector δ vT = [δ vT 1 , δ vT 2 , . . . , δ v

T n ].

Finally, recalling that the internal equivalent forces are nonlinear func-

tions of the current nodal positions xa and defining a complete vector of

unknowns x as the array containing all nodal position as,

x =

x1

x2...

xn

(7.21)

enables the complete nonlinear equilibrium equations to be symbolically

assembled as,

R(x) = T(x) − F(x) = 0 (7.22)

These equations represent the finite element discretization of the pointwise

differential equilibrium Equation (4.16).

7.3.2 DERIVATION IN MATRIX NOTATION

The discretized equilibrium equations will now be recast in the more famil-

iar matrix-vector notation. To achieve this requires a reinterpretation of

the symmetric stress tensor as a vector comprising six independent compo-

nents as,

σ = [σ11, σ22, σ33, σ12, σ13, σ23]T (7.23)

Similarly, the symmetric rate of deformation tensor can be re-established in

a corresponding manner as,

d = [d11, d22, d33, 2d12, 2d13, 2d23]T (7.24)

where the off-diagonal terms have been doubled to ensure that the product

dT σ gives the correct internal energy as, v

σ : d dv =

v

dT σ dv (7.25)

The rate of deformation vector d can be expressed in terms of the usual


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B matrix and the nodal velocities as,

d =na=1

Bava; Ba =

∂N a∂x1

0 0

0 ∂N a∂x2

0

0 0 ∂N a∂x3

∂N a∂x2∂N a∂x1

0

∂N a∂x3

0 ∂N a∂x1

0 ∂N a∂x3∂N a∂x2

(7.26)

Introducing Equation (7.26) into Equation (7.25) for the internal energy

enables the discretized virtual work Equation (7.13) to be rewritten as,

δW (φ, N aδ va) =

v(e)

(Baδ va)T σ dv −

v(e)

f · (N aδ va) dv

−

∂v(e)

t · (N aδ va) da (7.27)

Following the derivation given in the previous section leads to an alternative

expression for the element equivalent nodal forces T (e)a for node a as,

T (e)a = v(e)

BT aσ dv (7.28)

Observe that because of the presence of zeros in the matrix Ba, Expres-

sion (7.15a) is computationally more efficient than Equation (7.28).

7.4 DISCRETIZATION OF THE LINEARIZED

EQUILIBRIUM EQUATIONS

Equation (7.22) represents a set of nonlinear equilibrium equations with

the current nodal positions as unknowns. The solution of these equations

is achieved using a Newton–Raphson iterative procedure that involves thediscretization of the linearized equilibrium equations given in Section 6.2.

For notational convenience the virtual work Equation (7.12) is split into

internal and external work components as,

δW (φ, δ v) = δW int(φ, δ v) − δW ext(φ, δ v) (7.29)

which can be linearized in the direction u to give,

DδW (φ, δ v)[u] = DδW int(φ, δ v)[u] − DδW ext(φ, δ v)[u] (7.30)


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7.4 LINEARIZED EQUILIBRIUM EQUATIONS 11

where the linearization of the internal virtual work can be further subdividedinto constitutive and initial stress components as,

DδW int(φ, δ v)[u] = DδW c(φ, δ v)[u] + DδW σ(φ, δ v)[u]

=

v

δ d : c : ε dv +

v

σ : [(∇u)T ∇δ v] dv (7.31)

Before continuing with the discretization of the linearized equilibrium Equa-

tion (7.30), it is worth reiterating the general discussion of Section 6.2 to

inquire in more detail why this is likely to yield a tangent stiffness matrix.

Recall that Equation (7.15), that is, δW (e)(φ, N aδ va) = δ va · (T (e)a −F

(e)a ),

essentially expresses the contribution of the nodal equivalent forces T (e

)a

and F (e)a to the overall equilibrium of node a. Observing that F

(e)a may be

position-dependent, linearization of Equation (7.15) in the direction N bub,

with N aδ va remaining constant, expresses the change in the nodal equiv-

alent forces T (e)a and F

(e)a , at node a, due to a change ub in the current

position of node b as,

DδW (e)(φ, N aδ va)[N bub] = D(δ va · (T (e)a − F

(e)a ))[N bub]

= δ va ·D(T (e)a − F

(e)a )[N bub]

= δ va ·K (e)ab ub (7.32)

The relationship between changes in forces at node a due to changes in

the current position of node b is furnished by the tangent stiffness matrix

K (e)ab

, which is clearly seen to derive from the linearization of the virtual

work equation. In physical terms the tangent stiffness provides the Newton–

Raphson procedure with the operator that adjusts current nodal positions so

that the deformation-dependent equivalent nodal forces tend toward being

in equilibrium with the external equivalent nodal forces.

7.4.1 CONSTITUTIVE COMPONENT – INDICIAL FORM

The constitutive contribution to the linearized virtual work Equation (7.31)for element (e) linking nodes a and b is,

DδW (e)c (φ, N aδ va)[N bub]

=

v(e)

1

2(δ va⊗∇N a + ∇N a⊗ δ va) : c :

1

2(ub⊗∇N b + ∇N b⊗ub) dv

(7.33)

In order to make progress it is necessary to temporarily resort to indicial


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notation, which enables the above equation to be rewritten as,

DδW (e)c (φ, N aδ va)[N bub]

=3i,j,k,l=1

v(e)

1

2

δva,i

∂N a∂x j

+ δva,j∂N a∂xi

c ijkl

1

2

ub,k

∂N b∂xl

+ ub,l∂N b∂xk

dv

=3i,j,k,l=1

δva,i

v(e)

∂N a∂x j

1

4(c ijkl +c ijlk +c jikl +c jilk)

∂N b∂xl

dv

ub,k

= δ va ·K (e)c,ab ub (7.34)

where the constitutive component of the tangent matrix relating node a to

node b in element (e) is,

[K c,ab]ij =

v(e)

3k,l=1

∂N a∂xk

csymikjl

∂N b∂xl

dv; i, j = 1, 2, 3 (7.35)

in which the symmetrized constitutive tensor is,

csymikjl =

1

4(c ikjl +c iklj +c kijl +c kilj) (7.36)

EXAMPLE 7.4: Constitutive component of tangent matrix

[K c, ab]

The previous example is revisited in order to illustrate the calcula-

tion of the tangent matrix component connecting nodes two to three.

Omitting zero derivative terms the summation given by Equation (7.35)

yields,

[K c,23]11 =

1

8

c

sym1112

16

−

1

6

c

sym1212

16

(24t)

[K c,23]12 =

18c sym

11221

6−1

6c sym

12221

6

(24t)

[K c,23]21 =

1

8

c

sym2112

16

−

1

6

c

sym2212

16

(24t)

[K c,23]22 =

1

8

c

sym2122

16

−

1

6

c

sym2222

16

(24t)

(continued)


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EXAMPLE 7.4 (cont.)

where t is the thickness of the element. Substituting for c symijkl from

Equations (7.36) and (5.37–8) yields the stiffness coefficients as,

[K c,23]11 = −2

3λt; [K c,23]12 =

1

2µt; [K c,23]21 =

1

2µt;

[K c,23]22 = −2

3(λ + 2µ)t

where λ = λ/J and µ = (µ − λ ln J )/J .

7.4.2 CONSTITUTIVE COMPONENT – MATRIX FORM

The constitutive contribution to the linearized virtual work Equation (7.31)

for element (e) can alternatively be expressed in matrix notation by defining

the small strain vector ε in a similar manner to Equation (7.26) for d as,

ε = [ε11, ε22, ε33, 2ε12, 2ε13, 2ε23]T ; ε =

na=1

Baua (7.37a,b)

The constitutive component of the linearized internal virtual work – see

Equation (7.31) – can now be rewritten in matrix-vector notation as,

DδW c(φ, δ v)[u] =

V

δ d : C : εdv =

v

δ dT Dε dv (7.38)

where the spatial constitutive matrix D is constructed from the components

of the fourth-order tensor c by equating the tensor product δ d : c : ε to the

matrix product δ dT Dε to give, after some algebra,

D = 1

2

2c 1111 2c 1122 2c 1133 c 1112 +c 1121 c 1113 +c 1131 c 1123 +c 11322c 2222 2c 2233 c 2212 +c 2221 c 2213 +c 2231 c 2223 +c 2232

2c 3333 c 3312 +c 3321 c 3313 +c 3331 c 3323 +c 3332

c 1212 +c 1221 c 1213 +c 1231 c 1223 +c 1232sym. c 1313 +c 1331 c 1323 +c 1332

c 2323 +c 2332

(7.39)


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In the particular case of a neo-Hookean material (see Equations (5.28–29)),D becomes,

D =

λ + 2µ λ λ 0 0 0

λ λ + 2µ λ 0 0 0

λ λ λ + 2µ 0 0 0

0 0 0 µ 0 0

0 0 0 0 µ 0

0 0 0 0 0 µ

;

λ = λ

J

; µ = µ − λ ln J

J

(7.40)

Substituting for δ d and ε from Equations (7.26) and (7.37b) respectively

into the right-hand side of Equation (7.38) enables the contribution from

element (e) associated with nodes a and b to emerge as,

DδW (e)c (φ, N aδ va)[N bub] =

v(e)

(Baδ va)T D(Bbub) dv

= δ va ·

v(e)

BT aDBb dv

ub (7.41)

The term in brackets defines the constitutive component of the tangent

matrix relating node a to node b in element (e) as,

K (e)c,ab =

v(e)

BT aDBb dv (7.42)

7.4.3 INITIAL STRESS COMPONENT

Concentrating attention on the second term in the linearized equilibrium

Equation (7.31), note first that the gradients of δ v and u can be interpolated

from Equations (7.3–4) as,

∇δ v =na=1

δ va⊗∇N a (7.43a)

∇u =nb=1

ub⊗∇N b (7.43b)

Introducing these two equations into the second term of Equation (7.31)

and noting Equation (2.51b), that is, σ : (u⊗ v) = u ·σv for any vectors

u,v, enables the initial stress contribution to the linearized virtual work


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Equation (7.31) for element (e) linking nodes a and b to be found as,

DδW σ(φ, N aδ va)[N bub] =

v

σ : [(∇ub)T ∇δ va] dv

=

v(e)

σ : [(δ va ·ub)∇N b⊗∇N a] dv

= (δ va ·ub)

v(e)∇N a ·σ∇N b dv (7.44)

Observing that the integral in Equation (7.44) is a scalar, and noting that

δ va ·ub = δ va · Iub, the expression can be rewritten in matrix form as,

DδW σ(φ, N aδ va)[N bub] = δ va ·K (e)σ,abub (7.45a)

where the components of the so-called initial stress matrix K (e)σ,ab are,

K (e)σ,ab =

v(e)

(∇N a ·σ∇N b)I dv

[K (e)σ,ab]ij =

v(e)

3k,l=1

∂N a∂xk

σkl∂N b∂xl

δ ij dv; i, j = 1, 2, 3

EXAMPLE 7.5: Initial stress component of tangent matrix

[K σ,ab]

Using the same configuration as in examples 7.1–7.4 a typical initial

stress tangent matrix component connecting nodes 1 and 2 can be found

from Equation (7.45) as,

[K σ,12] =

v(e)

∂N 1∂x1

∂N 1∂x2

σ11 σ12

σ21 σ22

∂N 2∂x1

∂N 2∂x2

1 0

0 1

dv

=

1

8

8

1

8

+

−1

8

4

−1

6

1 0

0 1

24t

=−1t 00 −1t

7.4.4 EXTERNAL FORCE COMPONENT

As explained in Section 6.5 the body forces are invariably independent of the

motion and consequently do not contribute to the linearized virtual work.

However, for the particular case of enclosed normal pressure discussed in


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Section 6.5.2, the linearization of the associated virtual work term is givenby Equation (6.23) as,

DδW pext(φ, δ v)[u] = 1

2

Aξ

p∂ x

∂ξ ·

∂δ v

∂η × u

−

δ v ×

∂ u

∂η

dξdη

− 1

2

Aξ

p∂ x

∂η ·

∂δ v

∂ξ × u

−

δ v ×

∂ u

∂ξ

dξdη

(7.46)

Implicit in the isoparametric volume interpolation is a corresponding surface

representation in terms of ξ and η as (see Figure 7.1),

x(ξ, η) =na=1

N axa (7.47)

where n is the number of nodes per surface element. Similar surface in-

terpolations apply to both δ v and u in Equation (7.46). Considering, as

before, the contribution to the linearized external virtual work term, in

Equation (7.30), from surface element (e) associated with nodes a and b

gives,

DδW p(e)ext (φ, N aδ va)[N bub]

= (δ va × ub) · 1

2 Aξ

p∂ x

∂ξ

∂N a

∂η N b −

∂ N b

∂η N adξdη

− (δ va × ub) · 1

2

Aξ

p∂ x

∂η

∂N a∂ξ

N b − ∂ N b

∂ξ N a

dξdη

= (δ va × ub) · k p,ab (7.48)

where the vector of stiffness coefficients k p,ab is,

k p,ab = 1

2

Aξ

p∂ x

∂ξ

∂N a∂η

N b − ∂ N b

∂η N a

dξdη

+ 1

2

Aξ

p∂ x

∂η

∂N a∂ξ

N b − ∂ N b

∂ξ N a

dξdη (7.49)

Equation (7.48) can now be reinterpreted in terms of tangent matrix com-

ponents as,

DδW p(e)ext (φ, N aδ va)[N bub] = δ va ·K

(e) p,ab ub (7.50a)

where the external pressure component of the tangent matrix is,

K (e) p,ab = E k

(e) p,ab;

K

(e) p,ab

ij

=3k=1

E ijkk

(e) p,ab

k

; i, j = 1, 2, 3


45/122


where E is the

fem implementation

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