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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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520 26 Micro-Mechanical Finite Element Models for Crystal Plasticity
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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522 26 Micro-Mechanical Finite Element Models for Crystal Plasticity
(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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524 26 Micro-Mechanical Finite Element Models for Crystal Plasticity
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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526 26 Micro-Mechanical Finite Element Models for Crystal Plasticity
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.
References
ABAQUS (2003). Reference Manuals, Hibbit, Karlsson, and Sorensen, Inc.
Asaro, R. J. and A. Needleman (1985). “Texture development and strain hardening in rate dependent polycrystals.”
Acta Metall. 33: 923-953.
Asaro, R. J. and J. R. Rice (1977). “Strain localization in ductile single crystals.” J. Mech. Phys. Solids 25: 309-338.
Bachu, V. and S. R. Kalidindi (1998). “On the accuracy of the predictions of texture evolution by the finite element
technique for FCC polycrystals.” Materials Science and Engineering A 257: 108-117.
Barrett, C. S. and L. H. Levinson (1940). “The structure of aluminum after compression.” Trans. Metall. Soc. AIME
137: 112-127.
Beaudoin, A. J., P. R. Dawson, K. K. Mathur and U. F. Kocks (1995). “A hybrid finite element formulation for
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
Simulation With Explicit Use of Polycrystal Plasticity Models.” International Journal of Plasticity 9(7): 833-860.
Bhattacharyya, A., E. El-Danaf, S. R. Kalidindi and R. D. Doherty (2001). “Evolution of grain-scale microstructure
during large strain simple compression of polycrystalline aluminum with quasi-columnar grains: OIM measure-
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-
tion in Columnar Polycrystalline Aluminum Using Orientation Image Mapping and Crystal Plasticity Models.”
Proceedings of Royal Society of London.
Bronkhorst, C. A., S. R. Kalidindi and L. Anand (1992). “Polycrystalline Plasticity and the Evolution of Crystallo-
graphic Texture in FCC Metals.” Philosophical Transactions of the Royal Society of London A 341: 443-477.
Delannay, L., S. R. Kalidindi and P. Van Houtte (2002). “Quantitative prediction of textures in aluminium cold rolledto moderate strains.” Materials Science and Engineering A 336(1-2): 233-244.
Kalidindi, S. R. (1998). “Incorporation of Deformation Twinning in Crystal Plasticity Models.” Journal of the Mecha-
nics and Physics of Solids 46: 267-290.
Kalidindi, S. R., C. A. Bronkhorst and L. Anand (1992). “Crystallographic texture evolution in bulk deformation
processing of FCC metals.” Journal of the Mechanics and Physics of Solids 40: 537-569.
Kalidindi, S. R. and S. E. Schoenfeld (2000). “On the prediction of yield surfaces by the crystal plasticity models for
fcc polycrystals.” Materials Science and Engineering A 293: 120-129.
Molinari, A., G. R. Canova and S. Ahzi (1987). “Self consistent approach of the large deformation polycrystal vis-
coplasticity.” Acta Metallurgica 35(12): 2983-2994.
Panchanadeeswaran, S., R. D. Doherty and R. Becker (1996). “Direct observation of orientation changes by channel
die compression of polycrystalline aluminum-use of split sample.” Acta Mater. 44: 1233.
Raabe, D., M. Sachtleber, Z. Zhao, F. Roters and S. Zaefferer (2001). “Micromechanical and macromechanical effects
in grain scale polycrystal plasticity experimentation and simulation.” Acta Materialia 49(17): 3433-3441.
Sarma, G. B. and P. R. Dawson (1996). “Texture predictions using a polycrystal plasticity model incorporating neigh-bor interactions.” International Journal of Plasticity 12(8): 1023-1054.
Taylor, G. I. (1938). “Plastic strain in metals.” J. Inst. Met. 62: 307-324.
Toth, L. S., A. Molinari and P. D. Bons (1994). “Self consistent modelling of the creep behavior of mixtures of
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Van Houtte, P., L. Delannay and S. R. Kalidindi (2002). “Comparison of two grain interaction models for polycrystal
plasticity and deformation texture prediction.” International Journal of Plasticity 18: 359-377.
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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.
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172 Implementation of plasticity models
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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174 Implementation of plasticity models
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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176 Implementation of plasticity models
(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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178 Implementation of plasticity models
(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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180 Implementation of plasticity models
–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 .
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182 Implementation of plasticity models
(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
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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
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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
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4 DISCRETIZATION AND SOLUTION
(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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6 DISCRETIZATION AND SOLUTION
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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8 DISCRETIZATION AND SOLUTION
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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10 DISCRETIZATION AND SOLUTION
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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12 DISCRETIZATION AND SOLUTION
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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7.4 LINEARIZED EQUILIBRIUM EQUATIONS 13
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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14 DISCRETIZATION AND SOLUTION
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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7.4 LINEARIZED EQUILIBRIUM EQUATIONS 15
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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16 DISCRETIZATION AND SOLUTION
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
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7.4 LINEARIZED EQUILIBRIUM EQUATIONS 17
where E is the