Microplane Model for Finite Strain
In some applications of the microplanc model, for example, the impact of missiles into hardened concrete structures or nuclear reactor containments, or the analysis of energy absorption of a highly confined column in an earthquake, very large strains, ranging from 10% to 200%, and shear angles up to 40°, have been encountered in calculations. For such situations, the microplane model must be generalized to finite strain. However, a thorough exposition of the finite strain generalization would require introducing advanced mathematical apparatus that has not yet appeared in this book. Therefore, only a summary of the main results will be given here. The interested reader can find the details in Bazant, Xiang and Prat (1996) and Bazant, Xiang et al. (1996) for the case of moderately large strains (up to about 10%), and in Bazant (1997b) for the case of very large strains (100% or more, with shear angles up to 40°).
The simplest finite strain tensor to use is Green’s Lagrangian strain tensor E — (FTF — l)/2 where F is the deformation gradient and 1 the unit tensor (see, e. g., Bazant and Cedolin 1991, Chapter 11). Its conjugate stress tensor, that is the tensor for which Green’s Lagrangian strain tensor gives a correct work expression dW — T ■ ГІЕ, is a tensor called the second Piola-Kirchhoff stress tensor, T = F_l JSF-7 where J — det F is the Jacobian of the transformation (giving the relative volume change); F"7 -■ (F"1)7 =(F7)-1.
Difficult problems arise in the modeling of very large strains. In finite-strain generalization of the microplane model, a definite physical meaning needs to be attached to the normal and shear strain components on the microplanes. In this regard, the following two conditions must be met:
Condition I. The normal and shear components of the stress tensor used in the constitutive relation must uniquely characterize the normal and shear components of the tensor of true stress S in the deformed material, called the Cauchy stress tensor.
Condition II. The normal strain component ел/, characterizing the stretch Ал’ of a material line segment in the directionli initially normal to the microplane, must be independent of the stretches of material line segments in other initial directions. Furthermore, the shear strain component е/ум (or e/vi), characterizing the change of angle or #/vl between two initially orthogonal material line segments with initial unit vectors n and m (or n and (), must be independent of the stretches and angle changes in planes other than (n, m) or (n, £).
Consider first only condition I. It turns out that, in this regard, the use of the second Piola-Kirchhoff stress tensor is possible only if the largest magnitude of the principal strains is less than about 7% to 10%, i. e., if the strain is only moderately large.. It has been shown by numerical examples (Bazant 1997b) that, for large isochoric deformations, the shear components of the second Piola-Kirchhoff stress tensor T strongly depend on the volumetric component of the Cauchy (true) stress tensor S (i. e., the true hydrostatic pressure), and the volumetric component of the second Piola-Kirchhoff stress tensor strongly depends on the shear components of the Cauchy stress tensor. This indicates that the projections of the second Piola-Kirchhoff stress tensor on the microplanes have no physical meaning. They cannot be used to characterize the strength, yield limit and damage on the microplane, nor the phenomenon of friction.
The stress tensor must be referred to the initial configuration of the material (as required for the modeling of a solid remembering the initial state). The only such tensor whose microplanc components have a physical meaning is the rotated Kirchhoff stress tensor r = Rr JSR, where JS represents the Kirchhoff stress tensor, and R is the material rotation tensor defined in the polar decomposition of the deformation gradient F — RU = VR. Here, U and V are the right and left stretch tensors. When the principal stress axes do not rotate against the material, the rotated Kirchhoff stress tensor is equal to the Cauchy (true) stress tensor scaled by a scalar factor, J. Only this tensor is free of the aforementioned problems revealed by numerical examples.
A variational procedure can be used to obtain an expression for the finite strain tensor 7 that is conjugated by work with the rotated Kirchhoff stress tensor r. If the principal strain axes do not rotate against the material, this tensor is found to be identical to Hencky’s (logarithmic) strain tensor. However, when the principal strain axes rotate, one obtains an incremental expression for d’y that cannot be integrated. This means that the strain tensor conjugate to the rotated Kirchhoff stress tensor is nonunique, path-dependent (nonholomonic).
The aforementioned path-dependence is strong and unacceptable, except for moderately large strains less than about 1% to 10%. For such strains, and for larger strains for which the rotations of principal strain axes is small, the use of Hencky’s strain tensor is advantageous. However, there is also the problem of the efficient calculation of the Hencky tensor. This tensor is defined by the spectral representation, which is computationally demanding for large finite element programs in which this tensor may have to be calculated up to a billion times. Nevertheless, an easy-to-compute very close approximation of the Hencky strain tensor has recently been found (Bazant 1997c).
Consider now condition II. The relative length change of a segment normal to the microplane from length dS (in the initial configuration) to length ds (in the deformed configuration) is characterized, e. g., by ejv = (ds — dS)/dS (called Biot strain or engineering strain). The change of angle between the microplane normal vector n and vector m in the microplane represents the shear angle вмм – When Green’s Lagrangian strain tensor E is used, C;-j can be expressed (exactly) in terms of the normal component Em, and 0mm can be expressed (exactly) in terms of the shear component Emm and the normal components Ем, Em (see, e. g., Malvern 1969, pp. 165-166). In other words, the exact change of length in normal direction and of shear angle for a microplane can be expressed solely in terms of the strain tensor components on the same microplane. This is not true, however, for all the other strain tensors, including Hencky’s (logarithmic) strain tensor and Biot’s strain tensor. For them, the exact ед> and Omm depend also on the ratio of the principal strains (which seems an inconvenient feature for the programming of microplane model and would increase demands on computer time). This dependence can be neglected only when the maximum principal strain is less than about 25% (Bazant 1997d).
It thus appears that, for large strains (i. e., when the maximum principal strain exceeds 7% to 10%) the only suitable strain and stress tensors are Green’s Lagrangian strain tensor and the rotated Kirchhoff stress tensor. These two tensors are not conjugate.
It has normally been considered a taboo to use nonconjugate stresses and strains. However, due to the special character of the present microplane model, the use of nonconjugate stresses and strains in formulating a constitutive relation is admissible if certain precautions are taken (see Bazant 1995d and Bazant, Adley and Xiang 1996). One point to note in this regard is that the constitutive relation in terms of the aforementioned nonconjugate stress and strain tensors is a transformation of the constitutive relation in terms of conjugate stress and strain tensors such that the transformation depends only Green’s Lagrangian strain tensor (or the stretch tensor) but, importantly, is independent of the material rotation tensor. Such a transformation is perfectly admissible. The second point to note is that nonnegativeness of energy dissipation is ensured for two reasons: (l)The elastic parts of strains are always small (which ensures that the elastic part of the nonconjugate stress-strain relation preserves energy), and (2) the drop of stress to the boundary surface is carried out in each load step at constant strain and cannot cause negative energy dissipation.
A further precaution that must be taken is that the work done by the stresses (or by the nodal forces on displacements) cannot be directly calculated from the stresses and strains used in the constitutive law, because the areas under the stress-strain curves for the nonconjugate constitutive law do not correctly
characterize energy dissipation. If the work needs to be calculated, one can easily obtain the second Piola-Kirchhoff stress tensor from the rotated Kirchhoff stress tensor and evaluate the work that way. Also, elastic response cannot be described as a functional relation between nonconjugate stresses and strains.
For moderately large strains, of course, the conjugate pair or Green’s Lagrangian strain tensor and second Piola-Kirchhoff strain tensor can be used, and has been used by Bazant, Xiang and Prat (1996) in a finite strain generalization of the microplane model. There is, however, a gap in the experimental data for very large strains of concrete. To (ill this gap, one must get reconciled with the fact that it is next to impossible to keep the specimen deformation uniform when triaxial deformations become large. Triaxial test data on concrete at strains up to shear angle of 35° at very high hydrostatic pressures (several times the uniaxial compression strength) have recently been obtained by Bazant and Kim (1996b) using a novel type of test, called the ‘tube-squash’ test. In this test, a thick-walled tube of very ductile steel is filled with concrete, and after curing, it is compressed axially to about half the initial length. The concrete undergoes shear angles over 30°. Due to high confining pressure (which exceeds 1000 MPa), the concrete in the tube retains integrity and small cores can be drilled out from the concrete. These cores show uniaxial compression strength between 20% and 50% of the virgin concrete, both for normal and high strength concretes. In the evaluation of the ‘tube-squash test’ one must fit the measured load-displacement curves with a finite element program incorporating finite strain constitutive models for both the concrete and the steel.
Finite strain tests need to be also carried out at small hydrostatic (confining) pressures, at which concrete turns into rubble when large deformations occur. A constitutive relation for such rubbcliz. ed concrete at finite strain needs to be developed.
Another problem that needs to be resolved for the inicroplane model is the split of mtal normal strain into deviatoric and volumetric components. The decomposition of large deformations into their volumetric and deviatoric (strictly speaking, isochoric) parts is, in general, multiplicative. Specifically, it has the form U = FpUz (Flory 1961; Sidoroff 1974; Simo 1988; Sirno and Ortiz. 1985; Lubliner 1986; Bell 1985) where U is the right stretch tensor, Ui/ the volumetric riglu-stretch tensor, and Fg> = the deviatoric transformation tensor. An additive volumetric-deviatoric decomposition exists only for the Hencky (logarithmic) strain tensor H.
For any type of finite strain tensor, however, an approximate additive decomposition in terms of volumetric strain tensor Ер — ер 1 and deviatoric strain tensor Eg> is possible for materials that can exhibit only large deviatoric strains but not large volumetric strains (Baz. ant 1996c), as is the case for concrete. Unlike Fв, the components of Ед depend on J, i. e., the relative volume change (unless the Hencky strain tensor is used). However, their dependence on J is, in the case of Green’s Lagrangian strain tensor, negligible if the volume change is less than about 3% in magnitude (Baz. ant, Xiang and Prat 1996; Bazant, Xiang et al. 1996). For Biot strain tensor E6 = U — 1 (Biot 1965; Ogden 1984; Bazant and Cedolin 1991), the limit is about 8% (for concrete, the volume change is -3% al highest pressure tested so far, which is 300000 psi or 2069 MPa; Bazant, Bishop and Chang 1986). Thus, the classical multiplicative decomposition, which is not as convenient for calculations as the additive decomposition, seems to be inevitable only for materials exhibiting very large volume changes, such as stiff foams. An additive decomposition of the aforementioned kind, developed in Bazant (1996c), was used by Bazant, Xiang and Prat (1996) in the generalization of the microplane model for moderately large finite strains of concrete.
The multiplicative decomposition could nevertheless be implemented in the microplane model by decomposing each loading step into two substeps, pure volumetric deformation followed by pure isochoric deformation, but that would greatly complicate the analysis, especially if the solution is not explicit.