Journal of Engineering Mechanics ASCE, 126 (2000), 971-980
LARGE STRAIN GENERALIZATION OF
MICROPLANE MODEL FOR CONCRETE AND APPLICATION
Zdenek P. Bazant, Mark D. Adley, Ignacio Carol, Milan
Jirásek, Stephen A. Akers, Bob Rohani, J. Donald Cargile, and
The preceding formulation of the microplane model for concrete and development
of model M4 is extended into large strains. After giving examples
of certain difficulties with the second Piola-Kirchhoff stress tensor in
the modeling of strength and frictional limits on weak planes within the
material, the back-rotated Cauchy (true) tensor is introduced as the stress
measure. The strain tensor conjugate to the back-rotated Cauchy (or
Kirchhoff) stress tensor is unsuitable because it is nonholonomic (i.e.,
path-dependent) and because its microplane components do not characterize
meaningful deformation measures. Therefore Green's Lagrangian tensor
is adopted even though it is not conjugate. Only for this strain
measure the microplane components of the strain tensor suffice to characterize
the normal stretch and shear angle on that microplane. Using such
nonconjugate strain and stress tensors is admissible because, for concrete,
the elastic parts of strains as well as the total volumetric strains are
always small, and because the algorithm used guarantees the energy dissipation
by large inelastic strains to be non-negative. Examples of dynamic structural
analysis are given.
For the formulation of microplane model as well as other sophisticated
finite-strain constitutive relations for frictional (pressure-sensitive)
materials such as concretes, soils, rocks and some composites, it is convenient
to use a stress tensor referred to the initial configuration that has a
clear physical meaning, such that the stress components on any plane give
in a simple way the true (Cauchy) normal and shear stresses on that plane,
and that the hydrostatic pressure is easily controlled. Only then
is it easy to model friction, yield or strength limit, and hardening or
softening on that plane. The stress tensor referred to the initial
configuration that satisfies these conditions is the back-rotated Cauchy
stress tensor, which has been adopted in this study. Another, only
slightly less convenient, would be the back-rotated Kirchhoff stress tensor.
The strain tensors conjugate to the back-rotated Cauchy and Kirchhoff stress
tensors are nonholonomic, i.e., their values depend on the strain path.
This property makes them unsuitable for quasibrittle materials such as
concrete which have a memory of their initial state.
For the sake of simplicity and clarity, it is desirable that the strain
tensor components on one microplane suffice to fully characterize the stretch
(relative change of length) of a material line segment initially normal
to that microplane and the true shear angle on that microplane (change
of the angle between the initial normal and the microplane). This
condition is satisfied only by Green's Lagrangian finite strain tensor.
For any other strain tensor, the aforementioned stretch and shear angle
on one microplane depend also on the strain tensor components on planes
of other orientation, which would greatly complicate the formulation of
microplane constitutive model.
Consequently, it is best to base the finite-strain microplane model on
the Green's Lagrangian strain tensor and the back-rotated Cauchy stress
tensor, even though they are not conjugated by work. Transformation
of the constitutive relation to conjugate strain and stress tensors (e.g.,
the Green's Lagrangian strain tensor and the second Piola-Kirchoff stress
tensor) is of course generally possible, but the resulting complexity would
defeat the purpose of microplane approach---the conceptual simplicity and
The adoption of nonconjugate strain and stress tensors for a microplane
model of concrete is admissible because the following four conditions are
The adopted nonconjugate constitutive law on the microplane level is in
unique correspondence to a certain conjugate constitutive law on the macro-level.
The micro-macro kinematic constraint of microplane model imposed in terms
of one type of strain tensor implies the kinematic constraint to hold also
for any other finite strain tensor.
The elastic parts of strains are always small, which ensures the energy
dissipation caused by elastic deformations formulated in terms of the nonconjugate
stress and strain tensors to be negligible.
The inelastic stress drops to the boundary surface, which are made in the
numerical algorithm at constant strain, always dissipate energy (i.e.,
a negative energy dissipation is ruled out).
Because the volume changes of concrete can never be large, the split of
strains and stresses into their volumetric and deviatoric components can
be considered as additive if the volumetric and deviatoric strain definitions
from Bazant (1996) are employed. This greatly simplifies the finite-strain
generalization of the microplane model.
Extended to the rate effect and finite strain, model M4 gives realistic
predictions in vectorized explicit finite element simulations of missile
impact and ground shock on concrete structures.
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EPFL / 25 September 2000 / Milan.Jirasek@epfl.ch