Abstract

Conclusions

• 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 intuitive clarity.
• The adoption of nonconjugate strain and stress tensors for a microplane model of concrete is admissible because the following four conditions are satisfied:
• 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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