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Concept
Microplane model for constitutive laws of materials
Résumé
The microplane model, conceived in 1984, is a material constitutive model for progressive softening damage. Its advantage over the classical tensorial constitutive models is that it can capture the oriented nature of damage such as tensile cracking, slip, friction, and compression splitting, as well as the orientation of fiber reinforcement. Another advantage is that the anisotropy of materials such as gas shale or fiber composites can be effectively represented. To prevent unstable strain localization (and spurious mesh sensitivity in finite element computations), this model must be used in combination with some nonlocal continuum formulation (e.g., the crack band model). Prior to 2000, these advantages were outweighed by greater computational demands of the material subroutine, but thanks to huge increase of computer power, the microplane model is now routinely used in computer programs, even with tens of millions of finite elements.
The basic idea of the microplane model is to express the constitutive law not in terms of tensors, but in terms of the vectors of stress and strain acting on planes of various orientations called the microplanes. The use of vectors was inspired by G. I. Taylor's idea in 1938 which led to Taylor models for plasticity of polycrystalline metals. But the microplane models differ conceptually in two ways.
Firstly, to prevent model instability in post-peak softening damage, the kinematic constraint must be used instead of the static one. Thus, the strain (rather than stress) vector on each microplane is the projection of the macroscopic strain tensor, i.e.,
where and are the normal vector and two strain vectors corresponding to each microplane, and and where and are three mutually orthogonal vectors, one normal and two tangential, characterizing each particular microplane (subscripts refer to Cartesian coordinates).
Secondly, a variational principle (or the principle of virtual work) relates the stress vector components on the microplanes ( and ) to the macro-continuum stress tensor , to ensure equilibrium.
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