Elastic Microplane Formulation for Transversely Isotropic Materials
  This contribution investigates the extension of the microplane formulation to the description of transversely isotropic materials such as shale rock, foams, unidirectional composites, and ceramics. Two possible approaches are considered: 1) the spectral decomposition of the stiffness tensor to define the microplane constitutive laws in terms of energetically orthogonal eigenstrains and eigenstresses; and 2) the definition of orientation-dependent microplane elastic moduli. It is shown that the first approach provides a rigorous way to tackle anisotropy within the microplane framework whereas the second approach represents an approximation which, however, makes the formulation of nonlinear constitutive equations much simpler. The efficacy of the second approach in modeling the macroscopic elastic behavior is compared to the thermodynamic restrictions of the anisotropic parameters showing that a significant range of elastic properties can be modeled with excellent accuracy. Further, it is shown that it provides a very good approximation of the microplane stresses provided by the first approach, with the advantage of a simpler formulation.   It is concluded that the spectral stiffness decomposition represents the best approach in such cases as for modeling unidirectional composites, in which accurately capturing the elastic behavior is important. The introduction of orientation-dependent microplane elastic moduli provides a simpler framework for the modeling of transversely isotropic materials with remarked inelastic behavior, as in the case, for example, of shale rock. 
Congrui Jin, Marco Salviato, Weixin Li, Gianluca Cusatis
cond-mat.mtrl-sci