Particular aspects of internal length scales in strain localization analysis of multiphase porous materials

Following the work by Zhang et al. [Mech. Cohes.-Frict. Mater. Struct. 4 (1999) 443] we discuss some special features of the internal length scale found in multiphase materials such as saturated and partially saturated porous media, where viscous terms are introduced in the fluid mass balance equations naturally by Darcy's law. Particular attention is focused on the two cases of wave number K=0 and K→∞ of the perturbation waves. The internal length scale properties corresponding to the two cases called "lower and upper critical hardening modulus φ" mentioned in Zhang and Schrefler [Int. J. Numer. Anal. Methods Geomech. 25 (2001) 29], are discussed in detail. It is shown that for the case of the upper value of the critical hardening modulus there will be also a wave number domain for which the material model is dispersive when strain softening behavior occurs for solid skeleton. However, this kind of dispersive waves may not supply enough energy to activate the internal length scale in strain localization analysis. This is true also for the quasi-static case, where it has been found that for both zero and infinite wave numbers the internal length scale associated with permeability disappears. However, it will be pointed out in this paper that the consideration of fluid interaction is necessary for the prediction of the internal length when regularization through a constitutive model is introduced in the numerical model to overcome mesh dependence in a finite element solution. This problem is discussed by considering a gradient dependent model.