饱和多孔介质分析解的唯一性与应变局部化分岔

UNIQUENESS AND LOCALIZATION BIFURCATION ANALYSIS SOLUTION OF SATURATED POROUS MEDIA

基于不连续性分岔基本理论导出了静态非渗流状态下弹塑性饱和多孔介质应变局部化发生时的临界硬／软化模量：利用二阶功正定性原理研究了两相问题分析解的唯一性问题，并给出了基于主轴空间下解的显式表达式．研究工作表明，在静态非渗流状态下，弹塑性饱和多孔介质分析的唯一性与应变局部化发生的临界条件除了在量值上与单相介质有着明显的不同外两者之间还有许多一致的特性，这些一致的特性对问题的分析是十分重要的．

Strain localization and related material instabilities are of considerable interest because of their importance in failure prediction of materials. A suitable tool for describing localization in solid mechanics is based on the strain rate discontinuity in continuum theory. The basic theory can be found in the works by Mandel (1964) 5 Rudnicki and Rice (1975), Rice (1975, 1976), Vardoulakis (1976) and Rice and Rudnicki (1980). On the other hand, strain localization also means both a non-uniqueness in the incremental elastoplastic response of the deformable continuum and a vanishing speed of the acceleration waves (see Hadamard (1903), Thomas (1961), Rice (1976), Hill (1962) and Mandel (1964)). Some recent works for quite general classes of elastic-plastic solids are given by Ottosen and Runesson (1991), Runesson, Ottosen and Peric (1991), and Bigoni and Hueckel (1991). Most work published so far is related to the behaviour of single phase materials. However, strain localization phenomena are also relevant for elastic-plastic porous solids, the pores of which are filled with a fluid such as water, oil, etc. Strain localization analysis for this problem is generally performed under locally undrained conditions where net inflow or outflow is prevented. Some classical and recent results can be found in the papers by Rice (1975), Rice and Cleary (1976), Rudnicki (1983), Han and Vardoulakis (1991), Runesson, et al. (1996). Rudnicki (1983) and Zhang et al. (1999). Zhang et al. (1999) discussed the relevant localization condition for the more general situation of drained behaviour due to finite permeability of the media, i.e. internal flow is not inhibited. In this paper, the conditions for localization of deformation into a planar (shear) band and loss of uniqueness in the incremental response of elastic-plastic saturated porous media were systematically studied. The critical modulus for shear band localization of undrained condition are studied in terms of the discontinuous bifurcation analysis of the problems. Loss of uniqueness of the response of the coupled problem is investigated by means of the positiveness of the second order work density which has been broadly used in solid mechanics. Based on the general solutions, the explicit solutions of the critical hardening modulus for both kinds of material instability problems, i.e. strain localization and loss of uniqueness, are found in the principle axial space. The existence of the explicit solutions makes it possible to directly inspect possible localization models. Some important results such as that the critical haxdening modulus for strain localization is never positive for associative plasticity and coincides with the critical hardening modulus for zero second order work in saturated porous media, are obtained. It has been shown from the present research that the loss of uniqueness and the critical conditions for strain localization at static undrained conditions are different in quantity with the results obtained for single phase materials. However, it has been also noticed that there are still some common features between the results for the two different media, i.e. single and two phase problems. These common features are quite useful for the study of the problems discussed here.