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SUMMARYAn inverse micromechanics approach allows interpretation of nanoindentation results to deliver cohesivefrictional strength behavior of the porous clay binder phase in shale. A recently developed strength homogenization model, using the Linear Comparison Composite approach, considers porous clay as a granular material with a cohesive-frictional solid phase. This strength homogenization model is employed in a Limit Analysis Solver to study indentation hardness responses and develop scaling relationships for indentation hardness with clay packing density. Using an inverse approach for nanoindentation on a variety of shale materials gives estimates of packing density distributions within each shale and demonstrates that there exists shale-independent scaling relations of the cohesion and of the friction coefficient that vary with clay packing density. It is observed that the friction coefficient, which may be interpreted as a degree of pressure-sensitivity in strength, tends to zero as clay packing density increases to one. In contrast, cohesion reaches its highest value as clay packing density increases to one. The physical origins of these phenomena are discussed, and related to fractal packing of these nanogranular materials.

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This paper presents a variational formulation for the analysis of plastic collapse conditions for a class of hardening materials that accounts for some non-associated flow laws such as the modified Cam-clay model of soils. In this framework, classical statical and kinematical principles of limit analysis do not hold. The variational principle is formulated for the general class of materials whose flow equations are derived from a kind of generalized potentials named bipotentials by de Saxcé.The plastic collapse phenomenon for hardening materials is considered first and formulated as a system of equations. In particular, the case of the usual modified Cam-clay model is analyzed. The paper follows with the proposal of a minimization principle whose solution is then related to the solution of the plastic collapse problem. We demonstrate the use of this minimum principle in a simple example of triaxial compression of a modified Cam-clay material. Finally, we discuss the particular form of the proposed variational formulation for the case of associated plasticity.

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