13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Strength properties of a Drucker-Prager porous medium reinforced by rigid particles Zheng He1,*, Luc Dormieux1, Kondo Djimedo2 1 E.N.P.C, Institut Navier, 77455 Marne-la-Vallee, France 2 U.M.P.C, Institut d’Alembert, 75252 PARIS CEDEX 05, France * Corresponding author: hez@lmsgc.enpc.fr Abstract In the present study, we investigate the strength properties of ductile porous materials reinforced by rigid particles. The microporous medium is constituted of a Drucker-Prager solid phase containing spherical voids; its behaviour is described by means of an elliptic criterion (issued from a modified secant moduli approach) whose corresponding support function is determined. The later is then implemented in a limit analysis approach in which a careful attention is paid for the contribution of the inclusion matrix-interface. This delivers parametric equations of the effective strength properties of the porous material reinforced by rigid particles. The predictions are compared to available results obtained by means of variationnal homogenization methods successively applied for micro-to-meso and then for meso-to-macro scales transitions. Moreover, we discuss in detail the predictions of the material strength under isotropic mechanical loadings. To this end, additional static solutions are derived and compared to the kinematics limit analysis ones. Finally, we derive an approximate closed-form expression of the macroscopic strength which proves to be very accurate. Keywords Porous; Strength; Drucker-Prager; inclusion-matrix, interfaces. 1. Introduction Being hard clayey rocks, COx Argillite is a porous clay matrix in which quartz or silica inclusions are embedded. In the present study, we mainly aim to derive new closed-form results for the strength of the COx argillite, under the assumption that the solid phase of the clay is a Drucker-Prager perfectly plastic material. Therefore, the behaviour of the microporous clay is described by means of an elliptic criterion [2] whose corresponding support function is determined in this paper. Then by using this support function, we explore an alternative approach which can be viewed as an extension of the original Gurson model. Instead of a spherical cavity surrounded by a matrix, the proposed ’rigid core sphere model’ consists of a rigid spherical core surrounded by the homogeneous porous material. The failure criterion of this ’rigid core sphere model’ is derived in the framework of the cinematic approach of limit analysis (LA). It is worth noting that from the LA point of view the failure mechanism can include a strain concentration at the core-matrix interface which can be described mathematically. ([1-3],[6],[9],[11],[12]). Notations: 1andI are the second and fourth order identity tensors. (1/3) , = ⊗ = − 1 1 J K I J are respectively the spherical and deviatoric projectors of isotropic fourth order symmetric tensor. 2. The micro-to-meso transition: support function of porous matrix The first homogenization step approach starts at the microscopic scale. At this scale, the porous clay matrix is described as a heterogeneous material being made up of a Drucker-Prager perfectly plastic solid in which pores are embedded. Let 1 d mσ = − σ σ denote the deviatoric part of the stress tensor σat the microscopic scale. The scalar deviatoric stresses mσ are defined as : : dσ = K σ σ and ( : : ) /3 1 mσ = σ J and the Drucker-Prager criterion reads:
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