# Dual integrals in non-linear fracture mechanics

## The analysis of the dissipation shows that the thermodynamical force associated with the evolution of a crack is an energy release rate, form of which depends on the presence or not of mechanical discontinuities. This energy release rate is given as an integral based on free or complementary energy. We analyse the invariance of such integrals and we discuss the obtained results in elastoplasticity. | Vietnam Journal of Mechanics, VAST, Vol. 31, No. 3 &4 (2009), pp. 221 – 231 DUAL INTEGRALS IN NON-LINEAR FRACTURE MECHANICS Claude Stolz Laboratoire de Mécanique des Solides, CNRS UMR7649 École polytechnique, Palaiseau 91128 Cedex-France Abstract. We propose a dual approach in fracture mechanics based on complementary energy. The analysis of the dissipation shows that the thermodynamical force associated with the evolution of a crack is an energy release rate, form of which depends on the presence or not of mechanical discontinuities. This energy release rate is given as an integral based on free or complementary energy. We analyse the invariance of such integrals and we discuss the obtained results in elastoplasticity. 1. INTRODUCTION Consider a body Ω with a crack, represented by a straight line. Around the crack tip we distinguish three domains determined by the distance from the tip. Fig. 1. The description of the behaviour near the crack tip • The nearest zone I, where all physical processes of rupture occur, that is the process zone. • Zone II, where the mechanical fields are described by the singular field. • Zone III, where the mechanical fields match with boundary conditions. 222 Claude Stolz At our scale crack is represented by a line along Ox. The normal is ey in the plane, and ez normal to the plane. The crack length is l(t) and a = l˙ is the crack speed. If the singularities of mechanical fields govern the crack propagation, it is not necessary to take into account the rupture process. This is an approximation which leads to a global approach of rupture. This description is powerfull and constitutes the key point for describing classical fracture mechanics. In this case, the singularities represent the loading applied on the process zone. In non linear mechanics these notions must be revisited. The dissipation can be represented by several manner, depending on the local constitutive law and on the mechanisms of rupture. The dissipation can be

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