13th International Conference on Fracture June 16-21, 2013, Beijing, China (l = ∂v/∂x) are deformation gradient, velocity of a material point, and velocity gradient tensors, respectively An updated Lagrangian formulation is established. u* XA xi FiA t * Γ Ω S O = t t = 0 t Initial configuration current configuration Figure 1. Boundary value problem of elastic body including discontinuous planes 2.1. Updated-Lagrangian Formulation St.Venant-Kirchhoff material is assumed to be analyzed. The second Piola-Kirchhoff stress tensor S is expressed as a function of Green-Lagrange strain Eas following: SAB =C (0) ABCDECD (1) where C(0) is the elastic tensor. Truesdell’s stress rate ˘T is defined as follows: ˘T = ˙T −lT −Tl T +Ttrl, (2) where T and ˙T are Cauchy stress tensor and its material derivative. Then, we assume the linearity between the Truesdell’s stress rate ˘T and strain rate tensor dwith a tensor C: ˘Ti j =Ci jkldkl (3) Using an approximation of the updated Lagrangian formulation, we obtain Ci jkl = 1 J FiAFjBFkCFlDC (0) ABCD. (4) where J =det|F|. (5) For the UL formulation, the principle of virtual power is written as follows: ∫Ω δdTCddv +∫ Ω δlTTldv =∫ Ω ρ˙b· δvdv +∫ ∂Ω ˙t∗ · δvda, (6) where ρ, ˙b, and ˙t∗ are mass density, body force, and surface traction, respectively. -2-
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