PSI - Issue 33

Domenico Ammendolea et al. / Procedia Structural Integrity 33 (2021) 858–870 Domenico Ammendolea et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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In Eqs. (1)-(2), u and T are the displacement and temperature fields, C is the isotropic elastic tensor,  is the coefficient of thermal expansion, T 0 is the reference temperature, f the body force vector, k is the thermal conductivity, and Q is the inner body heat source. In the proposed scheme, the thermo-mechanics problem (Eqs. (1) -(2)), must be solved together with conditions imposed by fracture mechanics. Because of the fracture problem is crack-path dependent and, considering that the crack path is unknown before the analysis, the entire problem must be stated in an incremental form (Greco et al. (2021b), Greco et al. (2020c), Scuro et al. (2021)). In particular, the fracture mechanics problem is expressed by using the classic Karush-Kuhn-Tucker conditions, as follows: 0, 0, 0 F F F F f f   =   (3) where, f F = ( K I , K II ,  c , K IC ) represents a fracture function that identifies crack onset conditions and F  is the incremental displacement of the crack tip. In particular, f F depends on the stress intensity factors at the crack front, the kinking angle (  c ), and the fracture toughness of the material ( K IC ). Eqs. (1)-(3) must be solved together to find the incremental crack tip displacement by assuming f F = 0. Notice that f F and  c can be defined using standard fracture criteria, such as the Maximum Hoop Stress (Erdogan and Sih (1963)) or the Maximum Strain Energy Release Rate (Hussain et al. (1974)) one. In particular, in the present study, the former is adopted. The Arbitrary Lagrangian-Eulerian formulation (ALE) In the proposed modeling strategy, the ALE formulation serves as an effective technique to handle the geometry variations caused by growing cracks. The basic idea is to move the nodes of the mesh frame according to the crack trajectory. This capability finds its best utility to simulate random crack propagation mechanisms, such as those induced by non-uniform temperature gradients. Indeed, non-uniform thermal gradients create irregular thermal strain fields that affect the stress distribution around the crack tip, and so the crack trajectory. Such a condition makes it difficult to predict the crack trajectory in advance. The ALE involves the definition of two reference frames for the nodes of the mesh frame. The first is a Referential or fixed system (R) that serves to describe the motion of the mesh nodes (Fig. 1-b). The second is a Moved or current system (M) that identifies the i- th varied configuration of the computational domain because of nodes motion ( i.e. , crack advance). Such reference systems are linked to each other through a continuous and bijective mapping function : R M C C  → with   1 2  =   , which implies: ( ) ( ) 1 M R R M X X X X − =  =  (4) 2.2









T M M M X X Y = and

T R R R X X Y = the current and referential position vector functions of the mesh nodes,

being

respectively.

Fig. 1. (a) A two-dimensional fractured domain under mechanical and thermal loadings. (b) The Arbitrary Lagrangian-Eulerian formulation: Referential (R) and Moved (M) configurations