تجزیه و تحلیل حساسیت بر اساس معیار انتشار ترک برای تراکم پذیر و مواد الاستیک بالای (تقریبا) تراکم ناپذیر

Sensitivity analysis of an XFEM crack propagation model is developed for shape and material parameters, where the direct differentiation method is applied to large strain problems with hyperelastic neo-Hookean materials. The presence of level set functions to describe the crack position requires the development of a proper differentiation technique which is also addressed. In order to compute the analytical derivatives of such a complex numerical model the capabilities of the symbolic system AceGen are employed. A crack propagation criterion based on the sensitivity formulation is developed, allowing the direct calculation of the crack growth length and direction without post-processing. Special attention is paid to the ability of satisfying incompressibility and near-incompressibility conditions. The performance of the XFEM sensitivity analysis is assessed by the Cook's Membrane and Pre-crack Plate benchmark tests where sensitivities of displacements and crack propagation criteria based on potential energy have been analysed with respect to crack length and crack growth parameters. The techniques presented in this paper can be extended to anisotropic materials and non-linear materials exhibiting plasticity and viscoplasticity. Additionally, this formulation constitutes a base for further analysis of crack branching and crack joining problems.

Numerical simulations have been gaining importance in industry as a valuable decision support tool in the design of engineering materials, components and systems, while saving resources. Nevertheless, to develop accurate and feasible numerical models it is essential to understand the influence of the different parameters on the overall response of a model. Through the sensitivity analyses it is possible to compute a rate of performance change with respect to the model design parameters. In general, the derivatives of an arbitrary response functional are calculated with respect to chosen parameters, such as model inputs (material constants, shape parameters, etc.) or intermediate or final results of the analysis (solution vectors, stress tensor, damage factors, etc.) [19], [20] and [7]. Among the different methods available for sensitivity analysis, the direct differentiation technique, intensively developed in the work of Michaleris [25], is considered in this paper. Sensitivity analyses are often applied in the context of shape optimisation problems, in which the behaviour of an energy functional is analysed when the shape of a body is modified [14] and [15]. This approach may be equally applied to fracture problems, as crack growth may be interpreted as a change of the shape of the body. First works in this direction have been developed by Hellen [16] and Parks [33] which evaluate the energy release change in a finite element problem when a crack is extended by a small amount. Nevertheless these works were based on specific examples, with an analytical or experimental solution available, and not on a fully developed sensitivity analysis formulation. Later, Zumwalt and El-Sayed [11] incorporated sensitivity analysis by calculating directly the derivative of the finite element stiffness matrix, however their work was restricted to specific finite element types. Another interesting approach was developed by Feijóo et al. [12] in which sensitivity analysis is used to obtain the expression for the energy release rate in a three-dimensional cracked body. As a drawback, this model requires the construction of an approximation for the velocity vector field, depending on some additional parameters. This work is related to the described approaches, as one of the main objectives is to build a robust crack criterion based on sensitivity analysis. Nevertheless, it differs from the existing works, as the sensitivity of the potential energy with respect to the crack length is obtained through direct differentiation with exact derivatives. Moreover, the formulation is not restricted to a particular element type, is able to handle non-linear materials and large deformations and pays special attention to the satisfaction of incompressibility and near-incompressibility conditions. The cracks are described through the XFEM and the crack path is tracked by level set functions [31] and [30]. The presence of these kind of functions demands careful handling, as the derivatives required for the sensitivity calculations are not correctly evaluated by the simple application of the chain rule. The XFEM [1] and [26] is one of the most prominent numerical techniques used in the field of fracture, which avoids the computationally expensive task of remeshing, each time a crack propagates. A large amount of literature concerning the XFEM is available in these days from applications to linear elastic fracture [1], [26], [9] and [43], continuum–discontinuum transition [39], [2], [5] and [3] and ductile fracture [35] and [36] to name a few. Nevertheless, in general, the sensitivity of different parameters used in the various XFEM formulations is only qualitatively evaluated. Problems with an available analytical solution are often used to determine the variations of the response of a certain numerical model under particular conditions, as the derivation of general sensitivity terms may become quite complex in the presence of non-linearities. Here tools are provided for efficient quantitative evaluation of the sensitivities associated with crack propagation problems modelled with the XFEM, taking advantage of the modern symbolic and algebraic computer systems. In particular, the AceGen [23] system allows the treatment of the equations associated with a crack propagation problem at a high abstract level. Furthermore, it includes an advance automatic differentiation tool, with simultaneous stochastic simplification of numerical code [45], [24], [22] and [21], allowing the generation of highly efficient FEM codes for analysis of both the primal problem and the subsequent sensitivity analysis. This paper is organised as follows. In Section 2 the general sensitivity problem is described. The specific crack propagation problem is formulated in Section 3, focussing on the kinematic quantities and crack description through the XFEM. The crack propagation criterion is developed in Section 4. In Section 5 some numerical implementation details are given. Finally in Section 6 some numerical examples are presented and in Section 7 the main conclusions are outlined.

Sensitivity analysis is a powerful tool to analyse the influence of the parameters of a certain model in the general response of the system. In particular in fracture problems, the comprehension of the role played by material constants, crack length or loading scenarios in a crack propagation problem may bring considerable advantages in the design of structures and parts. In this work, a direct differentiation technique is applied to a crack problem, described through the XFEM, in order to derive the quantitative sensitivities of the model parameters. Moreover, it was possible to develop a sensitivity-based crack propagation criterion. In elastic materials, the condition for crack propagation may be given by the Griffith crack grow energy. Nevertheless, in its traditional version, Griffith's criteria only gives information on the crack incremental length and not on its direction. In this work, the Griffith's condition was re-interpreted and its information combined with sensitivity analysis. Therefore, by calculating two orthogonal sensitivities of the potential energy with respect to the crack length change is possible to determine the crack propagation direction and final length. The developed methodology was successfully applied to hyperelastic materials (and thus involving finite strains) with cracks described through the XFEM. Special attention was giving to sensitivity calculations of enrichment functions involving level sets and care was taken to handle incompressibility and near incompressibility conditions. Results suggest that this approach may be extended to other materials, which include plasticity and/or anisotropy, for instance. Additionally, this approach constitutes a solid base for future analysis of crack branching and crack joining, as the most favourable scenario may be determined using sensitivity calculations.