انطباق در مکانیزم های شکستن الاستیک خطی بر اساس تجزیه و تحلیل حساسیت شکلی

If crack growth of an elastic body is viewed as a shape change we can use the well known concept of shape sensitivity analysis to compute the energy release rate. To do this, we adopt as cost function the total potential energy and as state equation the equilibrium equation. The shape derivative of the total potential energy stored in the cracked body View the MathML sourceΠ˙ depends on the displacement field u and on the shape change velocity field V which characterize the crack growth. Following this procedure the present paper deals with the derivation of a novel a posteriori error estimator which is an upper bound of the global error View the MathML source|Π˙-Π˙h|. This error estimator has been specifically designed to evaluate the energy release rate in mesh refinement or re-meshing procedures so as to obtain improved meshes for which the optimal rate of convergence is recovered even in a case of singularities. This novel estimator is capable to capture all source of errors for the energy release rate View the MathML sourceΠ˙: the ones from stress concentration and the errors from the sensitivity of the solution to shape changes due to crack growth. Finally, well known three-dimensional examples of un-cracked and cracked body are considered in order to illustrate the potentiality of the proposed methodology.

An upstanding concept in linear elastic fracture mechanics is the total potential energy release rate. This concept was introduced by Griffiths [6] applying the energy conservation principle in the analysis of cracked bodies. As known, the energy release rate means the rate of change with respect to the crack advance of the energy available for fracture. In other words, the derivative of the total potential energy with respect to the crack growth. Due to the difficulty to obtain expressions for the potential energy as an explicit function of the crack length, which enable us to obtain derivatives in a direct form, several numerical and experimental procedures, have been developed in fracture mechanics. However, the Shape Sensitivity Analysis (proposed originally by Cèa [4], Zolésio [20], and widely discussed by Haug et al. [8]) has been successfully applied as a systematic methodology to obtain the expression of the potential energy release rate in cracked bodies. As stated in [10], the crack growth can be interpreted as a shape change of the body. Then, using well known results of Shape Sensitivity Analysis, the general expression for the energy release rate as an integral over the initial configuration of a cracked body was obtained. Moreover, this expression is a function of a velocity field describing the change of the initial shape and which simulates the crack advance. Further, using the Finite Element Method and post processing techniques, several applications in three-dimensional linear elastic cracked bodies were presented in [10] showing the potentiality of this approach. Following this technique, one question arise: is it possible to find an a posteriori error estimator which can be applied in an adaptive finite element energy release rate analysis? Before answering this question, some considerations are in order. The adaptive method for linear elasticity equations, which is based on the standard a posteriori error estimator for the total potential energy, ignore the particular velocity field used to simulate the crack advance. However, for each velocity field, an adequate adaptive procedure might be able to find the best sequence of meshes to compute sharply the energy release rate. The present paper deals with the derivation of an a posteriori error estimator for shape sensitivity analysis, which has been specifically designed to evaluate the energy release rate in an adaptive finite element scheme. This estimator can be used to adapt the mesh automatically so as to increase the accuracy of the energy release rate finite element computations. To this end, and using the well known terminology of shape optimization, we adopt as a cost function the total potential energy of an isotropic linear elastic cracked body under infinitesimal deformation and, as a restriction, the variational formulation (weak form) of the static equilibrium. We also use the so-called (continuous) direct method for the evaluation of the sensitivity of the total potential energy with respect to the shape change of the cracked body under consideration. The outline of the paper is as follows. In Section 2 we recall the formulation of the elasticity problem and the total potential energy properly posed to apply the shape sensitivity analysis. In Section 3, we shall briefly describe the shape sensitivity analysis (and its finite element approximations) for both the displacement and the total potential energy by mean of a velocity field. We shall use this velocity field to simulate the changes in the shape due to the crack growth. In Section 4 we will introduce the a posteriori error estimator and will prove that this estimator provide upper estimates for the energy release rate. In order to show the potentiality of the present approach, several numerical experiments are presented in Section 5 where the characterization of the velocity field and the adaptive scheme are also discussed (Sections 5.1 and 5.2, respectively). Finally, Section 6 will be devoted to conclusions.

high values of ∇V together with error in the evaluation of View the MathML sourceu˙) are significant. Full-size image (18 K) Fig. 10. Problem 3. Final mesh using the θ-Adaptive Algorithm (8th step, 7358 nodes). Figure options 6. Conclusions Shape sensitivity analysis has been strongly related and traditionally applied to shape optimization problems. However, we have shown in this paper how this approach allows us to obtain an exact expression for the energy release rate in a three-dimensional cracked body. Therefore, shape sensitivity analysis could be used in fracture mechanics as an alternative method to establish the general expression for View the MathML sourceG=-Π˙(u,V). Also, shape sensitivity analysis, together with numerical methods, such as the finite element method, can provide reliable numerical results of this important parameter in real fracture mechanical problems. Then, the accuracy of the energy release rate finite element computation takes fundamental importance. The question now is how to perform the mesh refinement in order to obtain such accuracy. To answer this, in Section 4 we present the derivation of a novel error estimator θ for the energy release rate View the MathML sourceΠ˙. At element level, this estimator depends on two terms: the term η T, associated to errors in the evaluation of the displacement field u, and the term ξ T related to errors in the evaluation of the sensitivity to shape change due to crack growth of the displacement field, i.e. View the MathML sourceu˙. Therefore, this error estimator gives an upper bound for View the MathML source|Π˙-Π˙h| and can be used in an adaptive finite element procedures recovering the optimal rate of convergence even in case of strong singularities. Moreover and for different velocity fields V characterizing the same crack growth, similar values for the associated approximate energy release rate View the MathML sourceΠ˙h=Πh(uh,V) are obtained when this error estimator is used. Finally, the above characteristics and the numerical results of Section 5 clearly demonstrate the additional advantage one has using this estimator in place of the traditional energy-type estimators.