تجزیه و تحلیل حساسیت شکل در مکانیک شکست الاستیک خطی
|کد مقاله||سال انتشار||مقاله انگلیسی||ترجمه فارسی||تعداد کلمات|
|25534||2000||16 صفحه PDF||سفارش دهید||7070 کلمه|
Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : Computer Methods in Applied Mechanics and Engineering, Volume 188, Issue 4, August 2000, Pages 697–712
Shape sensitivity analysis of an elastic solid in equilibrium with a known load system applied over its boundary is presented in this work. The domain and boundary integral expressions of the first- and second-order shape derivatives of the total potential energy are established, by using an arbitrary change of the domain characterized by a velocity field defined over the initial body configuration. In these expressions we recognize free divergence tensors that are denoted in this paper as energy shape change tensors. Next, shape sensitivity analysis is applied to cracked bodies. For that purpose, a suitable velocity distribution field is adopted to simulate the crack advance of a unit length in a two-dimensional body. Finally, the corresponding domain and the equivalent path-independent integral expressions of the first- and second-order potential energy release rate of fracture mechanics are also derived.
The interest in structural optimization and sensitivity analysis has increased greatly during the last decades due to the advent of reliable general numerical methods and computer power. More recently, researchers have focused their attention on one important and perhaps more complex class of engineering design problem, in which the shape, or more specifically, the domain over which the problem is defined is to be determined ,  and . For such problem, known in the literature as shape sensitivity analysis and shape optimization, the domain becomes the design variable. Contributions to this field have been made using two fundamentally different approaches. The first approach uses a discretized numerical model based either on the finite element method, the boundary element method, or other available numerical method, to carry out a shape design sensitivity analysis controlling either finite element or boundary element node motion and reducing the shape derivatives into differentiations of algebraic equations  and . The second approach, which is adopted in the present work, resorts to a variational formulation of continuum mechanics  and . In this case, well-known as continuum formulation, the shape design sensitivity analysis is carried out by controlling a velocity field which is introduced to simulate the shape change of the initial domain  and . The variational equilibrium equation associated with the kinematical model is taken as the state equation and the total potential energy stored in an elastic body is chosen as the cost function, using the optimization nomenclature. The immediate difficulty in the application of this latter procedure is that the domain of integration is variable and both the state equation and cost function do not exhibit an explicit dependence on the project variable. To obviate this mathematical difficulty, the domain variation is parameterized and an analogy is established between the shape change and the motion of the deforming solid using the concept of material derivative of continuum mechanics to account for these changes in shape . Although the second approach presented and adopted in this work poses major complexity regarding both physical and mathematical aspects, it nevertheless possesses the advantage of yielding the exact general expressions of the derivatives, which are independent of the approximate method used for the analysis. Through this approach, integral expressions are derived for first- and second-order shape sensitivity analysis in terms of changes in the shape of the domain , , , ,  and . Furthermore, the condition of null divergence of the energy shape change tensors is employed to transform the domain integrals to boundary integrals, so that shape design sensitivity expressions can be obtained in terms of a shape perturbation of the boundary. Once the derivatives of the total potential energy stored in an elastic body have been obtained, this analysis can be easily extended to the study of fracture mechanics  and . For simplicity, we consider a two-dimensional body with a straight crack, under plane-strain state condition, subjected to a loading system on its boundary and simulate the initiation of crack advance as a suitable shape change of the body. By selecting a distribution pattern of adequate velocities in the initial domain of the cracked body, we can specialize the expressions of the first and second derivatives of the potential energy to obtain the total potential energy release rates and the corresponding first- and second-order path-independent integrals.
نتیجه گیری انگلیسی
Shape sensitivity analysis, which is strongly related to shape optimization, is a useful tool to establish domain and boundary expressions for first and higher derivatives of the potential energy stored in an elastic body with respect to the change of shape. These expressions allow us to implement in Finite Element or Boundary Element codes, numerical procedures to compute shape derivatives. Then, by means of a post-processing technique and numerical integration, the expressions of Π derivatives can be automatically evaluated. To perform the first derivative of the potential energy we must carry out a first analysis in order to find the fields and T in equilibrium with the external loads. In the case of second-order derivative, the expression of indicates that a second analysis to evaluate is also required. If we compare the second analysis with the first one , we verify that both bilinear forms of the equilibrium equations, that represent the internal work, are the same, while the linear form associated with the external work, differ. That is, if we resort to an approximation method to transform the variational equilibrium equations in algebraic equations, the resulting matrices are the same. This feature considerably simplifies the numerical evaluation of and consequently the second-order shape derivative estimation. In the analysis of cracked bodies, as shown in this work, we easily arrive at the expressions of first- and second-order potential energy release rate of fracture mechanics if the crack advance is simulated by change of shape. In the particular case, when we assume no body forces and null traction over the faces of the crack, sensitivity analysis of bidimensional problems easily leads to the equivalence between first- and second-order potential energy release rates and the corresponding path independent integrals. This procedure could be extended to analyze three-dimensional cracked bodies. Similar approaches could also be used to obtain, in a systematic way, higher-order energy release rates and their equivalent path-independent integrals. Furthermore, crack propagation is essentially a non-linear phenomenon and presents a typical instability behavior, even occurring in the case of the so-called linear fracture mechanics. Thus, the first- and higher-order potential energy release rates play an outstanding role in the analysis of crack initiation and crack advance stability  and . Finally we would like to stress that shape sensitivity analysis associated with numerical methods, such as the Finite Element Method, are able to yield reliable numerical results in fracture mechanics ,  and .