To confirm the reliability of the results obtained on the basis of the semi-analytical finite element method using the approaches suggested in [1-2, 4-6], a wide range of test problems is considered. Within the framework of the elastic approach, the spatial problem of an unevenly loaded shell, the middle surface of which has the shape of an elliptical paraboloid, is considered. The validation of the reliability of solutions obtained on the basis of the semi-analytical finite element method for physically nonlinear problems is carried out using the example of elastic-plastic calculation of an unevenly heated cube, the physical and mechanical characteristics of the material of which depend on temperature. To substantiate the reliability of the results obtained when solving problems of large plastic deformations, the elastic-plastic deformation of a parallelepiped between plane-parallel plates in the absence of contact friction is considered. The efficiency of application of the semi-analytical finite element method to the calculation of curvilinear inhomogeneous prismatic objects is shown. The solution of the control problems of the theory of elasticity, thermoelasticity and thermoplasticity, as well as the problems of shape change, makes it possible to draw a conclusion about the reliability of the results of the study of the selected class of objects on the basis of the developed methodology and the applied software package that implements it.
We present theoretical approaches and a procedure for the FEM computation of parameters of nonlinear fracture mechanics in prismatic bodies with a crack. The efficiency of the proposed approaches and the veracity of the results obtained have been analyzed.Keywords: fracture mechanics, invariant J-integral, path of integration, finite element method, linear and nonlinear problems.The majority of problems of fracture mechanics can only be tackled using numerical methods, in particular a finite element method (FEM). The most difficult tasks in these cases are to state the problem and elaborate a procedure for solving three-dimensional problems of fracture mechanics. The efficiency of an FEM procedure for solving such problems can be improved through special FEM modifications, among which there is a semi-analytic finite element method (SA-FEM) [1,2]. The method has proved high efficient in solving a wide range of problems of determination of stress intensity factor (SIF) on the basis of a direct method under mechanical [1, 2] and thermomechanical loading conditions [3] as well as the problems of modeling a crack growth in three-dimensional bodies [4]. However, the field of applicability of direct methods the SIF values determined by such methods is limited to the problems of linear fracture mechanics to treat elastic deformation of solids. Meanwhile, considering an unambiguous relation between the J-integral and SIF in elastic deformation, the possibility of performing a parallel computation of parameters of fracture mechanics by both the direct methods and the strain-energy methods permits validation of the results obtained. In the case of significant plastic strains, the bearing capacity of cracked bodies should be assessed by the strain-energy approaches to the determination of fracture-mechanics parameters, in particular, the Cherepanov-Rice J-integral [5, 6] -the most universal parameter that can be used in nonlinear fracture problems.The objective of the present work is to provide a theory for and practically implement the procedure of computation of the contour J-integral for three-dimensional prismatic bodies using SA-FEM and to analyze the veracity of the results obtained, including satisfaction of the fundamental properties of invariance of the J-integral, as well as to compare the efficiency of the proposed procedure versus other available approaches. We also intend to implement a new J-integral calculation procedure which would ensure the integral invariance in the discrete FEM models in both linear and nonlinear three-dimensional problems of fracture mechanics.According to the basic definition of the J-integral [5,6], to calculate this integral at some point of the crack front (point C in Fig. 1) in the vicinity of this point we pick out a surface F F F F k = + + 1 2 of an arbitrary configuration, which covers the crack front and has a characteristic dimension D along the front.
The initial relations of thermo elastic-plastic deformation of prismatic bodies are given in the paper. The basic concepts, indifference of deformation tensors, with the condition of energy conjunction in description of the shaping process are laid out on the basis of classical works.
A new procedure for computing the J-integral for FEM discrete models, which is based on the use of magnitudes of nodal reactions and displacements, has been implemented. The paper presents theoretical substantiation and practical evidence that confirm the invariance of the J-integral values found by this procedure for the case of cracks of Mode I and mixed fracture mode. A threedimensional elastic-plastic deformation problem for a compact specimen has been solved.Keywords: fracture mechanics, invariant J-integral, path of integration, finite element method, linear and nonlinear problems.In Part 1 [1], we undertook a study of reliability of computation of the J-integral for discrete models of the finite element method (FEM) on the basis of the conventional approach to magnitudes of stresses and displacement gradients. The results obtained demonstrate that the procedure provides a higher efficiency of the J-integral computation, in particular, if compared to that of the equivalent volume integration method [2]. However, during the computation of the J-integral for an edge-notched prismatic body in both elastic and elastic-plastic deformation cases, it was found out that the results depend significantly on the dimensions of the path of integration. With the finite element models being the same, the J-integral computation error in the case of elastic deformation turned out to be twice that for the elastic-plastic deformation, and a considerable densening of the finite element mesh is required in both cases in order to achieve reliable results. Also, the implementation of the conventional approach for the mixed fracture mode, as proposed by other researches, involves additional computational costs. Therefore, an important task is to elaborate a new efficient procedure for computing the J-integral for FEM discrete models, which would ensure the J-integral invariance.Let us consider a closed path of an arbitrary shape in the general case, which is constructed by a discrete model of the semi-analytic finite element method (SA-FEM) for a body with a longitudinal crack. The path represents a cross section of the volume chosen in the vicinity of a crack-front point for the J-integral computation [1], passes through the middle of finite elements (FE) along the x 1 axis direction and at the FE boundaries parallel to axis x 2 (Fig. 1). This closed path can be arbitrarily located in the discrete model; in an extreme case, one of its sides parallel to the axis x 2 can coincide with one of the crack edges (faces).The J-integral along this path can be written as J Wn n t dS t i ik k l l S = -¢ ¢ ¢ ¢ ò ( ) . s z (1) 122 0039-2316/11/4302-0122
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.