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Numerical solution of the integral equations of elasto-plasticity for a homogeneous medium with several heterogeneous inclusions S. Kanaun , R. Martinez
Abstract
The work is devoted to the calculation of stress and strain fields in a homogeneous elasto-plastic medium with a finite number of heterogeneous inclusions. The medium is subjected to an arbitrary external stress field. Elasto-plastic behavior of the medium is described by the equations of the incremental theory of plasticity with isotropic hardening. For the numerical solution, the external stress field applied to the medium is divided on a consequence of small steps, and the problem is linearized at every step. The linearized problem is reduced to the solution of the volume integral equations for the stress field increment inside the inclusions and in the regions involved in the plastic deformations. Then, these equations are discretized using Gaussian approximating functions. For such functions, the elements of the matrix of the discretized problems are calculated in explicit analytical forms. If the approximating nodes compose a regular grid, the matrix of the discretized problem obtains the Toeplitz properties, and the product of such a matrix and a vector can be calculated by the Fast Fourier Transform technique. The latter accelerates essentially the process of iterative solution of the discretized problems. The proposed method is mesh free, and the coordinates of the approximating nodes and elasto-plastic properties of the material at the nodes are the only information required for carrying out the method. Distributions of stresses and plastic strains in the media with isolated inclusions are compared with the finite element calculations. The influence of the number of approximating nodes and the rate of hardening of the material on the convergence of the numerical solutions is analyzed.
... Thus, the problem is reduced to the calculation of the stress field increment Δσ(x) inside the region W 0 that contains a finite number of inclusions of arbitrary shapes and properties. The numerical method of the solution of this problem was developed in Kanaun [18] and Kanaun and Martinez [23] and presented in "Effective Properties of Composite Materials, Reinforced Structures and Smart Composites. Asymptotic Homogenization Approach". ...
... Note that for a regular cubic node grid, the coefficients Δσ where the vector X is expressed through the components of the tensor Δσ (Δε e ) at the nodes, and F is the vector which components expressed through the increment of the external field Δσ 0 (Δε 0 ), the matrix A is reconstructed from Eqs. (5.48) or (5.49). The detailed forms of X, F, and A are presented in Kanaun and Martinez [23]. The matrix A of the discretized problem is non-sparce and have large dimensions if high accuracy of the numerical solution is required. ...
... The number of iterations and, as a result, the time of calculations increases with the decreasing of the value of the hardening exponent n in the plastic deformation law (5.67). The material of this Chapter is based on the work of Kanaun and Martinez [23], Kanaun [19]. ...
The work is devoted to the effective field method and its application in the theory of heterogenous materials. For many years, various versions of the method have been used for the calculation of effective physical and mechanical properties of composite materials (the homogenization problem). In the historical survey, the most important steps of the development of the method are indicated starting from nineteenth century. The main attention is focused on the combination of the effective field and numerical methods that yields efficient numerical algorithms for the calculation of effective properties and detailed fields in periodic and random composite materials. Examples of the application of the method to prediction of conductive, elastic, and elasto-plastic properties of composites are presented.
... The latter accelerates substantially the iterative process of the solution of the discretized problem. For the calculation of the stresses and elasto-plastic deformations in a medium with a finite number of inclusions, a similar numerical method was developed by Kanaun (2011) and Kanaun and Martinez (2012). Application of the method to the homogenization problem for elastic composites was done by Kanaun and Pervago (2011). ...
... Here Rot ijkl = imk jnl @ m @ n is the operator of incompatibility ( ijk is the Levi-Civita tensor), and e e ij ðxÞ þ e p ij ðxÞ ¼ e ij ðxÞ is the total strain tensor. As shown by Kanaun and Levin (2008) and Kanaun and Martinez (2012), the stress tensor r ij (x) in the heterogeneous elasto-plastic medium satisfies the following integral equation: ...
... Thus, the problem is reduced to the construction of the stress field increment Dr(x) inside the region W 0 that contains a finite number of inclusions of arbitrary shapes and properties. An efficient numerical method of the solution of this problem was developed by Kanaun (2011) and Kanaun and Martinez (2012). ...
a b s t r a c t The homogenization problem for elasto-plastic media with arrays of isolated inclusions (matrix composite) is considered. A combination of self-consistent and numerical methods is used for calculation of the overall response of such composites under quasi-static load-ing. Elasto-plastic properties of the medium and the inclusions are described by the equa-tions of the incremental theory of plasticity with isotropic hardening. For the construction of the average stress–strain relations of the composites, the process of external loading is divided into a sequence of small steps, and the problem is linearized at every step. The self-consistent effective field method allows reducing the homogenization problem at every step to the calculation of stresses and elasto-plastic deformations in a composite cell that contains a finite number of inclusions. The linearized problems are formulated in terms of volume integral equations for the stress or elastic strain field increments in the cell. For the numerical solution, these equations are discretized by Gaussian approximating functions concentrated in a set of nodes that cover the composite cell. For such functions, elements of the matrix of the discretized problems are calculated in explicit analytical forms. If the approximating nodes form a regular grid, the matrix of the discretized problem has Toeplitz's properties, and the matrix–vector products of such matrices can be calculated by the fast Fourier transform technique. The latter accelerates substantially the process of iterative solution of the discretized problems. The dependencies of the overall stress–strain curves on the number of inclusions inside the cell are studied in the 2D and 3D cases. The inclusions that are stiffer or softer then the matrix are considered. The predictions of the method are compared with the finite element calculations available in the literature.
... It is widely accepted that inclusions with low melting point have good deformability. Most of the research on plasticization of inclusions is based on the concept of lowering their melting point [7][8][9][10][11][12][13][14][15][16][17][18][19]. Guo et al. found that non-deformable Al2O3 inclusions in aluminum killed steel can be modified into Al2O3-CaO-MgO-CaS inclusions with a lower melting point and better deformability by calcium treatment [7]. ...
... Most of the research on plasticization of inclusions is based on the concept of lowering their melting point [7][8][9][10][11][12][13][14][15][16][17][18][19]. Guo et al. found that non-deformable Al2O3 inclusions in aluminum killed steel can be modified into Al2O3-CaO-MgO-CaS inclusions with a lower melting point and better deformability by calcium treatment [7]. Qin et al. noted that the increase of MgO content enlarges the liquid area of the CaO-SiO2-Al2O3-MgO system based on the phase diagram obtained from FactSage calculations. ...
In this study, a new method is proposed for the plasticization of inclusions by taking advantage of the behavior of alkali metals in lowering the inclusion melting point. A series of experiments with NaF additions to molten steel were carried out using a carbon tube furnace followed by simulated rolling tests using the solidified ingots and characterization of inclusions with the help of automated scanning electron microscopy. Compositional changes of the steel and evolution of gaseous species were evaluated using thermodynamic software FactSage 7.2 (ThermFact Inc., Montreal, QC, Canada). Based on this approach, NaF/steel/inclusion interactions and the effects of NaF addition on the melting point, size and deformability of inclusions were clarified. The modification of MnO-SiO2 inclusions by NaF also promoted the removal of inclusions and improved the cleanliness of steel. The results show that with the addition of NaF, the melting point of inclusions is greatly reduced, the deformability is improved, and the removal of inclusions is enhanced, all of which indicates a good prospect for industrial application.
... But the two models are mostly applied to the case of homogeneous materials in the open literature. For multiphase material problems, quite limited work can be found (Kanaun and Martinez, 2012;Yi et al., 2014). ...
Elastic-plastic stress analysis on a radial crack interacting with a coated-circular inclusion in a matrix has been carried out with the aid of a generalized Irwin plastic zone correction. The crack line is assumed to be at the angle of 90° − θ from a remote tensile loading. In the mathematical formulation, the distributed dislocation method is used to simulate the crack. By solving a set of singular integral equations, three quantities, the effective stress intensity factor, the plastic zone size and the crack tip opening displacement (CTOD), are evaluated with the generalized Irwin model proposed. Numerical examples are given to show the influence of the key parameters such as the crack orientation angle θ, the normalized crack distance, the normalized coating phase thickness and the shear modulus ratio (, coating phase/matrix) on the fracture behavior. The results indicate that the influence of angle θ is the greatest, while the effect of shear modulus ratio is relatively small. A validation checking is performed by the finite element method (FEM) for one case. The result obtained from the FEM simulation matches well with that from the current method.
The other important type of problem in science and engineering is integral equations. Thus, developing precise numerical algorithms for approximating the solution to these problems is one of the main questions of scientific computing. In this chapter, the least squares support vector algorithm is utilized for developing a numerical algorithm for solving various types of integral equations. The robustness and also the convergence of the proposed method are discussed in this chapter by providing several numerical examples.KeywordsIntegral equationsGalerkin LS-SVRCollocation LS-SVRNumerical simulation
The problem of the overall, or effective, properties of heterogeneous materials is a
classical one that started to attract attention in the first half of the nineteenth
century, in works of Poisson and Faraday. In the last half-a-century, the field has
experienced rapid growth, due to several factors. One of them relates to growing
needs of materials science, due to appearance of new materials such as composites,
as well as the necessity to model naturally occurring heterogeneous materials such
as rocks. Another one was the development of continuum mechanics foundation of
the field that started with works of Hill and Eshelby. Computational micromechanics,
fueled by increasing computer powers, emerged as a separate field, and
has experienced rapid advance in the last two decades.
The present book contains five state-of-the-art reviews on the analytical and
computational aspects of the problem.
a b s t r a c t The work is devoted to the calculation of static elastic fields in 3D-composite materials consisting of a homogeneous host medium (matrix) and an array of isolated heterogeneous inclusions. A self-consistent effective field method allows reducing this problem to the problem for a typical cell of the composite that contains a finite number of the inclusions. The volume integral equations for strain and stress fields in a heterogeneous medium are used. Discretization of these equations is performed by the radial Gaussian functions cen-tered at a system of approximating nodes. Such functions allow calculating the elements of the matrix of the discretized problem in explicit analytical form. For a regular grid of approximating nodes, the matrix of the discretized problem has the Toeplitz properties, and matrix–vector products with such matrices may be calculated by the fast fourier trans-form technique. The latter accelerates significantly the iterative procedure. First, the method is applied to the calculation of elastic fields in a homogeneous medium with a spherical heterogeneous inclusion and then, to composites with periodic and random sets of spherical inclusions. Simple cubic and FCC lattices of the inclusions which material is stiffer or softer than the material of the matrix are considered. The calculations are per-formed for cells that contain various numbers of the inclusions, and the predicted effective constants of the composites are compared with the numerical solutions of other authors. Finally, a composite material with a random set of spherical inclusions is considered. It is shown that the consideration of a composite cell that contains a dozen of randomly dis-tributed inclusions allows predicting the composite effective elastic constants with suffi-cient accuracy.
The work is devoted to calculation of static elastic and thermo-elastic fields in a homogeneous medium with a finite number of isolated heterogeneous inclusions. Firstly, the problem is reduced to the solution of in-tegral equations for strain and stress fields in the medium with inclusions. Then, Gaussian approximating functions are used for discretization of these equations. For such functions, the elements of the matrix of the discretized problem are calculated in explicit analytical forms. The method is mesh free, and only the coor-dinates of the approximating nodes are the geometrical information required in the method. If such nodes compose a regular grid, the matrix of the discretized problem obtains the Toeplitz properties. By the calcula-tion of matrix-vector products with such matrices, the Fast Fourier Transform technique may be used. The latter accelerates essentially the process of the iterative solution of the disretized problem. The results of cal-culations of elastic fields in 3D-medium with an isolated spherical heterogeneous inclusion are compared with exact solutions. Examples of the calculation of elastic and thermo-elastic fields in the medium with several inclusions are presented.
Nonlinear stress analysis (a branch of solid mechanics) is an essential feature in the design of such diverse structures as aircraft, bridges, machines, and dams. Computational techniques have become vital tools in dealing with the complex, time-consuming problems associated with nonlinear stress analysis. Although finite element techniques are widely used, boundary element methods (BEM) offer a powerful alternative, especially in tackling problems of three-dimensional plasticity. This book describes the application of BEM in solid mechanics, beginning with basic theory and then explaining the numerical implementation of BEM in nonlinear stress analysis. The book includes a state-of-the-art CD-ROM containing BEM source code for use by the reader. This book will be especially useful to stress analysts in industry, research workers in the field of computational plasticity, and postgraduate students taking courses in engineering mechanics.
The unknown strains in the inclusions are expressed in terms of a series of radial basis functions (RBF) and polynomials in global coordinates. Based on the radial integration method proposed by Gao [J. Appl. Mech., Trans. ASME 69 (2002) 154, Engng Anal. Bound. Elem. 26 (2002) 905], the volume integrals for the evaluation of strains can be transformed into contour integrals on the inclusion boundaries. As a result of this transformation, there is no need to discretize the inclusions into finite elements. For the determination of the strains, collocation points are distributed at the interior of the inclusions to form a system of linear equations. Numerical results are compared with available analytical solutions and those based on a finite element discretization of the volume integrals.
A volume integral equation technique developed by Lee and Mal (J. Appl. Mech. Trans. ASME 64 (1997) 23) has been extended to investigate three-dimensional stress problems with multiple inclusions of various shapes. Based on the volume integral formulation, displacement continuity and traction equilibrium along the interfaces between the matrix and the inclusions are automatically satisfied. While the embedding matrix is represented by an integral formulation, only the inclusion parts are discretized into finite elements (isoparametric quadratic 10-node tetrahedral or 20-node hexahedral elements are used in the present study). A number of numerical examples are given to show the accuracy and effectiveness of the proposed method.
This work is devoted to the calculation of static elastic fields in a homogeneous medium with a finite number of isolated heterogeneous inclusions. First, the problem is reduced to the solution of integral equations for strain fields inside the inclusions. Then, Gaussian approximating functions are used for discretization of these equations. For such functions, the elements of the matrix of the discretized problem are calculated in explicit analytical forms. The method is mesh-free, and the coordinates of the approximating nodes is the only geometrical information required in the method. If such nodes compose a regular lattice, the matrix of the discretized problem will have Toeplitz structure. By the calculation of matrix-vector products with such matrices, the fast Fourier transform technique may be used. The latter essentially accelerates the process of the iterative solution of the disretized problem. The results of calculations of elastic fields in a 2-D medium with an isolated heterogeneous inclusion and with several inclusions are presented.
A critical analysis of foundations and domain of applicability of the theory of elastic media with microstructure and nonlocal elasticity is given.
The product of a unique collaboration among four leading scientists in academic research and industry, Numerical Recipes is a complete text and reference book on scientific computing. In a self-contained manner it proceeds from mathematical and theoretical considerations to actual practical computer routines. With over 100 new routines bringing the total to well over 300, plus upgraded versions of the original routines, the new edition remains the most practical, comprehensive handbook of scientific computing available today.
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and Monographs, American Mathematical Society, 2007. 141.
Methods of Tensor Analysis in the Theory of Dislocations
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Department of Commerce, Clearing House for Fed. Sci. Techn. Information,
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Distributions of the plastic strain intensity J p (x 1 ,0) along the x 1 -axis in the elasto-plastic 3D-medium with a spherical cavity for various values of the grid step h (left) and the tangent modulus E t in the plastic region (right). The medium is
- Fig
Fig. 16. Distributions of the plastic strain intensity J p (x 1,0) along the x 1 -axis in the elasto-plastic 3D-medium with a spherical cavity for various values of the grid step h (left)
and the tangent modulus E t in the plastic region (right). The medium is subjected to a uniaxial stress along the x 1 -axis.