Formulation of the plastic source method for plane inelastic problems Part 2: Numerical implementation for elastoplastic problems

Summary In Part 1 of this paper, Green's function representation for the residual stress, arising from any plane inelastic strain distribution under no applied load, has been given. In Part 2 (i.e., the current paper) we focus our attention to the plastic strain arising in plane elastoplastic problems subject to applied loads. A numerical procedure, based on the Green's function approach, to determine the unknown plastic strain distribution under the applied load is established. The numerical result for an elastoplastic steady crack growth problem demonstrates, overwhelmingly, the advantage of the proposed method over the finite element method approach for the same problem.With 5 FiguresThis paper is dedicated to the memory of Aris Phillips, founding Co-Editor of Acta Mechanica, and was presented at the Aris Phillips Memorial-Symposium, Gainesville, Fla., 1987.     

A complex variable Green's function representation of plane inelastic deformation in isotropic solids

Summary A Green's function representation of the plane inelastic deformation in isotropic solids is given using a complex variable method of Muskhelishvili. Based on the fact that the inelastic deformation in a plane infinitesimal region (which we call a plastic source) can be represented by a double couple, its Green's functions are derived in terms of the complex potential functions; these Green's functions, then, are used as the kernel functions in an area integral representation of the complex potential functions for the inelastic deformation of a finite extent. Emphasis is placed on deriving the area integral representation of the two basic complex potential functions (i.e., and in Muskhelishvili's notation); once they are obtained, any physical quantities such as the displacement, the stress, and the traction can be calculated by simply following the formulae of Muskhelishvili.With 4 Figures     

An analysis procedure for plane strain contained plastic deformation problems

An analysis procedure for the plane strain contained plastic deformation problem is proposed. It is shown that, when piecewise-linear elastic–plastic laws are adopted, the plane strain problem can be formulated in terms of exactly the same variables appearing in a plane stress problem, even if the transverse stress and plastic strain components do not vanish in general. The two problems are governed by the same equilibrium and compatibility equations and the only difference is in the elastic–plastic material constitutive laws, in complete analogy with the linear elastic case. The problem is discretized by using a compatible finite element model on which basic elastic-plastic constitutive laws for an element in plane strain conditions can be obtained. When the assemblage is performed, the resulting set of governing relations presents a mathematical structure which enable, for solution, techniques efficiently used for truss and frame problems. Some solutions for axisymmetric tubes subjected to pressure and temperature gradients are illustrated.     

Numerical Green's functions for some electroelastic crack problems

A plane electroelastic problem involving planar cracks in a piezoelectric body is considered. The deformation of the body is assumed to be independent of time and one of the Cartesian coordinates. The cracks are traction free and are electrically either permeable or impermeable. Numerical Green's functions which satisfy the boundary conditions on the cracks are derived using the hypersingular integral approach and applied to obtain a boundary integral solution for the electroelastic crack problem considered here. As the conditions on the cracks are built into the Green's functions, the boundary integral solution does not contain integrals over the cracks. It is used to derive a boundary element procedure for computing the crack tip stress and electrical displacement intensity factors.     

Green's functions in the solutions of heat-conduction problems for a hollow cylinder

We present solutions of heat-conduction problems for a hollow cylinder for mixed boundary conditions of the second and third kind, the solutions containing rapidly converging series. For the fundamental types of boundary conditions we obtain expressions for the Green's functions which enable us to improve the convergence of the series.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.21, No. 6, pp. 1096–1100, December, 1971.     

Relaxation element method in calculations of stress state of elastic plane with the plastic deformation band

On the basis of relaxation element method an analytical representation of the band of localized plastic deformation, as the defect with its own internal stress field in the plane under tensile loading, is given. The influence of orientation of the band with respect to tensile axis on the stress concentration in the plane is analyzed.     

Green's functions for dislocations in bonded strips and related crack problems

Green's functions are derived for the plane elastostatics problem of a dislocation in a bimaterial strip. Using these fundamental solutions as kernels, various problems involving cracks in a bimaterial strip are analyzed using singular integral equations. For each problem considered, stress intensity factors are calculated for several combinations of the parameters which describe loading, geometry and material mismatch.     

Elastoplastic BIE analysis of cracked plates and related problems. Part 1: Formulation

The boundary-integral equation formulation for two-dimensional, planar fracture mechanics based on the use of a special Green's function forms the basis of this analytical paper. The Green's function method is extended to problems of anelastic strain distributions (e.g. elastoplasticity, thermal gradients, residual strains) through a volume (area) integral. The role of the elastic Green's function for the crack problem on the distribution of elastoplastic strains is reviewed. Further, new analytical results for elastic stress intensity factor models for the residual strain and thermal gradient problems are presented. Part 2 of this paper outlines the numerical solution strategy and results for several test problems.     

Green's functions for boundary-value problems for the heat equation over regions with uniformly moving boundaries

Green's functions are obtained for a semi-infinite straight line with a uniformly moving boundary (10), (11), (12) and for a segment with boundaries moving uniformly and in parallel (16), (17), (18). For the solution a moving coordinate system is introduced and the method of Laplace transforms is applied.     

Prediction of the fracture of metals in the processes of cold plastic deformation. Part 1. Approximate model of plastic deformation and fracture of metals

We propose an approximate semiphenomenological model of the joint process of cold plastic deformation and fracture of metals. Within the framework of this model, for 10kp, 20kp, 20G2R, and 38KhGNM steels, we show that moving dislocations overcome barriers through a force process. The formation of nascent microcracks is also realized through a force process, i.e., local stresses in the “head” of an arrested dislocation pileup attain the levels of theoretical strength. We also suggest a general algorithm of the application of the proposed model to the prediction of fracture of metals in technological processes of plastic metal working. Ufa State Aviation Technical University, Ufa, Russia. Translated from Problemy Prochnosti, No. 1, pp. 76–85, January–February, 1999.     

Large inelastic deformation of glassy polymers. Part II: numerical simulation of hydrostatic extrusion

Oriented polymers may be successfully produced by the warm mechanical process of hydrostatic extrusion. This forming process is analyzed here using the constitutive model for the large inelastic deformation of glassy polymers developed in the companion paper. The model is numerically integrated and incorporated into a finite element code. The numerical results for the quantities of die swell, shrinkage stress, and pressure vs. velocity compare favorably with those obtained in documented experiments. The effects of strain rate, temperature, and friction on the process are also examined.     

An analytical treatment of a class of plane plastic strain problems

Conclusion In the foregoing sections a series of solutions of the problem of plastic plane strain have been found, all of which are of the following form: the Cartesian coordinates x and y of the physical plane are trigonometrical functions of , the direction of the major principal stress, multiplied by a power of s, a quantity directly connected with the isotropic stress. If the boundary condition can be described in the same form, the boundary value problem can be solved. In sec. 4 this was done for a special sort of boundary condition. There the shape of the boundary was arbitrary, but the surface traction was a constant normal pressure.The analytical method seems very suitable for the determination of the stresses in the plastic region around a hole. The method may also be applicable to other sorts of plasticity problems, but this is beyond the scope of the present paper.The possibilities are limited in the first place by the requirement that along the boundary the functions and s are continuous and further that there exists a one-to-one relation between the points (, s) and (x, y) of the boundary.One conclusion of practical importance to be drawn from sec. 3 is that the plastic stress distribution around a hole of general shape, and loaded by an arbitrary surface traction, will tend to circular symmetry at a great distance from the hole. An example of this behaviour is shown in fig. 8.     

Application of the Trefftz method in plane elasticity problems

Direct and indirect approximations using sets of non-singular, complete Trefftz functions, i.e. the complete systems of solutions, have been successfully applied to harmonic problems.¹ In this paper, the procedure is applied to a more complex situation—plane elasticity problems. The examples show that the present method can avoid the difficulties relating to singular integration and has good accuracy compared with traditional boundary elements.     

A Green's function time‐domain boundary element method for the elastodynamic half‐plane

The transient Green's function of the 2‐D Lamb's problem for the general case where point source and receiver are situated beneath the traction‐free surface is derived. The derivations are based on Laplace‐transform methods, utilizing the Cagniard–de Hoop inversion. The Green's function is purely algebraic without any integrals and is presented in a numerically applicable form for the first time. It is used to develop a Green's function BEM in which surface discretizations on the traction‐free boundary can be saved. The time convolution is performed numerically in an abstract complex plane. Hence, the respective integrals are regularized and only a few evaluations of the Green's function are required. This fast procedure has been applied for the first time. The Green's function BEM developed proved to be very accurate and efficient in comparison with analogue BEMs that employ the fundamental solution. Copyright © 1999 John Wiley & Sons, Ltd.     

Green's functions for Kirchhoff plates of irregular shape

A method is proposed for the construction of Green's functions for the Sophie Germain equation in regions of irregular shape with mixed boundary conditions imposed. The method is based on the boundary integral equation approach where a kernel vector function B satisfies the biharmonic equation inside the region. This leads to a regular boundary integral equation where the compensating loads and moments are applied to the boundary. Green's function is consequently expressed in terms of the kernel vector function B, the fundamental solution function of the biharmonic equation, and kernel functions of the inverse regular integral operators. To compute moments and forces, the kernel functions are differentiated under the integral sign. The proposed method appears highly effective in computing both displacements and stress components.