Wedge indentation into elastic-plastic single crystals, 1: Asymptotic fields for nearly-flat wedge

Asymptotic stress and deformation fields under the contact point singularities of a nearly-flat wedge indenter and of a flat punch are derived for elastic ideally-plastic single crystals with three effective in-plane slip systems that admit a plane strain deformation state. Face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal-close packed (HCP) crystals are considered. The asymptotic fields for the flat punch are analogous to those at the tip of a stationary crack, so a potential solution is that the deformation field consists entirely of angular constant stress plastic sectors separated by rays of plastic deformation across which stresses change discontinuously. The asymptotic fields for a nearly-flat wedge indenter are analogous to those of a quasistatically growing crack tip fields in that stress discontinuities can not exist across sector boundaries. Hence, the asymptotic fields under the contact point singularities of a nearly-flat wedge indenter are significantly different than those under a flat punch. A family of solutions is derived that consists entirely of elastically deforming angular sectors separated by rays of plastic deformation across which the stress state is continuous. Such a solution can be found for FCC and BCC crystals, but it is shown that the asymptotic fields for HCP crystals must include at least one angular constant stress plastic sector. The structure of such fields is important because they play a significant role in the establishment of the overall fields under a wedge indenter in a single crystal. Numerical simulations—discussed in detail in a companion paper—of the stress and deformation fields under the contact point singularity of a wedge indenter for a FCC crystal possess the salient features of the analytical solution.     

Galin’s problem for a spatial elastic wedge

We study a three-dimensional contact problem on the indentation of an elliptic punch into a face of a linearly elastic wedge. The wedge is characterized by two parameters of elasticity and its edge is subjected to the action of an additional concentrated force. The other face wedge is free from stresses. The problem is reduced to an integral equation for the contact pressure. An asymptotic solution of this equation is obtained which is effective for a given contact region fairly remote from the edge. Calculations are performed that allow one to evaluate the effect of a force applied outside the contact region on the contact pressure distribution. The problem under study is a generalization of L. A. Galin’s problem on a force applied outside a circular punch on an elastic half-space [1, 2]. In a special case of a wedge with an opening angle of 180° and zero contact ellipse eccentricity, the obtained asymptotic relation coincides with the expansion of Galin’s exact solution in a series. Problems of indentation of an elliptic punch into a spatial wedge with the face not loaded outside the contact region have been studied previously. For example, the paper [3] dealt with the case of a known contact region (asymptotic method) and the paper [4] considered the case of an unknown contact region (numerical method). The solution of Galin’s problem allowed the authors of [2] to reduce the contact problem on the interaction of several punches applied to a half-space to a system of Fredholm integral equations of the second kind (Andreikin-Panasyuk method). A topical direction in contact mechanics is the model of discrete contact as well as related problems on the interaction of several punches [2, 5–8]. The interaction of several punches applied to a face of a wedge can be treated in a similar manner and an asymptotic solution can be obtained for the case where a concentrated force is applied at an arbitrary point of this face beyond the contact region rather than on the edge.     

Variational multiscale analysis of elastoplastic deformation using meshfree approximation

A variational multiscale method has been presented for efficient analysis of elastoplastic deformation problems. Severe deformation occurs in plastic region and leads to high gradient displacement. Therefore, solution needs to be refined to properly capture local deformation in plastic region. In this work, scale decomposition based on variational formulation is presented. A coarse scale and a fine scale are introduced to represent global and local behavior, respectively. The displacement is decomposed into a coarse and a fine scale. Subsequently the problem is also decomposed into a coarse and a fine scale from the variational formulation. Each scale variable is approximated using meshfree method. Adaptivity can easily and nicely be implemented in meshfree method. As a method of increasing resolution, extrinsic enrichment of partition of unity is used. Each scale problem is solved iteratively and conversed results are obtained consequently. Iteration procedure is indispensable for the elastoplastic deformation analysis. Therefore iterative solution procedure of each scale problem is naturally adequate. The proposed method is applied to the Prandtl’s punch test and shear band problem. The results are compared with those of other methods and the validity of the proposed method is demonstrated.     

Wedge-shaped cut in three-dimensional elastic wedge

An asymptotic solution of the integro-differential equation of the problem of describing a low-angle wedge-shaped cut in a three-dimensional elastic wedge is found. On the basis of the solution, the asymptotic behavior of the contact pressures at the apex of the punch, which in horizontal projection forms one side of an elastic wedge that includes a low-angle wedge-shaped cut, is investigated. Scientific Research Institute for Mechanics and Applied Mathematics, Rostov University, Russia. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 27–32, March, 1999.     

Interaction Between a Slow Magnetohydrodynamic Shock Wave and a Tangential Discontinuity

The self-similar problem of the oblique interaction between a slow MHD shock wave and a tangential discontinuity is solved within the framework of the ideal magnetohydrodynamic model. The constraints on the initial parameters necessary for the existence of a regular solution are found. Various feasible wave flow patterns are found in the steady-state coordinate system moving with the line of intersection of the discontinuities. As distinct from the problems of interaction between fast shock waves and other discontinuities, when the incident shock wave is slow the state ahead of it cannot be given and must to be determined in the process of solving the problem. As an example, a flow in which the slow shock wave incident on the tangential discontinuity is generated by an ideally conducting wedge located in the flow is considered. The basic features of the developing flows are determined.     

Puckering instability phenomena in the hemispherical cup test

The problem investigated here is the plastic bifurcation of an initially flat circular plate held frictionlessly between a blankholder and a die and deformed by a spherically shaped punch.In view of the large deviations of the prebifurcation solution from proportional loading, a recently developed phenomenological corner theory has been employed and an appropriate bifurcation criterion has been developed. The effects of geometry and material properties on the onset of the (nonaxisymmetric) plastic instability have been investigated using a numerical solution of the resulting equations based on the finite element method.     

Flow stability of a flat plastic ring with free boundaries

The problem of two-dimensional unstable flow of an ideally plastic ring acted upon by internal pressure is formulated. The determination of the law of motion for the boundaries and of the time change of pressure is reduced to an ordinary nonlinear differential equation of the second order. For this equation a particular solution of the Cauchy problem is determined; this corresponds to a widening of the ring boundaries with a negative acceleration. For the field of initial velocities an estimate from above is available, expressed in terms of the original parameters. The very particular unstable flow obtained for an ideally plastic ring is also investigated with respect to stability to small harmonic perturbations of the velocity vector, the pressure, or the boundaries of the ring. It is shown that the fundamental flow is stable irrespective of the wave number. This result has been obtained by assuming that the inertial forces in the perturbed flow are small compared to the lasting ones.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 94–101, March–April, 1975.     

On Brinell and Boussinesq indentation of creeping solids

As an alternative to traditional tensile testing of materials subjected to creep, indentation testing is examined. Axisymmetric punches of shapes defined by smooth homogeneous functions are analysed in general at power law behaviour both from a theoretical and a computational point of view. It is first shown that by correspondence to nonlinear elasticity and self-similarity the problem to determine time-dependent properties admits reduction to a stationary one. Specifically it is proved that the creep rate problem posed depends only on the resulting contact area but not on specific punch profiles. As a consequence the relation between indentation depth and contact area is history independent. So interpreted, the solution for a flat circular cylinder (Boussinesq) is not only of intrinsic interest but serves as a reference solution to generate results for various punch profiles. This is conveniently carried out by cumulative superposition and in particular ball indentation (Brinell) is analysed in depth. A carefully designed finite element procedure based on a mixed variational principle is used to provide a variety of explicit results of high accuracy pertaining to stress and deformation fields. Universal relations for hardness at creep are proposed for Boussinesq and Brinell indentation in analogy with the celebrated formula by Tabor for indentation of strain-hardening plastic materials. Quantitative comparison is made with a diversity of experimental data attained by earlier writers and the relative merits of indentation strategies are discussed.     

Problems on elastic equilibrium of a wedge with cracks on the axis of symmetry

We use the Wiener-Hopf method to obtain exact solutions of plane deformation problems for an elastic wedge whose lateral sides are stress free and which has rectilinear cracks on its axis of symmetry. In problem 1, a finite crack issues from the wedge apex edge; in problem 2, a half-infinite crack originates at a certain distance from the wedge apex edge; and in problem 3, the wedge contains an internal finite crack.     

Extension of a V-notched bar and failure of plastic bodies

The effect of plastic strain localization near the domains of sharp variation in shape and transverse cross-section of bodies is well known. But such processes have not yet been studied analytically well enough. On the basis of the model of an ideally rigid-plastic body, we propose an approach for determining the strain fields near the concentrators on the basis the motion of the displacement velocity field (near surfaces or discontinuity lines in the form of rigid-plastic boundaries and centers of the fan of slip lines under plane strain). We consider the problem on plastic flow with failure for a V-notched bar. We show that the plastic flow is not unique (in the framework of the solution completeness).We propose to use the strain criterion for choosing the preferable plastic flow. On the basis of the solutions thus obtained, we state an approach to studying failure processes for more complicated models of bodies.     

Particle Motion in a Plastic Medium with Allowance for Particle Attrition

Mathematical simulation of impact-induced deep penetration of an absolutely rigid spherical particle into an ideally plastic medium is performed; the law of particle motion and the distance covered by the particle are determined. The problem for a particle whose size varies owing to attrition is solved.     

Edge crack opening in an elastic plane with a wedge-like cut in contact with a punch

The equilibrium of an elastic plane with a wedge-like cut and an internal or edge crack on the symmetry axis was studied in [1] in the case of punch indentation in the lateral faces of the cut at a distance from the cut tip. In [1], the systemof singular integral equations of the problemwas solved numerically by the mechanical quadrature method. In this paper, the generalized Wiener-Hopf method [2] is used to obtain the analytic solution of a similar problem in the case of an edge crack under punch pressure on parts of the cut lateral faces adjacent to the cut tip. Some special cases of this problem were considered earlier without a crack [3, 4] or a punch [5, 6].     

Inclined circular flat punch on a transversely isotropic piezoelectric half-space

Summary Utilizing the general solution of transversely isotropic piezoelectricity, the paper analyzes the problem of an inclined rigid circular flat punch indenting a transversely isotropic piezoelectric half-space. The potential theory method is employed and generalized to take into account the effect of the electric field in piezoelectric materials. Assuming that the punch is maintained at a constant electric potential, exact expressions for the elastoelectric field are derived in terms of elementary functions. It is noted that the solution corresponding to a flat circular punch centrally loaded by a concentrated force can be obtained as a special case. Received 15 December 1998; accepted for publication 9 March 1999     

Critical velocity in a contact elasticity problem for the case of transonic punch velocity

The paper deals with a dynamic contact problem in the presence of friction forces in the transonic range of punch velocities, where the punch velocity exceeds the transverse wave velocity but is still less than the longitudinal wave velocity. It is shown that there exists a critical velocity at which the solution structure and the character of its behavior on the boundary of the contact region change. This velocity is $\sqrt 2 $ times the transverse wave velocity. The existence of this velocity is possibly related to the surface wave velocity under restricted deformation conditions.     

Wave processes in an elastic half-plane impacted by a blunt-nosed rigid body

The problem of unsteady deformation of an elastic half-plane is considered whose surface is impacted, at an initial instant, by a blunt-nosed rigid body, which generates diverging unsteady elastic waves and deforms the medium. The corresponding initial-boundary-value problem is formulated whose solution is constructed for the early stage of the interaction. The integral Laplace transform in the time variable and the integral Fourier transform in the one of the spatial variables are used. The solution of the problem is obtained in terms of the transforms and a formal solution is constructed in terms of the original functions. For a body with a fixed contact region, an analytical expression of the normal stress at an arbitrary point of the half-plane as a function of time is obtained. For a body shaped as an obtuse-angled wedge, analytical expressions of the normal stress and displacement at an arbitrary point at the symmetry axis of the problem are obtained. Calculations are performed and used to analyze the characteristic features of the wave processes in the medium as functions of time, the surface distance, and the mechanical properties of the material.     

Action of an elliptic punch on a transversally isotropic half-space

The 3D contact problem on the action of a punch elliptic in horizontal projection on a transversally isotropic elastic half-space is considered for the case in which the isotropy planes are perpendicular to the boundary of the half-space. The elliptic contact region is assumed to be given (the punch has sharp edges). The integral equation of the contact problem is obtained. The elastic rigidity of the half-space boundary characterized by the normal displacement under the action of a given lumped force significantly depends on the chosen direction on this boundary. In this connection, the following two cases of location of the ellipse of contact are considered: it can be elongated along the first or the second axis of Cartesian coordinate system on the body boundary. Exact solutions are obtained for a punch with base shaped as an elliptic paraboloid, and these solutions are used to carry out the computations for various versions of the five elastic constants. The structure of the exact solution is found for a punch with polynomial base, and a method for determining the solution is proposed.     

Wrinkling of elastoplastic circular plates during stamping

Circular plates that are stamped into a shallow, biaxially curved die by a matching punch develop radial wrinkles near the periphery when the edge is not clamped. Thin ductile metal plates develop these wrinkles after some plastic deformation occurs at the center of the plate. In comparison with elastic wrinkling, the center deflection to thickness ratio for wrinkling is increased as a consequence of the plastic deformation. In elastoplastic plates, this critical deflection ratio is a decreasing function of the plate thickness parameter ξ_o     

Cylindrical void in a rigid-ideally plastic single crystal III: Hexagonal close-packed crystal

The analytical solution is derived for the plane strain stress field around a cylindrical void in a hexagonal close-packed single crystal with three in-plane slip systems oriented at the angle π/3 with respect to one another. The critical resolved shear stress on each slip system is assumed to be equal. The crystal is loaded by both internal pressure and a far-field equibiaxial compressive stress. The deformation field takes the form of angular sectors, called slip sectors, within which only one slip system is active; the boundaries between different sectors are radial lines. The stress fields are derived by enforcing equilibrium and a rigid, ideally plastic constitutive relationship, in the spirit of anisotropic slip line theory. The results show that each slip sector is divided into smaller regions denoted as stress sectors and the stress state valid within each stress sector is derived. It is shown that stresses are unique and are continuous within stress sectors and across stress sector boundaries, but the gradient of stresses is not continuous across the boundaries between stress sectors. The solution shows self-similarity in that the stresses over the entire domain can be determined from the stresses within a small region adjacent to the void by invoking certain scaling and symmetry properties. In addition, the stress state exhibits periodicity along logarithmic spirals which emanate from the void. The results predict that the mean value of in-plane pressure required to activate plastic deformation around a void in a single crystal can be higher than that necessary for a void in an isotropic material and is sensitive to the orientation of the slip systems relative to the void.     

Contact problem for an orthotropic half-space

Numerical and analytical solutions of the 3D contact problem of elasticity on the penetration of a rigid punch into an orthotropic half-space are obtained disregarding the friction forces.A numericalmethod ofHammerstein-type nonlinear boundary integral equations was used in the case of unknown contact region, which permits determining the contact region and the pressure in this region. The exact solution of the contact problem for a punch shaped as an elliptic paraboloid was used to debug the program of the numerical method. The structure of the exact solution of the problem of indentation of an elliptic punch with polynomial base was determined. The computations were performed for various materials in the case of the penetration of an elliptic or conical punch.