Elastoplastic torsion of a cylindrical rod for finite deformations

A solution of the problem of the torsion of a cylindrical rod was obtained in /1/ for a general, isotropic, incompressible elastic material. The present paper gives an analytical solution of the elastoplastic torsion problem for finite deformations, written in terms of quadratures of elliptic functions. The non-linear kinematics of elastoplastic deformation is introduced into the defining equations with the help of a multiplicative decomposition of the deformation gradient into elastic and plastic components /2, 3/. The elastic deformation and rate of plastic deformation are related to the state of stress of the body, in accordance with the defining Mooney-Rivlin equations /4/ and the law of flow for finite deformations associated with the Tresca yield condition /5/. A non-linear first-order partial differential equation and the initial data at the elastoplastic boundary are obtained in order to determine the angle of rotation within the plastic zone of the basis formed from the eigenvectors of the stress tensor, relative to the radial direction. The integration of the resulting equation is reduced to determining the general integral of the Ricatti equation with right-hand side determined from the angular velocity of flow of the material within the plastic zone. It is shown that neglecting the finiteness of the deformation leads to too high an estimate of the rigidity of the rod.     

Computer simulation of underthrusting and subduction due to collision of slabs

We come up with a mathematical simulation of a collision between lithospheric slabs (plates) where one slab is forced into the mantle beneath another. Problems of the Earth’s crust and mantle deformation are solved numerically: for spatial discretization of equations of deformable solid mechanics, a finite-element method is used, and for evolution of the collision process, a stepwise integration of quasistatic deformation equations is applied. Problems of plate motion are solved within a geometrically nonlinear setting in a two-dimensional approximation (plane deformation) with due regard for large deformations of bodies and contact interactions of slabs with the mantle. A numerical solution is obtained via a MSC.Marc 2005 code, encompassing formulations of equations with required types of nonlinearities. A part of the Earth’s crust that has no tendency to delving into the mantle is simulated by a prescribed motion of a rigid body. A part of the Earth’s crust that should sink by virtue of properties of initial geometry is simulated as a deformable solid made up of elastoplastic strain-hardening material. The mantle is simulated by an ideal elastoplastic material with a low yield stress value. We are concerned with parts of the Earth’s crust that have different geometric parameters. Computer simulation of plate collision shows that under standard conditions, underthrusting of one slab beneath another occurs; at sites of initial thickening of a slab in a contact zone, subduction (deep sinking) of the slab into the mantle is expected. In the latter case account should be taken of a well-known experimental fact, that of material compaction of the sunken piece of a slab.     

Elastoplastic deformation during projectile–wall collision

The Smoothed Particle Hydrodynamics method for elastic solid deformation is modified to include von Mises plasticity with linear isotropic hardening and is then used to investigate high speed collisions of elastic and elastoplastic bodies. The Lagrangian mesh-free nature of SPH makes is very well suited to these extreme deformation problems eliminating issues relating to poor element quality at high strains that limits finite element usage for these types of problems. It demonstrates excellent numerical stability at very high strains (of more than 200%). SPH can naturally track history dependent material properties such as the cumulative plastic strain and the degree of work hardening produced by its strain history. The high speed collisions modelled here demonstrate that the method can cope easily with collisions of multiple bodies and can also naturally resolve self-collisions of bodies undergoing high levels of plastic strain. The nature and the extent of the elastic and plastic deformation of a rectangular body impacting on an elastic wall and of an elastic projectile impacting on a thin elastic wall are investigated. The final plastically deformed shapes of the projectile and wall are compared for a range of material properties and the evolution of the maximum plastic strain throughout each collision and the coefficient of restitution are used to make quantitative comparisons. Both the elastoplastic projectile–elastic wall and the elastic projectile–elastoplastic wall type collisions have two distinct plastic flow regimes that create complex relationships between the yield stress and the responses of the solid bodies.     

Boundary integral equation method in rolling contact problems involving roughness,wear and frictional heating

The method of boundary integral equations is developed to the solution of the rolling contact problems for rough bodies. A model of boundary roughness is proposed. The method permits to involve wear and frictional heating due to the slip in rolling contact.     

Combined grid-characteristic method for the numerical solution of three-dimensional dynamical elastoplastic problems

A combined method blending the advantages of smoothed particles hydrodynamics (SPH) and the grid-characteristic method (GCM) is proposed for simulating elastoplastic bodies. Various grid methods, including the GCM, have long been used for the numerical simulation of elastoplastic media. This method applies to the simulation of wave processes in elastic media, including elastic impacts, in which case an advantage is the use of moving tetrahedral meshes. Additionally, fracture processes can be simulated by applying various fracture criteria. However, this is a technically complicated task with the accuracy of the results degrading due to the continual updating of the grid. A more suitable approach to the simulation of processes involving substantial fractures and deformations is based on SPH, which is a meshless method. However, this method also has shortcomings: it produces spurious modes, and the simulation of oscillations requires particle refinement. Thus, two families of methods are available that are optimal as applied to two different groups of problems. However, a realworld problem can frequently be a mixed one, which requires a substantial tradeoff in the numerical methods applied. Aimed at solving such problems, a combined GCM-SPH method is developed that blends the advantages of two constituting techniques and partially eliminates their shortcomings.     

Micro‐ and nanoscale modelling of dry friction with application to the wheel/rail contact

Friction is a phenomenon involving elastic interactions, plastic deformation and failure processes at different length scales. A model of dry friction is established based on the method of Movable Cellular Automata (MCA). The influence of material and loading parameters has been investigated within a large number of numerical simulations. The new friction law is applied to the calculation of stresses, deformations and tractive forces in wheel/rail contact with rough surfaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rotationless formulation for large deformations in flexible multibody dynamics

Especially for specific applications, such as contact problems, computer methods for flexible multibody dynamics that are able to treat large deformation phenomena are important. Classical formalisms for multibody dynamics are based on rigid bodies. Their extension to flexible multibody systems is typically restricted to linear elastic material behavior whereas large deformation phenomena are formulated in the framework of the nonlinear finite element method. In the talk we address computer methods that can handle large deformations in the context of multibody systems. In particular, the link between nonlinear continuum mechanics and multibody systems is facilitated by a specific formulation of rigid body dynamics [1]. It makes possible the incorporation of state-of-the-art computer methods for large deformation problems. In the talk we focus on the treatment of large deformation contact whithin flexible multibody dynamics [2]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Averaging of a finely laminated elastic medium with roughness or adhesion on the contact surfaces of the layers

The governing relations of a laminated elastic medium with non-ideal contact conditions in the interlayer boundaries are obtained by an asymptotic averaging method. The interaction of rough surfaces is described by a non-linear contact condition which simulates the local deformation of the microroughnesses using a certain penetration of the nominal surfaces of the elastic layers. The cohesive forces, caused by the thin adhesive layer, are described within the limits of the Frémond model which includes a differential equation characterizing the change in the cohesion function. A piecewise-linear approximation of the initial positive segment of the Lennard–Jones potential curve is proposed to describe of the adhesive forces between smooth dry surfaces. A comparison is made with the solution obtained within the limits of the Maugis–Dugdale model based on a piecewise-constant approximation. Solutions of the above problems are constructed taking account of the possible opening of interlayer boundaries.     

Thermodynamically consistent nonlinear model of elastoplastic Maxwell medium

The paper is devoted to new applications of the ideas underlying Godunov’s method that was developed as early as in the 1950s for solving fluid dynamics problems. This paper deals with elastoplastic problems. Based on an elastic model and its modification obtained by introducing the Maxwell viscosity, a method for modeling plastic deformations is proposed.     

On the formulation and investigation of a spatial contact problem for elastic bodies under mixed friction conditions

A spatial contact problem is formulated and investgated for rough elastic bodies which touch each other under mixed friction conditions: the elastic bodies are separated in one part of the contact domain by a layer of viscous incompressible liquid (lubricant), while in the other they are in direct contact (such conditions are characteristic for roller bearings, gear transmissions, etc.). The problem is reduced to a system of nonlinear integro-differential and integral equations and inequalities in the contact domain, part of the external boundary, and a number of inner boundaries that are unknown in advance, but separate the lubricated and unlubricated zones. Special cases are problems of dry and completely lubricated contact. A formulation is given for the problem for the case when the materials of the bodies are identical. The problem of mixed friction is considered in strongly drawn out contact. Sections of the contact domain in which the interaction between the bodies is direct or by means of the lubrication layer are investigated using asymptotic methods.     

Modelling of the deformation processes and the localization of plastic deformations in the torsion–tension of solids of revolution

Generalized two-dimensional problems of the torsion of elastoplastic solids of revolution of arbitrary shape for large deformations under non-uniform stress-strain conditions are formulated and a method for their numerical solution is proposed. The use of this method to construct strain diagrams of materials based on experiments on the torsion of axisysmmetric samples of variable thickness until fracture occurs is described. Experimental and numerical investigations of processes of elastoplastic deformation, loss of stability and supercritical behaviour of solid cylindrical steel samples of variable thickness under conditions of monotonic kinematic loading with a torque, a tension and a combined load are presented. The mutual influence of torsion and tension on the deformation process and the limit states is estimated, and the universality (the independence of the form of the stress-strain state) of the “stress intensity – Odqvist parameter” diagram for steel for large deformations is proved.     

Coupled FEM-BEM for elastoplastic contact problems using Lagrange multipliers

The coupling of the elastoplastic finite element and elastic boundary element methods for two-dimensional frictionless contact stress analysis is presented. Interface traction matching (boundary element approach), which involves the force terms in the finite element analysis being transformed to tractions, is chosen for the coupling method. The analysis at the contact region is performed by the finite element method, and the Lagrange multiplier approach is used to apply the contact constraints. Since the analyses of elastoplastic problems are non-linear and involve iterative solution, the reduced size of the final system of equations introduced by combining the two methods is very advantageous, especially for contact problems where the nature of the problem also involves an iterative scheme.     

Some qualitative effects in the exact solutions of the Lamé problem for large deformations

Qualitative effects in the solution of a number of radially symmetric and plane axisymmetric problems for bodies made of non-linearly elastic incompressible materials are analysed for large deformations. In the case of problems of the axisymmetric plane deformation of cylindrical bodies, the lack of uniqueness of the solution for a given follower load in the case of a Bartenev–Khazanovich material and the existence of a limiting load in the case of a Treloar (neo-Hookian) material have been studied in detail and the dependences of the limiting load on the ratio of the external and internal radii of a hollow cylinder in the undeformed state have been presented. A similar study has been carried out for constitutive relations of a special form that well describe the properties of rubber. For this material, the lack of uniqueness of the solution is revealed for fairly high loads. The axisymmetric problem of the plane stress state of a circular ring made of a Bartenev–Khazanovich material has been solved and a lack of uniqueness of the solution for a given follower load was discovered in the case when the dimensions of the ring are given in the undeformed state. Similar studies have been carried out for Chernykh and Treloar materials in the case of the problem of the radially symmetric deformation of a spherical shell. It was established that, in the case of a Chernykh material, the lack of uniqueness of the solution depends considerably on the constant characterizing the physical non-linearity. The limit case of the deformation of a spherical cavity in an infinitely extended body has been investigated. The effect of an unbounded increase in the boundary stresses is observed for finite external loads, that appears in the case of the problem of the plane axisymmetric deformation of a cylindrical cavity in an infinitely extended body made of a Bartenev–Khazanovich material and in the case of the problem of the radially symmetric deformation of an infinitely extended body made of a Chernykh material with a spherical cavity.     

Stability of hydrostatic-pressured FGM thick rings with material nonlinearity

Buckling behaviors of elastoplastic ceramic/metallic functionally graded material (FGM) rings are investigated by using the first order shear deformation theory. The hydrostatic-pressured rings are assumed to be in both the plane-stress case and the plane-strain case, which lead respectively to a uniaxial and a biaxial elastoplastic stress states in prebuckling stage. A uniform strain hypothesis helps to deal with the elastoplastic stress states. By introducing in the graded material properties, the constitutive model of FGMs is formulated under the framework of J₂ deformation theory. By considering the kinetic relations of von-Kárman type and employing the principle of virtual displacement, the equilibrium equations and the buckling governing equations of FGM circular rings are formulated, and the analytical solution of the anisotropic rings is obtained. Finally, the elastoplastic buckling problem is numerically solved through a semi-analytical method, which is proposed to seek the real circumferential strain of FGM rings at the buckling point and determinate the elastoplastic buckling critical hydrostatic pressure. The effects of the inhomogeneous and geometrical parameters on the buckling critical load and the position of the elastoplastic interface are discussed. Results show that, in both the plane-stress and the plane-strain cases, the elastoplastic critical loads are generally lower than their elastic counterparts due to material flow, and the plane-strain critical load is generally larger than the plane-stress one. The elastoplastic critical load does not always decrease monotonously with the increase of the inhomogeneous parameters, which is quite different from their elastic counterparts.     

Partial three dimensional deformations for the perfectly elastic Mooney material

Summary The governing equations for finite elastic deformations are highly nonlinear and there is still only a limited number of known exact solutions. In general for large elastic fully three dimensional deformations of the isotropic incompressible perfectly elastic neo-Hookean and Mooney materials, a non-trivial deformation for say the neo-Hookean strain-energy function, is frequently not well-defined for the general Mooney strain-energy function because the additional coupling imposes extra constraints on the deformation which are generally inconsistent with one another. Here we note two fully three dimensional deformations for which this is not the case. In both cases the resulting coupled systems of ordinary differential equations need to be integrated numerically but the deformations are nevertheless well-defined for the general Mooney material. The first deformation is simply noted because the details are given elsewhere. For the second deformation, the coupled system is derived and some new simple special solutions are given. Such deformations are important and noteworthy because of the scarcity of exact solutions in finite elasticity.     

Instable Waves between Rough Elastic Bodies in Sliding Contact

Most sliding bodies are not perfectly flat, but show a rough surface topography or tribological layer. This paper analyses a sliding system consisting of two elastic bodies with continuous contact and steady sliding. Surface topographies are taken into account by an inertia film on one of the sliding surfaces. A linear model is developed that allows an analytical solution. As a typical tribological system, the contact between pad and disc in a brake system is discussed. It can be shown that unstable elastic waves travel through the contact between both bodies. This instability is opened only when a tribological layer on the brake pad is taken into account. This microscopic excitation can cause a loss of stability of the entire (brake) system. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A method for the approximate solution of quasi-static problems for hardening bodies

A method for the approximate solution of quasi-static problems for hardening elastoplastic bodies is proposed. The constitutive relation of the model is taken in the form of a variational inequality. An approximate solution of the initial problem is constructed in time steps and, by means of the finite element method, is reduced to the solution of a system of two variational inequalities in corresponding finite-dimensional space. It is shown that the solution of this system is equivalent to finding the saddle point of the corresponding quadratic functional. To find the saddle point, Udzawa's algorithm is used, by means of which the process of finding the velocity vector and stress tensor reduces to the successive calculation of these quantities: the velocity vector is determined from the variational inequality corresponding to the equilibrium equation, and the stress tensor is determined from the variational inequality corresponding to the constitutive relation. The latter inequality is reduced to a certain non-linear equation containing the operation of projection onto a closed convex set corresponding to the elastic strains of the medium. In turn, the solution of the non-linear equation is constructed using the method of successive approximations. To illustrate the use of the proposed method, the one-dimensional problem of the quasi-static deformation of a cylindrical tube under a load applied to its internal surface is considered.     

The contact of a solid with an elastic half-space through a thin coating

More accurate equations of the deformation of thin plates, which are more convenient for solving contact problems for bodies with coatings and containing, as a special case, the equations of all known applied theories, are derived by an asymptotic analysis of the first fundamental problem of the theory of elasticity. The equations of the deformation of thin-walled elastic bodies are classified, their qualitative correspondence to the equations of the theory of elasticity is clarified, and the forms of the features that arise along the shift lines of the boundary conditions in the corresponding contact problems are established. A criterion for selecting approximate models to describe the properties of the coatings depending on the geometrical and mechanical characteristics of the coating and the substrate and also on their degree of adhesion is given.     

Modeling of nonlinear viscoelastic contact problems with large deformations

Contact problems are one of the most important engineering problems. These problems become much more tedious when one of the contacting bodies behaves nonlinear viscoelasticity and large deformations. This paper presents an incremental-iterative finite element model for the analysis of two dimensional quasistatic frictionless contact problems. Nonlinear viscoelastic behavior and large deformations are considered. The Schapery’s single-integral creep model with stress-dependent properties is used for nonlinear viscoelasticity. The constitutive equations are transformed into an incremental form resulting in a recursive relationship. Thereby, the need to store the entire strain histories is eliminated, except that from the previous time increment. The updated Lagrangian formulation is used to model the material and geometrical nonlinearities. Also, the Lagrange multiplier method is adopted to enforce the contact constraints. The converged solution is obtained using the Newton–Raphson iterative technique. The developed model has been verified with the previously published works and found a good agreement with them. To demonstrate the efficient capability of the developed computational model, three contact problems with different nature are analyzed.