The existence of riemann invariants in the one-dimensional equations of the non-linear theory of elasticity

The conditions for the existence of Riemann invariants of a one-dimensional system of equations of the non-linear theory of elasticity are investigated. Haantjes' diagonalization criterion is used to determine the form of the elastic potential for which the system has six Riemann invariants or three Riemann invariants (for waves which propagate in one direction). In particular, it is shown that the Haantjes criterion is satisfied and there are three Riemann invariants in the case of the elastic potential for slightly-non-linear weakly-anisotropic elastic media [1–3]. A procedure for computing Riemann invariants is described. The Riemann invariants are computed approximately for a form of elastic potential which satisfies the Haantjes criterion.     

Solution of the elastic/visco-plastic constitutive equations: A finite volume approach

A novel two dimensional finite volume (FV) approach to the solution of elastic / visco-plastic solid mechanics problems is described. The numerical procedure is an extension of that developed by Fryer et al. for linear elastic materials and utilizes an unstructured mesh. The procedure is compared favorably with conventional finite element (FE) formulations with respect to both accuracy and CPU time. A number of test problems demonstrate the ability of the FV procedure to model a range of boundary constraints, in addition to mechanical and thermal loads. It might be argued from the work reported here that the FV approach shows the potential to be at least as effective as conventional FE formulations in representing nonlinear solid mechanics behavior.     

Plane problem of interaction between an elastic longitudinal wave and a cylindrical shell with axial cut

We have proposed a procedure for investigating the stressed state of a thin-walled round cylindrical shell with a cut along its generatrix, which is placed into an elastic space. A longitudinal elastic wave is incident on the shell. The procedure is based on the use of the Rayleigh expansion in partial waves. We have obtained relations for determining the displacement jumps and angle of rotation at the lips of the shell cut, the radial component of the displacement vector and the normal force in the cylindrical shell, as well as the stresses and displacements in the elastic medium. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 131–137, January–March, 2008.     

The solution of the Hertz axisymmetric contact problem

The main terms of the asymptotic form of the solution of the contact problem of the compression without friction of an elastic body and a punch initially in point contact are constructed by the method of matched asymptotic expansions using an improved matching procedure. The condition of unilateral contact is formulated taking account of tangential displacements on the contact surface. An asymptotic solution of the problem for the boundary layer is constructed by the complex potential method. An asymptotic model is constructed, extending the Hertz theory to the case where the surfaces of the punch and elastic body in the vicinity of the contact area are approximated by paraboloids of revolution. The problem of determining the convergence of the contacting bodies from the magnitude of the compressive force is reduced to the problem of calculating the so-called coefficient of local compliance, which is an integral characteristic of the geometry of the elastic body and its fixing conditions.     

On the Propagation of Love Waves in Anisotropic Elastic Media

The Love waves concentrated near the surface of an anisotropic elastic body are studied. A uniform asymptotics of the wave field is constructed with the use of the nonstationary caustic expansion (Yu. A. Kravtsov's ansatz) in the form of a space-time ray series. Using three types of waves, which propagate along any direction in an elastic medium, as a vector basis, sufficient conditions for the existence of a nonzero asymptotic solution of the problem under study are obtained. The procedure for constructing asymptotic series is illustrated with the model of a transversely isotropic medium. Bibliography: 9 titles.     

Diffusion limits of the linear boltzmann equation in extended kinetic theory: Weak and strong inelastic collisions

We consider the asymptotic analysis for the linear Boltzmann equation with elastic and inelastic scattering. The physical model describes the motion of test particles propagating by elastic and inelastic collisions through a host medium in the Lorentz gas limit. The background is in thermodynamical equilibrium with only two internal energy levels. We apply the compressed Chapman-Enskog procedure to derive the diffusive-type approximations in the cases of dominant elastic and dominant inelastic collisions. Then we present numerical examples showing the time evolution of the distribution function in some physically relevant cases. In the appendix the successive overrelaxation method is briefly cutlined. Conferenza tenuta il giorno 27 Settembre 1999 da G. Frosali     

Elastic displacements of a cement-based composite beam reinforced with punched steel grids

Using Mohr's method, a procedure is proposed for calculating the elastic displacements of a cement-based composite beam reinforced with punched steel grid ribbons. The approach proposed takes into account a possible asymmetry of the structure, the geometry of grids, and the elastic properties of cement mortar and steel.     

Stress Integration of the Hoek-Brown Material Model Using Incremental Plasticity Theory

This paper presents the formulation of the stress integration procedure for the Hoek-Brown (HB) material model with non-associative yielding condition by using the incremental plasticity method. Main idea of this method is to reach the solution by calculating the plastic matrix according to the method of incremental plasticity (used for elastic constitutive matrix corrections), and with the use of the total strain increment. Computational procedure is implemented within the PAK program package. Results of this procedure were compared with the solutions obtained by other program that contains this material model. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The deformation of a half-space with a gradient elastic coating under arbitrary axisymmetric loading

An analytical-numerical method of solving the Neumann boundary-value problem for an elastic half-space with a gradient elastic coating is proposed. The problem is formulated and the construction of the fundamental solution (Green’s function) is described. The method enables a solution of the problem to be obtained for a fairly wide class of types of non-uniformity of the medium, and effects related to the non-uniformity are investigated analytically. A procedure for calculating the displacement, stress and strain fields is described. Particular attention is devoted to analysing the mechanical characteristics in the transition region from the coating to the elastic substrate.     

Structurally damped elastic waves in 2D

The goal of the paper is to study structurally damped elastic waves in 2D. A suitable diagonalization procedure allows to derive WKB representations for solutions to the Cauchy problems. As consequences, we obtain results on energy decay with and without additional regularity for the data, Gevrey smoothing, and propagation of singularities for the visco‐elastic damped elastic wave model. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical implementation of iteration processes applied in the study of nonlinear boundary value problems of the theory of elasticity and filtration

Nonlinear elastic problems for hardening media are solved by applying the universal iteration process proposed by A.I. Koshelev in his works on the regularity of solutions to quasilinear elliptic and parabolic systems. This requires numerically solving a linear elliptic system at each step of the iteration procedure. The method is numerically implemented in the MATLAB environment by using its PDE Toolbox. A modification of the finite-element procedure is suggested in order to solve a linear algebraic system at each iteration step. The computer model is tested on simple examples. The same nonlinear problems are also solved by the method of elastic solutions, which consists in replacing the Laplace operator in the universal iteration process by the Lamé operator of linear elasticity. As is known, this iteration process converges to a weak solution of the nonlinear problem, provided that the displacements are fixed on the boundary. The method is tested on examples with stresses on the boundary. The third part of the paper is devoted to the nonlinear filtration problem. General properties of the iteration process for nonlinear parabolic systems have been studied by A.I. Koshelev and V.M. Chistyakov. The numerical implementation is based on slightly modified PDE Toolbox procedures. The program is tested on simple examples.     

Vibrations of mechanical systems in the presence of uncertainty

The minimax state estimation problem is solved for a linear mechanical system with a finite number of degrees of freedom. The proposed procedure is extended to distributed-parameter systems. As an example, we consider an elastic rod that executes small longitudinal oscillations under uncertainty in the initial-boundary conditions.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 95–100, 1986.     

Numerical Estimation of the Elastic Properties of Thin-Walled Structures Manufactured from Short-Fiber-Reinforced Thermoplastics

A model which allows us to estimate the elastic properties of thin-walled structures manufactured by injection molding is presented. The starting step is the numerical prediction of the microstructure of a short-fiber-reinforced composite developed during the filling stage of the manufacturing process. For this purpose, the Moldflow Plastic Insight^® commercial program is used. As a result of simulating the filling process, a second-rank orientation tensor characterizing the microstructure of the material is obtained. The elastic properties of the prepared material locally depend on the orientational distribution of fibers. The constitutive equation is formulated by means of orientational averaging for a given orientation tensor. The tensor of elastic material properties is computed and translated into the format for a stress-strain analysis based on the ANSYSÒ finite-element code. The numerical procedure and the convergence of results are discussed for a thin strip, a rectangular plate, and a shell of revolution. The influence of manufacturing conditions on the stress-strain state of statically loaded thin-walled elements is illustrated.     

Rigorous justification of the short-pulse equation

We prove that the short-pulse equation, which is derived from a quasilinear Klein–Gordon equation with formal asymptotic methods, can be rigorously justified. The justification procedure applies to small-norm solutions of the short-pulse equation. Although the small-norm solutions exist for infinite times and include modulated pulses and their elastic interactions, the error bound for arbitrary initial data can only be controlled over finite time intervals.     

Topology Optimization of Members of Elastic Multibody Systems

Lightweight techniques are applied increasingly often for modern machine designs in order to reduce the moving masses and therewith the energy consumption. However, as a result the stiffness of the system decreases causing undesired elastic deformations and therewith end-effector deviations. In this work a topology optimization procedure for elastic multibody systems based on the solid isotropic material with penalization approach is presented to lower end-effector tracking errors. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of the three-dimensional micromechanical behavior of woven-fabric composites

A three-dimensional representative volume-element model is presented to study the micromechanical behavior of woven-fabric composites. The effects of the fiber undulation zone and the fiber braid angle on the elastic modulus of the composites are taken into account in the unit cell. Based on isostrain and isostress assumptions, a standard homogenization procedure is used to calculate the effective elastic properties of woven-fabric composites, and all the final stiffness components are expressed in an explicit form. The results obtained by the model considered agree with published experimental results very well. The relationship between the geometric parameters, such as fiber width, thickness, volume fraction, etc., and the macromechanical behavior of the composites can be obtained by this model. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 2, pp. 209–220, April–May, 2006.     

A model of the elastodynamic interaction of a thin-walled inclusion with a matrix under antiplanar shear

Using the example of antiplanar shear we give a generalized scheme for constructing models of elastodynamic interaction of an elastic matrix with a thin-walled elastic curvilinear inclusion lying in the matrix under conditions of ideal contact subject to steady vibrations, by introducing suitable conditions on the middle line of the inhomogeneity. The procedure for the study is based on the application of the methods of the theory of singular perturbations. We describe the behavior of the stress-strain state of the body near the edges of the inhomogeneity, where boundary-layer phenomena appear. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 54–61.     

A regional mixed refinement procedure for finite element mesh design

A practical algorithm is developed for automated mesh design in finite element stress analysis. A regional mixed mesh improvement procedure is introduced. The error control, algorithm implementation, code development, and the solution accuracy are discussed. Numerical examples include automated mesh designs for plane elastic media with singularities. The efficiency of the procedure is demonstrated.     

Elastic and inelastic local buckling of stiffened plates subjected to non-uniform compression using the Galerkin method

A solution for the elastic and inelastic local buckling of flat rectangular plates with centerline boundary conditions subjected to non-uniform in-plane compression and shear stress is presented. The loaded edges are simply supported, the longitudinal edges may have any boundary conditions and the centerline is simply supported with a variable rotational stiffness. The Galerkin method, an effective method for solving differential equations, is applied to establish an eigenvalue problem. In order to obtain plate buckling coefficients, combined trigonometric and polynomial functions that satisfy the boundary conditions are used. The method is programmed, and several numerical examples including elastic and inelastic local buckling, are presented to illustrate the scope and efficacy of the procedure. The variation of buckling coefficients with aspect ratio is presented for various stress gradient ratios. The solution is applicable to stiffened plates and the flange of the I-shaped beams that are subjected to biaxial bending or combined flexure and torsion and shear stresses, and is important to estimate the reduction in elastic buckling capacity due to stress gradient.     

应用广义阶梯函数解变刚度超静定梁的弯曲问题

变刚度超静定梁的弯曲问题,可以近似的以受分段均布荷载(包括集中荷载、集中力偶)作用下的阶梯梁来代替.本文推广Heaviside函数{x-a}0的概念,定义一个新的函数{x-a}n,n=0,1,2…,称为广义阶梯函数,并给出{x-a}n{x-b}0的运算法则.然后将抗弯刚度的倒数1/EJ和弯矩方程M(x)都用{x-a}n来表示.代入挠曲线近似微分方程,从而建立一套适用于各类直梁弯曲问题的统一解法,并给出一般情况下挠曲线方程的通式.