Nonlinear Elastic Half-space Research Articles

The secular equation for non-principal Rayleigh waves in deformed incompressible doubly fiber-reinforced nonlinearly elastic solids

The explicit and implicit secular equations for the speed of a (surface) Rayleigh wave propagating in a pre-stressed, doubly fiber-reinforced incompressible nonlinearly elastic half-space are obtained. Hence, the anisotropy is associated with two preferred directions, thereby modelling the effect of two families of fiber reinforcement. One of the principal planes of the primary pure homogeneous strain coincides with the free surface while the surface wave is not restricted to propagate in a principal direction. Results are illustrated with numerical examples. In particular, an isotropic material reinforced with two families of fibers is considered. Each family of fibers is characterized by defining a privileged direction. Furthermore, the fibers of each family are located throughout the half space and run parallel to each other and perpendicular to the depth direction, i.e. the free surface is a plane of symmetry of the anisotropy. The wave speed depends strongly on the anisotropic character of the material model as well as the direction of propagation.

Rayleigh wave in a quadratic nonlinear elastic half-space (Murnaghan model)

The nonlinear equations that underlie the analysis of classical Rayleigh waves are derived for the two-dimensional case of nonlinear elastic deformation described by the Murnaghan model. In addition to the case of presence of both geometrical and physical nonlinearities, two special cases are considered where one only type of nonlinearity is taken into account. It is shown that unlike the one-dimensional problems for plane waves where only three types of nonlinear interaction should be allowed for, the two-dimensional problems should include 24 types of nonlinear interaction. In the case of geometrical nonlinearity alone, a preliminary analysis of the nonlinear equations is carried out. Second-order equations are derived. The second approximation includes the second harmonics of the wave itself and its attenuating amplitude and is nonlinearly dependent on the initial amplitude of the Rayleigh wave and linearly increasing with the distance traveled by the wave

О косом ударе жестким телом, имеющим плоскую границу, по нелинейному упругому полупространству

О косом ударе жестким телом, имеющим плоскую границу, по нелинейному упругому полупространству

Surface waves and surface stability for a pre-stretched, unconstrained, non-linearly elastic half-space

An unconstrained, non-linearly elastic, semi-infinite solid is maintained in a state of large static plane strain. A power–law relation between the pre-stretches is assumed and it is shown that this assumption is well motivated physically and is likely to describe the state of pre-stretch for a wide class of materials. A general class of strain-energy functions consistent with this assumption is derived. For this class of materials, the secular equation for incremental surface waves and the bifurcation condition for surface instability are shown to reduce to an equation involving only ordinary derivatives of the strain-energy equation. A compressible neo-Hookean material is considered as an example and it is found that finite compressibility has little quantitative effect on the speed of a surface wave and on the critical ratio of compression for surface instability.

On an Incompressible Nonlinearly Elastic Half-Space under a Compressive Point Load

The problem of determining the deformation produced in an elastic half space due to a point force normal to the undeformed boundary has been a subject of considerable interest in the linear theory of elas-ticity and in a variety of engineering and geotechnical applications for well over a century. Recently, this inherently nonlinear problem has been posed within the framework of the nonlinear theory of elasticity for tensile point loads and results stemming from the exact governing equations have been obtained aymptotically In this paper, we consider a compressive point load and address this core problem within the context of incompressible finite elasticity. An asymptotic analysis of the exact governing equations is carried out, and asymptotic tests providing a simple way to determine if an isotropic homogeneous hyperelastic material is capable of sustaining a compressive point load are developed. The results are then applied to several particular constitutive models for incompressible nonlinearly elastic materials, and contrasts with the tensile load problem are made.

On an Incompressible Nonlinearly Elastic Half-Space under a Compressive Point Load

On an Incompressible Nonlinearly Elastic Half-Space under a Compressive Point Load

Methods for solving one-dimensional boundary-value problems in a nonlinear elastic medium

One-dimensional problems connected with a mechanical exposure to the boundary of a nonlinear elastic half-space, which leads to a constantly accelerated motion of this boundary, are considered. The value ɛ, equal to the square root of the ratio between the velocity of the boundary motion and the velocity of longitudinal wave propagation in a linear elastic medium, is used as the value characterizing the intensity of this exposure. It is shown that as a result of such an exposure shock waves of small or finite amplitude may propagate in the half-space. The asymptotic matching principle and the ray method are used as methods of solution. The merits and demerits of each method are analyzed. It has been inferred that the matched asymptotic method can be applied to waves of small amplitude, and the ray method is usable when investigating the propagation of shock waves not only of small amplitude, but of finite amplitude as well if the time of a consideration of the shock process is not long. The results obtained by the two methods for shock waves of small amplitude are in close agreement. It has been demonstrated that the ray method is adaptable for solving more intricate boundary-value problems resulting in the propagation of several shock waves of finite amplitude at a time. The problem connected with the constantly accelerated motion of one of the boundaries of an initially deformed elastic layer provides an example.

Long non-linear strain waves in layered elastic half-space

Elastic strain wave propagation in a thin non-linearly elastic layer superimposed on non-linearly elastic half-space is studied. Two layer-half-space contact models are considered. It is found that the Benjamin-Ono equation can be derived for description of longitudinal non-linear strain waves, when the contact between the layer and the half-space is provided only by means of the normal stresses and displacements. When the full contact problem is considered the more complicated integro-differential equation is derived. It is found that long non-linear periodical and solitary strain waves as well as envelope waves may exist in the first case, while only envelope wave solutions are found to the full contact problem. Linear wave analysis shows that the Korteveg-de Vries equation, often usable, is unlikely to be an adequate model for longitudinal surface strain waves. Application of the results obtained to experiments devoted to superconductivity threshold control in thin metal films as well as to generation of acoustic solitons in layered half-space is discussed.

Indentation of a power law creeping solid

The aim of this paper is to establish a rigorous theoretical basis for interpreting the results of hardness tests on creeping specimens. We investigate the deformation of a creeping half-space with uniaxial stress-strain behaviour ⋵ = ⋵ 0 (σ/σ 0 ) m , which is indented by a rigid punch. Both axisymmetric and plane indenters are considered. The shape of the punch is described by a general expression which includes most indenter profiles of practical importance. Two methods are used to solve the problem. The main results are found using a transformation method suggested by R. Hill. It is shown that the creep indentation problem may be reduced to a form which is independent of the geometry of the punch, and depends only on the material properties through m . The reduced problem consists of a nonlinear elastic half-space, which is indented to a unit depth by a rigid flat punch of unit radius (in the axisymmetric case), or unit semi-width (in the plane case). Exact solutions are given for m = 1 and m = ∞. For m between these two limits, the reduced problem has been solved using the finite element method. The results enable the load on the indenter and the contact radius to be calculated in terms of the indentation depth and rate of penetration. The stress, strain and displacement fields in the half-space may also be deduced. The accuracy of the solution is demonstrated by comparing the results with full-field finite element calculations. The predictions of the theory are shown to be consistent with experimental observations of hardness tests on creeping materials reported in the literature.

The refraction of a shear wave in a non-linearly elastic awd elastoplastic half-space

Dynamic equations describing antiplane deformation in a non-linearly elastic medium are studied. Selfsimilar solutions are analysed for the case when the displacement rates and stresses depend on two variables x = x 1 — ct, y = x 2. in the limiting case, when the τ - γ-plot for a non-linearly elastic material is the same as that for a perfect elastoplastic material, a solution is constructed for the problem of refraction of plane-polarized plane waves of pure shear in a non-linearly elastic half-space. The results obtained are compared with the solution constructed earlier /1/ in which the system of Prandtl-Reuss equations was used to study the refraction of pure shear waves in an elastoplastic half-space. Selfsimilar problems in which the displacement rates and stresses depend only on the ratio of the coordinates, were studied in /2–7/.

The dynamic contact stresses caused by the impact of a nonlinear elastic half-space by an axisymmetrical projectile

A numerical procedure is developed for the treatment of impact-contact problems in which mixed boundary conditions must be satisfied over a time-dependent region which is not known in advance but should be determined from the solution. This procedure involves an iterative process which is continued until the correct solution is obtained. This solution is determined by the equations of motion, the moving boundary conditions, the relation between the mass of the projectile and the reaction of the half-space, and the requirements that the contact normal stress is compressive and that no interpenetration occurs outside the contact region. The reliability of the method is checked in the dynamic indentation problem of a linearly elastic half-space by a conical wedge for which an analytical solution is known, and excellent agreement is obtained. The proposed method is applied to the problem of a rigid axisymmetrical projectile which impacts a nonlinearly elastic compressible half-space, and the problem of an impacted linear half-space is obtained as a special case.

Moving force on a nonlinearly elastic material

The problem of a nonlinearly elastic half-space and slab subjected to a steadily moving line force on the boundary is investigated. The conditions for the hyperbolicity of the governing equations are derived and shown to be dependent on the deformation gradients, particle velocities and the force speed. A numerical method of solution is applied whose stability criteria are derived and is shown to yield satisfactory accurate solutions. The conditions for the existence of generalized simple waves in the half-space are obtained and applied to produce primary dilatational and primary transverse simple waves. It is shown that an apparent anisotropy is introduced by the travelling force in the sense that the nonlinear response and velocities generated by a positive tangential force is different from that of a negative one. For a moving force over the surface of a slab, it is shown that waves propagate in opposite directions, from a point within the slab, at different velocities.

Two-dimensional wave propagation in a nonlinear elastic half-space

Abstract Finite amplitude wave propagation in an elastic, isotropic half-space is investigated. A numerical scheme that was previously developed and shown to yield satisfactory accurate results whenever smooth solutions occur is modified here for the cases in which steep solutions are obtained. The stability analysis of the proposed numerical procedure is carried out, and the stability criteria are given in terms of the spectral radii of the matrices involved in the equations of motion. The hyperbolicity conditions of the equations of motion are derived and shown to impose restrictions on the possible values of displacement gradients so that the range of variation of the strength of the applied load is limited. As a first chek of the accuracy of the numerical results, a propagating shock wave is produced numerically and compared with the analytical solution. In a second check, propagating circularly polarized waves are numerically simulated and compared with the corresponding analytical solution. In each case good agreement is obtained. For the “quadratic material” adopted in this paper, it is shown that a compressive normal line force yields propagating pulses having larger amplitudes, broader widths and larger arrival times, as compared with those caused by a tensile one. The linear response is also shown for comparison.