Uniqueness in the boundary-value problems for the static equilibrium equations of a mixture of two elastic solids occupying an unbounded domain

In this paper the Authors study the uniqueness of the solutions to the most important boundary-value problems for the static equilibrium equations of a mixture of two linear elastic solids. Some uniqueness theorems concerning the mixed boundary-value problem and the displacement problem are proved for unbounded domains. If the mixture is anisotropic, mild assumptions are imposed on the displacement fields at infinity. If the mixture is isotropic, uniqueness is proved for exterior domains without artificial restrictions upon the behavior of the unknown fields at infinity.     

Energy bounds for a mixture of two linear elastic solids occupying a semi-infinite cylinder

Summary The aim of this paper is to establish some forms of the Saint-Venant principle for a mixture of two linear elastic solids occupying a semi-infinite prismatic cylinder. We examine the behaviour of the energy for both static and dynamical problems. Under mild assumptions on the asymptotic behaviour of the unknown fields at infinity, we show that in the static case the elastic energy of the portion of the cylinder beyond a distancex ₃ from the loaded region decays exponentially withx ₃. For the dynamical problem we estimate through the data the total energy stored in that part of the cylinder whose minimum distance from the loaded end isx ₃; these estimates, which are based on the assumption that the initial total energy is finite, depend uponx ₃ but do not depend upon the timet.     

Thermostatics of an infinite mixture of two elastic solids and a spherical thermal inclusion problem

This work is concerned with the linear theory of a binary mixture of two elastic solids. With the help of displacement potentials, two differential equations that govern the displacement of each constituent are obtained. Then, more generalised forms of Betti's reciprocal theorem and Maysel's formula for the mixture are found. Finally, a solution of spherical thermal inclusion problem in an infinite mixture of two elastic solids is obtained by using generalised Maysel's formula and by direct integration of the governing differential equations.     

The Galerkin vector solution for a mixture of two elastic solids and Boussinesq problem

This work is concerned with the linear theory of a binary mixture of two elastic solids. Using the constitutive equations for the mixture which are given by Green and Steel, the displacement equations in the case of isotropic mixture of two elastic solids are derived. By use of the Galerkin vector solution, the displacement vector of each component in the mixture is obtained. Finally, an equilibrium solution for the Boussinesq problem of the mixture of two elastic solids in an infinite half-space is examined.     

Acceleration waves through an isotropic mixture of two non-linear elastic solids

Summary In this paper we study the propagation of acceleration waves through an isotropic isothermal mixture of two non-linear elastic solids. After giving the constitutive equations of the mixture, we calculate the possible normal speeds of propagation. Then we state that, in general, it is possible to distinguish between longitudinal and transverse acceleration waves. Finally, we establish the evolution law of the discontinuities along the normal trajectories associated with the wave front.Work performed under the auspices of C.N.R. (GNFM) and supported by M.P.I. of Italy.     

Acceleration waves through a mixture of two non-linear elastic solids

Summary In this paper we study the propagation of acceleration waves through an anisotropic isothermal mixture of two non-linear elastic solids. First of all, we show that, under suitable hypotheses on the constitutive equations, there exist twelve real normal speeds of propagation; then by means of Nariboli method, we state the evolution law of the discontinuities along the rays associated with the wave front.Work performed under the auspices of C.N.R. (GNFM) and supported by M.P.I. of Italy.     

On a boundary value problem for an elastic dielectric half-plane

Summary Using Papkovitchtype representations for the displacement and polarization vectors and Fourier transforms, a general solution to boundary value problems of a half plane subjected to an arbitrary charge distribution is constructed within Mindlin's theory of elastic dielectrics. Explicit expressions for various mechanical and electric potentials are obtained for a point charge located on the boundary of the elastic dielectric half-space.

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Über ein Randwertproblem für eine elastische, dielektrische Halbebene

Zusammenfassung Auf der Basis von Mindlins Theorie des elastischen Dielektrikums und mit Benutzung der Papkovitch-Darstellungen für Verschiebungs- und Polarisationsvektoren wird mit Hilfe von Fourier-Transformationen eine allgemeine Lösung für Randwertprobleme einer Halbebene mit einer beliebigen Ladungsverteilung gefunden. Es werden explizite Ausdrücke für verschiedene mechanische und elektrische Potentiale für eine Punktladung an der Berandung des elastischen, dielektrischen Halbraums erhalten.

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With 2 Figures     

On the spatial decay for the dynamical problem of thermo-microstretch elastic solids

This paper derives spatial decay bounds for a dynamical problem of thermo-microstretch elasticity defined on a semi-infinite cylindrical region. Previous results for isothermal elastodynamics and the parabolic heat equation lead us to suspect that the solution of the problem should tend to zero faster than a decaying exponential of the distance from the finite end of the cylinder. We prove that an energy expression is actually bounded above by a decaying exponential of a quadratic polynomial of the distance.     

Wave curves for the riemann problem of plane waves in isotropic elastic solids

A modified Riemann problem in which the initial and boundary conditions are constants is considered for plane waves in a half space occupied by an elastic solid. The governing quasilinear differential equations form a system of hyperbolic conservation laws which possesses three wave speeds c₁ ≥ c₂ ≥ c₃. The system is genuinely nonlinear with respect to c₁ and c₃ and linearly degenerate with respect to c₂. Thus it is sufficient to study a two-wave-speed system with c₁ and c₃. Wave curves for simple waves and shock waves are used to construct the solution. Second-order hyperelastic materials which contain four material constants are considered and the solution in the form of wave curves is obtained for all possible combinations of initial and boundary conditions. With a proper nondimensionalization, the wave curves depend only on one material parameter k. The solutions are thermodynamically correct because entropy effects do not come into the picture until the third-order terms in stresses are included in the constitutive laws. The two-wave-speed system has one umbilic point at which c₁ = c₃ and hence the system is not totally hyperbolic (or not strictly hyperbolic). Several interesting and unexpected results are obtained due to the existence of the umbilic point. In one example, we find that a shock wave satisfies the Lax stability condition for a V₁ shock as well as a V₃, shock. In another, a shock wave which involves only one stress component does not satisfy the Lax stability condition for either a V₁ shock or a V₃ shock. However, it satisfies the Lax stability condition if we consider it under the context of a one-wave-speed system. Finally we consider the effects on the solution when the third-order terms are included. We show that although the entropy affects the shock wave solution, it does not appear in the simple wave solution until the fourth-order terms are included. With the third-order terms, there may be as many as three umbilic points, one of which may be an umbilic line.     

Comparative solutions for a punch problem of an elastic sphere encapsulated in an elastic shell

One of the major causes of mechanical damage incurred in agricultural commodities is attributed to the frequent impacts they receive in harvesting and handling. In agricultural operations, in general, only the local contact phenomenon is considered and the effect of wave propagation is ignored. The punch problem is a special case in the class of contact problems that is of particular practical interest in the impact loading encountered in fruit handling and harvesting.Two potential methods are proposed for analysing the deformation of an elastic sphere encapsulated in an elastic shell and subjected to punch loading. Numerical evaluation of both models showed that the results are comparable. Although more examples related to different geometries and loadings would be required to substantiate the data, the results provide a useful tool for the selected specific geometry which is very common in agriculture. Because good agreement was obtained between the two methods, the choice of which to implement should be made according to the specific problem in question. The Boussinesq method is a more rigorous mathematical approach and, as such, offers a better insight into the actual behaviour of the domain under given boundary conditions. Its utilization is limited to well defined geometrics because of the complexity involved. The finite element method is capable of handling irregular shapes but requires large computer memory for an exact solution. Both methods have been successfully implemented in a practical agricultural problem in an attempt to decrease the mechanical damage encountered during mechanized fruit harvesting.     

In-plane problem for wave propagation through elastic solids with a periodic array of cracks

Summary In the context of wave propagation in damaged (elastic) solids, an analytical method previously introduced for scalar problems, is now applied to study the (vector) problem for normal penetration of a longitudinal plane wave into a periodic array of collinear cracks. Reduced the problem to an integral equation holding over the openings, an approximation of one-mode type leads to analytical solutions and then to explicit representations for the wave fields and the scattering parameters. Some graphs will finally compare our results with the numerical ones by other authors.     

Axisymmetric contact problem for an elastic layer and an elastic foundation

This paper is concerned with the axisymmetric problem of an elastic layer lying on a semi-infinite foundation. The layer is pressed against the foundation by a uniform clamping pressure applied over its entire surface and a uniform vertical body force due to the effect of gravity. In addition, an axisymmetric vertical line load is applied to the layer. It is assumed that the contact between the layer and the foundation is frictionless and that only compresive normal tractions can be transmitted through the interface. The contact along the interface will be continuous if the value of the line load is less than a critical value. However, interface separation takes place if it exceeds this critical value. The problem is formulated and solved for the cases of tensile and compressive line loads. Numerical results for contact stress distributions are given for different material combinations.     

Determination of the effective mode-I toughness of a sinusoidal interface between two elastic solids

A finite element model of crack propagation along a sinusoidal interface with amplitude A and wavelength λ between identical elastic materials is presented. Interface decohesion is modeled with the Xu and Needleman (J Mech Phys Solid 42(9):1397, 1994) cohesive traction–separation law. Ancillary calculations using linear elastic fracture mechanics theory were used to explain some aspects of stable and unstable crack growth that could not be directly attained from the cohesive model. For small aspect ratios of the sinusoidal interface (A/λ ≤ 0.25), we have used the analytical Cotterell–Rice (Intl J Fract 16:155–169, 1980) approximation leading to a closed-form expression of the effective toughness, K _Ic , given by where is the work of separation, E is Young’s modulus, and ν is Poisson’s ratio. For A/λ > 0.25, both the cohesive zone model and numerical J-integral estimates of crack tip stress intensity factors suggest the following linear relationship: Parametric studies show that the length of the cohesive zone does not significantly influence K _Ic , although it strongly influences the behavior of the crack between the initiation of stable crack growth and the onset of unstable fracture. An erratum to this article can be found at     

Axisymmetric contact problem for a frictionless elastic layer indented by an elastic cylinder

This paper is concerned with the elastostatic contact problem of a semi-infinite cylinder compressed against a layer lying on a rigid foundation. It is assumed that all the contacting surfaces are frictionless and that only compressive normal tractions can be transmitted through the interfaces. Upon loading the contact along the layer-foundation interface shrinks to a circular area whose radius is unknown. The analysis leads to a system of singular integral equations of the second kind. The integral equations are solved numerically and the contact pressures, extent of the contact area between the layer and the foundation, and the stress intensity factor round the edge of the cylinder are calculated for various material pairs.     

A problem of Reissner-Sagoci type for an elastic cylinder embedded in an elastic half-space

The problem considered is that of the torsion of an elastic cylinder which is embedded in an elastic half-space of different rigidity modulus. It is assumed that there is perfect bonding at the common cylindrical surface and also that the torque is applied to the cylinder through a rigid disk bonded to its flat surface. The problem is reduced, by means of the use of integral transforms and the theory of dual integral equations to that of solving a Fredholm integral equation of the second kind. The results obtained by solving this equation are exhibited graphically in Fig. 2.