On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials I: Plane Strain Deformations

In three recent papers [6–8], the present authors show that both plane strain and axially symmetric deformations of perfectly elastic incompressible Varga materials admit certain first integrals, which means that solutions for finite elastic deformations can be determined from a second order partial differential equation, rather than a fourth order one. For plane strain deformations there are three such integrals, while for axially symmetric deformations there are two. The purpose of the present papers is to present the general equations for small deformations which are superimposed upon a large deformation, which is assumed to satisfy one of the previously obtained first integrals. The governing partial differential equations for the small superimposed deformations are linear but highly nonhomogeneous, and we present here the precise structure of these equations in terms of a second-order linear differential operator D², which is first defined by examining solutions of the known integrals. The results obtained are illustrated with reference to a number of specific large deformations which are known solutions of the first integrals. For deformations of limited magnitude, the Varga strain-energy function has been established as a reasonable prototype for both natural rubber vulcanizates and styrene-butadiene vulcanizates. Plane strain deformations are examined in this present part while axially symmetric deformations are considered in Part II [16]. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials II: Axially Symmetric Deformations

In Part I of this article, we have formulated the general structure of the equations governing small plane strain deformations which are superimposed upon a known large plane strain deformation for the perfectly elastic incompressible 'modified' Varga material, and assuming only that the initial large plane deformation is a known solution of one of three first integrals previously derived by the authors. For axially summetric deformations there are only two such first integrals, one of which applies only to the single term Varga strain-energy function, and we give here the corresponding general equations for small superimposed deformations. As an illustration, a partial analysis for the case of small deformations superimposed upon the eversion of a thick spherical shell is examined. The Varga strain-energy functions are known to apply to both natural and synthetic rubber, provided the magnitude of the deformation is restricted. Their behaviour in both simple tension and equibiaxial tension, and in comparison to experimental data, is shown graphically in Part I of this paper [1]. This revised version was published online in August 2006 with corrections to the Cover Date.     

Finite-Amplitude Inhomogeneous Plane Waves of Exponential Type in Incompressible Elastic Materials

It is proved that elliptically polarized finite-amplitude inhomogeneous plane waves may not propagate in an elastic material subject to the constraint of incompressibility. The waves considered are harmonic in time and exponentially attenuated in a direction distinct from the direction of propagation. The result holds whether the material is stress-free or homogeneously deformed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Propagation of the Energy of Nonlinearly Elastic Plane Waves

The propagation of the energy of nonlinearly elastic plane waves in a Murnaghan material is simulated on a computer. The velocity of energy propagation is found in an explicit form. A procedure of determining the critical values of the time and space coordinates for the given material is described. The resultant plots are discussed and analyzed     

Waves in Constrained Linear Elastic Materials

This paper deals with the propagation of acceleration waves in constrained linear elastic materials, within the framework of the so-called linearized finite theory of elasticity, as defined by Hoger and Johnson in [12, 13]. In this theory, the constitutive equations are obtained by linearization of the corresponding finite constitutive equations with respect to the displacement gradient and significantly differ from those of the classical linear theory of elasticity. First, following the same procedure used for the constitutive equations, the amplitude condition for a general constraint is obtained. Explicit results for the amplitude condition for incompressible and inextensible materials are also given and compared with those of the classical linear theory of elasticity. In particular, it is shown that for the constraint of incompressibility the classical linear elasticity provides an amplitude condition that, coincidently, is correct, while for the constraint of inextensibility the disagreement is first order in the displacement gradient. Then, the propagation condition for the constraints of incompressibility and inextensibility is studied. For incompressible materials the propagation condition is solved and explicit values for the squares of the speeds of propagation are obtained. For inextensible materials the propagation condition is solved for plane acceleration waves propagating into a homogeneously strained material. For both constraints, it is shown that the squares of the speeds of propagation depend by terms that are first order in the displacement gradient, while in classical linear elasticity they are constant. This revised version was published online in July 2006 with corrections to the Cover Date.     

Evolution Equations for Plane Cubically Nonlinear Elastic Waves

Consideration is given to the nonlinear theory of elastic waves with cubic nonlinearity. This nonlinearity is separated out, and the interaction of four harmonic waves is studied. The method of slowly varying amplitudes is used. The shortened and evolution equations, the first integrals of these equations (Manley–Rowe relations), and energy balance law for a set of four interacting waves (quadruplet) are derived. The interaction of waves is described using the wavefront reversal scheme     

On Non-Symmetric Deformations of an Incompressible Nonlinearly Elastic Isotropic Sphere

A class of non-symmetric deformations of a neo-Hookean incompressible nonlinearly elastic sphere are investigated. It is found via the semi-inverse method that, to satisfy the governing three-dimensional equations of equilibrium and the incompressibility constraint, only three special cases of the class of deformation fields are possible. One of these is the trivial solution, one the solution describing radially symmetric deformation, and the other a (non-symmetric, non-homogeneous) deformation describing inflation and stretching. The implications of these results for cavitation phenomena are also discussed. In the course of this work, we also present explicitly the spherical polar coordinate form of the equilibrium equations for the nominal stress tensor for a general hyperelastic solid. These are more complicated than their counterparts for Cauchy stresses due to the mixed bases (both reference and deformed) associated with the nominal (or Piola-Kirchhoff) stress tensor, but more useful in considering general deformation fields. This revised version was published online in August 2006 with corrections to the Cover Date.     

Loss of Ellipticity in Plane Deformation of a Simple Directionally Reinforced Incompressible Nonlinearly Elastic Solid

Change of type in the governing equations of equilibrium is examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for strength of reinforcement. Plane deformations interpreted in terms of both local and global plane strain are considered. Loss of ordinary ellipticity is found to occur for sufficiently large strength of reinforcement under sufficiently severe deformation which necessarily involves contraction in the reinforcing direction. Loss of ellipticity in local plane strain is easily characterized, and its incipient breakdown is associated with the possible emergence of surfaces of weak discontinuity with orientation normals in the reinforcing direction. Loss of ellipticity in global plane strain is given a two-dimensional manifold characterization in a space involving 2 deformation parameters and the strength of reinforcing parameter. Orientation normals for the associated surfaces of weak discontinuity at incipient breakdown do not in general conform to the reinforcing direction. This revised version was published online in August 2006 with corrections to the Cover Date.     

Cubically Nonlinear Versus Quadratically Nonlinear Elastic Waves: Main Wave Effects

This paper is a review of studies on quadratically and cubically nonlinear elastic waves in elastic materials. The main methods for analysis of the wave equations are demonstrated. The main wave phenomena are described. The disproportion between the achievements in the analyses of quadratically and cubically nonlinear waves is pointed out—cubically nonlinear waves have been studied much less     

Equivalence Theorems on the Propagation of Small-Amplitude Waves in Prestressed Linearly Elastic Materials with Internal Constraints

By extending the procedure of linearization for constrained elastic materials in the papers by Marlow and Chadwick et al., we set up a linearized theory of constrained materials with initial stress (not necessarily based on a nonlinear theory). The conditions of propagation are characterized for small-displacement waves that may be either of discontinuity type of any given order or, in the homogeneous case, plane progressive. We see that, just as in the unconstrained case, the laws of propagation of discontinuity waves are the same as those of progressive waves. Waves are classified as mixed, kinematic, or ghost. Then we prove that the analogues of Truesdell"s two equivalence theorems on wave propagation in finite elasticity hold for each type of wave. This revised version was published online in August 2006 with corrections to the Cover Date.     

Nonlinear Waves and the Origin of Bubbles in Fluidized Beds

Early experiments in the mid-1940s established two different regimes of behavior of fluidized systems. These are broadly classified into systems that exhibit massive phase segregation, leading to particle-free regions called bubbles, and those that do not. Explaining the origin of bubbles and of these two regimes has represented both a technological and scientific challenge since then. The late 1960s through the 1970s saw a series of illuminating experiments that established many features of the flow regimes and their characteristics through both flow visualizations and quantitative measurements. Recent numerical and theoretical work has come close the resolving the problem. This paper represents the written version of the talk given at the Symposium in honor of Leen van Wijngaarden's retirement. In it, I review the history of progress on the problem in two giant 25-year steps.     

Plane Deformations in Incompressible Nonlinear Elasticity

The substantially general class of plane deformation fields, whose only restriction requires that the angular deformation not vary radially, is considered in the context of isotropic incompressible nonlinear elasticity. Analysis to determine the types of deformations possible, that is, solutions of the governing systems of nonlinear partial differential equations and constraint of incompressibility, is developed in general. The Mooney-Rivlin material model is then considered as an example and all possible solutions to the equations of equilibrium are determined. One of these is interpreted in the context of nonradially symmetric cavitation, i.e., deformation of an intact cylinder to one with a double-cylindrical cavity. Results for general incompressible hyperelastic materials are then discussed. The novel approach taken here requires the derivation and use of a material formulation of the governing equations; the traditional approach employing a spatial formulation in which the governing equations hold on an unknown region of space is not conducive to the study of deformation fields containing more than one independent variable. The derivation of the cylindrical polar coordinate form of the equilibrium equations for the nominal stress tensor (material formulation) for a general hyperelastic solid and a fully arbitrary cylindrical deformation field is also given. This revised version was published online in August 2006 with corrections to the Cover Date.     

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Plane-Strain State

A method is proposed for deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. The method is based on a rigorous approach of nonlinear continuum mechanics. Nonlinearity is introduced by means of metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. For a configuration (state) dependent on the radial and angle coordinates and independent of the axial coordinate, quadratically nonlinear wave equations for stresses are derived and stress-strain relationships are established. Four ways of introducing physical and geometrical nonlinearities to the wave equations are analyzed. For one of the ways, the nonlinear wave equations are written explicitly__________Translated from Prikladnaya Mekhanika,Vol. 41, No. 5, pp. 40–51, May 2005.     

Kink Surfaces in a Directionally Reinforced neo-Hookean Material under Plane Deformation: I. Mechanical Equilibrium

Stationary kinks (elastostatic shocks) are examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for the additional fiber reinforcing stiffness. Previous work has shown that such a transversely isotropic material can lose ellipticity in plane deformation if the reinforcing is sufficiently large and the fiber direction is sufficiently compressed. Here we show that the same reinforcing levels can give rise to piecewise smooth plane deformations separated by a plane stationary kink. Attention is restricted to deformations in which, on one side of the kink, the load axis is aligned with the fiber axis. Then the fiber stretch on this side of the kink is a natural load parameter. It is found that such a deformation can support a planar kink for a certain range of this load parameter. This range is dependent on the reinforcing parameter, and can even involve fiber extension if the reinforcing is sufficiently large. The set of all deformation states on the other side of the kink is precisely characterized in terms of a one-parameter family of (kink orientation, kink strength)-pairs. The results are interpreted in terms of the associated fiber alignment discontinuity and fiber stretch discontinuity.     

压电压磁复合材料中裂纹对反平面简谐弹性波的散射问题

分析了压电压磁复合材料中裂纹对反平面简谐弹性波的散射问题。利用傅立叶变换,使问题的求解转换为对一对以裂纹表面上的位移差为未知变量的对偶积分方程的求解。为了求解对偶积分方程,把裂纹面上的位移差展开为雅可比多项式形式,进而得到了裂纹长度、入射波波速及入射波频率对裂纹应力强度因子的影响。从数值结果可以看出,压电压磁复合材料中可导通裂纹的反平面问题的动应力奇异性与一般弹性材料中的反平面断裂问题动应力奇异性相同。     

Solitary Waves in Nonlinear Beam Equations: Stability,Fission and Fusion

We continue work by the second author and co-workers onsolitary wave solutions of nonlinear beam equations and their stabilityand interaction properties. The equations are partial differentialequations that are fourth-order in space and second-order in time.First, we highlight similarities between the intricate structure ofsolitary wave solutions for two different nonlinearities; apiecewise-linear term versus an exponential approximation to thisnonlinearity which was shown in earlier work to possess remarkablystable solitary waves. Second, we compare two different numericalmethods for solving the time dependent problem. One uses a fixed griddiscretization and the other a moving mesh method. We use these methodsto shed light on the nonlinear dynamics of the solitary waves. Earlywork has reported how even quite complex solitary waves appear stable,and that stable waves appear to interact like solitons. Here we show twofurther effects. The first effect is that large complex waves can, as aresult of roundoff error, spontaneously decompose into two simplerwaves, a process we call fission. The second is the fusion of twostable waves into another plus a small amount of radiation.     

Transformation of Long Nonlinear Waves in a Two-Layer Viscous Fluid Between a Gently Sloping Lid and Bottom

Dynamics of three-dimensional disturbances of the interface between two fluid layers of different densities is considered analytically and numerically. An evolutionary integrodifferential equation is derived, which takes into account long-wave contributions of inertia of the layers and surface tension, small but finite amplitude of disturbances of the interface between two incompressible immiscible fluids, gentle slopes of the lid and bottom, and nonstationary shear stresses at all boundaries. Numerical solutions of this model equation for several (most typical) nonlinear problems of transformation of two- and three-dimensional waves are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 45–57, November–December, 2005.     

Plane steady-state incompressible fluid flow through a porous medium with a nonlinear resistance laws in the presence of a sink

Plane nonlinear fluid flows through a porous medium which simulate a sink located at the same distance from the roof and floor of the stratum for two nonlinear flow laws are constructed. The following flow laws are taken: a power law and a law of special form reducing to analytic functions in the hodograph plane.     

On the Resonant Generation of Weakly Nonlinear Stokes Waves in Regions with Fast Varying Topography and Free Surface Current

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