H. Cohen

Nonlinear axisymmetric waves in compressible hyperelastic rods: long finite amplitude waves

SummaryThis paper investigates nonlinear axisymmetric waves in compressible hyperelastic circular cylindrical rods. We consider first a compressible Mooney-Rivlin material to obtain exact governing equations. To further study the problem, we introduce the notion of long finite amplitude waves and derive the corresponding simplified model equations, which gives the framework for studying problems like wave-interactions arising through collision or reflection. The asymptotically valid far-field equation is consequently deduced from the simplified model equations. Then, using a strained-coordinate method, we obtain the second-order solitary wave solution. The result is not only of interest itself, but also provides a suitable initial condition for wave interaction problems. Finally, the results for a general hyperelastic rod are presented.

Constitutive equations for curved and twisted, initially stressed elastic rods

SummaryThe effects of geometry and material properties on the linear constitutive equations for curved and twisted, initially stressed elastic rods are investigated. Forms of the constitutive equations are given for special materials in undistorted and natural states, special cross sectional symmetries and for slightly curved and thin rods.ZusammenfassungDie Einflüsse der Geometrie, sowie der Materialeigenschaften auf die linearen Materialgleichungen für gekrümmte und verdrehte, vorgespannte elastische Stäbe, werden untersucht. Die Formen der Materialgleichungen werden angegeben für spezielle Materialien im unverzerrten und natürlichen Zustand, für spezielle Querschnittssymmetrien, sowie für schwach gekrümmte und dünne Stäbe.

Transient waves in inhomogeneous anisotropic elastic plates

SummaryThis paper considers the problem of transient wave propagation in elastic Cosserat plates that may be anisotropic, inhomogeneous, or of variable thickness. The methods of rays and of singular wave curves are combined to find and integrate the transport equations governing growth-decay behaviour of the extensional and bending wave modes to derive a common general formula involving the material parameters and wave geometry. For inhomogeneous isotropic plates of variable thickness, conditions for the uncoupling of the wave modes are obtained and some special cases are worked out in detail.

Transient waves in inhomogeneous elastic shells of variable thickness

SummaryThis paper considers the problem of transient wave propagation in Cosserat shells of variable thickness, the inhomogeneous material of which is linearly elastic and isotropic. We do not say that the shells are isotropic because varying thickness causes behaviour characteristic of anisotropy despite the materials being isotropic. The methods of rays and of singular wave curves are combined to find and integrate the transport equations governing growth-decay behaviour of the six possible wave modes. Conditions on material parameters, thickness variation, and wave geometry are obtained for various different uncouplings of the wave modes. Some special cases of propagation conditions and of decay equations are worked out in detail.

Homogeneity conditions for elastic membranes

SummaryThis paper deals with the concept of homogeneity within the framework of hyperelastic anisotropic membranes. A frame field, i.e. an orthonormal set of vectors lying in the tangent plane at each point of the membrane, is used to represent observers regarded as equivalent for comparing material response. The notions of homogeneity and unidirectional homogeneity are formulated in this setting and conditions required for a given strain energy function to define a homogeneous material are derived. The paper concludes with a discussion of certain additional features which illustrate the concepts and which arise out of special choices of the strain energy function.

On dissipation induced by phase transformations

SummaryMotions of hyperelastic materials involving surfaces of strain or stress discontinuities are generally dissipative in the sense that, in any portion of the body that is traversed by a moving singular surface, the rate of work of the external forces differs from the rate of storage of the total energy (the strain energy and the kinetic energy) by the rate of work done in moving the singular surface. Hence, the corresponding continuum theory is capable of modeling dissipative behavior associated with phase transformations which has potential applications in the design ofadaptive structures. The present work indicates that this dissipative behavior is characterized by a material function, called the driving-traction-response function, which is uniquely determined by the strain energy potential of the material. The driving-traction-response function vanishes identically if and only if the Piola-Kirchhoff stress-response function depends upon the deformation gradient linearly.

Waves in thermo-viscoelastic rods

SummaryThis paper deals with the propagation and decay of shock and acceleration waves in nonhomogeneous linear thermo-viscoelastic rods. The growth-decay laws governing the propagation of such waves is obtained. These clearly exhibit the effects of nonhomogeneity, viscous damping and thermal damping on wave amplitude. An example is treated which brings into play thermo-mechanical coupling in bending.

Geometrical Crystal Acoustics

Publisher Summary This chapter discusses geometrical crystal acoustics. Acceleration waves in an inhomogeneous and anisotropic two-dimensional elastic solid are studied using compatibility relations generalized from those of T. Y. Thomas. Ray directions and an integral expression for decay are obtained in the case of differentiable inhomogeneity of density and elastic coefficients and anisotropy. Decay is further given in the homogeneous case as an algebraic function of time. The chapter discusses an integral expression for the decay of an acceleration wave obtained without the assumptions of isotropy or homogeneity; only differentiable inhomogeneity of density and elastic coefficients is required. The chapter discusses the other specialization, homogeneity, for full seven-constant anisotropy, giving an algebraic expression for the decay of an acceleration wave over a time domain, the limit of that depends ray by ray on the direction of propagation.