A plate and more generally a shell is a special three-dimensional body whose boundary surface has special features. Although we defer defining a shell-like body in precise terms until Sect. 4, for the purpose of these preliminary remarks consider a surface—called a reference surface—and imagine material filaments from above and below surrounding the surface along the normal at each point of the reference surface. Suppose further that the bounding surfaces formed by the end points of the material filaments are equidistant from the reference surface. Such a three-dimensional body is called a shell if the dimension of the body along the normals, called the thickness, is small. A shell is said to be thin if its thickness is much smaller than a certain characteristic length of the reference surface, e.g., the minimum radius of the curvature of the reference surface for initially curved shells.2

The Theory of Shells and Plates

Two main questions are at the center of this paper. The first one concerns the choice of a rheological model in the frequency range of transient elastography, sonoelasticity or NMR elastography for soft solids (20-1000 Hz). Transient elastography experiments based on plane shear waves that propagate in an Agar-gelatin phantom or in bovine muscles enable one to quantify their viscoelastic properties. The comparison of these experimental results to the prediction of the two simplest rheological models indicate clearly that Voigt's model is the better. The second question studied in the paper deals with the feasibility of quantitative viscosity mapping using inverse problem algorithm. In the ideal situation where plane shear waves propagate in a sample, a simple inverse problem based on the Helmholtz equation correctly retrieves both elasticity and viscosity. In a more realistic situation with nonplane shear waves, this simple approach fails. Nevertheless, it is shown that quantitative viscosity mapping is still possible if one uses an appropriate inverse problem that fully takes into account diffraction in solids.

Measurement of viscoelastic properties of homogeneous soft solid using transient elastography: An inverse problem approach

A summary of the recent developments in the analysis of laminated beams and plates with an emphasis on vibrations and wave propagations in presented. First, a review of the recent studies on the free-vibration analysis of symmetrically laminated plates is given. These studies have been conducted for various geometric shapes and edge conditions. Both analytical (closed-form, Galerkin, Rayleigh-Ritz) and numerical methods have been used. Because of the importance of unsymmetrically laminated structural components in many applications, a detailed review of the various developments in the analysis of unsymmetrical ly laminated beams and plates also is given. A survey of the nonlinear vibrations of the perfect and geometrically laminated plates is presented next. It is seen that due to the bending-stretching coupling, the nonlinear behavior of the unsymmetrically laminated perfect and imperfect plates, depending upon the boundary conditions, may be hardening or softening type. Similar behavior also is observed for imperfect isotropic and laminated plates. Lastly, the developments in studying the wave propagation in laminated materials are reviewed. It is seen that a significant effort has been spent on developing appropriate continuum theories for modeling the composite materials. Some recent studies on the linear and nonlinear transient response of laminated materials also are described.

Recent Advances in Analysis of Laminated Beams and Plates, Part II: Vibrations and Wave Propagation

Several methods have been proposed to estimate the viscoelastic properties of soft biological tissues using forced low-frequency vibrations (10-500 Hz). Those methods are based on the measurement of phase velocity of the shear waves (approximately 5 m/s). It is shown in this article that the measurements of velocity as well as attenuation are subjected to biases. These biases are related to reflected waves created at boundaries, to the nonnegligible size of the piston source which causes diffraction effects and to the influence of a low-frequency compressional wave. Indeed, a theoretical analysis of the field radiated by a point source explains how mechanical vibrations of a piston generate a shear wave with a longitudinal component and how this component can interfere with a low-frequency compressional wave. However, by using a low-frequency transient excitation, these biases can be avoided. Then the precise numerical values of elasticity and viscosity can be deduced. Experiments in phantoms and beef muscles are shown. Moreover, a relative hardness imaging of a phantom composed of two media with different elasticities is presented.

A solution to diffraction biases in sonoelasticity: The acoustic impulse technique

The reciprocity theorem has been used for over 100 years to establish interesting and useful relations between different loading states of a body. This book discusses current and novel uses of reciprocity relations for the determination of elastodynamic fields. The author, who is internationally distinguished for his contributions to theoretical and applied mechanics, presents a novel method to solve for wave fields, shedding new light on the use of reciprocity relations for dynamic fields in an elastic body. The material presented in the book is relevant to several fields in engineering and applied physics. Examples are ultrasonics for medical imaging and non-destructive evaluation, acoustic microscopy, seismology, exploratory geophysics, structural acoustics, and the response of structures to high-rate loads and the determination of material properties by ultrasonic techniques.

Reciprocity in Elastodynamics

Dynamics of elastic systems

The propagation of transient waves in an elastic half-space excited by a traveling normal point load is investigated. The load is suddenly applied and then it moves rectilinearly at a constant speed along the free surface. The displacements are computed for all points of the half-space as well as for all load speeds.

The disturbance is analyzed by using multi-integral transforms and an inversion scheme based on the well-known Cagniard technique. This reduces the displacements to single integral and algebraic contributions, each of which is identified as the disturbance behind a specific wave front. The same solution is valid for all load speeds, even though the wave front geometry varies greatly, depending on the speed of the load relative to the body wave speeds. Moreover, the surface displacements are obtained from the interior ones, but only after the Rayleigh waves are computed by a separate calculation. Then, by taking advantage of the form of the exact solution, wave front expansions and Rayleigh wave approximations are computed for all load speeds.

Several other analytical results are obtained for restricted values of the load speed. In particular, when it exceeds both of the body wave speeds the steady-state displacement field is separated from the transient one and reduced to algebraic form. Also, for the limit case of zero load speed a new representation of the interior displacements for Lamb's point load problem is displayed in terms of single integrals.

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Transient Excitation of an Elastic Half Space by a Point Load Traveling on the Surface

Plane-Stress Unloading Waves Emanating From a Suddenly Punched Hole in a Stretched Elastic Plate

Transient Compressional Waves in an Infinite Elastic Plate or Elastic Layer Overlying a Rigid Half-Space

: For some classes of anisotropic media and certain orientations of the axes of symmetry, plane-strain and torsional motions are possible in elastic plates. Using the linear equations of motion, the solutions to several time-dependent boundary value problems which generate such motions are derived, by means of multi-integral transform techniques. The problems considered are (i) sudden symmetric application of normal line loads on the surfaces of an infinite plate; (ii) sudden application of a normal displacement, together with zero shear stress, at the edge of a semi-infinite plate, whose lateral surfaces are stress-free; (iii) sudden application of a normal stress, together with zero lateral displacement, at the edge of a semi-infinite plate, whose lateral surfaces are stress-free and (iv) impulsive torsional loading of the surfaces of an infinite plate by means of concentrated torques. In problems (i), (ii), and (iii), far-field information is obtained by means of a low frequency, small wavenumber approximation, and numerical results are presented for ice, zinc, copper, and several isotropic media for comparison purposes. (Author)

Julius Miklowitz

Papers

Dynamics of elastic systems

Transient Excitation of an Elastic Half Space by a Point Load Traveling on the Surface

Plane-Stress Unloading Waves Emanating From a Suddenly Punched Hole in a Stretched Elastic Plate

Transient Compressional Waves in an Infinite Elastic Plate or Elastic Layer Overlying a Rigid Half-Space

Transient Elastic Waves in Anisotropic Plates