Finite amplitude one-dimensional wave propagation in a thermoelastic half-space

The problem of finite wave propagation in a nonlinearly thermoelastic half-space is considered. The surface of the half-space is subjected to a time-dependent thermal and normal mechanical loading. The solution is obtained by a numerical procedure, which is shown to furnish accurate results, and linear dynamic thermoelastic problems are obtained as special cases. The accuracy of the results is checked by comparison with some known analytical solutions which can be obtained in some special cases of both the linear and the nonlinear problems. In those cases where the solution contains shocks, it is shown that the numerical results satisfy the necessary jumps conditions which need to hold across such discontinuities.     

On an inhomogeneous deformation of a generalized Neo-Hookean material

We study the inhomogeneous deformation of a wedge of an incompressible generalized power-law Neo-Hookean material. We find solutions which have a boundary layer structure, in the sense that adjacent to the boundary the solution is inhomogeneous, while in the core region the solution is homogeneous. It is found that such solutions have an associated pressure field that is bounded. Inhomogeneous solutions are also possible when the pressure varies logarithmically with the radial coordinate. We also establish explicit exact solutions for specific values of the parameter. The results reduce to the Neo-Hookean solution when the power law exponent is set to unity.     

Finite amplitude vibrations of an orthotropic circular plate with an isotropic core

The free and forced non-linear vibrations of a fixed orthotropic circular plate, with a concentric core of isotropic material, are studied. Existence of harmonic vibrations is assumed and thus the time variable is eliminated by a Ritz-Kantorovich method. Hence, the governing non-linear partial equations for the axisymmetric vibration of the composite circular plate are reduced to a set of ordinary differential equations which form a non-linear eigen-value problem. Solutions are obtained by utilizing the related initial-value problems in conjunction with Newton's integration method. The results reveal the effects of finite amplitude and anisotropy of materials upon the dynamic responses. Further, the method developed in this paper, which is used to solve the title problem, is one of some generality. It can be applied to many differential eigenvalue problems with piecewise continuous functions.     

Wave propagation in an unconsolidated granular material: A micro-mechanical approach

We provide a theoretical analysis to support the presence of both slow and fast compression waves in an unconsolidated, fully saturated, granular material. We derive the constitutive relation for such an aggregate based upon a micro-mechanics analysis. In doing this, we take in account the coupling between the solid particles and fluid. As a consequence of this coupling, the lubrication layer provides a connection between particles, both when they are separating and when they are compressing. The predictions of the speed and attenuation of the fast compression waves compare well with experimental data over the range of frequencies for which the nonlinear dissipation associated with the relative velocities between solid and fluid is negligible. Slow waves are also predicted without comparison, because of the absence of clear experimental data. Predictions of the speed and attenuation for the shear wave are also provided and show a good agreement with experimental data when surface roughness is taken into account.     

Dynamics of one-dimensional granular arrays with pre-compression

Nonlinear Dynamics - Bifurcations of periodic orbits and band zones of a one-dimensional granular array are numerically investigated in this study. A conservative two-bead system is first...     

Two-dimensional model of a plate made of an anisotropic inhomogeneous material

We present an asymptotic derivation of the two-dimensional equations of equilibrium of a thin elastic inhomogeneous plate manufactured of an anisotropic material of general form with 21 moduli of elasticity. We also consider simplified models obtained under special assumptions on the moduli. We use test examples to illustrate the error estimate of the proposed model and discuss its scope. The model is compared with the classical Kirchhoff–Love and Timoshenko–Reissner models.     

Stress and fabric in granular material

It has been well recognized that, due to anisotropic packing structure of granular material, the true stress in a specimen is different from the applied stress. However, very few research efforts have been focused on quantifying the relationship between the true stress and applied stress. In this paper, we derive an explicit relationship among applied stress tensor, material-fabric tensor, and force-fabric tensor; and we propose a relationship between the true stress tensor and the applied stress tensor. The validity of this derived relationship is examined by using the discrete element simulation results for granular material under biaxial and triaxial loading conditions.     

Modeling of properties and motions of an inhomogeneous one-dimensional continuum of a complicated Cosserat-type microstructure

We consider an approach to modeling the properties of the one-dimensional Cosserat continuum [1] by using the mechanical modeling method proposed by Il’yushin in [2] and applied in [3]. In this method, elements (blocks, cells) of special form are used to develop a discrete model of the structure so that the average properties of the model reproduced the properties of the continuum under study. The rigged rod model, which is an elastic structure in the form of a thin rod with massive inclusions (pulleys) fixed by elastic hinges on its elastic line and connected by elastic belt transmissions, is taken to be the original discrete model of the Cosserat continuum. The complete system of equations describing the mechanical properties and the dynamical equilibrium of the rigged rod in arbitrary plane motions is derived. These equations are averaged in the case of a sufficiently smooth variation in the parameters of motion along the rod (the long-wave approximation). It was found that the average equations exactly coincide with the equations for the one-dimensional Cosserat medium [1] and, in some specific cases, with the classical equations of motion of an elastic rod [4–6]. We study the plane motions of the one-dimensional continuum model thus constructed. The equations characterizing the continuum properties and motions are linearized by using several assumptions that the kinematic parameters are small. We solve the problem of natural vibrations with homogeneous boundary conditions and establish that each value of the parameter distinguishing the natural vibration modes is associated with exactly two distinct vibration mode shapes (in the same mode), each of which has its own frequency value.     

Computation of compaction in compressible granular material

Equations modeling compaction in a mixture of granular high explosive and interstitial gas are solved numerically. Both phases are modeled as compressible, viscous fluids. This overcomes well known difficulties associated with computing shock jumps in the inviscid version of the equations, which cannot be posed in a fully conservative form. One-dimensional shock tube and piston-driven compaction solutions compare favorably with experiment and known analytic solutions. A simple two-dimensional extension is presented.     

Wave propagation in an inhomogeneous transversely isotropic material obeying the generalized power law model

The analytical solutions for body-wave velocity in a continuously inhomogeneous transversely isotropic material, in which Young’s moduli (E, E′), shear modulus (G′), and material density (ρ) change according to the generalized power law model, (a+b z) ^c , are set down. The remaining elastic constants of transversely isotropic media, ν, and ν′ are assumed to be constants throughout the depth. The planes of transversely isotropy are selected to be parallel to the horizontal surface. The generalized Hooke’s law, strain-displacement relationships, and equilibrium equations are integrated to constitute the governing equations. In these equations, utilizing the displacement components as fundamental variables, the solutions of three quasi-wave velocities (V _SV , V _P ,?V _SH ) are generated for the present inhomogeneous transversely isotropic materials. The proposed solutions are compared with those of Daley and Hron (Bull Seismol Soc Am 67:661–675, (1977)), and Levin (Geophysics 44:918–936, (1979)) when the inhomogeneity parameter c?=?0. The agreement between the present results and previously published ones is excellent. In addition, the parametric study results reveal that the magnitudes of wave velocity are remarkably affected by (1) the inhomogeneity parameters (a, b, c); (2) the type and degree of material anisotropy (E/E′, ν/ν′, G/G′); (3) the phase angle (θ); and (4) the depth of the medium (z). Consequently, it is imperative to consider the effects of inhomogeneity when investigating wave propagation in transversely isotropic media.     

The influence of particle fluctuations on the average rotation in an idealized granular material

Granular materials are a simple example of a Cosserat continuum in that the average particle rotations may differ from the rotation of the average deformation. In the absence of couple stress, this difference insures that the stress is symmetric. This has been shown in theories that assume that the displacement at particle contacts is given by the average deformation and spin. Here, we indicate how the difference between the average rotation of the particles and the average rotation of the deformation can be determined when fluctuations in particle displacements and rotations satisfy local force and moment equilibria in a random aggregate of identical spheres. The predictions based on this model are in better agreement with numerical simulation than that given by the simple average strain theory.