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Estimates Based on First Order Inequalities 2: Volume Integral and Other Methods
Published in James N. Flavin, Salvatore Rionero, Qualitative Estimates for Partial Differential Equations, 2020
James N. Flavin, Salvatore Rionero
Consider a homogeneous, isotropic (linear) elastic right cylinder whose cross- section (assumed simply connected) is R with boundary C; rectangular cartesian coordinates (x1,x2,x3) are chosen such that the x3 axis is parallel to the generators and the x1, x2 plane coincides with that of R. In order that the ‘antiplane shear displacement field’ with components u1, u2, u3 such that () u1=u2=0u3=u(x1,x2)
Growth of a Semi-infinite Crack at a Varying Velocity
Published in Arabinda Roy, Rasajit Kumar Bera, Linear and Non-Linear Deformations of Elastic Solids, 2019
Arabinda Roy, Rasajit Kumar Bera
Most studies of crack are assumed to be either stationary or moving at uniform speed. First such study on nonuniform crack motion was by Kostrov (1966) under antiplane shear deformation or Mode III cracks. Fossum and Freund (1975) used nonuniformly moving shear cracks as shallow earthquake models. Nonuniformly moving source were used by Roy (1979) and Freund (1972) to study the response of elastic solids. Kostrov (1975) first considered crack propagation at nonuniform motion under general loading by ingenious extension of the nonuniformly moving antiplane shear crack method in an infinite medium. However, usual restriction of propagation velocity must be less than the Rayleigh wave velocity is valid, as in the case of uniform motion. Willis (1992) also discussed the accelerating crack. Freund used an ingenious way of deriving the nonuniform velocity by incremental change from the solution in the uniform velocity case.
Antiplane Shear Waves in Fiber Composites with Structural Nonlinearity
Published in Igor V. Andrianov, Vladyslav Danishevskyy, Jan Awrejcewicz, Linear and Nonlinear Waves in Microstructured Solids, 2021
Igor V. Andrianov, Vladyslav Danishevskyy, Jan Awrejcewicz
We investigate the propagation of the elastic waves in plane perpendicular to the fibers axis. In this case the components of the stress-strain state depend only on two spatial co-ordinates x1, x2, and the system of equations of motion is split into two uncoupled problems governing propagation of plane waves and antiplane shear waves. Our further considerations are limited to consider only antiplane transverse waves. The governing wave equation has the following form: μn∂2un∂x12+∂2un∂x22=ρn∂2un∂t2,
Exact solution of two collinear cracks normal to the boundaries of a 1D layered hexagonal piezoelectric quasicrystal
Published in Philosophical Magazine, 2018
Consider a 1D hexagonal PQC with point group 6mm, where the quasiperiodic poling axis is denoted as the z-axis, and the isotropic periodic plane, denoted as the xoy-plane. In the current paper, the PQC is assumed to have finite width, that is, it is restricted within the domain and infinite along the quasiperiodic axis. Two through cracks of length are symmetrically distributed in , as shown in Figure 1. Under antiplane mechanical loading and inplane electric loading with reference to the xoy-plane, deformation involved is independent of the spatial variable z. Consequently, the problem under consideration is converted to a so-called antiplane shear problem. In other words, there are only non-vanishing out-of-plane displacements of phonon and phason fields and inplane electric fields. The total elastic displacement vector is denoted as (), , being the phonon and phason displacement components, respectively, and inplane electric field vector as (). That is,
Elasto-hydrodynamics of quasicrystals with a crack under sudden impacts
Published in Philosophical Magazine Letters, 2018
For 1D hexagonal QCs loaded suddenly by antiplane mechanical loading, deformation involved is independent of the spatial variable z. Consequently, the problem under scrutiny is converted to a problem of so-called antiplane shear. In other words, there are only non-vanishing out-of-plane displacements of phonon and phason fields. That is,