Anti-symmetric waves in pre-stressed imperfectly bonded incompressible elastic layered composites

The effect of an imperfect interface on the dispersive behavior of in-plane time-harmonic symmetric waves in a pre-stressed incompressible symmetric layered composite, was analyzed recently by Leungvichcharoen and Wijeyewickrema (2003). In the present paper the corresponding case for time harmonic anti-symmetric waves is considered. The bi-material composite consists of incompressible isotropic elastic materials. The imperfect interface is simulated by a shear-spring type resistance model, which can also accommodate the extreme cases of perfectly bonded and fully slipping interfaces. The dispersion relation is obtained by formulating the incremental boundary-value problem and using the propagator matrix technique. The dispersion relations for anti-symmetric and symmetric waves differ from each other only through the elements of the propagator matrix associated with the inner layer. The behavior of the dispersion curves for anti-symmetric waves is for the most part similar to that of symmetric waves at the low and high wavenumber limits. At the low wavenumber limit, depending on the pre-stress for perfectly bonded and imperfect interface cases, a finite phase speed may exist only for the fundamental mode while other higher modes have an infinite phase speed. However, for a fully slipping interface in the low wavenumber region it may be possible for both the fundamental mode and the next lowest mode to have finite phase speeds. For the higher modes which have infinite phase speeds in the low wavenumber region an expression to determine the cut-off frequencies is obtained. At the high wavenumber limit, the phase speeds of the fundamental mode and the higher modes tend to the phase speeds of the surface wave or the interfacial wave or the limiting phase speed of the composite. The bifurcation equation obtained from the dispersion relation yields neutral curves that separate the stable and unstable regions associated with the fundamental mode or the next lowest mode. Numerical examples of dispersion curves are presented, where when the material has to be prescribed either Mooney–Rivlin material or Varga material is assumed. The effect of imperfect interfaces on anti-symmetric waves is clearly evident in the numerical results.     

Dispersion effects of extensional waves in pre-stressed imperfectly bonded incompressible elastic layered composites

The effect of an imperfect interface, on time-harmonic extensional wave propagation in a pre-stressed symmetric layered composite is considered. The bimaterial composite consists of incompressible isotropic elastic materials. The shear spring type resistance model employed to simulate the imperfect interface can accommodate the extreme cases of perfect bonding and a fully slipping interface. The dispersion relation obtained by formulating the incremental boundary-value problem and the use of the propagator matrix technique, is analyzed at the low and high wavenumber limits. For the perfectly bonded and imperfect interface cases in the low wavenumber region, only the fundamental mode has a finite phase speed, while other higher modes have an infinite phase speed when the dimensionless wavenumber approaches zero. However, for the fully slipping interface in the low wavenumber region, both the fundamental mode and the next lowest mode have finite phase speeds. In the high wavenumber region, when the dimensionless wavenumber tends to infinity, the phase speeds of the fundamental mode and the higher modes depend on the phase speeds of the surface and interfacial waves and on the limiting phase speed of the composite. An expression to determine the cut-off frequencies is obtained from the dispersion relation. Numerical examples of dispersion curves are presented, where when the material has to be prescribed either Mooney–Rivlin material or Varga material is assumed. The effect of the imperfect interface is clearly evident in the numerical results.     

Uniqueness of Stoneley waves in pre-stressed incompressible elastic media

The main aim of this paper is to prove, for the general case, the uniqueness of Stoneley waves propagating along the bonded interface of two pre-stressed incompressible elastic half-spaces. In order to do that the authors have used the complex function method. By this approach, it is shown that the secular equation of Stoneley waves in pre-stressed incompressible elastic half-spaces has at most one solution in the complex plane. This says that if a Stoneley wave exists, then it is unique.     

Finite-amplitude Love waves in a pre-stressed compressible elastic half-space with a double surface layer

The dispersive behavior of finite-amplitude time-harmonic Love waves propagating in a pre-stressed compressible elastic half-space overlaid with two compressible elastic surface layers of finite thickness is investigated. The half-space and layers are made of different pre-stressed compressible neo-Hookean materials. The dispersion relation which relates wave speed and wavenumber is obtained in explicit form. Results for the energy density and energy flux of the waves are also presented. The special case where the interfaces between the layers and the half-space are principal planes of the left Cauchy–Green deformation tensor is also investigated. Numerical results are presented showing the variation of the Love wave speed with the pre-stress and the propagation angle.     

Surface waves in a stretched and sheared incompressible elastic material

In this paper, we analyze the effect of a combined pure homogeneous strain and simple shear in a principal plane of the latter on the propagation of surface waves for an incompressible isotropic elastic half-space whose boundary is normal to the glide planes of the shear. This generalizes previous work in which, separately, pure homogeneous strain and simple shear were considered. For a special class of materials, the secular equation is obtained in explicit form and then specialized to recover results obtained previously for the two cases mentioned above. A method for obtaining the secular equation for a general form of strain–energy function is then outlined. In general, this is very lengthy and the result is not listed, but, for the case in which there is no normal stress on the half-space boundary, the result is given, for illustration, in respect of the so-called generalized Varga material. Numerical results are given to show how the surface wave speed depends on both the underlying pure homogeneous strain and the superimposed simple shear. Further numerical results are provided for the Gent model of limiting chain extensibility.     

Rayleigh waves in an incompressible elastic half-space overlaid with a water layer under the effect of gravity

This paper is concerned with the propagation of Rayleigh waves in an incompressible isotropic elastic half-space overlaid with a layer of non-viscous incompressible water under the effect of gravity. The authors have derived the exact secular equation of the wave which did not appear in the literature. Based on it the existence of Rayleigh waves is considered. It is shown that a Rayleigh wave can be possible or not, and when a Rayleigh wave exists it is not necessary unique. From the exact secular equation the authors arrive immediately at the first-order approximate secular equation derived by Bromwich [Proc. Lond. Math. Soc. 30:98–120, 1898]. When the layer is assumed to be thin, a fourth-order approximate secular equation is derived and of which the first-order approximate secular equation obtained by Bromwich is a special case. Some approximate formulas for the velocity of Rayleigh waves are established. In particular, when the layer being thin and the effect of gravity being small, a second-order approximate formula for the velocity is created which recovers the first-order approximate formula obtained by Bromwich [Proc. Lond. Math. Soc. 30:98–120, 1898]. For the case of thin layer, a second-order approximate formula for the velocity is provided and an approximation, called global approximation, for it is derived by using the best approximate second-order polynomials of the third- and fourth-powers.     

Resonant-triad instability of a pre-stressed incompressible elastic plate

The dynamic stability properties of a pre-stressed incompressible elastic plate are studied in this paper with respect to perturbations in the form of one near-neutral mode and two non-neutral modes interacting resonantly. The pre-stresses are assumed to be an all-round pressure. With the aid of a novel derivation procedure, the evolution equations governing the scaled amplitudes of the three modes are found to be given by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaCa% aaleqabaGaaGOmaaaakiaadgeadaWgaaWcbaGaaGymaaqabaGccaGG% VaGaamizamaaBaaaleaacqaHepaDdaahaaadbeqaaiaaikdaaaaale% qaaOGaeyypa0JaeyOeI0Iaam4yamaaBaaaleaacaaIWaaabeaakiaa% dgeadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGJbWaaSbaaSqaai% aaigdaaeqaaOGaaiiFaiaadgeadaWgaaWcbaGaaGymaaqabaGccaGG% 8bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyAaiabeo7aNnaaBa% aaleaacaaIXaaabeaakiqadgeagaqeamaaBaaaleaacaaIYaaabeaa% kiqadgeagaqeamaaBaaaleaacaaIZaaabeaaaaa!5308!\[d^2 A_1 /d_{\tau ^2 } = - c_0 A_1 - c_1 |A_1 |^2 - i\gamma _1 \bar A_2 \bar A_3 \], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadg% eadaWgaaWcbaGaaGOmaaqabaGccaGGVaGaamizaiabes8a0jabg2da% 9iabeo7aNnaaBaaaleaacaaIYaaabeaakiqadgeagaqeamaaBaaale% aacaaIXaaabeaakiqadgeagaqeamaaBaaaleaacaaIZaaabeaaaaa!4324!\[dA_2 /d\tau = \gamma _2 \bar A_1 \bar A_3 \] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadg% eadaWgaaWcbaGaaG4maaqabaGccaGGVaGaamizaiabes8a0jabg2da% 9iabeo7aNnaaBaaaleaacaaIZaaabeaakiqadgeagaqeamaaBaaale% aacaaIXaaabeaakiqadgeagaqeamaaBaaaleaacaaIYaaabeaaaaa!4325!\[dA_3 /d\tau = \gamma _3 \bar A_1 \bar A_2 \], where a bar denotes complex conjugation, is a slow time variable and c ₀, c ₁, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS% baaSqaaiaaigdaaeqaaaaa!387B!\[\gamma _1 \], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS% baaSqaaiaaikdaaeqaaaaa!387C!\[\gamma _2 \], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS% baaSqaaiaaiodaaeqaaaaa!387D!\[\gamma _3 \] are real constants. These equations are solved exactly for the special case when A ₂ and A ₃ have constant amplitudes but time-dependent phases. A series of new post-buckling states, which does not exist when the perturbation is monochromatic, are found. We show that two nonneutral modes can interact resonantly to produce a much larger near-neutral mode, and in particular, two O() non-neutral modes may induce a much larger % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4taiaacI% cacqaH1oqzdaahaaWcbeqaamaalyaabaGaaGOmaaqaaiaaiodaaaaa% aOGaaiykaaaa!3B87!\[O(\varepsilon ^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} )\] oscillation or static post-buckling state. In this sense, resonant-triad interaction is a powerful mechanism in producing high levels of strain and stress in a pre-stressed elastic plate.     

Dynamics of a linearly inhomogeneous incompressible isotropic elastic half-space

The study presented in this paper treats the harmonic and transient wave motion of an incompressible isotropic semi-infinite elastic medium with a shear modulus increasing linearly with depth. The medium has a constant mass density and an initial hydrostatic stress distribution due to a constant gravity. In particular, attention is given to the case of a vanishing top rigidity. For this case it is shown that the governing equations resemble the equations governing the deep water motion, and that under normal loading the behaviour of the upper surface resembles that of the upper surface of (deep) water.     

Dynamics of a prestressed incompressible layered half-space under moving load

The paper addresses a plane problem: a concentrated force acts on a plate resting on an elastic half-space with homogeneous prestrain. The equations of motion of the plate incorporate shear and rotary inertia. The half-space is assumed to be incompressible and isotropic in the natural state. The elastic potential is given in general form and is only specified for numerical purposes. The dependence of the critical velocity of the load and the stress-strain state on the prestresses is analyzed for different ratios between the stiffnesses of the layer and half-space and different contact conditions. The calculations are carried out for a half-space with Bartenev-Khazanovich potential __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 36–54, March 2008.     

Surface waves in a rotating anisotropic elastic half-space

The Stroh formalism for surface waves in an anisotropic elastic half-space is extended to the case when the half-space rotates about an axis with a constant rotation rate. The sextic equation for the Stroh eigenvalues, the eigenvectors, the orthogonality and closure relations are obtained. The Barnett–Lothe tensors are no longer real, but two of them are Hermitian. Taziev’s equation is generalized and used to derive the polarization vector and the secular equation without computing the Stroh eigenvalues and eigenvectors. An alternative derivation using the method of first integrals by Mozhaev and Destrade yields new invariants that relate the displacement and stress and are independent of the depth from the free surface. Explicit expression of the polarization vector and the secular equation for monoclinic materials with the symmetry plane at x₃ = 0 with the rotation about the x₃-axis obtained by Destrade is re-examined, and new results are presented. Also presented is the one-component surface wave in the rotating half-space.     

The incremental shear modulus in an incompressible elastic material

Summary The incremental shear modulus is calculated for infinitesimal shear of an incompressible isotropic elastic material in a state of pure homogeneous deformation. Necessary and sufficient conditions that it be positive are obtained. These conditions are related to the Hadamard conditions for propagation of a wave of infinitesimal amplitude in the deformed material.

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Sommario Il modulo trasversale incrementale è calcolato per una deformazione trasversale infinitesima di un materiale elastico isotropico ed incompressibile in uno stato di deformazione pura ed omogenea. Si ottengono le condizioni necessarie e sufficienti per un modulo positivo. Si possono riferire queste condizioni alle condizioni di Hadamard per la propagazione di un'onda di ampiezza infinitesima in un materiale deformato.

     

Stroh formulation for a generally constrained and pre-stressed elastic material

The Stroh formalism is essentially a spatial Hamiltonian formulation and has been recognized to be a powerful tool for solving elasticity problems involving generally anisotropic elastic materials for which conventional methods developed for isotropic materials become intractable. In this paper we develop the Stroh/Hamiltonian formulation for a generally constrained and prestressed elastic material. We derive the corresponding integral representation for the surface-impedance tensor and explain how it can be used, together with a matrix Riccati equation, to calculate the surface-wave speed. The proposed algorithm can deal with any form of constraint, pre-stress, and direction of wave propagation. As an illustration, previously known results are reproduced for surface waves in a pre-stressed incompressible elastic material and an unstressed inextensible fibre-reinforced composite, and an additional example is included analyzing the effects of pre-stress upon surface waves in an inextensible material.