Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

Many composite materials, including biological tissues, are modeled as non-linear elastic materials reinforced with elastic fibers. In the current paper, the full set of dynamic equations for finite deformations of incompressible hyperelastic solids containing a single fiber family are considered. Finite-amplitude wave propagation ansätze compatible with the incompressibility condition are employed for a generic fiber family orientation. Corresponding non-linear and linear wave equations are derived. It is shown that for a certain class of constitutive relations, the fiber contribution vanishes when the displacement is independent of the fiber direction.Point symmetries of the derived wave models are classified with respect to the material parameters and the angle between the fibers and the wave propagation direction. For planar shear waves in materials with a strong fiber contribution, a special wave propagation direction is found for which the non-linear wave equations admit an additional symmetry group. Examples of exact time-dependent solutions are provided in several physical situations, including the evolution of pre-strained configurations and traveling waves.     

Degenerate weakly non-linear elastic plane waves

Weakly non-linear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically non-linear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is fourfold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third-order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.     

Nonlinear elastic effects in graphite/epoxy: An analytical and numerical prediction of energy flux deviation

Manipulating acoustic wave propagation through a material have several interdisciplinary applications. Here we predict shift in energy flux deviation for acoustic waves propagating in unidirectional graphite/epoxy due to applied normal and shear stresses using both an analytical model, using acoustoelastic continuum theory, and a finite element discrete numerical model. The acoustoelastic theory predicts that the quasi-transverse (QT) wave exhibits larger shifts in energy flux deviation compared to quasi-longitudinal (QL) or the pure transverse (PT) due to an applied shear stress for fiber orientation angle ranging from 0° to 60°. Due to an applied shear stress the QT wave exhibits a shift in energy flux deviation at 0° fiber orientation angle as compared to normal stress case where the flux deviation and its load induced shift are both zero. A finite element model (FEM) is developed where equations of motion include the effect of nonlinear elastic coefficients. Element equations were integrated in time using Newmark’s method to determine the shift in energy flux deviations in graphite/epoxy for different loading cases. The energy flux shift of QT waves predicted by FEM for fiber orientation angles from 0° to 60° for applied shear stress case is in excellent agreement with acoustoelastic theory. Because energy shift magnitudes are not small, it is possible to experimentally measure these shifts and calibrate shifts with respect to load type (normal/shear) and magnitude.     

Rayleigh waves in anisotropic porous media and the polarization vector method

In this paper, the propagation of Rayleigh waves in orthotropic non-viscous fluid-saturated porous half-spaces with sealed surface-pores and with impervious surface is investigated. The main aim of the investigation is to derive explicit secular equations and based on them to examine the effect of the material parameters and the boundary conditions on the propagation of Rayleigh waves. By employing the method of polarization vector the explicit secular equations have been derived. These equations recover the ones corresponding to Rayleigh waves propagating in purely elastic half-spaces. It is shown from numerical examples that the Rayleigh wave velocity depends strongly on the porosity, the elastic constants, the anisotropy, the boundary conditions and it differs considerably from the one corresponding to purely elastic half-spaces. Remarkably, in the fluid saturated porous half-spaces, Rayleigh waves may travel with a larger velocity than that of the shear wave, a fact that is impossible for the purely elastic half-spaces.     

Time-harmonic wave propagation in a pre-stressed compressible elastic bi-material laminate

The dispersive behaviour of time-harmonic waves propagating along a principal direction in a perfectly bonded pre-stressed compressible elastic bi-material laminate is considered. The dispersion relation which relates wave speed and wavenumber is obtained by formulating the incremental boundary value problem and the use of the propagator matrix technique. At the low wavenumber limit, depending on the pre-stress, both the fundamental mode and the next lowest mode may 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 higher modes tend to phase speeds of the surface wave, the interfacial wave or the limiting phase speed of the composite. For numerical examples, either a two-parameter compressible neo-Hookean material or a two-parameter compressible Varga material is assumed.     

Stress intensity factors for a moving cylindrical crack in a nonhomogeneous cylindrical layer in composite materials

Stresses are determined for a finite cylindrical crack that is propagating with a constant velocity in a nonhomogeneous cylindrical elastic layer, sandwiched between an infinite elastic medium and a circular elastic cylinder made from another material. The Galilean transformation is employed to express the wave equations in terms of coordinates that are attached to the moving crack. An internal gas pressure is then applied to the crack surfaces. The solution is derived by dividing the nonhomogeneous interfacial layer into several homogeneous cylindrical layers with different material properties. The boundary conditions are reduced to two pairs of dual integral equations. These equations are solved by expanding the differences in the crack surface displacements into a series of functions that are equal to zero outside the crack. The Schmidt method is then used to solve for the unknown coefficients in the series. Numerical calculations for the stress intensity factors were performed for speeds and composite material combinations.     

The effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid

In this paper the equations governing small amplitude motions in a rotating transversely isotropic initially stressed elastic solid are derived, both for compressible and incompressible linearly elastic materials. The equations are first applied to study the effects of initial stress and rotation on the speed of homogeneous plane waves propagating in a configuration with uniform initial stress. The general forms of the constitutive law, stresses and the elasticity tensor are derived within the finite deformation context and then summarized for the considered transversely isotropic material with initial stress in terms of invariants, following which they are specialized for linear elastic response and, for an incompressible material, to the case of plane strain, which involves considerable simplification. The equations for two-dimensional motions in the considered plane are then applied to the study of Rayleigh waves in a rotating half-space with the initial stress parallel to its boundary and the preferred direction of transverse isotropy either parallel to or normal to the boundary within the sagittal plane. The secular equation governing the wave speed is then derived for a general strain–energy function in the plane strain specialization, which involves only two material parameters. The results are illustrated graphically, first by showing how the wave speed depends on the material parameters and the rotation without specifying the constitutive law and, second, for a simple material model to highlight the effects of the rotation and initial stress on the surface wave speed.     

Elastic stability of inhomogeneous thin plates on an elastic foundation

We study the elastic stability of infinite inhomogeneous thin plates on an elastic foundation under in-plane compression. The elastic stiffness constants depend on the coordinate variable in the thickness direction of the plate. The elastic foundation is represented as a Winkler-type model characterized by linear and nonlinear spring constants. First we derive the Föppl–von Kármán equations by taking variations of the elastic strain energy. Next we develop the linear stability analysis of the plate under uniform in-plane compression and explicitly derive the critical loads and wave numbers for particular three cases. The effects of the material inhomogeneity, material orthotropy and loading orthotropy on the critical states are examined independently. Finally, we perform a weakly nonlinear analysis of the plate at the onset of the buckling instability. With the multiple scales method, the amplitude equations for the unstable modes that provide insight into the mode type and its amplitude are derived and then the effect of the material inhomogeneity on buckling modes are evaluated qualitatively.     

A mixture theory analysis for the surface-wave propagation in an unsaturated porous medium

The mixture theory is employed to the analysis of surface-wave propagation in a porous medium saturated by two compressible and viscous fluids (liquid and gas). A linear isothermal dynamic model is implemented which takes into account the interaction between the pore fluids and the solid phase of the porous material through viscous dissipation. In such unsaturated cases, the dispersion equations of Rayleigh and Love waves are derived respectively. Two situations for the Love waves are discussed in detail: (a) an elastic layer lying over an unsaturated porous half-space and (b) an unsaturated porous layer lying over an elastic half-space. The wave analysis indicates that, to the three compressional waves discovered in the unsaturated porous medium, there also correspond three Rayleigh wave modes (R1, R2, and R3 waves) propagating along its free surface. The numerical results demonstrate a significant dependence of wave velocities and attenuation coefficients of the Rayleigh and Love waves on the saturation degree, excitation frequency and intrinsic permeability. The cut-off frequency of the high order mode of Love waves is also found to be dependent on the saturation degree.     

A theory of composites modeled as interpenetrating solid continua

The differential equations and boundary conditions describing the behavior of a finitely deformable, heat-conducting composite material are derived by means of a systematic application of the laws of continuum mechanics to a well-defined macroscopic model consisting of interpenetrating solid continua. Each continuum represents one identifiable constituent of the N-constituent composite. The influence of viscous dissipation is included in the general treatment. Although the motion of the combined composite continuum may be arbitrarily large, the relative displacement of the individual constituents is required to be infinitesimal in order that the composite not rupture. The linear version of the equations in the absence of heat conduction and viscosity is exhibited in detail for the case of the two-constituent composite. The linear equations are written for both the isotropic and transversely isotropic material symmetries. Plane wave solutions in the isotropic case reveal the existence of high-frequency (optical type) branches as well as the ordinary low-frequency (acoustic type) branches, and all waves are dispersive. For the linear isotropic equations both static and dynamic potential representations are obtained, each of which is shown to be complete. The solutions for both the concentrated ordinary body force and relative body force are obtained from the static potential representation.     

孤立波与半无限复合材料体耦合作用研究

基于一维颗粒链中产生的高度非线性孤立波,研究孤立波与半无限复合材料体的耦合作用。根据赫兹定律推导了一维颗粒链中颗粒间相互作用的运动微分方程,建立了颗粒链与半无限复合材料体的接触模型。对于颗粒与复合材料的接触,采用已有文献中修正后的赫兹定律,研究了高度非线性孤立波与半无限复合材料体的耦合力学作用机理,推导了颗粒链与半无限复合材料体的相互耦合运动微分方程组,通过数值计算,得到了各颗粒的内力、速度、位移曲线。分析了材料属性对回弹孤立波出现的时间、幅值的影响。结果表明:随着纤维方向弹性模量的增大,次级回弹波出现的时间和波幅都逐渐增大,随着垂直纤维方向弹性模量的增大,次级回弹波出现的时间先减小后增大,次级回弹波的幅值逐渐减小直至消失。     

On the influence of a flexural biasing state on the velocity of piezoelectric surface waves

A system of linear electroelastic equations for small fields superposed on a bias is applied in the determination of the velocity of acoustic surface waves in piezoelectric substrates subject to flexural biasing stresses. The influence of the biasing stresses appears in the boundary conditions as well as the differential equations. Direct calculations performed for both quartz and lithium niobate when the spatial variation of the flexural biasing state is omitted indicate that the biasing stresses in the boundary conditions have an important influence of the surface wave velocity. In addition, perturbation calculations are performed which include the influence of the spatial variation of all flexural biasing terms and it is shown that, for substrate thickness-to-wavelength ratios well within the practical range, the spatial variation in the biasing state has an appreciable effect on the velocity of acoustic surface waves.     

Anisotropic wave propagation and its applications to NDE of composite materials

This paper utilized anisotropic wave propagation theory to measure the elastic constants of a unidirectional fiber-reinforced composite specimen. For plane waves propagating in the composite specimen, the deviation of the propagational directions between the energy and phase velocities were measured. It is found that in such a sample, the deviations may be as large as 60 degrees. The measured energy velocities were transformed to the phase velocities of the plane waves by employing a numerical scheme. It is demonstrated that the elastic constants of a unidirectional fiber-reinforced composite can be determined by conducting ultrasonic experiments in two principal symmetry planes.     

Interesting effects in harmonic generation by plane elastic waves

The harmonics of plane longitudinal and trans-verse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic. This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a pri-mary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.     

Wave propagation in layer with two preferred directions

This article is concerned with overall or macroscopic properties of a composite material with no distinction made between the fibres and the matrix which they are embedded in. All the properties with dimensions larger than the fibre diameter and spacing are regarded as averaged over a volume of material. The systems of particular interest here are in the fibre reinforced composites with the fibres being very much stiffer and stronger than the matrix.Laminated plates of fibre-reinforced material are often fabricated from prepreg tapes, laid up according to some specific arrangement of fibre orientation and then bonded together. An angle-ply laminate is formed by alternating plies so that the families in adjacent laminas are inclined by angle ϕ and −ϕ to given direction alternately. The process of fabricating a multilayered plate of this material gives rise to a laminate in which the plies are separated by resin rich layer, and when this layer is thin enough that its thickness is negligible it may be regarded as plate reinforced by two families of fibres. Problems shall be considered in three dimensions, but attention shall be restricted to linear elasticity theory. The plate under consideration is reinforced by two mechanically equivalent families of fibres, but with no other preferred directions, so that it is locally orthotropic with respect to the plane of the fibres and to the two planes that orthogonally bisect the fibres.In this article linear elastic stress–strain relation is employed to derive dispersion curves for plane harmonic waves propagating in a plate of finite thickness but of infinite lateral extent. Attention is restricted to waves propagating in the plane parallel to stress free plate faces where waves travelling at any angle to one of the families of very strong fibres are examined. The dispersion equations, relating the phase velocity to the wavelength, are obtained. The fundamental modes are examined for symmetric as well as for anti-symmetric deformations. This leads to full understanding of displacement field as well as stress field.     

Wave propagation along a non-principal direction in a compressible pre-stressed elastic layer

The dispersive behavior of small amplitude waves propagating along a non-principal direction in a pre-stressed, compressible elastic layer is considered. One of the principal axes of stretch is normal to the elastic layer and the direction of propagation makes an angle θ with one of the in-plane principal axes. The dispersion relations which relate wave speed and wavenumber are obtained for both symmetric and anti-symmetric motions by formulating the incremental boundary value problem for a general strain energy function. The behavior of the dispersion curves for symmetric waves is for the most part similar to that of the anti-symmetric waves at the low and high wavenumber limits. At the low wavenumber limit, depending on the pre-stress and propagation angle, it may be possible for both the fundamental mode and the next lowest mode to have finite phase speeds, while other higher modes have an infinite phase speed. At the high wavenumber limit, the phase speeds of the fundamental mode and the higher modes tend to the Rayleigh surface wave speed and the limiting wave speeds of the layer, respectively. Numerical results are presented for a Blatz–Ko material and the effect of the propagation angle is clearly illustrated.     

Love waves in functionally graded piezoelectric materials

To investigate the features of Love waves in a layered functionally graded piezoelectric structure, the mathematical model is established on the basis of the elastic wave theory, and the WKB method is applied to solve the coupled electromechanical field differential equation. The solutions of the mechanical displacement and electrical potential function are obtained for the piezoelectric layer and elastic substrate. The dispersion relations of Love waves are deduced for electric open and short cases on the free surface respectively. The actual piezoelectric layer–elastic substrate systems are taken into account, and some corresponding numerical examples are proposed comparatively. Thus, the effects of the gradient variation about material constants on the phase velocity, the group velocity, the coupled electromechanical factor and the cutoff frequency are discussed in detail. So the propagation behaviors of Love waves in inhomogeneous medium is revealed, and the dispersion and the anti-dispersion are analyzed. The conclusions are significant both theoretically and practically for the surface acoustic wave devices.     

曲面曲率对Rayleigh波传播特性的影响

对任意形状的均匀各向同性线弹性曲面物体,用 WKB～（1）方法求解了沿曲面传播的Rayleigh表面波的运动微分方程,同时考虑了波传播方向及其垂直方向曲面曲率对波的穿透性的影, 所获波动方程的势函数解答表明,在一般情况下垂直波传播方向的曲面曲率对波的穿透深度的影响是不容忽视的.进而以同种介质平面表面情况下的Rayleigh面波的传播特性为基准,给出了曲面曲率引起波数或波速变化的解析表达式.通过理论分析和数值算例,描述了曲面上Rayleigh面波传播行为的一些基本特征.     

在旋转的半无限LiNbO3上传播的声表面波

声表面波(SAW)陀螺具有无源、无线、单层平面结构等优点。以目前国外研究小组所用的铌酸锂为对象，对含陀螺效应的声表面波的波动方程进行求解。用编制的程序进行了数值计算并绘制了基体绕各不同坐标轴旋转时，陀螺效应对铌酸锂表面传播的声表面波速度及对机电耦合系数影响的相关曲线，并对结果进行了分析。     

Dispersion relations and mode shapes for waves in laminated viscoelastic composites by variational methods

The propagation of oscillatory waves through periodic elastic composites has been analysed on the basis of the Floquet theory. This leads to self-adjoint differential equation systems which it was proved convenient to solve by variational methods. Many composites, such as the light-weight high-strength boron-epoxy material, consist of strong reinforcing components in a plastic matrix. The latter can exhibit viscoelastic properties which can have a significant influence on wave propagation characteristics. Replacement of the elastic constant by the viscoelastic complex modulus changes the mathematical structure so that the differential equation system is no longer self-adjoint. However, a modification of the variational principles is suggested which retains formal self-adjointness, and yields variational principles which contain additional boundary terms. These are applied to the determination of wave speeds and mode shapes for a laminated composite made of homogeneous elastic reinforcing plates in a homogeneous viscoelastic matrix for plane waves propagating normally to the reinforcing plates. These results agree well with the exact solution which can be evaluated in this simple case. The variational principles permit solutions for periodic, but otherwise arbitrary variation of material properties.