Micromechanics modeling of viscoelastic properties of hybrid composites with shunted and arbitrarily oriented piezoelectric inclusions

A comprehensive micromechanics model is developed to estimate the effective viscoelastic properties of hybrid composites containing polymer matrix, conductive inclusions and shunted piezoelectric inclusions. The model is derived using the viscoelastic correspondence principle in conjunction with the Mori-Tanaka approach and the orientation averaging scheme. Three dimensional complex moduli are explicitly presented for hybrid composites with any orientation distribution. The model is first validated by comparison with available experimental results. Then, the loss factors are examined for hybrid composites with inclusions of various volume fractions and of shapes ranging from thin disks to long fibers. It is seen that hybrid composites with randomly oriented inclusions exhibit shear loss factors which are not possible with monolithic piezoelectric plate. Furthermore, the numerical results indicate that composites with long spheroid inclusions provide the best damping performance. The results recommend that aligned inclusion composites are good for alleviating longitudinal oscillations. If oscillation energy needs to be dissipated in all directions and for all modes, three dimensional random composites should be used. It is also observed that spherical inclusion composites cannot improve shear damping irrespective of the orientation and the volume fraction. In general, to achieve a pronounced damping piezoelectric inclusions that lie in aspect ratio range 0.1?α?2 should be avoided.     

Propagation of low-frequency elastic waves in double-porositymedia containing viscoelastic component in the pores

A self-consistent scheme named the effective field method (EFM) is applied for the calculation of the velocities and quality factors of elastic waves propagating in double-porosity media. A double-porosity medium is considered to be a heterogeneous material composed of a matrix with primary pores and inclusions that are represent by flat (crack-like) secondary pores. The prediction of the effective viscoelastic moduli consists of two steps. First, we calculate the effective viscoelastic properties of the matrix with the primary small-scale pores (matrix homogenization). Then, the porous matrix is treated as a homogeneous isotropic host where the large-scale secondary pores are embedded. Spatial distribution of inclusions in the medium is taken into account via a special two-point correlation function. The results of the calculation of the viscoelastic properties of double-porosity media containing isotropic fields of crack-like inclusions and double-porosity media with some non-isotropic spatial distributions of crack-like inclusions are presented.     

Effect of three-parameter viscoelastic medium on vibration behavior of temperature-dependent non-homogeneous viscoelastic nanobeams in a hygro-thermal environment

In this article, free vibration of functionally graded (FG) viscoelastic nanobeams resting on viscoelastic foundation subjected to hygrothermal loading is investigated employing a higher order refined beam theory which captures shear deformation influences needless of any shear correction factor. The three-parameter viscoelastic medium consists of parallel springs and dashpots as well as a shear layer. Temperature-dependent material properties of FGM beam are graded across the thickness via the power-law model. Employing non-local elasticity theory of Eringen and Hamilton's principle, non-local governing equations of a size-dependent viscoelastic nanobeam are obtained and solved analytically for various boundary conditions. To verify the reliability of the developed model, the results of the current work are compared with those available in literature. The effects of viscoelastic foundation parameters, internal damping coefficient, hygrothermal loading, non-local parameter, gradient index, mode number, and slenderness ratio on the vibrational characteristics of nanoscale viscoelastic FG beams are explored.     

Narrowband shear wave generation by a Finite-Amplitude radiation force: The fundamental component

A highly localized source of low-frequency shear waves can be created by the modulated radiation force resulting from two intersecting quasi-continuous-wave ultrasound beams of slightly different frequencies. In contrast to most other radiation force-based methods, these shear waves can be narrowband. Consequently, different frequency-dependent effects will not significantly affect their spectrum as they propagate within a viscoelastic medium, thereby enabling the viscoelastic shear properties of the medium to be determined at any given modulation frequency. This can be achieved by tracking the shear wave phase delay and change in amplitude over a specific distance. In this paper we explore the properties of short duration (dynamic) low-frequency shear wave propagation and study how the shear displacement field depends on the excitation conditions. Our investigations make use of the approximate Green's functions for viscoelastic media, and the evolution of such waves is studied in the spatiotemporal domain from a theoretical perspective. Although nonlinearities are included in our confocal source model, just the properties of the fundamental shear component are examined in this paper. We examine how the shear wave propagation is affected by the shear viscosity, the coupling wave, the spatial distribution of the force, the shear speed, and the duration of the modulated wave. A method is proposed for estimating the shear viscosity of a viscoelastic medium. In addition, it is shown how the Voigt model paremeters can be extracted from the frequency-dependent speed and attenuation.     

The problem of two plastic and heterogeneous inclusions in an anisotropic medium

We demonstrate an integral equation for the total local strain ε^T in an anisotropic heterogeneous medium with incompatible strain ε^p and which is at the same time submitted to an exterior field. The integral equation is solved in the case of an heterogeneous and plastic pair of inclusions, for which we calculate the average fields in each inclusion as well as the different parts of the elastic energy stocked in the medium.The solution is applied to the case of two isotropic and spherical inclusions in an isotropic matrix loaded in shear. The results are compared with those deduced from a more approximate method based on Horn's approximation of the integral equation. In appendix we give a numerical method for calculating the interaction tensors between anisotropic inclusions in an anisotropic medium as well as the analytic solution in the case of two spherical inclusions located in an isotropic medium.     

A time domain direct boundary integral method for a viscoelastic plane with circular holes and elastic inclusions

This paper considers the problem of an infinite, isotropic viscoelastic plane containing an arbitrary number of randomly distributed, non-overlapping circular holes and isotropic elastic inclusions. The holes and inclusions are of arbitrary size. All inclusions are assumed to be perfectly bonded to the material matrix but the elastic properties of the inclusions can be different from one another. The Kelvin model is employed to simulate the viscoelastic plane. The numerical approach combines a direct boundary integral method for a similar problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described in [Crouch SL, Mogilevskaya SG. On the use of Somigliana's formula and Fourier series for elasticity problems with circular boundaries. Int J Numer Methods Eng 2003;58:537–578], and a time-marching strategy for viscoelastic material analysis described in [Mesquita AD, Coda HB, Boundary integral equation method for general viscoelastic analysis. Int J Solids Struct 2002;39:2643–2664]. Several numerical examples are given to verify the approach. For benchmark problems with one inclusion, results are compared with the analytical solution obtained using the correspondence principle and analytical Laplace transform inversion. For an example with two holes and two inclusions, results are compared with numerical solutions obtained by commercial finite element software—ANSYS. Benchmark results for a more complicated example with 25 inclusions are also given.     

An extended Butterworth-Van Dyke model for quartz crystal microbalance applications in viscoelastic fluid media

An extended Butterworth-Van Dyke (EBVD) model with frequency-independent parameters for the characterization of a resonant compound formed by a quartz crystal in contact with a finite viscoelastic layer contacting a semi-infinite viscoelastic medium is extracted by analysis of the lumped element model. The formulation of the EBVD model is compared with the complete expression of the electrical admittance of the loaded quartz derived from the transmission line model (TLM). Relative deviations between them do not exceed 3% around 1% bandwidth near resonance. An extended Martin and Granstaff's model and an explicit expression for the frequency shift that supposes an extension of Kanazawa's model for viscoelastic media are obtained. An analysis of the errors associated with the extraction of shear parameters of the coating for different materials prove that, to obtain an error less than 5% in the shear parameters determination, the viscoelastic contribution, defined as the relative error in the thickness computed from the frequency shift by Sauerbrey equation, must be greater than a limit that depends on Q, which is defined as the ratio of the shear storage modulus (G') to shear loss modulus (G"). In the materials studied polymers in the transition range or in the rubbery state with Q=1 and 10, the viscoelastic contribution must be higher than 15% and 50%, respectively, for a 5% limit error in the shear parameters extraction. A criterion for a practical determination of the appropriate viscoelastic regimes is indicated     

The role of viscosity in the impulse diffraction field of elastic waves induced by the acoustic radiation force

Several ultrasound-based techniques for the estimation of soft tissue elasticity are currently being investigated. Most of them study the medium response to dynamic excitations. Such responses are usually modeled in a purely elastic medium using a Green's function solution of the motion equation. However, elasticity by itself is not necessarily a discriminant parameter for malignancy diagnosis. Modeling viscous properties of tissues could also be of great interest for tumor characterization. We report in this paper an explicit derivation of the Green's function in a viscous and elastic medium taking into account shear, bulk, and coupling waves. From this theoretical calculation, 3D simulations of mechanical waves in viscoelastic soft tissues are presented. The relevance of the viscoelastic Green's function is validated by comparing simulations with experimental data. The experiments were conducted using the supersonic shear imaging (SSI) technique which dynamically and remotely excites tissues using acoustic radiation force. We show that transient shear waves generated with SSI are modeled very precisely by the Green's function formalism. The combined influences of out-of-plane diffraction, beam shape, and shear viscosity on the shape of transient waves are carefully studied as they represent a major issue in ultrasound-based viscoelasticity imaging techniques.     

VPF粘性介质粘弹性本构关系研究

为研究粘性介质的力学性能对板材变形的影响,通过剪切蠕变-回复和松弛试验分析了甲基乙烯基粘性介质的流变性能.实验结果表明粘性介质可以简化为线性粘弹性材料.建立了粘性介质的积分型粘弹性本构方程,并结合剪切蠕变-回复过程的有限元分析确定了方程中的材料参数.利用该本构方程对粘性介质压力胀形过程进行了有限元分析,模拟结果与试验结果对比表明,所建立的本构方程可以较好的预测板材的变形过程.     

Postbuckling of viscoelastic micro/nanobeams embedded in visco-Pasternak foundations based on the modified couple stress theory

This paper investigates the postbuckling analysis of a viscoelastic microbeam embedded in a double layer viscoelastic foundation. This viscoelastic microbeam is modeled using the Kelvin–Voigt model and the modified couple stress theory. A material length scale parameter is utilized to describe the size-dependent behavior of the viscoelastic microbeam. The visco-Pasternak foundation used in this study contains a viscoelastic medium and a shear layer. This microbeam is subjected to an axial compressive load at the beam ends which can change as a function of time. According to the Euler–Bernoulli beam theory and von-Karman nonlinearity, the time-dependent equations of motion are derived by Hamilton’s principle. The nonlinear equations of motion are directly solved under the simply supported boundary condition. Both time-dependent deflection and viscoelastic buckling load are investigated. Finally, the influences of the material length scale parameter, parameters of the visco-Pasternak foundation and the material viscosity coefficient on the dynamic postbuckling response are studied.

     

Boundary element analysis for viscoelastic solids containing interfaces/holes/cracks/inclusions

With the aid of the elastic–viscoelastic correspondence principle, the boundary element developed for the linear anisotropic elastic solids can be applied directly to the linear anisotropic viscoelastic solids in the Laplace domain. Green's functions for the problems of two-dimensional linear anisotropic elastic solids containing holes, cracks, inclusions, or interfaces have been obtained analytically using Stroh's complex variable formalism. Through the use of these Green's functions and the correspondence principle, special boundary elements in the Laplace domain for viscoelastic solids containing holes, cracks, inclusions, or interfaces are developed in this paper. Subregion technique is employed when multiple holes, cracks, inclusions, and interfaces exist simultaneously. After obtaining the physical responses in Laplace domain, their associated values in time domain are calculated by the numerical inversion of Laplace transform. The main feature of this proposed boundary element is that no meshes are needed along the boundary of holes, cracks, inclusions and interfaces whose boundary conditions are satisfied exactly. To show this special feature by comparison with the other numerical methods, several examples are solved for the linear isotropic viscoelastic materials under plane strain condition. The results show that the present BEM is really more efficient and accurate for the problems of viscoelastic solids containing interfaces, holes, cracks, and/or inclusions.     

Harmonic wave propagation in viscoelastic heterogeneous materials Part I: Dispersion and damping relations

An analytico-numerical method is presented to study the propagation of plane harmonic waves in infinite periodic linear viscoelastic media. Part I considers only the dispersion and attenuation of acoustical longitudinal and shear waves. To show the accuracy of the method, examples of plane harmonic wave propagation in an infinite homogeneous medium and in a periodic layered viscoelastic medium are presented. The method is then used to calculate the damping and dispersion relations for a fibre-reinforced viscoelastic composite material. The results show clearly the influence of materials' viscoelastic properties and heterogeneities on the propagation of plane harmonic waves through the media.     

Mode II Macrocrack Initiation in Orthotropic Composite Viscoelastic Plate

Safe loads and initiation time for a straight macrocrack in viscoelastic orthotropic material that is intended to model a fiber composite plate under shear loads is investigated. The composite material is modeled by viscoelastic orthotropic medium. Determination of expression for crack shear displacement as function of time is based on the corresponding elastic solution and the method of operator continued fractions. Initiation time is obtained as a solution of integral equation for the incubation period. Numerical calculations are given for mode II macrocrack initiation.     

Elasto-plastic solution for a cylindrical inclusion in plane strain

Understanding the role of inclusions in the fracture process is of great importance and up to now there have been few continuum solutions which give elasto-plastic stress and strain fields around an inclusion. The incremental solution presented is for plane strain conditions, the cylindrical inclusion being assumed rigid, while the matrix obeys the Mises yield criterion and its associated flow rule and has a very small or zero linear work-hardening characteristic. On the infinite boundary, strains of up to nearly 16× yield strain are applied under various states of stress, varying from pure shear to the nearly purely hydrostatic state.The simultaneous governing equations in terms of displacements are solved using finite differences and successive over-relaxation.The results presented give distributions of radial stress and shear strain on the matrix-inclusion interface showing the variation with applied strain and stress state, and work-hardening characteristic: shear strains are found to be almost independent of stress state. Comparison is made with the linear viscoelastic solution and a slip-line solution is attempted from a plot of maximum shear strain trajectories.     

Stress Concentration in Microstructural Elements of Viscoelastic Composite Materials

We present solution of the problem of estimating the stress concentration in the constituents (matrix and inclusions) of a multiconstituent viscoelastic composite material, depending on the shape of inclusions, properties of the matrix and the whole composite. The matrix material is isotropic and viscoelastic. A wide range of properties of inclusions (e.g., pores, solid and viscous particles) is considered. To solve this problem, we use the method of integral transformations. __________ Translated from Problemy Prochnosti, No. 5, pp. 138 – 149, September – October, 2005.     

Theory of thermal expansion of composites

The problem discussed is to find the overall thermal expansion of a composite consisting of inclusions in a matrix of material with different expansion coefficients and elastic moduli. The volume mismatch causes strain fields. The total strain energy must be a minimum. This problem was solved previously for inclusions which are either spheres or randomly oriented long cylinders. In these simple cases the matrix strain field consists of a short-range shear component and a uniform expansion; the inclusion suffers uniform strain. The matrix is replaced by an effective medium having the average properties of the composite. The overall expansion coefficient could be obtained in closed form. This separation of the strain field into short-range shear and long-range uniform dilation is valid, at least to a good approximation, for all inclusion shapes. Simple expressions can thus be obtained in terms of coefficients which, although not calculated exactly, can be deduced approximately or can be determined empirically. Plasticity can be accounted for by allowing the shear modulus to depend on the temperature and on the maximum shear strain. The size of the inclusions does not enter the theory except through the yield strain, which depends on the extent of the strain field.Paper presented at the Ninth International Thermal Expansion Symposium, December 8–10, 1986, Pittsburgh, Pennsylvania, U.S.A.     

Direct boundary integral procedure for a Boltzmann viscoelastic plane with circular holes and elastic inclusions

A direct boundary integral method in the time domain is presented to solve the problem of an infinite, isotropic Boltzmann viscoelastic plane containing a large number of randomly distributed, non-overlapping circular holes and perfectly bonded elastic inclusions. The holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can, in general, be different. The method is based on a direct boundary integral approach for the problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described by Crouch and Mogilevskaya [1], and a time marching strategy for viscoelastic analysis described by Mesquita and Coda [2–8]. Benchmark problems and numerical examples are included to demonstrate the accuracy and efficiency of the method.     

Analytical relations between effective material properties and microporosity: Application to bone mechanics

This work deals with the problem of reconstruction of microstructural parameters of viscoelastic composite material from measurements of its effective properties. The Stieltjes representation of the effective shear complex modulus of two-component composite material is exploited to recover information about structural parameters. This representation is derived from problem of torsion of a heterogeneous cylinder using asymptotic expansions method. The microstructural information is contained in the spectral measure in this analytical representation. The spectral function can be recovered from the measurements over a range of frequencies. The problem of reconstruction of the spectral measure is very ill-posed. Regularized algorithm is derived to ensure stability of the results. We apply the proposed method to recovery of porosity of cancellous bone from measurements of its effective shear modulus. Bone is modeled as a medium with a microstructure composed of two viscoelastic isotropic or/and transversely isotropic materials, with the components representing trabeculae (elastic component) and bone marrow (viscoelastic component). Porosity is understood as volume fraction of bone marrow. To verify the approach we apply it to analytically and numerically simulated response of a cylinder filled with a composite material with hexagonal microstructure. The proposed method does not use any specific assumptions about the microgeometry of the composite and is applicable to any two-phase composite medium.     

Propagation of shear waves generated by a modulated finite amplitude radiation force in a viscoelastic medium

An effective way to generate localized narrow-band low-frequency shear waves within tissue noninvasively, is by the modulated radiation force, resulting from the interference of two confocal quasi-CW ultrasound beams of slightly different frequencies. By using approximate viscoelastic Green's functions, investigations of the properties of the propagated shear-field component at the fundamental modulation frequency were previously reported by our group. However, high-amplitude source excitations may be needed to increase the signal-to-noise-ratio for shear-wave detection in tissue. This paper reports a study of the generation and propagation of dynamic radiation force components at harmonics of the modulation frequency for conditions that generally correspond to diagnostic safety standards. We describe the propagation characteristics of the resulting harmonic shear waves and discuss how they depend on the parameters of nonlinearity, focusing gain, and absorption. For conditions of high viscosity (believed to be characteristic of soft tissue) and higher modulation frequencies, the approximate shear wave Green's function is inappropriate. A more exact viscoelastic Green's function is derived in k-space, and using this, it is shown that the lowpass and dispersive effects, associated with a Voigt model of tissue, are more accurately represented. Finally, it is shown how the viscoelastic properties of the propagating medium can be estimated, based on several spectral components of the shear wave spectrum.     

Propagation of a Mode-II crack in a viscoelastic medium with different bulk and shear relaxation

A solution for a crack propagating under shear-loading in an isotropic viscoelastic medium with different relaxation under volume and shear deformations is presented. The medium is infinite and the semi-infinite crack propagates along the x ₁-axis at constant speed V, which may take any value up to the speed of dilatational waves. The requisite Riemann–Hilbert problem for the steady-state case has been solved and the asymptotics of the stress component σ₁₂ directly ahead of the crack and at infinity have been obtained.