Correspondence Principle Of Linear Viscoelasticity Research Articles

A viscoelastic Eshelby inclusion model and analysis of the Cell-in-Gel system

We develop a viscoelastic generalization of the elastic Eshelby inclusion solution, where the inclusion and surrounding matrix are two different viscoelastic solids and the inclusion’s eigenstrain is a time-periodic oscillatory input. The solution exploits the Correspondence Principle of Linear Viscoelasticity and a Discrete Fourier Transform to efficiently capture the steady-state oscillatory behavior of the 3-D mechanical fields. The approach is illustrated here in the context of the recently-developed in vitro Cell-in-Gel system, where an isolated live cardiomyocyte (the inclusion) is paced to contract periodically within a soft hydrogel (the matrix), for the purpose of studying the effect of mechanical load on biochemical signals that regulate contractility. The addition of viscoelasticity improves the fidelity of our previous elastic Eshelby inclusion analysis of the Cell-in-Gel system by accounting for the time-varying fields and the resulting hysteresis and dissipated mechanical energy. This mathematical model is used to study the parametric sensitivities of the relative stiffness of the inclusion, the inclusion’s aspect ratio (slenderness), and the cross-link density of the hydrogel matrix.

A comparative study of deformation field due to center of dilatation and center of rotation in a viscoelastic half-space model

Closed-form analytic expressions for the displacement and stress fields due to two axially symmetric sources, namely, the center of dilatation, and center of rotation in an elastic half-space are obtained using Galerkin vector approach. The correspondence principle of linear viscoelasticity is used to obtain the viscoelastic displacement and stress fields. A comparative study of both these sources has been done graphically. Our analysis revealed that the center of dilatation is more efficient than the center of rotation in generating radial displacement. The results are valid for arbitrary values of the Poisson's ratio and relaxation time.

Deformation of two welded half-spaces caused by a centre of rotation source

Closed-form analytic expressions for Papkovich-Neuber displacement potentials, displacements, and stresses due to the centre of rotation source in an elastic half-space overlying an elastic half-space are obtained. The elastic results are valid for different values of Poisson's ratio and a change in the rigidity of the two half-spaces. Further, the correspondence principle of linear viscoelasticity is used to obtain the corresponding solutions in an elastic half-space overlying a viscoelastic half-space. The viscoelastic results are valid for different values of relaxation time and a change in the rigidity of the two half-spaces. The variation of the displacements and stresses with the epicentral distance as well as with the relaxation time are studied. The magnitude of displacement and normal stress components assume a maximum value for the Maxwell model and minimum value for the Kelvin model whereas the shear stresses assume a minimum value for the Maxwell model and maximum value for the Kelvin model.

Quasi-Static Strain and Stress Fields due to a Moment Tensor in Elastic–Viscoelastic Layered Half-Space

We derived explicit expressions in the time domain for 3-D quasi-static strain and stress fields, due to a point moment tensor source in an elastic surface layer overlying viscoelastic half-space under gravity. The expressions of strain in the elastic surface layer were directly obtained from the expressions of displacement in our previous paper. The conversion of strain into stress is easy, because the stress–strain relation of elastic material is linear. In the viscoelastic substratum, the expressions of strain were obtained by applying the correspondence principle of linear viscoelasticity to the associated elastic solution. The strain–stress conversion is not straightforward, as the stress–strain relation of viscoelastic material is usually given in a differential form. To convert strain into stress, we used an integral form of the stress–strain relation instead of the usual differential form. The expressions give the responses of elastic half-space at \( t = 0 \), and the responses of an elastic plate floating on non-viscous liquid at \( t = \infty \). The moment tensor is rationally decomposed into the three independent force systems, corresponding to isotropic expansion, shear faulting and crack opening, and so the expressions include the strain and stress fields for these force systems as special cases. As the first numerical example, we computed the temporal changes in strain and stress fields after the sudden opening of an infinitely long vertical crack cutting the elastic surface layer. Here, we observe that the stress changes caused by the sudden crack opening gradually decay with time and vanish at \( t = \infty \) everywhere. After the completion of stress relaxation, a characteristic pattern of shear strain remains in the viscoelastic substratum. Since the strain and stress fields at \( t = \infty \) can be read as the strain- and stress-rate fields caused by steady crack opening, respectively, this numerical example demonstrates the realization of a steady stress state supported by steady viscous flow in the asthenosphere, associated with steady seafloor spreading at mid-ocean ridges. For the second numerical example, we computed the temporal changes in strain and stress fields after the 2011 Tohoku-oki mega-thrust earthquake, which occurred at the North American-Pacific plate interface. In this numerical example, the stress changes caused by coseismic fault slip vanish at \( t = \infty \) in the viscoelastic substratum, but remain in the elastic surface layer. The coseismic stress changes (and also strain changes) in the elastic surface layer diffuse away from the source region with time, due to gradual stress relaxation in the viscoelastic substratum.

Surface viscoelasticity and effective properties of thin-walled structures at the nanoscale

We discuss the influence of surface viscoelasticity effects on the effective properties of materials such as effective bending stiffness of plates or shells. Viscoelastic properties in the vicinity of the surface can differ from the properties of the bulk material. This difference influences the behavior of nanostructural elements. In particular, the surface viscoelastic stresses are responsible for the size-depended dissipation of nanosized structures. Using the extension of the Gurtin–Murdoch model and the correspondence principle of linear viscoelasticity we derive the expressions of the stress and couple stress resultant tensors for shear deformable plates and shells.

General expressions for internal deformation due to a moment tensor in an elastic/viscoelastic multilayered half-space

SUMMARY In the framework of elasticity theory any indigenous source can be represented by a moment tensor. We have succeeded in obtaining general expressions for internal deformation due to a moment tensor in an elastic/viscoelastic multilayered half-space under gravity. First, starting from Stokes’ classical solution, we obtained the expressions for static displacement fields due to a moment tensor in an infinite elastic medium. Then, performing the Hankel transformation of the static solution in Cartesian coordinates, we derived static displacement potentials for a moment tensor in cylindrical coordinates. Second, representing internal deformation fields by the superposition of a particular solution calculated from the displacement potentials and the general solution for an elastic multilayered half-space without sources, and using the generalized propagator matrix method, we obtained exact expressions for internal elastic deformation fields due to a moment tensor. Finally, applying the correspondence principle of linear viscoelasticity to the elastic solution, we obtained general expressions for quasi-static internal deformation fields due to a moment tensor in an elastic/viscoelastic multilayered half-space. The moment tensor can be generally decomposed into the three independent force systems corresponding to isotropic expansion, crack opening and shear faulting, and so the general expressions include internal deformation fields for these force systems as special cases. As numerical examples we computed the quasi-static internal displacement fields associated with dyke intrusion, episodic segmental ridge opening and steady plate divergence in an elastic‐viscoelastic two-layered half-space. We also demonstrated the usefulness of the source representation with moment tensor through the numerical simulation of deformation cycles associated with the periodic occurrence of interplate earthquakes in a ridge-transform fault system.

Displacement, strain and stress fields due to shear and tensile dislocations in a viscoelastic half-space

Okada (1992) provided expressions for the displacement and strain fields due to a finite rectangular source in an elastic, homogeneous and isotropic half-space. Starting with these results, we applied the correspondence principle of linear viscoelasticity to derive the quasi-static displacement, strain and stress fields in a viscoelastic, homogeneous and isotropic half-space. We assume that the medium deforms viscoelastically with respect to both the shear and the normal stresses but keeps a constant bulk modulus; in particular, the shear modulus relaxes as Maxwell fluid. We presented the viscoelastic effect on displacement, displacement gradient and stress fields, for a choice of parameter values. The viscoelastic effect due to the sudden dislocation reaches a limit value after about 10 times the Maxwell time. The expressions obtained here provide tools for the study of viscoelastic relaxation of lithosphere associated with seismic and volcanic phenomena.

Constitutive modeling of porous viscoelastic materials

The effect of porosity on the constitutive response of an isotropic linearly viscoelastic solid that obeys a constitutive law of the standard differential form is investigated under small strain deformation conditions. The correspondence principle of linear viscoelasticity is used to solve the viscoelastic boundary value problem at a unit cell containing a spherical void and loaded axisymmetrically by macroscopic stresses. The results are used to devise a constitutive potential for the description of the porous material for any arbitrary combination of hydrostatic and deviatoric loadings, and the associated 3-D constitutive relationship is determined in the Laplace transform domain. Inversion to the time domain yields the constitutive law of the porous material as a function of porosity in the standard form of convolution integrals. The presence of porosity establishes relaxation time scales for the porous body that differ from the relaxation time of the pure matrix material and brings about a viscous character to the overall hydrostatic response. The numerical implementation of the model in a general purpose finite element code is outlined. The model is used to predict the response of a porous solid propellant material in uniaxial tension and cyclic loading at room temperature.

Quasi-static internal deformation due to a dislocation source in a multilayered elastic/viscoelastic half-space and an equivalence theorem

We have obtained general expressions for quasi-static internal deformation fields due to a dislocation source in a multilayered elastic/viscoelastic half-space under gravity by applying the correspondence principle of linear viscoelasticity to the associated elastic solution (Fukahata & Matsu'ura 2005). The use of the upgoing propagator matrix for the region below the source and the downgoing propagator matrix for the region above the source in the derivation of mathematical expressions guarantees the numerical stability of the obtained viscoelastic solution over the whole region. The viscoelastic deformation fields due to a dislocation source tend to a certain steady state with the progress of viscoelastic stress relaxation. The completely relaxed viscoelastic solution can be directly obtained from the associated elastic solution by taking the rigidity of the elastic layer corresponding to a Maxwell viscoelastic layer to be zero. We gave an explicit mathematical proof of this theoretical relationship, which we named the equivalence theorem, on the basis of the correspondence principle of linear viscoelasticity and the limiting value theorem of the Laplace transform. The equivalence theorem is applicable not only to the elastic-viscoelastic stratified medium but also to general elastic and linear-viscoelastic composite media. As numerical examples we show the quasi-static internal displacement fields due to strike-slip motion on a vertical fault and dip-slip motion on a subduction plate boundary in an elastic surface layer overlying a viscoelastic half-space. The temporal variation of the computed deformation fields shows that the effective relaxation time of the elastic-viscoelastic system is much longer than the Maxwell relaxation time defined by the ratio of viscosity to rigidity in the viscoelastic layer.

General expressions for internal deformation fields due to a dislocation source in a multilayered elastic half-space

SUMMARY We have obtained general expressions for internal displacement and stress fields due to a point dislocation source in a multilayered elastic half-space under gravity. Most previous expressions for the internal deformation fields were obtained by applying one of two different types of Thomson‐Haskell propagator matrix, namely the up-going propagator matrix proposed by Singh (1970) and the down-going propagator matrix proposed by Sato (1971). The solution derived with the up-going propagator matrix is stable below the source, but becomes unstable above the source. In contrast, the solution derived with the down-going propagator matrix is stable above the source, but becomes unstable below the source. We succeeded in unifying the up-going and the down-going propagator matrices into a generalized propagator matrix, and applied it to obtain general expressions that are stable at any depth. By integrating the effects of point sources distributed along an infinitely long horizontal line, we also obtained general expressions for a line dislocation source. We give some examples of internal displacement fields computed with these expressions to examine the effects of layering. Applying the correspondence principle of linear viscoelasticity to the derived elastic solutions, we can obtain the internal viscoelastic displacement and stress fields due to dislocation sources.

Micromechanical modeling of viscoelastic composites with compliant fiber–matrix bonding

The micromechanical method of cells is used to calculate the average time-dependent constitutive properties of the homogenized substitute continuum from the viscoelastic material properties and volume fractions of the individual phases as well as from the interphase behavior of polymeric fiber composites. Two approaches are investigated to establish and solve the micromechanical model equations of the rate-dependent material phases. The first one is based on the L aplace-transformation of the time-dependent material functions and the application of the correspondence principle of linear viscoelasticity to the governing equations of the micromechanical model. In the second approach a numerical time integration scheme is developed to compute the convolution integrals. For the study of the influence of compliant fiber–matrix bonds alternative models are proposed: A representative volume element with three constituents accounts for the properties of the flexible interphase between the fibers and the matrix. The three-phase model is then compared to a two-phase model with a time-dependent flexible interface to simulate the compliant fiber–matrix bond. Various examples show the performance of both models and the numerical algorithms presented.

Effective Electroelastic Properties of a Piezocomposite with Viscoelastic and Dielectric Relaxing Matrix

We use the correspondence principle of linear viscoelasticity and the Mori-Tanaka averaging method to determine the effective electroviscoelastic moduli of a piezocomposite made of piezoceramic (PZT) inclusions and a viscoelastic matrix whose dielectric constants also vary with time t. For elliptic cylindrical PZT inclusions, closed form expressions for the electroviscoelastic moduli are derived. It is found that for the 1-3 piezocomposite, the relaxation of the dielectric constants of the matrix affects only the dielectric moduli and the shear components of the piezoelectric and the viscoelastic moduli of the piezocomposite. However, the viscoelasticity of the matrix influences every effective electroelastic modulus of the piezocomposite. The relaxation times of any two effective moduli are different, and the piezocomposite is an orthotropic material. For time-harmonic loading of the piezocomposite studied, only frequencies in the range of 10 2- 102 Hz strongly influence the effective storage and the effective loss moduli. Whereas thin PZT ribbons provide the most mechanical strength to the piezocomposite, they give the least piezoelectric effect.

Concentration dependence of the linear viscoelastic properties of particle suspensions

The response under small amplitude oscillatory deformations of a suspension of non-Brownian spheres dispersed in a viscoelastic fluid is investigated. The correspondence principle of linear viscoelasticity is used to derive a simple constitutive model from a model for a suspension in a Newtonian liquid. The theory predicts that for a specific particulate system the concentration dependence of the viscoelastic properties should collapse to a single master curve when the values are normalized with those of the carrier fluid alone. Measurements with the micro-Fourier rheometer using oscillatory squeeze flow are carried out on two suspensions of 60 and 80 μm sized particles dispersed in polymeric fluid and in silicon oil, and the master curve is verified.

Ground deformation in a viscoelastic medium composed of a layer overlying a half-space: a comparison between point and extended sources

SUMMARY We obtain and compare analytical and numerical solutions for ground displacement caused by an overpressurized magma chamber placed in a linear viscoelastic medium composed of a layer over a half-space. DiVerent parameters such as size, depth and shape of the chamber, crustal rheology and topography are considered and discussed. Numerical solutions for an axisymmetric extended source are computed using a finite element method (FEM). Analytical solutions for a point source are obtained using the dislocation theory and the propagator matrix technique. In both cases, the elastic solutions are used together with the correspondence principle of linear viscoelasticity to obtain the solution in the Laplace transform domain. Viscoelastic solutions in the time domain are derived inverting the Laplace transform using the Prony series method. The diVerences between the results allow us to constrain the applicability of the point source and the flat surface hypothesis, which are usually implicitly assumed when analytical solutions are derived. The eVect of the topography is also considered. The results obtained show that neglecting the topographic eVects may, in some cases, introduce an error greater than that implicit in the point-source hypothesis. Therefore, for an adequate modelling and interpretation of the time-dependent displacements, topography must be considered.

Measurement of the viscoelastic properties of bituminous materials using an oscillating needle technique

An embedded oscillating needle is used to measure the dynamic viscoelastic properties of a stiff bituminous material. A Micro-Fourier Rheometer was used to cause the embedded needle to undergo pseudorandom small amplitude oscillations in the axial direction with measurement of the instantaneous resistance force. The phase and magnitude of the force signal are used to calculate the storage and loss moduli. A theoretical framework for this technique is developed from the Mindlin solution coupled with slender body theory, and the correspondence principle of linear viscoelasticity. Experiments are performed on neat bitumen binders as well as mixtures of glass spheres in bitumen; the results show that the presence of the glass spheres dramatically increases the viscoelastic response functions. The results agree reasonably well with those obtained using the parallel plate squeezing mode.

Antisymmetric vibrations of multilayered conical shells with constrained viscoelastic layers

The governing equations of motion for antisymmetric vibrations of a general multilayered conical shell consisting of an arbitrary number of specially orthotropic material layers have been derived using the variational principles. The analysis considers bending, extension, in-plane shear and transverse shear deformations in each of the layers and also includes its rotary, longitudinal translatory and transverse inertias. The Galerkin method has been applied for finding the approximate solution of the shell. The correspondence principle of linear viscoelasticity has been used for evaluating the damping effectiveness of the shells consisting of elastic and viscoelastic layers. A computer program has been developed for antisymmetric vibrations of a general multilayered conical shell consisting of an arbitrary number of elastic and viscoelastic layers. Variation of resonance frequencies and the associated system loss factors with shear parameter and thickness ratio parameter for all families of modes of vibration has been reported for three-, five- and seven-layered conical shells with three sets of classical end conditions: simply supported at both ends, clamped-clamped and free-free.

VIBRATION AND DAMPING ANALYSIS OF FIBER REINFORCED COMPOSITE MATERIAL CONICAL SHELLS

A vibration and damping analysis based on individual layer deformation is presented for axisymmetric vibrations of laminated composite and material conical shells. The analysis includes consideration of bending, extension and transverse shear deformations in each of the laminate layers and also includes its rotary, meridional and circumferential translatory and transverse inertias. The Galerkin method has been applied for finding the approximate solutions for the simply supported conical shells. The correspondence principle of linear viscoelasticity has been used to evaluate the damping effectiveness of the shell. The variations of the resonance frequencies and of the associated system loss factors with fiber orientation angles and cone apex angles for axisymmetric vibrations of cross-ply and angle-ply fiber reinforced plastic conical shells are reported.

Surface deformation due to a strike‐slip fault in an elastic gravitational layer overlying a viscoelastic gravitational half‐space

We extend a technique previously used to model surface displacements resulting from thrust faulting in an elastic‐gravitational layer over a viscoelastic‐gravitational half‐space to the case of strike‐slip faulting. The method involves the calculation of the Green's functions for a strike‐slip point source contained in an elastic‐gravitational layer over a viscoelastic‐gravitational half‐space. The correspondence principle of linear viscoelasticity is applied to introduce time dependence. The resulting Green's functions are then integrated over the source region to obtain the near‐field displacements. Several sample calculations are presented involving 90° and 30° dipping faults and ruptures completely and partially through the elastic layer. We also illustrate the time dependent deformation due to a buried fault. Results show that the use of a viscoelastic half‐space underlying an elastic layer introduces a long wavelength component into the deformation field [Cohen and Kramer, 1984], even in cases of non vertical strike‐slip fault and inclusion of gravitational effect, that cannot be modeled by purely elastic techniques. Calculations have shown that vertical postseismic displacement is insignificant and that the horizontal movement is about the same magnitude as the coseismic strike‐slip displacement. The inclusion of gravity affects the horizontal displacement due to vertical strike‐slip faulting in far field and the vertical displacement for dipping strike‐slip faulting in near‐field. The computed results have been fit to the Global Positioning System measurements of the Landers earthquake taken shortly after the main shock, assuming a relaxation time of the order of days. This relaxation time is considerably shorter than times of the order of years to decades found in previous studies. The major differences between this detailed three‐dimensional and simplified two‐dimensional model are the decay of magnitude in displacement field and the distinct displacement pattern in the regions beyond the fault tip. The displacement field due to the cyclic earthquakes was constructed by considering the finite fault length and inclusion of gravity. It is found that the displacement field is dominated by the plate motion in the case of short recurrence time. On the other hand, a “looping” and migrating pattern in the displacement field is found in the case of very long recurrence time, which is not seen in those 2D simplified models.

On the Prediction of Damping and Vibratory Behaviour of Free-Free Multilayered Conical Shells

An analysis is presented for the prediction of resonance frequencies and the associated system loss factors for all families of modes of vibrations in free-free multilayered conical shells, which are frequently used in aircraft, missiles and other allied systems. Because of the spin of a high order, missiles and gun-launched shells are subjected to torsional vibrations (Axisymmetric). The governing equations of motion for the axisymmetric vibrations of a general multilayered conical shell have been derived using Hamilton's vibrational principle. The solution is obtained by utilising simple trignometric series modal assumptions in Galerkin's procedure. The Correspondence principle of linear viscoelasticity for harmonic motions is used for evaluating the damping effectiveness of shells with elastic and viscoelastic layers. The resonance frequencies and the associated system loss factors for three-, five-and seven-layered conical shells with free edges are evaluated and their variations with geometric parameters are investigated. An increase in the number of layers (N) increases maximum obtainable ( system loss factor) for most of the modes with proper selection of geometric parameters, but increasing N beyond 5 yields only marginal increase in (system loss factor). Uniformly high values of (system loss factor) are obtained for all modes of vibration for high values of thickness ratio parameter. For thick shells, more layers are advisable for relatively high values of (system loss factor) for all modes.

Finite Element Viscoelastic Analysis of Temperature and Moisture Effects in Electronic Packaging

Several workers have demonstrated the presence of singular stress fields near the edge of a bimaterial interface subject to thermal gradients. Many of the analyses in the literature are limited to linear elastic materials and are useful as first estimates of stresses in such assemblies. However, a time-dependent stress analysis is necessary when viscoelastic materials such as polymer films are bonded to elastic substrates. This paper shows the relevance of viscoelastic modeling in demonstrating the effects of combined temperature and moisture loading on bimaterial interfaces and viscoelastic films sandwiched between elastic substrates. A 2-D finite element method for linear hygrothermoviscoelasticity based on an incremental creep strain rate form is developed. The correspondence principle of linear viscoelasticity is invoked and comparisons between FEA and analytical solutions are demonstrated. Moisture is modeled using Fick’s law and the thermorheologically simple material (TSM) postulate is extended to incorporate a moisture shift function in analogy with temperature. The analysis of a polyimide film sandwiched between elastic substrates subject to a thermal cycle with diffusion from the free edge shows stress reversals for the peel stresses that may explain failure modes not anticipated by an elastic analysis.