Linear Viscoelastic Analysis Research Articles

On fundamental solutions in time-domain boundary element analysis of linear viscoelasticity

This paper presents an efficient method for the computation of the fundamental solutions in time-domain boundary integral equation for viscoelastic problems. The method proposed is based on the Laplace transform and the correspondence principle. The relaxation function is expanded in a sum of exponentials and the transformed fundamental solutions are inverted numerically into real time space. The proposed procedure requires a small computational effort and it is applicable in time-domain boundary element analysis of realistic viscoelastic problems. Numerical results of an example problem show the effectiveness and applicability of the proposed method.

Prevention of Stress Relaxation in Viscoelastic Structures

This paper presents a new problem in linear viscoelastic structural analysis. Homogeneous structures with rigid restraints, where stress relaxation is prevented at a time after loading, are studied. This restriction produces time-dependent increments in the initial displacements. In addition, if the structure has more than one initial redundant reaction, a significant stress redistribution also occurs. Assuming the increments of displacement of the cross section where the stress relaxation is prevented as unknown, the equation of equilibrium assumes the form of an integral Volterra equation. This mathematical formulation leads to the definition of an integral function (which only takes into account creep effects) that make resolving the problem by affinity to elastic quantities easy. The problem examined is related by duality to the well-known problem of delayed restraints (also called change of structural system), in which the additional restraints, introduced after loading, prevent the creep increments of displacements at their points of application. Finally, the relation existing between the relaxation function and the integral function obtained is shown.

Linear viscoelastic analysis with random material properties

Analytical studies are presented which extend the elastic-viscoelastic analogies to stochastic processes caused by random linear viscoelastic material properties. Separation of variable as well as integral transform correspondence principles are formulated and discussed in detail. The statistical differential equation of the moment characteristic functional is derived, but rather than solving the highly complex functional equation, solutions are formulated in terms of first and second order statistical properties. Both Gaussian and beta distributions are considered for the probability density distributions of creep and relaxation functions and their effectiveness is evaluated. In order to illustrate the developed general theory, specific examples of beam bending and pressurized hollow cylinders are solved. The influence of various parameters contributing to the creep and relaxation correlation functions is evaluated and the relationship between deterministic and stochastic bounds is also investigated. It is shown that deterministic bounds based on material data spread are unrealistic in the presence of random viscoelastic properties, since the do not correctly predict the limits of this stochastic process.

Boundary integral equation analysis in linear viscoelasticity: variational and saddle point formulations

This paper presents a symmetric, double integration boundary integral equation (BIE) approach to linear viscoelastic analysis. The time evolution of the boundary unknowns is shown to be characterized by a variational theorem over an arbitrary time interval and by a min-max theorem from the time origin to infinite (minimum for displacements, maximum for tractions). Boundary element (BE) discretization in space and time leads to algebraic equations with symmetric coefficient matrix.

Thermal Stress in Semiconductor Encapsulating Materials

To examine molding-compound-induced stresses, and to determine the influence of the time-dependent and temperature-dependent thermomechanical properties of the molding compound, a linear viscoelastic analysis has been applied to a simple bilayer structure. Stress generation is examined over the temperature range from -150 degrees C to the molding temperature (180 degrees C), based on a complete consideration of the thermomechanical properties of a molding compound. Stress calculations are compared to experimentally determined radii of curvature of the bilayer structure as a function of temperature. Both calculated and experimental results demonstrate the necessity to consider a complete materials description in order to adequately describe stress generation. The importance of the viscoelastic contribution of the transition range and the temperature-dependent elastic contribution of the glassy state to stress generation are highlighted.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Thermal stress in semiconductor encapsulating materials

To examine molding-compound-induced stresses, and to determine the influence of the time-dependent and temperature-dependent thermomechanical properties of the molding compound, a linear viscoelastic analysis has been applied to a simple bilayer structure. Stress generation is examined over the temperature range from -150 degrees C to the molding temperature (180 degrees C), based on a complete consideration of the thermomechanical properties of a molding compound. Stress calculations are compared to experimentally determined radii of curvature of the bilayer structure as a function of temperature. Both calculated and experimental results demonstrate the necessity to consider a complete materials description in order to adequately describe stress generation. The importance of the viscoelastic contribution of the transition range and the temperature-dependent elastic contribution of the glassy state to stress generation are highlighted. >

Linear viscoelastic analysis in time domain by boundary element method

Isotropic, linear viscoelastic problems are formulated in the time domain by a boundary element method. The viscoelastic fundamental solutions are represented in terms of the constant coefficients of relaxation functions. Starting from the reciprocal work theorem, an alternative form of boundary integral equation is derived by integration by parts. This form requires the regularity of field variables to be one order less than that in the usual formulation. A time marching process is incorporated in the numerical method. The matrices in the resulting algebraic equations are constant over time, except for the matrix for past history. Four test problems in two dimensional space are solved for displacement and stress and checked against exact solutions. It is shown that the formulation is economical and generates stable solutions of good accuracy. The effect of time steps with regard to spatial discretizations is also discussed.

Analysis of Linear Viscoelasticity of a Crosslinking Polymer at the Gel Point

We suggest a very simple memory integral constitutive equation for the stress in crosslinking polymers at their transition from liquid to solid state (gel point). The equation allows for only a single (!) material parameter, the strength S[Pas1/2], and it is able to describe every known viscoelastic phenomenon at the gel point. Measurements were performed on polydimethylsiloxane model networks with balanced stoichiometry for which the crosslinking reaction has been stopped at different degrees of conversion. At the gel point, the loss and storage moduli were found to be congruent and proportional to ω1/2 over a wide range of temperature (−50°C to +180°C) and five decades of frequency ω. The hypothesis is made that this behavior is valid in the entire range 0<ω<∞. This congruence hypothesis is consistent with the Kramers‐Kronig relation and leads to a constitutive equation which shows that, for our polymer, congruent functions G′(ω)=G″(ω) are as much a rheological property at the gel point as are infinite ...

Linear viscoelastic analysis in rock-support system of roadway : Liu, B; Lin, D In: Application of Analytical Methods to Mining Geomechanics (paper to the Proceedings of the 7th Plenary Session of the International Bureau of Rock Mechanics, World Mining Congress, Katowice, 24–26 June 1981)P117–123. Publ Rotterdam: A. A. Balkema, 1982

Linear viscoelastic analysis in rock-support system of roadway : Liu, B; Lin, D In: Application of Analytical Methods to Mining Geomechanics (paper to the Proceedings of the 7th Plenary Session of the International Bureau of Rock Mechanics, World Mining Congress, Katowice, 24–26 June 1981)P117–123. Publ Rotterdam: A. A. Balkema, 1982

Response of polycarbonate, poly(methylmethacrylate), and epoxy resins to dynamic contact loading I. Experimental results

AbstractMeasurements of the resistance force to a wide range of contact loadings have been performed on poly(methylmethacrylate), polycarbonate, and epoxy specimens. Rigid spherical indenters of 4.5, 9, and 18 mm diameter, as well as cones of various apex angles, were driven into thick specimens at constant velocity over the range 0.005 to 150000 mm/min. The objective of this study was to develop a data base useful for the understanding and analysis of the phenomena involved when polymeric solids are impacted by small hard bodies. Spheres and cones were used because these symmetrical shapes offer the best chance for analysis in terms of viscoelastic extensions of the classical Hertz elastic contact problem. Measurements covered penetration depths from first detection of contact, where the strains are smallest and the material response should most likely be interpretable in terms of linear viscoelastic analysis, to much greater depths where the strains far exceed the yield values. The observations are tabulated, as are the values of parameters that allow the data to be modeled to within experimental error. These parameters fit into a specially developed empirical law designed to facilitate subsequent treatment of the data using Laplace transform methods. An analysis is offered to try to sort out time‐dependent force relaxation from depth‐dependent force increase during the progress of the penetration. The results of the response at small penetration depths are in broad accord with published relaxation moduli. However, the dependence of force upon indenter geometry does not follow simple expectation.

Boundary element method for linear visco-elastic stress analysis in a rock mass and its application in rock engineering

Boundary element method for linear visco-elastic stress analysis in a rock mass and its application in rock engineering

Boundary element method applied to linear viscoelastic analysis

This paper analyses linear viscoelastic problems using the boundary element method together with the numerical inversion of the Laplace transform. This method reduces the amount of manual and computing time as compared with the finite element method. Two numerical examples are included to demonstrate the validity and accuracy of the method.

Analysis of the Viscoelastic Response of Composite Laminates During Hygrothermal Exposure

The linear viscoelastic response of polymeric matrix laminated com posites to hygrothermal exposure is investigated within the framework of lamination theory. Experimentally measured curvature changes in non- symmetric GY70/339 composite laminates during long term exposures to elevated hygrothermal conditions are shown to be accurately predicted by the linear viscoelastic analysis. Application of the analysis is made to prob lems involving 1) dimensional stability and alteration of residual stresses, 2) the strain rate dependence of tensile response, and 3) the effect of residual thermal stress relaxation upon creep in laminated composite plates under mechanically applied loads.

Linear Viscoelastic Analysis by Using Numerical Inversion of the Laplace Transform (By Takashi Kusama, Yasushi Mitsui and Shun-ya Yoshida)

Linear Viscoelastic Analysis by Using Numerical Inversion of the Laplace Transform (By Takashi Kusama, Yasushi Mitsui and Shun-ya Yoshida)

Application of Catastrophe Theory to Draw Resonance

Catastrophe theory was successfully applied to the analysis of draw resonance phenomena. It was shown that the potential of cusp catastrophe was appropriate to the analysis of oscillating viscoelasticity appearing in the above-mentioned phenemena. Moreover, basic behavior such as stress relaxation is discussed in regard to this potential. In these analyses, no relaxation spectrum is introduced. However, an apparent spectrum can be obtained if the results of stress relaxation are analyzed by a procedure similar to that used in the analysis of linear viscoelasticity. But a spectrum may not always exist even if derived analytically.

Estimating settlements in dry sands during earthquakes

An analytical method for predicting the settlement of a horizontal stratum of dry sand subjected to earthquake excitation is presented. An equivalent linear viscoelastic dynamic analysis is used to determine the time-history of shear strains at various levels within the stratum. The vertical strains caused by these shear strains are calculated by an empirical relationship shown to be in close agreement with laboratory test data over a wide range of relative densities. This allows the settlement distribution with depth to be calculated as a function of time.The method was applied to a 50 ft (15 m) stratum of sand. A base acceleration corresponding to 0–10 s of the N–S component of El Centro earthquake was used. Settlement distribution vs. depth, as well as surface settlements for various relative densities and earthquake scaling factors are shown. The effect of surcharge loading on settlements is investigated in an approximate manner.The results may be used to estimate differential settlement of a structure due to an earthquake. The results show that while the compaction of the upper layers may be sufficient to reduce the settlements due to static loads to tolerable amounts, deep-seated settlements will still occur during an earthquake.

A finite element analysis for time‐dependent problems

AbstractThe finite element method is applied to dynamic linear viscoelastic analysis. The arising system of differential equations is integrated by means of the Laplace transformation. The direct numerical evaluation of the inversion integral seems to be appropriate in case of vibrational problems. The method is applied to examples of transient vibrations of elastic bodies. Initial conditions and arbitrary damping matrices can be considered.

Analysis of centrifugal stresses in anisotropic viscoelastic cylinder.

HE stress analysis for anisotropic viscoelastic materials has become the subject of increasing concern in the past few years. The ever increasing use of composites and oriented plastics as structural members has made it necessary to study the effect of anisotropy as well as viscoelasticity on the performance of these components. Problems in linear isotropic viscoelasticity have been analyzed by many authors, such as Lee, 1 Hilton,2 and Bland,3 under various loading conditions. However, similar attempts in the analysis of anisotropic viscoelasticity have been frustrating because of the difficulties faced in the use of Laplace transform, particularly in the inversion of solutions. As a result, even the simple problems which can be solved explicitly, become important in understanding the interaction of anisotropy and the viscoelastic behavior of materials.4 ~ 6 In this paper, some elementary solutions of stress analysis problems for an anisotropic viscoelastic cylinder are given. Specifically, analysis for a long hollow cylinder, rotating about its axis, is considered. In the course of the analysis, the constitutive equations of the theory of infinitesimal viscoelasticity exhibiting a general anisotropic behavior are assumed. These are expressed in terms of different time-dependent modulus functions. A particular type of spacial nonhomogeneity, where the material moduli are assumed to be representable as a production of functions of time and coordinates, is considered. Through the application of integral (Laplace) transform the problem is then solved. Although the analysis is made for a general linear viscoelastic medium, for simplicity, numerical results are presented for a material having simple viscoelastic behavior. Anisotropic Viscoelasticity The basic problem in quasi-static linear viscoelastic stress analysis is to determine a stress field atj(xk, t) and the displacement field wf(xfc, t) as functions of space variables xk and time t. The governing equilibrium equations, which are independent of material constitution, are the displacement-strain relations and the principles of linear momentum and moment of momentum. They are in the usual notation

An experimental study of the effects of prestrain on the propagation of small disturbances in neoprene filaments

A small disturbance was caused to propagate along a long, slender, prestrained Neoprene filament. The particle velocity of the pulse was measured at two stations along the length of the filament by means of electromagnetic transducers which operate on the Faraday principle. Particle velocity vs. time data were obtained from oscilloscope photographs of the transducer outputs for each level of prestrain from 0 percent up to 400 percent engineering strain. The two particle-velocity records for each level of prestrain were subjected to linear viscoelastic analysis which employed the use of numerical Fourier transforms of the particle-velocity records. Computer programs were written which allowed computation of the numerical transforms and from them the computation of the phase velocity and attenuation coefficients of the material over the narrow frequency bandwidth of the Fourier spectra of the particlevelocity pulses. Data analysis revealed that, at a given frequency, the phase velocity increases significantly and that the attenuation coefficient decreases significantly with an increase in prestrain level over the range of prestrains of the tests.

Strain sensitive response of filled elastomers

The strain dependence of the elastic and the anelastic behavior of filled elastomers is discussed in the light of recent experiments on carbon-black filled rubber as well as on sodium chloride filled polyurethane. The latter experiments suggest that the apparent creep of filled elastomers is the manifestation of the gradual change of the elastic moduli resulting from the loss with strain of effective filler matrix interaction, rather than a truly viscoelastic phenomenon. N recent years linear viscoelastic analysis of an assumedly homogeneous, isotropic and thermorheologically simple medium has been accepted as a standard procedure for evaluating the structural integrity of solid-propellant rocket motors.1 Although it is generally recognized that, on the microscale, the solid propellant is neither continuous nor homogeneous, the previously mentioned idealization is assumed to be permissible with important exceptions, such as stress induced anisotropy and fracture origins, on the basis of the purported similarity of the effect of microstructure in such materials with those in concrete, asphalt or for that matter metals7' (Ref. 1, p. 785). Coupling this assumption with the desire to introduce the least amount of analytical complexity, the way is opened for the application of the classical methods of linear viscoelastic analysis using the various well established procedures of material characterization and the correspondence principle between elastic and linear viscoelastic solutions of boundary value problems, the implicit assumption being that the relevant range of structural performance coincides, for all practical purposes, with the range of infinitesimal or, at least, of very small strains. Once these premises of the structural integrity analysis have been accepted, the further development is exclusively concerned with increasingly sophisticated analytical methods within the given framework, with their computerization and with increasingly elaborate generalizations. Unfortunately, the assumptions which have produced an accessible analytical theory have, at the same time, dissociated the analytical results from reality, simply because they have become irrelevant to the principal engineering purpose of the analysis, which is a realistic evaluation of structural integrity. The main reason for this dissociation is the fact that the highly filled elastomeric propellant can be represented by the linear viscoelastic medium only within the range of infinitesimal strains, which is not the range that is of particular relevance in structural integrity evaluation. The structurally relevant strain ranges are those of a) shorttime straining with strain rates of the order of up to 10 in./ in./sec during firing and flight, with maximum strains at the inner bore of the order of 10 to 60% reached in tens of milliseconds, b) of thermal strain cycling at rates of the order of 10~3 to 10~ 5 in./in./sec and maximum strains in the one to several percent range, and c) of long-time storage strains of a similar order of magnitude but considerably lower strain rates and material degradation. Although in materials other than solid propellants, the structural integrity analysis of which requires the consideration of