On the direct estimation of creep and relaxation functions

Two alternative approaches for estimating linear viscoelastic material functions from a single experiment under random excitation are derived and analyzed. First, Boltzmann’s superposition integral is discretized into a system of linear equations. Due to the ill-posedness of the resulting matrix equation, Tikhonov’s regularization is introduced. Second, the integral is transformed into a recursive formula, using a Prony series representation of viscoelastic material functions, in which gradient-based optimization is applied. Numerical results are provided to compare and verify the applicability of the presented numerical procedures.     

Determination of the relaxation function for viscoelastic materials at low velocity impact

The work discusses impact microindentation and its possible application for determination and control of physico-mechanical characteristics of viscoelastic materials. Technique and algorithm for determination of the relaxation function at the dynamic intrusion of the indenter into the viscoelastic half-space are proposed. Experimental dependencies for intrusion of spherical and conical indenter into rubber in the range of initial velocities 0.3–3 m/s were established. It was shown that obtained results do not depend on the shape of the indenter tip and its velocity. The analysis of the results was carried out from the point of view of time and temperature dependencies that are typical for elastomers. Obtained results can be used for analysis and prediction of elastomeric material's response to the applied load with consideration of the loading history.     

A relationship between non-exponential stress relaxation and delayed elasticity in the viscoelastic process in amorphous solids: Illustration on a chalcogenide glass

Inorganic glasses are viscoelastic materials since they exhibit, below as well as above their glass transition temperature, a viscoelastic deformation under stress, which can be decomposed into a sum of an elastic part, an inelastic (or viscous) part and a delayed elastic part. The delayed elastic part is responsible for the non-linear primary creep stage observed during creep tests. During a stress relaxation test, the strain, imposed, is initially fully elastic, but is transformed, as the stress relaxes, into an inelastic and a delayed elastic strains. For linear viscoelastic materials, if the stress relaxation function can be fitted by a stretched exponential function, the evolution of each part of the strain can be predicted using the Boltzmann superposition principle. We develop here the equations of these evolutions, and we illustrate their accuracy by comparing them with experimental evolutions measured on GeSe₉ glass fibers. We illustrate also, by simple equations, the relationship between any kind of relaxation function based on additive contribution of different relaxation processes and the delayed elastic contribution to stress relaxation: the delayed elasticity is directly correlated to the dispersion of relaxations times of the processes involved during relaxation.     

粘弹性介质中球形空腔嵌入层的共振特性研究

潜艇吸声覆盖层在低频对抗主动声纳的探测是目前面临的一个重要技术难题，利用空腔共振时的吸收是解决低频吸声问题的一种有效途径，文中在假设入射球形纵波完全被腔壁反射的情形下，对球形空腔嵌入层的共振特性进行了理论分析，结果表明，在粘弹性介质中嵌入适当厚度的低剪切波速包覆层能有效地降低空腔共振的频率。     

On the nonlinear viscoelastic behaviour of nylon fiber

Nonlinear time-dependent stress relaxation was determined experimentally in nylon fiber in the small strain domain. This process is accounted for on the basis of a quasi-linear viscoelastic theory. A nonlinear modified constitutive equation for the viscoelastic medium is deduced. In processing the experimental data it turned out to be necessary to keep the nonlinear terms up to cubic in the constitutive equation for nylon fiber. The elastic and rheological constants in the constitutive equation for nylon fiber are evaluated.     

Application of causality check and of the reduced variables method for experimental determination of Young''s modulus of a viscoelastic material

An accurate determination of the complex dynamic Young's modulus of a viscoelastic material, in a broad frequency range, is presented in this paper. Curves of Young's modulus of the tested material (a mixture of polypropylene and calcium carbonate), at different temperatures, are experimentally obtained by means of a laser sensor. The experimental curves are then gathered into a unique master curve, by applying the reduced variables method and a causality check on the curves. The master curve represents Young's modulus of the viscoelastic material over a much broader frequency range, with respect to the range of a single experimental curve.     

Determination of the shear modulus of orthotropic materials from off-axis tension tests

The paper is concerned with the study of the possibility to determine the shear modulus of orthotropic materials using testing under tension in the direction making some angle with the principal material axis. Conditions that should be imposed on the material properties providing the possibility of this test are derived and analyzed in application to composite materials.     

Representative indentation elastic modulus evaluated by unloading of nanoindentation made with a point sharp indenter

The conventional method to extract elastic modulus from the nanoindentation on isotropic linearly elastic solids is based on Sneddon’s solution (1965). However, it is known that the solution is valid only for incompressive elastic solids with the Poisson’s ratio ν of 0.5. This paper first proposes the modification of the solution in a wide range of ν from 0 to 0.5 through the numerical analysis on the unloading behavior of a simulated conical nanoindentation with a finite element method. As a result of the modification, the coefficient of linearity between the indentation elastic parameter k_e and Young’s modulus E is empirically given as a function of ν and the inclined face angle of the indenter, β, where k_e is defined as k_e ≡ P/h² with the indentation load P and penetration depth of the indenter h. According to the linear relationship between k_e and E, it is found that elastic rebound during unloading of a nanoindentation is uniquely characterized by a representative indentation elastic modulus E¹ defined in terms of E, ν and β, and that the value of E¹ can be evaluated from the P–h relationship with k_e and β. For an isotropic elastoplastic solid, the indentation unloading parameter k₂ defined as k₂ ≡ P/(h–h_r)² for a residual depth h_r is different from k_e even though a linearly elastic solid with k_e and elastoplastic solid with k₂ have a common E¹. In order to evaluate E¹ of an elastoplastic solid, the corresponding k_e is estimated from k₂ with an empirical equation as a function of the relative residual depth ξ defined as ξ ≡ h_r/h_max for the maximum penetration depth h_max. A nanoindentation experiment confirmed the validity of the numerical analysis for evaluating the elastic modulus.     

Estimation of model errors in the calibration of viscoelastic material models

In this paper we discuss a posteriori error estimation in the context of parameter identification problems. In particular, we pay attention to the errors arising from the discretization of the parameters, which is perceived as a model error. A previously developed method for goal‐oriented a posteriori error estimation is employed for two fundamentally different types of model hierarchies. The numerical results show rather good agreement between estimated and actual errors. Copyright © 2008 John Wiley & Sons, Ltd.     

Experimental investigation of the viscoelastic deformation of PC, ABS and PC/ABS alloys

Dynamic Mechanical Analysis (DMA) tests are carried out to investigate the viscoelastic deformation of PC, ABS and PC/ABS alloys with ratio of PC to ABS being 80/20, 60/40, 50/50, 40/60, respectively. Storage and loss moduli and loss angle of PC, ABS and PC/ABS alloys are measured from DMA tests from 30 °C in 160 °C at 1 Hz. Glassy temperature of PC/ABS alloys is determined with loss modulus and loss angle curves. Also, static tests are carried out to measure the relaxation moduli of PC, ABS and PC/ABS alloys. Effect of ABS fraction on the glassy transition temperatures and relaxation moduli of PC/ABS alloys is discussed.     

Predicting the tensile relaxation modulus of asphalt mixes based on the mix design and environmental factors

The main objective of this study was to predict the tensile relaxation modulus of asphalt mixes, without having to perform the common relaxation modulus tests, by developing a predictive model based on the mix characteristics, ageing condition, temperature and loading time. To this end, cylindrical asphalt mixture specimens containing crushed stone aggregates with 60/70 penetration asphalt binder were fabricated using two aggregate gradations, two binder contents, two air void levels and three ageing conditions with four replicates. Uniaxial tensile relaxation modulus tests were conducted on the specimens at four temperatures using the trapezoidal loading pattern at a low level of strain. Tensile relaxation modulus master curves of all the experimental combinations were constructed by the sigmoidal model. Statistical analysis of variance and regression analysis was performed on the test data and a predictive model was developed. Finally, the predictive model was verified using a group of measured values other than those used for the development of the model, and it was found that the predicted values correlated well with the measured ones.     

Creep and relaxation functions of a heterogeneous viscoelastic porous medium using the Mori-Tanaka homogenization scheme and a discrete microscopic retardation spectrum

In this paper the macroscopic creep and relaxation functions of a heterogeneous viscoelastic porous medium are derived by using Mori-Tanaka homogenization scheme. Analytical and semi-analytical solutions can then be determined with a parametric number of heterogeneous phases embedded in a viscoelastic matrix whose behavior is described with a parametric number of analogical units. Under some simplifying assumptions, a solution strategy is presented in order to make explicit how the microscopic retardation and relaxation times of the viscoelastic matrix control the distribution of the retardation and relaxation times of the homogenized medium.     

Validation of time temperature superposition principle for high modulus asphalt concrete in the linear viscoelastic and fatigue domains

Validation of time–temperature superposition principle (TTSP) in the fatigue domain for a high modulus asphalt concrete (HMAC) is presented in this paper. All tests were performed in tension-compression under strain control mode. First, TTSP was validated in the linear viscoelastic domain. Then, fatigue tests were performed under three loading conditions, 9.2°C and 5 Hz, 11.0°C and 10 Hz and 12.9°C and 20 Hz, which are equivalent regarding TTSP. Two fatigue protocols were adopted: continuous fatigue test (FT) and fatigue test with rest period (FTRP). For FT, three samples were tested at 180μm/m for each loading condition whereas for FTRP, one sample was tested at 100 μm/m. The data were analysed by comparing the dynamic modulus evolution as a function of time or the fatigue life duration. The results showed that HMAC with fatigue damage remains thermorheologically simple (i.e., respects the TTSP) in the studied temperatures range.     

The relaxation effects of the volume properties of electrically conducting viscoelastic material

A new model of the equations of generalized thermo-viscoelasticity for an electrically conducting isotropic media permeated by a primary uniform magnetic field, taking into consideration the rheological properties of the volume, is given. The formulation is applied to both generalizations, Lord–Shulman theory and the Green–Lindsay theory, as well as to the coupled theory.

The state space approach is adopted for the solution of one-dimensional problems in the absence or presence of heat sources. The Laplace-transform technique is used. A numerical method is employed for the inversion of the Laplace transforms. Numerical results for the stress distribution are given and illustrated graphically for each problem.

Comparisons are made with the results predicted by the three theories, or ignoring the viscous effects of the volume. Also, the effect of the magnetic field is studied. It is found that the consideration of these effects is to decrease the thermal stresses.     

Fractional-derivative viscoelastic model of the shock interaction of a rigid body with a plate

The impact of a rigid body upon an elastic isotropic plate is investigated for the case when the equations of motion take rotary inertia and shear deformation into account. The impactor is considered as a mass point, and the contact between it and the plate is established through a buffer involving a linear-spring–fractional-derivative dashpot combination, i.e., the viscoelastic features of the buffer are described by the fractional-derivative Maxwell model. It is assumed that a transient wave of transverse shear is generated in the plate, and that the reflected wave has insufficient time to return to the location of the spring’s contact with the plate before the impact process is completed. To determine the desired values behind the transverse-shear wave front, one-term ray expansions are used, as well as the equations of motion of the impactor and the contact region. As a result, we are led to a set of two linear differential equations for the displacements of the spring’s upper and lower points. The solution of these equations is found analytically by the Laplace-transform method, and the time-dependence of the contact force is obtained. Numerical analysis shows that the maximum of the contact force increases, tending to the maximal contact force when the fractional parameter is equal to unity.     

Sensitivity analysis of potential tests for determining the interlaminar shear modulus of fibre reinforced composites

A sensitivity analysis using finite element (FE) simulations was conducted as part of an overall attempt to develop a new and robust parameter identification method for determining the interlaminar shear moduli G₁₃ and G₂₃ of laminated composite materials. It is proposed that the new method will use an integrated experimental and numerical technique. Six different shear and bending tests were investigated numerically using three-dimensional FE models to determine their suitability for this integrated technique. The sensitivity of the potential tests to changes in the different material properties, especially the interlaminar shear moduli, G₁₃ and G₂₃, and the elastic modulus in the through-thickness direction, E₃, was determined. It was discovered that several configurations within three of the six potential tests considered are suitable for the new proposed parameter identification method. This is based on the criteria that they are more sensitive to the interlaminar shear moduli than to other material constants. When manufacturing factors were considered, the two most suitable tests were identified for use in the new proposed method.     

Further developments in testing and analysis of the plate twist test for in-plane shear modulus measurements

The plate twist test for testing the in-plane shear modulus of composite materials is analysed. In the proposed test set-up the loads are not applied to the corners of the plate. This makes the test easier to perform than the original plate twist test. Based on a strength of materials approach, an analytical expression is found for the correction factor that accounts for the shifted position of the loading points. The analysis is made without making the assumption of equal bending stiffness in the diagonal directions of the plate, and can thus be applied for off-axis tests on orthotropic or lower symmetry composite materials.     

Numerical studies on the effective shear modulus of particle reinforced composites with an inhomogeneous inter-phase

Numerical studies on the effective shear modulus of particle reinforced composites with an inhomogeneous inter-phase are performed. The influences of many parameters to the equivalent shear modulus of composites are carefully analyzed, including the inter-phase thickness, variation of interfacial properties, boundary conditions and volume fraction of particles, etc. Numerical results show that the Poisson ratio can be assumed as a constant across the whole inter-phase zone in the computation. The form of property variation across the inter-phase also greatly affects the effective shear modulus of composite. Numerical results predicted by the rigid boundary conditions are remarkably higher than those by the free boundary conditions and the exact solutions. The reasonability and exactness of the available models for predicting the effective shear modulus of composites are accessed by the numerical results in the present work.     

Comparing various fitting models to construct the tensile relaxation modulus master curve of asphalt mixes

This study was to compare the relative ability of seven common fitting models, i.e. Pure Power Law (PPL), Generalized Power Law (GPL), Modified Power Law (MPL), Modified Power Law Series (MPLS), Standard Sigmoid (SS), Generalized Logistic Sigmoid (GLS) and Prony Series (PS), to construct the tensile relaxation modulus master curve of dense graded asphalt mixes. To this end, cylindrical asphalt mixture specimens containing crushed stone aggregates with 60/70 penetration asphalt binder were fabricated using two aggregate gradations, two binder contents, two air void levels and three ageing conditions with three replicates. Direct tension relaxation modulus tests were conducted on the specimens at four different temperatures using the trapezoidal loading pattern at a low level of input strain. Tensile relaxation modulus master curves were constructed using all the fitting models together with the numerical shifting technique. Finally, both the graphical and statistical comparisons were made among the fitting models, and the best one was found to be PS, followed by MPLS, GLS, MPL, SS, GPL and PPL.