Standard Linear Solid Research Articles

Optimal decay rate for the Cauchy problem of the standard linear solid model with Gurtin–Pipkin thermal law

In this paper, we consider the Cauchy problem for the standard linear solid model in the whole space Rn, where the heat conduction is given by the Gurtin–Pipkin thermal law. The energy method in the Fourier space is used to obtain the optimal decay rate of the L2–norm of the solution. More precisely, we prove that the decay rate of the solution is of the form (1+t)−n/4 and of regularity–loss type. Hence, we improve and extend the result obtained in [20].

Identification of the Fractional Zener Model Parameters for a Viscoelastic Material over a Wide Range of Frequencies and Temperatures.

The paper presents an analysis of the rheological properties of a selected viscoelastic material, which is dedicated to the reduction of vibrations in structures subjected to dynamic loads. A four-parameter, fractional Zener model was used to describe the dynamic behavior of the tested material. The model parameters were identified on the basis of laboratory tests performed at different temperatures and for different vibration frequencies. After proving that the material is thermoreologically simple, the so-called master curves were created using a horizontal shift factor. The Williams–Landel–Ferry formula was applied to create graphs of the master curves, the constants of which were determined for the selected temperature. The resulting storage and loss module functions spanned several decades in the frequency domain. The parameters of the fractional Zener model were identified by fitting the entire range of the master curves with the gradientless method (i.e., Particle Swarm Optimization), consisting in searching for the best-fitted solution in a set of feasible solutions. The parametric analysis of the obtained solutions allowed for the formulation of conclusions regarding the effectiveness of the applied rheological model.

Testing and Modelling of Elastomeric Element for an Embedded Rail System.

Modelling of elastomeric elements of railway components, able to represent stiffness and damping characteristics in a wide frequency range, is fundamental for simulating the train–track dynamic interaction, covering issues such as rail deflection as well as transmitted forces and higher frequency phenomena such as short pitch corrugation. In this paper, a modified non-linear Zener model is adopted to represent the dependences of stiffness and damping of the rail fastening, made of elastomeric material, of a reference Embedded Rail System (ERS) on the static preload and frequency of its deformation. In order to obtain a reliable model, a proper laboratory test set-up is built, considering sensitivity and frequency response issues. The equivalent stiffness and damping of the elastomeric element are experimentally characterised with force-controlled mono-harmonic tests at different frequencies and under various static preloads. The parameters of the non-linear Zener model are identified by the experimental equivalent stiffness and damping. The identified model correctly reproduces the frequency- and preload-dependent dynamic properties of the elastomeric material. The model is verified to be able to predict the dynamic behaviour of the elastomeric element through the comparison between the numerically simulated and the experimentally measured reaction force to a given deformation time history. Time domain simulations with the model of the reference ERS demonstrate that the modelled frequency- and preload-dependent stiffness and damping of the elastomeric material make a clear difference in the transient and steady-state response of the system when distant frequency contributions are involved.

Stability analysis of viscoelastic pipes conveying fluid with different boundary conditions described by fractional Zener model

The stability of viscoelastic pipes conveying fluid with the Zener fractional-order constitutive relationship is investigated in this paper. The pipe is subjected to the axial follower and directed forces with uniform distribution. The pipe’s beginning and end are restrained with lateral and rotational springs to satisfy a vast range of boundary conditions. The fluid flow effect is taken into account as a lateral distributed force. Based on the Euler-Bernoulli beam theory, the equation of motion is derived using the extended Hamilton’s principle. The Laplace transform and Galerkin method are used to obtain a set of algebraic equation. The pipe conveying fluid’s stability boundary will be obtained by calculating the roots of the determinant of the obtained equations coefficients. Effects of the fractional Zener viscoelastic model parameters are considered on the stability margin of pipes conveying fluid with different boundary conditions, and some conclusions are drawn.

Relationship between the Deformation Processes Occurring in Rubbers and Their Molecular Structure

Based on measurements of rubber hysteresis curves, the relationship between the network structure of elastomers and their mechanical behavior is analyzed. On the basis of the classical theory of rubber elasticity, the Flory corrections, which take into account physical cross-links in the elastomer network structures in the course of straining a rubber sample at a constant rate, were calculated. To this end, the Frenkel-Eyring thermal-fluctuation theory was used, also taking into account the activation energy of the polymer viscosity, which depends on the entropic elasticity of macromolecules in the processes of their flow and deformation. Flory corrections were calculated within the standard linear solid model.

Exact Solutions for Time-Varying Bending Behavior of Multilayer Orthotropic Beams Bonded by Viscoelastic Interlayers

This study proposed exact solutions for the purpose of investigating and understanding the time-varying bending behavior of a multilayer orthotropic beam bonded by viscoelastic interlayers. In the analytical model, each orthotropic layer involved four independent elastic constants and was modeled on the basis of the two-dimensional (2D) elasticity theory. The interlayer exhibited viscoelastic properties and was described by the standard linear solid model. By means of Fourier series expansion, the general solutions of stresses and displacements were derived with unknown coefficients, which were analytically determined by interfacial and loading conditions, in which the convolution was converted by the Laplace transform. The results of numerical examples showed that the present solutions converged rapidly and were consistent with the finite element solutions. Comparison of the present 2D solutions with the one-dimensional (1D) solutions based on the Euler–Bernoulli beam theory indicated that the 1D solutions were inaccurate for thick beams. The effects of viscoelastic material and orthotropic constants on the long-term stresses and displacements in the beam were thoroughly examined. These results provided a reference for the optimization of a multilayer orthotropic beam design.

Steady-State Harmonic Vibrations of Viscoelastic Timoshenko Beams with Fractional Derivative Damping Models

Due to growing demands on newly developed products concerning their weight, sound emission, etc., advanced materials are introduced in the product designs. The modeling of these materials is an important task, and a very promising approach to capture the viscoelastic behavior of a broad class of materials are fractional time derivative operators, since only a small number of parameters is required to fit measurement data. The fractional differential operator in the constitutive equations introduces additional challenges in the solution process of structural models, e.g., beams or plates. Therefore, a highly efficient computational method called Numerical Assembly Technique is proposed in this paper to tackle general beam vibration problems governed by the Timoshenko beam theory and the fractional Zener material model. A general framework is presented, which allows for the modeling of multi-span beams with general linear supports, rigid attachments, and arbitrarily distributed force and moment loading. The efficiency and accuracy of the method is shown in comparison to the Finite Element Method. Additionally, a validation with experimental results for beam systems made of steel and polyvinyl chloride is presented, to illustrate the advantages of the proposed method and the material model.

Finite Element Formulation of Fractional Constitutive Laws Using the Reformulated Infinite State Representation

In this paper, we introduce a formulation of fractional constitutive equations for finite element analysis using the reformulated infinite state representation of fractional derivatives. Thereby, the fractional constitutive law is approximated by a high-dimensional set of ordinary differential and algebraic equations describing the relation of internal and external system states. The method is deduced for a three-dimensional linear viscoelastic continuum, for which the hydrostatic and deviatoric stress-strain relations are represented by a fractional Zener model. One- and two-dimensional finite elements are considered as benchmark problems with known closed form solutions in order to evaluate the performance of the scheme.

Fractional differential constitutive model for linear viscoelasticity of asphalt and asphalt mastic

To accurately describe the linear viscoelastic properties of asphalt and asphalt mastic, the authors proposed a fractional differential constitutive model called the generalized fractional differential Zener (GFDZ) model. Then, by means of the GFDZ model, the dynamic and static viscoelastic properties of two kinds of asphalt and two kinds of asphalt mastic were described and predicted. Finally, the GFDZ model was compared with the modified CAM (mCAM) model and 2S2P1D model. The results show that the GFDZ model can simultaneously and accurately characterize the dynamic and static viscoelastic properties of asphalt or asphalt mastic using a set of model parameters. The GFDZ model overcomes the deficiencies of the mCAM model and 2S2P1D model, and provides a reference for the characterization and constitutive description of the linear viscoelastic properties of asphalt and asphalt mastic.

Numerical Modelling of Multicellular Spheroid Compression: Viscoelastic Fluid vs. Viscoelastic Solid

Parallel-plate compression of multicellular spheroids (MCSs) is a promising and popular technique to quantify the viscoelastic properties of living tissues. This work presents two different approaches to the simulation of the MCS compression based on viscoelastic solid and viscoelastic fluid models. The first one is the standard linear solid model implemented in ABAQUS/CAE. The second one is the new model for 3D viscoelastic free surface fluid flow, which combines the Oldroyd-B incompressible fluid model and the incompressible neo-Hookean solid model via incorporation of an additional elastic tensor and a dynamic equation for it. The simulation results indicate that either approach can be applied to model the MCS compression with reasonable accuracy. Future application of the viscoelastic free surface fluid model is the MCSs fusion highly-demanded in bioprinting.

Free vibrations of laminate plates with viscoelastic layers using the refined zig-zag theory – Part 1. Theoretical background

The refined zig-zag theory is used to solve the dynamic problems for composite sandwich plates made from the elastic and the viscoelastic (VE) layers. The linear constitutive relations are used to model physical properties of the VE layers separately for volumetric and deviatoric strains. The fractional Zener model is used in the formulation of VE material. The Laplace transformation and the finite element method are used to derive the discretized nonlinear eigenvalue problem. The required dynamic characteristics of the plates – the frequency and the damping ratio, fundamental in design of structural elements in many fields of engineering, can be obtained using the procedure described in the companion paper Litewka P. Lewandowski R, Wielentejczyk P. Free vibrations of laminate plates with viscoelastic layers using the refined zig-zag theory – Part 2. Numerical analysis.

Modeling elastic wave propagation through a partially saturated poroviscoelastic interlayer by fractional order derivatives

The reflection and transmission problems of elastic waves through a partially saturated porous interlayer sandwiched between two elastic solid half-spaces are investigated. The partially saturated porous interlayer contains solid phase, gas phase and liquid phase. The solid frame is modeled by the fractional Zener model where the fractional order derivatives are used to describe the complex history-dependent viscoelastic behavior. Due to the coupled effects of the multiple fields, there exist three longitudinal waves and one transverse wave in the porous medium. The numerical results of reflection and transmission coefficients and the energy dissipation ratios are provided and shown graphically. The effects of the gas saturation, the interlayer thickness and the viscoelasticity of the solid frame are discussed based upon the numerical results. It is observed that the reflection and transmission coefficients and the energy dissipation ratios are affected noticeably by the gas saturation, the interlayer thickness and the viscoelasticity of the solid frame. In particular, there exists the characteristic gas saturation where the energy dissipation increases drastically.

A transient and time lag deformation alternating-coupling micro elastohydrodynamic lubrication model

As transient and time lag deformations occur alternately and influence each other for viscoelastic materials of dynamically loaded journal bearings, the time lag deformation per unit time is defined based on the finite element stiffness equation and the standard linear solid model, accumulating it by loop and superposing transient deformation to obtain actual deformation at every moment. Thereby, a novel temporal association deformation equation of bearing is constructed. It is used to improve the film thickness calculation method of the traditional micro elastohydrodynamic lubrication (MEHD) model. Finally, a novel transient and time lag deformation alternating-coupling micro elastohydrodynamic lubrication (TTL-MEHD) model is established. The comparison between lubrication performance parameters calculated by the new model and the MEHD model under the eccentric motion of the journal proves the new model is correct, and can effectively analyze the knocking and rough rubbing characteristics at the early stage of bearing instability. The new model is implemented to explore the effect of the time lag degree of material deformation on mixed lubrication performance. The application of the new model in connecting rod bearing of internal-combustion engine proves that it can effectively compensate for the impact lag deviation caused by the complete elasticity assumption.

Free vibration of three-layer sandwich plate with viscoelastic core modelled with fractional theory

In the present study, natural frequencies and damping behavior of three-layer sandwich plates with viscoelastic core modelled with fractional theory are studied. Governing equations of motion are derived using Lagrange's equation and then solved via Rayleigh-Ritz method. In the first section of the analysis, the coefficients of the fractional and classical viscoelastic models are determined utilizing the curve fitting method on existed experimental data in the literature, and the best viscoelastic model is selected for free vibration analysis of sandwich plate. The method of least squares for fitting smooth curves to experimental data is applied. The results show that the fractional Zener model gives the best fit curve to the experimental data. After that, the effects of some geometrical parameters such as length to width ratio and viscoelastic core thickness on the natural frequency and loss factor of the sandwich plate with viscoelastic core are investigated, and some conclusions are drawn.

Experimental identification of the material standard linear solid model parameters by means of dynamical measurements

A procedure for the experimental identification of the material standard linear solid model parameters by means of dynamic mechanical analysis test instrument measurements is presented. Since the standard linear solid material stress–strain functional D(ω) relationship in the frequency domain formally depends on the standard linear solid material parameters, a procedure able to identify these parameters from test measurement estimates is proposed in this work. Nevertheless, a critical, nonlinear and non-parametric approach is to be followed since the number of the material standard linear solid block components is generally unknown, and the material D(ω) shows a highly nonlinear dependency on the unknown standard linear solid material parameters. For these reasons, measurement and test model noise is expected to strongly influence the accuracy of the identification results. A multi-step procedure is presented, consisting first in the non-parametric identification of a frequency dependent, two degrees of freedom model instrument frame by means of a polynomial rational function, where polynomial order and parameters, such as polynomial coefficients and pole-residue couples, are optimally identified by means of an algebraic numerical technique and of an iterative stabilization procedure. Another procedure able to identify the material D(ω) polynomial rational functional relationship in the frequency domain is also proposed, taking into account the dynamic contribution of the instrument frame, of the inertial contribution of the distributed mass of the beam and of the lumped mass of the instrument force measuring system. An effective procedure, able to identify the standard linear solid material model parameters in the time domain from the identified material physical poles, is finally proposed. Some application examples, concerning the identification of the standard linear solid model of a known material and of an unknown composite material, are shown and discussed as well.

Free vibrations of laminate plates with viscoelastic layers using the refined zig-zag theory – Part 2. Numerical analysis

In this paper the numerical procedure to solve the non-linear eigen-value problem describing the free vibrations of the laminate plates with viscoelastic (VE) layers is presented. The derivation of that main equation based on the refined zig-zag theory and using fractional Zener material model to describe VE material is presented in the companion paper Lewandowski R, Litewka P. Wielentejczyk P. Free vibrations of laminate plates with viscoelastic layers using the refined zig-zag theory – Part 1. Theoretical background. The solution obtained using the continuation method provides the required dynamic characteristics of the plates – the frequency and the damping ratio. In the iterative part of each continuation step either the full Newton or the quasi-Newton iterative procedure are used. Several numerical examples are solved to verify accuracy, efficiency and versatility of the proposed formulation. Numerous parametric studies are carried out to determine the influence of various solution variants and combinations of material parameters onto the results describing the damping properties of laminate plates.

Exploring the Impact of Thermally Controlled Crustal Viscosity on Volcanic Ground Deformation

AbstractVolcanoes undergoing unrest often produce displacements at the ground surface, providing an important window to interpret the dynamics of the underlying magmatic system. The thermomechanical properties of the surrounding host rock are expected to be highly heterogeneous, with key physical parameters having a strong dependence on temperature. Deformation models that incorporate nonelastic rheological behaviors are therefore heavily reliant on the assumed thermal conditions, and so it is critical to understand how the thermomechanical crustal structure affects the observed deformation field. Here, we use a series of thermo‐viscoelastic Finite Element models to explore how variations in thermal constraints (i.e., reservoir temperature and background geothermal gradient) affect surface displacement patterns when using the Maxwell and Standard Linear Solid (SLS) viscoelastic configurations. Our results demonstrate a strong variability in the viscoelastic deformation response when changing the imposed thermal constraints, caused by the partitioning of deformation and the dissipation of induced stresses. When using the SLS rheology, we identify that cumulative long‐term displacements can vary by over 20%, relative to a reference model with a reservoir temperature of 900°C and background geothermal gradient of 30 K km−1. The relative change increases to a maximum of 35% when thermal weakening of the Young's modulus is also considered. Contrastingly, the deformation patterns of the Maxwell rheology are governed by unbounded displacements and complete stress relaxation. Ultimately, we outline that uncertainties in the thermal constraints can have a significant impact on best‐fit source parameters (e.g., size and depth) and overpressure/volume‐change loading histories inferred from thermo‐viscoelastic models.

A new approach to modelling the ground reaction force from a runner

The impact force experienced by a runner when his/her foot makes contact with the ground has been the subject of much research. This force is called the ground reaction force (GRF), and has been measured by several groups. In parallel with this, mathematical models have been developed to simulate GRFs in order to investigate various effects on this, such as the parameters of the human body and types of running shoe soles. Lumped parameter models have been developed by several researchers with limited success, because they are either constrained to model translational motion, or become complicated if they include rotational motion. This paper proposes a new approach based on modes of vibration, which encompasses the simplicity of the lumped parameter approach, without the motion constraints. The GRF is decomposed into contributions due to the various vibration modes of the system. To achieve this, a linear system is required, so a Zener model, which is used to model viscoelastic materials, is employed as the ground reaction model. The modal modelling approach is described in detail using established lumped parameter models used to predict the GRF. It is then applied to four experimental data sets from the literature, where it is shown that at most three modes are required to model GRF data accurately. Two of these modes are oscillatory modes and one is a non-oscillatory exponentially decaying mode. In general, it is shown that the modal model can capture the dynamics of each measured GRF independently of speed and running style.

Elastic waves propagation through a liquid-saturated poroelastic interlayer based on the fractional viscoelastic BISQ model

The reflection and transmission problems of elastic waves, through a liquid-saturated poroelastic interlayer sandwiched between two elastic solid half-spaces, are investigated. The reflection and transmission coefficients are derived based on the Biot/squirt (BISQ) model, which incorporates both the Biot and squirt-flow mechanisms. Moreover, the viscoelasticity of the solid frame and the pore fluid are also considered. The fractional Zener model and the fractional dashpot model (Abel dashpot) are adopted for the solid frame and the pore fluid, respectively. The numerical results are provided and shown graphically. The dependences of the attenuation, the reflection and transmission coefficients upon the characteristic squirt flow length are numerically discussed. Comparing with that predicted by Biot theory without considering the squirt flow effects, it is observed that the reflection and transmission coefficients and the attenuation due to the sandwiched interlayer are affected noticeably by the effect of the squirt flow.

Instrumentation and methods for rapid estimation of selected viscoelastic parameters in foods.

A new method to estimate the selected viscoelastic parameters of foods using damped vibration analysis is presented for the evaluation of fruits and baked products. A flat disk is attached to the flat sample surface using a laser rangefinder that measures the sample thickness in advance, and it is locked by a trigger. Next, the trigger is released to allow the probe to press the sample through the force of gravity. The damped vibration of the probe caused by the deformation of the food is measured by monitoring the displacement of the probe via a linear encoder. The bulk modulus and viscosity are estimated using the fractional Zener model and mass. Young's modulus (E) is estimated independently by determining the maximum velocity of the probe using Hooke's law. Poisson's ratio (ν), and the shear modulus and viscosity are calculated by employing the estimated E and bulk modulus. The bulk modulus, bulk viscosity, shear modulus, shear viscosity, and E of apples were found to be higher than those of bananas. The bulk modulus, bulk viscosity, E, and shear modulus for white bread were lower than those for pound cake, but the ν values were higher, whereas those of sponge cake were intermediate. After drying the baked products for 1 day, most of the parameters of the samples increased, but the value of ν for white bread decreased. The proposed free-falling device estimated the four viscoelastic coefficients, Poisson's ratio, and Young's modulus of the food sample in less than 1s.