Viscoelastic Material Model Research Articles

Geometrically nonlinear modelling of pre-stressed viscoelastic fibre-reinforced composites with application to arteries.

Mechanical behaviour of pre-stressed fibre-reinforced composites is modelled in a geometrically exact setting. A general approach which includes two different reference configurations is employed: one configuration corresponds to the load-free state of the structure and another one to the stress-free state of each material particle. The applicability of the approach is demonstrated in terms of a viscoelastic material model; both the matrix and the fibre are modelled using a multiplicative split of the deformation gradient tensor; a transformation rule for initial conditions is elaborated and specified. Apart from its simplicity, an important advantage of the approach is that well-established numerical algorithms can be used for pre-stressed inelastic structures. The interrelation between the advocated approach and the widely used "opening angle" approach is clarified. A full-scale FEM simulation confirms the main predictions of the "opening angle" approach. A locking effect is discovered: in some cases the opening angle of the composite is essentially smaller than the opening angles of its individual layers. Thus, the standard cutting test typically used to analyse pre-stresses does not carry enough information and more refined experimental techniques are needed.

Fractional derivative group shear wave speeds for model-based viscoelastic characterization

We present a method for model-based viscoelastic characterization using shear wave elastography. The shear wave displacement signal is differentiated to a series of fractional derivative orders, and the group speed of the signal at each derivative order is calculated. These speeds are compared to a look up table built from simulations for a given viscoelastic material model. The best-fit material parameters are chosen to minimize the mean square difference between the group speeds from the look up table and experimental measurements across all fractional derivative orders. Various two- and three-parameter viscoelastic material models are tested. The technique is validated in simulations with ultrasonic tracking and demonstrated in three viscoelastic phantoms with material properties that match those associated with different degrees of liver fibrosis. Comparatively, phase velocity curves are computed, and parameters determined by fitting analytic expressions for phase velocity for a given material model. The fractional derivative group speed-based technique gives model parameters with lower mean square error than fitting material model parameters directly to phase velocities. Additionally, we conclude these viscoelastic phantoms are sufficiently characterized by a two-parameter material model and are better characterized by the Linear Attenuation model than the more commonly used Voigt model.

Constitutive modeling of visco‐hyperelastic behavior of double‐network hydrogels using long‐term memory theory

AbstractDouble‐network hydrogels with viscoelastic behavior are appropriate materials for biomechanical applications. In this article, the standard linear solid (SLS) rheological model for the linear viscoelastic materials is generalized to the viscoelastic materials with large nonlinear deformations. Based on this viewpoint, the constitutive equation is proposed as sum of two parts including the strain‐dependent elastic stress, and the viscous stress, which depends on the strain and strain rate. The elastic part of the stress is modeled via considering a hyperelastic strain energy function, while the main core of the viscous stress part requires a time‐dependent weight function to satisfy the long‐term memory fading principle. In addition, the weight function is proposed such that it can capture the mechanical behavior trend corresponding to the strain and strain rate for a double‐network hydrogel in the relaxation test. Finally, to evaluate the performance of the proposed constitutive equation for the mechanical behavior modeling of double‐network hydrogels, the tests on these materials have been used, and the material parameters are determined from fitting the experimental results to the theory. The agreement of test and theory results showed that the proposed model is capable to model the mechanical behavior of double‐network hydrogels.

Vibration attenuation analysis of periodic underground barriers using complex band diagrams

Vibrations propagating underground may have harmful effects on adjacent buildings and human health. Recently, the notion of phononic crystals has been introduced and applied to isolate vibrations below the ground. In this paper, considering the viscoelastic material model, the complex band diagrams are suggested to analyse the transportation of vibrations in the composite foundation consist of soil and different underground structures. In the complex band diagram, the minimum imaginary part of the wave vectors is taken as the index to measure the ability of the pile group to weaken vibrations. Several types of underground barriers, including circular piles, continuous walls, and multi-layer piles, are analysed. The result shows that multi-layer piles can induce local resonance and achieve an attenuation zone at the low-frequency level. The attenuation effect of periodic barriers is then verified by transmission models. The results show that the complex band diagrams can quantitatively evaluate the isolation effect of periodic barriers, which is beyond the capability of classic band diagrams.

Circular arc rules of complex plane plot for model parameters determination of viscoelastic material

We present a new approach to determine the rheological parameters of a mechanical model of viscoelastic materials. The fractional derivative solid model composed of a spring in series with a fractional derivative Kelvin–Voigt element has been employed to characterize the dynamic mechanical response of a real viscoelastic material. In dynamic mechanical analysis (DMA) measurements, the frequency-dependent loss modulus is plotted against the storage modulus in the form of complex plane plot. We find that the complex plane plot of the fractional derivative solid model is a depressed or distorted semicircle with its center below the real axis. The model parameters could be identified graphically via its complex plane curve, from which the spring constants can be obtained through the two intercepts of the extrapolated circular arc with the real axis, whereas the order of fractional dashpot can be estimated by the displacement of the semicircle center. The graphical method allows us to easily find rheological parameters, without using complicated calculation and special algorithms.

Gradient-Enhanced Modelling of Damage for Rate-Dependent Material Behaviour-A Parameter Identification Framework.

The simulation of complex engineering components and structures under loads requires the formulation and adequate calibration of appropriate material models. This work introduces an optimisation-based scheme for the calibration of viscoelastic material models that are coupled to gradient-enhanced damage in a finite strain setting. The parameter identification scheme is applied to a self-diagnostic poly(dimethylsiloxane) (PDMS) elastomer, where so-called mechanophore units are incorporated within the polymeric microstructure. The present contribution, however, focuses on the purely mechanical response of the material, combining experiments with homogeneous and inhomogeneous states of deformation. In effect, the results provided lay the groundwork for a future extension of the proposed parameter identification framework, where additional field-data provided by the self-diagnostic capabilities can be incorporated into the optimisation scheme.

A nonlinear uniaxial stress-strain constitutive model for viscoelastic membrane materials

Membrane materials with the excellent thermal, optical, electrical and chemical properties have attracted significant attention in numerous research fields recently. However, while being used to construct the membrane structures, the mechanical behaviors of membrane materials are more foundational than the other properties in evaluating the structure safety. This paper thus proposes a nonlinear stress-strain constitutive model for revealing the viscoelastic behaviors of membrane materials under uniaxial tensile loading. To this end, the constitutive equations for expressing the uniaxial tensile stress-strain relationships of viscoelastic materials are established gradually from the kinematic equations of the generalized Maxwell model that includes several basic Maxwell models and one basic spring element. Meanwhile, the uniaxial tensile tests of two typical viscoelastic membrane materials were carried out in order to examine the proposed constitutive model. The constitutive model parameters of the stress-strain properties of both membrane materials are accurately identified using the least square method. By comparing the true stress-strain curves between experimental results and constitutive models, good agreements with the maximum differences of 4.67% and 3.41% are acquired for the two employed viscoelastic membrane materials, respectively. These observations are able to validate the accuracy and efficiency of this proposed constitutive model in predicting the uniaxial stress-strain behaviors of viscoelastic membrane materials, which are significant in the nonlinear structural analysis of membrane structures.

Gradient Chain Structure Model for Characterizing Frequency Dependence of Viscoelastic Materials

AbstractThe frequency dependence modeling of viscoelastic (VE) materials faces the challenge of high modeling accuracy over a wide frequency range and double-objective optimization in model paramet...

Derivation of von Kármán Plate Theory in the Framework of Three-Dimensional Viscoelasticity

We apply a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin's-Voigt's rheology to derive a viscoelastic plate model of von K\'arm\'an type. We start from time-discrete solutions to a model of three-dimensional viscoelasticity where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. Combining the derivation of nonlinear plate theory by Friesecke, James and M\"{u}ller, and the abstract theory of gradient flows in metric spaces by Sandier and Serfaty we perform a dimension-reduction from 3D to 2D and identify weak solutions of viscoelastic form of von K\'arm\'an plates.

TheFinite Element Modeling and Experimental Study of Sandwich Plates with Frequency-Dependent Viscoelastic Material Model.

Athree-layer composite plate element is developed for finite element modeling and vibration analysis of sandwich plate with frequency-dependent viscoelastic material core. The plate element is quadrilateral element bounded by four-node with 7-degree-of-freedom per node. The frequency-dependent characteristics of viscoelastic material parameters are described using the Biot model. The method of identifying the parameters of the Biot model is given. By introducing auxiliary coordinates, the Biot model is combined with the finite element equation of the viscoelastic sandwich plate. Through a series of mathematical transformations, the equation is transformed into a standard second-order steady linear system equation form to simplify the solution process. Finally, the vibration characteristics of the viscoelastic sandwich plate are analyzed and experimentally studied. The results show that the method in this paper is correct and reliable, and it has certain reference and application value for solving similar engineering vibration problems.

Influence of Waveforms on Fatigue Crack Growth Characteristics of Tire Tread Rubber using Finite Element Analysis

ABSTRACT Rubber products are typically subjected to cyclic fatigue loading in service. During prolonged exposure to cyclic loading, damages initiate at intrinsic defect sites at microscopic levels and subsequently propagate, leading to catastrophic failure. Therefore, the material that offers better resistance to fatigue crack growth (FCG) is suitable for a durable product. FCG characteristics of rubber compounds depend on many factors, such as constituent material (rubber, filler, etc.), environment, and operational conditions (loading amplitude, loading pattern, etc.). To simulate the realistic service condition of a product, the choice of loading pattern is a key factor and has emerged as a very important research topic in recent times. The present work focuses on the effect of loading pattern on FCG characteristics of tire tread rubber compounds. In the present study, FCG characteristics of a 100% natural rubber (NR) compound were measured on a tear and fatigue analyzer (TFA, Coesfeld, Germany) using double edge notched pure shear specimen. Fatigue loading was applied using sinusoidal and pulse waveforms over a wide range of tearing energy levels. Pulse mode recorded a very high crack growth rate (∼2 times) compared to sine mode at equivalent peak energy levels. In order to understand the mechanics of the higher crack growth rate in pulse mode, finite element analysis (FEA) of a pure shear specimen was performed wherein FCG experimental conditions were used as boundary conditions. FE analyses were carried out using both linear and nonlinear viscoelastic material models. Nonlinear viscoelastic FEA results revealed that viscous energy dissipation at the crack tip is much lower in the case of pulse mode, which is in support of higher the FCG rate in pulse mode as observed in the experiments.

Application of computationally efficient TR-BDF2 scheme for the finite element implementation of explicit non-linear viscoelastic models for filled elastomers

This paper presents the implementation of a nonlinear viscoelastic multi-configurational rate type material model within the general framework of the commercial finite element package Abaqus (Implicit). A user-defined subroutine, UMAT for continuum elements, is employed to implement a model similar to the one developed and extensively validated experimentally by Devendiran et al. One of the major challenges of using an implicit finite element (FE) solver is the description of material Jacobian, which may be difficult to obtain in the form of an analytical equation for multi-configurational models. Hence, numerical approximations are used to implement the consistent material Jacobian used in the global Newton iterations. We have used a Jacobian that is formulated by perturbing the deformation gradient, by extending an idea developed by Miehe and widely implemented in hyperelasticity. In order to determine the multiple intermediate configurations that keep evolving with time, this study also examines the computational efficiency of a fully second order integrator in comparison to traditional first order BDF integrator. A highly efficient TR-BDF2 integrator, originally developed for simulating transients of silicon VLSI devices, is utilised to determine the “current relaxed configuration”, which to the author’s knowledge is the first implementation in literature for rate type viscoelastic constitutive equations. Coupling the numerically approximated Jacobian and TR-BDF2 integrator, the UMAT is generic and can easily be modified for various combinations of stored energy potential functions and rate of dissipation equations. The UMAT is validated against material point solution of the constitutive model in MATLAB; and is found to be exact within a tight tolerance. The implemented UMAT is used to determine the rolling resistance of a Grosch wheel which demonstrates the practical application of such a material model focussed towards the workflows for tyre analysis.

Simulation and optimization of roll to roll hot embossing based on viscoelastic material PMMA

The roll to roll hot embossing(R2RHM) is an innovative technique for rapid fabrication of polymer microcomponents. In this paper, ABAQUS simulation software is used to establish the roller model in R2RHM. Moreover, the polymer material model PMMA viscoelastic material model is given by Prony series and WLF equation. Based on these models, the influence of five factors, as pattern radius, velocity, interference, temperature and duty cycle, on the quality of R2RHM is studied by a control variable method, and the influence trend of eachsingle factor on the quality of hot impression is obtained. It is resulted that the radius of the pattern has little effect on the quality of the hot impression, while the velocity, interference, temperature and duty cycle exhibited a significant effect on the quality of the hot impression.

Modeling selected mechanical properties of magnetorheological elastomers

Magnetorheological elastomers are an important group of non-classical engineering materials. The aim of the works carried out is to study the influence of magnetic field on a series of mechanical properties of composite materials in this group. Based on prior analyses, a limited usability of the viscoelastic material model was identified for the purpose of describing the stress-strain relationship in magnetorheological elastomers. Therefore, it has become necessary to propose changes to this rheological model. The present article includes theoretical considerations in relation to the mathematical derivation of functions allowing to account for the influence of strain amplitude on the progressive character of the mechanical hysteresis loops in the materials under discussion. It also provides a detailed analysis of the derived formulas together with a discussion, supplemented with the description of employed simplifying assumptions.

Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models

The visco-elastic dampers can be economically designed for response reduction of dynamical systems. Recent developments in fractional calculus have affected the modeling of visco-elastic materials. The fractional models for visco-elastic dampers, which are parsimonious, require fewer parameters in comparison with other models. In this paper, we use the visco-elastic dampers for control of structural responses. The visco-elastic damper utilizes three important parameters including damping coefficient, stiffness, and fractional order. The dynamic model of the building with visco-elastic dampers can be acquired by a system of fractional differential equations, which includes both fractional and ordinary derivatives. We apply spline collocation methods for obtaining the numerical solution of this complex system. The collocation method is implemented in a graded mesh. The discrete data of earthquake are smoothed by spline interpolation of higher degree. The introduced method is used to simulate the response of a structure with active and passive controllers. A comparison of using different dampers is considered. Also, norm of the transform function is used for response analysis. The results of simulations for four-story and 10-story buildings show significant reductions in structural responses. Moreover, our analysis presents that the large variety of values can be used for parameters of the visco-elastic damper to provide flexibility in designing appropriate visco-elastic-dampers.

Control of MIMO mechanical systems interacting with actuators through viscoelastic linkages

This paper proposes a new method for control of nonlinear multi input – multi output (MIMO) mechanical systems that incorporate viscoelastic dampers (VED) for reducing undesired vibrations of actuators. To this end, a control algorithm is proposed based on considering various characteristics of the described dynamical systems (namely mechanical dynamics, viscoelasticity and actuator dynamics) in generation of control inputs guaranteeing convergence of system response to desired reference signals. This procedure features three consecutive parts within the control loop which are conducted iteratively at each control sample. At each sample, initially necessary forces and moments exerted to mechanical system are calculated as virtual control inputs generated based on a MIMO discrete-time sliding mode control (DSMC) algorithm. As the aim of control model is obtaining a closed-loop system without resulting in notable vibrational effects, undesired chattering effects should be eliminated from inputs generated by DSMC. This objective is attained by calculation of appropriate input bounds. Next, an additional virtual input is assigned corresponding to viscoelastic strain such that virtual mechanical input from previous part of the control loop is generated. To this end, Maxwell model for viscoelastic material is considered. Finally, actual controller input is generated such that all virtual control objectives are satisfied. The effectiveness of control procedure is numerically illustrated for a 3-PRR manipulator.

Viscoelastic model with complex rheological behavior (VisCoR): incremental formulation

A thermo-rheologically complex linear viscoelastic material model, accounting for temperature and degree of cure (DoC), is developed starting with series expansion of the Helmholtz free energy and systematically implementing simplifying assumptions regarding the material behavior. In addition to the temperature and DoC dependent shift factor present in rheologically simple models, the derived novel model contains three cure and temperature dependent functions. The first function is identified as the rubbery modulus. The second is a weight factor to the transient integral term in the model and reflects the current temperature and cure state, whereas the third function is under the sign of the convolution integral, thus affecting the “memory” of the material. An incremental form of this model is presented which, due to improved approximation inside the time increment, has better numerical convergence than most of the similar forms. Parametric analysis is performed simulating stress development in a polymer, geometrically constrained in the mold during curing and cool-down. The importance of using proper viscoelastic model is shown, and the role of parameters in the model is revealed and discussed.

Time-Temperature Dependency of Laminated Glass Subjected to Blast Load – A Numerical Study

The behavior of laminated glass has strong time-temperature dependency. Viscoelastic material models are often employed to define mechanical properties of Polyvinyl Butyral (PVB), the most common interlayer for structural glass applications. However, it is an apparent notion to simplify the high complexity of such material models, as only specific software is capable of considering this behavior. Most studies in blast design of laminated glass have focused on room temperature condition and recommend the use of elastic material models for PVB with high modulus of elasticity for simplification. The main purpose of this study is to develop an understanding of time and temperature dependency of interlayers in real building application. On the basis of empirical weather data, a range of interlayer temperatures is proposed to be considered for blast design situation in Germany for vertical double glazed and triple glazed units in accordance with Eurocode 0 and Eurocode 1. The results obtained from this analysis are further investigated within a transient structural parametric study of laminated glass to identify the effect of winter interlayer temperature and summer interlayer temperature in difference to simplified monolithic glass approach. As a result, significant increase of maximum principal glass stress and maximum deformation is observed for laminated glass subjected to blast load under summer temperature condition.

ГРАНИЧНО-ЭЛЕМЕНТНЫЙ АНАЛИЗ РАСПРОСТРАНЕНИЯ ВОЛН В ПОРОВЯЗКОУПРУГОМ СЛОИСТОМ ПОЛУПРОСТРАНСТВЕ И ПОЛУПРОСТРАНСТВЕ С ПОЛОСТЬЮ

The paper is dedicated to the wave propagation a porous-viscoelastic material. As a mathematical model of a fully saturated poroelastic medium, we consider the Biot model with four basic functions – pore pressure and skeleton movements. The Biot model is supplemented by the principle of elastic and viscoelastic reaction correspondence. The skeleton of a porous material is assumed to be viscoelastic material. A model of a standard viscoelastic solid is spplied to describe the viscoelastic properties of a skeleton. The initial boundary-value problem is reduced to a boundary-value problem by formal application of the Laplace transform. To solve boundary integral equations, the boundary element method is performed. Quadrangular eight-node biquadratic elements are used for boundary element discretization. Numerical integration is carried out according to Gaussian quadrature formulas using algorithms for lowering the order and eliminating features. To obtain a solution in explicit time, numerical inversion of the Laplace transform is applied based on the Durbin algorithm with a variable frequency step. This study is a development of the existing boundary-element technique for solving problems on layered porous-elastic half-spaces. This will allow you to take into account the heterogeneity of the soil in depth. The problem of the action of a vertical force in the form of the Heaviside function on the surface of a layered porous-elastic half-space and a half-space with a cavity is considered. Variants of a homogeneous and heterogeneous half-space are considered. Under the model of heterogeneity we understand the piecewise homogeneous solid. The responses of the boundary displacements on the surface of the half-space are presented. The effect of the viscoelastic material model parameter on the dynamic response of displacements is demonstrated. It is established that the viscosity parameters have a significant effect on the nature of the distribution of parameters of wave processes.

Towards asphalt concrete modeling by the multiscale finite element method

Reliable numerical modeling of asphalt concrete (AC) is a complex problem due to a non periodic random structure of this material and a nonlinear behavior of its interacting constituents. Phenomena observed at the lower resolution highly influence the overall response of pavement layers made of asphalt concrete. In this paper, we focus on the selected aspects of its efficient numerical modeling using the viscoelastic Burgers material model. In this paper, we focus on two important components of the modeling process. Firstly, we present a straightforward algorithm for the random AC structure generation, which is automatically linked with the finite element (FE) mesh generation. It is based on the predefined shapes of inclusions and the relocation of nodes of the regular grid. Such an approach provides definitely coarser (but related to the microstructure) FE meshes than other ones used for this purposes. Thus, it saves a lot of computer resources at this level of the modeling process. Our next development concerns further reduction of the number of degrees of freedom (NDOF) that are necessary to tackle the heterogeneous domain. We take advantage of the multiscale finite element method (MsFEM) to solve given problems using a coarse macroscale mesh. Within its elements, we construct special shape functions of order 1, 2 or 3 revealing the complexity of the microresolution. We propose a novel extension of the linear elastic analysis to the viscoelastic one in terms of the MsFEM. Combining the aforementioned approaches, we provide a versatile tool for the numerical modeling of asphalt concrete. The proposed framework is illustrated with the examples of the special shape function constructions and the solutions to the compression tests of the samples made of AC. Obtained results confirm the efficiency and applicability of the proposed methodology.