Hereditary Integral Research Articles

Modelling of viscoelastic materials using non-ordinary state-based peridynamics

This paper proposes a framework for implementing viscoelastic constitutive model from the classical continuum mechanics (CCM) theory within non-ordinary state-based peridynamics (NOSBPD). The motivation stems from the inadequacy of CCM to model very complex material behaviours such as initiation and propagation of cracks and nonlocal behaviour due to size effects. The proposed formulation leverages on the constitutive correspondence between NOSBPD and CCM to incorporate a CCM viscoelastic constitutive model based on hereditary integral into NOSBPD. The combination of hereditary constitutive model and NOSBPD effectively makes this formulation a nonlocal time–space viscoelastic framework where temporal nonlocality is incorporated by a hereditary viscoelastic model which stipulates that the behaviour of a material at any point in time depends on both the present action and the complete history of previous actions on the material, and spatial nonlocality on the other hand is incorporated via the nonlocal mechanism provided by the NOSBPD. For model validation, three benchmark problems were solved using the proposed framework. Results obtained were compared to results from analytical solution and solutions from referenced literature. In addition, parametric study was conducted to determine the influence of nonlocality on numerical prediction. Conclusions drawn from the validation studies presented are that the proposed framework is able to predict viscoelastic responses that agree well with local macro models as well as nonlocal micromodels/nanomodels as reported in the literature.

Analytical-based exact-kernel vibration and long-term creep stress and large deformation redistributions of the suddenly pressurized incompressible visco-hyperelastic thick cylinders

The long-term creep and short-term damped dynamic redistributions of the large deformations and stresses are investigated here for long, suddenly pressurized, rubber-like visco-hyperelastic thick-walled cylinders, for the first time. The extension of the Mooney–Rivlin strain energy density function to incompressible visco-hyperelastic materials is accomplished by using a hereditary integral based on the Prony series and an exponential relaxation kernel. To extract a more accurate/efficient formulation, the relaxation kernel is processed instead of the common approximate discretization of the time derivative of the stress tensor in the convolution integral. The resulting nonlinear integrodifferential equations whose number of terms grows with time are solved by a novel analytical approach in conjunction with the second-order Runge–Kutta time-marching, bound-varying trapezoidal technique, and an iterative/updating scheme. In comparison to the commercial finite element analysis codes, the present analytical formulation and solution algorithm lead to higher accuracies from the mathematical point of view. The visco-hyperelastic properties are extracted from previously reported experimental results, using an error minimization fitting technique. Results reveal that in contrast to the hyperelastic vessel, the oscillations of the displacements and stresses of the visco-hyperelastic cylinder occur around levels that grow with time. Moreover, smaller relaxation parameters lead to smaller displacements and stresses and the responses intend earlier to the steady-state condition for higher values of the stiffness elements of the Prony series. In long term, the mean stresses and mean radial displacements of the inner and outer layers of the cylinder grow asymptotically with time but the largest overshoots and amplitudes of the responses occur at the early times. However, smaller values of the relaxation parameter expedite the convergence to the steady-state condition.

Multi-scale constitutive model of human trabecular bone

The present study aims to formulate a new multiscale constitutive model of human trabecular bone. The trabecular bone was modelled as a nonlinear viscoelastic material. The viscoelastic effects of single trabeculae were considered by means of a hereditary integral in which stress depends on time and strain, while the elastic response was described by the hyperelastic Mooney–Rivlin model. The cuboid bone sample was extracted from the femoral head during the hip replacement surgery. The material constants in the constitutive equation were identified based on the stress relaxation test performed on the cuboid sample and the microindentation tests performed on trabeculae using the curve-fitting procedure. The microindentation tests were performed using a spherical tip instead of Vickers or Berkovich tip to minimize plastic effects during trabecular deformation. In order to validate formulated constitutive model, results from a FE simulation of stress relaxation test and uniaxial compression test were compared to the results of the corresponding experiments conducted on a macroscopic bone sample. Good agreement was observed between numerical and experimental results. The viscoelastic behaviour predicted by the proposed constitutive equation corresponds well to the response of human trabecular bone under various types of load conditions. This demonstrates the high ability of our constitutive model to simulate the behaviour of trabecular bone on a micro- and macroscopic scale. Thus, we conclude that the model, which was formulated for a single trabecula, can be successfully applied to simulate mechanical behaviour of the tissue in a macroscale.

Multifield constitutive identification of non-conventional thermo-viscoelastic periodic Cauchy materials

The present work proposes a variational-asymptotic homogenization technique for non-conventional thermo-viscoelastic periodic microstructured materials. According to second sound theories, the heat flux vector linearly depends upon the history of temperature gradient and the heat conduction tensor represents the kernel of the relative hereditary integral, thus overcoming the paradox inferred from the usual Fourier’s law of thermal waves propagating at infinite speed. Down-scaling relations have been provided in the frequency domain, relating the transformed micro displacement and relative temperature fields to the corresponding macro variables and their gradients through frequency-dependent perturbation functions which convey the influence of the underlying microstructural heterogeneity. Average field equations of infinite order have been derived. Transformed field equations of the equivalent first-order medium have been obtained as Euler–Lagrange equations of a suitable functional whose first variation is properly truncated. They are characterized by frequency-dependent overall constitutive tensors, whose closed form has been provided. A benchmark test assesses the capabilities of the proposed homogenization method, where the solutions relative to a periodic, bi-phase, thermo-viscoelastic material and to the equivalent homogenized medium match under the hypothesis of periodic source terms.

ТОЧНОЕ РЕШЕНИЕ ЗАДАЧИ О РЕЛАКСАЦИИ НАПРЯЖЕНИЙ В ТОЛСТОСТЕННОЙ ТРУБЕ ИЗ НЕЛИНЕЙНО ВЯЗКОУПРУГОГО МАТЕРИАЛА, ПОДЧИНЯЮЩЕГОСЯ СООТНОШЕНИЮ РАБОТНОВА

We constructed and studied analytically the exact solution of the quasi-static boundary value problem for a hollow cylinder (a tube) made of physically nonlinear homogeneous isotropic viscoelastic material obeying the Rabotnov constitutive equation with two arbitrary material functions (a creep compliance and a function which governs physical nonlinearity). A time-dependent displacement and zero shear stresses are given on the inside cylinder surface, a pressure (normal stress) is given on outside surface and zero axial displacements and zero shear stresses are preset on the end cross sections of the tube (thus, cylinder stress and plain strain are realized in a tube). We supposed that a material is incompressible and that given boundary displacement and pressure vary slowly enough with time to neglect inertia terms in the equilibrium equations. We obtained explicit expressions for strains and stresses at any point via the ratio of a tube radii, given external pressure and integral operators involving composition of two material functions of the constitutive relation and displacement preset on the internal cylindrical surface. In particular, assuming given boundary displacement and pressure are constant we considered the stress relaxation problem. For arbitrary material functions, we calculated all the hereditary integrals involved in the general representation for the stress field and reduced it to simple algebraic formulas convenient for analysis and use. We studied analytically general properties of the stress relaxation curves at any point of a tube and features of stress distributions along radius. We proved that the first material function (relaxation modulus) of the constitutive equation governs completely stresses dependence on time and the second one (non-linearity function) governs only stresses dependence on radial coordinate and found out that hoop stress and axial stress can change sign and can increase, decrease and be non-monotone with respect to radial coordinate. In case of zero external pressure, we proved that all stress relaxation curves at any point of a tube are proportional to each other (ratio of stresses at different points doesn't depend on time), decrease with time and have no flexure points. We also calculated integral mean values of axial and hoop stresses and discovered that their ratio depends on thickness/radius ratio only and doesn't depend on time and material functions although mean stresses do.

Visco-elastic-plastic model to represent the compression behaviour of sugarcane agricultural residue

The objective was to develop a visco-elastic-plastic model for the prediction and characterisation of the compaction behaviour of sugarcane agricultural residues during the production of bales under different moisture contents (27%, 37% and 48%) and different particle sizes (∼300 mm, 150 mm and 80 mm). Data of confined-compaction and stress relaxation experiments were used to calibrate the model parameters. The proposed constitutive equation was composed of a generalised Maxwell model with three branches in series with a plastic component. The procedure was developed to characterise model parameters and used a numeric approach to calculate the hereditary integral when fitting the experimental curves, whilst the stress due to the plastic strains was fitted with a polynomial. The model provided good accuracy ( R 2 > 0.92) to fit the compaction and relaxation stages with the same set of parameters. Thus, it predicted the energy required to produce bales as well as the final relaxed density of this sustainable material that may become an important source of energy. • Biomass was characterised using a new visco-elasto-plastic model. • Fit of compaction and relaxation stages with unique set of parameters achieved. • Model could be applied to study the compaction of any type of biomass.

New Mechanistic Procedure to Predict the Critical Cracking Temperature of Asphalt Concrete from Bending Beam Rheometer and Indirect Tension Test Data

This study presents a new mechanistic procedure for determining the critical cracking temperature of asphalt concrete (AC) using data from bending beam rheometer (BBR) test of asphalt binder and indirect tension (IDT) test of AC. This new procedure uses BBR creep data to generate the mixture relaxation modulus mastercurve by utilizing the Hirsch model, time-temperature superposition principle, and Prony series-based interconversion method. The Hirsch model parameters are calibrated by comparing creep data from BBR and IDT creep tests performed at the same temperature. Boltzmann hereditary integral and second-order heat equation are then used to calculate thermal stress from the developed relaxation modulus mastercurve. IDT strength data is transferred from test strain rate to thermal strain rate using the viscoelastic continuum damage model. Since a strain gauge is not attached for traditional laboratory IDT strength testing, this study derived an analytical equation based on the Hondros solution to compute the horizontal strain rate from the applied vertical displacement rate. Finally, the critical cracking temperature is determined by coupling the thermal stress and strength profiles. Using the procedure presented in this paper, the critical cracking temperatures of four AC mixtures were predicted from BBR and IDT data. Their actual critical cracking temperatures were measured using thermal stress restrained specimen test performed in the laboratory to validate the method. The predicted critical cracking temperatures are found to be very close to the laboratory measured values. The developed procedure has substantial practical and technical importance in predicting the critical cracking temperature of AC because it utilizes widely available BBR and IDT tests.

Free volume based nonlinear viscoelastic model for polyurea over a wide range of strain rates and temperatures

A series of quasi-static and dynamic uniaxial compression experiments over a wide range of temperatures were conducted on polyurea elastomer. The stress-strain response of polyurea exhibits extreme nonlinearity, temperature and strain-rate sensitivity. How to comprehensively describe these complex behaviors regulated by the unique micro-structure of polyurea is still an open question. In this paper, a nonlinear viscoelastic constitutive model for polyurea is developed. The nonlinear viscoelasticity, entropic and energetic elasticity are taken into account, correlated with the interaction of chain segments, the stretch of molecular chains and covalent bonds from the perspective of molecular structure. The viscoelastic response is formulated by the hereditary integral of relaxation modulus and strain rate with time. The real time is accelerated by shift factor, which depends on the free volume. A new mechanism of shear-induced increase of free volume, which gives rise to the nonlinearity of viscoelasticity, is proposed, and the corresponding rate-dependent evolution equation of fractional free volume is developed. Parameters in the model are determined through the compression experiments and DMA test. Comparison with experimental data, as well as the data from references, verifies the ability of the model to predict the nonlinear stress-strain response of polyurea over a wide range of temperatures and strain rates.

Viscoelastic analysis of bistable composite shells via absolute nodal coordinate formulation

Material viscoelasticity may play a significant role in the bistable composite shells, especially after long-term stowage. This article investigates the influence of viscoelastic effects on the mechanical behaviors of bistable composite cylindrical shells in coiled state, for which a novel modeling approach for viscoelastic analysis in integral form of laminated composite shells is proposed based on the nonlinear continuum mechanics and the absolute nodal coordinate formulation. The model presented herein is also very appropriate for the dynamics analysis of deployment of bistable tape springs. Furthermore, an efficient calculation scheme of the highly nonlinear viscoelastic forces with hereditary integral is presented. By comparing the predictions of viscoelastic and elastic analyses for the radii, stresses distributions, and configurations of coiled shells, the effect of viscoelasticity on the second stable equilibrium is illustrated. The results show that relaxation behaviors of bistable composite shells at high strains are evident.

Modeling and analysis of the transient response of viscoelastic solids

ABSTRACT In this paper, an incremental finite element model (IFEM) for the transient analysis of viscoelastic solids is developed. The viscoelastic constitutive response is modeled based on the hereditary integral. Two new incremental constitutive forms suitable for finite element implementation are derived; creep-based (CB) and relaxation-based (RB) and, as a consequent, the difficulties associated with the interconversion from the relaxation modulus to the creep compliance and vice versa are avoided. Creep compliance and relaxation modulus are, respectively, modeled using generalized Kelvin–Voigt model and generalized Wiechert’s model. To check the validity, the computational procedure, semi-analytical solution is derived for simple viscoelastic cases using Laplace and inverse Laplace transform techniques. The obtained results from the computational procedure are compared with that obtained from the derived semi-analytical solution. To show the validity and applicability of the developed model, two different applications under different excitation patterns are presented. In each application, the time response for displacements and stresses, especially for the dynamic stress concentration factor for both elastic and viscoelastic analyses are investigated. Moreover, the dissipative recoverable nature of the viscoelastic solids under different loading-unloading excitation patterns is illustrated.

Response functions in linear viscoelastic constitutive equations and related fractional operators

This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators.

Application of a Viscoelastic Model to Creep Settlement of High‐Fill Embankments

In order to predict the creep settlement of high‐fill embankments, the time‐dependent viscoelastic model of Poynting–Thomson (the standard linear solid) has been chosen to represent the creep behavior of soils. In the present study, the hereditary integral was applied to calculate the strain while the load increase is varied with time. Calculation expressions of the creep settlement of an embankment during and after construction were obtained under one‐dimensional compression conditions. Using this approach, the three parameters of every layer can be determined and adjusted to accommodate in situ monitoring data. The calculated results agreed well with those from the field, which imply that the method proposed in this paper can give a precise prediction of creep settlement of high‐fill embankments.

Thermo-mechanical coupling of a viscoelastic-viscoplastic model for thermoplastic polymers: Thermodynamical derivation and experimental assessment

In this paper, a new constitutive model is proposed for the behavior of thermoplastic polymers under non-isothermal conditions. The model couples linear viscoelasticity, viscoplasticity and thermal effects. It is formulated within the framework of irreversible thermodynamics. The total strain is the sum of viscoelastic, viscoplastic and thermal strains. General hereditary integrals describe the thermo-viscoelastic response. The viscoplastic part accounts for both isotropic and kinematic hardenings. The stress-strain response and the material self-heating are predicted and compared to experimental data on Polyamide 66 (PA66) and Polypropylene (PP). Good agreement between the numerical simulations and experimental data was obtained for the two materials.

A Comparison Between Fractional-Order and Integer-Order Differential Finite Deformation Viscoelastic Models: Effects of Filler Content and Loading Rate on Material Parameters

Elastomers or rubber-like materials exhibit nonlinear viscoelastic behavior such as creep and relaxation upon mechanical loading. Differential constitutive models and hereditary integrals are the main frameworks followed in the literature for modeling the viscoelastic behavior at finite deformations. Regular differential operators can be replaced by fractional-order derivatives in the standard models in order to make fractional viscoelastic models. In the present paper, the relaxation behavior of elastomers is formulated both in terms of ordinary (integer-order) and fractional differential viscoelastic models. The derived constitutive equations are fitted to several experimental data to compare their efficiency in modeling the stress relaxation phenomenon. Specifically, a fractional viscoelastic model with one fractional dashpot (FD) is compared with two ordinary models including respectively one and two ordinary dashpots (OD). The models are compared in fitting accuracy, number of required material parameters and also variation of parameters from one compound to another to clarify the effects of filler content and deformation rate. It is shown that, the results of the ordinary model with one OD is not good at all. The fractional model with one FD and the ordinary model with two ODs provide good fittings for all compounds whereas the former uses only three parameters and the latter uses five material parameters. For the fractional model, the order of the Maxwell element and the associated relaxation time approximately remain the same for different compounds of each material at certain loading rates, but it is not the case for the ordinary differential models.

Multiregional viscoelastic characterization of the corona radiata in the sagittal plane of the porcine brain.

Detailed finite element (FE) models are used as promising tools to investigate traumatic brain injuries, although their accuracy is strongly dependent on the characterization of the mechanical behaviors of the different anatomic structures in the brain. In some cases, when the FE models require finer spatial resolution, the heterogeneous and anisotropic corona radiata cannot be taken as a homogeneous whole body. In this work, indentation experiments were conducted on the anterior, superior, and posterior regions of the corona radiata in the sagittal plane. To determine the parameters available for computational modeling purposes, a linear viscoelastic model using the Boltzmann hereditary integral was fitted to the force-time data of the three regions. In the indentation tests, the superior region appeared to be the stiffest, while no significant differences were observed between the anterior and posterior regions until the viscoelastic tissue reached equilibrium. During the period of relaxation, statistical comparisons among the different regions indicated significant differences between the superior and anterior regions, and between the superior and posterior regions. This work complements existing investigations into the anatomic heterogeneity of the brain, and contributes toward improving the spatial resolution of future computational models. Graphical abstract Relaxation functions of different regions based on the Prony series parameters and the multiregional Kolmogorov-Smirnov comparisons (*p< 0.017). The anisotropy and interregional differences of the corona radiata observed in this study are supplementary to the previous explorations of the mechanical properties of different brain anatomic structures.

New microscale constitutive model of human trabecular bone based on depth sensing indentation technique.

A new constitutive model for human trabecular bone is presented in the present study. As the model is based on indentation tests performed on single trabeculae it is formulated in a microscale. The constitutive law takes into account non-linear viscoelasticity of the tissue. The elastic response is described by the hyperelastic Mooney-Rivlin model while the viscoelastic effects are considered by means of the hereditary integral in which stress depends on both time and strain. The material constants in the constitutive equation are identified on the basis of the stress relaxation tests and the indentation tests using curve-fitting procedure. The constitutive model is implemented into finite element package Abaqus® by means of UMAT subroutine. The curve-fitting error is low and the viscoelastic behaviour of the tissue predicted by the proposed constitutive model corresponds well to the realistic response of the trabecular bone.

Application of fractional time derivatives in modeling the finite deformation viscoelastic behavior of carbon-black filled NR and SBR

Mechanical behavior of elastomers includes nonlinear and time dependent phenomena such as creep and relaxation where due to low stiffness, these materials usually undergo large deformations. The classical approaches of viscoelasticity,i.e. differential constitutive models and hereditary integrals have been widely used to model the viscoelastic behavior of these materials even at finite deformations. However, from the viewpoint of computational analysis, the integral based methods are often time and memory consuming and the differential methods usually need a large number of material parameters whose identification is a real challenge. A potential solution is to introduce fractional order derivatives instead of regular differential operators in the standard spring-dashpot mechanical models which remarkably reduces the number of required material parameters. In the present paper, relaxation behavior of NR and SBR is investigated in finite strains. Due to the remarkable effects of filler particles and deformation rate on the mechanical properties of elastomers, compounds of each material with different amounts of filler content are investigated at two strain rates. A main interest is to examine efficiency of the fractional derivative models in modeling the finite deformation relaxation behavior of elastomers with different filler contents. The other goal is to investigate the effect of loading rate on the obtained model parameters. It is observed that, the fractional Maxwell chain is able to well fit the stress relaxation response of all compounds with only three parameters. Analyzing the obtained parameters and the experimental data, it is concluded that, order of the associated Maxwell element is not much affected with the filler content. Seemingly, it depends on the matrix material and more noticeably on the deformation rate. Moreover, as much as the deformation rate, the associated relaxation time is low. Decrease of the relaxation time with the deformation rate is more evident for compounds of SBR compared with NR compounds.

Rate-dependent behavior of connective tissue through a micromechanics-based hyper viscoelastic model

In this paper, a micromechanical study on rate-dependent behavior of connective tissues is performed. To this end, a hyper viscoelastic constitutive model consisting a hyperelastic part for modeling equilibrium response of tissues and a viscous part using a hereditary integral is proposed to capture the time-dependent behavior of the tissues. With regard to the hierarchical structure of the tissue, strain energy function are developed for modeling elastic response of the tissue constituents i.e. collagen fibers and ground matrix. The rate-dependency is incorporated into the model using a viscous element with rate-dependent relaxation time. The proposed constitutive model is implemented into ABAQUS using the UMAT subroutine. Results of the presented micromechanics model are compared and validated with the available experimental data and also the quasi-linear viscoelastic (QLV) theory at different strain rates. It was found that the proposed model could well predict the behavior of the connective tissues in wide range of strain rates. Results show that, the presented model contains less constitutive parameters and leads to more accurate results in comparison with the QLV.

A nonlinear viscoelastic bend stiffener steady-state formulation

Bend stiffeners are essential components of a flexible riser system, employed to ensure a smooth transition at the upper connection and to protect the riser against over bending and from accumulation of fatigue damage. The highly nonlinear rate dependent behavior of these structures directly affects the integrity assessment of the riser in one of its most critical regions, the top connection. A steady-state formulation (disregarding inertial forces) and numerical solution procedure is developed in this work employing the perturbation method for a nonlinear viscoelastic bend stiffener large deflection beam model subjected to harmonic loading conditions. For stochastic loading conditions, the response is calculated employing the superposition principle by summing up the steady-state result of a number of individual frequency components. A time domain formulation is also derived employing the state-variable approach for the numerical solution of the resulting hereditary integral in the governing equations. A case study is presented for the top connection system of a 4″ ID flexible riser using relaxation and tensile experimental data obtained from a typical class of bend stiffener polyurethane. Harmonic and stochastic input loading conditions are employed for time and frequency domain model comparison/validation and to assess loading history and frequency influence in the curvature response.

Alternative Calculation of the Retardation and Relaxation Spectra for the Viscoelastic Response of the Solid Phase Materials

It is a well-known fact that rheology operates within the material science. Viscoelastic models represent various materials’ mechanical behavior as building materials, polymers, tissues. The constitutive relation of the model can be used for the prediction of the materials behavior. As each viscoelastic model consists of several Hookean elastic (H) and Newton viscous (N) matters, interconnected serially or in parallel, for constitutive equation deriving of the resulting model, the coupling of the elementary constitutive equations of (H) and (N) is performed with regard to configuration of the entire model. Within the creep and relaxation tests, the spectra of the retardation and relaxation are yielded. These spectra are important in hereditary integrals and Prony's series for materials that perform the hysteresis. As the mathematical tool the theory of ordinary differential equations systems is used.