Maxwell Elements In Parallel Research Articles

Numerical Investigation of Fracture Behaviour of Polyurethane Adhesives under the Influence of Moisture.

The mechanical behaviour of polymer adhesives is influenced by the environmental conditions leading to ageing and affecting the integrity of the material. The polymer adhesives have hygroscopic behaviour and tend to absorb moisture from the environment, causing the material to swell without applying external load. The focus of the work is to investigate the viscoelastic material behaviour under ageing conditions. The constitutive equations and the governing equations to numerically investigate the fracture in swollen viscoelastic material are discussed to describe the numerical implementation. Phase-field damage modelling has been used in numerical studies of ductile and brittle materials for a long time. The finite-strain phase-field damage model is used to investigate the fracture behaviour in aged viscoelastic polymer adhesives. The finite-strain viscoelastic model is formulated based on the continuum rheological model by combining spring and Maxwell elements in parallel. Commercially available post-cured crosslinked polyurethane adhesives are used in the current investigation. Post-cured samples of crosslinked polyurethane adhesives are prepared for different humidity conditions under isothermal conditions. These aged samples are used to perform tensile and tear tests and the test data are used to identify the material parameters from the curve fitting process. The experiment and simulation are compared to relate the findings and are the first step forward to improve the method to model crosslinked polymers.

Experimental characterization and numerical modeling of creep in woven textiles

In this study, Digital Image Correlation (DIC) is used to characterize creep in textiles. The creep of a plain-woven cotton fabric was studied over a relatively long period of time (32 days), which is not feasible with traditional apparatus. The results were compared with two models based on a Maxwell element in parallel with an elastic element. Good agreement was shown when the viscous law was chosen to follow the Norton-Hoff-like power law. Fitting the experimental results with a numerical scheme for this power law allowed to determine the material’s viscous parameters. In parallel, time-independent parameters were found using static testing apparatus. The obtained results will allow a better comprehension of creep in textiles when subjected to long-term stress, such like ancient textiles in museums, and hence to predict their behavior using modeling tools.

Force chains, creep and acoustic reflection in marine sediments

Marine sediments are water-saturated granular media. A feature of granular media is that the stress is not uniformly distributed but concentrated in force chains. Several force chains acting in parallel transmit external stresses through the medium. A force chain is a chain of particles through which the external stresses are concentrated and transmitted. Given that the contact stiffness between particles may be approximated as a Maxwell element, i.e. a spring in series with a damper, then the stiffness of any force chain may be modeled as several Maxwell elements in series. It is known that any number of springs and dampers in series may be reduced to just one spring and one damper in series, i.e., one equivalent Maxwell element. Therefore, several force chains in parallel may be modeled as parallel Maxwell elements. However, parallel Maxwell elements may not be reduced to one Maxwell element. The stiffness of a distribution of parallel Maxwell elements falls under the generalized creep model. This model is used to represent the skeletal frame of a larger porous medium model. The inversion of model parameters from the reflection coefficient is explored in the context of a muddy sediment. [Work supported by ONR, Ocean Acoustics Program.]

Connecting the grain-shearing, creep, and squirt flow models for wave propagation in the seabed.

The generalized creep and squirt flow models are shown to be stationary creep processes. A fractional exponent is used to develop a new generalized squirt flow model. The responses of the grain-shearing and generalized creep models are identical, although based on entirely different concepts: a single spring and time-varying damper versus a continuous distribution of parallel Maxwell elements. The random structure of marine sand is more consistent with the latter, implying absence of strain-hardening. The generalized squirt flow model has a high frequency cutoff suited to practical systems, limited by the speed with which underlying physical changes can occur.

Nonlinear Finite Element Modelling of Thermo-Visco-Plastic Styrene and Polyurethane Shape Memory Polymer Foams

This paper presents nonlinear finite element (FE) models to predict time- and temperature-dependent responses of shape memory polymer (SMP) foams in the large deformation regime. For the first time, an A SMP foam constitutive model is implemented in the ABAQUS FE package with the aid of a VUMAT subroutine to predict thermo-visco-plastic behaviors. A phenomenological constitutive model is reformulated adopting a multiplicative decomposition of the deformation gradient into thermal and mechanical parts considering visco-plastic SMP matrix and glass microsphere inclusions. The stress split scheme is considered by a Maxwell element in parallel with a hyper-elastic rubbery spring. The Eyring dashpot is used for modelling the isotropic resistance to the local molecular rearrangement such as chain rotation. A viscous flow rule is adopted to prescribe shear viscosity and stress. An evolution rule is also considered for the athermal shear strengths to simulate macroscopic post-yield strain-softening behavior. In order to validate the accuracy of the model as well as the solution procedure, the numerical results are compared to experimental responses of Styrene and Polyurethane SMP foams at different temperatures and under different strain rates. The results show that the introduced FE modelling procedure is capable of capturing the major phenomena observed in experiments such as elastic and elastic-plastic behaviors, softening plateau regime, and densification.

An RVE‐based investigation of thermoplastic vulcanizates exemplified by EPDM/PP

AbstractIn this contribution, the material behaviour of thermoplastic vulcanizate is investigated using the example of ethylene‐propylene‐diene monomer rubber (EPDM) and polypropylene (PP). For modelling of viscoelastic effects, the generalized Maxwell element is applied, whereby the isochoric stresses of parallel Maxwell elements are determined via evolution equations. In order to take the morphological composition of TPV into account, a representative volume element (RVE) consisting of PP as matrix and EPDM as included material is considered. In a following numerical example, a cyclic shear test is presented, where the resulting force and stress distribution is evaluated. Thus, the results show the macroscopic material behaviour under the consideration of microstructure.

Identification of hyper-viscoelastic material parameters of a soft member connected to another unidentified member by applying a dynamic load

Due to wide applications of hyper-viscoelastic material models in medicine and industry, many researchers have focused on determination of the parameters of these models. In problems for identification of material parameters of hyper-elastic and hyper-viscoelastic members, detailed information about boundary conditions is necessary. In reality, for an in vivo soft tissue or an engineering part in service, detailed information about the supports of the member may be unavailable. In such cases, assuming some kind of boundary conditions that may be far from truth will result in unacceptable solutions. In this work, for the first time, an inverse analysis-based method for the identification of hyper-viscoelastic parameters of a member with arbitrary shape under plane strain conditions is presented. The unknowns are found using displacement measurements in an elasto-dynamic loading, while details about the support of the member are unclear. The proposed algorithm minimizes the differences between measured and calculated displacements. The minimization algorithm needs some sensitivity analyses, which are performed via finite difference approximation and analytic differentiation. For modeling the hyper-viscoelastic behavior, a model consisting of some Maxwell elements in parallel with a neo-Hookean hyper-elastic spring is considered. The effectiveness of the proposed method is evaluated by presenting several numerical examples.

Micro-sphere based viscoplastic constitutive model for uncured green rubber

Uncured rubber exhibits strong viscoplastic flow without a distinct yield point accompanied by hardening. Classical hyperelastic models developed for crosslinked rubber do not apply to uncured rubber due to the lack of crosslinks which endow the material its elasticity. This paper presents a new constitutive model for the isothermal response of uncured green rubber. The kinematic structure of the proposed approach is based on the affine micro-sphere model. The computation of the stretch in the orientation direction follows the Cauchy-Born rule. The micro-sphere enables numerical integration over the unit sphere via finite summation over the orientation directions corresponding to the integration points over the sphere. This structure replaces the complex three-dimensional formulations of finite inelasticity based on the multiplicative split of the deformation gradient by a simpler and more attractive one-dimensional rheological representation at the orientation directions. The rheology of the model consists of two parts: (i) a part responsible for the rate-independent reponse and (ii) a part responsible for the rate-dependent response, respectively. The first branch consists of a spring connected to a modified Kelvin element, where the latter spring models the kinematic hardening. The dashpot describes a time-independent endochronic flow rule based solely on the deformation history. The second branch consists of a spring connected to a Maxwell element in parallel to a dashpot. The two dashpots in the latter branch model the ground-state viscoelasticity and rate-dependent hardening phenomenon. Albeit its apparent complexity, the proposed rheology and its numerical implementation are straightforward. The proposed model shows favorable results suitable for large scale finite element based simulations for forming process of uncured rubber components.

Relaxation base seismic isolator

SummaryThis paper deals with the use, for seismic applications, of a Maxwell element in parallel with a low damping isolator. The study of the properties of the frequency response function shows that this isolator is capable to reduce the base displacement of isolated structures with no considerable amplification of the non‐isolated modes. This is, also, confirmed by the floor response spectra under earthquake excitations. Hence, the previously mentioned isolator does not present the drawbacks met when base displacement is reduced by increasing damping. Moreover, it seems that its performance is comparable with that of more elaborated and expensive techniques combining passive and semi‐active devices. Copyright © 2016 John Wiley & Sons, Ltd.

A Gauss–Newton full-waveform inversion for material profile reconstruction in viscoelastic semi-infinite solid media

An inversion framework employing a Gauss–Newton method is developed to reconstruct material profiles in heterogeneous, viscoelastic, semi-infinite domains. In particular, a full-waveform inversion approach is investigated to image the elastic and attenuating parameters of a layered media. To account for the viscoelasticity of the medium, a Generalized Maxwell Body with one spring and two Maxwell elements in parallel (GMB2) is adopted in the forward and inverse wave propagation problems. Perfectly-matched-layers were introduced as wave absorbing buffers to simulate the semi-infinite extent of the domain. Using transient wave equations endowed with the GMB2 constitutive relation and the PML, a partial-differential-equations-constrained optimization scheme was implemented that lead to classic KKT (Karush–Kuhn–Tucker) conditions including time-dependent state, adjoint, and time-invariant control problems. An optimal solution of the viscoelastic parameters was obtained using a reduced-space approach based on a line search algorithm where the search direction was computed by the Gauss–Newton method. Considerable improvements on the accuracy and convergence rate of solutions were made by the developed Gauss–Newton inversion procedure compared to previous research using the Fletcher–Reeves method.

Comparison of approaches to model viscoelasticity based on fractional time derivatives

Two approaches to describe a constitutive fractional Zener model at large strain are presented. Both viscoelastic Zener models consist of a nonlinear elastic spring and a fractional Maxwell element in parallel. The fractional Maxwell element represents the viscoelastic behaviour of the formulation. Here, development of a new fractional viscoelastic material model under consideration of finite strain theory is presented. Additionally, the constitutive equations based on two different algorithmic approaches to capture the fractional time integration within this material model are derived. The consideration of fractional elements enables the characterization of highly inelastic, time dependent materials with relatively few material parameters. For the fractional element, a material parameter α determines the transition of the rheological element’s behaviour between spring (α=0) and dashpot (α=1). Accuracy and efficiency of a classical (non-recursive) and a new recursive algorithm to handle the fractional elements have been verified and validated by the comparison of several finite element (FE) simulations to material test results. The simulations found on finite strains and an implicit time integration scheme. Finally, a FE moulding simulation, found the extended constitutive model for the explicit time integration scheme has been carried out to illustrate the performance for large scale simulations in comparison to a real forming process and found to be quite efficient.

An orthotropic viscoelastic material model for passive myocardium: theory and algorithmic treatment

This contribution presents a novel constitutive model in order to simulate an orthotropic rate-dependent behaviour of the passive myocardium at finite strains. The motivation for the consideration of orthotropic viscous effects in a constitutive level lies in the disagreement between theoretical predictions and experimentally observed results. In view of experimental observations, the material is deemed as nearly incompressible, hyperelastic, orthotropic and viscous. The viscoelastic response is formulated by means of a rheological model consisting of a spring coupled with a Maxwell element in parallel. In this context, the isochoric free energy function is decomposed into elastic equilibrium and viscous non-equilibrium parts. The baseline elastic response is modelled by the orthotropic model of Holzapfel and Ogden [Holzapfel GA, Ogden RW. 2009. Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos Trans Roy Soc A Math Phys Eng Sci. 367:3445–3475]. The essential aspect of the proposed model is the account of distinct relaxation mechanisms for each orientation direction. To this end, the non-equilibrium response of the free energy function is constructed in the logarithmic strain space and additively decomposed into three anisotropic parts, denoting fibre, sheet and normal directions each accompanied by a distinct dissipation potential governing the evolution of viscous strains associated with each orientation direction. The evolution equations governing the viscous flow have an energy-activated nonlinear form. The energy storage in the Maxwell branches has a quadratic form leading to a linear stress–strain response in the logarithmic strain space. On the numerical side, the algorithmic aspects suitable for the implicit finite element method are discussed in a Lagrangian setting. The model shows excellent agreement compared to experimental data obtained from the literature. Furthermore, the finite element simulations of a heart cycle carried out with the proposed model show significant deviations in the strain field relative to the elastic solution.

Introducing objective power law rate dependence into a visco-elastic material model of bovine cortical bone

The strain rate dependent behaviour of some visco-elastic materials can be modelled accurately in 1-D by including a stress contribution which depends non-linearly on strain rate. A visco-elastic model by Shim et al. (2005) [3] comprises a Voigt and Maxwell element in parallel and provides an effective representation of cancellous bone from the human cervical spine. The present study demonstrates that the model by Shim et al. is also suitable for modelling the strain rate dependent compression of cortical bone from bovine femurs. Shim et al. found that the model requires a Voigt dash-pot contribution which is proportional to ε˙1/2 in order to model specimen response accurately over a large range of strain rates. Shim et al. proposed an expansion of the 1-D formulation to 3-D where the 1-D strain rate is replaced with a function of the strain rate tensor. This paper provides a frame invariant version of the model by Shim et al. which allows general power law rate dependence for the 3-D case. The response of the model is investigated under a load condition which comprises of an axial deformation and a shear twist. The model is implemented in a commercial finite element package and is used to simulate quasi-static and dynamic bovine bone compression experiments.

Creep of a Twaron®/Natural Rubber Composite

The creep behavior of a Twaron CT709® fabric/natural rubber composite under a uniaxial constant stress is studied using three viscoelasticity models with different levels of complexity and a newly developed para-rheological model. The three viscoelasticity models employed are a one-term generalized Maxwell (GM n =1) model (consisting of one Maxwell element and an additional spring in parallel), a two-term generalized Maxwell (GM n =2) model (including two parallel Maxwell elements and an additional spring in parallel), and a four-parameter Burgers model. The values of the parameters involved in each model are extracted from the experimental data obtained in this study. The numerical results show that the initial creep response of the composite is predicted fairly well by the GM n =2 model, while the secondary creep is more accurately described by the GM n =1 model. An implicit solution, together with a retardation time spectrum, obtained using the para-rheological model is found to provide more accurate predictions of the composite creep response than the three viscoelasticity models in both the primary and secondary stages. The long-time response of the composite is also studied using the four models, with the para-rheological model providing the best predictions.

Modelling mechanical characteristics of microbial biofilms by network theory

In this contribution, we present a constitutive model to describe the mechanical behaviour of microbial biofilms based on classical approaches in the continuum theory of polymer networks. Although the model is particularly developed for the well-studied biofilms formed by mucoid Pseudomonas aeruginosa strains, it could easily be adapted to other biofilms. The basic assumption behind the model is that the network of extracellular polymeric substances can be described as a superposition of worm-like chain networks, each connected by transient junctions of a certain lifetime. Several models that were applied to biofilms previously are included in the presented approach as special cases, and for small shear strains, the governing equations are those of four parallel Maxwell elements. Rheological data given in the literature are very adequately captured by the proposed model, and the simulated response for a series of compression tests at large strains is in good qualitative agreement with reported experimental behavior.

Time-domain forward and inverse modeling of lossy soils with frequency-independent Q for near-surface applications

In this paper, we are concerned with a full-waveform-based methodology that allows the simultaneous imaging of the soil's stiffness and attenuating properties, using solely the soil's surficial response to probing waves.To date, field observations of small-strain wave attenuation in geomaterials at moderate spatial scales suggest that a commonly used metric of intrinsic and apparent attenuation, the seismic quality factor Q, is frequency-independent for a wide part of the frequency spectrum, including the frequency range of interest to seismic applications. We discuss first the forward simulation of waves in near-surface soil deposits directly in the time-domain using simplified models that adequately approximate nearly frequency-independent Q. To this end, we first review various attenuation models that aim at reproducing the frequency-independent Q behavior, and conclude, supported by site analyses, that, even though a generalized Maxwell body with eight Maxwell elements in parallel (GMB8) provides the best fit to frequency-independent Q, we favor a version of it with fewer parameters (GMB2), in order to reduce modeling complexity, while still retaining good agreement with the GMB8 model.We report on forward site analyses that lend credence to the choice of the GMB2 simplified model. We, then, use the GMB2 constitutive relation in the context of full-waveform inversion, and report on numerical experiments that lead to the imaging of the soil's properties in heterogeneous semi-infinite domains.

The interfacial fracture behavior of foam core composite sandwich structures by a viscoelastic cohesive model

A sandwich beam model consisting of two face sheets and a foam core bonded by a viscoelastic adhesive layer is considered in order to investigate interfacial fracture behavior. Firstly, a cohesive zone model in conjunction with a Maxwell element in parallel, or with a Kelvin element in series, respectively, is employed to describe the characteristics of viscoelasticity for the adhesive layer. The models can be implemented into the implicit finite element code. Next, the parametric study shows that the influences of loading rates on the cohesive zone energy and strength are quite different for different models. Finally, a sandwich double cantilever beam model is adopted to simulate the interface crack growth between the face sheet and core. Numerical examples are presented for various loading rates to demonstrate the efficacy of the rate-dependent cohesive models.

Viscoelastic Properties of Tarocco Orange Fruit

In this work, the rheometrical behaviour of Tarocco orange fruit was assessed via stress–relaxation testing under constant engineering strain (ez) in the range of 5–20% of the initial fruit height, each test being prolonged for 15 min and replicated 20 times. The time course of the experimental dimensionless relaxation modulus (G*) was typical of a viscoelastic solid and was reconstructed by considering two Maxwell elements in parallel to an elastic spring element. In a more restricted ez range (i.e. 5–10%), the dimensionless viscoelastic constants (Ai) were found to be approximately constant, the elastic component responsible for the equilibrium residual stress representing about 42% of the initial relaxation modulus. After any orange fruit had been submitted to stress–relaxation testing and decompressed for 5 min, it exhibited a practically negligible mass loss (0.04 ± 0.03%) and a permanent volume contraction rising from 1.7 ± 1.0% to 3.0 ± 0.9% as ez was increased from 5% to 10%. Thus, by tasting the orange fruit firmness at ez ≤5%, fruit response, being of the viscoelastic type, is expected to give rise to nearly irrelevant deformations.

Thixotropic elasto-viscoplastic model for structured fluids

A constitutive model for structured fluids is presented. Its predictive capability includes thixotropy, viscoelasticity and yielding behavior. It is composed by two differential equations, one for the stress and the other for the structure parameter—a scalar quantity that represents the structuring level of the fluid. The equation for stress is obtained in accordance with a simple mechanical analog composed by a structuring-level-dependent Maxwell element in parallel with a Newtonian element, leading to an equation of the same form as the Jeffreys (or Oldroyd-B) equation. The relaxation and retardation times that arise are functions of the structure parameter. The ideas found in de Souza Mendes, J. Non-Newtonian Fluid Mech., 2009, 164, 66 are employed for the structure parameter equation as well as for the dependencies on the structure parameter of the structural viscosity and structural shear modulus. The model is employed in constant-rate, constant-stress, and oscillatory shear flows, and its predictive capability is shown to be excellent for all cases.

Finite element implementation including sensitivity analysis of a simple finite strain viscoelastic constitutive law

This article describes the implementation of a finite strain viscous hyper-elastic material law into a finite element code. The material model is made up of a Maxwell element in parallel with a spring. Both springs represent hyper-elastic material behaviour, and compressibility is taken into account. In addition to that, the sensitivity analysis using direct differentiation is described extensively. The implementation is verified by three test cases, uniaxial tension, pure shear and indentation testing. The sensitivity analysis by direct differentiation is validated by comparison against results obtained by a finite difference scheme.