Viscoelastic Constitutive Law Research Articles

Optimisation of process-induced residual stresses in composite laminates by different genetic algorithm and finite element simulation coupling methods

To address the residual stress induced during the cure of fibre reinforced thermoset polymer composites, two different approaches were suggested for coupling a non-dominated sorting genetic algorithm (NSGA-II) with finite element (FE) simulations based on a viscoelastic constitutive law. These two approaches were proposed with consideration of different ways of integrating NSGA-II and the FE model. In Approach A, NSGA-II was performed based on results from a series of simulations under various combinations of cure variables. Alternatively, Approach B employed NSGA-II to iteratively update and optimise the cure profile for subsequent simulations. Results indicated that both approaches achieved simultaneous reductions in cure time and macroscale residual stress, with Approach B showing further improvements due to the direct coupling between the NSGA-II and simulations. Specifically, the maximum residual stress and cure time optimised by Approach A were reduced by 5%–9% and 22%–50% respectively, while those obtained by Approach B were reduced by 7%–10% and 32%–49% respectively, compared to those based on the manufacturer recommended cure profile. The evolution of stress in composites based on optimised cure profiles from these two approaches was also elucidated. Additionally, microscale modelling further revealed a 3%–5% reduction in the average residual stress within a representative volume element (RVE) model was also shown, depending upon the approach adopted. Ultimately, by combining a NSGA-II and FE simulations, the optimisation of cure time and residual stress at the macroscale and cure time together with a reduction of microscale stress could be realised.

Convergence Results for History-Dependent Variational Inequalities

We consider a history-dependent variational inequality in a real Hilbert space, for which we recall an existence and uniqueness result. We associate this inequality with a gap function, together with two additional problems: a nonlinear equation and a minimization problem. Then, we prove that solving these problems is equivalent to solving the original history-dependent variational inequality. Next, we state and prove a convergence criterion, i.e., we provide necessary and sufficient conditions which guarantee the convergence of a sequence of functions to the solution of the considered inequality. Based on the equivalence above, we deduce various consequences that present some interest on their own, and, moreover, we obtain convergence results for the two additional problems considered. Finally, we apply our abstract results to the study of an inequality problem in solid mechanics. It concerns the study of a viscoelastic constitutive law with long memory and unilateral constraints, for which we deduce a convergence result and provide the corresponding mechanical interpretations.

Simulation of viscoelastic free-surface flows with the Particle Finite Element Method

AbstractViscoelastic fluids are central in numerous applications from polymer manufacturing to the pharmaceutical industry and biological research. However, since analytical solutions are generally not available or too complex, it is common practice to study free-surface viscoelastic flows through numerical simulation techniques. This work proposes the use of the so-called particle finite element method (PFEM), a Lagrangian approach combining standard FEM techniques with a remeshing strategy. The PFEM is able to efficiently handle mesh distortion and to accurately track the free-surface evolution. Therefore, it is exploited in this work to deal with large displacements problems in the context of nonlinear viscoelasticity. An implementation of the Oldroyd-B constitutive model in the PFEM framework is here presented including details regarding how to deal with the transfer of the internal variables during remeshing events. Additionally, an innovative approach to impose unilateral Dirichlet boundary conditions ensuring optimal mass conservation is presented. The implementation is verified with two free-surface highly viscous benchmark flows: the impacting drop and the jet buckling problems. The results show perfect agreement with those obtained with other numerical techniques. The proposed framework opens the way for using PFEM in various applications, ranging from polymer extrusion to more sophisticated scenarios involving viscoelastic and viscoelasto-plastic constitutive laws.

NUMERICAL IMPLEMENTATION OF THE TSCHOEGL–CHANG–BLOCH NONLINEAR VISCOELASTIC CONSTITUTIVE LAW FOR UNVULCANIZED CARBON BLACK–FILLED RUBBERS

ABSTRACT Rubber engineering design analysis requires a fundamental understanding of the mechanical behavior of polymers, especially in their unvulcanized state. It necessitates the establishment of a three-dimensional constitutive material to account for the observed mechanical behavior. This law is required to produce realistic descriptions of the viscoelastic performance in a mathematically simple form that is easy to implement in engineering applications. This article describes the theory of the Tschoegl–Chang–Bloch time-dependent nonlinear viscoelastic constitutive law. The experimental verification of this law is provided under different deformation fields and multiple load steps. Laboratory test procedures to obtain the parameters required to describe the material under consideration are provided in detail. A recursive form of the constitutive law, suitable for finite element application, is derived and coded in the finite element commercial code Abaqus via the user subroutine UMAT. Comparisons between the experimental observations, the theoretical results, and the numerical data are drawn for simple test models examined under creep or shear relaxation conditions. The excellent agreements observed indicate the suitability of the governing law in analyzing viscoelastic problems of unvulcanized carbon black–filled rubbers.

NUMERICAL IMPLEMENTATION OF THE TSCHOEGL–CHANG–BLOCH NONLINEAR VISCOELASTIC CONSTITUTIVE LAW FOR UNVULCANIZED CARBON BLACK–FILLED RUBBERS

ABSTRACT Rubber engineering design analysis requires a fundamental understanding of the mechanical behavior of polymers, especially in their unvulcanized state. It necessitates the establishment of a three-dimensional constitutive material to account for the observed mechanical behavior. This law is required to produce realistic descriptions of the viscoelastic performance in a mathematically simple form that is easy to implement in engineering applications. This article describes the theory of the Tschoegl–Chang–Bloch time-dependent nonlinear viscoelastic constitutive law. The experimental verification of this law is provided under different deformation fields and multiple load steps. Laboratory test procedures to obtain the parameters required to describe the material under consideration are provided in detail. A recursive form of the constitutive law, suitable for finite element application, is derived and coded in the finite element commercial code Abaqus via the user subroutine UMAT. Comparisons between the experimental observations, the theoretical results, and the numerical data are drawn for simple test models examined under creep or shear relaxation conditions. The excellent agreements observed indicate the suitability of the governing law in analyzing viscoelastic problems of unvulcanized carbon black–filled rubbers.

A novel refined Caputo kernel and constitutive concepts for semi-exact nonlinear dynamic and creep analyses of suddenly pressurized hollow fractional-order visco-hyperelastic cylinders

Some reports have confirmed that discretizing the mass, damping, and stiffness characteristics of a continuum can result in remarkable deviations from the experimental results; as all these properties are intertwined and coupled within a single material. The viscoelastic constitutive laws with fractional-order time derivatives can model this integrity much more accurately, due to their capability of using a set of an infinite number of orders. Here, an innovative constitutive model with a nonlinear hyperelastic memory element that treats simultaneously the fractional-order viscoelasticity and Mooney-Rivlin hyperelasticity of the continuum is proposed to track the suppression of the short-term dynamic stresses and displacements and the redistribution of the long-term creep stress and deformations of the suddenly pressurized thick cylinders. While the spatial variations of the quantities of the incompressible cylinder are treated by an exact analytical approach, the resulting time-dependent system of equations is converted by a novel numerical solution scheme to an integrodifferential system that contains integer-order partial derivatives and a time-growing number of integral terms. An enhancement to Caputo's singular-kernel integral definition of the fractional-order derivatives is proposed here to replace these derivatives with equivalent integrals whose integer derivatives are computed based on second-order Taylor's expansion. The integer-order and fractional-order time derivatives are then computed numerically using the second-order Runge-Kutta and accumulation-based trapezoidal techniques, respectively. Finally, comparisons are made among the dissipative pattern and time variations of the dynamic/vibration stresses and large-deformations of the hyperelastic and fractional-order visco-hyperelastic cylinders in addition to the assessment of the effects of the constitutive parameters of the fractional-order model. Results show that while the larger magnitudes of both the nonlinear viscosity coefficients and fractional-order time derivatives diminish the resulting deformations and stresses, they respectively lead to smaller and larger frequencies of oscillation.

One-dimensional haemodynamic model of a vascular network with fractional-order viscoelasticity

Abstract We propose a computational framework for a one-dimensional haemodynamic model with the arterial walls described by the fractional-order viscoelastic material constitutive law. This framework is used to compare blood flow characteristics for simulations with elastic and fractional-order viscoelastic walls. We use three well-established benchmark tests: a single pulse wave in a long vessel, flow in a 37-segment network of elastic tubes, and flow in anatomically detailed arterial network consisting of 61 arterial segments. All results for elastic model are in a good agreement with analytical solutions, in vitro data and other well-established approaches. Fractional-order model demonstrates noticeable differences in pulse wave propagation speed and minor differences in pressure and flow profiles. Differences in profiles are negligible in major vessels, but more profound in vessels beyond the third or fourth generation.

Decoupled two-scale simulation of damping vibration of composite laminates using image-based periodic representative volume element

This study investigates the damping vibration mechanism of composites using a decoupled two-scale simulation with an image-based periodic representative volume element (RVE). The parameters of the macroscopic viscoelastic constitutive law for unidirectional composites were determined using a homogenization method based on the properties of the constituent and microscale image-based periodic RVE. Using the determined viscoelastic properties, a macroscale modal simulation was conducted to predict the natural frequencies, modal damping ratios, and mode shapes of composite laminates. Then, a macroscale transient vibration simulation was performed using the calculated modal shapes and damping ratios. This simulation scheme was verified by comparing the predicted natural frequencies, modal damping ratios, mode shapes, and transient vibration responses with the experimental results. Following the homogenization simulations conducted from micro to macro scales, localization simulations were carried out from macro to micro scales to explore how microscopic deformation affects macroscopic damping properties. By utilizing both bottom-up and top-down simulations, the mechanism of vibration damping is examined by considering the microscopic structure and matrix deformation.

Metric-based anisotropic mesh adaptation for viscoelastic flows

The computation of flows of viscoelastic fluids is expensive compared to standard fluids. Besides degrees-of-freedom for velocity and pressure, the viscoelastic stresses introduce an additional unknown to the system. While adaptive meshing techniques are commonly used for simulations of standard fluids, their application is seldom explored in the context of viscoelastic fluids. In this work, we apply an anisotropic mesh adaptation algorithm to solve the viscoelastic flow problem. Our flow solver is based on a logarithmic reformulation of the viscoelastic constitutive laws, which greatly reduces the stringent stability requirements on the mesh size and allows adaptive meshing in this context. For discretization, we use a stabilized finite element method. We present numerical results for a flow past a cylinder. The adaptive method produces near mesh-independent results up to Weissenberg number Wi=0.7. Our results indicate that with an anisotropic refinement strategy, fewer mesh elements are needed to maintain the same error level as with an isotropic refinement strategy. In a stick-slip transition flow, we demonstrate the capability of the adaptive method to produce optimal meshes even in the presence of a singularity in the solution.

On viscoelastic drop impact onto thin films: axisymmetric simulations and experimental analysis

This study investigates the effect of fluid elasticity on axisymmetric droplets colliding with pre-existing liquid films, using both numerical and experimental approaches. The numerical simulations involve solving the incompressible flow momentum equations with viscoelastic constitutive laws using the finite volume method and the volume of fluid (VOF) technique to track the liquid’s free surface. Here, the Oldroyd-B model is used as the constitutive equation for the viscoelastic phase. Experiments are also performed for dilute viscoelastic solutions with 0.005% and 0.01% (w/w) polyacrylamide in 80:20 glycerin/water solutions, in order to ensure the validity of the numerical solution and to investigate the elasticity effect. The formation and temporal evolution of the crown parameters are quantified by considering the flow parameters, including the fluid’s elasticity. The results indicate that the axisymmetric numerical solutions reasonably agree with the experimental observations. Generally, the fluid’s elasticity can enlarge the crown dimension at different thicknesses of the fluid film. Moreover, at intermediate values of the Weissenberg number, the extensional force in the crown wall can control the crown propagation. Furthermore, the results reveal that the effects of the Weber number and the viscosity ratio on this problem are more significant at higher values of the Weissenberg number.

Experimental and analytical studies on shrinkage and creep behavior of RC walls and prisms

AbstractA component level experimental study has been carried out on four reinforced concrete (RC) walls subjected to a sustained compressive load for a period of 1 year and shrinkage tests on 10 prisms under ambient environmental conditions. Two extreme longitudinal reinforcement percentages and two grades of concrete were considered and their influence on the time‐dependent behavior studied. It was observed that both longitudinal percentage of steel and concrete grade have a significant influence on the time‐dependent strains. The evolution of time‐dependent strain in RC member was also predicted, adequately accounting for the effect of reinforcement, using a theoretical model which can employ any linear viscoelastic constitutive law for concrete and a linear elastic constitutive law for reinforcing steel. The ACI 209 and fib MC 10 recommendations for creep and shrinkage of plain concrete have been used for the prediction of long‐term strains. It is demonstrated that the analysis predicts the time‐dependent strains reasonably well (with a statistical mean deviation error of 1.16 and 1.00) for the creep tests on RC walls, when the compliance function proposed by ACI and fib is used. However, in the case of shrinkage tests, the accuracy with both ACI and fib models was limited (1.64 and 1.66, respectively). It is further demonstrated that by suitably recalibrating the compliance and shrinkage parameters in both ACI and fib models, the accuracy of shrinkage prediction in the companion RC specimens improves significantly with mean deviation error of 1.07 and 1.05 for ACI and fib models, respectively.

Automated identification of linear viscoelastic constitutive laws with EUCLID

We extend EUCLID, a computational strategy for automated material model discovery and identification, to linear viscoelasticity. For this case, we perform a priori model selection by adopting a generalized Maxwell model expressed by a Prony series, and deploy EUCLID for identification. The methodology is based on four ingredients: i. full-field displacement and net force data; ii. a very wide material model library — in our case, a very large number of terms in the Prony series; iii. the linear momentum balance constraint; iv. the sparsity constraint. The devised strategy comprises two stages. Stage 1 relies on sparse regression; it enforces momentum balance on the data and exploits sparsity-promoting regularization to drastically reduce the number of terms in the Prony series and identify the material parameters. Stage 2 relies on k-means clustering; starting from the reduced set of terms from stage 1, it further reduces their number by grouping together Maxwell elements with very close relaxation times and summing the corresponding moduli. Automated procedures are proposed for the choice of the regularization parameter in stage 1 and of the number of clusters in stage 2. The overall strategy is demonstrated on artificial numerical data, both without and with the addition of noise, and shown to efficiently and accurately identify a linear viscoelastic model with five relaxation times across four orders of magnitude, out of a library with several hundreds of terms spanning relaxation times across seven orders of magnitude.

Pseudo-bistability of viscoelastic shells.

Viscoelastic shells subjected to a pressure loading exhibit rich and complex time-dependent responses. Here we focus on the phenomenon of pseudo-bistability, i.e. a viscoelastic shell can stay inverted when pressure is removed, and snap to its natural shape after a delay time. We model and explain the mechanism of pseudo-bistability with a viscoelastic shell model. It combines the small strain, moderate rotation shell theory with the standard linear solid as the viscoelastic constitutive law, and is applicable to shells with arbitrary axisymmetric shapes. As a case study, we investigate the pseudo-bistable behaviour of viscoelastic ellipsoidal shells. Using the proposed model, we successfully predict buckling of a viscoelastic ellipsoidal shell into its inverted configuration when subjected to an instantaneous pressure, creeping when the pressure is held, staying inverted after the pressure is removed, and eventually snapping back after a delay time. The stability transition of the shell from a monostable, temporarily bistable and eventually back to the monostable state is captured by examining the evolution of the instantaneous pressure-volume change relation at different time of the holding and releasing process. A systematic parametric study is conducted to investigate the effect of geometry, viscoelastic properties and loading history on the pseudo-bistable behaviour. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

Mesh‐size independent rigid‐body spring network for simulation of the dynamic fracture behavior of concrete

AbstractIn this study, a numerical approach is developed for simulating the rate‐dependent behaviors of concrete without mesh sensitivity. The rheological units (e.g., dashpot) are integrated with the six directional springs in the rigid‐body spring network (RBSN) elements to reflect the rate‐dependent behavior of concrete. Previously, the viscoplastic damage model was associated with the elemental degrees of freedom. However, in the present approach, a viscoelastic constitutive law is newly defined for the normal direction as a function of the strain rate, from which the internal forces can be updated from the regularized elemental stresses. Such improvements are validated through the numerical simulations of the direct tensile test and spalling test using the modified split‐Hopkinson pressure bar (SHPB). The simulated results show consistent structural responses without mesh dependence on the strain rate. In addition, the influences of the softening curve shapes on the crack localizations are examined. It is shown that the rheological parameters can be optimized and determined for consistent applications through virtual experiments. The proposed numerical approach enables the mesh construction procedure without concerning the mesh‐size sensitivity due to the strain rate, and various applications are expected for the simulations of detailed numerical models of concrete materials and structures under high loading rates.

Visco-elastic constitutive law to simulate large creep deformations using particle based simulation approach

So far there are only very few publications which consider large creep deformations of solids using particle based approaches, however large creep deformations will occur in underground salt or soft rock pillars after decades even hundred years. This paper is dedicated towards a specific constitutive model called BurgerFG which is based on SSPM (Stiff Spherical Particle Method). The implementation is performed in the particle based software PFC3D. A solution for time-step optimization to simulate long-term creep processes is proposed. The model allows to simulate creep including damage and healing processes like typically observed for salt material under loading.

Subcutaneous Drug Delivery: A Review of the State-of-the-Art Modeling and Experimental Techniques.

Delivery of drug formulations through the subcutaneous route is a widely used modality for the treatment of several diseases, such as diabetes and auto-immune conditions. Subcutaneous injections are typically used to inject low-viscosity drugs in small doses. However, for new biologics, there is a need to deliver drugs of higher viscosity in large volumes. The response of subcutaneous tissue to such high-volume doses and higher viscosity injections is not well understood. Animal models have several drawbacks such as relevance to humans, lack of predictive power beyond the immediate population studied, cost, and ethical considerations. Therefore, a computational framework that can predict the tissue response to subcutaneous injections would be a valuable tool in the design and development of new devices. To model subcutaneous drug delivery accurately, one needs to consider: (a) the deformation and damage mechanics of skin layers due to needle penetration and (b) the coupled fluid flow and deformation of the hypodermis tissue due to drug delivery. The deformation of the skin is described by the anisotropic, hyper-elastic, and viscoelastic constitutive laws. The damage mechanics is modeled by using appropriate damage criteria and damage evolution laws in the modeling framework. The deformation of the subcutaneous space due to fluid flow is described by the poro-hyperelastic theory. The objective of this review is to provide a comprehensive overview of the methodologies used to model each of the above-mentioned aspects of subcutaneous drug delivery. We also present an overview of the experimental techniques used to obtain various model parameters.

On the use of Stochastic Bouc–Wen model for simulating viscoelastic internal variables from a finite element approximation of steady-rolling tire

Reducing tire rolling resistance and energy loss is a topic of interest to the tire industry. Understanding and modeling these phenomena are essential to approach this problem and propose robust solutions. This work suggests a reduced-order model based on the Bouc–Wen model to simulate internal variables from viscoelastic constitutive laws. Furthermore, sensitivity analysis is performed on the Bouc–Wen parameters to evaluate their influence on the system response and capture the full range of possible values that improve the predictive ability of the reduced-order model. This task is accomplished by calculating the Sobol’s indices estimated from a Polynomial-Chaos expansion. Once the range of feasible model solutions is established, the reduced-order model is calibrated through Bayesian inference. Finally, the uncertainties are propagated, and the reduced-order model is validated using data of viscoelastic internal variables from the finite element approximation of a steady-rolling tire. Satisfactory results are obtained, as the reduced-order model can simulate viscoelastic internal variables with a reduced computational cost for some branches of interest. Its responses are in agreement with the experimental data.

Stabilized finite element methods for a fully-implicit logarithmic reformulation of the Oldroyd-B constitutive law

Logarithmic conformation reformulations for viscoelastic constitutive laws have alleviated the high Weissenberg number problem, and the exploration of highly elastic flows became possible. However, stabilized formulations for logarithmic conformation reformulations in the context of finite element methods have not yet been widely studied. We present stabilized formulations for the logarithmic reformulation of the Oldroyd-B model by Saramito (2014) based on the Variational Multiscale framework and Galerkin/Least-Squares method. The reformulation allows the use of Newton’s method due to its fully-implicit nature for solving the steady-state problem directly. The proposed stabilization methods cure instabilities of convection and compatibility while preserving a three-field problem. The spatial accuracy of the formulations is assessed with the four-roll periodic box, revealing comparable accuracy between the methods. The formulations are validated with benchmark flows past a cylinder and through a 4:1 contraction. We found an excellent agreement to benchmark results in the literature. The algebraic sub-grid scale method is highly robust, indicated by the high limiting Weissenberg numbers in the benchmark flows.

Solvability of Conformable Type Frictionless Contact Problem via Hemivariational Inequalities

In this paper, we study a class of conformable frictionless contact problems with the surface traction driven by the conformable impulsive differential equation. The existence of a mild solution for conformable impulsive hemivariational inequality is obtained by the Rothe method, subjectivity of multivalued pseudomonotone operators and the property of the conformable derivative. Notice that we imply some new fractional viscoelastic constitutive laws.

Dynamic viscoelastic unilateral constrained contact problems with thermal effects

A new model that describes a dynamic frictional contact between a viscoelastic body and an obstacle is investigated in this paper. We consider a nonlinear viscoelastic constitutive law which involves a convex subdifferential inclusion term and thermal effects. The contact condition is modeled with unilateral constraint condition for a version of normal velocity. The boundary conditions that describe the contact, friction and heat flux are govern by the generalized Clarke multivalued subdifferential. We derive a coupled system of two nonlinear first order evolution inclusions problems, which consists of a parabolic variational-hemivariational inequality for the displacement and a hemivariational inequality of parabolic type for the temperature. Then, the unique weak solvability of the contact problem is obtained by virtue of a fixed point theorem and the surjectivity result of multivalued maps. Finally, we deliver a continuous dependence result on a coupled system when the data are subjected to perturbations.