Nonlinear Viscoelastic Constitutive Relations Research Articles

Numerical techniques for solving truss problems involving viscoelastic materials

We develop a methodology for solving truss problems involving viscoelastic materials where, of all the member forces that satisfy the nodal force equilibrium equation and nodal displacements that satisfy the displacement boundary conditions, those member forces and nodal displacements that satisfy the constitutive relation are sought. Since a rate type viscoelastic constitutive relation involves the rate of the stress or strain, this study explores the use of member forces, nodal displacements, and support reactions or their rates as independent variables. Assuming small deformations, the nodal force equilibrium and the displacement boundary condition results in a linear equality constraint between the independent variables. Then we find the unknown independent variables such that the root mean squared error in the constitutive relation of the members of the truss is minimized subject to the satisfaction of the linear constraint at selected times. The objective function is evaluated at selected times or integrated over subintervals of time. We explore six possible solution methods and benchmark them for their accuracy and efficiency. We study statically determinate and indeterminate truss whose members are modeled using rate and integral type viscoelastic constitutive relations for creep and oscillatory loading. For the standard linear solid model, we find that the proposed methods are more accurate than ABAQUS and, at times, require lesser computational wall time. We also demonstrate the applicability of the proposed methodology to fractional order and nonlinear viscoelastic constitutive relations.

Large deformations of nonlinear viscoelastic and multi-responsive beams

This study presents analyses of deformations in nonlinear viscoelastic beams that experience large displacements and rotations due to mechanical, thermal, and electrical stimuli. The studied beams are relatively thin so that the effect of the transverse shear deformation is neglected, and the stretch along the transverse axis of the beams is also ignored. It is assumed that the plane that is perpendicular to the longitudinal axis of the undeformed beam remains plane during the deformations. The nonlinear kinematics of the finite strain beam theory presented by Reissner [27] is adopted, and a nonlinear viscoelastic constitutive relation based on a quasi-linear viscoelastic (QLV) model is considered for the beams. Deformation in beams due to mechanical, thermal, and electric field inputs are incorporated through the use of time integral functions, by separating the time-dependent function and nonlinear measures of field variables. The nonlinear measures are formulated by including higher order terms of the field variables, i.e. strain, temperature, and electric field. Responses of beams under mechanical, thermal, and electrical stimuli are illustrated and the effects of nonlinear constitutive relations on the overall deformations of the beams are highlighted.

Modelling and subject-specific validation of the heart-arterial tree system.

A modeling approach integrated with a novel subject-specific characterization is here proposed for the assessment of hemodynamic values of the arterial tree. A 1D model is adopted to characterize large-to-medium arteries, while the left ventricle, aortic valve and distal micro-circulation sectors are described by lumped submodels. A new velocity profile and a new formulation of the non-linear viscoelastic constitutive relation suitable for the {Q, A} modeling are also proposed. The model is firstly verified semi-quantitatively against literature data. A simple but effective procedure for obtaining subject-specific model characterization from non-invasive measurements is then designed. A detailed subject-specific validation against in vivo measurements from a population of six healthy young men is also performed. Several key quantities of heart dynamics-mean ejected flow, ejection fraction, and left-ventricular end-diastolic, end-systolic and stroke volumes-and the pressure waveforms (at the central, radial, brachial, femoral, and posterior tibial sites) are compared with measured data. Mean errors around 5 and 8%, obtained for the heart and arterial quantities, respectively, testify the effectiveness of the model and its subject-specific characterization.

Numerical implementation of strain rate dependent thermo viscoelastic constitutive relation to simulate the mechanical behavior of PMMA

In this work, a nonlinear viscoelastic constitutive relation was implemented to describe the mechanical behavior of a transparent thermoplastic polymer polymethyl methacrylate (PMMA). The quasi-static and dynamic response of the polymer was studied under different temperatures and strain rates. The effect of temperature was incorporated in elastic and relaxation constants of the constitutive equation. The incremental form of constitutive model was developed by using Poila–Kirchhoff stress and Green strain tensors theory. The model was implemented numerically by establishing a user defined material subroutine in explicit finite element (FE) solver LS-DYNA. Finite element models for uniaxial quasi-static compressive test and high strain rate split Hopkinson pressure bar compression test were built to verify the accuracy of material subroutine. Numerical results were validated with experimental stress strain curves and the results showed that the model successfully predicted the mechanical behavior of PMMA at different temperatures for low and high strain rates. The material model was further engaged to ascertain the dynamic behavior of PMMA based aircraft windshield structure against bird impact. A good agreement between experimental and FE results showed that the suggested model can successfully be employed to assess the mechanical response of polymeric structures at different temperature and loading rates.

Crack analysis in PVB laminated windshield impacted by pedestrian head in traffic accident

A damage-modified nonlinear viscoelastic constitutive relations model of PVB laminated windshield is suggested based on updated Lagrangian method to study the crack evolutions on windshield plate subject to pedestrian head. After verified by a classical example, the constitutive relations are implemented into FEA software. With proper material properties, the FEA model describing the pedestrian head impact with vehicle windshield in traffic accident is established. The computation results of FEA model and classical Hertzian pressure model are in good agreement. Moreover, the FEA model is used to analyse the internal stresses. Shear stress, compressive stress and tensile stress are main causes of plastic deformation, radial cracks and circumferential cracks, respectively. Internal stresses decrease as the distance away from the impacted point increases. In addition, parametric studies were carried out to study the effects of Poisson's ratio and impact velocity.

Time-Dependent Mechanical Behaviors of Polyamide 6/Nano-SiO<sub>2</sub> Composite

The uniaxial tension under various strain rates, creep under various sustained loads, and equalamplitude- strain loading and unloading tests are carried out at room temperature with polyamide 6/nano- SiO2 composite specimens. According to the elasticity recovery correspondence principle, the recovered elastic stresses (strains) in the case of prescribed strain (stress) history are calculated, and the instantaneous elastic constitutive equations are deduced. The nonlinear viscoelastic constitutive relations in single integral form on the basis of the instantaneous elastic constitutive equations are constructed and applied to model the current stress (strain) responses of polyamide 6/nano-SiO2 composite. The theoretic results agree well with the experimental data, which demonstrates that the single integral constitutive relations used in this work can accurately simulate the physical nonlinear viscoelastic properties of polyamide 6/nano-SiO2 composite. Finally, the creep curve at higher stress level is horizontally shifted along logarithmic timescale using a stress shift factor in terms of the time-stress superposition principle and superposed on that at relative lower stress level to form a master creep compliance curve that spans a longer timescale interval than the short-term test curve does, which suggests that TSSP provides an accelerated characterization method for the long-term creep performance of polyamide 6/nano-SiO2 composite.

Predictions of two nonlinear viscoelastic constitutive relations for polymers under multiaxial loadings

AbstractTwo viscoelastic constitutive relations in differential form are further developed here to include material nonlinearity and distinction between loading and unloading regimes, which is a characteristic of polymers. The effects of hydrostatic pressure and anisotropy in tension and compression on the deformation response of polymers are accounted for through the definition of a pressure‐dependent equivalent stress. In the uniaxial stress state, these constitutive relations reduce to the two well‐known mechanical analogue representations: “Kelvin–Voigt‐type” and “Maxwell‐type” rheological models. The predictive capabilities of these constitutive relations are then assessed against a wide range of experimental results, which include both uniaxial and biaxial stress states subjected to quasi‐static and cyclic‐loading conditions. The predictions of both models are found to be in good agreement with the test data. POLYM. ENG. SCI., 47:593–607, 2007. © 2007 Society of Plastics Engineers.

235 生体軟組織の引張試験による非線形粘弾性構成関係同定

235 生体軟組織の引張試験による非線形粘弾性構成関係同定

Stress Relaxation in Prestressed Composite Laminates

Viscoelastic deformation caused in symmetric laminated plates by release of fiber prestress and by uniform thermomechanical loads is analyzed on the constituent, ply and overall laminate scales with the Transformation Field Analysis (TFA) method (G. J. Dvorak, Proc. R. Soc. Lond., 1992, A437, pp. 311–327). Fiber prestress is applied in individual plies prior to matrix cure and released after matrix consolidation. Linear or nonlinear viscoelastic constitutive relations are used to evaluate the inelastic deformation rates in terms of current constituent stress averages. The TFA method regards both thermal and inelastic strains as piecewise uniform eigenstrains acting in superposition with mechanical loads and fiber prestress release on an elastic laminate. Interactions between the eigenstrains at the three different size scales are described by certain influence functions derived from micromechanical analysis of the plies and laminates. Applications describe stress relaxation in two carbon/epoxy laminates after cooling from the curing temperature and release of optimized fiber prestress, that allows maximum tensile load application while keeping both interior and free-edge stresses within prescribed strength limits. Subsequent viscoelastic deformation under constant rate loading, and stress relaxation caused by a sustained application of an elevated temperature to a laminate without prestress are also analyzed. Results are presented in the form of initial failure maps that identify overall stress states which may or may not initiate a specific damage mode in the laminate.

Nonlinear viscoelastic stress singularities near free edges of unsymmetrically laminated composites

Nonlinear viscoelastic stress singularities near free edges of laminated composites are evaluated based on Pipes and Pagano's displacement field for laminates under a generalized plane deformation state and Schapery's nonlinear viscoelastic constitutive relations. The interlaminar stresses in unsymmetrically laminated composites due to axial strain and bending have been studied as a function of time and loading magnitude. Numerical studies show that the stacking sequence significantly affects the distributions of interlaminar stresses and the stress symmetric conditions about the midplane for an unsymmetrically laminated composite are different than those for a symmetric laminate. The results indicate a strong sensitivity to the nonlinearities of the viscoelastic constitutive relations. Increasing the load significantly increases nonlinear behavior. The rate of stress relaxation also varies with the magnitude of loading.

Creep Characterization of a Fiber Reinforced Plastic Material

Creep behavior of [±45°]s composite material is characterized by using uniaxial creep and recovery tests. The well-known Schapery nonlinear viscoelastic consti tutive relation was modified to make it suitable for characterizing the creep behavior of this material. Then, using this modified Schapery constitutive equation, by which the vis coplastic and creep damage can be taken into consideration, the creep behavior of [±45°]. glass fiber reinforced epoxy laminate was studied. The constitutive parameters of the material were determined experimentally, and the procedure and method of determination of the material parameters are proved to be valid.

On a multiaxial non-linear hereditary constitutive law for non-ageing materials with fading memory

A hereditary type complementary power potential is postulated which permits definition of the strain rate tensor in terms of the history and current magnitude of a certain combination of stress and stress deviator components. The strain rate tensor is decomposed into elastic and viscous portions, the latter being further separated into reversible and irreversible parts with the reversible portion being represented by Volterra type integrals. For a material obeying Norton's power law, the three-dimensional non-linear constitutive equations are derived in terms of two elastic constants and five viscous material parameters which may be determined from creep tests. These equations are applied to the derivation of general non-linear viscoelastic constitutive relations for a plate element which, in turn, are used to solve the cylindrical time-deflection problem of a long simply supported ice plate subjected to in-plane compressive forces applied along the longitudinal edges. The governing equations are solved numerically, using an incremental approach. An approximate method for the effective numerical treatment of problems involving hereditary type constitutive relations is discussed in detail.

Nonlinear stress field in blood vessels under the action of connective tissues.

Finite deformations and stresses in vascular walls are studied analytically in the paper. Nonlinear viscoelastic constitutive relations suitably modified for the consideration of the anisotropic behavior of the vessel walls are used in the analysis. The time dependence of the inner surface and the force exerted by the surrounding connective tissues are also taken into account. Using approximations for short time ranges, a quantitative analysis is made in order to illustrate the applicability of the analytical study.