Modeling of nonlinear viscoelasticity at large deformations

A constitutive model of finite strain viscoelasticity, based on the multiplicative decomposition of the deformation gradient tensor into elastic and inelastic parts, is presented. The nonlinear response of rubbers, manifested by the rate effect, cycling loading and stress relaxation tests was captured through the introduction of two internal variables, namely the constitutive spin and the back stress tensor. These parameters, widely used in plasticity, are applied in this work to model the nonlinear viscoelastic behaviour of rubbers. The experimental results, obtained elsewhere, related with shear deformation in monotonic and cyclic loading, as well as stress-relaxation, were simulated with a good accuracy.     

Micro-Macro Analysis of Viscoelastic Unidirectional Laminated Composite Plates Using DR Method

The Dynamic Relaxation (DR) technique together with finite difference discritization is used to study the bending behavior of Mindlin composite plate including geometric nonlinearity. The overall behavior of the unidirectional composite is obtained from a three-dimensional (3D) micromechanical model, in any combination of normal and shear loading conditions, based on the assumptions of Simplified Unit Cell Method (SUCM). The composite system consists of nonlinear viscoelastic matrix reinforced by transversely isotropic elastic fibers. A recursive formulation for the hereditary integral of the Schapery viscoelastic constitutive equation in multiaxial stress state is used to model the nonlinear viscoelastic matrix material in the material level. The creep tests data is used for verification of the predicted response of the current approach. Under uniform lateral pressure, the laminated plate deformation with clamped and hinged edged constraints is predicted for various time steps.     

The use of stress sweep tests for asphalt mixtures nonlinear viscoelastic and fatigue damage responses identification

Asphaltic materials are known to present a behavior that can be approximated by the theory of viscoelasticity. For these materials it is essential to characterize fatigue damage. An important aspect therein is the separation between nonlinear viscoelastic and fatigue damage responses. This is a complex issue, since both nonlinearity and damage have a similar effect on the overall material mechanical behavior, i.e. decrease in the stiffness and increase in the phase angle. This paper presents an experimental and a mathematical procedure to separate the nonlinear viscoelastic from the fatigue damage response for asphaltic materials. Stress sweep tests were used to characterize a hot mixture asphalt at nine conditions (three temperatures and three frequencies). Once all strain values were obtained in a stress controlled sweep test, a statistical analysis was used to find the maximum stress that can be applied to the material without invoking the damage response. The results showed that the transition stress value is directly associated with material properties, the stiffness being an important factor in this result. Consequently, stress, temperature and frequency determine together the mechanical response of the material (linear or nonlinear viscoelastic, fatigue damage and/or plastic deformation). Results from this study can be associated with other fatigue damage approaches in order to better select the stress or strain amplitude that should be used in fatigue tests, and to eliminate the amount of energy that is dissipated in the nonlinear viscoelastic region.     

Effective properties of power-law solids containing elliptical inhomogeneities. Part I: Rigid inclusions

The plane strain deformation of a power-law material containing a dispersion of rigid elliptical inclusions is investigated. Accurate constitutive relations are established for dilute concentrations of inclusions over a wide range of martrix hardening exponents and inclusion aspect ratios. The essential step in the analysis is the solution of the kernel problem for an isolated inclusion, and this is obtained using a Ritz procedure in which trial displacements are derived from a displacement potential in elliptic-cylindrical coordinates. Approximate constitutive relations for nondilute concentrations of inclusions are then established using a differential self-consistent scheme. The study is primarily concerned with randomly oriented inclusions, but limited results are also presented for aligned inclusions. As part of the investigation, the procedure introduced by Ponte Castaneda (J. Mech. Phys. Solids (1991)) is evaluated. This is a variational procedure which exploits information about a linear solid to obtain bounds or estimates for the overall response of nonlinear solids with the same microgeometry of second phase. Finally, connection is made with recent results for spheroidal inclusions given by Lee and Mear (J. Mech. Phys. Solids (1991); Int. J. Solids Struct. (1991)), and an ad hoc estimate for the overall response of solids containing randomly oriented rigid, spheroidal inclusions is proposed.     

单个蛋白质气泡的振动特性分析

根据粘弹性材料有限变形的应变能密度函数、Maxwell模型的松弛函数及气泡的变形梯度张量,推导出蛋白质气泡有限变形的应力方程.并结合气泡的动力学方程,得到气泡在内外压力差、弹性有限变形应力及粘性耗散应力共同作用下内径的非线性振动方程.利用该方程,通过数值模拟方法,对蛋白质气泡有限变形时的振动特性进行了分析,研究了气泡内外压力差、膜的厚度、膜的粘性以及气泡大小对气泡振动特性的影响.结果表明,蛋白质气泡的振动具有非线性特性,当初始压力差不同时,气泡的振动频率、振幅、速度的变化是不同的,停止振动时的大小也不相同;增加膜的厚度和膜的粘性会抑制气泡的振动,增强气泡承受载荷的能力;对于大小不同的气泡,尺寸较小的气泡振动频率高,速度衰减慢.     

On the influence of preloading in the nonlinear viscoelastic–viscoplastic response of carbon–epoxy composites

On an ongoing research for the nonlinear viscoelastic response of composites and polymers, a study of the influence of preloading applied to composite laminates subjected to creep–recovery loading is performed. In cases where high stress levels are applied, this response becomes highly nonlinear and has to be taken into account when designing composite parts. A major problem encountered in the experimental investigation of the nonlinear viscoelastic behaviour is the mode of the initial applied loading and its effect in the overall viscoelastic response of the test sample. The damage that occurs due to the instantaneous application of the load leads to an additional viscoelastic/viscoplastic strain component. In order to investigate this effect as well as to compare different preloading modes, as far as viscoelastic/viscoplastic response is concerned, a test program was initiated and the experimental data were investigated in the current study. A preloading mode is applied in each specimen prior to the creep–recovery testing at different applied stress levels. Useful results concerning the effect of preloading in the time dependent response of the material are concluded. Variation of the values of viscoplastic strain in respect to the preloading mode is also of great concern.     

Geometrically nonlinear analysis of elastic membranes with embedded rigid inclusions

In this paper the deformation of membranes containing rigid inclusions is analyzed. These rigid inclusions can significantly change the entire stress distribution in the membrane and therefore create major difficulties for the design. The initially flat membrane, which may be prestretched by boundary in-plane tractions or displacements, is subjected to externally applied loads and to the weight of the rigid inclusions. The composite system is examined in cases where its deformation reaches a state for which the undeformed and deformed shapes are substantially different. In such cases large deflections of membranes are considered, which result from nonlinear kinematic relations. The three coupled nonlinear equations in terms of the displacements governing the response of the membrane are solved using the analog equation method, which reduces the problem to the solution of three uncoupled Poisson's equations with fictitious domain source densities. The problem is strongly nonlinear [Katsikadelis JT, Nerantzaki MS. The boundary element method for nonlinear problems. Eng Anal Boundary Elements 1999;23:365–73]. In addition to the geometrical nonlinearity, the problem is itself nonlinear, since the membrane's reactions on the boundary of the rigid inclusions are not a priori known as they depend on the produced deflection surface. Iterative schemes are developed for calculation of deformed membrane's configuration, which converge to the final equilibrium state of the membrane with the given external applied loads. Several example problems are presented, which illustrate the method and demonstrate its accuracy and efficiency. The method employed for the solution is boundary only with all the advantages of the pure BEM.     

An Assessment of Nonlinearly Viscoelastic Constitutive Models for Cyclic Loading: The Effect of a General Loading/Unloading Rule

Predicting the unloading and/or cyclic deformation behavior of polymers is a challenge for most nonlinear viscoelastic constitutive models. Experimental data of an epoxy polymer under uniaxial loading/unloading and two other types of cyclic loadings are used to assess the predictive capabilities of three types of nonlinear viscoelastic models. A general loading/unloading criterion and a switching rule, proposed recently by the authors, are further modified and incorporated into each of the three models. For each model, predictions by both the original formulations and that incorporating the proposed loading/unloading rule are compared with the test data. It is clearly shown that such a rule is essential to correctly simulate the unloading and cyclic loading behavior of polymers. By introducing such a rule to constitutive models, the quantitative predictions can be improved, to various degrees of success, with respect to cyclic deformation features such as ratcheting under cyclic loading with a mean stress and stress relaxation under cyclic straining with a mean strain.     

Stress Concentration in Microstructural Elements of Viscoelastic Composite Materials

We present solution of the problem of estimating the stress concentration in the constituents (matrix and inclusions) of a multiconstituent viscoelastic composite material, depending on the shape of inclusions, properties of the matrix and the whole composite. The matrix material is isotropic and viscoelastic. A wide range of properties of inclusions (e.g., pores, solid and viscous particles) is considered. To solve this problem, we use the method of integral transformations. __________ Translated from Problemy Prochnosti, No. 5, pp. 138 – 149, September – October, 2005.     

The overall elastopIastic behavior of multiphase materials with isotropic components

Summary A multiphase material is considered, which consists of a homogeneous elastoplastic matrix containing a homogenous statistically uniform random set of ellipsoidal elastic inclusions. An approach based on the multiparticle effective field method is introduced for determining the overall elastoplastic behavior of the material under monotonic loading. A secant modulus concept is employed, and linearized problems are solved at each step of an iterative procedure. Physically consistent assumptions are used for linearizing nonlinear functions which depend on the phase averages of the second invariant of stress and on the stress deviator. Exact expressions for the second moments of the microstresses are employed.     

An algorithm for stress and mixed control in Galerkin-based FFT homogenization

A new algorithm is proposed to impose a macroscopic stress or mixed stress/deformation gradient history in the context of nonlinear Galerkin-based fast Fourier transform homogenization. The method proposed is based on the definition of a modified projection operator in which the null frequencies enforce the type of control (stress or strain) for each component of either the macroscopic first Piola stress or the deformation gradient. The resulting problem is solved exactly as the original variational method, and it does not require additional iterations compared to the strain control version, neither in the linear iterative solver nor in the Newton scheme. The efficiency of the proposed method is demonstrated with a series of numerical examples, including a polycrystal and a particle-reinforced hyperelastic material.     

高斯激励黏弹性夹层梁的非线性随机响应特性

黏弹性夹层梁的随机振动控制是一个重要的实际问题。基于性能可控黏弹性体的夹层梁具有无需改变结构设计的可优化性而倍受关注。虽然关于该可控黏弹性夹层梁的振动已有一定研究,但所用的动力学模型在几何或物理上是线性的,而对于较强激励情况则需要考虑非线性因素。首次考虑该黏弹性体的物理非线性,建立黏弹性夹层梁及其支承质量系统的非线性运动微分方程,并离散化为多模态耦合的非线性振动方程;对于平稳随机激励,运用统计线性化法推导等价拟线性系统,并计算系统的随机响应,得到黏弹性夹层梁非线性随机振动的均方位移,及等价的频响函数和功率谱,用以评价可控黏弹性夹层梁的响应抑制性能。     

蛋白质气泡有限变形研究

根据粘弹性材料有限变形的应变能密度函数、Maxwell模型的松弛函数及气泡的变形梯度张量,推导出蛋白质气泡有限变形的应力方程。并结合气泡的平衡方程,得到气泡在动态压力作用下有限变形时内径相对变化率随时间变化的表达式。运用该表达式,通过数值模拟方法,对蛋白质气泡有限变形的非线性特性、径向变形随气泡内外压力差、膜的厚度以及膜的粘性的变化规律进行了计算分析。结果表明:在不同载荷作用下,蛋白质气泡径向变形不但具有明显的非线性特性,而且气泡变形达到平衡时的变形大小和时间也不相同。增加气泡膜的厚度和膜的粘性既可以延长气泡变形达到平衡的时间,又可以大大增强气泡承受载荷的能力。     

Viscoelastic response of thin membranes with application to red blood cells

The rate-type constitutive analysis of viscoelastic response of thin membranes, which includes an instantaneous elastic response and viscous behavior in both shear and dilatation, is developed with the aim to study the mechanical response of red blood cells. A convenient set of generalized stress and strain variables is introduced, which facilitates the derivation and integration of the governing differential equations. Gradual or sudden loading and stepwise unloading histories are considered. The performed parametric study of the mechanical response illustrates the effects of the introduced material parameters on the coefficient of viscoelastic lateral contraction and the overall membrane deformation. A closed form solution to the problem of radial stretching of a viscoelastic hollow circular membrane is derived without referral to the correspondence principle, which is of interest for the micropipette aspiration experiment of the red blood cell. The effects of the material parameters on the instantaneous elastic response and the subsequent rate of creep are discussed.     

An investigation on strain/stress distribution around the overlap end of laminated composite single-lap joints

The damages of laminated composite single-lap joints often begin from their overlap ends because strain/stress concentrations often occur at the overlap ends. This paper presents the results of a combined experimental and finite element (FE) investigation on the strain/stress distributions around the overlap ends of laminated composite single-lap joints. Digital image correlation (DIC) technique is used to measure the strain fields near the overlap ends. A three-dimensional geometrically nonlinear FE model based on the submodel technique is developed to predict the deformation of single-lap joints. A reasonable agreement is achieved between the results from experimental measurements and FE analysis. Additionally, some FE models are built to investigate the effects of the mesostructures at the overlap ends on the stress concentrations around the overlap ends.     

Deformation of composites with physically nonlinear matrix

Summary. A method and an algorithm for determining the effective deformational properties of granular material with a physically nonlinear matrix and linearly elastic inclusions are elaborated based on the stochastic differential equations of the physically nonlinear theory of elasticity. Their transformation to integral equations and the application of the method of conditional moments reduce the problem to a system of nonlinear algebraic equations, whose solution is constructed by the iteration method. The deformation diagrams as functions of the volume content of inclusions are investigated.     

Description of depth-dependent nonlinear viscoelastic behavior for articular cartilage in unconfined compression

An optimized digital image correlation (DIC) technique was applied to investigate the depth-dependent nonlinear viscoelastic properties of articular cartilage and simultaneously the biphasic nonlinear viscoelastic relaxation model of cartilage was proposed and validated. The stress relaxation tests were performed with different strain levels and it is found that the initial stress and relaxed stress at any time increase with increasing strain levels. The depth-dependent strain of cartilage was obtained by analyzing the images acquired using the optimized DIC technique and moreover the inhomogeneous relaxation modulus distributions within the tissues were determined at different relaxation time points under strain of 11.35, 19.35 and 30% respectively. The strain rate dependent nonlinear stress and strain curves were obtained for articular cartilage through uniaxial compression tests. It is noted that the Young's modulus exhibits a slight increase near the cartilage surface, and then increases fast with depth and both the magnitude and the variation of the Young's modulus are affected by increasing strain rates. A biphasic nonlinear viscoelastic relaxation model was proposed to predict the depth-dependent relaxation behavior of cartilage under unconfined compression and the results show that there are good agreements between the experimental data and predictions.     

求解一类非线性粘弹性问题的弹性回复对应原理

本文提出了一个全新的粘弹性理论体系,它把传统的线粘弹性理论作为特殊情况包括在内。其主要结果是两个用于求解一类物理非线性粘弹性问题的弹性回复对应原理。利用它,只要知道相应的非线性弹性问题的解,就可以求出非线性粘弹性问题的解答。对应原理不是基于本构关系的相似性,而是基于从粘弹性现时响应到瞬时弹性响应的可回复性[1]。首先找到非线性粘弹性与非线性弹性本构关系之间的对应,然后导出了两个弹性回复对应原理。通过对改性聚丙烯材料的实验验证了该对应原理的正确性和对此类材料的实用性。     

Nonlinear Shear Behavior of Poly(vinyl acetate) Material

Because of the strong environmental sensitivity of poly(vinyl acetate), PVAc, especially with respect to moisture, and the fact that shear deformation is essentially equivoluminal up to moderate strain levels, little has been reported in the literature on the nonlinear mechanical creep behavior of this polymeric material loaded in shear. This paper presents the results of torsional tests which establish the shear response through the linear zone and well into the nonlinear region. Test specimens were thin-walled cylinders giving an approximately uniform deformation field. Because of carefully chosen wall thickness to length ratio, it is considered that these measurements represent some of the most accurate nonlinear shear results to date in the strain range above 1%. Measurements of stress, strain and creep compliance were made at temperatures near the glass transition temperature and somewhat below it. Isochronal shear stress-strain dependence into the nonlinear range was used to establish limits of viscoelastic linearity during creep. As temperature is increased toward the glass transition, the limit shows a greater dependence on stress than on strain. The stored distortional strain energy at the limit of linearity was not a constant but varied with temperature and load. Thus, these results appear not to support the concept of stored energy as a material property defining the threshold for nonlinear viscoelastic behavior. Strain during the short-time load-up period gives evidence that PVAc is also subject to nonlinear elasticity in the glassy response region.     

Adaptive reproducing kernel particle method using gradient indicator for elasto-plastic deformation

An adaptive meshless method based on the multi-scale Reproducing Kernel Particle Method (RKPM) for analysis of nonlinear elasto-plastic deformation is proposed in this research. In the proposed method, the equivalent strain, stress, and the second invariant of the Cauchy–Green deformation tensor are decomposed into two scale components, viz., high- and low-scale components by deriving them from the multi-scale decomposed displacement. Through combining the high-scale components of strain and the stress update algorithm, the equivalent stress is decomposed into two scale components. An adaptive algorithm is proposed to locate the high gradient region and enrich the nodes in the region to improve the computational accuracy of RKPM. Using the algorithm, the high-scale components of strain and stress and the second invariant of the Cauchy–Green deformation tensor are normalized and used as criteria to implement the adaptive analysis. To verify the validity of the proposed adaptive meshless method in nonlinear elasto-plastic deformation, four case studies are calculated by the multi-scale RKPM. The patch test results show that the used multi-scale RKPM is reliable in analysis of the regular and irregular nodal distribution. The results of other three cases show that the proposed adaptive algorithm can not only locate the high gradient region well, but also improve the computational accuracy in analysis of the nonlinear elasto-plastic deformation.