Modeling for piezoelectric-shape memory alloy composites

A new concept of a piezoelectric ceramic/shape memory alloy (SMA) composite is proposed with aim of using this as a new actuator material with fast actuation speed and large strain. To prove the new concept, a new model is constructed based on Eshelby formulation where linear piezoelectric constitutive equations and bi-linear superelastic equations of SMA are used. The predictions of the strain induced by applied stress and electric field are made for two simple designs of piezo–SMA composites, 1-D series and 1-D parallel laminated composites. The proposed model indicated that 1-D parallel laminate provides the highest strain induced under bias stress and applied electric field among other composite geometries.     

An exact nonlinear hybrid-coordinate formulation for flexible multibody systems

The previous low-order approximate nonlinear formulations succeeded in capturing the stiffening terms, but failed in simulation of mechanical systems with large deformation due to the neglect of the high-order deformation terms. In this paper, a new hybrid-coordinate formulation is proposed, which is suitable for flexible multibody systems with large deformation. On the basis of exact strain–displacement relation, equations of motion for flexible multibody system are derived by using virtual work principle. A matrix separation method is put forward to improve the efficiency of the calculation. Agreement of the present results with those obtained by absolute nodal coordinate formulation (ANCF) verifies the correctness of the proposed formulation. Furthermore, the present results are compared with those obtained by use of the linear model and the low-order approximate nonlinear model to show the suitability of the proposed models. The project supported by the National Natural Science Foundation of China (10472066, 50475021).     

Integral formulation and self-consistent modelling of elastoviscoplastic behavior of heterogeneous materials

Summary Based on projection operators, an integral formulation is proposed for elastoviscoplastic heterogeneous materials. The problem requires a complete mechanical formulation, including the static equilibrium property concerning the known field σ, in addition to the classical field equations concerning the unknown fields ɛ˙ and σ˙. The formulation leads to an integral equation, in which elasticity and viscoplasticity effects interact through an homogeneous elastoviscoplastic medium with elastic moduli C and viscoplastic moduli B. To approximate the integral equation, the self-consistent scheme is followed. In order to obtain consistent approximation conditions, we introduce fluctuations of elastic and viscoplastic strain rate fields with respect to known kinematically compatible fields. It results in a strain rate concentration relation connecting the strain rate at each point to the macroscopic loading conditions and the local stress field. The results are presented and compared with other models and with experimental data in the case of a two-phase material. Received 26 August 1997; accepted for publication 2 July 1998     

Finite viscoplasticity of semicrystalline polymers

Summary  A series of uniaxial tensile tests is performed on isotactic polypropylene at room temperature with cross-head speeds ranging from 5 to 100 mm/min. Prior to mechanical testing, injection-molded samples are annealed for 24 h at the temperature 160 ^°C that ensures that the specimens can be drawn up to the Hencky strain of 0.6 without necking. A constitutive model is derived for the viscoplastic behavior of a semicrystalline polymer at finite strains. The stress–strain relations are determined by five adjustable parameters that are found by matching the observations. Fair agreement is demonstrated between the experimental data and the results of numerical simulation. It is shown that the material parameters in the constitutive equations are strongly affected by the strain rate. Received 9 July 2002; accepted for publication 12 November 2002     

An interface damage model accounting for in-plane effects

The present work deals with the formulation of a kinematic enriched model for cohesive interface. In fact, the interface kinematics is defined by the relative displacement occurring between the two surfaces of the interface and, even, by the strain arising in the plane of the interface. A damage model which accounts for the mode I and mode II and for the axial deformation of the interface is proposed starting from the Drucker–Prager failure criterion. A numerical procedure is developed implementing the proposed interface model into a new finite element. The nonlinear evolutive problem is solved adopting a predictor–corrector technique within the backward time integration scheme. Simple numerical simulations are presented in order to assess the features of the model. Moreover, numerical applications are carried out in order to demonstrate the ability of the proposed model in reproducing the mechanical behavior of the cohesive elements strengthened with external FRP reinforcements. Comparisons between available experimental data and numerical results obtained using the proposed model show the effectiveness of the presented formulation.     

A dual-porosity model for gas reservoir flow incorporating adsorption behaviour—Part II. Numerical algorithm and example applications

In the present study, an algorithm is presented for the dual-porosity model formulated in Part I of this series. The resultant flow equation with the dual-porosity formulation is of an integro-(partial) differential equation involving differential terms for the Darcy flow in large fractures and integrals in time for diffusion within matrix blocks. The algorithm developed here to solve this equation involves a step-by-step finite difference procedure combined with a quadrature scheme. The quadrature scheme, used for the integral terms, is based on the trapezoidal method which is of second-order precision. This order of accuracy is consistent with the first- and second-order finite difference approximations used here to solve the differential terms in the derived flow equation. In an approach consistent with many petroleum reservoir and groundwater numerical flow models, the example formulation presented uses a first-order implicit algorithm. A two-dimensional example is also demonstrated, with the proposed model and numerical scheme being directly incorporated into the commercial gas reservoir simulator SIMED II that is based on a fully implicit finite difference approach. The solution procedure is applied to several problems to demonstrate its performance. Results from the derived dual-porosity formulation are also compared to the classic Warren–Root model. Whilst some of this work confirmed previous findings regarding Warren–Root inaccuracies at early times, it was also found that inaccuracy can re-enter the Warren–Root results whenever there are changes in boundary conditions leading to transient variation within the domain.     

Modelling large deformation behaviour under loading–unloading of semicrystalline polymers: Application to a high density polyethylene

In this work, the large deformation behaviour under monotonic loading and unloading of a high density polyethylene (HDPE) is studied. To analyze the nonlinear time-dependent response of the material, mechanical tests were conducted at room temperature under constant true strain rates and stress relaxation conditions. A physically-based inelastic model written under finite strain formulation is proposed to describe the mechanical behaviour of HDPE. In the model, the inelastic mechanisms involve two parallel elements: a visco-hyperelastic network resistance acting in parallel with a viscoelastic–viscoplastic intermolecular resistance where the amorphous and crystalline phases are explicitly taken into consideration. The semicrystalline polymer is considered as a two-phase composite. The influence of the crystallinity on the loading and unloading behaviour is investigated. Numerical results are compared to experimental data. It is shown that the model is able to accurately reproduce the experimental observations corresponding to monotonic loading, unloading and stress relaxation behaviours at different strain levels. Finally, the model capabilities to capture cyclic loading–unloading behaviour up to large strains are discussed. To demonstrate the improved modelling capabilities, simulations are also performed using the original model of Boyce et al. [Boyce, M.C., Socrate, S., Llana, P.G., 2000. Constitutive model for the finite deformation stress–strain behavior of poly(ethylene terephthalate) above the glass transition. Polymer 41, 2183–2201] modified by Ahzi et al. [Ahzi, S., Makradi, A., Gregory, R.V., Edie, D.D., 2003. Modeling of deformation behavior and strain-induced crystallization in poly(ethylene terephthalate) above the glass transition temperature. Mechanics of Materials 35, 1139–1148].     

Constitutive modeling of biodegradable polymers: Hydrolytic degradation and time-dependent behavior

A large range of biodegradable polymers has been used to produce implantable medical devices. Apart from biological compatibility, these devices shall be also functional compatible and perform adequate mechanical temporary support during the healing process. However, the mechanical behavior of biodegradable materials during its degradation, which is an important aspect of the design of these biodegradable devices, is still an unexplored subject. Based on the literature, the mechanical behavior of biodegradable polymers is strain rate dependent and exhibits hysteresis upon cyclic loading. On the other hand, ductility, toughness and strength of the material decay during hydrolytic degradation. In this work, it is considered a three-dimensional time-dependent model adapted from the one developed by Bergström and Boyce to simulate the performance of biodegradable structures undergoing large deformations incorporating the hydrolysis degradation. Since this model assumes that the mechanical behavior is divided into a time independent network and a non-linear time-dependent network, it enables to simulate the monotonic tests of a biodegradable structure loaded under different strain rates. The hysteresis effects during unloading–reloading cycles at different strain levels can be predicted by the model. A parametric study of the material model parameters evolution during the hydrolytic degradation was conducted to identify which parameters are more sensible to this degradation process. The investigated model could predict very well the experimental results of a blend of polylactic acid and polycaprolactone (PLA–PCL) in the full range of strains until rupture during hydrolytic degradation. From these results and analyses, a method is proposed to simulate the three-dimensional mechanical behavior during hydrolytic degradation.     

Development and validation of a thermo-mechanical finite element model of the steel quenching process including solid–solid phase changes

A computational thermo-metallographic and thermoelastoplastic model for the analysis of the quenching process is developed and validated. The diffusive transfor-mations are modeled according to the Johnson–Mehl–Avrami–Kolmogorov model and the Scheil’s additivity rule. Two different models are investigated for the non-diffusive transformation—the Koistinen–Marburger model and the Yu model. A large displacement formulation is assumed for the deformation analysis, modeling the plastic behavior of the material according to the Prandtl–Reuss model. Two different bilinear hardening models—the isotropic and the kinematic hardening model—are used and compared. The model allows to evaluate the transient stress and strain distributions during the quenching process, the final phases and hardness distributions, and to predict the residual stress and the final deformation of the processed part. A good agreement between computational results and reference data is found     

Coupled thermoviscoplasticity of glassy polymers in the logarithmic strain space based on the free volume theory

The paper outlines a constitutive model for finite thermo-visco-plastic behavior of amorphous glassy polymers and considers details of its numerical implementation. In contrast to existing kinematical approaches to finite plasticity of glassy polymers, the formulation applies a plastic metric theory based on an additive split of Lagrangian Hencky-type strains into elastic and plastic parts. The analogy between the proposed formulation in the logarithmic strain space and the geometrically linear theory of plasticity, makes this constitutive framework very transparent and attractive with regard to its numerical formulation. The characteristic strain hardening of the model is derived from a polymer network model. We consider the particularly simple eight chain model, but also comment on the recently developed microsphere model. The viscoplastic flow rule in the logarithmic strain space uses structures of the free volume flow theory, which provides a highly predictive modeling capacity at the onset of viscoplastic flow. The integration of this micromechanically motivated approach into a three-dimensional computational model is a key concern of this work. We outline details of the numerical implementation of this model, including elements such as geometric pre- and post-transformations to/from the logarithmic strain space, a thermomechanical operator split algorithm consisting of an isothermal mechanical predictor followed by a heat conduction corrector and finally, the consistent linearization of the local update algorithm for the dissipative variables as well as its relationship to the global tangent operator. The performance of the proposed formulation is demonstrated by means of a spectrum of numerical examples, which we compare with our experimental findings.     

A URI 4-Node quadrilateral element by assumed strain

In this paper one-point quadrature ““““assumed strain““““ mixed element formulation based on the Hu-Washizu variational principle is presented. Special care is taken to avoid hourglass modes and volumetric locking as well as shear locking. The assumed strain fields are constructed so that those portions of the fields which lead to volumetric and shear locking phenomena are eliminated by projection, while the implementation of the proposed URI scheme is straightforward to suppress hourglass modes. In order to treat geometric nonlinearities simply and efficiently, a corotational coordinate system is used. Several numerical examples are given to demonstrate the performance of the suggested formulation, including nonlinear static/dynamic mechanical problems.     

A robust hyper-elastic beam model under bi-axial normal-shear loadings

In this paper, a simple and robust constitutive model is proposed to simulate mechanical behaviors of hyper-elastic materials under bi-axial normal-shear loadings in the finite strain regime. The Mooney–Rivlin strain energy function is adopted to develop a two-dimensional (2D) normal-shear constitutive model within the framework of continuum mechanics. A motion field is first proposed for combined normal and shear deformations. The deformation gradient of the proposed field is calculated and then substituted into right Cauchy–Green deformation tensor. Constitutive equations are then derived for normal and shear deformations. They are two explicit coupled equations with high-level polynomial non-linearity. In order to examine capabilities of the developed hyper-elastic model, uniaxial tensile responses and non-linear stability behaviors of moderately thick straight and curved beams undergoing normal axial and transverse shear deformations are simulated and compared with experiments. Fused deposition modeling technique as a 3D printing technology is implemented to fabricate hyper-elastic beam structures from soft poly-lactic acid filaments. The printed specimens are tested under tensile/compressive in-plane and compressive out-of-plane forces. A finite element formulation along with the Newton–Raphson and Riks techniques is also developed to trace non-linear equilibrium path of beam structures in large defamation regimes. It is shown that the model is capable of predicting non-linear equilibrium characteristics of hyper-elastic straight and curved beams. It is found that the modeling of shear deformation and finite strain is essential toward an accurate prediction of the non-linear equilibrium responses of moderately thick hyper-elastic beams. Due to simplicity and accuracy, the model can serve in the future studies dealing with the analysis of hyper-elastic structures in which two normal and shear stress components are dominant.     

Simulation of planar flexible multibody systems with clearance and lubricated revolute joints

Modeling of clearance joints plays an important role in the analysis and design of multibody mechanical systems. Based on the absolute nodal coordinate formulation (ANCF), a new computational methodology for modeling and analysis of planar flexible multibody systems with clearance and lubricated revolute joints is presented. A planar absolute nodal coordinate formulation based on the locking-free shear deformable beam element is implemented to discretize the flexible bodies. A continuous contact-impact model is used to evaluate the contact force, in which energy dissipation in the form of hysteresis damping is considered. A force transition model from hydrodynamic lubrication forces to dry contact forces is introduced to ensure continuity in the joint reaction force. A comprehensive study with different lubrication force models has also been carried out. The generalized-α method is used to solve the equations of motion and several efficient methods are incorporated in the proposed model. Finally, the methodology is validated by two numerical examples.     

一种适合橡胶类材料的非线性粘弹性本构模型

借助非线性流变模型建立大变形情况非线性粘弹材料的本构关系,考虑到大多数橡胶类材料具有的几乎不可压缩性,以及体积响应和剪切响应的流变性能不同,将变形梯度乘法分解为等容部分和体积变形部分,给出了一种适合橡胶类材料的非线性粘弹性本构模型,并模拟了粘滞效应。对于极快或极慢的过程,该模型退化为橡胶弹性理论;在小变形情况下退化为经典的广义Maxwell粘弹性材料。模型与热力学第二定律相容,适合于大规模数值分析。     

A viscoelastic constitutive model of rubber under small oscillatory load superimposed on large static deformation

Summary  A viscoelastic constitutive equation of rubber that is under small oscillatory load superimposed on large static deformation is proposed. The model is derived through linearization of Simo's nonlinear viscoelastic constitutive model and reference configuration transformation. Most importantly, in this model, static deformation correction factor is introduced to consider the influence of pre-strain on the relaxation function. Natural statically pre-deformed state is served as reference configuration. The proposed constitutive equation is extended to a generalized viscoelastic constitutive equation that includes widely used Morman's model as a special case using objective stress increment. The proposed constitutive model is tested for dynamic behavior of rubber specimens with different carbon black content. It is concluded from the test that the assumption that the effects of static deformation can be separated from time effects, which is the basis of Morman's model, is only applicable to unfilled rubber. The viscoelastic constitutive equation for filled rubber must include, therefore, the influence of the static deformation on the time effects. The suggested constitutive equation with static deformation correction factor shows good agreement with test values. Received 4 January 2001; accepted for publication 13 June 2001