A thermodynamic consistent damage and healing model for self healing materials

Thermodynamics of the damage and the healing processes for viscoplastic materials is discussed in detail and constitutive equations for coupled inelastic-damage-healing processes are proposed in a thermodynamic consistent framework. Small deformation state is utilized and the kinematic and the isotropic hardening effects for the damage and healing processes are introduced into the governing equations. Two new yield surfaces for the damage and healing processes are proposed that take into account the isotropic hardening effect. The computational aspect for solving the coupled elasto-plastic-damage-healing problem is investigated, and the mechanical behavior of the proposed polymeric based self healing system is obtained. Uniaxial compression tests are implemented on a shape memory polymer based self healing system and the damage and the healing are captured by measurement of the changes in the modulus of elasticity. It is concluded that the proposed constitutive equations can model the damage and healing effectively and the mechanical behavior of a shape memory polymer based self healing system can be precisely modeled using this formulation.     

Rate equations for viscoplastic materials subjected to finite strains

The constitutive equations for plasticity proposed by Voyiadjis [1984] and Voyiadjis & Kiousis [1987] are modified here in order to introduce rate sensitivity in the plastic region. Some of the basic concepts of the theory of viscoplasticity outlined by Naghdi & Murch [1963], Perzyna & Wojno [1966], and Eisenberg & Yen [1981] are used in this work in order to obtain the proposed viscoplastic constitutive model for finite strain deformation analysis.Uniaxial loading-reverse loading tests are conducted so as to check the validity of the proposed constitutive model as well as to determine its material parameters. The model is effectively used in simulating numerically the obtained experimental results at finite strains.     

A thermo-viscoelastic–viscoplastic–viscodamage constitutive model for asphaltic materials

A temperature-dependent viscodamage model is proposed and coupled to the temperature-dependent Schapery’s nonlinear viscoelasticity and the temperature-dependent Perzyna’s viscoplasticity constitutive model presented in Abu Al-Rub et al., 2009, Huang et al., in press in order to model the nonlinear constitutive behavior of asphalt mixes. The thermo-viscodamage model is formulated to be a function of temperature, total effective strain, and the damage driving force which is expressed in terms of the stress invariants of the effective stress in the undamaged configuration. This expression for the damage force allows for the distinction between the influence of compression and extension loading conditions on damage nucleation and growth. A systematic procedure for obtaining the thermo-viscodamage model parameters using creep test data at different stress levels and different temperatures is presented. The recursive-iterative and radial return algorithms are used for the numerical implementation of the nonlinear viscoelasticity and viscoplasticity models, respectively, whereas the viscodamage model is implemented using the effective (undamaged) configuration concept. Numerical algorithms are implemented in the well-known finite element code Abaqus via the user material subroutine UMAT. The model is then calibrated and verified by comparing the model predictions with experimental data that include creep-recovery, creep, and uniaxial constant strain rate tests over a range of temperatures, stress levels, and strain rates. It is shown that the presented constitutive model is capable of predicting the nonlinear behavior of asphaltic mixes under different loading conditions.     

A physically-based constitutive model for anisotropic damage in rubber-toughened glassy polymers during finite deformation

The present work focuses on the development of a physically-based model for large deformation stress-strain response and anisotropic damage in rubber-toughened glassy polymers. The main features leading to a microstructural evolution (regarding cavitation, void aspect ratio, matrix plastic anisotropy and rubbery phase deformation) in rubber-toughened glassy polymers are introduced in the proposed constitutive model. The constitutive response of the glassy polymer matrix is modelled using the hyperelastic-viscoplastic model of [Boyce et al., 1988] and [Boyce et al., 2000]. The deformation mechanisms of the matrix material are accounted for by two resistances: an elastic-viscoplastic isotropic intermolecular resistance acting in parallel with a visco-hyperelastic anisotropic network resistance, each resistance being modified to account for damage effects by void growth with a variation of the void aspect ratio. The effective contribution of the hyperelastic particles to the overall composite behaviour is taken into account by treating the overall system in a composite scheme framework. The capabilities of the proposed constitutive model are checked by comparing experimental data with numerical simulations. The deformation behaviour of rubber-toughened poly(methyl methacrylate) was investigated experimentally in tension at a temperature of 80 °C and for different constant true strain rates monitored by a video-controlled technique. The reinforcing phase is of the soft core-hard shell type and its diameter is of the order of one hundred nanometers. The particle volume fraction was adjusted from 15% to 45% by increments of 5%. The stress-strain response and the inelastic volumetric strain are found to depend markedly on particle volume fraction. For a wide range of rubber volume fractions, the model simulations are in good agreement with the experimental results. Finally, a parametric analysis demonstrates the importance of accounting for void shape, matrix plastic anisotropy and rubber content.     

Plastic flow in oriented glassy polymers

A manufactured product often possesses residual texture which was either incidentally or deliberately acquired during its processing history. This is particularly true for the case of polymers, where the ability to easily preferentially preorient the material in specific directions is exploited in order to obtain a higher strength product. Specific examples include synthetic fibers, and biaxially-oriented films and containers. The response of the preoriented/textured product to normal service life loading conditions will differ considerably from that of a product composed of isotropic material. This paper addresses the issue of the effects of texture on the deformation behavior of glassy polymers. Here, the physically-based constitutive model of Boyce, Parks, and Argon describing the rate, temperature, and pressure dependent inelastic deformation of initially isotropic glassy polymers is used to model the effects of preorientation (i.e., initial texture), via the use of appropriate initial conditions on internal state variables. The model is then utilized in an analysis of the effects of texture on the yield of glassy polymers and the shear localization which normally follows yielding in oriented polymers. These results are compared with trends found in experiments as reported in the literature. The effectiveness of the proposed model for the present application is also compared with earlier models of yielding of anisotropic materials such as Hill's criterion.     

An effective temperature theory for the nonequilibrium behavior of amorphous polymers

Amorphous polymers lack an organized microstructure, yet they exhibit structural evolution, where physical properties change with time, temperature, and inelastic deformation. To describe the influence of structural evolution on the mechanical behavior of amorphous polymers, we developed a thermomechanical theory that introduces the effective temperature as a thermodynamic state variable representing the nonequilibrium configurational structure. The theory couples the evolution of the effective temperature and internal state variables to describe the temperature-dependent and rate-dependent inelastic response through the glass transition. We applied the theory to model the effect of temperature, strain rate, aging time, and plastic pre-deformation on the uniaxial compression response and enthalpy change with temperature of an acrylate network. The results showed excellent agreement with experiments and demonstrate the ability of the effective temperature theory to explain the complex thermomechanical behavior of amorphous polymers.     

Rate (time)-dependent deformation behavior: an overview of some properties of metals and solid polymers

The inelastic deformation behaviors of metals and polymers are discussed with the aim of finding a common base that would simplify academic and engineering analyses. Only monotonic loading conditions at room temperature are considered. For loading at different rates, nonlinear relations between loading rate and stress level, creep stress level and creep strain, and relaxation rate and stress were common to both type of materials. There are, of course, significant differences in elastic properties, strength levels and the strains involved. Special properties such as relaxation behaviors and creep anomalies can be qualitatively and quantitatively reproduced by the state variable model VBO (viscoplasticity theory based on overstress). Since experimental investigations typically concentrate on one particular aspect of inelastic deformation behavior such as creep or strain-rate dependence, it is often difficult to gather a comprehensive data set for a given material. In spite of this, considerable similitude in the deformation behavior of metals and polymers in various test conditions has nevertheless been established.     

Nonlocal continuum theory of anisotropically damaged metals

The paper deals with a consistent and systematic general framework for the development of anisotropic continuum damage in ductile metals based on thermodynamic laws and nonlocal theories. The proposed model relies on finite strain kinematics based on the consideration of damaged as well as fictitious undamaged configurations related via metric transformation tensors which allow for the interpretation of damage tensors. The formulation is accomplished by rate-independent plasticity using a nonlocal yield condition of Drucker–Prager type, anisotropic damage based on a nonlocal damage growth criterion as well as non-associated flow and damage rules. The nonlocal theory of inelastic continua is established to be able to take into account long-range microstructural interaction. The approach incorporates macroscopic interstate variables and their higher-order gradients which properly describe the change in the internal structure and investigate the size effect of statistical inhomogeneity of the heterogeneous material. The idea of bridging length-scales is made by using higher-order gradients in the evolution equations of the equivalent inelastic strain measures which leads to a system of elliptic partial differential equations which is solved using the finite difference method at each iteration of the loading step and the displacement-based finite element procedure is governed by the standard principle of virtual work. Numerical simulations of the elastic–plastic deformation behavior of damaged solids demonstrate the efficiency of the formulation. Tension tests undergoing large strains are used to investigate the damage growth in high strength steel. The influence of various model parameters on the prediction of the deformation and localization of ductile metals is discussed.     

预延伸非晶聚合物材料各向异性变形局部化

给出一种可描述预延伸各向异性特性的背应力张量三维表达式，引入大变形弹塑性有限元驱动应力法，结合ＢＰＡ８ 链细观分子网络模型，模拟了预延伸各向异性非晶聚合物材料平面应变拉伸变形局部化力学行为．详细讨论了预延伸比（ＩｎｉｔｉａｌＤｒａｗｉｎｇＲａｔｉｏ；ＩＤＲ）和预延伸方向（ＩｎｉｔｉａｌＤｒａｗｉｎｇＤｉｒｅｃｔｉｏｎ；ＩＤＤ）对变形抗力、颈缩规律、剪切带方向以及试件中心部位链延伸比的影响．     

On modeling the micro-indentation response of an amorphous polymer

Interest in instrumented indentation experiments as a means to estimate mechanical properties has grown rapidly in recent years. Although numerous nano/micro-indentation experimental studies on polymeric materials have been reported in the literature, a corresponding methodology for extracting material property information from the experimental data does not exist. This situation for polymeric materials exists primarily because baseline numerical analyses of sharp indentation using appropriate large deformation constitutive models for the nonlinear viscoelastic–plastic response of these materials appear not to have been previously reported in the literature. An existing, widely used theory for amorphous polymers (e.g. [Boyce, M., Parks, D., Argon, A.S., 1988. Large inelastic deformation of glassy polymers. Part 1: Rate dependent constitutive model. Mechanics of Materials 7, 15–33; Arruda, E.M., Boyce, M.C., 1993. Evolution of plastic anisotropy in amorphous polymers during finite straining. International Journal of Plasticity 9, 697–720]) has been recently found to lack sufficient richness to enable one to quantitatively reproduce the major features of the indentation load-versus-depth curves for some common amorphous polymers [Gearing, B.P., 2002. Constitutive equations and failure criteria for amorphous polymeric solids. Ph.D. thesis, Massachusetts Institute of Technology].This study develops a new continuum model for the viscoelastic–plastic deformation of amorphous polymeric solids. We have applied the constitutive model to capture salient features of the mechanical response of the amorphous polymeric solid poly(methyl methacrylate) (PMMA) at ambient temperature and stress states under which this material does not exhibit crazing. We have conducted compression-tension strain-controlled experiments, as well as stress-controlled compression-creep experiments, and these experiments are used to calibrate the material parameters in the constitutive model for PMMA.We have implemented our constitutive model in a finite-element computer program, and using this finite-element program we have simulated micro-indentation experiments on PMMA. We show that our constitutive model and finite element simulations reproduce the experimentally-measured indentation load-versus-depth response with reasonable accuracy.     

Finite viscoplasticity of amorphous glassy polymers in the logarithmic strain space

The paper outlines a new constitutive model and experimental results of rate-dependent finite elastic–plastic behavior of amorphous glassy polymers. In contrast to existing kinematical approaches to finite viscoplasticity of glassy polymers, the formulation proposed is constructed in the logarithmic strain space and related to a six-dimensional plastic metric. Therefore, it a priori avoids difficulties concerning with the uniqueness of a plastic rotation. The constitutive framework consists of three major steps: (i) A geometric pre-processing defines a total and a plastic logarithmic strain measures determined from the current and plastic metrics, respectively. (ii) The constitutive model describes the stresses and the consistent moduli work-conjugate to the logarithmic strain measures in an analogous structure to the geometrically linear theory. (iii) A geometric post-processing maps the stresses and the algorithmic tangent moduli computed in the logarithmic strain space to their nominal, material or spatial counterparts in the finite deformation space. The analogy between the formulation of finite plasticity in the logarithmic strain space and the geometrically linear theory of plasticity makes this framework very attractive, in particular regarding the algorithmic implementation. The flow rule for viscoplastic strains in the logarithmic strain space is adopted from the celebrated double-kink theory. The post-yield kinematic hardening is modeled by different network models. Here, we compare the response of the eight chain model with the newly proposed non-affine micro-sphere model. Apart from the constitutive model, experimental results obtained from both the homogeneous compression and inhomogeneous tension tests on polycarbonate are presented. Besides the load–displacement data acquired from inhomogeneous experiments, quantitative three-dimensional optical measurements of the surface strain fields are carried out. With regard to these experimental data, the excellent predictive quality of the theory proposed is demonstrated by means of representative numerical simulations.     

Thermomechanical formulation of ductile damage coupled to nonlinear isotropic hardening and multiplicative viscoplasticity

In this paper, we present a thermomechanical framework which makes use of the internal variable theory of thermodynamics for damage-coupled finite viscoplasticity with nonlinear isotropic hardening. Damage evolution, being an irreversible process, generates heat. In addition to its direct effect on material's strength and stiffness, it causes deterioration of the heat conduction. The formulation, following the footsteps of Simó and Miehe (1992), introduces inelastic entropy as an additional state variable. Given a temperature dependent damage dissipation potential, we show that the evolution of inelastic entropy assumes a split form relating to plastic and damage parts, respectively. The solution of the thermomechanical problem is based on the so-called isothermal split. This allows the use of the model in 2D and 3D example problems involving geometrical imperfection triggered necking in an axisymmetric bar and thermally triggered necking of a 3D rectangular bar.     

混凝土的弹塑性损伤双面本构模型

针对混凝土材料拉压应变空间下损伤机制的不同,结合连续损伤力学和塑性理论建立了一个全新的本构模型。该模型中损伤和塑性变形的演变由应变空间的同一个非弹性曲面来控制,但对拉压应变空间中非弹性曲面的演变分别采用了随动强化法则和各向同性演化规律。计算结果表明,该模型能较好地描述混凝土材料在单轴及多轴单调加载和低周反复荷载下的典型非线性特征。     

高分子材料平面应变拉伸变形局部化

将基于应变软化玻璃状高分子材料微观特征建立的ＢＰＡ８－链分子网络模型引入ＵｐｄａｔｉｎｇＬａｇｒａｎｇｅ有限元方法，建立了适于变形局部化分析的大变形弹塑性有限元驱动应力法．在此基础上，数值模拟了初始各向同性高分子材料平面应变拉伸变形局部化的传播过程．探讨了ＢＰＡ模型对具有加工硬化特性的结晶性高分子材料变形分析的适应性；分析了局部化传播过程中颈缩截面的非均匀应力三轴效应；最后，讨论了网格尺寸以及初始几何不均匀性对颈缩扩散以及应力三轴效应的影响     

Micromechanics, macromechanics and constitutive modeling of the elasto-viscoplastic deformation of rubber-toughened glassy polymers

Under certain conditions, such as sufficiently low temperatures, high loading rates and/or highly triaxial stress states, glassy polymers display an unfavorable characteristic—brittleness. A technique used for reducing the brittleness (increasing the fracture toughness) of these materials is rubber toughening. While there is significant qualitative understanding of the mechanical behavior of rubber-toughened polymers, quantitative modeling tools for the large-strain deformation of rubber-toughened glassy polymers are largely lacking.In this paper, we develop a suite of numerical tools to investigate the mechanical behavior of rubber-toughened glassy polymers, with emphasis on rubber-toughened polycarbonate. The rubber particles are modeled as voids in view of their deformation-induced cavitation early during deformation. A three-dimensional micromechanical model of the heterogeneous microstructure is developed to study the effects of initial rubber particle (void) volume fraction on the underlying elasto-viscoplastic deformation mechanisms in the material, and how these mechanisms influence the macroscopic response of the material. A continuum-level constitutive model is developed for the large-strain elasto-viscoplastic deformation of porous glassy polymers, and it is calibrated against micromechanical modeling results for porous polycarbonate. The constitutive model can be used to study various boundary value problems involving rubber-toughened (porous) glassy polymers. As an example, the case of an axisymmetric notched bar is simulated for the case of polycarbonate with varying levels of initial porosity. The quality of the constitutive model calibration is assessed using a multi-scale modeling approach.     

一种高聚物银纹损伤演化的实验研究

采用光学显微技术对高聚物蠕变条件下的银纹损伤引发和演化进行了实验观测．通过对银纹面密度的测量,定义一种损伤变量,得到了聚甲基丙烯酸甲酯不同蠕变时间和不同应力水平下的银纹损伤值,并给出了一种损伤演化模型．     

Necking in glassy polymers: Effects of intrinsic anisotropy and structural evolution kinetics in their viscoplastic flow

Two recently proposed developments of the Glass–Rubber constitutive model for glassy polymers treat the viscoplastic deformation as intrinsically anisotropic, and incorporate the kinetics of structural evolution. These features enable the model to capture better the distinctive features of glassy polymers’ constitutive response: post-yield strain-softening and strain-hardening and effects of pre-existing molecular orientation. They have been combined to form a new variant of the model, and the consequences for necking have been explored. Uniaxial extension of prismatic bars was simulated using the finite element method, employing a numerical implementation of the new model, with material parameters of polystyrene. Strain localization predicted with the new model was found to be systematically retarded as compared to predictions with the original (intrinsically isotropic) version of the model, for the same conditions. In particular, the effect of frozen-in molecular orientation was examined. This was found to retard strain localization for stretching parallel to the orientation direction, for both models. But the localization predicted with the new model was always significantly less pronounced than with the original model. Indeed, for sufficiently high pre-orientation (e.g. a uniaxial stretch of 2.2), localization could be effectively prevented with the new model, under conditions when otherwise failure by necking is predicted. Such results can all be explained in terms of a linear stability analysis. They suggest that all previous simulations of necking in glassy polymers made using intrinsically isotropic representations of polymer viscoplasticity may have over-predicted the rate of strain localization.     

Modeling deformation behavior of polymers with viscoplasticity theory based on overstress

The nonlinear strain rate sensitivity, multiple creep and recovery behavior of polyphenylene oxide (PPO), which were explored through strain rate-controlled experiments at ambient temperature by Khan [The deformation behavior of solid polymers and modeling with the viscoplasticity theory based overstress, Ph.D. Thesis, Rensselaer Polytechnic Institute, New York], are modeled using the modified viscoplasticity theory based on overstress (VBO). In addition, VBO used by Krempl and Ho [An overstress model for solid polymer deformation behavior applied to Nylon 66, ASTM STP 1357, 2000, p. 118] and the classical VBO are used to demonstrate the improved modeling capabilities of VBO for solid polymer deformation. The unified model (VBO) has two tensor valued state variables, the equilibrium and kinematic stresses and two scalar valued states variables, drag and isotropic stresses. The simulations include monotonic loading and unloading at various strain rates, multiple creep and recovery at zero stress. Since creep behavior has been found to be profoundly influenced by the level of the stress, the tests are performed at different stresses above and below the yield point. Numerical results are compared to experimental data. It is shown that nonlinear rate sensitivity, nonlinear unloading, creep and recovery at zero stress can be reproduced using the modified viscoplasticity theory based on overstress.     

Broken beams

A series of experiments has been conducted, utilizing sheet explosive opplied to clamped aluminum beams, with a neoprene buffer. As the load is monotonically increased, three damage modes are identified, which respecitively are major inelastic deformation, tearing at the extreme fiber, and transverse shear at the support. Satisfactory correlation is reported for the extent of inelastic deformation using a lumped parameter, finite-difference code; thresholds for tearing and shear failure based on empirical criteria are presented. Using a Timoshenko beam theory, the shear threshold appears to be dependent on the section velocity, rather than upon the shear stress.     

Modelling of the elasto-viscoplastic damage behaviour of glassy polymers

Constitutive equations are proposed in order to describe the elasto-viscoplastic damage behaviour of polymers. The behaviour is well accounted for by a modified Bodner–Partom model comprising hydrostatic and void evolution terms. The true stress–strain and volumetric strain behaviour of typical rubber-toughened glassy polymers (RTPMMA and HIPS) were experimentally determined at constant local true strain rate by using a video-controlled technique. Successful agreement is obtained between experimental results and the proposed model.