Physically-based Constitutive Model Research Articles

An electromechanically coupled micro-sphere framework: application to the finite element analysis of electrostrictive polymers

The number of industrial applications of electroactive polymers (EAPs) is increasing and, consequently, the need for reliable modelling frameworks for such materials as well as related robust simulation techniques continuously increases. In this context, we combine the modelling of non-linear electroelasticity with a computational micro-sphere formulation in order to simulate the behaviour of EAPs. The micro-sphere approach in general enables the use of physics-based constitutive models like, for instance, the so-called worm-like chain model. By means of the micro-sphere formulation, scalar-valued micromechanical constitutive relations can conveniently be extended to a three-dimensional continuum setting. We discuss several electromechanically coupled numerical examples and make use of the finite element method to solve inhomogeneous boundary value problems. The incorporated material parameters are referred to experimental data for an electrostrictive polymer. The numerical examples show that the coupled micro-sphere formulation combined with the finite element method results in physically sound simulations that mimic the behaviour of an electrostrictive polymer.

Thermomechanical behavior and constitutive modeling of tungsten-based composite over wide temperature and strain rate ranges

Thermomechanical behavior of tungsten-based composite 93W–4.9Ni–2.1Fe is investigated systematically over strain rates ranging from 0.001 to 3000s−1, and temperatures ranging from 173 to 873K. Different micromechanisms are found in the evolution of microstructures between quasi-static and dynamic tests. The deformation of the tungsten particles is sensitive not only to strain rates, but also to plastic strain levels; the interaction between the grains is found to be the determining factor that cracks the grains, regardless of strain rates. Based on experimental results, two phenomenological and five physically-based constitutive models are established through a procedure of regression analysis and constrained optimization. Descriptive and predictive capabilities of these models are examined and compared. The performance of the models in characterization of work-hardening, temperature, and strain rate effects of the material is also investigated separately.

Granular element method (GEM): linking inter-particle forces with macroscopic loading

We present a new method capable of inferring, for the first time, inter-particle contact forces in irregularly-shaped natural granular materials (e.g., sands), using basic Newtonian mechanics and balance of linear momentum at the particle level. The method furnishes a relationship between inter-particle forces and corresponding average particle stresses, which can be inferred, for instance, from measurements of average particle strains emanating from advanced experimental techniques (e.g., 3D X-ray diffraction). Inter-particle forces are the missing link in understanding how forces are transmitted in complex granular structures and the key to developing physics-based constitutive models. We present two numerical examples to verify the method and showcase its promise.

A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity

This paper presents the development of a physical-based constitutive model for the representation of viscous effects in rubber-like materials. The proposed model originates from micromechanically motivated diffusion processes of the highly mobile polymer chains described within the formalism of Brownian motion. Following the basic assumption of accounting for the elastic and the viscous effects in rubber viscoelasticity by their representation through a separate elastic ground network and several viscous subnetworks, respectively, the kinetic theory of rubber elasticity is followed and extended to represent also the viscous contribution in this work. It is assumed that the stretch probability of certain chain segments within an individual viscous subnetwork evolves based on the movement of the chain endpoints described by the Smoluchowski equation extended in this work from non-interacting point particles in a viscous surrounding to flexible polymer chains. An equivalent tensorial representation for this evolution is chosen which allows for the closed form solution of the macroscopic free energy and the macroscopic viscous overstress based on a homogenization over the probability space of the introduced micro-objects. The resulting model of the viscous subnetwork is subsequently combined with the non-affine micro-sphere model and applied in homogeneous and non-homogeneous tests. Finally, the model capacity is outlined based on a comparison with in the literature available experimental data sets.

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, 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.

Numerical modeling of response of monolithic and bilayer plates to impulsive loads

In this paper, we present and discuss the results of our numerical simulation of the dynamic response and failure modes of circular DH-36 steel plates and DH-36 steel–polyurea bilayers, subjected to impulsive loads in reverse ballistic experiments. In our previous article, we reported the procedure and results of these experiments [MR Amini, JB Isaacs, S Nemat-Nasser. Experimental investigation of response of monolithic and bilayer plates to impulsive loads. accepted]. For the numerical simulations, we have used physics-based and experimentally-supported temperature- and rate-sensitive constitutive models for steel and polyurea, including in the latter case the pressure effects. Comparing the simulation and the experimental results, we focus on identifying the potential underpinning mechanisms that control the deformation and failure modes of both monolithic steel and steel–polyurea bilayer plates. The numerical simulations reveal that the bilayer plate has a superior performance over the monolithic plate if the polyurea layer is cast on its back face (opposite to the blast-receiving side). The presence of the polyurea layer onto the front face (blast-receiving side) amplifies the initial shock loading and thereby enhances the destructive effect of the blast, promoting (rather than mitigating) the failure of the steel plate. In addition, the interface bonding strength between polyurea and steel is examined numerically and it is observed that the interface bonding strength has a significant effect on the performance of the steel–polyurea bilayer plates. The numerical simulations support the experimentally observed facts provided the entire experiment is simulated, employing realistic physics-based constitutive models for all constituents.

Experimental investigation of response of monolithic and bilayer plates to impulsive loads

This article presents the results of a series of experiments performed to assess the dynamic response of circular monolithic steel and steel–polyurea bilayer plates to impulsive loads. A convenient technique to enhance the energy absorption capability of steel plates and to improve their resistance to fracturing in dynamic events, is to spray-cast a layer of polyurea onto the plates. Since polyurea readily adheres to metallic surfaces and has a short curing time, the technique may be used to retrofit existing metallic structures to improve their blast resistance. We have examined the effectiveness of this approach, focusing on the question of the significance of the relative position of the polyurea layer with respect to the loading direction; i.e., we have explored whether the polyurea layer cast on the front face (the impulse-receiving face) or on the back face of the steel plate would provide a more effective blast mitigating composite. The experimental results suggest that the polyurea layer can have a significant effect on the response of the steel plate to dynamic impulsive loads, both in terms of failure mitigation and energy absorption, if it is deposited on the back face of the plate. And, remarkably, when polyurea is placed on the front face of the plate, it may actually enhance the destructive effect of the blast, promoting (rather than mitigating) the failure of the steel plate, depending on the interface bonding strength between the polyurea and steel layers. These experimental results are supported by our computational simulations of the entire experiment, employing realistic physics-based constitutive models for the steel (DH-36, in the present work) and polyurea [Amini MR, Amirkhizi AV, Nemat-Nasser S. Numerical modeling of response of monolithic and bilayer plates to impulsive loads. Int J Impact Eng, submitted for publication].

Analysis of plastic flow localization under strain paths changes and its coupling with finite element simulation in sheet metal forming

Formability of sheet metal is usually assessed by the useful concept of forming limit diagrams (FLD) and forming limit curves (FLC) represent a first safety criterion for deep drawing operations. The level of FLC is strongly strain path dependent as observed by experimental and numerical results and therefore non-proportional strain paths need to be incorporated when analyzing formability of sheet metal components. Simulations using finite element method allow accurate predictions of stress and strain distributions in complex stamped parts. However, the prediction of localized necking is a difficult task and the combination of forming limit diagram analysis with finite element simulations often fail to give the right answer, if complex strain paths are not included in these predictions. In this work a code is presented aimed at formability prediction in sheet metal forming, with a concept and structure which allows the implementation of any hardening law, yield function or constitutive equation without major difficulty. The model incorporates both approaches of the theory of plasticity, namely the phenomenological one and the physical one. Several phenomenological constitutive equations, such as, Swift hardening power law and Voce saturation hardening law, the isotropic von Mises yield criterion, the quadratic Hill yield criterion (Hill’48), the non-quadratic Hill yield criterion (Hill’79) and the Yld’96 Barlat yield criterion as well as a physics-based constitutive model accounting for the texture and strain path induced anisotropy, specifically based on the Van Houtte's anisotropic texture strain-rate plastic potential and Teodosiu and Hu microstructural hardening model, are implemented in the new model. The necking phenomenon is carried out in the framework of heterogeneous materials using the Marciniak–Kuczincki (M–K) analysis coupled with the theory of plasticity. Such code may be used to obtain forming limit curves under linear and complex strain paths as well as being used to be coupled with finite element results, as a post-processing tool, to predict occurrence of necking. Studies are presented to test and validate implemented models including some sensitivity analysis to defined variables. The influence of strain path change is presented through the consideration of several non-proportional loading sequences and theoretical results are compared with experimental ones. Also a selected sheet metal component is considered to test and validate developed code as a post-processing tool for finite element analysis and such results are compared with those obtained experimentally.

Strain localization analysis using a large deformation anisotropic elastic–plastic model coupled with damage

Sheet metal forming processes generally involve large deformations together with complex loading sequences. In order to improve numerical simulation predictions of sheet part forming, physically-based constitutive models are often required. The main objective of this paper is to analyze the strain localization phenomenon during the plastic deformation of sheet metals in the context of such advanced constitutive models. Most often, an accurate prediction of localization requires damage to be considered in the finite element simulation. For this purpose, an advanced, anisotropic elastic–plastic model, formulated within the large strain framework and taking strain-path changes into account, has been coupled with an isotropic damage model. This coupling is carried out within the framework of continuum damage mechanics. In order to detect the strain localization during sheet metal forming, Rice’s localization criterion has been considered, thus predicting the limit strains at the occurrence of shear bands as well as their orientation. The coupled elastic–plastic-damage model has been implemented in Abaqus/implicit. The application of the model to the prediction of Forming Limit Diagrams (FLDs) provided results that are consistent with the literature and emphasized the impact of the hardening model on the strain-path dependency of the FLD. The fully three-dimensional formulation adopted in the numerical development allowed for some new results – e.g. the out-of-plane orientation of the normal to the localization band, as well as more realistic values for its in-plane orientation.

A framework for numerical integration of crystal elasto-plastic constitutive equations compatible with explicit finite element codes

In this paper we summarize the elements of a numerical integration scheme for elasto-plastic response of single crystals. This is intended to be compatible with large-scale explicit finite element codes and therefore can be used for problems involving multiple crystals and also overall behavior of polycrystalline materials. The steps described here are general for anisotropic elastic and plastic response of crystals. The crystallographic axes of the lattice are explicitly stored and updated at each time step. A plastic predictor–elastic corrector scheme is used to calculate the plastic strain rates on all active slip systems based on a rate-dependent physics-based constitutive model without the need of further auxiliary assumptions. Finally we present the results of numerous calculations using a physics-based rate- and temperature-dependent model of copper and the effect of elastic unloading, elastic crystal anisotropy, and deformation-induced lattice rotation are emphasized.

Influence of temperature, strain rate and thermal aging on the structure/property behavior of uranium 6 wt% Nb

A rigorous experimentation and validation program is being undertaken to create constitutive models that elucidate the fundamental mechanisms controlling plasticity in uranium-6 wt.% niobium alloys (U-6Nb). These models should accurately predict high-strain-rate large-strain plasticity, damage evolution and failure. The goal is a physically-based constitutive model that captures 1) an understanding of how strain rate, temperature, and aging affects the mechanical response of a material, and 2) an understanding of the operative deformation mechanisms. The stress-strain response of U-6Nb has been studied as a function of temperature, strain-rate, and thermal aging. U-6Nb specimens in a solution-treated and quenched condition (ST/Q) and after subsequent aging at 473K for 2 hours were studied. The constitutive behavior was evaluated over the range of strain rates from quasi-static (0.001 sec -1 ) to dynamic (∼2000 sec -1 ) and temperatures ranging from 77 to 773K. The yield stress of U-6Nb was exhibited pronounced temperature sensitivity. The strain hardening rate is seen to be less sensitive to strain rate and temperature beyond plastic strains of 0.10. The yield strength of the aged material is less significantly affected by temperature and the work hardening rate shows adiabatic heating at lower strain rates (1/s).

Mechanical response of dibenzyl-based polyurethanes with diol chain extension

A systematic investigation was made of the effects of varying hard and soft segment chemistry, crosslinking and preparation procedures, on the mechanical response of melt-cast polyurethane elastomers. In particular, two hard segments were compared, based on the diisocyanates: 4,4′-methylene bis(phenyl isocyanate) (MDI) and 4,4′-dibenzyl diisocyanate (DBDI). Rotation around the central –CH2–CH2– bridge in DBDI allows alignment of aromatic rings and hence crystallization within the hard phase, which is not available with MDI in melt-cast polyurethanes. Thus, new polymers were achieved, with a controlled ordering of copolymer hard segment blocks on the macromolecular chain. Wide angle X-ray diffraction of the as-moulded polymers revealed the presence of crystallinity in some cases, in the DBDI-based PU materials. Mechanical tests included load–unload cycles at constant rate of extension, with measurement of hysteresis and strain recovery, and stress relaxation tests. The presence of DBDI hard segments instead of MDI led systematically to increases in: the input strain energy to a given elongation, hysteresis and residual strain under cyclic loading, and stress relaxation. The results were interpreted in terms of a physically-based constitutive model framework previously proposed. This revealed that the observed effects of varying hard segment could all be explained by the hard domains having a higher flow stress in the presence of DBDI relative to MDI, associated with increased hydrogen bonding in DBDI-based polymers, which is enhanced in some cases by hard segment crystallinity. Materials with mixed MDI and DBDI hard segments were found to give the optimum combination of high input strain energy, but minimum residual strain, compared to equivalent materials based on MDI or DBDI alone.

Physically-based constitutive modelling of residual stress development in welding of aluminium alloy 2024

A finite element model has been developed to predict the evolution of residual stress and distortion which takes into account the history-dependence of the yield stress–temperature response of heat-treatable aluminium alloys during welding. The model was applied to TIG welding of 2024-T3 aluminium alloy, and the residual strain predictions validated using high resolution X-ray synchrotron diffraction. The goal was to capture the influence of the permanent evolution of the microstructure during the thermal cycle with a straightforward numerical procedure, while retaining a sound physical basis. Hardness and resistivity measurements after isothermal hold-and-quench experiments were used to identify salient temperatures for zero, partial and full dissolution of the initial hardening precipitates, and the extent of softening – both immediately after welding, and after natural ageing. Based on these data, a numerical procedure for weld modelling was proposed for tracking the different yield responses during heating and cooling based on the peak temperature reached locally. This history-dependent model was superior to a conventional model in predicting the peak tensile strains, but otherwise the effect of temperature history was weak for 2024-T3. Predictions of the hardness profile immediately after welding compared with the post-weld naturally aged hardness provided insight into the competition between dissolution and coarsening of the precipitates in the heat-affected zone.

A physically-based constitutive model for bcc crystals with application to polycrystalline tantalum

Based on the results of an extensive series of systematic experiments on commercially pure tantalum (bcc crystals), a physically-based, rate- and temperature-dependent constitutive model is proposed for bcc single crystals and is applied to simulate the experimental results, using the Taylor averaging method. The model calculation is based on a new efficient algorithm for the numerical solution of the finite deformation of bcc single crystals, involving up to 48 potentially active slip systems. The accuracy and efficiency of the proposed algorithm are checked through comparison with the results of the conventional explicit Euler time-integration scheme, using a very large number of timesteps. The model effectively simulates a large body of experimental data, over a broad range of strain rates (10 −3 − 4 × 10 4/s), and temperatures (77 to 1300 K), with strains exceeding 100%, using very few adjustable parameters whose values are fixed at the outset for a given material. All other involved constitutive parameters are estimated based on the crystal structure and the physics of plastic flow.

Development of an Integrated Approach to Modelling of Polymer Film Orientation Processes

An integral part of many manufacturing processes for polymer products is the deliberate introduction of preferred molecular orientation by means of uniaxial or biaxial stretching. An integrated scheme is under development for the modelling of biaxial orientation processes and has been applied to the case of poly(ethylene terephthalate) (PET) films. It consists of: experimentation under process-relevant conditions using a purpose-built biaxial testing machine; development of a physically-based constitutive model based on the data obtained; incorporation of the constitutive model within a Finite Element continuum model; and validation tests of the model. A first pass has been made through the complete sequence, and the application of the model to a section of the process for production of biaxially oriented PET film has been demonstrated.

Modeling very large plastic flows at very large strain rates for large-scale computation

There are many important technological problems where accurate modeling of very large plastic deformations at very high strain rates is essential for effective and reliable design and manufacture of metal components. Examples are crashworthy vehicles, automated manufacturing, and automated metal forming and cutting. Another example is the understanding of the deformation field at the tip of a fast-running crack, where strain rates can easily exceed 10 5s −1 with local strains of several hundred percent. This paper summarizes some of the recent efforts by the authors and coworkers at the University of California, San Diego to develop physically-based constitutive models for mono- and polycrystalline solids, and to implement these models in large-scale finite element codes in the form of constitutive modules with efficient and robust computer algorithms. In the case of single crystal viscoplasticity, the constitutive integration is complicated by the fact that the essential set of ordinary differential equations is highly stiff for realistic levels of instantaneous rate-sensitivity. The algorithm that was developed for the integration of these equations in an explicit finite element environment is presented. Two example finite element solutions are given to demonstrate features of the overall finite element code. Also, a phenomenological viscoplasticity model that includes noncoaxiality between the stress and the plastic deformation rate is presented. This model is thought to be appropriate for high-strain-rate problems involving polycrystalline metals. Since the essential equation in this model is a single ordinary differential equation, numerical stiffness is not a factor; however, the integration is still complicated by an abrupt transition from the elastic regime to the plastic flow regime. A simple, robust “predictor-corrector” algorithm for the integration of this equation is presented.

Plastic flow in oriented glassy polymers

Abstract 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.

A physically-based constitutive model for metal deformation

A set of constitutive relations that describe the deformation behavior of metals under complex loading or hold time histories is needed for predictive applications in metal forming, creep, creep-fatigue interactions and other situations. It is desirable for such a constitutive model to be based on physical processes that occur within the metal over wide ranges of temperatures and strain rates, and yet the model must contain parameters measurable from mechanical tests. The objective of this work is to combine into a generalized, unified model the existing level of understanding of the various physical processes, such as strain hardening, recovery, dislocation velocity dependence of effective stress, anelastic and grain boundary effects. Part II of this two-part paper uses this model of incremental deformation, incorporating strain history dependence, for predicting material behavior in various laboratory experiments. Based on a constitutive model developed in Part I of this paper, numerical calculations have been made here to check the capabilities of the model for predicting various aspects of material deformation behavior. Of particular interest are common laboratory experiments such as the tensile test, creep test, and other variations, such as, load relaxation, strain-rate change, decrementai unloading, unloading holding reloading and reverse loading tests. Since the various inputs for the model, such as basic strain hardening behavior, recovery rate constants and effective stresses, are not well known in their desired forms for the present model for any given material, no attempt has been made to simulate the stress level for a particular material. However, insights into the deformation phenomena are obtained through simulation of the overal material response over wide ranges of strain rates and temperatures. Confidence gained through this verification now opens the avenue for conducting suitable experiments necessary to determine the various parameters.