Finite Deformation Constitutive Model Research Articles

A 3D finite deformation constitutive model for anisotropic shape memory polymer composites integrating viscoelasticity and phase transition concept

The phase transition theory of shape memory polymers (SMPs) often involves a phenomenological assumption that the reference configuration of the newly transformed phase deviates from that of the initial phase. This distinction serves as a crucial mechanism in the manifestation of the shape memory effect. However, elucidating the precise definition of the reference configuration of the transformed phase poses a significant challenge in the formulation of the constitutive model. To tackle this challenge, a three-dimensional (3D) finite deformation constitutive model incorporating effective phase evolution for SMPs has been developed. This model merges insights from the classical viscoelastic framework with the phase transition theory. The anisotropic thermo-viscoelastic constitutive model is further developed by introducing hyperelastic fibers, which integrate the anisotropy of the fibers into a continuous thermodynamic framework through structure tensors. Implemented within the ABAQUS software via a user material (UMAT) subroutine, the proposed model has been meticulously validated against experimental data, showcasing its prowess in simulating stress-strain responses and shape memory characteristics of SMPs and their composites (SMPCs). This innovative model stands as an invaluable instrument for the design and of sophisticated SMP and SMPC structures.

A thermo-mechanical constitutive model for triple-shape and two-way shape memory polymers

Semicrystalline polymers often have diverse molecular configurations in response to the variation of temperature, making it easier to realize multiple or two-way shape memory effects (SMEs). In order to provide guidance for the design of relevant smart structures, it is necessary to develop corresponding constitutive models to better characterize the thermal-mechanical behavior of such shape memory polymers (SMPs). For the first time, a thermo-mechanical finite deformation constitutive model is presented to reveal the deformation mechanism of the triple-shape two-way SME of semicrystalline SMPs in our work, based on the theory of thermodynamics with internal state variables. To verify the validity of the model, the model results are compared with the test data for a typical shape memory cycle. It is found that the model can be employed to describe the non-equilibrium response of semicrystalline SMPs with two types of crystallites in the vicinity of crystallization and melting. Since the model results fit well with the test data, the effectiveness of the model in predicting the triple two-way shape memory behavior is demonstrated.

Is the Jaumann Stress Rate Applicable? — Discussions on Choosing Proper Rate Expressions of a Finite Deformation Constitutive Model

The rate form constitutive expression plays an important role in finite deformation theory and is widely used in the finite element method (FEM) software. However, the choice of objective stress and strain rates in constitutive expressions has caused many controversies. This paper aims to clarify these controversies by distinguishing the related concepts: constitutive behavior, constitutive model and constitutive expression, and discuss the influence of constitutive model choices and constitutive expression choices separately. As a certain constitutive model may correspond to many different rate form constitutive expressions, the conversion relationships among them are derived, which can help clarify many controversies. First, it is proven that three different stress–strain curves in the controversial simple shear example actually correspond to three constitutive models. For each constitutive model, the corresponding different rate form constitutive expressions yield consistent calculation results. We also demonstrate that the applicability of a constitutive model is completely independent of the choice of objective stress rates in the constitutive expressions. Second, we find that for FEM simulation, the element tangent stiffness matrix can serve as an indicator to evaluate the validity of different constitutive expressions and their corresponding FEM formulations. Moreover, even though non-work-conjugate stress-strain rates are adopted, such as the Jaumann stress rate/deformation rate, different rate form constitutive expressions can correctly reflect the constitutive model with the proper tangent modulus tensors. All these conclusions have been verified by examples. For convenience, we recommend using the constitutive expressions in terms of the second Piola–Kirchhoff stress versus Green’s strain tensor and Truesdell stress rate versus deformation rate for experimental measurements and FEM simulations. Of course, other objective stress rates are all applicable and have no effect on the calculation results as long as the correct conversion relationships are used.

A temperature/strain-rate-dependent finite deformation constitutive and failure model for solid propellants

A temperature/strain-rate-dependent finite deformation constitutive and failure model for solid propellants

A thermoviscoelastic finite deformation constitutive model based on dual relaxation mechanisms for amorphous shape memory polymers

ABSTRACT This paper proposes a new thermoviscoelastic finite deformation model incorporating dual relaxation mechanisms to predict the complete thermo-mechanical response of amorphous shape memory polymers. The model is underpinned by the detailed microscopic molecular mechanism and effectively reflects the current understanding of the glass transition phenomenon. Novel evolution rules are obtained from the model to characterize the viscous flow, and a new theory named an internal stress model is introduced and combined with the dual relaxation mechanisms to capture the stress recovery. The rationality of the constitutive model is verified as the theoretical results agree well with the experimental data. Moreover, the constitutive model is further simplified to facilitate engineering applications, and it can roughly capture the characteristics of shape memory polymers.

A constitutive model and its numerical implementation for reversible behavior of shape memory hydrogels

Shape memory hydrogels (SMHs) are kinds of smart materials with great importance in many fields, such as drug release and soft robotics. In order to design the structures base on SMHs, it is necessary to reveal the mechanism of the shape memory effect and establish the constitutive model of SMHs. However, the existing constitutive models can not describe some of important mechanical behavior of SMHs, such as reversible shape memory effects. In this paper, a three-dimensional finite deformation constitutive model is developed for SMHs with reversible shape memory effects caused by the conformation transition of N-isopropylacrylamide (PNIPAM). In order to well capture its reversible shape memory effect, the polymer network for PNIPAM is decomposed into two parts, coil PNIPAM and globule PNIPAM, with different reference configurations. Then this model is implemented into a user material subroutine (UMAT) and is used for simulating the equilibrium swelling, isothermal uniaxial tension, reversible shape memory behavior and multiple shape memory cycles of SMHs. Our model is validated by comparing the simulation results with experiments. The deformation of a more complex 3D structure and a bilayer structure containing SMHs are also numerically studied which demonstrates great potential of our model in exploring the application of SMHs.

A constitutive model for metallic glasses based on two-temperature nonequilibrium thermodynamics

We develop a new finite deformation constitutive model for metallic glass within the framework of irreversible nonequilibrium thermodynamics. To consider the intrinsically out-of-equilibrium characteristics of metallic glass, its total internal energy is divided into two weakly coupled configurational and kinetic subsystems, and configurational temperature coupling with configurational degrees of freedom is introduced as a thermodynamic state variable to characterize the evolution of the disordered structure. Furthermore, the classical shear transformation zone theory is extended by reasonably considering the reverse shear transformation as a form of the relaxation of the strain field, which is stored in the elastic matrix and produced by the constraint of the matrix on shear transformation. With the help of the finite element implementation for the new model, the effectiveness of the proposed model is validated by comparing the modeling stress–strain responses of the macroscopic deformations under different temperatures and strain rates with the experimental results, and the utility of the model for predicting the shear banding behavior of metallic glasses is examined as well. The results therefore show that the constructed constitutive model can not only effectively predict the deformations of metallic glass under different ambient temperatures and applied strain rates, but also reasonably explain the mechanisms of deformation mode evolution and shear band formation.

A finite-deformation constitutive model of particle-binder composites incorporating yield-surface-free plasticity

Abstract We present a constitutive model for particle-binder composites that accounts for finite-deformation kinematics, nonlinear elasto-plasticity without apparent yield, cyclic hysteresis, and progressive stress-softening before the attainment of stable cyclic response. The model is based on deformation mechanisms experimentally observed during quasi-static monotonic and cyclic compression of mock Plastic-Bonded Explosives (PBX) at large strain. An additive decomposition of strain energy into elastic and inelastic parts is assumed, where the elastic response is modeled using Ogden hyperelasticity while the inelastic response is described using yield-surface-free endochronic plasticity based on the concepts of internal variables and of evolution or rate equations. Stress-softening is modeled using two approaches; a discontinuous isotropic damage model to appropriately describe the softening in the overall loading-unloading response, and a material scale function to describe the progressive cyclic softening until cyclic stabilization. A nonlinear multivariate optimization procedure is developed to estimate the elasto-plastic model parameters from nominal stress-strain experimental compression data. Finally, a correlation between model parameters and the unique stress-strain response of mock PBX specimens with differing concentrations of aluminum is identified, thus establishing a relationship between model parameters and material composition.

Modeling the one-way and two-way shape memory effects of semi-crystalline polymers

Polymers of two-way shape memory effects (SMEs) are of great application potential ranging from intelligent actuators to artificial muscles due to their reversible shape response on the basis of an on-off stimulus such as a cooling-heating cycle, which motivates the development of theoretical tools to support their design. In the paper, a three-dimensional phase transition finite deformation constitutive model is proposed to phenomenologically describe the underlying mechanisms of the one-way and two-way SMEs in thermal-responsive semi-crystalline polymers. A unified transition equation is proposed to depict the volume changes of crystalline phase during both crystallization and melting transitions. The crystallization-induced elongation and melting-induced contraction behaviours in two-way SMEs are assumed to be realized by an elongation-contraction element and its temperature-dependent strain evolution rules. The overall constitutive model is then implemented into Mathematica, and the model predictions are compared to and agree well with the experimental results. The paper provides an efficient method on modeling the thermal-induced two-way SMEs under stress conditions in semi-crystalline polymers and detailed discussion about the influence of several parameters such as the crosslinking density, the applied external load and the cooling/heating rate.

A Dynamic Finite-Deformation Constitutive Model for Steels Undergoing Slip, Twinning, and Phase Changes

Depending on composition, processing, microstructure, and loading mode, steels may demonstrate various inelastic deformation mechanisms, notably dislocation glide, deformation twinning, and solid-solid phase changes. Damage in the form of voids, often leading to failure by macro-cracking, is also of current interest. A finite-deformation constitutive model is constructed in order to address such mechanisms in isotropic polycrystals under static and dynamic loading, where the latter encompasses high pressures and extreme strain rates pertinent to ballistic penetration. Slip and twinning are isochoric, and their relative contributions to kinematics are not explicitly distinguished in the present application. Phase changes among, e.g., face-centered-cubic (FCC), body-centered-cubic (BCC), body-centered-tetragonal (BCT), and hexagonal (HCP) structures are admitted, with corresponding deviatoric and volumetric strains. Porosity from voids contributes to volumetric strain. A new consistent thermodynamic framework incorporating an Eulerian strain tensor and internal state variables is developed, whereby kinetic equations for slip–twinning, phase changes, and damage evolution result in contributions to dissipation. An objective rate form of the model, derived assuming small deviatoric elastic strain, is implemented numerically. The model is applied to three different primarily austenitic, medium-high Mn steels. Specifically, representations of a slip-dominated (SLIP), a TRIP (transformation-induced plasticity) and a TWIP (twinning-induced plasticity) steel are calibrated to quasi-static tension, quasi-static compression, and dynamic compression data, at both room and high temperatures. A novel functional form of material strength distinguishes hardening profiles of the different alloys. Experimental data are reasonably well represented. Model extrapolations for dynamic strength and pressure in regimes pertinent to shock compression are analyzed. Predictions for multi-axial loading of shear with simultaneous expansion or contraction demonstrate competing physical mechanisms among the alloys that could be leveraged for optimal ballistic performance.

A hygro-thermo-mechanical constitutive model for hygrothermally activated shape memory polymers under finite deformations

With regard to the application of shape memory polymers (SMPs) in smart structures, it is difficult to accurately predict their shape memory effects (SMEs) triggered by various external environments. Therefore, developing constitutive models for SMPs in multi-field environments is crucial. In this paper, a hygro-thermo-mechanical finite deformation constitutive model incorporated with structural relaxation and moisture diffusion is proposed for hygrothermally activated amorphous SMPs. One of the main novelties is that for the first time an internal state variable modeling approach is used in the model to describe the structural relaxation of hygrothermally activated SMPs in the vicinity of the glass transition. The other novelty is that the theory of moisture diffusion is applied to split the absorbed moisture into the mobile (free) and bound phases, and their respective effects on the hygro-thermo-mechanical behavior are also considered in the model. Comparisons between the model results and the test data for both, thermally activated and moisture-activated SMEs, present a good agreement.

Constitutive modeling of the rate- and temperature-dependent macro-yield behavior of amorphous glassy polymers

The rate- and temperature-dependence of the macro-yield behavior, as one of the important concerns for the investigation and application of amorphous glassy polymers, is very significant below and near the glass transition temperature (Tg). Two molecular chain microstructures, i.e., sub-entanglement (SE) and topological entanglement (TE) and their evolution, are introduced as the inherent micro-mechanisms for the rate- and temperature-dependent macro-yield performance. With the introduced entanglement microstructures-based internal variables, an elasto-viscoplastic finite deformation constitutive model of amorphous glassy polymers is proposed to describe its rate- and temperature-dependent macro-yield response. The rate-dependence of Tg and its effect on the microstructures’ thermo-mechanical state and evolution are also taken into account. Comparing with the experimental results of PMMA in literature, the proposed model shows a good capability to simulate and predict the macro-yield behavior at various strain rates (from 0.0003 s−1 to 0.1 s−1) and temperatures (from room temperature to Tg). This work illustrated that the introduction of SE and TE as the inherent micro-mechanisms is a reasonable interpretation for the macro-yield behavior of amorphous glassy polymers.

Computational shape optimisation for a gradient-enhanced continuum damage model

An isotropic gradient-enhanced damage model is applied to shape optimisation in order to establish a computational optimal design framework in view of optimal damage distributions. The model is derived from a free Helmholtz energy density enriched by the damage gradient contribution. The Karush–Kuhn–Tucker conditions are solved on a global finite element level by means of a Fischer–Burmeister function. This approach eliminates the necessity of introducing a local variable, leaving only the global set of equations to be iteratively solved. The necessary steps for the numerical implementation in the sense of the finite element method are established. The underlying theory as well as the algorithmic treatment of shape optimisation are derived in the context of a variational framework. Based on a particular finite deformation constitutive model, representative numerical examples are discussed with a focus on and application to damage optimised designs.

Modeling the thermomechanical behaviors of short fiber reinforced shape memory polymer composites

Short fiber reinforced shape memory polymer composites (SF-SMPCs) possess better mechanical properties compared to pure shape memory polymers, and in the meantime, keep the high deformability and the good shape memory properties. In the paper, a thermoviscoelastic finite deformation constitutive model is developed for thermally activated SF-SMPCs. The key of the model is the modified temperature-dependent laminate analogy theory for the prediction of the thermal-dependent effective elastic properties of the composites of different fiber volume fractions. The constitutive theory is based on a generalized three-element model consisting of a hyperelastic branch and a viscoelastic branch, in which the volume fraction dependent yielding properties are described by a modified Eying model. The developed constitutive model is then implemented into Mathematica to simulate the thermomechanical behaviors of the materials under different loadings, and the simulation results show great agreements with the experimental data. Finally, the shape memory effects of the SF-SMPCs are studied by a typical free-recovery thermomechanical cycle. The shape fixity and shape recovery speed are slightly influenced by the reinforcements, and the shape recovery ratios are almost 100% for all SF-SMPCs. The paper offers an efficient method to theoretically describe and predict the thermomechanical behaviors of SF-SMPCs.

Finite deformation constitutive model for macro-yield behavior of amorphous glassy polymers with a molecular entanglement-based internal-state variable

To describe the macro-yield behavior of amorphous glassy polymers which is significantly affected by thermo-mechanical history and deformation state, an elasto-viscoplastic constitutive model incorporating a molecular entanglement-based internal-state variable is proposed. While the classical topological entanglement governs the benchmark mechanical performances such as the pre-yield linearity and the post-yield softening plateau, the secondary short-range microstructure, named as the sub-entanglement owing to the local interaction between neighboring molecular chains, is taken as the intrinsic source of yield peak. The microstructure related deformation resistance can be decomposed into the topological entanglement related one, s˜te, and the sub-entanglement related one, s˜se, respectively. During the macro-yield procedure, s˜te is assumed as constant because the topological entanglement rarely disentangles, while s˜se evolves with the gradually dissociating sub-entanglement. The evolution equation of s˜se is presented and its physical meaning is clarified. Validated with the experimental data of amorphous glassy polymers in literature, the proposed constitutive model can not only well describe the macro-yield behavior, but also reasonably explain the effects of thermo-mechanical history and deformation state on the macro-yield behavior.

The third Sandia Fracture Challenge: peridynamic blind prediction of ductile fracture characterization in additively manufactured metal

In the context of the third Sandia Fracture Challenge (SFC3), the details of the blind predictions performed by the University of Texas team are provided in this article. Over the past two decades, the peridynamic theory has shown great promise in modeling autonomous crack nucleation and growth in materials. While peridynamics has been commonly applied to simulate failure of brittle materials, its ability in predicting ductile fracture has remained mostly untested. This fracture challenge was seen as an opportunity to assess the state of the art of the peridynamic theory in predicting the response of an additively manufactured 316L stainless steel bar with a complex geometry under the dynamic tensile experiments performed by Sandia National Laboratories. The performance of a recently proposed, generalized, ordinary finite deformation constitutive correspondence model, coupled with a recent state-based damage correspondence model was explored over this problem. For finite deformation material modeling, the classical elastoplastic framework of Simo was implemented within the ordinary correspondence theory. Damage modeling was achieved by incorporating the Johnson-Cook failure criterion using the damage correspondence framework. An iterative inverse technique was applied to calibrate the model parameters using the longitudinal and notched tensile tests data provided by Sandia National Laboratories. A blind prediction of the deformation and failure behavior of the SFC3 geometry was performed by embedding the calibrated model in a peridynamic simulation. Uncertainty was introduced into the model parameters to quantify material variability. The results are compared to the experiments conducted at Sandia National Laboratories. While our modeling approach led to qualitatively good results and a correctly predicted crack path, it underpredicted the load-carrying capacity of the structure and simulated an early fracture. Our post-experiment analysis identifies material instability issues associated with the model as the primary sources of error.

Modeling the thermomechanical behaviors of particle reinforced shape memory polymer composites

Shape memory polymer composites (SMPCs) possess superior thermomechanical properties while keep good shape memory capabilities, so SMPCs attract great research interest from academia and industry. In the paper, a modified thermoviscoelastic finite deformation constitutive model is developed to investigate the thermomechanical properties and shape recovery behaviors of particle reinforced SMPCs. The model is incorporated with the modified Adam–Gibbs model and the Eyring model to describe the structural relaxation and yielding behaviors. Also, a non-Gaussian chain molecular network model is used to capture the hyperelastic properties in the large deformation situation. Then, the constitutive model is employed to estimate the free recovery behaviors of SMPCs with different particle fractions, and the comparison between the simulation results and the test data shows good agreement. Furthermore, the model is applied to predict the constrained recovery properties and uniaxial tensile stress–strain response of the composites. The thermomechanical model provides an efficient method for designing and optimizing particle reinforced SMPCs.

Thermomechanical constitutive modeling of fiber reinforced shape memory polymer composites based on thermodynamics with internal state variables

Shape memory polymer composites (SMPCs) can be used to improve the mechanical properties of the shape memory polymers (SMPs). They also have the potential to enhance or enable new stimulus approaches and novel shape memory effects (SMEs). In this paper, a thermoviscoelastic finite deformation constitutive model is developed for thermally activated unidirectional continuous elastic fiber reinforced SMPCs. Since the structural relaxation and viscous flow mainly exist in the SMP matrix, an internal state variable modeling approach is used to describe the thermomechanical behavior of the SMPCs. Recent works mainly focus on the thermomechanical behavior of carbon fiber reinforced SMPCs in the small strain range, for the tensile tolerant strain of carbon fiber is small. The model that is developed here shows that the unidirectional continuous carbon fiber reinforced SMPCs with proper fiber inclination angle and volume fraction can also be used in finite deformation range for the first time. The finite deformation thermomechanical response of unidirectional continuous carbon fiber reinforced SMPCs with different inclination angles and volume fractions is addressed here. Therefore, this study provides useful guidance for reasonable design of unidirectional continuous elastic fiber reinforced SMPCs.

A generalized, ordinary, finite deformation constitutive correspondence model for peridynamics

A finite deformation constitutive framework for peridynamics that uses generalized Kirchhoff stress measures as intermediate quantities in computing the peridynamic material point interactions is proposed. It is shown to be a generalization of a previous finite deformation constitutive correspondence theory and has additional useful properties. The new model is shown to be an ordinary material model and therefore conserves angular momentum without additional restrictions. Additionally, it does not suffer from well-known “surface effects” present in some peridynamic material models, has improved material stability properties, and allows for inclusion of elasto-plastic behavior in a straightforward manner.

Simulation of finite-strain inelastic phenomena governed by creep and plasticity

Inelastic mechanical behavior plays an important role in many applications in science and engineering. Phenomenologically, this behavior is often modeled as plasticity or creep. Plasticity is used to represent the rate-independent component of inelastic deformation and creep is used to represent the rate-dependent component. In several applications, especially those at elevated temperatures and stresses, these processes occur simultaneously. In order to model these process, we develop a rate-objective, finite-deformation constitutive model for plasticity and creep. The plastic component of this model is based on rate-independent \(J_2\) plasticity, and the creep component is based on a thermally activated Norton model. We describe the implementation of this model within a finite element formulation, and present a radial return mapping algorithm for it. This approach reduces the additional complexity of modeling plasticity and creep, over thermoelasticity, to just solving one nonlinear scalar equation at each quadrature point. We implement this algorithm within a multiphysics finite element code and evaluate the consistent tangent through automatic differentiation. We verify and validate the implementation, apply it to modeling the evolution of stresses in the flip chip manufacturing process, and test its parallel strong-scaling performance.