Finite Strain Constitutive Model Research Articles

A 3D finite strain constitutive model for shape memory polymers combined viscoelasticity and storage strain

A 3D finite strain constitutive model combined viscoelasticity and storage strain for shape memory polymers (SMPs) is proposed. SMPs are phenomenally regarded as a homogeneous material of the viscoelastic glassy phase and hyperelastic rubbery phase in this model. Based on energy decomposition, the constitutive equation is derived using Clausius inequality and unified Helmholtz free energy. Then, an analytical expression for storage strain is presented, significantly simplifying the prediction of shape memory recovery process. The model is implemented in MATLAB and the commercial finite element software package ABAQUS using the UMAT subroutine. Our additional DMA tests and experimental data from the literature for various SMPs with different shape structures are used to verify the proposed model. All simulation results demonstrate the effectiveness of the proposed model in predicting the shape memory effect and complex thermomechanical behaviors of various SMPs.

A finite strain integral model for the creep behavior of vaginal tissue

Vaginal delivery causes significant stretching of the vagina and surrounding muscles, potentially leading to the development of pelvic floor disorders and other maternal morbidities. Despite the extended duration of labor, little experimental and theoretical work has been done to characterize the long-term viscoelastic behavior of vaginal tissue. To mathematically capture the creep response of rat vaginal tissue measured, this study presents a new anisotropic finite-strain constitutive model within the single integral Pipkin-Rogers viscoelastic framework. The constitutive parameters are computed by curve-fitting the model to strain versus time data collected from ex vivo inflation testing along the two primary anatomical directions of the vagina, the longitudinal and circumferential directions. The results showed good agreement between theory and experiments, suggesting that the proposed model could be used to advance our understanding of the time-dependent deformations that are experienced by the vagina during delivery. This modeling framework represents a first step toward the development of accurate computational tools that can predict the safety of vaginal deliveries, reducing unnecessary Cesarean sections and their related complications.

Numerical simulation of three dimensional concrete printing based on a unified fluid and solid mechanics formulation

Deformation control constitutes one of the main technological challenges in three dimensional (3D) concrete printing, and it presents a challenge that must be addressed to achieve a precise and reliable construction process. Model-based information of the expected deformations and stresses is required to optimize the construction process in association with the specific properties of the concrete mix. In this work, a novel thermodynamically consistent finite strain constitutive model for fresh and early-age 3D-printable concrete is proposed. The model is then used to simulate the 3D concrete printing process to assess layer shapes, deformations, forces acting on substrate layers and prognoses of possible structural collapse during the layer-by-layer buildup. The constitutive formulation is based on a multiplicative split of the deformation gradient into elastic, aging and viscoplastic parts, in combination with a hyperelastic potential and considering evolving material properties to account for structural buildup or aging. One advantage of this model is the stress-update-scheme, which is similar to that of small strain plasticity and therefore enables an efficient integration with existing material routines. The constitutive model uses the particle finite element method, which serves as the simulation framework, allowing for modeling of the evolving free surfaces during the extrusion process. Computational analyses of three printed layers are used to create deformation plots, which can then be used to control the deformations during 3D concrete printing. This study offers further investigations, on the structural level, focusing on the potential structural collapse of a 3D printed concrete wall. The capability of the proposed model to simulate 3D concrete printing processes across the scales—from a few printed layers to the scale of the whole printed structure—in a unified fashion with one constitutive formulation, is demonstrated.

Insight of electro-chemo-mechanical process inside integrated configuration of composite cathode for solid-state batteries

The complicated electro-chemo-mechanical process that occurs inside the composite cathode for solid-state batteries (SSBs), is of first importance to be insighted for the development of SSBs to seek higher energy density. Herein, exampled with layered transition-metal oxide of LiNixCoyMn1-x-yO2 (NCM), an electro-chemo-mechanical model containing electrochemical kinetics, finite-strain constitutive model and cohesive zone model was built to uncover the impact of ionic conductivity and Young's modulus (E) of solid-state electrolyte (SE) on the electro-chemo-mechanical process inside composite cathode and the intergranular failure of single cathode particle. The intergranular failure of NCM particles is powerfully determined by the Young's modulus of SE and the primary particle size, which is postponed by the coarse-primary NCM with soft SE of E=∼2 GPa. Compared with Young's modulus, increasing the ionic conductivity can uniform the distribution of both Li-ion and stress in the whole composite NCM cathode, realizing improved electrochemical performance with larger normalized capacity and lower the interfacial impendence. Hence, high-adequate ionic conductivity of 5 × 10−4 S cm−1 and soft mechanical property of E=∼2 GPa can be proposed as the guideline of SE for great electrochemical performance with prolongated lifespan of composite NCM cathode, paving an avenue to foster the application of SSBs.

A constitutive model of cyclic plasticity with Lode dependence for structural steels

A finite strain constitutive model is proposed to represent the cyclic elastoplastic response of structural steels. The model simulates the Lode angle dependence of plastic response, which has been recently observed in experimental studies. The model combines Frederick-Armstrong kinematic hardening rules with an exponential isotropic hardening rule, and a non-associated plastic flow rule. A Lode angle-dependent coefficient is defined to modify the isotropic hardening response and determine the shape of yield surface, with respect to the conventional von Mises surface. Stabilization of hysteresis for cyclic plasticity is achieved through a memory surface, defined in the plastic strain space. Analytical formulas for an implicit Return-Mapping algorithm are derived to realize the numerical implementation of this model via a UMAT user subroutine. The effect of Lode dependence of the plasticity response of structural steel Q235 is investigated through monotonic and cyclic experiments on flat shear specimens. A process for parameter identification is outlined, and model predictions are validated through comparison with experimentally-measured force–displacement curves on uniaxial and shear specimens.

An Extended Three-Dimensional Finite Strain Constitutive Model for Shape Memory Alloys

AbstractA 3D finite-strain constitutive model for shape memory alloys (SMAs) is proposed. The model can efficiently describe reversible phase transformation from austenite to self-accommodated and/or oriented martensite, (re)orientation of martensite variants, minor loops, latent heat effects, and tension–compression asymmetry based on the Eulerian logarithmic strain and the corotational logarithmic objective rate. It further accounts for smooth thermomechanical response; temperature dependence of the critical force required for (re)orientation, temperature, and load dependence of the hysteresis width; and asymmetry between forward and reverse phase transformation, and it is flexible enough to address the deformation response in the concurrent presence of several phases, i.e., when austenite, self-accommodated, and oriented martensite co-exist in the microstructure. The ability of the proposed model to describe the aforementioned deformation response characteristics of SMAs under multiaxial, thermomechanical, and nonproportional loading relies on the set of three independent internal variables, i.e., the average volume fraction of martensite variants, their preferred direction, and the magnitude of the induced inelastic strain, which further allow for an implicit description of a fourth internal variable, the volume fraction of oriented as opposed to self-accommodated martensite. The calibration of the model and its numerical implementation in an efficient scheme are presented. The model is validated against experimental results associated with complex thermomechanical paths, including tension/compression/torsion experiments, and the efficiency of its numerical implementation is verified with simulations of the response of a biomedical superelastic SMA stent and an SMA spring actuator.

Finite Strain Constitutive Modelling of Shape Memory Alloys Considering Partial Phase Transformation with Transformation-Induced Plasticity

This paper presents a unified modelling effort to describe partial phase transformation during cyclic thermo-mechanical loading in Shape Memory Alloys (SMA). To this purpose, a three-dimensional (3D) finite strain constitutive model considering TRansformation-Induced Plasticity (TRIP) is combined with a modified hardening function to enable the accurate and efficient prediction of partial transformations during cyclic thermo-mechanical loading. The capabilities of the proposed model are demonstrated by predicting the behavior of the material under pseudoelastic and actuation operation using finite element analysis. Numerical results of the modified model are presented and compared with the original model without considering the partial transformation feature as well as with uniaxial actuation experimental data. Various aspects of cyclic material behavior under partial transformation are analyzed and discussed for different SMA systems.

A finite strain thermomechanically-coupled constitutive model for phase transformation and (transformation-induced) plastic deformation in NiTi single crystals

A 3D finite-strain constitutive model for the deformation response of NiTi at the single crystal level is proposed. The model accounts for reversible phase transformation from austenite to martensite habit plane variants, plastic deformation in the austenite phase, and rate effects induced from latent heat. It is developed within the formalism of irreversible thermodynamics with internal state variables based on the Eulerian logarithmic strain and its corrotational objective rate. The inelastic deformation is defined as an average over a representative volume element as classical in the micromechanics-based modeling approach. Transformation-induced plastic deformation is viewed as a mechanism for accommodation of the local deformation incompatibility at the austenite–martensite interface. It is accounted for by introducing an interaction term in the free energy, which is described through the Eshelby tensor by regarding the habit plane variants as ellipsoidal inclusions embedded in the austenite matrix, in order to accurately reflect the internal stress states that contribute to dislocation slipping. The numerical implementation of the model in an efficient scheme and its calibration are described in detail. The proposed model is validated by comparing simulations with available experimental data in single NiTi crystals. Numerical simulations of polycrystals are performed to obtain an insight into the interaction between phase transformation and plastic deformation induced by intergrannular constraints. The efficiency of the numerical implementation of the model is verified by simulations of indentation tests.

A finite-strain thermomechanical behavior model for iron-based shape memory alloys accounting for coupling between phase transformation and plastic slip

Shape Memory Alloys (SMAs) have the main interesting property to recover an inelastic strain induced by martensitic transformation. The initial shape can be recovered directly after unloading or with the application of an additional heating. Iron-based SMAs (Fe-SMAs) are characterized by a high coupling between phase transformation and plastic slip at low temperatures under small stress levels. Thermomechanical constitutive models describing such coupling developed based on small-strains are not suitable for higher loading levels. This motivates the proposed development of a finite-strain constitutive model for Fe-SMAs considering thermomechanical coupling between phase transformation and plastic slip, and by extending the small-strain model within a finite-strain thermodynamical framework in order to describe large strains mainly induced by plastic hardening in Fe-SMA material point. The model here has two internal variables (volume fraction of martensite and the accumulative plastic strain). It is based on the assumption of the local multiplicative split of the deformation gradient into elastic and inelastic parts with a total Lagrangian formulation. The inelastic deformation gradient splits also into a transformation and a plastic parts. The developed model is implemented into the commercial software Matlab. The results obtained for thermomechanical loadings are discussed and, a good agreement with experimental results is also observed.

Fiber-constrained dielectric elastomer composites: Finite deformation response and instabilities under non-aligned loadings

This paper is concerned with the development of finite-strain constitutive models for electro-active composites consisting of initially aligned, rigid dielectric fibers of elliptical cross-section that are distributed randomly in a dielectric elastomer matrix. For this purpose, we make use of a variational approach that partially decouples the mechanical and electrostatic contributions to the overall energy and leads to a minimum principle for the average orientation of the fibers. The resulting macroscopic electroelastic constitutive model accounts for electric and mechanical torques on the fibers, as well as for the microstructure and its evolution under finite deformations. In particular, the model characterizes the rotation of the fibers for general externally applied electromechanical loadings, predicting bifurcation instabilities for the special case of aligned loadings. To elucidate the main features of the model, we consider the application to a dielectric elastomer composite actuator and investigate the effects of fiber aspect ratio and initial fiber orientation on its macroscopic response. In addition, the new results are compared with the predictions of an earlier model approximating the fiber rotations by purely mechanical effects.

Compressibility, Damage, and Age-Hardening Effects of Solid Propellants Using Finite Strain Constitutive Model

Abstract In this paper, the finite strain viscoelastic constitutive model for particulate composite solid propellants is proposed considering strain rate, large deformation/large strain, thermorheological behavior, stress softening due to microstructural damage, compressibility, and chemical age hardening. The compressible Mooney–Rivlin hyperelastic strain energy density function is used along with the standard model of viscoelasticity. To model the compressibility, the dilatational strain energy is taken as the hyperbolic function of the determinant of deformation gradient. The stress-softening phenomenon during cyclic loading (Mullin's effect) due to microstructural damage is described by an exponential function of the current magnitude of intensity of strain and its previous maximum value. The variation of material properties with time are studied using the isothermal accelerated aging technique through simulation and experimental investigation. The comparison of predictions based on the proposed model with the uniaxial experimental data demonstrates that the proposed model successfully captures the observed behavior of the solid propellants.

Modeling Dynamic Anisotropic Behaviour and Spall Failure in Commercial Aluminium Alloys AA7010

This paper presents a finite strain constitutive model to predict a complex elastoplastic deformation behaviour involves very high pressures and shockwaves in orthotropic materials of aluminium alloys. The previous published constitutive model is used as a reference to start the development in this work. The proposed formulation that used a new definition of Mandel stress tensor to define Hill’s yield criterion and a new shock equation of state (EOS) of the generalised orthotropic pressure is further enhanced with Grady spall failure model to closely predict shockwave propagation and spall failure in the chosen commercial aluminium alloy. This hyperelastic-plastic constitutive model is implemented as a new material model in the Lawrence Livermore National Laboratory (LLNL)-DYNA3D code of UTHM’s version, named Material Type 92 (Mat92). The implementations of a new EOS of the generalised orthotropic pressure including the spall failure are also discussed in this paper. The capability of the proposed constitutive model to capture the complex behaviour of the selected material is validated against range of Plate Impact Test data at 234, 450 and 895 ms−1 impact velocities.

An anisotropic elastoplastic constitutive formulation generalised for orthotropic materials

This paper presents a finite strain constitutive model to predict a complex elastoplastic deformation behaviour that involves very high pressures and shockwaves in orthotropic materials using an anisotropic Hill’s yield criterion by means of the evolving structural tensors. The yield surface of this hyperelastic–plastic constitutive model is aligned uniquely within the principal stress space due to the combination of Mandel stress tensor and a new generalised orthotropic pressure. The formulation is developed in the isoclinic configuration and allows for a unique treatment for elastic and plastic orthotropy. An isotropic hardening is adopted to define the evolution of plastic orthotropy. The important feature of the proposed hyperelastic–plastic constitutive model is the introduction of anisotropic effect in the Mie–Gruneisen equation of state (EOS). The formulation is further combined with Grady spall failure model to predict spall failure in the materials. The proposed constitutive model is implemented as a new material model in the Lawrence Livermore National Laboratory (LLNL)-DYNA3D code of UTHM’s version, named Material Type 92 (Mat92). The combination of the proposed stress tensor decomposition and the Mie–Gruneisen EOS requires some modifications in the code to reflect the formulation of the generalised orthotropic pressure. The validation approach is also presented in this paper for guidance purpose. The $${\varvec{\psi }}$$ tensor used to define the alignment of the adopted yield surface is first validated. This is continued with an internal validation related to elastic isotropic, elastic orthotropic and elastic–plastic orthotropic of the proposed formulation before a comparison against range of plate impact test data at 234, 450 and $${\mathrm {895\,ms}}^{\mathrm {-1}}$$ impact velocities is performed. A good agreement is obtained in each test.

A multi-scale homogenization model for fine-grained porous viscoplastic polycrystals: I – Finite-strain theory

We make use of the recently developed iterated second-order homogenization method to obtain finite-strain constitutive models for the macroscopic response of porous polycrystals consisting of large pores randomly distributed in a fine-grained polycrystalline matrix. The porous polycrystal is modeled as a three-scale composite, where the grains are described by single-crystal viscoplasticity and the pores are assumed to be large compared to the grain size. The method makes use of a linear comparison composite (LCC) with the same substructure as the actual nonlinear composite, but whose local properties are chosen optimally via a suitably designed variational statement. In turn, the effective properties of the resulting three-scale LCC are determined by means of a sequential homogenization procedure, utilizing the self-consistent estimates for the effective behavior of the polycrystalline matrix, and the Willis estimates for the effective behavior of the porous composite. The iterated homogenization procedure allows for a more accurate characterization of the properties of the matrix by means of a finer “discretization” of the properties of the LCC to obtain improved estimates, especially at low porosities, high nonlinearties and high triaxialities. In addition, consistent homogenization estimates for the average strain rate and spin fields in the pores and grains are used to develop evolution laws for the substructural variables, including the porosity, pore shape and orientation, as well as the “crystallographic” and “morphological” textures of the underlying matrix. In Part II of this work has appeared in Song and Ponte Castañeda (2018b), the model will be used to generate estimates for both the instantaneous effective response and the evolution of the microstructure for porous FCC and HCP polycrystals under various loading conditions.

Modelling Dynamic Behaviour and Spall Failure of Aluminium Alloy AA7010

A finite strain constitutive model to predict the dynamic deformation behaviour of Aluminium Alloy 7010 including shockwaves and spall failure is developed in this work. The important feature of this newly hyperelastic-plastic constitutive formulation is a new Mandel stress tensor formulated using new generalized orthotropic pressure. This tensor is combined with a shock equation of state (EOS) and Grady spall failure. The Hill’s yield criterion is adopted to characterize plastic orthotropy by means of the evolving structural tensors that is defined in the isoclinic configuration. This material model was developed and integration into elastic and plastic parts. The elastic anisotropy is taken into account through the newly stress tensor decomposition of a generalized orthotropic pressure. Plastic anisotropy is considered through yield surface and an isotropic hardening defined in a unique alignment of deviatoric plane within the stress space. To test its ability to describe shockwave propagation and spall failure, the new material model was implemented into the LLNL-DYNA3D code of UTHM’s. The capability of this newly constitutive model were compared against published experimental data of Plate Impact Test at 234m/s, 450m/s and 895m/s impact velocities. A good agreement is obtained between experimental and simulation in each test.

A thermomechanically coupled finite-strain constitutive model for cyclic pseudoelasticity of polycrystalline shape memory alloys

This paper presents a new 3D thermomechanical finite-strain constitutive model for cyclic pseudoelasticity of polycrystalline shape memory alloys (SMAs). The model considers four primary characteristics related to the cyclic behavior of SMA that have not been integrally addressed within the finite-strain framework: (i) large accumulated residual strain that results from the residual martensite and dislocations slipping during cycling; (ii) degeneration of pseudoelasticity and hysteresis loop due to the increase of dislocation density and internal stresses with the number of cycles; (iii) rate dependence that can be attributed to the thermomechanical coupling effect; (iv) evolution of the phase transformation from abrupt to smooth transition, as a consequence of the diversified crystallographic orientations of the grains, the heterogeneity of internal stresses, and the presence of non-transforming precipitates during cycling. Based on the decomposition of finite Hencky strain into elastic, transformation, residual and thermal components, the model is constructed within a thermodynamically consistent framework. Evolution equations associated with the internal variables are derived from the reduced form of energy balance, the Clausius-Duhem form of entropy inequality, and a Helmholtz free energy function that includes elastic, thermal, interaction and constraint energies. The model is used to simulate the cyclic tensile experiments on NiTi wire at different loading rates. The good agreement of the model predictions against the experimental data demonstrates the capabilities of the proposed model to well describe cyclic pseudoelasticity of polycrystalline SMAs, and to capture the aforementioned characteristics. Furthermore, in order to demonstrate the capability of the cyclic model to solve multi-axial problems, a finite elements simulation of a SMA torsion spring undergoing large strains and rotations resulting in local multi-axial non-proportional stress and strain evolutions is performed.

A finite-strain constitutive model for anisotropic shape memory alloys

This paper presents a three-dimensional (3D) model to simulate self-accommodation, anisotropic martensitic transformation/orientation, reorientation of martensite variants, asymmetry in tension-compression and phase-change-dependent elastic properties in shape memory alloys (SMAs) within a finite-stain regime. The model is developed based on a multiplicative decomposition of the deformation gradient into elastic and inelastic parts by satisfying the second law of thermodynamics in sense of Clausius-Duhem inequality. The mathematical equations are derived in terms of symmetric tensors simplifying the constitutive relations. The finite-strain model is linearized into the small-strain regime preserving the materially non-linear feature. A description of the time-discrete form of the proposed model and its associated solution algorithm is presented. Numerical simulations of the mechanical behaviors of highly-textured NiTi 3D printed parts, wires and helical springs subjected to simple and complex loadings are performed and compared with experiments. Qualitative and quantitative correlation is observed between simulations and experiments to verify the predictive capabilities of the model and the solution procedure. It is also shown that the finite-strain modeling is essential for accurate prediction of SMA behaviors when deformations are prominent. Due to the absence of similar models in the specialized literature, this paper will fill a gap in the state of the art of this problem, and provide a computationally efficient tool for design and analysis of highly-textured SMA devices under complex loadings.

A 3D finite-strain-based constitutive model for shape memory alloys accounting for thermomechanical coupling and martensite reorientation

The paper presents a finite-strain constitutive model for shape memory alloys (SMAs) that accounts for thermomechanical coupling and martensite reorientation. The finite-strain formulation is based on a two-tier, multiplicative decomposition of the deformation gradient into thermal, elastic, and inelastic parts, where the inelastic deformation is further split into phase transformation and martensite reorientation components. A time-discrete formulation of the constitutive equations is proposed and a numerical integration algorithm is presented featuring proper symmetrization of the tensor variables and explicit formulation of the material and spatial tangent operators involved. The algorithm is used for finite element analysis of SMA components subjected to various loading conditions, including uniaxial, non-proportional, isothermal and adiabatic loading cases. The analysis is carried out using the FEA software Abaqus by means of a user-defined material subroutine, which is then utilized to simulate a SMA archwire undergoing large strains and rotations.

Modelling inelastic behaviour of orthotropic metals in a unique alignment of deviatoric plane within the stress space

A finite strain constitutive model to predict the deformation behaviour of orthotropic metals is developed in this paper. The important features of this constitutive model are the multiplicative decomposition of the deformation gradient and a new Mandel stress tensor combined with the new stress tensor decomposition generalized into deviatoric and spherical parts. The elastic free energy function and the yield function are defined within an invariant theory by means of the structural tensors. The Hill’s yield criterion is adopted to characterize plastic orthotropy, and the thermally micromechanical-based model, Mechanical Threshold Model (MTS) is used as a referential curve to control the yield surface expansion using an isotropic plastic hardening assumption. The model complexity is further extended by coupling the formulation with the shock equation of state (EOS). The proposed formulation is integrated in the isoclinic configuration and allows for a unique treatment for elastic and plastic anisotropy. The effects of elastic anisotropy are taken into account through the stress tensor decomposition and plastic anisotropy through yield surface defined in the generalized deviatoric plane perpendicular to the generalized pressure. The proposed formulation of this work is implemented into the Lawrence Livermore National Laboratory-DYNA3D code by the modification of several subroutines in the code. The capability of the new constitutive model to capture strain rate and temperature sensitivity is then validated. The final part of this process is a comparison of the results generated by the proposed constitutive model against the available experimental data from both the Plate Impact test and Taylor Cylinder Impact test. A good agreement between experimental and simulation is obtained in each test.

A finite strain constitutive model for non-quadratic yield criteria and nonlinear kinematic/isotropic hardening: application to sheet metal forming

In this paper, a finite strain material model for complex plastic anisotropy with nonlinear isotropic and kinematic hardening is consistently derived. The model is based on the classical multiplicative decomposition of the deformation gradient and derived in a thermodynamically consistent way. An important new aspect of the work is the straightforward implementation of general and anisotropic yield criteria into a constitutive model, which is entirely formulated in the reference configuration. Nevertheless, and for the sake of illustrating the potential of the model, in this work a Barlat-type (Yld2004-18p) yield criterion is employed. The kinematic hardening formulation follows the continuum mechanical extension of the classical rheological model of Armstrong–Frederick hardening. The numerical integration of the evolution equations is carried out using the exponential map approach, which is able to preserve plastic incompressibility. For numerical efficiency purposes, the exponential tensor functions are evaluated in a closed form using the spectral decomposition, and special attention is given to the preservation of the internal variables’ symmetry. The model is assessed by means of several numerical simulations for anisotropic materials at finite strains, including sheet metal forming processes with comparison to experimental data.