Finite Strain Framework Research Articles

3D crystal plasticity analyses on the role of hard/soft inclusions in the local slip formation

The present work aims to investigate the phenomenon of local plastic slips formation in the surrounding of inclusions by means of 3D finite element analyses from a mesoscale point of view. Isotropic elastic spherical Al2O3 and MnS inclusions are modelled inside a cylindrical ferritic body subjected to tensile loading conditions with the superimposition of hydrostatic pressure. A crystal plasticity constitutive model within the framework of finite strains is adopted for the description of the matrix together with an anisotropic elasticity tensor. Several crystallographic orientations are considered, in relation to the loading direction, to observe the stress distribution and the formation of the plastic deformation in the surrounding of the inclusion. Two modelling strategies of the matrix-defect interface bond are investigated.

Finite Element Method for 3D Deformation of Hyperelastic Materials

We describe a finite element method for solving 3D problems of nonlinear elasticity theory in the framework of finite strains for a hyperelastic material. Constitutive equations written with the use of the polar and upper triangular (QR) decompositions of the deformation gradient are considered. Our method permits developing an efficient, easy-to-implement technique for the numerical analysis of the stress—strain state of any hyperelastic material.

Phase-field simulation of martensite microstructure in low-carbon steel

We present three-dimensional phase-field simulations of martensite microstructure formation in low-carbon steel. In this study, a full set of 24 Kurdjumov-Sachs symmetry variants of martensite is considered. Three different carbon compositions are investigated in order to reveal the effect of carbon content on the martensite microstructure formation. The simulations are performed using the finite strain framework which allows considering real martensite transformation strains. Using Neuber elasto-plastic approximation to the mechanical equilibrium solution, realistic stresses and strains can be obtained during martensite formation resulting in realistic mechanical driving forces for the transformation. The simulated microstructures are compared to experimental results for three carbon compositions. Good agreement between simulated and experimental results is achieved.

Electro-Chemo-Mechanics of Si-C Composite Anode Particles Using Cyclic Voltammetry Simulations

Lithium-ion batteries are being used extensively due to various beneficial characteristics such as the absence of memory effect, wider range of operating temperatures and superior operating voltages. However, they still suffer from drawbacks such as low power density and capacity fade due to the failure of electrode particles during cyclic charging and discharging. Recently, high energy density anode particles such as silicon are being considered due to their potential for enhancing battery performance. However, these materials are associated with large volumetric changes on lithiation and de-lithiation. Furthermore, the stresses associated with these deformations can interfere with the electrode’s electrochemical performance. Composite anode particles made of graphite and Si can be alternatives to enhance performance, while at the same time retain the structural integrity of the anodes.In this work, a core-shell configuration of the anode particle is modeled with a silicon core and graphite shell. A two-way coupling between mechanics and diffusion is taken into account. Since silicon undergoes a large volumetric expansion on lithiation, a finite strain framework is employed for mechanics. The influence of hydrostatic stresses and higher order stress terms on the chemical potential arising from the changing material properties is also incorporated to fully couple the deformation with the electrochemical response. The interactions between mechanics and diffusion are switched on and off through a parameter. Cyclic voltammetry simulations are then carried out to study the effect of stresses on the electrochemical response of the composite particle. Specifically, two experiments have been simulated. In the first, the composite particle was first charged and then discharged, while in the second, the opposite was done. It was found that when the stress effects were included, the response of the bulk of the particle to the influx and efflux of the particles at the surface was delayed, leading to a lower observed current density. This delayed response of the bulk is due to the accumulated charge within the silicon core, arising from its relatively lower diffusivity as compared to graphite. A sharp stress gradient occurs at the material interface, and this affects the nature of the concentration profile at it. When the stresses were considered a sharp decrease was observed at the interface, whereas the concentration profile was smooth when the effect of the stresses was neglected. These experiments were also conducted for various core radius to shell thickness ratios. It was found that for larger particles, the effect of the stresses on the current density was negligible, since the volume fraction of the silicon core, responsible for the delayed bulk response, was lesser. However, the current density in smaller particles varies significantly with and without the consideration of the stresses. These observations were the same for the 42 particle sizes that were studied for the charge-discharge and discharge-charge voltammetry simulation. Smaller charge particles allow for a greater available surface, resulting in higher current density. Therefore, it can be concluded that is essential that we consider the effect of the stress on the electrochemistry while modeling anode particles for lithium-ion batteries. Figure 1

Modelling the stress-induced multi-step martensite transformation of single crystal NiMnGa ferromagnetic shape memory alloys

Existing experimental results have shown that multi-step martensite transformation can be induced by the applied stress in single crystal NiMnGa ferromagnetic shape memory alloys (FSMAs) and the transformation property depends on the ambient temperature strongly. In this work, a crystal-plasticity based constitutive model is established to quantify such a phenomenon. Based on the crystallographic symmetry, the lattice structures and eigenstrains of austenite (A) and three different martensite phases, i.e., the modulated martensite phases with a periodicity of 5 lattice cells (5M) and a periodicity of 7 lattice cells (7M), and a non-modulated (NM) martensite phase are analyzed. A three-dimensional (3D) single crystal constitutive model is constructed in the framework of finite strain by considering five different martensite transformation processes, i.e., the transformations between A and 5M, A and 7M, A and NM, 5M and 7M, and 7M and NM phases. Thermodynamically consistent kinetic equations for the five different martensite transformation processes are proposed based on the constructed Helmholtz free energy and Clausius's dissipative inequality. The capability of the proposed model to describe the tensile stress-induced multi-step martensite transformation of single crystal NiMnGa FSMA in [100] crystallographic orientation is verified by comparing the predictions with corresponding experimental results. Further, the proposed model is used to predict the deformations of the single crystal under the tension and compression in various crystallographic orientations.

Thermomechanical Modeling of Amorphous Glassy Polymer Undergoing Large Viscoplastic Deformation: 3-Points Bending and Gas-Blow Forming.

Polymeric products are mostly manufactured by warm mechanical processes, wherein large viscoplastic deformation and the thermomechanical coupling effect are highly involved. To capture such intricate behavior of the amorphous glassy polymers, this paper develops a finite-strain and thermomechanically-coupled constitutive model, which is based on a tripartite decomposition of the deformation gradient into elastic, viscoplastic, and thermal components. Constitutive equations are formulated with respect to the spatial configuration in terms of the Eulerian Hencky strain rate and the Jaumann rate of Kirchhoff stress. Hyperelasticity, the viscoplastic flow rule, strain softening and hardening, the criterion for viscoplasticity, and temperature evolution are derived within the finite-strain framework. Experimental data obtained in uniaxial tensile tests and three-point bending tests of polycarbonates are used to validate the numerical efficiency and stability of the model. Finally, the proposed model is used to simulate the gas-blow forming process of a polycarbonate sheet. Simulation results demonstrate well the capability of the model to represent large viscoplastic deformation and the thermomechanical coupling effect of amorphous glassy polymers.

Finite Element Models of Hyperelastic Materials Based on a New Strain Measure

To construct constitutive equations for hyperelastic materials, one increasingly often proposes new strain measures, which result in significant simplifications and error reduction in experimental data processing. One such strain measure is based on the upper triangular (QR) decomposition of the deformation gradient. We describe a finite element method for solving nonlinear elasticity problems in the framework of finite strains for the case in which the constitutive equations are written with the use of the QR-decomposition of the deformation gradient. The method permits developing an efficient, easy-to-implement tool for modeling the stress–strain state of any hyperelastic material.

Ductile damage modelling with locking‐free regularised GTN model

SummaryThe major goal of this work is to develop a robust modelling strategy for the simulation of ductile damage development including crack initiation and subsequent propagation. For that purpose, a Gurson‐type model is used. This model class, as many other damage models, leads to significant material softening and must be used within a large deformation framework due to the ductile character of the materials. This leads to 2 main difficulties that should be dealt with carefully: mesh dependency and volumetric locking. In this work, a logarithmic finite strain framework is adopted in which the Gurson‐Tvergaard‐Needleman constitutive law is reformulated. Then a nonlocal formulation with regularisation of hardening variable is applied so as to solve mesh dependency and strain localization problem. In addition, the nonlocal model is combined with mixed “displacement‐pressure‐volume variation” elements to avoid volumetric locking. Thereby, a mesh‐independent and locking‐free finite strain framework suitable for the modelling of ductile rupture is established. Attention is paid to mathematical properties and numerical performance of the model. Finally, the model parameters are identified on an experimental database for a nuclear piping steel. Simulations of standard test specimens (notched tensile bars and compact tension and single edge notched tensile cracked specimens) are performed and compared to experimental results.

A multisurface model for gradient‐enhanced damage coupled to finite plasticity

AbstractA gradient‐enhanced damage formulation is coupled to isotropic plasticity in the framework of finite strains. Within the finite element method, an additional field variable representing nonlocal damage is introduced and linked to its local counterpart in order to allow a standard local formulation at the material point level. The onset of damage and plasticity is governed by damage and yield criteria respectively. This multisurface approach requires the determination of the two Lagrange multipliers. By using logarithmic strains, a formulation in principal axes in combination with a von Mises yield criterion is implemented. The damage functions are defined by means of exponential functions in order to avoid further constraints on the local damage variable. The model is able to capture a wide range of material responses, ranging from brittle to ductile damage and plasticitiy dominated behaviour. The mesh independence of the model is shown by means of representative finite element examples. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Critical stresses estimation by crystal viscoplasticity modeling of rate-dependent anisotropy of Al-rich TiAl alloys at high temperature

Determining critical stresses for different slip systems is one of the most important parts in crystal plasticity modeling of anisotropy. However, the task of finding individual critical resolved shear stress (CRSS) for every single slip system, if not impossible, is formidable and a delicate one especially if the microstructure is very complex. Slip family-based, mechanism-based and morphology-based (e.g., phase interface) slip systems classification and hence determining CRSS consistent with experimental measurements are often used in crystal plasticity. In this work, a novel approach to determining CRSS at high homologous temperature has been proposed by crystal plasticity modeling of rate-dependent anisotropy. Two-internal-variable-based phenomenological crystal viscoplasticity model is adopted for simulating isothermal, two-phase, single-crystal-like Al-rich lamellar Ti–61.8at.%Al binary alloy at high-temperature compression state ( $$1050\,^\circ \hbox {C}$$ ) by employing finite strain and finite rotation framework. To the best of authors’ knowledge, this is the first micromechanical modeling attempt with long-period superstructures. Conventional approaches related to CRSS estimation are also compared with the proposed one. Our material parameters are based on calibrating three different sets of compressive stain rate-controlled plasticity data taken from the loading of two different lamellar directions. It is revealed that the proposed approach works fine for rate-dependent anisotropy modeling, while other conventional approaches highly under- or overestimate available anisotropic experimental behavior of this alloy.

Thermodynamics-based finite strain viscoelastic-viscoplastic model coupled with damage for asphalt material

A thermodynamics based thermo-viscoelastic-viscoplastic model coupled with damage using the finite strain framework suitable for asphalt material is proposed in this paper. A detailed procedure for model calibration and validation is presented, utilizing a set of experimental measurements such as creep-recovery, constant creep, and repeated creep-recovery tests under different loading conditions. The calibrated constitutive model is able to predict the sophisticated time- and temperature- dependent responses of asphalt material, both in tension and in compression. Moreover, a scenario case study on permanent deformation (rutting) prediction of a practical asphalt pavement structure is presented in this work. This paper presents the main features of this new constitutive model for asphalt: (1) A thermodynamics-based framework developed in the large strain context to derive the specific viscoelastic, viscoplastic and damage constitutive equations; (2) A viscoelastic dissipation potential involving deviatoric and volumetric parts, in which Prony series representations of the Lamé constants are used; (3) A modified Perzyna’s type viscoplastic formulation with non-associated flow rule adopted to simulate the inelastic deformation, using a Drucker–Prager type plastic dissipation potential; (4) A specific damage model developed for capturing the evolution disparity between tension and compression. As such, the developed model presents a robust, fully coupled and validated constitutive framework that includes the major behavioral components of asphalt materials, enabling thus an optimized simulation of predicted performance under various conditions. Further development improvements to the model in continued research efforts can be to include further environmental and physico-chemical material behavior such as ageing, healing or moisture induced damage.

Viscoelastic modeling of articular cartilage under impact loading

Recent unconstrained impact tests on articular cartilage indicate that under high strain rate the tissue behaves as a nonlinear viscoelastic material and hysteresis increases with impact velocity. As the dissipation of severe energies limits the potential damage in cartilage microstructure, the deep insight into the hysteretic properties of the tissue under impact loading represents a crucial issue. A quasilinear viscoelastic approach has been recently presented under simplified assumptions, in particular, the small strain hypothesis; the Kelvin–Voigt relaxation function was used besides. The current paper aims at extending this viscoelastic formulation into the framework of finite strain, in order to thoroughly investigate its accuracy to model impact loading on articular cartilage, taking into account the large deformation arising. Paralleling many hypotheses in the small strain approach, we describe the unconstrained impact test as uniaxial compression, assuming an average Cauchy stress in cartilage that obeys Fung’s model of viscoelasticity with Kelvin–Voigt relaxation function. The comparison between the experimental data available and the theoretical predictions on the basis of the current finite strain and the original small strain approaches shows a remarkable improvement in the descriptions of both stress–strain response and energy dissipation. Finally, the model formulated allows to single out some crucial physical aspects characterizing the behavior of articular cartilage under high strain-rates.

Aspects of power law flow rules in crystal plasticity with glide-climb driven hardening and recovery

In this work we will examine three power law type flow rules that are commonly used in crystal plasticity. Glide and climb driven combined hardening/recovery effect has been considered within their respective internal variables. With finite strain framework, a highly complex intermetallic material (single crystal like Al-rich TiAl lamellar binary alloy) has been analyzed at high homologous temperature. We have adopted a novel approach of estimating critical stresses and modified evolution equation for the combined hardening and recovery. From a variety of options in identifying modeling parameters, which set of flow rule dependent parameters are meaningful is discussed here. Different related numerical aspects including slip activities are also outlined. The investigation employs three sets of compressive strain rate controlled experimental data in two lamellar directions. It is revealed that, irrespective of the type of the flow rules, two internal variables based flow rules including slip-system-level kinematic hardening variable makes significant improvement in simulating high temperature medium stress viscoplasticity with estimating reasonable material parameter set, and a specific choice of the flow rule provides better prediction capability. It is also found that the most active slip system dictates the overall plastic deformation in reproducing experimental data.

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.

Rate dependent tension-compression-asymmetry of Ti-61.8at%Al alloy with long period superstructures at 1050 °C

Al-rich TiAl alloy system has recently become a focus of interest due to its higher oxidation resistance, 20% lower density and higher (200°C more) operating temperature application possibility over conventional Ti-rich TiAl alloys. Tension-compression asymmetry of such type of high temperature material is an integral part of modern engineering problems. Rate dependent asymmetric phenomena of such alloys is specially important at high homologous temperature because of the appearance of one or more long period superstructures. However, micromechanical modeling endeavor of Al-rich family has not been attempted so far. In this work, we have presented two internal variable based phenomenological crystal viscoplasticity modeling approach for predicting rate dependent tension-compression asymmetry of single crystal like lamellar Ti-61.8at%Al binary alloy at hot compression state (1050°C) by employing finite strain and finite rotation framework. Our material parameters were based on calibrating three different sets of compressive stain rate controlled plasticity data in two lamellar directions. Based on the set of identified material parameters we have predicted qualitatively the evolution of tension compression asymmetry of this alloy. It is found that tension-compression-asymmetry evolution is anisotropic and highly pronounced throughout the deformation process influenced by the lamellar morphology and long period superstructures. Slip system level operative stresses show that the slip domination and activity, more specifically, major contributing systems are different in tension and compression.

Intercalation Driven Porosity Effects in Coupled Continuum Models for the Electrical, Chemical, Thermal and Mechanical Response of Battery Electrode Materials

We present a coupled continuum formulation for the electrostatic, chemical, thermal and mechanical processes in battery materials. Our treatment applies on the macroscopic scale, at which electrodes can be modeled as porous materials made up of active particles held together by binders and perfused by the electrolyte. Starting with the description common to the field, in terms of reaction-transport partial differential equations for ions, variants of the classical Poisson equation for electrostatics, and the heat equation, we add mechanics to the problem. Our main contribution is to model the evolution of porosity as a consequence of strains induced by intercalation, thermal expansion and mechanical stresses. Recognizing the potential for large local deformations, we have settled on the finite strain framework. We present a detailed computational study of the influence of the dynamically evolving porosity, upon ion distribution, electrostatic potential fields, charge-discharge cycles and mechanical force generated in the cell.

On the Choice of the Power Law Flow Rule and its Consequences in Crystal Plasticity

Three types of power law flow rules are commonly used in classical crystal plasticity. These laws are purely phenomenological. The foremost point is how to define operative or effective stress and drag or slip system resistance. Specific choice of the definition leads to a unique number of implications including lattice rotation and slip activities, and we will highlight a few of them. We examined these three flow rules within finite strain framework with a single crystalline Al-rich TiAl binary alloy at very high homologous temperature with three strain rate controlled experimental data . It is revealed that two internal variables based flow rules give better results with a wide variety of applicability in plasticity and related phenomena.

Extension of TOUGH-FLAC to the finite strain framework

The TOUGH-FLAC simulator for coupled thermal-hydraulic-mechanical processes modeling has been extended to the finite strain framework. In the approach selected, this extension has required modifications to the flow simulator (TOUGH2) and to the coupling scheme between the geomechanics and the flow sub-problems. In TOUGH2, the mass and energy balance equations have been extended to account for volume changes. Additionally, as large deformations are computed by FLAC3D, the geometry is updated in the flow sub-problem. The Voronoi partition needed in TOUGH2 is computed using an external open source library (Voro++) that uses the centroids of the deformed geomechanics mesh as generators of the Voronoi diagram. TOUGH-FLAC in infinitesimal and finite strain frameworks is verified against analytical solutions and other approaches to couple flow and geomechanics. Within the finite strain framework, TOUGH-FLAC is also successfully applied to a large-scale case. The extension of TOUGH-FLAC to the finite strain framework has little impact to the user as only one additional executable is needed (for Voro++), and the input files and the workflow of a simulation are the same as in standard TOUGH-FLAC. With this new provision for finite strains, TOUGH-FLAC can be used in the analysis of a wider range of engineering problems, and the areas of application of this simulator are therefore broadened.

Three-dimensional finite element modeling of ductile crack initiation and propagation

A crack initiation and propagation algorithm driven by non-local ductile damage is proposed in a three-dimensional finite strain framework. The evolution of plastic strain and stress triaxiality govern a non-local ductile damage field via constitutive equations. When the damage reaches a critical threshold, a discontinuity in the form of a crack surface is inserted into the three-dimensional continuum. The location and direction of the introduced discontinuity directly result from the damage field. Crack growth is also determined by the evolution of damage at the crack tip and the crack surface is adaptively extended in the computed direction. Frequent remeshing is used to computationally track the initiation and propagation of cracks, as well as to simultaneously maintain a good quality of the finite elements undergoing large deformations. This damage driven remeshing strategy towards fracture allows one to simulate arbitrary crack paths in three-dimensional evolving geometries. It has a significant potential for a wide range of industrial applications. Numerical examples are solved to demonstrate the ability of the proposed framework.

Free and forced vibration nonlinear analysis of a microbeam using finite strain and velocity gradients theory

A nonlinear finite strain and velocity gradient framework is formulated for the Euler–Bernoulli beam theory. This formulation includes finite strain and the strain gradient within the strain energy generalization as well as velocity and its gradient within the kinetic energy generalization. Consequently, static and kinetic internal length scales are developed to capture size effects. The governing equation with initial and boundary conditions is obtained using the variational approach. Free and forced vibration of a simply supported nanobeam is studied for different values of static and kinetic length scales using the method of multiple scales.