Decomposition Of Deformation Gradient Research Articles

A Multiphysics Thermoelastoviscoplastic Damage Internal State Variable Constitutive Model including Magnetism.

We present a macroscale constitutive model that couples magnetism with thermal, elastic, plastic, and damage effects in an Internal State Variable (ISV) theory. Previous constitutive models did not include an interdependence between the internal magnetic (magnetostriction and magnetic flux) and mechanical fields. Although constitutive models explaining the mechanisms behind mechanical deformations caused by magnetization changes have been presented in the literature, they mainly focus on nanoscale structure-property relations. A fully coupled multiphysics macroscale ISV model presented herein admits lower length scale information from the nanoscale and microscale descriptions of the multiphysics behavior, thus capturing the effects of magnetic field forces with isotropic and anisotropic magnetization terms and moments under thermomechanical deformations. For the first time, this ISV modeling framework internally coheres to the kinematic, thermodynamic, and kinetic relationships of deformation using the evolving ISV histories. For the kinematics, a multiplicative decomposition of deformation gradient is employed including a magnetization term; hence, the Jacobian represents the conservation of mass and conservation of momentum including magnetism. The first and second laws of thermodynamics are used to constrain the appropriate constitutive relations through the Clausius-Duhem inequality. The kinetic framework employs a stress-strain relationship with a flow rule that couples the thermal, mechanical, and magnetic terms. Experimental data from the literature for three different materials (iron, nickel, and cobalt) are used to compare with the model's results showing good correlations.

An anisotropic finite strain elastoplastic model considering different plastic spin effects on the intermediate configuration

Large deformation anisotropic elastoplastic models have important application background and research value in engineering and materials fields. In this paper, a macroscopic phenomenological model of elastoplastic anisotropic deformation considering plastic spin is proposed. The model is based on the multiplication decomposition of deformation gradient. The free energy function is expressed as the isotropic function of the strain and the structural tensor on the intermediate configuration, which is a push-forward of initial configuration by using plastic deformation gradient. Advantageously, the non-equilibrium free energy function remains invariant under the superimposed rigid body rotation on the intermediate configuration, which is due to the non-uniqueness of the multiplicative decomposition of deformation gradient. However, the rate of superimposed rigid body rotation has effects on the model. The effects are discussed by considering three different spin assumptions. Numerical simulations show that the plastic spin has influence on the calculation results. Therefore, the plastic spin assumption should be carefully selected in the practical application of elastoplastic anisotropic model.

Eulerian finite element implementations of a dislocation density-based continuum model

In Eulerian finite element simulations, the mesh moves relative to the material. After every change of position between the mesh and the material, the state variables are interpolated to the new mesh position, which is referred to as advection. Large strain crystal plasticity models are based on the multiplicative decomposition of the total deformation gradient. The stress is evaluated as a function of the thermoelastic strain, temperature, and other state variables. Advection of tensor quantities, such as the strain, is coupled with possibly significant advection errors. In an effort to reduce the advection errors, we develop two rate forms of an established dislocation density-based continuum model. To that end, we replace the multiplicative decomposition of the deformation gradient with the additive decomposition of the velocity gradient, and define the stress rate instead of the total stress. The Eulerian implementation is compared with Lagrangian calculations, and two numerical examples with severe deformation levels are presented.

Modelling intralaminar damage mechanisms in fibre reinforced polymers at finite strains

An approach to predict intralaminar failure mechanisms in laminate fibre-reinforced composites, at finite strains, is proposed. Transverse matrix cracking is described by a smeared crack model, activated by an invariant-based failure criterion, where an embedded cohesive crack is considered in the continuum description. A deformation gradient decomposition based on homogenisation considerations is developed for a more objective finite strain kinematic description of transverse cracks. The local equilibrium problem is established in the reference configuration to naturally include re-orientation of the fracture plane, and the employed cohesive law is able to deal with non-monotonic loading. Moreover, a continuum damage model formulated in terms of the Green–Lagrange strain tensor is used for longitudinal failure. The simulation of an open-hole multi-directional specimen subjected to tension illustrates the ability of the proposed constitutive equations to capture experimental observations.

Mechanics and modeling of cold rolling of polymeric films at large strains — A rate-independent approach

Recently a new phenomenon of bonding polymeric films in solid-state, via symmetric rolling, at ambient temperatures (≈20 °C) well below the glass transition temperature (Tg≈78°C) of the polymer has been reported. In this new type of bonding, polymer films are subject to plane strain active bulk plastic compression between the rollers during deformation. Here, we analyze these plane strain cold rolling processes, at large strains but slow strain rates, by finite element modeling. We find that at low temperatures, slow strain rates, and moderate thickness reductions during rolling (at which the Bauschinger effect can be neglected for the particular class of polymeric films studied here), the task of material modeling is greatly simplified and enables us to deploy a computationally efficient, yet accurate, finite deformation rate-independent elastic–plastic material behavior (with the inclusion of isotropic-hardening). The finite deformation elastic–plastic material behavior based on (i) the additive decomposition of stretching tensor ( D = De+Dp, i.e., a hypoelastic formulation) with incrementally objective time integration and, (ii) multiplicative decomposition of deformation gradient (F=FeFp) into elastic and plastic parts, are programmed and carried out for cold rolling within Abaqus Explicit. Frictional interactions are modeled using a consistent rate-independent Coulombic law. Predictions from both formulations, i.e., hypoelastic and multiplicative decomposition, closely match the experimentally observed rolling loads. No specialized hyperelastic/visco-plastic material model is required to describe the behavior of the particular blend of polymeric films under the conditions described here, thereby significantly speeding up the computation for steady-state rolling simulations. It is revealed that under the deformation conditions when principal axes show negligible rotation, hypoelastic formulation can be valid at large elastic stretches. Moreover, the use of classical rigid-plastic modeling (often applicable to metals) is found to greatly underestimate the rolling loads for polymers due to large elastic stretches in the polymer films at large strains. Deformation aspects of solid-state polymeric sheets presented here are expected to facilitate the development of new processes involving (or related to) cold roll-bonding of polymers.

Isomorphism Continuum Stored Energy Functional for Finite Thermoelastic Deformation

Continuum mechanics for isotropic finite thermoelastic deformations have been reviewed. Thermal effects on mechanical responses of rubbers have been captured by the isomorphism continuum stored energy (CSE) functional with the multiplicative decomposition of deformation gradient while preserving the structure of symmetry for finite structural deformation. The CSE finite thermoelastic model fits and predicts experimental data of SR and NR-C60 rubbers at different external temperatures. For internal temperature effects of both NR and NR-SIC rubbers, the CSE finite thermoelastic model of stored energy and entropy, along with the newly developed CTE and CI models, fits both nominal stress-stretch and temperature change-stretch experimental data in uniaxial extension tests.

On the direct and reverse multiplicative decompositions of deformation gradient in nonlinear anisotropic anelasticity

In this paper we discuss nonlinear anisotropic anelasticity formulated based on the two multiplicative decompositions F=FeFa and F=FaFe. Using the Bilby–Kröner–Lee decomposition F=FeFa one can define a Riemannian material manifold (the natural configuration of an anelastic body) whose metric explicitly depends on the anelastic deformation Fa. We call this the global material intermediate configuration. Deformation is a map from this Riemannian manifold to the flat ambient space. Using the reverse decomposition F=FaFe, the reference configuration is a (flat) submanifold of the Euclidean ambient space, while the global intermediate configuration is a Riemannian manifold whose metric explicitly depends on the elastic deformation Fe. We call this the global spatial intermediate configuration. We show that the direct F=FeFa and reverse F=FaFe decompositions correspond to the same anelastic motion if and only if Fe and Fe are equal up to local isometries of the reference configuration. We discuss the constitutive equations of anisotropic anelastic solids in terms of both intermediate configurations. It is shown that the two descriptions of anelasticity are equivalent in the sense that the Cauchy stresses calculated using them are identical. We note that, unlike isotropic solids, for an anisotropic solid the material metric is not sufficient for describing the constitutive behavior of the solid; the energy function explicitly depends on Fa (or Fa) through the structural tensors.

Constitutive modeling of damage-induced stress softening in electro-magneto-viscoelastic materials

This work presents a continuum physics-based constitutive model for a damaged electro-magneto-viscoelastic (EMV) material class. An extended theory of electro-magneto-viscoelasticity is employed by incorporating the balance laws of micro-forces for continuum damage in physical and material space. Multiplicative decomposition of the deformation gradient tensor into elastic and viscous parts is incorporated to characterize the rheological behavior of EMV materials. The incorporated deformation gradient decomposition enables the intermediate configuration to set up the evolution equation of EMV deformation. The overall stress is disintegrated into equilibrium stress and viscosity-induced overstresses in the proposed modeling framework. In addition, an alternative invariant-based damage function is introduced to incorporate the damage-induced stress softening into the proposed model. The findings of the model agree well with the existing experimental results. At last, the present study successfully enables the damage-induced stress softening in electro-magneto-viscoelasticity with the least material parameters.

Load step reduction for adjoint sensitivity analysis of finite strain elastoplasticity

In this paper, load step reduction techniques are investigated for adjoint sensitivity analysis of path-dependent nonlinear finite element systems. In particular, the focus is on finite strain elastoplasticity with typical hardening models. The aim is to reduce the computational cost in the adjoint sensitivity implementation. The adjoint sensitivity formulation is derived with the multiplicative decomposition of deformation gradient, which is applicable to finite strain elastoplasticity. Two properties of adjoint variables are investigated and theoretically proved under certain prerequisites. Based on these properties, load step reduction rules in the sensitivity analysis are discussed. The efficiency of the load step reduction and the applicability to isotropic hardening and kinematic hardening models are numerically demonstrated. Examples include a small-scale cantilever beam structure and a large-scale conrod structure under huge plastic deformations.

Thermo-mechanical FDEM model for thermal cracking of rock and granular materials

A coupled thermo-mechanical (TM) model based on the combined finite-discrete element method (FDEM) was proposed to simulate the thermal cracking of rock and granular materials. For heat conduction, it links continuous and contact heat transfer simultaneously through the continuum and discrete thermal elements, and can handle thermal contacts explicitly. For thermally induced stresses, the new TM model uses the concept of multiplicative decomposition of deformation gradient, which is consistent with the finite strain theory used in FDEM – for handling large strains and rotations. For thermal cracking, the coupling of the new TM model occurs in two directions, such as cracking in mechanical calculation affects thermal conduction, and friction caused by contact interactions generates heat, thus can capture the dynamic process of thermally induced fracturing. Four verification examples were given to validate its performance in simulating the thermal expansion in the steady-state, thermal stress in transient states, thermal cracking, and complex-shaped multibody thermal contact. The new TM model exhibits attractive features: robust to capture heat transfer for multibody thermal contact problems; better handle large strains and thermal cracking; better capture the progressive process of fracturing affects thermal conductivity; applicable to not only the steady-state but also the transient TM response.

A total‐Lagrangian material point method for coupled growth and massive deformation of incompressible soft materials

AbstractA total‐Lagrangian material point method (TLMPM) with a mixed formulation is proposed in this article for the mechanical analysis of incompressible soft materials coupling the mass growth and massive deformation. In this work, the growth‐induced large deformation behavior is handled within the framework of the material point method (MPM) based on the multiplicative decomposition of deformation gradient. To overcome the volumetric locking caused by the incompressibility of soft materials, a weak‐form equation for hydrostatic pressure is implemented into a mixed MPM framework. To deal with the massive deformation caused by growth, the total‐Lagrangian formulation is further introduced into the mixed MPM. The TLMPM discretized via the B‐spline basis functions is then developed in the proposed solution procedure. The efficiency and accuracy of the proposed method are demonstrated by several representative two‐ and three‐dimensional numerical examples with large deformation such as the free growth of blocks and constrained growth of rings. The challenging problems including the volumetric growth of bilayers with complex geometry and the strain‐driven growth of skin are further investigated using the proposed TLMPM to illustrate its ability for evaluating the growth phenomena and behaviors as observed in nature and engineering.

Modeling of electro–viscoelastic dielectric elastomer: A continuum mechanics approach

The present study deals with the rheological behavior modeling of a dielectric elastomeric (DE) material class. The DE materials are commonly used to develop soft actuators and energy harvesters in soft robotics. DE is a class of electroactive polymers that produces large strains with an electrically induced load application. In this article, a micro mechanics-based thermodynamically consistent rheological model is derived to capture the electro–viscoelastic behavior of the DE material class. A multiplicative deformation gradient decomposition into elastic and viscous parts is incorporated to attain a transitional configuration. The decomposition is being used for setting up the constitutive evolution equations for an electro–viscoelastic deformation. The derived rheological model agrees well with existing experimental results. We succeed here to enable the effect of viscosity and electric field on the deformation simultaneously with a micro mechanics-based analytical finding with the least possible material parameters.

A temperature-dependent 3D anisotropic visco-hyperelastic constitutive model for jute woven fabric reinforced poly (butylene succinate) biocomposite in thermoforming

Based on multiplicative decomposition of deformation gradient and energy decomposition, a temperature-dependent 3D anisotropic visco-hyperelastic constitutive model was developed for jute woven fabric reinforced poly (butylene succinate) (PBS) biocomposites for thermoforming simulation. The melted PBS matrix was characterized by a viscoelastic model, while the woven jute fabric reinforcement was modelled as an anisotropic hyperelastic material. Temperature influence on interaction between fabric reinforcement and PBS matrix was considered. A specific jute plain weave fabric reinforced PBS biocomposite was prepared and tested under thermoforming temperature range to provide a complete set of data for model parameter identification. Based on the proposed model, finite element simulation of a hemispherical thermoforming process of the biocomposite sheet was implemented. Comparison with experiment results validated the proposed constitutive model.

MAP123-EPF: A mechanistic-based data-driven approach for numerical elastoplastic modeling at finite strain

Direct numerical simulation based on experimental stress–strain data without explicit constitutive models is an active research topic. In this paper, a mechanistic-based, data-driven computational framework is proposed for elastoplastic materials undergoing finite strain. Harnessing the physical insights from the existing model-based plasticity theory, multiplicative decomposition of deformation gradient and the coaxial relationship between the logarithmic trial elastic strain and the true stress is employed to perform stress-update, driven by two sets of the specifically measured one dimensional (1D) stress–strain data. The proposed approach, called MAP123-EPF, is used to solve several Boundary-Value Problems (BVPs) involving elastoplastic materials undergoing finite strains. Numerical results indicate that the proposed approach can predict the response of isotropic elastoplastic materials (characterized by the classical J2 plasticity model and the associative Drucker–Prager model) with good accuracy using numerically/experimentally generated data. The proposed approach circumvents the need for the several basic ingredients of a traditional finite strain computational plasticity model, such as an explicit yielding function, a hardening law and an appropriate objective stress rate. Demonstrative examples are shown and strengths and limitations of the proposed approach are discussed.

A mixed isogeometric analysis approach for the transient swelling of hydrogel

In this paper, a mixed isogeometric analysis approach is developed to realize accurate and efficient simulations of the transient swelling of hydrogel strongly coupled with large deformation and fluid diffusion. In this method, the mixed isogeometric element is presented for the coupled multi-physics problems accompanied with the volumetric locking caused by the incompressibility of polymeric chains and solvent molecules. Such type of element is based on the subdivision properties of the Non-Uniform Rational B-Spline (NURBS) basis functions, in which the fine and coarse basis functions with the same order are used to discretize the displacement and chemical potential fields, respectively. Meanwhile, an auxiliary configuration for the multiplicative decomposition of deformation gradient is introduced to eliminate the difficulty due to certain specific cases, such as a negative infinite chemical potential for the hydrogel in a dry state. Then, the virtual work equations are derived based on the strain energy function with the ability of describing the constitutive relation of the hydrogel for large deformation from any swollen state and the mass conservation. The discrete formulations of the linearized virtual work equations for two fields on the base of the mixed isogeometric element are obtained. The efficiency, accuracy and validity of the proposed mixed isogeometric analysis approach are illustrated by solving several representative problems. The applicability of the proposed method is further demonstrated through two practical applications, which indicates that it has great potential to serve as an effective numerical tool for mechanical analysis and design of many hydrogel-based bionic devices.

Riemannian and Euclidean material structures in anelasticity

In this paper, we discuss the mechanics of anelastic bodies with respect to a Riemannian and a Euclidean geometric structure on the material manifold. These two structures provide two equivalent sets of governing equations that correspond to the geometrical and classical approaches to non-linear anelasticity. This paper provides a parallelism between the two approaches and explains how to go from one to the other. We work in the setting of the multiplicative decomposition of deformation gradient seen as a non-holonomic change of frame in the material manifold. This allows one to define, in addition to the two geometric structures, a Weitzenböck connection on the material manifold. We use this connection to express natural uniformity in a geometrically meaningful way. The concept of uniformity is then extended to the Riemannian and Euclidean structures. Finally, we discuss the role of non-uniformity in the form of material forces that appear in the configurational form of the balance of linear momentum with respect to the two structures.

A generalized decoupling numerical framework for polymeric gels and its element‐free implementation

SummaryCoupled diffusion of solvent molecules and mechanical stretching of polymeric networks induced swelling/deswelling in polymeric gels contributes to the ability of its “smart response” and “environmental sensitivity.” In this article, the coupled diffusion and large deformation of polymeric gels are clarified into three kinds of deformation conditions which all involve swelling/deswelling. The real numerical challenge is identified for simulating swelling/deswelling induced deformation of gels by solving the partial deferential control equation of a popular monophase model for gels in steady equilibrium state with a standard process of numerical discretization. To address the numerical challenge, a generalized decoupling numerical framework is developed based on a multiplicative decomposition of deformation gradient. The proposed framework is more flexible and works for almost all cases of deformation of gels no matter involving swelling/deswelling or not. Its element‐free implementation is also developed using the element‐free IMLS‐Ritz method and is validated through analyzing the three‐dimensional stable and unstable large deformation of the constrained gel cube and tube.

Compatibility conditions from a Gram–Schmidt decomposition of deformation gradient in two dimensions

Planar deformations are analyzed in the context of a Gram-Schmidt decomposition of deformation gradient into a rotation and upper triangular stretch, where the latter consists of up to three possibly nonzero terms. Necessary—and sufficient under suitable restrictions—conditions for integrability of deformation gradient and stretch (geometrically, vanishing torsion and curvature of a certain affine connection) reduce to a system of three partial differential equations (PDEs). This system appears more convenient than corresponding compatibility conditions from polar decompositions, especially for motions involving simple shear. Admissible families of triangular stretch corresponding to simple shear, pure shear, dilation, and uniaxial strain are analyzed. Also considered are simultaneous simple shear and axial stretch, for which a non-trivial solution with non-vanishing rotation gradient is constructed.

High-order NURBS elements based isogeometric formulation for swellable soft materials

This paper presents an isogeometric formulation and the corresponding algorithm to predict the mechanical behaviors of swellable soft materials by the high-order NURBS elements. To deal with the difficulties aroused from the growth and incompressibility of soft materials, a numerical scheme based on two multiplicative decompositions of deformation gradient is developed. For introducing the growth effect, the deformation gradient is decomposed into a growth tensor and an elastic deformation gradient. For circumventing the volumetric locking caused by incompressibility of soft materials, the elastic deformation gradient is further decomposed into its volume-preserving and volumetric-dilatational parts. Meanwhile, the volumetric-dilatational part is modified by a linear projection operator. Then, the incremental equations of the weak form are derived by an effective linearization method for this nonlinear problem. After introducing the discrete technique in isogeometric analysis, the formulations for numerical calculations, i.e., the stiffness matrices and the force vectors, are derived. The efficiency and accuracy of the proposed isogeometric method are demonstrated by several examples discretized by the high-order NURBS elements exactly, including conic sections and other complex geometries. The proposed method is also proved to have great potentials for analyzing the mechanical behaviors of swellable soft materials as well as bionic structures and designing the bionic devices.

Thermodynamic properties and thermoelastic constitutive relation for cubic crystal structures based on improved free energy

This study proposes new models for thermodynamic properties and thermoelastic constitutive relation of materials with cubic crystal structures. The motion equation of atoms in cubic crystal structures is decomposed into structural deformation and thermal vibration at first. Then based on the thermo-mechanical coupling mechanism, both the thermal and mechanical aspects of structural deformation are investigated. And the thermal vibration equation is built at structural deformation positions by considering the non-harmonic approximation of interatomic potentials. Further, the improved formula of free energy is established as a function of the structural deformation and thermal vibration frequencies, which includes the thermo-mechanical coupling and non-harmonic effects. And the thermodynamic properties, including internal energy, entropy, heat capacity and thermal expansion, are derived from the free energy. Besides, based on the multiplicative decomposition of deformation gradient, the thermoelastic constitutive relation is constructed at finite deformation for cubic crystal materials. Finally, numerical results of the thermodynamic properties, thermoelastic stress–strain relations and elastic constants for face-centered cubic metals Cu, Au and Ag from 0 K to the melting points at 0–50 GPa are provided by comparing with the experimental data to demonstrate the usability of the present models.