A constitutive model for fibrous tissues considering collagen fiber crimp

A micromechanically based constitutive model for fibrous tissues is presented. The model considers the randomly crimped morphology of individual collagen fibers, a morphology typically seen in photomicrographs of tissue samples. It describes the relationship between the fiber endpoints and its arc-length in terms of a measurable quantity, which can be estimated from image data. The collective mechanical behavior of collagen fibers is presented in terms of an explicit expression for the strain-energy function, where a fiber-specific random variable is approximated by a Beta distribution. The model-related stress and elasticity tensors are provided. Two representative numerical examples are analyzed with the aim of demonstrating the peculiar mechanism of the constitutive model and quantifying the effect of parameter changes on the mechanical response. In particular, a fibrous tissue, assumed to be (nearly) incompressible, is subject to a uniaxial extension along the fiber direction, and, separately, to pure shear. It is shown that the fiber crimp model can reproduce several of the expected characteristics of fibrous tissues.     

A multi-axial constitutive model for metal matrix composites

Metal matrix composites (MMCs) generally do not follow the classical plasticity theory, even though the matrix metals do deform plastically. A tension-compression yield asymmetry is typically observed in MMCs. For particulate-reinforced MMCs, this non-classical response is mainly due to the variation of damage evolution with loading modes. In this paper, a viscoplastic multi-axial constitutive model for plastic deformation of MMCs is constructed using the Mises-Schleicher yield criterion. The subsequent plastic flow is characterized by an associated and decomposed flow rule considering effects from both deviatoric and hydrostatic stresses. This model is capable of describing the multi-axial yield and flow behavior of MMCs by using simulated or measured asymmetric tensile and compressive stress-strain responses as input. As an example, the influence of damage evolution in terms of interfacial debonding in MMCs (obtained from FEM simulations) is incorporated through the different tensile and compressive stress-strain behaviors. Applying this model to predict the torsion and the pressure-dependant tensile responses of some commonly used MMCs provides good agreement with experimental data.     

A constitutive model of aluminum foam for crash simulations

A new constitutive model for metallic foams is developed to overcome the deficiency of existing models in commercial finite element codes. The proposed constitutive model accounts for volume changes under hydrostatic compression and combines the hydrostatic pressure and von Mises stress into one yield function. The change of the compressibility of the metallic foam is handled in the constitutive model by allowing for shape changes of the yield surface in the hydrostatic pressure-von Mises stress space. The backward Euler method is adopted to integrate the constitutive equations to achieve numerical accuracy and stability. The model is implemented into LS-DYNA as a user-defined subroutine, verified with existing solutions, and validated with foam testing data. The verified and validated model is then utilized in the crushing simulations of foam-filled columns with square and hexagonal cross-sections. Two constitutive models are studied: the first using an exponential function to describe the relationship of plastic Poisson's ratio with respect to true strain and the second using linear interpolation function as an alternative approximation. The new foam model provides satisfactory prediction of crushing forces and deformed shapes compared to experimental results. Additionally, the new foam model was shown to have better numerical stability and accuracy than existing LS-DYNA built-in material models.     

Dislocation density based constitutive model for ultrasonic assisted deformation

The mechanical behaviour of metallic materials subjected to plastic deformation is altered with the superposition of ultrasonic vibrations. A significant effect is the reduction of flow stress or acoustic softening. This phenomenon is utilized in metal forming and other deformation based manufacturing processes. Experimental investigations on ultrasonic assisted tensile tests focus on the effect of ultrasonic vibrations along the longitudinal axis of the specimen, whereas the manufacturing processes employs in transverse directions. In the present work, transverse ultrasonic vibrations are imposed during a uniaxial tensile test using an aluminium alloy. The trend of acoustic softening due to transverse direction vibrations is similar to that along longitudinal direction. A dislocation density based constitutive model is extended to model the softening due to ultrasonic effect. The predicted results agree well with the experimental observations.     

Thermo-damage-viscoelastic constitutive model of HTPB composite propellant

The main goal of this work is to establish a thermo-damage-viscoelastic model for Hydroxyl-Terminated Polybutadiene (HTPB) composite propellant based on the thermodynamic theory and elastic–viscoelastic correspondence principles. The model will also consider the influence of temperatures. The parameter α which represent the damage evolution rate and the material constants a, b are defined as exponential functions of temperature T, i.e. α(T), a(T) and b(T). Relaxation tests and uniaxial constant rate tensile tests are used to acquire the model parameters, and C(S) curves of different rates under the same temperature states are considered to be overlapped in this paper, while noncoincidence under the different temperature states. Then, uniaxial constant rate tensile tests and multi-step tensile-relaxation tests are used to verify the accuracy of the model. The results show that, the model is highly accurate in describing the mechanical property of HTPB under various loading conditions, but some drawback in describing the relaxation property inside of the HTPB’s nonlinear viscoelastic segments.     

A constitutive model for the formation of martensite in austenitic steels under large strain plasticity

A constitutive model for diffusionless phase transitions in elastoplastic materials undergoing large deformations is developed. The model takes basic thermodynamic relations as its starting point and the phase transition is treated through an internal variable (the phase fractions) approach. The usual yield potential is used together with a transformation potential to describe the evolution of the new phase. A numerical implementation of the model is presented, along with the derivation of a consistent algorithmic tangent modulus. Simulations based on the presented model are shown to agree well with experimental findings. The proposed model provides a robust tool suitable for large-scale simulations of phase transformations in austenitic steels undergoing extensive deformations, as is demonstrated in simulations of the necking of a bar under tensile loading and also in simulations of a cup deep-drawing process.     

A micromechanics-inspired constitutive model for shape-memory alloys that accounts for initiation and saturation of phase transformation

A constitutive model to describe macroscopic elastic and transformation behaviors of polycrystalline shape-memory alloys is formulated using an internal variable thermodynamic framework. In a departure from prior phenomenological models, the proposed model treats initiation, growth kinetics, and saturation of transformation distinctly, consistent with physics revealed by recent multi-scale experiments and theoretical studies. Specifically, the proposed approach captures the macroscopic manifestations of three micromechanial facts, even though microstructures are not explicitly modeled: (1) Individual grains with favorable orientations and stresses for transformation are the first to nucleate martensite, and the local nucleation strain is relatively large. (2) Then, transformation interfaces propagate according to growth kinetics to traverse networks of grains, while previously formed martensite may reorient. (3) Ultimately, transformation saturates prior to 100% completion as some unfavorably-oriented grains do not transform; thus the total transformation strain of a polycrystal is modest relative to the initial, local nucleation strain. The proposed formulation also accounts for tension–compression asymmetry, processing anisotropy, and the distinction between stress-induced and temperature-induced transformations. Consequently, the model describes thermoelastic responses of shape-memory alloys subject to complex, multi-axial thermo-mechanical loadings. These abilities are demonstrated through detailed comparisons of simulations with experiments.     

Three-dimensional constitutive model for shape memory alloys based on microplane model

A new model for the behavior of polycrystalline shape memory alloys (SMA), based on a statically constrained microplane theory, is proposed. The new model can predict three-dimensional response by superposing the effects of inelastic deformations computed on several planes of different orientation, thus reproducing closely the actual physical behavior of the material. Due to the structure of the microplane algorithm, only a one-dimensional constitutive law is necessary on each plane. In this paper, a simple constitutive law and a robust kinetic expression are used as the local constitutive law on the microplane level. The results for SMA response on the macroscale are promising: simple one-dimensional response is easily reproduced, as are more complex features such as stress-strain subloops and tension-compression asymmetry. A key feature of the new model is its ability to accurately represent the deviation from normality exhibited by SMAs under nonproportional loading paths.     

A chemo-electro-mechanical model for simulation of responsive deformation of glucose-sensitive hydrogels with the effect of enzyme catalysis

A multi-effect-coupling glucose-stimulus (MECglu) model is developed and solved numerically for the swelling behavior of soft smart hydrogels responding to changes in the environmental glucose concentration. The model considers the effect of the glucose oxidation reaction catalyzed by enzymes including glucose oxidase and catalase. It is composed of the Nernst-Planck equation for the mobile species in the solvent, the Poisson equation for the electric potential, and a nonlinear mechanical equation for the large deformations of the hydrogel that arise due to the conversion of chemical energy to mechanical. Based on the theory of the chemo-electro-mechanical-coupled fields, the formulation of the fixed charge groups bound onto the cross-linked polymer network is associated with the change of the ambient solution pH. The MECglu model is validated by comparison between the steady-state computation and experimental equilibrium swelling curves, and good agreement is obtained. A parameter study is then conducted by steady-state simulations to ascertain the impact of various solvent parameters on the responsive swelling behavior of the hydrogel. One key parameter is the glucose concentration, which is varied within the range of practical physiological glucose concentrations from 0 to 16.5 mM (300 mg/ml) to support the design and optimization of an insulin delivery system based on a glucose-sensitive hydrogel with immobilized glucose oxidase and catalase. The influence of oxygen and glucose concentrations in the solvent is then further studied for the distributive profiles of reacting and diffusive species concentrations, the electric potential, the displacement, as well as the swelling ratio of the glucose-sensitive hydrogel.     

A coupled elastoplastic-damage constitutive model with Lode angle dependent failure criterion

A coupled elastoplastic-damage constitutive model with Lode angle dependent failure criterion for high strain and ballistic applications is presented. A Lode angle dependent function is added to the equivalent plastic strain to failure definition of the Johnson–Cook failure criterion. The weakening in the elastic law and in the Johnson–Cook-like constitutive relation implicitly introduces the Lode angle dependency in the elastoplastic behaviour. The material model is calibrated for precipitation hardened Inconel 718 nickel-base superalloy. The combination of a Lode angle dependent failure criterion with weakened constitutive equations is proven to predict fracture patterns of the mechanical tests performed and provide reliable results. Additionally, the mesh size dependency on the prediction of the fracture patterns was studied, showing that was crucial to predict such patterns.     

A nonlinear constitutive model for magnetostrictive materials

A general nonlinear constitutive model is proposed for magnetostrictive materials, based on the important physical fact that a nonlinear part of the elastic strain produced by a pre-stress is related to the magnetic domain rotation or movement and is responsible for the change of the maximum magnetostrictive strain with the pre-stress. To avoid the complicity of determining the tensor function describing the nonlinear elastic strain part, this paper proposes a simplified model by means of linearizing the nonlinear function. For the convenience of engineering applications, the expressions of the 3-D (bulk), 2-D (film) and 1-D (rod) models are, respectively, given for an isotropic material and their applicable ranges are also discussed. By comparison with the experimental data of a Terfenol-D rod, it is found that the proposed model can accurately predict the magnetostrictive strain curves in low, moderate and high magnetic field regions for various compressive pre-stress levels. The numerical simulation further illustrates that, for either magnetostrictive rods or thin films, the proposed model can effectively describe the effects of the pre-stress or residual stress on the magnetization and magnetostrictive strain curves, while none of the known models can capture all of them. Therefore, the proposed model enjoys higher precision and wider applicability than the previous models, especially in the region of the high field.The project supported by the National Natural Science Foundation of China (10132010 and 90405005)     

Structural bonding-breakage constitutive model for natural unsaturated clayey soils

The natural clayey soils are usually structural and unsaturated,which makes their mechanical properties quite different from the remolded saturated soils.A structural constitutive model is proposed to simulate the bonding-breakage micro-mechanism.In this model,the unsaturated soil element is divided into a cementation element and a friction element according to the binary medium theory,and the stress-strain coordination for these two elements is obtained. The cementation element is regarded as elastic,whereas the friction element is regarded as elastoplastic which can be described with the Gallipoli＇s model.The theoretical formulation is verified with the comparative experiments of isotropic compressions on the saturated and unsaturated structural soils.Parametric analyses of the effects of damage variables on the model predictions are further carried out,which show that breakage deformation of natural clayey soils increases with the rising amount of initial defects.     

A multiaxial constitutive model for concrete in the fire situation: Theoretical formulation

This paper aims to develop a multiaxial concrete model for implementation in finite element software dedicated to the analysis of structures in fire. The need for proper concrete model remains a challenging task in structural fire engineering because of the complexity of the concrete mechanical behavior characterization and the severe requirements for the material models raised by the development of performance-based design. A fully three-dimensional model is developed based on the combination of elastoplasticity and damage theories. The state of damage in concrete, assumed isotropic, is modeled by means of a fourth order damage tensor to capture the unilateral effect. The concrete model comprises a limited number of parameters that can be identified by three simple tests at ambient temperature. At high temperatures, a generic transient creep model is included to take into account explicitly the effect of transient creep strain. The numerical implementation of the concrete model in a finite element software is presented and a series of numerical simulations are conducted for validation. The concrete behavior is accurately captured in a large range of temperature and stress states. A limitation appears when modeling the concrete post-peak behavior in highly confined stress states, due to the coupling assumption between damage and plasticity, but the considered levels of triaxial confinement are unusual stress states in structural concrete.     

A novel constitutive model for semiconductors: The case of silicon

The photovoltaic industry relies heavily on solar-grade silicon multicrystals. Understanding their mechanical behavior requires the development of adequate constitutive models accounting for the effects of both high dislocation densities and complex loading situations in a wide range of temperature, strain rate, and impurity contents. The traditional model of Alexander and Haasen poses several limitations. We introduce in this work a novel constitutive model for covalent single crystals and its implementation into a rate-dependent crystal plasticity framework. It is entirely physically based on the dislocation generation, storage and annihilation processes taking place during plastic flow. The total dislocation density is segmented according to the dislocation mobility potential and their character. A dislocation multiplication law for the yield region more accurate than the one of Alexander and Haasen is proposed. The influence of additional dislocation sources created on forest trees, usually disregarded in models for semiconductors, is assessed. The dislocation velocity law combines three potentially rate-limiting mechanisms: the standard double kink mechanism, jog dragging and the influence of localized obstacles. The model is valid at finite strains, in multiple slip conditions and captures accurately the high temperature- and strain rate sensitivity of semiconductors. The experimental stress overshoot is qualitatively reproduced only when jog dragging is accounted for. Localized obstacles are shown not to have any significant effect on dislocation motion in silicon. The broader case of extrinsic semiconductors is discussed and the influence of dissolved oxygen on the upper yield stress of silicon monocrystals is successfully reproduced.     

Rate dependent finite strain constitutive model of polyurea

Continuous loading and unloading experiments are performed at different strain rates to characterize the large deformation behavior of polyurea under compressive loading. In addition, uniaxial compression tests are carried out with multistep strain history profiles. The analysis of the experimental data shows that the concept of equilibrium path may not be applied to polyurea. This finding implies that viscoelastic constitutive models of the Zener type are no suitable for the modeling of the rate dependent behavior of polyurea. A new constitutive model is developed based on a rheological model composed of two Maxwell elements. The soft rubbery response is represented by a Gent spring while nonlinear viscous evolution equations are proposed to describe the time-dependent material response. The eight material model parameters are identified for polyurea and used to predict the experimentally-measured stress-strain curves for various loading and unloading histories. The model provides a good prediction of the response under monotonic loading over wide range of strain rates, while it overestimates the stiffness during unloading. Furthermore, the model predictions of the material relaxation and viscous dissipation during a loading-unloading cycle agree well with the experiments.     

A constitutive model for plastically anisotropic solids with non-spherical voids

Plastic constitutive relations are derived for a class of anisotropic porous materials consisting of coaxial spheroidal voids, arbitrarily oriented relative to the embedding orthotropic matrix. The derivations are based on nonlinear homogenization, limit analysis and micromechanics. A variational principle is formulated for the yield criterion of the effective medium and specialized to a spheroidal representative volume element containing a confocal spheroidal void and subjected to uniform boundary deformation. To obtain closed form equations for the effective yield locus, approximations are introduced in the limit-analysis based on a restricted set of admissible microscopic velocity fields. Evolution laws are also derived for the microstructure, defined in terms of void volume fraction, aspect ratio and orientation, using material incompressibility and Eshelby-like concentration tensors. The new yield criterion is an extension of the well known isotropic Gurson model. It also extends previous analyses of uncoupled effects of void shape and material anisotropy on the effective plastic behavior of solids containing voids. Preliminary comparisons with finite element calculations of voided cells show that the model captures non-trivial effects of anisotropy heretofore not picked up by void growth models.     

A quasi-flow constitutive model with strain-rate dependence

In this paper, the proposed is a quasi-flow constitutive model with strain-rate sensitivity for elastic plastic large deformation. The model is based on the Quasi-flow Corner theory, and is suitable for the sheet metal forming process simulation with a variable punch machine velocity. Uniaxial tensile tests and deep-drawing tests of a circular blank with square punch are carried out and numerically simulated. The consistency between the experimental and the numerically simulated results shows the validity of the present new constitutive model. The project supported by the Scientific Foundation of National Outstanding Youth of China (10125208), the National Natural Science Foundation of China (19832020), and the National Education Committee of China     

A new constitutive model for monodispersed suspensions of spheres at high concentrations

We describe a constitutive model for monodispersed concentrated suspensions of spheres at high volume fractions. In this model the motion of a pair of generic spheres is represented by that of a pair of force-free and torque-free spheres tumbling along with the flow. The interaction with the surrounding spheres is modelled by an anisotropic diffusion process. Viscometric and elongational properties of the model are obtained by a Brownian dynamic simulation. Received: 24 September 1998 Accepted: 1 December 1998     

A continuum mechanics framework and a constitutive model for remodelling of collagen gels and collagenous tissues

Collagen is a very important protein of the human body and is responsible for the structural stability of many body components. Furthermore, collagen fibre networks are able to grow and remodel themselves, which enables them to adjust to varying physiological conditions. This remodelling is accomplished by fibre-producing cells, such as fibroblasts. The ability to adjust to new physiological conditions is very important, for example in wound healing. In the present paper, a theoretical framework for modelling collagenous tissues and collagen gels is proposed. Continuum mechanics is employed to describe the kinematics of the collagen, and affine deformations of fibres are assumed. Biological soft tissues can be approximated as being hyperelastic, and the constitutive model for the collagen fabric is therefore formulated in terms of a strain energy function. This strain energy function includes a density function that describes the distribution of the collagen fibre orientation. The density function evolves according to an evolution law, where fibres tend to reorient towards the direction of maximum Cauchy stress. The remodelling of the collagen network is also assumed to include a pre-stretching of collagen fibres, accomplished by fibroblasts. The theoretical framework is applied to experiments performed on collagen gels, where gels were exposed to remodelling under both biaxial and uniaxial constraints. The proposed model was able to predict both the resulting collagen distribution and the resulting stress-strain relationships obtained for the remodelled collagen gels. The influence of the most important model parameters is demonstrated, and it appears that there is a fairly unique set of model parameters that gives an optimal fit to the experimental data.