Macroscopic Constitutive Model Research Articles

A macroscopic constitutive model of temperature-induced phase transition of polycrystalline Ni2MnGa by directional solidification

Directional solidification technology has been widely used to improve the properties of polycrystalline Ni2MnGa materials. Mechanical training can adjust the internal organizational structures of the materials, reduce the stress of twin boundaries motion, and then result in larger strain at lower outfield levels. In this paper, we test the microscopic structure of Ni2MnGa polycrystalline ferromagnetic shape memory alloy produced by directional solidification and compress it along two axes successively for mechanical training. The influences of pre-compressive stresses on the temperature-induced strains are analyzed. The macroscopic mechanical behaviors show anisotropy. According to the generating mechanism of the macroscopic strain, a three-dimensional constitutive model is established. Based on thermodynamic method, the kinetic equations of the martensitic transformation and inverse transformation are presented considering the driving force and energy dissipation. The prediction curves of temperature-induce strains along two different directions are investigated. And the results coincide well with the experiment data. It well explains the macroscopic anisotropy mechanical behaviors and fits for using in engineering.

A macroscopic consitutive model on induced anisotropy for polymers with weighting functions

AbstractThe alignment of polymer chains is a well known microstructural evolution effect due to straining of polymers. This has a drastic influence on the macroscopic properties of the initially isotropic material, such as a pronounced strength in the loading direction of stretched films. Experiments on strain induced anisotropy at room temperature are analyzed by optical measurements. For modeling the effect of strain induced anisotropy a macroscopic constitutive model is presented. As a key idea, weighting functions are introduced to represent a strain‐softening/hardening‐effect to account for induced anisotropy. These functions represent the ratio between the total strain rate and a structural tensor. In this way, material parameters are used as a sum of weighted direction related quantities. In the finite element examples we simulate the cold‐forming of amorphous thermoplastic films below the glass transition temperature subjected to different re‐loading directions. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Damage characteristics and constitutive modeling of the 2D C/SiC composite: Part II – Material model and numerical implementation

In this work, a macroscopic non-linear constitutive model accounting for damage, inelastic strain and unilateral behavior is proposed for the 2D plain-woven C/SiC composite. A set of scalar damage variables and a new thermodynamic potential expression are introduced in the framework of continuum damage mechanics. In the deduced constitutive equations, the material’s progressive damage deactivation behavior during the compression loading is described by a continuous function, and different deactivation rates under uniaxial and biaxial compression loadings are also considered. In damage evolution laws, the coupling effect among the damage modes and impediment effect of compression stress on the development of shear damage in different plane stress states are taken into account. Besides, the general plasticity theory is applied to describing the evolution of inelastic strain in tension and/or shear stress state. The Tsai–Wu failure criterion is adopted for strength analysis. Additionally, the material model is implemented as a user-defined material subroutine (UMAT) and linked to the ABAQUS finite element software, and its performance is demonstrated through several numerical examples.

A mixed finite element procedure of gradient Cosserat continuum for second-order computational homogenisation of granular materials

A mixed finite element (FE) procedure of the gradient Cosserat continuum for the second-order computational homogenisation of granular materials is presented. The proposed mixed FE is developed based on the Hu---Washizu variational principle. Translational displacements, microrotations, and displacement gradients with Lagrange multipliers are taken as the independent nodal variables. The tangent stiffness matrix of the mixed FE is formulated. The advantage of the gradient Cosserat continuum model in capturing the meso-structural size effect is numerically demonstrated. Patch tests are specially designed and performed to validate the mixed FE formulations. A numerical example is presented to demonstrate the performance of the mixed FE procedure in the simulation of strain softening and localisation phenomena, while without the need to specify the macroscopic phenomenological constitutive relationship and material failure model. The meso-structural mechanisms of the macroscopic failure of granular materials are detected, i.e. significant development of dissipative sliding and rolling frictions among particles in contacts, resulting in the loss of contacts.

Applicability of micro–macro decoupling scheme to two-scale analysis of fiber-reinforced plastics

The numerical study is made to demonstrate the applicability of the method of decoupling multi-scale analysis to the micro–macro evaluation of the mechanical behavior of fiber-reinforced plastics (FRP) that exhibits inelastic deformations and internal damage of the matrix material. During the course of this demonstration, it is confirmed that the reliability of the decoupling method can be guaranteed if the macroscopic constitutive model is introduced so as to inherit the microscopic material behavior. To this end, with reference to the results of the numerical material testing on the periodic microstructures of FRP, we propose an anisotropic elastoplastic-creep-damage combined constitutive model to represent the macroscopic material behavior and illustrate the characteristics of the inelastic deformations that resemble the material behavior assumed for plastics at micro-scale. With the identified macroscopic material parameters, the macroscopic structural analysis, which is followed by the localization analysis consistently, can be an actual proof of the utility value of the decoupling method in practice.

Impact of anisotropy and viscosity to model the mechanical behavior of Ti–6Al–4V alloy

This paper compares the predictions of an isotropic thermo-elasto-viscoplastic approach and of an anisotropic thermo-elastoplastic one with experimental results representative of the mechanical behavior of Ti–6Al–4V at moderate temperatures and low strain rates. The first model is the well known Norton–Hoff viscoplastic constitutive law with isotropic von Mises yield locus identified by using monotonic tension tests performed at strain rates from 10−3s−1 to 10−1s−1 and at temperatures up to 400°C. The second model is a thermo-elasto-plastic one defined by the orthotropic yield criterion CPB06. It takes into account the anisotropy and the strength differential (SD) effect in tension–compression of Ti–6Al–4V at RT, 150°C and 400°C. The identification of the SD effect is done by using tension and compression tests and the anisotropy behavior is identified by using shear, plane strain, tension and compression tests performed in three orthogonal material directions. The accuracy of the load and displacement predictions of the two macroscopic constitutive models are compared to experimental results obtained from tests performed on specimens with multiaxial loadings and large strain at several temperatures.

Simulation of strain-induced anisotropy for polymers with weighting functions

The alignment of polymer chains is a well-known microstructural evolution effect due to straining of polymers. This has a drastic influence on the macroscopic properties of the initially isotropic material, such as a pronounced strength in the loading direction of stretched films. For modeling the effect of strain-induced anisotropy, a macroscopic constitutive model is developed in this paper. Within a thermodynamic framework, an additive decomposition of the logarithmic Hencky strain tensor into elastic and inelastic parts is used to formulate the constitutive equations. As a key idea, weighting functions are introduced to represent a strain-softening/hardening effect to account for induced anisotropy. These functions represent the ratio between the total strain rate (representing the actual loading direction) and a structural tensor (representing the stretched polymer chains). In this way, we introduce material parameters as a sum of weighted direction-related quantities. The numerical implementation of the resulting set of constitutive is used to identify material parameters based on experimental data, exhibiting strain-induced anisotropy. In the finite-element examples, we simulate the cold-forming of amorphous thermoplastic films below the glass transition temperature subjected to different re-loading directions.

Anisotropic yielding of injection molded polyethylene: Experiments and modeling

The anisotropic yielding of injection molded polyethylene is discussed, specifically focusing on the differences between the tensile and compressive yield stress as a function of loading angle and strain rate. For the first time, it is demonstrated that a strong Bauschinger effect exists in polymers that possess molecular orientation due to melt-processing. A macroscopic constitutive model is proposed to capture the yielding phenomena observed in the experiments. This model features two sources of anisotropy, the physical significance of which is discussed: a frozen-in stress originating from the oriented elastic network, and an intrinsically anisotropic viscoplastic flow rule based on the yield function of Hill, extended to incorporate the asymmetry between the tensile and compressive response. Model simulations demonstrate that the constitutive relation proposed accurately captures the important features of the experimental data. Also, it is able to predict the anisotropy in the failure kinetics of injection molded polyethylene subjected to a constant tensile load in different directions without the requirement of additional parameters.

Homogenised constitutive model coupling damage and debonding for reinforced concrete structures under cyclic solicitations

A new stress resultant constitutive model for reinforced concrete plates under cyclic solicitations is presented. This model is built by the periodic homogenisation approach using the averaging method and couples damage of concrete and periodic debonding between concrete and steel rebar. In one-dimensional situations, we derive a closed-form solution of the local problem useful to verify and set up the plate problem. The one dimensional macroscopic constitutive model involves a limited number of parameters, the sensibility of which is studied. Comparison to experimental results underlines the pertinence of the model by considering internal debonding in order to properly represent the mechanical dissipation occurring during cyclic loadings on reinforced concrete panels.

Micro–macro homogenization of granular materials based on the average-field theory of Cosserat continuum

This paper performs further study on the micro–macro homogenization approach of granular materials (Li et al., 2010) based on the advancement of Hill’s lemma for Cosserat continuum (Liu, 2013). Firstly, the average couple stress tensor, expressed as the volume integration of quantities over the representative volume element (RVE) in the average-field theory of Cosserat continuum, is further deduced and expressed in terms of discrete quantities on the discrete particle assembly RVE of granular materials. The expression is also discussed and compared with other typical definitions of the effective couple stress tensor for granular materials in the literature. Then, rate forms of micromechanically based constitutive models consistent with different types of RVE boundary conditions are derived and discussed. Since the presented micro–macro homogenization approach is used, not only the micro–macro energy equivalence is satisfied, but also the microstructure and its evolution can be taken into account in the constitutive formulation with no need of specifying macroscopic phenomenological constitutive model.

Computational modelling of constrained sintering in EB-PVD thermal barrier coatings

A micromechanical model is developed to simulate the evolution of microstructure during in-service sintering and eventual inter-columnar cracking in coatings made using electron beam vapour deposition (EB-PVD) route. The coating is idealized with a discrete distribution of axisymmetric asperities across interfaces between columnar grains. The model assumes that inter-columnar sintering is driven by changes in interface free energy of columns and the potential energy of the applied stress. Much faster diffusion that occurs over the free surfaces of the asperities is neglected. It is further assumed that the rate of sintering of the contacting asperities is determined by diffusion along the interface between the contacting asperities. Time evolution of contact modulus of the coating is accounted for as a function of sintering strain. The developed macroscopic constitutive model is employed to evaluate the sensitivity of the sintering response to imperfections and examine the conditions under which inter-columnar cracks can develop within the coating.

Coupled rate-dependent superelastic behavior of shape memory alloy bars induced by combined axial-torsional loading: a semi-analytic modeling

In this article, the coupled thermomechanical response of superelastic shape memory alloy bars and tubes in combined tension and torsion is studied both analytically and experimentally. Using the Gibbs free energy as the thermodynamic potential and choosing appropriate internal state variables, a three-dimensional phenomenological macroscopic constitutive model for shape memory alloys is derived. Taking into account the effect of latent heat during the forward and reverse martensitic phase transformation, the appropriate form of the energy balance relation is obtained. The three-dimensional coupled relations for the energy balance in the presence of the internal heat flux and the constitutive equations are reduced to a two-dimensional form for the tension-torsion case. An explicit finite difference approach is utilized to discretize the governing boundary conditions of bars. An empirical expression for the free heat convection from the surface of a shape memory alloy bar was proposed and experimentally validated. Several sets of experiments were then carried out to evaluate the mechanical and thermal responses of the model for a shape memory alloy tube subjected to uniaxial, pure torsion and non-proportional tension and torsion loading–unloading conditions. The approach could be used in the design of shape memory alloy devices undergoing combined loads with high strain rates or in the fatigue design of shape memory alloy devices subjected to cyclic loading.

A macroscopic constitutive model of shape memory alloy considering cyclic effects

This paper presents a macroscopic constitutive model reproducing the hysteretic behaviors of the superelastic shape memory alloy (SMA) under cyclic loading. The progressive increase of residual strain with the increased cycle number in such materials is assumed to be a consequence of the progressive increase of residual stress-induced martensitic volume fraction upon the cyclic effects. The progressive decrease of phase transformation critical stresses with the increased cycle number in such materials is assumed to be a result from the progressive increase of phase transformation critical temperatures upon the cyclic effects. A cyclic evolution equation is supposed to describe the influences of cycle effects on the material properties of the SMA under cyclic loading. A phase transformation equation expressing the phase transformation behaviors of the SMA under cyclic loading is established based on the differential relationship between martensitic volume fraction and the free energy increment of phase transformation. A mechanical constitutive equation predicting the mechanical characteristics of the SMA under cyclic loading is developed on the basis of thermodynamics and continuum mechanics. The cyclic evolution equation, phase transformation equation, and mechanical constitutive equation together compose the presented macroscopic constitutive model considering cyclic effects. Results of the numerical simulations illustrate that it can well reproduce the superelastic hysteretic behaviors of the SMA under cyclic loading.

Thermodynamic consistent modeling of polymer curing coupled to visco–elasticity at large strains

We develop a macroscopic constitutive model for temperature-dependent visco–elastic effects accompanied by curing, which are important phenomena in production processes. Within a thermodynamic framework we use an additive ternary decomposition of the logarithmic Hencky strain tensor into mechanical, thermal and chemical parts. Based on the concept of stoichiometric mass fractions for resin, curing agent and solidified material the bulk compression modulus as well as the bulk heat- and shrinking dilatation coefficients are derived and compared with ad hoc assumptions from the literature. Moreover, we use the amount of heat generated during differential scanning calorimetry until completion of the chemical reactions, to define the chemical energy. As a major result, the resulting latent heat of curing occurring in the heat-conduction equation derived in our approach reveals an ad hoc approach from the literature as a special case. In addition, thermodynamic consistency of the model will be proved, and the numerical implementation of the constitutive equations into a finite-element program is described. In the examples we illustrate the characteristic behaviour of the model, such as shrinking due to curing and temperature dependence and simulate the deep drawing of a spherical part with the finite-element-method.

Assessment of unreinforced masonry cross sections under eccentric compression accounting for strain softening

The mechanical behaviour of unreinforced masonry cross sections subjected to eccentric compressive loading has been studied through laboratory tests, analytical procedures, and numerical models. Several macroscopic constitutive models have been implemented in beam-like analytical approaches in order to derive axial force – bending moment limit domains. This paper investigates the influence of different macroscopic constitutive models and strain softening on the strength domain evolution from the elastic limit state to the ultimate limit state. Simplified equations are proposed for the Eurocode 6 constitutive model and strength-degrading stress–strain relationships. The corresponding limit strength domains and those derived from other constitutive models are compared to experimental data available in the literature. It is shown that more conservative estimates of ultimate moment capacity are obtained if some of the proposed equations, or those corresponding to the elastic-perfectly brittle constitutive model, are used. Finally, evolutionary strength domains defined in a dimensionless axial force – bending moment – strain ductility space were developed to investigate how bending moment capacity changes with masonry ductility in compression. Such domains could be implemented in equivalent frame models to simulate moment capacity variations with deformation demands on individual macro-elements as static pushover analysis proceeds.

Edge on Impact Simulations and Experiments

In the quest to understand damage and failure of ceramics in ballistic events, simplified experiments have been developed to benchmark behavior. One such experiment is known as edge on impact (EOI). In this experiment, an impactor strikes the edge of a thin square plate, and damage and cracking that occur on the free surface are captured in real time with high speed photography. If the material of interest is transparent, additional information regarding damage and wave mechanics within the sample can be discerned. Polarizers can be used to monitor stress wave propagation, and photography can record internal damage. This information serves as an excellent benchmark for validation of ceramic and glass constitutive models implemented in dynamic simulation codes. In this paper, recent progress towards predictive modeling of EOI is discussed. Time-dependent crack propagation and damage front evolution in silicon carbide (SiC) and aluminum oxynitride (AlON) ceramics are predicted using the Kayenta macroscopic constitutive model. Aspects regarding modeling material failure, variability, and volume scaling are noted. Mesoscale simulations of dynamic failure of anisotropic ceramic crystals facilitate determination of limit surfaces entering the macroscopic constitutive model, offsetting limited available experimental data.

A Thermo-Dynamical Constitutive Model Based on Kinetic Approach for Shape Memory Materials

The technological application of nickel–titanium shape memory alloys (SMA) requires a constitutive model that can be easily implemented into numerical methods. For these reasons, macroscopic constitutive models have gained ground in SMA designs. A new model is developed that encompasses all the characteristics of these materials over the whole range of transformation temperatures, several macro-mechanical properties and evolution of martensite fraction. A finite element scheme is proposed to solve a semi-inverse dynamic problem. For a prescribed temperature range and external stress-boundary conditions, the outcomes are the possibility to identify the required electrical current density.

A finite-strain constitutive model for magnetorheological elastomers: Magnetic torques and fiber rotations

This paper is concerned with the development of constitutive models for a class of magnetoelastic composites consisting of stiff, aligned cylindrical fibers of a magnetizable material that are embedded firmly in a soft elastomeric matrix. The fibers have elliptical cross section and their (transverse) in-plane axes are also aligned, but their distribution is random and characterized by “elliptical” two-point correlations. Estimates are obtained for the macroscopic response and stability of this new type of magnetorheological elastomer (MRE) under combined in-plane mechanical and magnetic loading by means of the finite-strain homogenization framework and “partial decoupling approximation” of Ponte Castañeda and Galipeau (2011). The resulting macroscopic magnetoelastic constitutive model accounts for the microstructure of the composite and its evolution under finite strains and rotations, as well as for the nonlinear magnetic behavior of the fibers, including the effect of magnetic saturation. When the loading directions are not aligned with the fiber axes, the model predicts magnetic and mechanical torques on the fibers, leading to their in-plane rotation, which is found to have significant effects on the coupled magnetoelastic response of the composite, including the possible development of macroscopic torques on a given finite-size sample of the composite. To eliminate these macroscopic torques, while maintaining the advantageous effects of the fiber rotations, we also investigate the response of a laminated composite consisting of plus/minus orientations of the fibers relative to the layering direction, and subjected to magnetic and mechanical loadings along the layering direction. The results for the actuation tractions, magnetostrictive strain and magnetoelastic moduli demonstrate that the microstructure of these laminated MRE samples can be designed optimally for significantly enhanced magnetoelastic effects. In particular, the actuation tractions and magnetostrictive strains can be made several times larger than the corresponding tractions and strains for isotropic MREs with spherical (circular) particles.

The mechanical behavior of porous metal fiber sintered sheets

Porous metal fiber sintered sheets (MFSSs) are a type of low density cellular materials promising for functional and structural applications. A micromechanics random beam model is proposed to investigate the elasto-plastic behavior of MFSSs. The relative density dependence of the elastic constants and yield strength of MFSSs is predicted and found to agree well with available experimental results. Fiber stretching is identified as the dominant deformation mechanism under uniaxial and multiaxial loading. When compared with two-dimensional Voronoi foams and honeycombs, the stretching deformation dominated MFSSs exhibit higher stiffness and tensile strength, but lower compressive strength due to long fiber buckling. With the developed micromechanics model, the multiaxial elasto-plastic responses of MFSSs are simulated. A macroscopic phenomenological constitutive model with a segmented yield function is proposed to describe the predicted multiaxial responses. The yield function and its evolution can be fully calibrated in terms of the uniaxial tension and compression responses rather than complex multiaxial loading responses, which can greatly facilitate practical applications of the model. This constitutive model is also expected to be applicable to other fiber sintered materials with hydrostatic pressure sensitive and asymmetric tension–compression yielding behaviors.

Numerical modeling and analysis of ECC structures

The objective of this paper is to develop a tool for the numerical analysis of full-scale ECC structures. For this purpose, a macroscopic cyclic constitutive model for engineered cementitious composite (ECC) materials is developed based on the response of the material at the stress–strain level under different loading regimes. Various features specific to ECC material such as the unloading and reloading characteristics, different backbone curves in tension and compression, and residual strains are taken into account in the model development. The input parameters are limited to those that can be obtained from conventional monotonic compression and tension tests, thus facilitating its use with minimum information. The model is first validated at the stress–strain level and then implemented into fiber-based finite element analysis software for structural level simulation. The results from simulation of ECC members under cyclic and static time history loading are compared to experimental data for model validation at the structural level. Finally, a parametric study is conducted at the member level to investigate the effect of ECC tensile strength and ductility on the structural level response metrics: stiffness, strength, ductility, and energy dissipation capacity. It is observed that the structural level response metrics change considerably depending on the material properties. With its sensitivity to the main behavioral features of ECC, its simplicity, and its sufficient accuracy, the model is suited for use in predicting the behavior of ECC structures under monotonic, cyclic, and dynamic loading scenarios.