Current Constitutive Models Research Articles

Stress-Strain Curves and Modified Material Constitutive Model for Ti-6Al-4V over the Wide Ranges of Strain Rate and Temperature.

The mechanical properties of Ti-6Al-4V alloy are sensitive to strain rate and temperature load. The finite element simulation results of high-speed machining Ti-6Al-4V alloy depend on the accurate description of dynamic deformation. However, it is hard to describe the flow stress behavior in current constitutive models in a complex high-speed machining process for Ti-6Al-4V alloy. In this paper, the stress-strain curves of Ti-6Al-4V alloy under the wide ranges of strain rate and temperature are obtained by high-velocity uniaxial impact tests. The apparent coupling between temperature and strain is observed, which proves that the temperature is dependent on a hardening effect for Ti-6Al-4V alloy. A function describing the coupling between temperature and strain is then introduced into the modification for the original Johnson-Cook (JC) constitutive model. The maximum deviation between the predicted data from using the proposed modified JC constitutive model and experimental data is reduced from 10.43% to 4.19%. It can be concluded that the modified JC constitutive model is more suitable to describe the temperature-dependent hardening effect, which provides strong support for accurate finite element simulation of high-speed machining Ti-6Al-4V alloy.

A method for three-dimensional quantification of vascular smooth muscle orientation: application in viable murine carotid arteries.

When studying in vivo arterial mechanical behaviour using constitutive models, smooth muscle cells (SMCs) should be considered, while they play an important role in regulating arterial vessel tone. Current constitutive models assume a strictly circumferential SMC orientation, without any dispersion. We hypothesised that SMC orientation would show considerable dispersion in three dimensions and that helical dispersion would be greater than transversal dispersion. To test these hypotheses, we developed a method to quantify the 3D orientation of arterial SMCs. Fluorescently labelled SMC nuclei of left and right carotid arteries of ten mice were imaged using two-photon laser scanning microscopy. Arteries were imaged at a range of luminal pressures. 3D image processing was used to identify individual nuclei and their orientations. SMCs showed to be arranged in two distinct layers. Orientations were quantified by fitting a Bingham distribution to the observed orientations. As hypothesised, orientation dispersion was much larger helically than transversally. With increasing luminal pressure, transversal dispersion decreased significantly, whereas helical dispersion remained unaltered. Additionally, SMC orientations showed a statistically significant (p < 0.05) mean right-handed helix angle in both left and right arteries and in both layers, which is a relevant finding from a developmental biology perspective. In conclusion, vascular SMC orientation (1) can be quantified in 3D; (2) shows considerable dispersion, predominantly in the helical direction; and (3) has a distinct right-handed helical component in both left and right carotid arteries. The obtained quantitative distribution data are instrumental for constitutive modelling of the artery wall and illustrate the merit of our method.

Probabilistic predictive modelling of carbon nanocomposites for medical implants design

Modelling of the mechanical properties of carbon nanocomposites based on input variables like percentage weight of Carbon Nanotubes (CNT) inclusions is important for the design of medical implants and other structural scaffolds. Current constitutive models for the mechanical properties of nanocomposites may not predict well due to differences in conditions, fabrication techniques and inconsistencies in reagents properties used across industries and laboratories. Furthermore, the mechanical properties of the designed products are not deterministic, but exist as a probabilistic range. A predictive model based on a modified probabilistic surface response algorithm is proposed in this paper to address this issue. Tensile testing of three groups of different CNT weight fractions of carbon nanocomposite samples displays scattered stress–strain curves, with the instantaneous stresses assumed to vary according to a normal distribution at a specific strain. From the probabilistic density function of the experimental data, a two factors Central Composite Design (CCD) experimental matrix based on strain and CNT weight fraction input with their corresponding stress distribution was established. Monte Carlo simulation was carried out on this design matrix to generate a predictive probabilistic polynomial equation. The equation and method was subsequently validated with more tensile experiments and Finite Element (FE) studies. The method was subsequently demonstrated in the design of an artificial tracheal implant. Our algorithm provides an effective way to accurately model the mechanical properties in implants of various compositions based on experimental data of samples.

Parametric Study on the Nonlinear Dynamics of a Three-Stay Cable Network under Stochastic Free Vibration

AbstractCrossties and cable networks are used on cable-stayed bridges to mitigate wind-induced stay vibration. A nonlinear free-vibration analysis of a cable network with random oscillation amplitudes is presented in this study. Specifically, a nonlinear restoring-force spring model is introduced at the crosstie to simulate in-plane network free vibration at large amplitudes. The current constitutive model of the crosstie is also combined with taut-cable theory to simulate the dynamics in the stays and is solved by the equivalent linearization method. Stochastic functions are introduced in the model because the measure of the amplitudes produced by wind- and rain-wind–induced vibrations in the network can be affected by various uncertainties. The stochastic approximation (SA) algorithm is applied to find the roots of the characteristic polynomials associated with a stochastic vibration amplitude parameter. Brute force Monte Carlo methods are also used to analyze the SA convergence properties.

Three-dimensional vascular smooth muscle orientation as quantitatively assessed by multiphoton microscopy: mouse carotid arteries do show a helix.

Smooth muscle cells (SMCs) play a pivotal role in regulating vascular tone in arteries, and are therefore an essential part of constitutive models of the artery wall. In the present study, we developed a method to quantify in 3D the orientation of SMCs in the intact artery wall. We stained cell nuclei in excised mouse carotid arteries mounted between micropipettes and imaged these in 3D using two-photon laser scanning microscopy. A clustering method was used to identify individual nuclei. Orientations of these nuclei (as a representative of the SMC orientations) were found by calculating the inertia matrix eigenvectors. Subsequently, SMC locations and orientations were converted to cylindrical and spherical coordinate systems, respectively. We found SMCs to be arranged in two distinct layers. For each of these layers, SMC orientations were described by a Bingham distribution. Distributions showed a statistically significant helical and transversal angular component in both inner and outer layers. In conclusion, this study demonstrates that SMC orientation can be quantified in 3D, and shows a distinct helical as well as transversal orientation. The acquired distribution data are essential to improve current constitutive models of the artery wall, by describing physiological SMC orientation and dispersion.

Role of phase transformation in chip segmentation during high speed machining of dual phase titanium alloys

Chip segmentation during machining of titanium alloys is primarily due to adiabatic shear localization associated with thermally driven α–β phase transformation at extremely high speeds. Current constitutive material models used in simulating the machining process ignore the role of phase transformation in shear localization and its influence on the material associated dynamic response. This research presents a new phase approach to chip segmentation that includes a recently developed constitutive material model based on the self-consistent method (SCM) that accounts for material composition, as well as α–β phase transformation, during machining. This SCM-based model is implemented in the finite element framework to validate and predict the effects of starting material property, cutting speeds, uncut chip thicknesses, rake angles, tool radius, and friction coefficients on the strains, temperatures and β volume fractions in chip segmentation. It confirms that cutting speed and uncut chip thickness have great impact, rake angle has less effect, tool radius and friction coefficient have the least effects on chip segmentation. However, tool geometry as well as machining parameters have great influence on the machined surface in terms of temperature magnitude, affected depth and the associated α–β phase transformation.

Ordered rate constitutive theories in Lagrangian description for thermoviscoelastic solids with memory

This paper presents ordered rate constitutive theories in Lagrangian description for compressible as well as incompressible homogeneous, isotropic thermoviscoelastic solid matter with memory in which the material derivative of order m of the deviatoric stress tensor and heat vector are functions of temperature, temperature gradient, time derivatives of the conjugate strain tensor up to any desired order n, and the material derivatives of up to order m−1 of the stress tensor. The thermoviscoelastic solids described by these theories are called ordered thermoviscoelastic solids with memory due to the fact that the constitutive theories are dependent on orders m and n of the material derivatives of the conjugate stress and strain tensors. The highest orders of the material derivative of the conjugate stress and strain tensors define the order of the thermoviscoelastic solid. The constitutive theories derived here show that the material for which these theories are applicable have fading memory. As is well known, the second law of thermodynamics must form the basis for deriving constitutive theories for all deforming matter (to ensure thermodynamic equilibrium during evolution), since the other conservation and balance laws are independent of the constitution of the matter. The entropy inequality expressed in terms of Helmholtz free energy density \({\Phi}\) does not provide a mechanism to derive a constitutive theory for the stress tensor when its argument tensors are stress and strain rates in addition to others. With the decomposition of the stress tensor into equilibrium and deviatoric stress tensors, the constitutive theory for the equilibrium stress tensor is deterministic from the entropy inequality. However, for the deviatoric stress tensor, the entropy inequality requires a set of inequalities to be satisfied but does not provide a mechanism for deriving a constitutive theory. In the present work, we utilize the theory of generators and invariants to derive rate constitutive theories for thermoviscoelastic solids with memory. This is based on axioms and principles of continuum mechanics. However, we keep in mind that these constitutive theories must satisfy the inequalities resulting from the second law of thermodynamics. The constitutive theories for heat vector q are derived: (i) strictly using conditions resulting from the entropy inequality; (ii) using the theory of generators and invariants with admissible argument tensors that are consistent with the stress tensor as well as the theories in which simplifying assumptions are employed which yield much simplified theories. It is shown that the rate theories presented here describe thermoviscoelastic solids with memory. Mechanisms of dissipation and memory are demonstrated and discussed, and the derivation of memory modulus is presented. It is shown that simplified forms of the general theories presented here result in constitutive models that may resemble currently used constitutive models but are not the same. The work presented here is not to be viewed as extension of the current constitutive models; rather, it is a general framework for rate constitutive theories for thermoviscoelastic solids with memory based on the physics and derivations that are consistent within the framework of continuum mechanics and thermodynamics. The purpose of the simplified theories presented in the paper is to illustrate possible simplest theories within the consistent framework presented here.

Toward the thermodynamic modeling of reacting ionic mixtures

Based on the Muller–Liu entropy principle and the axioms of constitutive theory, a continuum model for reacting ionic mixtures is presented. The influence of microscopic structure on the mixture dynamics is taken into account through the thermodynamics of polar materials. Moreover, mechanical balance laws for classical mixtures under influence of electromagnetic fields and quasi-electrostatic Maxwell’s equations are briefly shown. With an appropriate constitutive model for a diluted and isotropic mixture of non-volatile solutes and by considering the same temperature field for all constituents, constraints on constitutive quantities are imposed, and the conditions for the thermodynamic equilibrium are established from the entropy principle. Furthermore, the nonlinear nature of chemical reactions as well as the reciprocal nature of some irreversible processes is highlighted. Unlike the classical approach for electrolyte solutions, the current constitutive model incorporates thermoelectric and electro-kinetic phenomena into the phenomenological equations, providing a more comprehensive approach of electrolyte solutions dynamics.

Micromechanical modelling of the arterial wall: influence of mechanical heterogeneities on the wall stress distribution and the peak wall stress

Previous tensile tests have shown the high anisotropy and heterogeneity of the arterial wall mechanical properties as a function of age, pathology (e.g. abdominal aortic aneurysm, AAA) and location (Vande Geest et al. 2006). The issues of this anisotropic behaviour and the AAA geometrical shapes have been tackled in many numerical studies. However, the wall mechanical heterogeneity has been very sparsely considered (Tierney et al. 2012). This work aims to study the effects of such heterogeneity on the stress distribution and the peak wall stress during a static pressurisation. Thence, a micromechani- cal-based model was used for the wall, and finite element analyses (FEA) were carried out on idealised AAAs. Unlike many previous phenomenological models, the current constitutive model depends on five material parameters only, which also allow us to control the wall microarchitecture.

Steel Fiber-Reinforced Concrete Panels in Shear: Analysis and Modeling

Finite element (FE) studies are undertaken to investigate the accuracy of currently available concrete constitutive models in predicting the behavior of steel fiber-reinforced concrete (SFRC) panels tested in pure shear. The tension-stiffening, tension-softening, and compression-softening behaviors of the panels are evaluated and compared with predictions made using the disturbed stress field model and currently available concrete constitutive models. In addition, the effects of crack slip and crack spacing parameters on the modeling accuracy are assessed. The FE analysis results indicate that although current constitutive models are able to simulate the behavior of conventionally reinforced concrete panels accurately, the models overestimate the strength and deformation capacity of SFRC elements. Three factors are found to significantly influence the calculation accuracy: the tension stiffening/softening model, the consideration of shear slip on crack surfaces, and the crack spacing parameters.

Modeling and experimental validation of trabecular bone damage, softening and densification under large compressive strains

Vertebral fractures represent a major health problem and involve the progressive collapse of trabecular bone over large compressive strains. This collapse is driven by local failure and interaction of the trabecular rod and plate elements, which translates into stress softening and densification at the material level. Current constitutive models for trabecular bone are essentially limited to infinitesimal strains. Accordingly, the aim of this work was to extend our current phenomenological model of trabecular bone (Garcia et al., 2009) for the simulation of large compressive strains by including post-yield softening and densification. A constitutive model of trabecular bone based on both volume fraction and trabecular orientation was formulated in a proper theoretical framework, implemented in commercial FE software and validated with human vertebral sections subjected to large compressive strains. As it is for infinitesimal strains, the evolution of plastic strains and damage is described by local internal variables. An isotropic softening rule was controlled by the cumulated plastic strain and a non-linear elastic spring was added to account for densification of the porous material in moderate-to-large compressive strains beyond a given threshold. To avoid convergence problems occurring as a result of softening, a consistent visco-plastic regularization approach was adopted.The experimental results for 37 vertebral sections from previous work (Dall'Ara et al., 2010) were used to validate the constitutive model for compressive loading up to 45% of the average axial deformation. This validation study showed that the model provides both qualitative predictions of damage localization on the cortex and quantitative predictions of dissipated energy (ρC=0.912) of vertebral body behavior under large compressive strains. Since the evolution of the internal variables was considered in local manner, a mesh sensitivity analysis of the finite element model was conducted via two different mesh sizes and revealed that strain localization was dominated by trabecular bone heterogeneity. To our knowledge, this model is the first to simulate collapse of trabecular bone and may help improve the biomechanical understanding of several musculoskeletal conditions such as vertebral fractures or orthopedic implant migration.

Evolution of microstructure in compacted London Clay during wetting and loading

The influence of fabric on mechanical behaviour is explicitly considered in some of the current constitutive models for unsaturated soils. These are based on assumptions regarding the interaction between different levels of structure, which still require experimental validation. A study has been carried out to investigate the evolution of fabric in a compacted natural clay during wetting and loading, and the results are presented in this paper. Samples of London Clay were statically compacted to the same initial conditions, dry of optimum moisture content in a Proctor plot, and subsequently taken along complex stress paths involving wetting, loading, or a combination of both. Mercury intrusion porosimetry (MIP) and environmental scanning electron microscopy (ESEM) microstructure investigations were carried out to observe and quantify the change in fabric associated with each path. The soil fabric was observed to change from an aggregate to a matrix type structure along all wetting paths. This transition was found to take place only when the suction was reduced to a value close to zero kPa. Results also showed that fabric changes during yielding were stress path-dependent. It was not possible to find a correspondence between the volume of free porosity and the volume of intra-aggregate pores, as suggested by some authors. Finally, the volume of pore water was found to agree closely with the volume of intra-aggregate pores, providing support to the assumption that in an unsaturated aggregate microstructure the clay aggregations are saturated.

A constitutive model of the human vocal fold cover for fundamental frequency regulation

The elastic as well as time-dependent mechanical response of the vocal fold cover (epithelium and superficial layer of the lamina propria) under tension is one key variable in regulating the fundamental frequency. This study examines the hyperelastic and time-dependent tensile deformation behavior of a group of human vocal fold cover specimens (six male and five female). The primary goal is to formulate a constitutive model that could describe empirical trends in speaking fundamental frequency with reasonable confidence. The constitutive model for the tissue mechanical behavior consists of a hyperelastic equilibrium network in parallel with an inelastic, time-dependent network and is combined with the ideal string model for phonation. Results showed that hyperelastic and time-dependent parameters of the constitutive model can be related to observed age-related and gender-related differences in speaking fundamental frequency. The implications of these findings on fundamental frequency regulation are described. Limitations of the current constitutive model are discussed.

Finite element forming simulation for non-crimp fabrics using a non-orthogonal constitutive equation

The forming behaviour of non-crimp fabric (NCF) was simulated using finite element (FE) analysis incorporating a non-orthogonal constitutive model. NCFs feature asymmetric shear behaviour caused by the stitching used to hold the tows together. This asymmetric shear property causes an asymmetric draping pattern of NCF, even when formed over a symmetrical hemispherical forming tool. Current work focuses on the feasibility of a continuum mechanics model to simulate the asymmetric forming behaviour of NCF. The constitutive equation consists of two parts: the tensile contribution from fibre reinforcement and the shear stiffness. For the fibre directional properties, a non-orthogonal equation originally developed for woven fabric was adopted. The shear stiffness was modelled through a constitutive equation incorporating picture-frame shear data. Both a picture-frame shear test and forming of NCF over a hemisphere tool were simulated by commercial finite element software with the current constitutive model implemented within a user material subroutine. The virtual picture-frame test confirmed the validity of the constitutive equation in simulating planar deformation behaviour of NCF. Furthermore, the numerical analysis of hemispherical forming suggests that increasing blank-holder force decreases the asymmetry of the draped pattern.

Application of a Temperature Dependent Viscoplastic Hierarchical Single Surface Model for Asphalt Mixtures

Asphalt mixtures exhibit both temperature and rate dependencies. Most current constitutive models used in design and prediction of the performance of asphalt pavements do not incorporate both characteristics into their equations. Therefore, they do not effectively reflect the actual pavement responses under traffic loads and environmental conditions. This paper presents a temperature dependent viscoplastic model that incorporated temperature and loading rate into the Hierarchical Single Surface plasticity based model. The model presented in this paper was able to reflect the nonlinear plasticity, as well as the temperature and loading rate dependencies of the asphalt mixtures. Triaxial compression, triaxial extension, and axial creep tests at three temperatures (28, 40 and 60°C) were performed to calibrate the material properties. A comparison of numerical analysis obtained from the proposed model and experimental results was conducted based on the algorithm proposed by the writers. A reasonable agreement was observed from the back calculation and the experimental results. Further studies of the applicability of this model in predicting pavement performance were recommended.

軽水炉開発における混相流技術

Research and development activities for light water rector power plant have been reviewed focusing on those of multiphase technology in last ten years. In various fields such as reactor safety research, development of thermal-hydraulic analysis codes, improvement of reactor components, design of advanced, nextgeneration, or innovative light water reactors, researches related to multiphase technology have been performed. In future, demands for cost reduction and prevention for severe accident will be increased in development of future light water reactors. There will be increasing interest in replacing the current constitutive relation models with more mechanically or micro-mechanically-based models to make developmental work more efficient.

Constitutive Modeling of 17-12 Mo SPH Stainless Steel Behavior at High Temperature

A systematic comparison of the detailed behaviors predicted by current constitutive models under various loadings and the actual behavior of austenitic steels under the same lodings was carried out. This comparison leads us to reject these models and to propose a new model whose predictions seem more realistic for these loadings: the unified multilayers model in which the material is represented by a composite of layers of various hardnesses. The details of the model are given in the paper.