Anisotropic Constitutive Equations Research Articles

A developed energy-dependent model for studying thermal shock damage and phase transition of composite reinforced panel subjected to lightning strike

Lightning strikes generate large amounts of energy. Thus, composite structures subjected to lightning strikes undergo significant physicochemical changes. In this study, the thermal shock damage and three-phase transition of a composite reinforced panel were investigated through a numerical simulation, an experiment, and ultrasonic C-scanning. An anisotropic constitutive model and PUFF equation of state (EOS) were proposed to study the damage behaviors of composite materials. The dynamical damage behaviors and three-phase transition of the composite reinforced panel were simulated using the proposed constitutive model and PUFF EOS. Finally, a lightning experiment and ultrasonic C-scanning were conducted to validate the numerical results. Obvious fiber upwarping and swelling were observed in the numerical simulation. The melted and vaporized materials caused a reverse thermal shock effect, which led to concave pits, buckling and internal damages in the composite reinforced panel. The numerical results were compared with experimental results and scanning results to validate that the proposed constitutive model and modified PUFF EOS can be well used to simulate the dynamical damage behaviors of composite materials. The simulated damage behaviors of the composite reinforced panel agreed well with the behaviors observed in the experiment.

A strongly objective, robust integration algorithm for Eulerian evolution equations modeling general anisotropic elastic-inelastic material response

A background to the constitutive modeling of elastic-inelastic material response is provided to highlight the uniqueness of the Eulerian formulation of general nonlinear fully anisotropic thermoelastic-inelastic materials proposed in Rubin (1994) [1]. This model introduced Eulerian evolution equations for a triad of microstructural vectors that characterize elastic deformations and anisotropic orientations. Components of tensors which transform like the Cauchy stress referred to these vectors are insensitive to superposed rigid body motions so they can be used to formulate general elastically and inelastically anisotropic constitutive equations. This paper develops a strongly objective, robust numerical algorithm for integrating the evolution equations for the microstructural vectors. This algorithm can easily be implemented into computer codes to simplify the use of general anisotropic constitutive equations for thermoelastic-inelastic material response.

A Co-Rotational Based Anisotropic Elasto-Plastic Model for Geometrically Non-Linear Analysis of Fibre Reinforced Polymer Composites: Formulation and Finite Element Implementation.

Geometrical non-linearity is one of the aspects to be taken into account for accurate analysis of fibre reinforced polymers (FRPs), since large displacements and rotations may be observed in many of its structural applications such as in aircraft wings and wind turbine blades. In this paper, a co-rotational formulation and implementation of an invariant-based anisotropic plasticity model are presented for geometrically non-linear analysis of FRPs. The anisotropic constitutive equations are formulated in the format of isotropic tensors functions. The model assumes an anisotropic pressure-dependent yield function, and in addition to this, a non-associated plastic potential function in order to model realistic plastic deformations in FRPs. The formulation is then cast in the co-rotational framework to consider the geometrical non-linear effects in an efficient manner. The developed model is implemented in the commercial finite element (FE) software ABAQUS/Implicit via the means of the user-defined material subroutine (UMAT). The kinematics within the co-rotational frame is explained briefly while the important aspects regarding the numerical treatment and implementation are discussed in detail. Representative numerical examples at different scales are presented to demonstrate the applicability and robustness of the proposed development.

Plastic anisotropic constitutive equation based on stress-rate dependency related with non-associated flow rule for bifurcation analysis

In metal forming, progress in material models is required to construct a general and reliable fracture prediction framework because of the increased use of advanced materials and growing demand for higher prediction accuracy. In this study, a fracture prediction framework based on bifurcation theory is constructed. A novel material model based on the stress-rate dependence related to a non-associated flow rule is presented. This model is based on a non-associated flow rule with an arbitrary higher-order yield function and a plastic potential function for any anisotropic material. This formulation is combined with the stress-rate-dependent plastic constitutive equation, which is known as the Ito–Goya rule, to construct a generalized plastic constitutive model in which non-normality and non-associativity are reasonably included. Then, by adopting three-dimensional bifurcation theory, which is referred to the 3D theory, a new theoretical framework for fracture prediction based on the initiation of a shear band is constructed. Using virtual material data, a numerical simulation is carried out to produce a fracture limit diagram, which is used to investigate the characteristics of the proposed methodology.

A new inverse method for estimation of in vivo mechanical properties of the aortic wall

The aortic wall is always loaded in vivo, which makes it challenging to estimate the material parameters of its nonlinear, anisotropic constitutive equation from in vivo image data. Previous approaches largely relied on either computationally expensive finite element models or simplifications of the geometry or material models. In this study, we investigated a new inverse method based on aortic wall stress computation. This approach consists of the following two steps: (1) computing an “almost true” stress field from the in vivo geometries and loading conditions, (2) building an objective function based on the “almost true” stress fields, constitutive equations and deformation relations, and estimating the material parameters by minimizing the objective function. The method was validated through numerical experiments by using the in vivo data from four ascending aortic aneurysm (AsAA) patients. The results demonstrated that the method is computationally efficient. This novel approach may facilitate the personalized biomechanical analysis of aortic tissues in clinical applications, such as in the rupture risk analysis of ascending aortic aneurysms.

Experimental and numerical studies of AL7020 formability under orthogonal loading paths with considering yield surface distortion

In metal forming, material points are often subject to complex loading paths even if the applied loading imposed by the forming tools is simple monotonic loading paths. Including the yield surface distortion effect in constitutive equations is expected to highly enhance the capability of the model in simulating the material forming behavior under complex loading paths. In the present work, fully coupled anisotropic constitutive equations with mixed isotropic and kinematic nonlinear hardening strongly coupled with isotropic ductile damage are improved, considering the subsequent yield surfaces distortion. A model based on the framework of non-associative plasticity theory has been developed recently. In this model, a distortion stress tensor S̲d replaces the usual deviator stress tensor S̲ in the yield criterion and dissipation potential. Three parameters Xl1c,Xl1p, and Xl2 are introduced in the coupled constitutive equations, to be used to control the subsequent yield surfaces distortion. This model is applied and compared with the experiments specifically performed for that purpose. Series of experiments are conducted using the material AL7020 to identify and validate the proposed model. Through the combined loading tests performed on Twente biaxial testing machine, the results of non-proportional loading paths demonstrate that the proposed model can fully represent the forming behavior when complex loading paths. The influence of the yield surface distortion on the plastic flow and on the ductile damage evolution is also discussed in this study.

Modeling of the induced plastic anisotropy fully coupled with ductile damage under finite strains

The introduction of the kinematic hardening induced distortion of the yield surface is expected to enhance the capability of the model in simulating the material behavior under complex loading paths occurring during sheet metal forming processes.In this work non-associative elastoplastic anisotropic constitutive equations accounting for mixed nonlinear hardening and fully coupled with isotropic ductile damage, is improved by considering the distortion of the yield subsequent surfaces. A convenient “distortion” deviatoric stress tensor S̲d is used in both yield criterion and plastic potential instead of the usual classical deviatoric stress tensor S̲. Accordingly, the yield surface distortion is controlled by three adequate material parameters named Xl1c, Xl1p and Xl2c. A parametric study is conducted under simple loading paths (tension and torsion) in order to explore the capabilities of the model in describing the yield surface distorsion. Comparisons with experimental results on aluminum alloy, taken from the literature, are also given. Moreover, the influence of the subsequent yield surfaces distortion under complex non-proportional loading paths on the evolution of the plastic flow with hardening and the ductile damage are also discussed and highlighted.

Ductile damage prediction in sheet and bulk metal forming

Abstract This paper is dedicated to the presentation of an advanced 3D numerical methodology for virtual sheet and/or bulk metal forming simulation to predict the anisotropic ductile defects occurrence. First, the detailed formulation of thermodynamically-consistent fully coupled and fully anisotropic constitutive equations is given. The proposed constitutive equations account for the main material nonlinearities as the anisotropic plastic flow, the mixed isotropic and kinematic hardening and the anisotropic ductile damage under large inelastic strains. Second, the related numerical aspects required to solve the initial and boundary value problem (IBVP) are very briefly presented in the framework of the 3D finite element method. The global resolution schemes as well as the local integration schemes of the fully coupled constitutive equations are briefly discussed. Finally, some typical examples of sheet and bulk metal forming processes are numerically simulated using the proposed numerical methodology.

Anisotropic ductile damage fully coupled with anisotropic plastic flow: Modeling, experimental validation, and application to metal forming simulation

The main goal of this paper is the modeling, numerical simulation, and experimental validation of the anisotropic ductile damage effects on initially anisotropic plastic flow with mixed (isotropic and kinematic) nonlinear hardening under large plastic strains for metal forming processes simulation. A symmetric second-rank damage tensor together with a symmetrized fourth-rank damage-effect tensor is used to describe the anisotropic ductile damage evolution and its effect on the large plastic flow with hardening. Following the concept of effective state variables in the framework of the total energy equivalence assumption, the “Murakami” fourth-rank damage-effect tensor is chosen to describe the anisotropic damage effect on the elastic-plastic behavior including the mixed hardening. The “Lemaitre” ductile anisotropic damage evolution relationships, where the principal directions of the damage rate tensor are governed by those of the plastic strain rate tensor, are used. As difference with the works cited above, the nonlinear mixed isotropic and kinematic hardening is taken into account considering the full and strong damage effects through the effective state variables deduced from the total energy equivalence assumption initially proposed by Saanouni et al. The non-associative plasticity theory is considered, and the “ Hill 1948 ” quadratic (equivalent) stress norm is used to describe the large plastic anisotropic flow accounting for mixed isotropic and kinematic hardening with anisotropic damage effects. The formulation is performed assuming finite plastic strains and small elastic strains through the so-called rotated frame formulation. The obtained model was implemented into ABAQUS/Explicit® FE software thanks to the user’s developed subroutine VUMAT. The numerical aspects related to the time discretization of the fully coupled anisotropic constitutive equations are carefully described. Finally and for the validation purpose, the model is identified using an appropriate experimental data base concerning the grade 316L stainless steel to simulate numerically some metal forming processes.

Three-dimensional numerical simulation of rock deformation in bolt-supported tunnels: A homogenization approach

Despite fully grouted bolts are nowadays widely used for tunnel support, bolting design is still based in many cases on empirical or semi-empirical considerations. This paper describes a three-dimensional theoretical and numerical model for the behavior of tunnels reinforced by bolts. From a theoretical viewpoint, the elastoplastic constitutive equations for the reinforced rock are derived in the framework of homogenization method. Emphasis is given to the formulation of the homogenized strength criterion with account for the bolt/rock interface properties. The anisotropic constitutive equations are then implemented in a 3D finite element computer code in which the processes of excavation, installation of bolts and lining placement are simulated by means of the “activation/deactivation” technique. Due to the multi-potential nature of the plastic flow, a specific iterative algorithm for plastic integration is proposed. The finite element model is applied to the analysis of the Kielder experimental tunnel for which in situ measured data are available. The accuracy of the numerical predictions based on homogenization method is also assessed by comparison with the results derived from implementation of the so-called embedded model.

Effect of base metal anisotropy on the spot weld characteristics of interstitial free steels

An isotropic weld nugget is generally assumed in modelling spot weld behaviour. The present investigation is aimed at understanding the influence of anisotropy of base metal in the characteristics of the spot weld. The spot welding experiments were carried out on specimens along three different orientations with respect to rolling direction. The spot weld section is analysed and the influence of the base metal anisotropy on the microstructure and hardness profile was established. Lap-shear tensile testing was carried out to establish the influence of anisotropy in the weld failure strength. It has been inferred that the base metal anisotropy has considerable influence on the width and hardness of the heat affected zone. The base metal anisotropy also influences the weld failure strength. The variation of weld strength with orientation correlates well with the base metal monotonic strength. An anisotropic constitutive equation is used to predict the variation of weld strength with orientation. The theoretical prediction correlates with the experimental data.

Frontiers in the Constitutive Modeling of Anisotropic Shock Waves

Studies of anisotropic materials and the discovery of various novel and unexpected phenomena under shock loading has contributed significantly to our understanding of the behavior of condensed matter. The variety of experimental studies for isotropic materials displays systematic patterns, giving basic insights into the underlying physics of anisotropic shock wave modeling. There are many similarities and significant differences in the phenomena observed for isotropic and anisotropic materials under shock-wave loading. Despite this, the anisotropic constitutive equations must represent mathematical and physical generalization of the conventional constitutive equations for isotropic material and reduce to the conventional constitutive equations in the limit of isotropy. This article presents the current state of the art in the constitutive modeling of this fascinating field.

Thermodynamically consistent anisotropic constitutive relations for a poroelastic material saturated by two immiscible fluids

This paper presents thermodynamically consistent, fully anisotropic linear poroelastic constitutive equations for porous material saturated by two immiscible fluids formulated with measurable material coefficients. General formulations of fully anisotropic constitutive equations were derived from results of the existing thermodynamic framework. Presented formulations are independent with the averaged pore pressure approach or any assumptions on the relation between Bishop's effective stress coefficient and the saturation of the wetting phase fluid. Series of thought experiments are conducted to reformulate the derived equations into those expressed by measurable material coefficients under the assumption of homogeneous solid constituent. As a result, all the material coefficients were found to be described by porosity, drained elastic constants and the Biot–Willis coefficient tensor, which are common to saturated poroelasticity and Bishop's effective stress coefficient.

Development of Inelastic Analysis for Ni Base Single Crystal Superalloy under Multiaxial Stress States

This paper proposes a method of inelastic structural analysis for YH61 nickel base single crystal superalloy. Hill's anisotropic constitutive equation was applied to describe the elastic-plastic deformation of the superalloy. The equation well described the static tensile curves of the solid bar specimen with and axes at 1173K. The equation also well expressed the torsion curves of hollow cylinder specimens with axis at 1173K. The Hill's equation was applied to simulate the creep deformation of the superalloy. The equation gave satisfactory results for predicting the creep deformation of solid bars having and axes. The Hill's equation was used to predict the creep-fatigue deformation of a cruciform specimen in biaxial loading, resulting in a good agreement with the experimental deformation.

Modelling and Numerical Simulation of Ductile Damage in Bulk Metal Forming

A complete set of fully coupled constitutive equations accounting for both combined isotropic and kinematic hardening as well as the ductile damage under anisothermal conditions at finite (visco)plastic strain is developed and implemented into the general purpose Finite Element code for metal forming simulation. First, the fully coupled anisotropic constitutive equations in the framework of Continuum Damage Mechanics are presented. Attention is paid to the strong coupling between the main thermomechanical fields as thermal effects, elasto-viscoplasticity, mixed hardening, ductile isotropic damage and contact with friction. The associated numerical aspects concerning both the local integration of the coupled constitutive equations as well as the (global) equilibrium integration schemes are presented. The local integration is outlined thanks to the Newton iterative scheme applied to a reduced system of two differential equations. For the global resolution of the equilibrium problem, the classical dynamic explicit (DE) scheme with an adaptive time step control is used. A fully adaptive 2D methodology with mesh and loading sequences adaptation based on some appropriate error estimates is used. For 3D simulations only a constant appropriately refined 3D mesh is used. Various 2D and 3D examples are given in order to show the capability of the methodology to predict the ductile damage initiation and growth during metal forming processes.

3D numerical simulation of anisotropic thin sheet metal slitting process using fully coupled constitutive equations including ductile damage

In this work, an anisotropic plastic constitutive equation accounting for both isotropic and kinematic hardening fully coupled with isotropic ductile damage is presented from both theoretical and numerical standpoints. A radial return algorithm applied to a reduced number of constitutive equations is used in order to compute the stress tensor as well as all the internal variables at the end of time steps. The proposed model is implemented into ABAQUS/EXPLICIT FE code using the Vumat user’s subroutine and the resultant initial and boundary value problem is solved thanks to the classical Dynamic Explicit scheme. Application is made to slitting of 3D anisotropic sheet by studying the influence of some process parameters as the shape of the knives, the sheet thickness and the initial residual stress fields. The proposed methodology is shown to be very useful and helpful when dealing with a “virtual” improvement of any slitting process.

Study for Rectangular Box Drawing of Segmented Variable Blank Holder Force

The sheet forming simulation of rectangular box was conducted by finite element method (FEM), the forming process experiment, further, was investigated, so as to understand the effect of the form of blank holder and the manner of blank holder force (BHF) on the complicated parts forming. In my study, a kind of low carbon steel was investigated, and its mechanical properties were obtained by simple tension tests. The outermost contour of the blank shape was determined by “one step method”. Furthermore, the finite element model was constructed by ANSYS parametric design language (APDL), which has characteristic of grid meshing by Belytschko-Tsay (BT) shell element, applying anisotropic constitutive equation of Barlat yield criterion, dealing contact by penalty function method and using adaptive mesh algorithm in the simulation process. Then the forming process simulation of rectangular box with segmented variable BHF was conducted. On the basis of analyzing the work principle and technical parameters of XP3CEF-100 hydraulic press, the rectangular box drawing system of segmented VBHF was established, which was made of hydraulic press, rectangular box drawing die of segmented blank holder, hydraulic part of blank holder and control part. Finally the low carbon steel forming tests were fulfilled by the rectangular box drawing system on the basis of the simulation result.

Advances in Virtual Metal Forming Including the Ductile Damage Occurrence: Application to 3D Sheet Metal Deep Drawing

An advanced numerical methodology to simulate virtually any sheet or bulk metal forming including various kinds of initial and induced anisotropies fully coupled to the isotropic ductile damage is presented. First, the fully coupled anisotropic constitutive equations in the framework of continuum damage mechanics under large plastic deformation are presented. Special care is paid to the strong coupling between the main mechanical fields such as elastoplasticity, mixed nonlinear isotropic and kinematic hardenings, ductile isotropic damage, and contact with friction in the framework of nonassociative and non-normal formulation. The associated numerical aspects concerning both the local integration of the coupled constitutive equations as well as the (global) equilibrium integration schemes are presented. The local integration is outlined, thanks to the Newton iterative scheme applied to a reduced system of ordinary differential equations. For the global resolution of the equilibrium problem, the classical dynamic explicit (DE) scheme with an adaptive time step control is used. This fully coupled procedure is implemented into the general purpose finite element code for metal forming simulation, namely, ABAQUS/EXPLICIT. This gives a powerful numerical tool for virtual optimization of metal forming processes before their physical realization. This optimization with respect to the ductile damage occurrence can be made either to avoid the damage occurrence to have a nondamaged part as in forging, stamping, deep drawing, etc., or to favor the damage initiation and growth for some metal cutting processes as in blanking, guillotining, or machining by chip formation. Two 3D examples concerning the sheet metal forming are given in order to show the capability of the proposed methodology to predict the damage initiation and growth during metal forming processes.

A numerical method for rapid estimation of drawbead restraining force based on non-linear, anisotropic constitutive equations

Numerical procedures to predict drawbead restraining forces (DBRF) were developed based on the semi-analytical (non-finite-element) hybrid membrane/bending method. The section forces were derived by equating the work to pull sheet material through the drawbead to the work required to bend and unbend the sheet along with frictional forces on drawbead radii. As a semi-analytical method, the new approach was especially useful to analyze the effects of various constitutive parameters with less computational cost. The present model could accommodate general non-quadratic anisotropic yield function and non-linear anisotropic hardening under the plane strain condition. Several numerical sensitivity analyses for examining the effects of process parameters and material properties including the Bauschinger effect and the shape of yield surface on DBRF were presented. Finally, the DBRFs of SPCC steel sheet passing a single circular drawbead were predicted and compared with the measurements.

An Anisotropic Constitutive Equation for the Stress Tensor of Blood Based on Mixture Theory

Based on ideas proposed by Massoudi and Rajagopal (M‐R), we develop a model for blood using the theory of interacting continua, that is, the mixture theory. We first provide a brief review of mixture theory, and then discuss certain issues in constitutive modeling of a two‐component mixture. In the present formulation, we ignore the biochemistry of blood and assume that blood is composed of red blood cells (RBCs) suspended in plasma, where the plasma behaves as a linearly viscous fluid and the RBCs are modeled as an anisotropic nonlinear density‐gradient‐type fluid. We obtain a constitutive relation for blood, based on the simplified constitutive relations derived for plasma and RBCs. A simple shear flow is discussed, and an exact solution is obtained for a very special case; for more general cases, it is necessary to solve the nonlinear coupled equations numerically.