J2 Flow Theory Research Articles

Direct modeling of non-uniform strain field of heterogeneous materials

Engineering materials have a heterogeneous microstructure that induces non-uniform deformations (NUDs) even under uniform stress. Therefore, the development of NUDs in heterogeneous materials under a macroscopically non-uniform stress is characterized by the interactions between micro- and macroscopic NUDs. In this study, we attempt to directly describe microscopic NUDs by introducing a constitutive equation for the strain field. Using polycrystalline metal, which is a well-known heterogeneous material, we performed the uniaxial tensile tests on specimens with curved gauge sections. During modeling, the constitutive equation without nonlocal effect was first fitted based on the distribution of the local equivalent stress and strain. The strain calculated using the fitted equation was regarded as the strain for the homogeneous material, whereas the strain deviating from the fitted equation was assumed to be induced by the microscopic heterogeneity of the material. The respective strains were represented using different constitutive models, and the parameters employed in the models were fitted separately using the experimentally measured strain field. The plastic compliance gradient was employed as a material parameter quantifying the microscopic NUDs. Subsequently, a multiaxial elastoplastic nonlocal constitutive equation was formulated based on the nonlocally extended J2 flow theory. Finite element method simulations introducing the proposed nonlocal effect indicated that a scale-dependent NUD characterized by the interactions between micro- and macro-NUDs could be represented using the proposed model. Furthermore, the proposed model could predict the experimentally observed strain field characterized by a polycrystalline microstructure, such as the dispersion of the strain concentration at the curved gauge section and the appearance of NUDs even under uniform stress.

Burst pressure of thin-walled pipes with arbitrarily orientated isolated surface corrosion defect

Accurate prediction of the burst pressure is essential for structural integrity assessment of pipelines. Current design codes and models for residual burst pressure assessment of pipelines with surface corrosion defects are only applicable to axially orientated defects. Theoretical and empirical models for the prediction of burst pressure of thin-walled pipes with pre-existing localised and arbitrarily oriented surface corrosion defect are proposed. The theoretical model is based on power-law hardening and J2 flow theory while the empirical model is derived from nonlinear finite element analysis results of the burst pressure. Burst pressure experiments are used to validate the model predictions. The theoretical and empirical models are found to be reliable in predicting the effect of defect width, length, and orientation on the burst pressure.

Micromechanical modeling of damage evolution and fracture behavior in particle reinforced metal matrix composites based on the conventional theory of mechanism-based strain gradient plasticity

The particle/matrix interface of metal matrix composites (MMCs) can give rise to extra strength and then affect the local deformation behavior. This strengthening effect originates from the plastic strain gradients due to the incompatibility of plastic deformation between the particle and matrix. However, only limited researches utilized the strain gradient plasticity to study the damage evolution and fracture behavior, in which only one or two of the damage mechanisms (i.e., matrix damage, interface debonding, and particle fracture) was considered. In this work, all of these damage mechanisms were coupled into the finite element model under conventional theory of mechanism-based strain-gradient (CMSG) plasticity. Besides, a new numerical algorithm of geometrically necessary dislocations (GNDs) was proposed for a multi-scale model, and then this model was used to analyze the damage evolution and failure behavior of SiCp/AZ91 composite. The results show that the strengthening effects of plastic strain gradients can describe interface debonding, break the monotonicity of the effective plastic strain with the effective stress at the local area close to the particle/matrix interface, and give a more reasonable distribution of stress and plastic strain compared to the classical J2 flow theory. If the probability distribution of the interfacial strength is considered, the CMSG model has the potential capacity to capture the crack initiation in the matrix. When using the continuum damage mechanics approach to describe the fracture process based on the multi-scale model, the weakening exponent value should be considered.

A framework for formulating and implementing non-associative plasticity models for shell buckling computations

In modelling the behavior of thick-walled metal shells under compressive loads, the use of J2 flow theory can lead to unrealistic buckling estimates, while alternative ‘corner’ models, despite offering good predictions, have not been widely adopted for structural computations due to their complexity. The present work develops a new and efficient plasticity model for predicting the structural response of compressed metal shells. It combines the simplicity of the Von Mises yield surface, with a non-associative flow rule, mimicking the effect of a yield surface corner. This allows for tracing the equilibrium path of the loaded shell and identifying consistently structural instability, employing a single constitutive model. A robust backward-Euler integration scheme, suitable for both three-dimensional (solid) and shell elements is developed, along with the corresponding consistent algorithmic moduli for nonlinear isotropic hardening materials, accounting rigorously for the nonlinear dependence of plastic straining on the direction of strain increments. The model is implemented in ABAQUS as a user material subroutine. Simulations of thick-walled metal cylinders under compression predict structural instability in good agreement with experimental data.

Thermal debonding in compacted graphite iron: effect of interaction of graphite inclusions

This paper focuses on investigation of the interaction effect of graphite inclusions on failure of compacted graphite iron (CGI) subjected to thermal load. Although CGI has been used extensively in industrial applications and studied for many years, its failure under pure thermal load has not yet been fully understood, nor has the influence of interacting graphite particles on it. To analyse this phenomenon, a finite-element two-dimensional unit cell is constructed, consisting of two graphite particles presented as ellipses embedded in a metallic matrix. Each inclusion is surrounded by a thin layer that accounts for its interface in order to study its decohesion from the surrounding matrix. An elastoplastic behaviour is assumed for both graphite and matrix material, described by classical J2 flow theory of plasticity, whereas a traction-separation law is applied to the interface. The model is analysed numerically, with an emphasis on the influence of different parameters such as the inclusion size, the distance between the particles, and the combination of their shapes on thermal debonding, identifying critical combinations that either deteriorate or relieve the effect. The obtained results can provide significant knowledge on the response of CGI under thermal load at the microscale, contributing also to understanding of its macroscopic behaviour at elevated temperatures.

A crystal plasticity based finite element framework for RVE calculations of two-phase materials: Void nucleation in dual-phase steels

A crystal plasticity based finite element (CPFE) framework is developed for performing representative volume element (RVE) calculations on two-phase materials. The present paper investigates the mechanical response and the evolution of microstructure of dual-phase (DP) steels under uniaxial tensile loading, with a special focus on void nucleation. The spatial distribution and morphology of the ferrite and martensite grains in DP steels are explicitly accounted for by generating three-dimensional RVEs with Voronoi tessellations. The effects of microstructural parameters—the volume fraction, morphology and spatial distribution of the martensite phase, and the grain size and orientation distribution of the ferrite phase—on the initiation and spread of localized plastic deformation (leading to void nucleation) are investigated in detail. The soft ferrite phase is modelled by a local crystal plasticity theory with anisotropic elasticity, and the hard martensite phase by the J2 flow theory with isotropic elasticity. The developed CPFE framework successfully predicts both the overall and the local mechanical response, and is perfectly capable of distinguishing between different void nucleation mechanisms experimentally observed for DP steels.

Plastic forming processes of transverse non-homogeneous composite metallic sheets

Abstract In this work an analysis of the radial stress and velocity fields is performed according to the J 2 flow theory for a rigid/perfectly plastic material. The flow field is used to simulate the forming processes of sheets. The significant achievement of this paper is the generalization of the work by Nadai & Hill for homogenous material in the sense of its yield stress, to a material with general transverse non-homogeneity. In Addition, a special un-coupled form of the system of equations is obtained where the task of solving it reduces to the solution of a single non-linear algebraic differential equation for the shear stress. A semi-analytical solution is attained solving numerically this equation and the rest of the stresses term together with the velocity field is calculated analytically. As a case study a tri-layered symmetrical sheet is chosen for two configurations: soft inner core and hard coating, hard inner core and soft coating. The main practical outcome of this work is the derivation of the validity limit for radial solution by mapping the “state space” that encompasses all possible configurations of the forming process. This configuration mapping defines the “safe” range of configurations parameters in which flawless processes can be achieved. Several aspects are researched: the ratio of material's properties of two adjacent layers, the location of layers interface and friction coefficient with the walls of the dies.

Coupled numerical simulation of low-cycle fatigue damage in metal components

A coupled cyclic plasticity-damage model is implemented for simulating low-cycle fatigue in metal components. Constitutive relations account for J2-flow theory with nonlinear kinematic/isotropic hardening, coupled with isotropic continuum damage mechanics. The damage potential is written in a general form, allowing for implementing different damage models. An implicit numerical integration scheme is developed and the incremental update of the internal variables is achieved through the solution of a single scalar equation. Consistent linearisation of the integration algorithm is provided explicitly to guarantee robustness of the proposed algorithm. The algorithm is implemented in a user subroutine and is inserted into a commercial finite element software. Its accuracy and computational efficiency are demonstrated through numerical simulation of large-scale experiments on metal piping components that failed under low-cycle fatigue loading. The numerical analyses are conducted using finite element models of different mesh density, implementing an appropriate simulation methodology. A simple and efficient damage evolution function is employed, regularised with respect to the element’s size, so that the numerical results present negligible mesh dependency. Excellent comparison is observed between experimental and numerical results in terms of global structural response, local strains and the number of cycles for developing through-thickness crack, indicating that the present formulation can be used as an efficient numerical tool for simulating inelastic damage and low-cycle fatigue in large-scale metal structural components.

Numerical study of stationary cracks in bulk metallic glass composites under Mode I, small scale yielding conditions

In this study, finite element analyses of Mode I loading of stationary cracks in in-situ bulk metallic glass composites (BMGCs) are performed under plane strain, small scale yielding (SSY) conditions. In the first part of this work, a thermodynamically consistent finite strain constitutive theory for BMGCs is employed to represent the entire domain. In this homogenized model, the soft crystalline dendrites are considered to obey J2 flow theory of plasticity with isotropic power law hardening, while the bulk metallic glass (BMG) matrix is taken to follow a free-volume based constitutive model. The effects of volume fraction Vf and hardening of the crystalline dendrites on the macroscopic stress and plastic strain fields are systematically studied. The predicted trend of the fracture toughness versus Vf corroborates well with experiments. In the second part, multi-scale SSY analyses are performed, with discrete soft dendrites in the BMG matrix close to notch tip explicitly simulated, while the surrounding region is represented by the homogenized BMGC model. It is found that low strength and/or low hardening dendrites with high Vf dissipate more plastic work in the neighborhood of the notch tip, thereby shielding the relatively brittle BMG matrix.

Strain gradient plasticity in gradient structured metals

Structural gradients in metallic materials can give rise to substantial extra strengths compared to their non-gradient counterparts. This strengthening effect originates from the plastic strain gradients arising in plastically deformed gradient structures. Here we develop a gradient theory of plasticity by incorporating the strengthening effect of plastic strain gradients into the classical J2 flow theory. This gradient theory is numerically implemented to study the strain gradient plasticity in gradient nanotwinned (GNT) Cu with different twin thickness gradients and accordingly different built-in strength gradients. Numerical simulations reveal the primary features of gradient plasticity in GNT Cu under uniaxial tension, including progressive yielding, gradient distributions of plastic strain and extra plastic flow resistance. The combined simulation and experimental results show that the extra strength of GNT Cu depends on the square root of the built-in strength gradient and also on the square root of the saturated plastic strain gradient in fully yielded GNT samples. This finding enables the predictions of optimal gradient structures and associated gradient strength distributions, which suggest possible routes for achieving the maximum strength of GNT Cu.

Numerical and Experimental Fracture Study for 7003 Aluminum Alloy at Different Triaxialities

This paper investigates the fracture process featured wide triaxiality range (− 0.1, 2/3), with varied fracture mode observed by fractography. The micro fracture mode changes from shear fracture in low triaxiality to dimple rupture in high triaxiality. The fracture locus (fracture strain vs. triaxiality) exhibits complex trends, containing increased fracture strain in negative triaxiality. Hardening behavior is calculated by classic J2 flow theory and the Ludwig hardening law. Coupling process of softening and fracture are represented by the GISSMO model. The deforming and the coupling process are solved by 2D and 3D finite element method by LS-Dyna/Explicit. Good agreements are observed between experimental and numerical results in tensile tests, punch tests and three points bending tests, verifying that the GISSMO model and Ludwig hardening law can be used in deforming and fracture process for 7003 alloy. Numerical predictions on perforated beam bending shows that the opening can change the cracking position, increases the ultimate strength and decreases the deflection.

Prediction of Wrinkling of a Beverage Can Subjected to the Redrawing Process by J2 Deformation Theory

Wrinkling of beverage cans is one of the problems faced by can manufacturers and aluminum suppliers. The bottom of an aluminum can is wrinkled by compression during the forming process. In this study, to predict the occurrence of wrinkles during the redrawing process of AA3104 (t = 0.265 mm), which is the material used to fabricate aluminum cans, the classical plasticity J2 deformation theory (J2D) and flow theory (J2F) were considered. J2F considers only the deformation perpendicular to the yield locus, whereas J2D considers the deformation perpendicular to the yield locus and that tangential to the yield locus. Wrinkles are predicted using finite element (FE) analyses based on J2D and J2F, and the results are compared. J2F could not predict the number and amplitude of wrinkles. By contrast, the wrinkles predicted using J2D exhibited good agreement with sample data obtained for a real can. To find the difference between the results obtained using J2F and J2D, evolutions of stress path in a wrinkled element are compared. It was confirmed that compressive stress is more dominant in the J2D case than in the J2F case. Moreover, the measured effective strain of the element is small, under 0.04. In conclusion, J2D is more suitable for predicting the wrinkling behavior of aluminum cans than J2F. In addition, ANOVA and ANOM analysis are performed to evaluate the influence of the design parameters, namely friction coefficient, thickness, and outer profile angle, and the parameters are optimized to reduce wrinkles by combining the Taguchi method with FE simulation based on the J2D theory.

Compact and unified elasto-plastic formulation to study isotropic plates

We introduce a compact and unified shear deformation theory for plates with elasto-plastic behavior. We formulate the kinematics of the two-dimensional structure in a compact and unified manner using the Carrera Unified Formulation. This formulation allows for generalized expansions of the primary variables and through-the-thickness functions. We obtain the governing equations using the principle of virtual work and a finite element discretization. We solve the nonlinear equations using a Newton–Raphson linearization scheme, and linearize the constitutive equations using the algorithmic tangent moduli. We consider the J2 flow theory of plasticity, and use a backwards Euler scheme to update the stresses. We analyze the convergence, and compare the effectiveness of the Mixed Interpolation of Tensorial Components technique in contrasting the shear locking phenomenon in the nonlinear regime to the use of full and uniform reduced integration. We also conduct numerical assessments for plates under uniform and line loads. We compare the present results to those obtained by finite element commercial software, and demonstrate the computational efficiency of the present method.

Effective behavior of composites with combined kinematic and isotropic hardening based on additive tangent Mori–Tanaka scheme

The goal of the present work is to propose a multi-scale approach for composite materials which accounts for kinematic hardening in the phases. For that purpose, the additive/sequential interaction rule and tangent linearization of viscoplastic response proposed for elastic–viscoplastic material can be extended in a straightforward manner. A two phase composite where each phase is elastic–viscoplastic is considered. The viscoplastic flow is governed by a J2 flow theory with an overstress. To find the overall behavior of the composite, a Mori–Tanaka model is applied. Numerical validation of the proposition is carried out by considering a representative volume element with 30 inclusions. Various configurations have been tested: hard or soft inclusion cases with or without isotropic hardening. It is shown that the quality of the model predictions is not affected by the introduction of the kinematic hardening component in the local constitutive behavior. Namely, in most cases considered in the paper the overall stress–strain response as well as the average stress–strain response per phase is accurately estimated. It has been also verified that the obtained backstress components are consistent with the ones predicted by Finite element calculations with ABAQUS Software.

A unified criterion for void growth and coalescence under combined tension and shear

An analytical micromechanics-based yield criterion is developed to describe both void growth and coalescence under combined tension and shear, with smooth transition between growth and coalescence, thus its name unified. The model is obtained by limit analysis over a cylindrical elementary cell embedding a coaxial cylindrical void of finite height. The velocity field employed is an extended counterpart of the discontinuous, yet kinematically admissible trial field utilized in a recent work. Plasticity in the deformable matrix is modeled using rate-independent J2 flow theory, and the effective dissipation function is calculated by exact as well as approximate integration techniques, the latter generating a simpler flow potential. The model is aimed to predict void growth as well as coalescence by internal necking or shearing. The complete yield surface, being function of normal as well as shear stresses, exhibits curved and planar parts signifying void coalescence. The transition between the curved and planar parts is cornerless. The analytical predictions are compared to results of FEM single-step cell-model calculations of limit analysis executed on an identical geometry exposed to quasi-periodic boundary conditions.

A Micromechanics Based Numerical Investigation of Dual Phase Steels

The aim of this paper is to investigate the effects of microstructural parameters such as the volume fraction, morphology and spatial distribution of the martensite phase and the grain size of the ferrite phase on the plasticity and localized deformation of dual-phase (DP) steels. For this purpose, Voronoi based representative volume elements (RVEs) are subjected to proportional loading with constant stress triaxility. Two alternative approaches are employed in a comparative way to model the plastic response of the ferrite phase, namely, micromechanically motivated crystal plasticity and phenomenological J2 flow theory. The plastic response of the martensite phase, however, is modeled by the J2 flow theory in all the calculations. The predictions of both approaches closely agree with each other in terms of the overall macroscopic response of the DP steels, while clear differences are observed in the localized deformation patterns. The results of the present study are also compared with experimental and computational findings from the literature.

A finite strain incremental-secant homogenization model for elasto-plastic composites

This paper presents a finite strain extension of the incremental-secant mean-field homogenization (MFH) formulation for two-phase elasto-plastic composites. The formulation of the local finite strain elasto-plastic constitutive equations of each phase is based on a multiplicative decomposition of the deformation gradient as suggested by Simo in (Computer Methods in Applied Mechanics and Engineering, 99(1):61–112, 1992.). The latter has proposed algorithms which preserve the classical return mapping schemes of the infinitesimal theory by using principal Kirchhoff stresses and logarithmic eigenvalues of the left elastic Cauchy–Green strain. Relying on this property, we show that, by considering a quadratic logarithmic free energy and J2-flow theory at the local level, infinitesimal strain incremental-secant MFH is readily extended to finite strains. The proposed formulation and corresponding numerical algorithms are then presented. Finally, the predictions are illustrated with several numerical simulations which are verified against full-field finite element simulations of composite cells, demonstrating that the micro-mechanically based approach is able to predict the influence of the micro-structure and of its evolution on the macroscopic properties in a very cost-effective manner.

Constitutive analysis of pressure-insensitive metals under axisymmetric tensile loading: A stress triaxiality-dependent plasticity damage model

In this paper, by separately characterizing the influence of stress triaxiality (ratio of hydrostatic stress to von Mises stress) on the stress response to plastic deformation and to damage evolution, the constitutive equation of the pressure-insensitive metal under axisymmetric tensile loading is defined as a new stress triaxiality-dependent plasticity damage model. For this purpose, experimental testing and finite element modelling are performed based on smooth and notched round bars, machined from a high-strength rail steel. The test results show that mechanical behaviour of both types of tensile specimens can be closely simulated based on the same constitutive equation calibrated following the classical J2 flow theory; even though stress triaxiality and damage evolution generated in these two types of specimens are very different. Such an issue is investigated by implementing the new constitutive model to finite element modelling of smooth and notched specimens using the ABAQUS user subroutine VUMAT. It is found that the rates of both damage-free stress response to deformation and damage evolution increase with increasing stress triaxiality. Also, the increase of damage-free stress response to deformation, caused by the increase of stress triaxiality, is offset by the decrease of stress due to the increase of damage evolution. This results in the constitutive equation insensitive to the change of stress triaxiality.

Aluminium plates with pre-formed slits subjected to blast loading

The dynamic response of thin, perforated aluminium plates subjected to blast loading was studied both experimentally and numerically. Two different blast intensities were used and the plates were pre-cut with four horizontal and vertical slits prior to testing. The applied AA6016-T4 plates had an exposed area of 0.3 m x 0.3 m and a thickness of 1.5 mm. Special focus was placed on the dynamic response and failure characteristics of the plates. Uniaxial tensile tests were conducted in three different directions to determine the material behaviour and material parameters were found by inverse modelling using the optimization tool LSOPT. Finally, numerical simulations were performed in the finite element code Abaqus/Explicit where the plates were uniformly loaded with time-dependent pressure histories from similar tests on massive plates. The material behaviour was assumed to follow the J2 flow theory of plasticity and an uncoupled damage model was used in combination with element erosion to predict material failure. The numerical results were in good agreement with the experimental observations and predicted both the dynamic response and the complete tearing of the centre part of the plates.

A Lode-dependent Gurson model motivated by unit cell analyses

In this study, the effects of including a dependency on the third deviatoric stress invariant in the void evolution equation of the Gurson model are examined using unit cell calculations and imperfection band analyses. Finite element analyses of a unit cell model are conducted to approximate the behaviour of the material microstructure. The unit cell was modelled as a cube made from an elastic-plastic matrix governed by J2 flow theory with a spherical void located at the centre. The results of the unit cell calculations show a monotonic decrease in void growth when the stress state changes from generalized tension to generalized compression. To mimic the resulting evolution of the void volume fraction, an extension of the Gurson model based on the shear modification proposed by Nahshon and Hutchinson (2008) is proposed. This Lode-dependent void evolution term is further qualitatively assessed through comparisons with the unit cell simulations and through strain localization predictions using imperfection band analyses. The assessment demonstrates that the proposed modification of the void evolution equation is consistent with the evolution of the unit cell in the case of moderate and high stress triaxiality ratios. Furthermore, the imperfection band analyses exhibit a greater difference between the failure strain values in generalized tension and generalized compression using this Lode-dependent void evolution term compared to similar analyses that employ the original Gurson model or the shear-modified Gurson model. The Lode-dependent void evolution term thus renders the ductility predictions more consistent with previously reported studies based on unit cell calculations for the set of material parameters employed in this study.