Associative Flow Rule Research Articles

Numerical simulation of hydro-mechanical constraints on the geometry of a critically tapered accretionary wedge

A critically tapered active accretionary wedge was simulated using a numerical analysis of plastic slip-line theory to understand the mechanics of morphologic evolution. The concept of critical state soil mechanics was applied to describe the entire wedge area overlying a basal decollement fault. Presuming a condition of two-dimensional plane strain along the compressional direction, we obtained the numerical solution of conjugate plastic slip lines at a critical state of stress defined by the Coulomb yield criterion. The velocity vectors were obtained by applying the associate flow rule with the boundary conditions at the upper surface of the wedge. Finally, the detachment was determined from the effective stress condition inside the wedge and the sliding friction coefficient along the fault. Our numerical simulations demonstrate that the morphology of a critically tapered wedge is dependent on the frictional strengths of both the wedge materials and the basal fault. The critical taper angle decreases with increasing internal friction angle and decreasing basal friction coefficient. The results also revealed that the pore pressure controls the morphology of the accretionary wedge for cohesive sediments but not for non-cohesive materials. The effect of pore pressure on the morphology of a critically tapered accretionary wedge becomes more significant as the cohesion increases. Assuming that the cohesion is very low, we could infer the ranges of strengths that most observed wedge geometry data have 0.3–0.6 for the basal friction coefficient and ~35–45° for the internal friction angle of the wedge materials.

Anisotropic plasticity characterization of 6000- and 7000-series aluminum sheet alloys at various strain rates

This study aims to characterize the high strain rate constitutive behavior and the anisotropic plasticity of three high strength aluminum sheet alloys, AA6013-T6, AA7075-T6 and a developmental AA7xxx-T76 alloy. A comprehensive series of mechanical characterization tests were performed under uniaxial tension, simple shear and through-thickness compression (representing equal-biaxial tension) at room temperature and various strain rates ranging from 10−3 to 103 s−1. It was observed that all three alloys exhibited appreciable plastic anisotropy in terms of the measured r-values along various sheet orientations, although they exhibited minimal anisotropy in the stress response. For all three alloys, the flow stress slightly increases with increasing strain rate, which indicates mild positive strain rate sensitivity. However, the flow stress ratios and r-values do not show any explicit relation with respect to strain rate suggesting a negligible effect of strain rate on in-plane plastic anisotropy. Measurement of the constitutive response to large strains (beyond onset of diffuse necking) was achieved using shear specimens that are not subject to onset of necking. To model the flow stress dependence on equivalent plastic strain and strain rate, an extended Hockett–Sherby constitutive model is proposed and shown to fit accurately the experimental data over the studied range of strain and strain rates. The yield behavior of the sheet alloys was predicted using the Barlat YLD2000-2D plane stress yield function (Barlat et al., 2003) assuming both associative and non-associative (NA) flow rules, as well as for general stress states employing two 3D anisotropic models, Barlat YLD2004 (Aretz and Barlat, 2004) and modified–Yoshida (Lou and Yoon, 2018). An associative flow rule was assumed for both of the higher order 3D yield functions considered herein. The measured flow stress ratios and r-values from the uniaxial tension and equal-biaxial tension experiments, along with the stress ratios from the shear tests, were used to determine the coefficients of the yield functions. The calibrated yield functions exhibited good agreement with the experimental data for most loading conditions considered. The higher order Barlat YLD2004 and modified–Yoshida yield functions provided the better predictive accuracy. The non-associative Barlat YLD2000 yield function also performed reasonably well, while the associative Barlat YLD2000 yield function had the lowest accuracy. The constitutive models were implemented into the FE simulation, in order to evaluate the accuracy of the model parameters calibrated in this work.

Investigation of evolving yield surfaces of dual-phase steels

Abstract The aim of this paper is to describe the evolving yield behavior of dual-phase steels during plastic deformation characterized for ten loading paths using a series of mechanical tests including uniaxial tension, uniaxial compression, in-plane torsion and cruciform biaxial tension with the aid of digital image correlation techniques for strain measurement. Large plastic strains in the gauge area of cruciform specimens tested were enabled by a laser deposition process to strengthen the arms in order to measure deformation behavior of the sheet without arbitrarily thinning the gauge section. Experimental yield loci were determined for three dual phase steels with different strength levels up to equivalent plastic strains of ˜0.11 for DP590, ˜0.07 for DP780, ˜0.05 for DP980, respectively. Several existing anisotropic yield criteria under both associated flow rule (AFR) and non-associated flow rule (non-AFR) were applied to describe the anisotropic yield behavior of these DP steels. A comparative study was preformed to validate prediction accuracy of yield criteria with experimental measurements including yield loci, yield stresses and r φ -values under uniaxial tension in seven orientations as well as yield stresses and r b -value under equi-biaxial tension. The results show that non-AFR significantly improved prediction accuracy of both stresses and r-values simultaneously. Under non-AFR, an order of two in the yield stress function is sufficient to accurately predict flow stresses. The evolution of both yield stress and plastic potential surfaces of DP steels were illustrated by changing parameters in the yield criterion as functions of equivalent compliance λ ¯ .

Analysis of maximum arching conditions in active plane-strain trapdoors in sand

The trapdoor problem is a useful model to understand the stress distribution around geo-structures. This paper focuses on evaluating the conditions of maximum arching (minimum loads on trapdoors) developing during the lowering of plane-strain active trapdoors in cohesionless granular materials. A parametric study using finite element analysis has been performed to investigate various factors affecting the maximum arching conditions in active trapdoors, with a particular focus on the effect of soil dilatancy. The paper also presents rigorous upper bound limit analysis solutions. Previously published solutions dealing with soil non-associativity have been discussed and compared with the finite element results. The finite element analysis shows that using a Mohr Column model with the associative flow rule and reduced strength parameters, overestimates the load reduction on trapdoors compared with a non-associative model with full soil strength parameters.

Constitutive modeling related uncertainties: Effects on deformation prediction accuracy of sheet metallic materials

Constitutive modeling is the most fundamental issue for nonlinearity characterization of sheet metallic materials. However, uncertainties from multiple sources inevitably involved in material characterizing make the high-order accurate constitutive modeling of sheet materials a non-trivial and challenging issue for reliable finite element (FE) simulations of forming processes. In this study, taking multi-stage cupping of aluminum and steel sheet materials as the case, from four respects of the uncertainty sources, viz., stress updating algorithms, yield criteria, flow rules and hardening laws, the epistemic uncertainties involved in constitutive modeling is articulated and the uncertainties induced errors in deformation prediction are quantitatively evaluated. Within the implicit integration framework, the differences in the updating of normal strain for plane stress and three-dimensional (3D) stress states are elaborated, and typical constitutive models are numerically implemented into explicit FE code. By combining the Hill’48-s (characterized by yield stresses), Hill’48-r (characterized by r-values), YLD2004, CPB06 or Yoon's yield model, with the associated flow rule (AFR) or the non-associated flow rule (NAFR), and Swift, Voce, Hollomon or Ludwik hardening law, the effects of the constitutive modeling related uncertainties on deformation prediction accuracy of sheet metals are investigated. The results show that the constitutive modeling related uncertainties have discrepant effects on deformation prediction of two materials for multiple forming indexes. For AA5352 alloy, the employed models cannot provide accurate prediction with the prediction errors of greater than 1%. While, for TH330 steel, the combination of the NAFR Hill’48 model and Swift hardening law can significantly reduce the uncertainties in the deformation prediction and the prediction errors are less than 1%. By introducing forming limit diagram (FLD), the above uncertainties induced errors in the deformation prediction result in different prediction of necking/fracture failure of the above alloys upon the multi-stage forming processes, viz., the failure of the AA5352 is captured with large prediction error in deformation, while the failure cannot be predicted for the TH330 with minor errors of deformation prediction.

Non associative damage interface model for mixed mode delamination and frictional contact

The present paper proposes a new interface constitutive model based on the non-associative damage mechanics and frictional plasticity. The model is developed in a thermodynamically consistent framework, with three independent damage variables. The non associative flow rules drive the concurrent evolution of the three damage variables. The interface model provides two independent values for the mode I fracture energy and the mode II fracture energy and it is able to accurately reproduce arbitrary mixed mode fracture conditions. The model can also take into account the presence of frictional effects both at the fully debonded zones and at the partially debonded ones. The experimental tests developed by Benzeggagh and Kenane with seven different mixed mode ratios have been numerically simulated with a unique set of constitutive parameters. The split shear torsion, for the evaluation of the mode III delamination toughness, has been analysed by a three-dimensional numerical simulation.

Effect of tangential plasticity on structural response under non-proportional cyclic loading

In an attempt to correct the unrealistic material stiffness predicted by elastoplastic models which adopt an associative flow rule, this paper introduces an innovative technique for the computation of the inelastic contributions generated in a non-proportional loading path. The formulation of these inelastic contributions takes into account the plastic response along the direction normal to the plastic potential, neglecting the irreversible stretch caused by the tangential component of the stress rate. Here, the introduction of tangential plasticity, in combination with the return mapping technique, eliminates this drawback, allowing fast and accurate computation. The present paper focuses on the evaluation of the load-carrying capacity of a steel bridge pier, indicating the necessity of considering the additional tangential plasticity term for a correct description of the structural response.

Active earth pressure for retaining structures in cohesive backfills with tensile strength cut-off

The Mohr-Coulomb failure criterion, which is typically used to represent soil strength, has been found to overestimate the tensile strength of most soils. Considering this fact, the concept of tensile strength cut-off was adopted to reduce or eliminate the tensile strength from the criterion. Based on that concept, this study proposes a kinematical framework for the estimation of active earth pressure on retaining structures in cohesive backfills. According to the associative flow rule, a valid rotational collapse mechanism is established. The resultant active earth pressure is determined from the work rate balance equation and is expressed as a dimensionless active earth pressure coefficient. The proposed approach is validated by a numerical simulation. Design charts are presented for a parametric study and for simple use in engineering. The results show that the presence of soil cohesion has a favorable influence on the stability of retained soil masses. Accounting for the impact of tension cut-off leads to a more conservative result, and this impact tends to be more obvious with the increase of cohesion. Finally, the assumptions that were made about the validity of the associative flow rule and the action point of the resultant earth pressure are discussed.

Pseudo-Spring smoothed particle hydrodynamics (SPH) based computational model for slope failure

Large deformation and strain localization are the common physical processes that a soil slope may encounter when it becomes unstable and fails. Numerical modelling of such phenomenon is generally difficult through mesh-based methods. While the Smoothed Particle Hydrodynamics (SPH), a particle-based method, has emerged as a potential alternative in modelling failure, it still suffers some computational pitfalls mostly ascribed to the use of material independent kernel function. Moreover, in the standard implementation of SPH, the support of the kernel function may significantly affect the computation resulting in unphysical prediction. In this study, an improved SPH based computational framework has been developed for studying stability and failure of soil slopes. Herein, unlike the existing practice, the kernel function is continuously modified and so is the particle interaction depending on the deformation and failure state of the material. The varying particle interaction has been achieved via a pseudo-spring analogy. The soil has been modelled as an elastic-plastic material with Drucker–Prager plasticity and associative flow rule. The factor of safety of the slope, determined by the algorithm has been found to remain unaffected even with different choices of the smoothing length unlike the standard SPH implementation.

Analysis of Anisotropic Plastic Properties of Perforated Sheet Metal Using Homogenization Method

In this study, the anisotropic plastic properties of perforated sheet metal with regular staggered holes were analyzed by the homogenization method. A hexagonal unit cell with periodic boundary conditions was selected for the analysis. To validate the analysis results, comparisons were made with the experimental results obtained from a uniaxial tensile test. The analyzed stress-strain curves obtained from directional tensile tests match the corresponding experimental stress-strain curves. The in-plane variations of the analyzed r-values also agree qualitatively with the experimental results. In the biaxial tensile test, the macro yield surface was evaluated as an equal plastic work contour in the proportional stress path. Obtained results indicate differential (anisotropic) hardening and slight tension-compression asymmetry of the subsequent macro yield surface. In the case of the proportional loading path, the associated flow rule (AFR) for the initial and subsequent yield surfaces is applicable. However, regarding the non-proportional loading path, the AFR may not be applicable in certain cases because the distribution patterns of the strain rate around the holes depend on the history of local work hardening.

A micromechanical inspired model for the coupled to damage elasto-plastic behavior of geomaterials under compression

We propose an elasto-plastic model coupled with damage for the behavior of geomaterials in compression. The model is based on the properties, shown in [S. Andrieux, et al., Un modèle de matériau microfissuré pour les bétons et les roches, J. Mécanique Théorique Appliquée 5 (1986) 471?513], of microcracked materials when the microcracks are closed with a friction between their lips. That leads to a macroscopic model coupling damage and plasticity where the plasticity yield criterion is of the Drucker–Prager type with kinematical hardening. Adopting an associative flow rule for the plasticity and a standard energetic criterion for damage, the properties of such a model are illustrated in a triaxial test with a fixed confining pressure.

Numerical simulation of hydraulic fracture propagation in deep reservior

To investigate the large-scale yielding of rock in the process of the hydraulic fracturing in deep reservoir, the mathematical model of elasto-plastic hydraulic fracture propagation was established. Rock deformation was modeled with the Drucker-Prager yield criterion with associative flow rule. Fluid flow in the fracture was modeled by the lubrication theory. The cohesive zone model was used as the fracture propagation criterion. The problem was modeled numerically with the finite element method to obtain the solution. The correctness of mathematical model and algorithm were verified by comParing its results with the classic model. Research results showed that the created elasto-plastic hydraulic fracture was much shorter and wider than the elastic hydraulic fracture of the same volume of injecting fluid, and higher pressure was needed to propagate an elasto-plastic hydraulic fracture than an elastic hydraulic fracture. Elasto-plastic hydraulic fracture remained open after the applied load was released as a result of the presence of plastic strain. With the increase of the friction angle of the rock, the width of the elasto-plastic hydraulic fractures decreased and the required fluid pressure increased. The hydraulic fracture in the hardened rock was narrower than that in the ideal elasto-plastic rock, and higher pressure was needed to propagate a hydraulic fracture in the hardened rock than in the ideal elasto-plastic rock.

A simple nonlinear constitutive model based on non-associative plasticity for UD composites: Development and calibration using a Modified Arcan Fixture

A simple nonlinear constitutive model based on non-associative plasticity for unidirectional (UD) composites is developed and calibrated using a Modified Arcan Fixture (MAF) and Digital Image Correlation (DIC). The plasticity model accounts for the nonlinear response of unidirectional composites subjected to multiaxial loading, assuming transverse isotropy and negligible plasticity in the fibre direction. The different responses in transverse tension and compression are accounted for by a Drucker–Prager type yield function to include transverse pressure sensitivity. It is shown that using an associative flow rule leads to the prediction of non-physical plastic strain components, whilst the use of a non-associative flow rule resolves the deficiency of the associative model. The non-associative model is calibrated and verified against biaxial test data obtained from glass/epoxy specimens using the MAF, as well as against off-axis test data available in the literature for two additional composite material systems. The nonlinear stress-strain curves for unidirectional composites subjected to multiaxial stress states predicted by the model are in good agreement with all three sets of experimental data, thus demonstrating the predictive capabilities of the proposed model.

Application of an Evolving Non-Associative Anisotropic-Asymmetric Plasticity Model for a Rare-Earth Magnesium Alloy

Magnesium sheet metal alloys have a hexagonal close packed (hcp) crystal structure that leads to severe evolving anisotropy and tension-compression asymmetry as a result of the activation of different deformation mechanisms (slip and twinning) that are extremely challenging to model numerically. The low density of magnesium alloys and their high specific strength relative to steel and aluminum alloys make them promising candidates for automotive light-weighting but standard phenomenological plasticity models cannot adequately capture the complex plastic response of these materials. In this study, the constitutive plastic behavior of a rare-earth magnesium alloy sheet, ZEK100 (O-temper), was considered at room temperature, under quasi-static conditions. The CPB06 yield criterion for hcp materials was employed along with a non-associative flow rule in which the yield function and plastic potential were calibrated for a range of plastic deformation levels to account for evolving anisotropy under proportional loading. The non-associative flow rule has not previously been applied to magnesium alloys which require the use of flexible constitutive models to capture the severe anisotropy and its evolution with plastic deformation. The non-associative flow rule can provide the required flexibility by decoupling the yield function and plastic potential. For the associative flow rule, such flexibility can only be achieved by multiple linear transformations of the stress tensor resulting in expensive models for calibration and simulations. The constitutive model was implemented as a user material subroutine (UMAT) within the commercial finite element software, LS-DYNA, for general 3-D stress states along with an interpolation technique to consider the evolution of anisotropy based upon the plastic work. To evaluate the accuracy of the implemented model, predictions of a single-element model were compared with the experimental results in terms of flow stresses and plastic flow directions under various proportional loading conditions and along different test directions. Finally, to assess the predictive capabilities of the model, full-scale simulations of coupon-level formability experiments were performed and compared with experimental results in terms of far-field load-displacement and local strain paths. Using these experiments, the constitutive model was evaluated across the full range of representative stress states for sheet metal forming operations. It was shown that the predictions of the model were in very good agreement with experimental data.

Modelling Thermo‐Viscoplasticity of Case‐Hardening Steels Over Wide Temperature Ranges

AbstractThe aim of this work is the development of a thermodynamically consistent fully coupled thermo‐viscoplastic material model for metals. The model is based on a split of the free energy into a thermo‐elastic, a thermo‐plastic and a purely thermal part and covers nonlinear cold‐work hardening and thermal softening. Nonlinear temperature dependent effects are accounted for the elastic moduli, the plastic hardening moduli, the thermal expansion, the heat capacity and the heat conductivity. Furthermore, strain rate‐dependency of the current yield stress is realized using a temperature dependent nonlinear Perzyna‐type viscoplastic model based on an associative flow rule. The model and its parameters are fitted against experimental data for case hardening steel 16MnCr5 (1.7131).

A nonlinear simulation of a bi-failure specimen through improved discretisation methods: A validation study

A robust and efficient scheme is rendered to elastoplastically study the material nonlinearity of structural components. In this investigation, a specimen manufactured from the aluminium alloy AA6061-T6 is considered. It is mechanically loaded under a uniaxial tensile state and the experimental strain datum is analysed by three-dimensional digital image correlation. Due to specific specimen geometry, complex stress states will occur. However, the specimen yields due to an approximated uniaxial stress state. The obtained remote stress/strain from experimental data is used to validate the computational solutions using advanced discretisation approaches. Therefore, as a preliminary numerical study, the model is simulated through the finite element method formulation. Afterwards, another numerical strategy is adopted – the radial point interpolation method. The Newton–Raphson initial stiffness method is thereby adapted to complete the nonlinear solutions algorithm. Furthermore, the elastoplastic demeanour of aluminium alloys is determined with the von Mises yield criterion, an isotropic hardening rule and an associative flow rule. Obtained computational results fit the experimental digital image correlation solution, which allow to conclude that the proposed meshless methodology is efficient and reliable.

On the plastic flow rule formulation in anisotropic yielding aluminium alloys

In the last decade, an increasing trend has been observed in the use of advanced anisotropic material models to improve the numerical predictions about sheet metal-forming processes. Among the actions following this trend, the use of non-associated flow rule (NAFR) models has been presented as an alternative to the classic associated flow rule (AFR) models. In this work, the influence of the flow rule in advanced anisotropic models has been analysed, following a systematic methodology. To evaluate both advantages and disadvantages of the flow rule, models with similar flexibility (i.e. with 8 and 16 parameters) have been compared by performing a finite element (FE) simulation of a cup-drawing process to study earing phenomena regarding accuracy, computational time and complexity. From the present study, it has been concluded that the main drivers on the final cup-drawing profile prediction are the model flexibility and the coefficient selection rather than the flow rule. Even though the NAFR formulation is complex when compared to the AFR regarding numerical implementation, this study shows that the total simulation time in a cup-drawing process while using explicit FE solvers is reduced in some cases, this one being strongly dependent on the material parameter definition.

Evolution of multiple Martensite variants in a SMA thick-walled cylinder loaded by internal pressure

The stress and deformation fields in a SMA ring or a thick-walled cylinder loaded by internal pressure at constant temperature (over the start temperature of the martensitic transformation) are determined in closed form under plane stress loading conditions. The phenomenological SMA constitutive model incorporates the volume fractions of multi-variants Martensite, which are assumed to evolve linearly with the Tresca effective stress, according to the associative flow rule and the corner flow rule. Initially, the cylinder is everywhere in a state of Austenite. The application of an internal pressure then triggers the martensitic transformation starting from the inner radius of the cylinder wall and extending towards the outer radius. If the wall thickness is large enough, the tangential stress may vanish at the inner radius and correspondingly the stress state may reach a corner of the Tresca transformation condition, thus originating two different Martensite variants according to the corner transformation rule. The admissible phase partitions within the wall thickness originating during the loading process have been systematically investigated according to the ratio between the outer and inner radii. The results obtained here suggest that the loading process should be interrupted soon after the complete martensitic transformation is achieved at the inner radius of the cylinder to avoid permanent plastic deformations.

A new material model for concrete subjected to intense dynamic loadings

The development of material models for concrete subject to intense dynamic loadings, e.g., impact, penetration and blast loadings, is a topic of current research. A high-fidelity physics-based concrete material model should not only capture the high strain-rate, high pressure and large strain generated by intense dynamic loadings, but also the typical failure phenomena of concrete material, e.g., spalling and cracking. This paper presents a new concrete material model, which includes the advantages and discards the disadvantages of frequently used commercial material models. In this model, a three-invariant failure surface is proposed by interpolating the maximum strength surface and the residual strength surface based on the level of current damage. The compressive damage and tensile damage are separately treated. The compressive damage is a function of the modified equivalent plastic strain, while the tensile damage is based on observations of recent dynamic tensile tests which demonstrated that, at high strain-rate, the fracture strain is a constant and the fracture energy increases with increase of strain-rate. Strain-rate effect is taken into account through the radial enhancement approach where the failure surface is enhanced equally in radial direction. To capture the shear dilation of concrete material that is ignored by most commercial concrete material models, a partially associative flow rule is introduced. Automated generation of parameters is suggested for convenience of practical application. The new concrete material model is implemented into the finite element code LS-DYNA through user defined material model. To validate the new material model, comprehensive single element tests including the uniaxial, biaxial and triaxial compressions and tensions are firstly conducted, where the improved performances are demonstrated by comparisons of corresponding predictions by the commercial concrete material models. Then selected numerical examples covered a wide range of loading intensity are presented, and numerical predictions are found to be in excellent agreement with corresponding experimental data.

A thermoelastoplastic theory for special Cosserat rods

A general framework is presented to model coupled thermoelastoplastic deformations in the theory of special Cosserat rods. The use of the one-dimensional form of the energy balance in conjunction with the one-dimensional entropy balance allows us to obtain an additional equation for the evolution of a temperature-like one-dimensional field variable, together with constitutive relations for this theory. Reduction to the case of thermoelasticity leads us to the well-known nonlinear theory of thermoelasticity for special Cosserat rods. Later on, additive decomposition is used to separate the thermoelastic part of the strain measures of the rod from their plastic counterparts. We then present the most general quadratic form of the Helmholtz energy per unit undeformed length for both hemitropic and transversely isotropic rods. We also propose a prototype yield criterion in terms of forces, moments, and hardening stress resultants, as well as associative flow rules for the evolution of plastic strain measures and hardening variables.