Viscoplastic Flow Rule Research Articles

Fractional order viscoplastic modeling of anisotropically overconsolidated clays with modified isotach viscosity

Current overstress typed elastic viscoplastic models fall short in describing some time-dependent mechanical behaviors of anisotropically overconsolidated clays comprehensively. This paper presents a rigorous fractional order anisotropic elastic viscoplastic two-surface model for such clays, based on the principles of fractional consistency viscoplasticity and bounding or subloading surface theory. First, a three-dimensional formulation of isotach viscosity is proposed and integrated into two rate-dependent surfaces, i.e., the loading surface and yield surface. Then, by incorporating the stress-fractional operator of the rate-dependent loading surface into isotropic, progressive, and rotational hardening rules, the incremental form of stress-strain-time model with a fractional order viscoplastic flow rule is developed by meeting the consistency condition on the loading surface. Accordingly, the proposed model cannot only maintain the predictive capabilities of a classic bounding surface model but also describe the general features of the time-dependent behavior under various stress conditions. Validation and versatility of the proposed fractional order elastic viscoplastic model are successfully evaluated against constant strain-rate and stress relaxation tests on anisotropically overconsolidated resedimented Boston Blue clay.

Ratcheting assessment of pressurized elbow pipes using the visco-plastic Chen-Jiao-Kim hardening rule frameworks

In this study, the ratcheting responses of pressurized long and short radius 90° elbow pipes made of 304L austenitic stainless steel under displacement-controlled cyclic loading is evaluated mainly using the Chen-Jiao-Kim (CJK) kinematic hardening rule at room temperature. Visco-plastic flow rule and isotropic hardening are introduced into the CJK model to better reflect the time-dependent behavior and cyclic hardening characteristics of the 304L stainless steel material. The model is implemented employing the user material subroutine (UMAT) in the ABAQUS finite element software. The predictions show that the maximum ratcheting strain occurs at the flank of the long and short radius 90° elbow pipes. The influence of internal pressure, cyclic loading, and bending radius on the ratcheting responses of the elbow pipes is respectively investigated. It is found that the predictions of the CJK model are very nearly the same as the test data. In addition, the ratcheting responses of two long radius elbow pipe specimens under force-controlled cyclic bending are evaluated. Further, the C-TDF (The Committee of Three Dimensional Finite Element Stress Evaluation) combined with CJK model and Linear Matching Method (LMM) is respectively used to determine the shakedown and ratcheting boundaries of the pressurized long and short radius 90° elbow pipes under displacement-controlled and force-controlled bending. The comparison with the test data in this study shows that the shakedown and ratcheting boundaries determined by LMM, divides the shakedown and ratcheting zone well.

A bounding surface viscoplastic damage model for time-dependent rock deformation

A novel bounding surface viscoplastic damage (BS-VPD) framework is proposed in this study for capturing the time-dependent rock deformation driven by subcritical cracking. Our constitutive creep model has been established via imposing effective control over stress-state evolution with respect to the characteristic surfaces while enforcing strictly the thermodynamic constraint. In particular, a new viscoplastic flow rule is first developed by upscaling the microscopic subcritical cracking towards macroscopic creep characterization based on the bounding surface concept. A continuum damage criterion has then been deduced from the maximum dissipation principle for capturing realistically the strain hardening and damage softening during sustained creep loading. By further completing the model with a novel strategy for regulating the interactive surface evolution, we have thus reproduced successfully the progressive transition from stable subcritical cracking characteristic of decelerating creep into unstable propagation during accelerating rupture. In addition, a hierarchical procedure has also been formulated in this study to facilitate straightforward calibration via conventional laboratory testing, and the applicability of our BS-VPD model is then verified against constant strain-rate and creep tests of brittle rocks representing distinct lithologies (i.e., granite, sandstone, and gneiss). The close agreement between model simulation and experimental measurement suggests that the proposed BS-VPD framework is able to capture faithfully the trimodal creep behaviour of brittle rocks across primary, secondary, and tertiary regimes.

Rock shear creep modelling: DEM – Rate process theory approach

Understanding the rock creep behavior is necessary to determine the long-term strength and safety of several geotechnical designs. There are several formulations to study the rock creep; however, most of them do not properly capture the tertiary creep. To overcome such limitation, model improvements have been made and new creep models (e.g., creep models with an associated viscoplastic flow rule) have been proposed. As an alternative, the Rate Process Theory (RPT) has been recently used to study the soil/rock creep behavior. This article expands previous works by analyzing the applicability of the Discrete Element Method (DEM) with RPT implementation to simulate Rock Shear Creep (RSC). To do that, (i) 2D DEM direct shear creep tests under Constant Normal Load (CNL) conditions are used, (ii) DEM specimens are built by a combination of the Flat-Joint Contact Model (FJCM) and the Linear Model (LM), and (iii) the DEM + RPT approach is calibrated by using experimental tests from the literature. DEM results presented here illustrate the suitability of DEM–RPT methodology to reproduce all stages of RSC, including tertiary creep. The effect of the applied shear stress and normal stress on RCS is also analyzed. Finally, the most important novelties of this paper are: (1) the DEM–RPT methodology can be easily calibrated by using a laboratory direct shear creep test; (2) the calibrated DEM models are suitable to analyze the main aspects of RSC; and (3) DEM results qualitatively agree with the overall experimental trend published in the literature.

An anisotropic elasto-viscoplastic-damage model for Selective Laser Sintered food

Selective Laser Sintered food exhibits anisotropic mechanical properties that are characteristic for its layer-by-layer manufacturing process. A constitutive model that is capable of describing the anisotropic elasto-viscoplastic-damaging behavior of the material, which was experimentally characterized by compression testing in two distinct principal orientations, is presented. Damage is modeled by applying the effective stress concept in combination with a viscoplastic flow rule that effectively regularizes damage localization. The model allows for quantitative predictions of the mechanical response of the printed food and is a valuable tool for assessing texture properties of food with Selective Laser Sintering.

Uniaxial and Biaxial Response of Anisotropic Polypropylene

AbstractThe response of uniaxial and biaxial anisotropic polypropylene is discussed. A 3D elasto‐viscoplastic constitutive model is developed to account for material anisotropy. The famous Hill’s anisotropic yield criterion is combined with the Eyring relation and implemented in a finite element framework to model the response of the polymer during uniaxial loading. An associated viscoplastic flow rule that describes the magnitude and the direction of the viscoplastic flow is incorporated in the model to simulate complex loading conditions. The model quantitatively captures the yield stresses for uniaxial deformation at a given anisotropic state and material orientation. In addition, the results of simulations demonstrate that the constitutive relations qualitatively describe the material deformation during biaxial loading for both isotropic and anisotropic cases.

Creep constitutive model considering the overstress theory with an associative viscoplastic flow rule

Abstract Tunnel excavation in squeezing ground is very challenging due to the difficulty in making reliable predictions at the preliminary design stage. Tunnel response in squeezing ground is made possible by employing creep constitutive models. However, literature outlines the limitations of the conventional creep constitutive models in estimating delayed deformations due to the squeezing mechanism. Hence this paper presents, a fractional-order derivative viscoelastic viscoplastic (FDVP) constitutive model capable of estimating delayed deformations characterized by squeezing. The FDVP constitutive equations are derived as an extension to the Burgers model and adjusted Perzyna overstress function with an associated viscoplastic flow rule. The constitutive model validation and verification are conducted by using the experimental data obtained from literature and monitored tunnel convergence data, respectively. Thereafter, the constitutive equations are implemented in FLAC3D and applied to simulate deformations responsible for squeezing within a tunnel employing in-built constitutive models for verification purposes. The constitutive model shows very good agreement with experimental data and yields close results with monitored tunnel convergence data. The model can be successfully used in numerical code for tunnel stability analysis in squeezing ground.

Uniaxial Ratcheting Assessment of 304 Stainless Steel Samples Undergoing Step-Loading Conditions at Room and Elevated Temperatures

AbstractThe present study evaluates ratcheting response of 304 stainless steel samples subjected to various step-loading conditions at room and elevated temperatures using the kinematic hardening rules of Ohno–Wang (O–W), AbdelKarim–Ohno (AK–O), and Ahmadzadeh–Varvani (A–V). The hardening rules were employed along with the visco-plastic flow rule to account for the time-dependent response of 304 stainless steel samples. Ratcheting over low–high–low loading sequences consistently showed a small drop in ratcheting strain over the third loading step. This is mainly due to plastic strain accumulation over the first two loading steps preventing ratcheting strain to drop significantly with a drop in the mean stress. Moreover, dynamic recovery terms in these models were further modified through the inclusion of an exponential function developed by Kang to address the dynamic strain aging phenomenon. Low ratcheting rate and shakedown shortly after a few stress cycles within loading steps as operating temperatures varied between 400 and 600 °C were attributed to dynamic strain aging phenomenon in SS304 steel alloy. Progressive ratcheting response and their stress–strain hysteresis loops were highly influenced at various operating temperatures, stress levels, and stress rates. Coefficients in the dynamic recovery term of the A–V model controlled ratcheting progress and hysteresis loops agreeable with those of experimental data over consecutive loading steps. Choices of material constants and the number of segments defined from stress–strain curve based on the O–W and AK–O models noticeably influenced the ratcheting response of steel samples. Predicted ratcheting values by means of the A–V, O–W, and AK–O models were discussed and compared with those of the experimental data.

Viscoplastic Ratcheting Response of Materials under Step-Loading Conditions at Various Cyclic Stress Levels

The present study intends to assess ratcheting response of rate-dependent SS316L, SS316L(N), and 63Sn37Pb alloys under various step-loading conditions at room temperature through use of the Ahmadzadeh-Varvani (A-V) model developed based on a viscoplastic flow rule. The choice of viscoplastic description enabled the model to control plastic strain increments over loading progress at various stress rates. Predicted ratcheting over asymmetric loading cycles and multi-step cycles was influenced by cyclic stress levels, stress rates, and sequence of loading steps. Predicted ratcheting in steel samples SS316L and SS316L(N) over the second loading step in Low-High and High-Low loading sequences was primarily influenced with the stress amplitude in the second step. Multi-step-loading sequences High-Low-High and Low-High-Low in solder samples revealed how noticeable the impact of mean stress on ratcheting is. Predicted ratcheting values by means of the A-V model closely agreed with experimental data of alloys examined in this study.

Experimental and numerical investigation of the mechanical behavior of the AA5383 alloy at high temperatures

Because of its excellent properties, such as good corrosion resistance, high specific strength and important ductility, the AA5383 aluminum alloy is largely employed for naval applications. In this work, the mechanical behavior of the AA5383 alloy at elevated temperatures, which is an important aspect for the control of forming operations, is investigated. For this purpose, an experimental campaign, including uniaxial tension, biaxial tension, and shear tests, is performed to cover an important range of temperatures (623∼723 K) and strain rates (10−4∼10-1 s-1). A constitutive model for the description of the high temperature behavior of the AA5383 alloy is then proposed. For the deformation behavior, this model combines a viscoplastic flow rule with the BBC2003 anisotropic yield criterion. Also, the prediction of ductile fracture, which is an important aspect for formability, relies on an extended version of the modified Mohr-Coulomb criterion. The extension allows including the impact of temperature and strain rate on ductile fracture as well as a cut-off value for stress triaxiality. Finally, numerical simulations of the experimental tests are performed to identify the flow rule, yield criterion and fracture criterion parameters by combining different optimization methods. The numerical and experimental results of the different tests are in good agreement, which indicates that the proposed constitutive model is well suited for investigating the impact of process conditions on the formability of the AA5383 alloy at high temperatures.

A thermodynamically consistent formulation of the Johnson–Cook model

The model of Johnson and Cook, which includes a viscoplastic flow rule and a damage criterion, is widely used to describe the mechanical behaviour of metallic materials subjected to severe loading conditions, such as those encountered during fabrication operations or impact. This model has been built on empirical, rather than physical, grounds. The present paper therefore aims at revisiting the model of Johnson and Cook from the view point of thermodynamics with internal variables. The interest of this approach is twofold. First, it provides a guide for the construction of a complete thermomechanical constitutive model, with some constitutive relations not only for the stress tensor but also specific internal energy, specific entropy and heat flux vector. Second, it allows highlighting some possible limitations of the original model of Johnson and Cook. Such limitations can be circumvented with an alternative model, which is described in the present work. For illustration purpose, some applications of both the original and alternative models are presented in the final section.

Thermomechanical Modeling of Amorphous Glassy Polymer Undergoing Large Viscoplastic Deformation: 3-Points Bending and Gas-Blow Forming.

Polymeric products are mostly manufactured by warm mechanical processes, wherein large viscoplastic deformation and the thermomechanical coupling effect are highly involved. To capture such intricate behavior of the amorphous glassy polymers, this paper develops a finite-strain and thermomechanically-coupled constitutive model, which is based on a tripartite decomposition of the deformation gradient into elastic, viscoplastic, and thermal components. Constitutive equations are formulated with respect to the spatial configuration in terms of the Eulerian Hencky strain rate and the Jaumann rate of Kirchhoff stress. Hyperelasticity, the viscoplastic flow rule, strain softening and hardening, the criterion for viscoplasticity, and temperature evolution are derived within the finite-strain framework. Experimental data obtained in uniaxial tensile tests and three-point bending tests of polycarbonates are used to validate the numerical efficiency and stability of the model. Finally, the proposed model is used to simulate the gas-blow forming process of a polycarbonate sheet. Simulation results demonstrate well the capability of the model to represent large viscoplastic deformation and the thermomechanical coupling effect of amorphous glassy polymers.

A new visco-plastic self-consistent formulation implicit in dislocation-based hardening within implicit finite elements: Application to high strain rate and impact deformation of tantalum

Modeling deformation processes of materials under high strain rate and impact conditions in which the strain rates vary spatiotemporally over several orders of magnitude is challenging, especially in terms of constitutive description. Visco-plastic power-law flow rule, commonly used within crystal plasticity constitutive models, usually introduces superfluous strain rate effects, entering numerically via the slip activation criterion, which result in artificially high flow stress predictions under high strain rate deformation conditions. This paper presents a novel, implicit finite element implementation of a visco-plastic self-consistent polycrystal model that is implicit in dislocation-based hardening with removed strain rate sensitivity introduced by the power-law flow rule. In the model, the strain rate dependence of the flow stress is defined solely by the rate sensitivity of Peierls stress and the evolution of dislocation density within the hardening law, enabling the prediction of high strain rate and impact deformation. As a result, the predicted flow stress is accurate in its magnitude. The physically based strain rate and temperature sensitive model also features, in addition to Schmid, the non-Schmid activation contribution for slip, which is necessary for understanding and modeling the plastic deformation of body-centered cubic metals. The model is calibrated to simulate the strain rate- and temperature-sensitive monotonic deformation of tantalum and is subsequently applied to a Taylor impact test of the same material. Since the impact simulation required remeshing and interpolation of state variables, the spherical linear interpolation algorithm in the space of quaternions is implemented to facilitate the interpolation of texture. Predictions of the simulation were found to compare favorably with experimental measurements of post-test geometrical changes and texture evolution. The implementation and insights from these predictions are presented and discussed in the paper.

Micromechanical modeling of the effect of elastic and plastic anisotropies on the mechanical behavior of β-Ti alloys

Near β-titanium alloys like Ti-5553 or Ti-1023 often exhibit bimodal α−phase constituents embedded in a retained β-phase matrix, which represents up to 40% of the volume. The highly elastic anisotropic β-phase may strongly influence the mechanical behavior of these alloys. The present work models the effect of the coupled role of β-phase elastic and plastic anisotropies on the local and overall responses of a fully β-phase polycrystalline aggregate like the Ti-17 alloy. The model is based on an advanced elasto-viscoplastic self-consistent (EVPSC) homogenization scheme solved by the “translated field” method together with an affine linearization of the viscoplastic flow rule. The effects of elastic anisotropy, crystallographic texture and grain morphology are theoretically studied during uniaxial tensile tests, tension-compression tests as well as multiaxial plastic yielding.First, it is shown that different sets of elastic constants taken from literature give rise to similar effective responses but to widely scattered incompatibility stresses. During uniaxial tensile loading, the highest local incompatibility stresses are achieved in <111> oriented grains at the end of the elastic regime. Likewise, the effect of the β-grain morphology for realistic grain aspect ratios is seen to be weak on the overall behavior but strong on incompatibility stresses. In addition, the elastic anisotropy can have a significant influence on yield surfaces for β-forged textured polycrystals. Finally, the simulated Bauschinger stress monotonically increases with the elastic anisotropy coefficient for a random texture while it may be reduced in case of β-forged texture due to a competition between elastic and plastic sources of incompatibility stresses.

Transitioning rate sensitivities across multiple length scales: Microstructure-property relationships in the Taylor cylinder impact test on zirconium

A finite-element based plasticity model is developed for polycrystals deformed to high-strain-rates. The model is multiscale, covering from thermally activated dislocation motion on a specific crystallographic slip system (nm), to single crystal plasticity (μm), to polycrystalline aggregate plasticity (mm), and ultimately heterogeneous deformation of the macroscale test sample (m). Within the model, the rate dependence in macroscale response arises solely from the microscopic characteristic stress to activate dislocation motion. This is accomplished by introduction of a novel methodology, used at the intermediate length scales, to relax the extraneous rate dependencies occurring as a result of the visco-plastic rate sensitive flow rule commonly associated with single crystal plasticity formulations. The multi-scale model developed here also permits simulations to be carried out in stress-imposed, strain-rate imposed, and mixed stress/strain-rate-imposed boundary conditions, another advancement over previous techniques. Simulation results are presented for the deformation of high-purity Zr in a Taylor impact cylinder test. The variation in sample shape changes, texture evolution, and deformation twin fraction after the test are experimentally examined. These same quantities are calculated with the model and good agreement is achieved in all aspects. We show that without adjustment of material parameters that the thermally activated hardening model applies to much higher strain-rates (104/s–105/s) than the strain-rates used previously to characterize it. This model can be broadly applied to understanding microstructure-property relationships in high-strain-rate deformation processes that generate spatially and temporarily heterogeneous mechanical fields.

Logarithmic rate based elasto-viscoplastic cyclic constitutive model for soft biological tissues

Based on the logarithmic rate and piecewise linearization theory, a thermodynamically consistent elasto-viscoplastic constitutive model is developed in the framework of finite deformations to describe the nonlinear time-dependent biomechanical performances of soft biological tissues, such as nonlinear anisotropic monotonic stress–strain responses, stress relaxation, creep and ratchetting. In the proposed model, the soft biological tissue is assumed as a typical composites consisting of an isotropic matrix and anisotropic fiber aggregation. Accordingly, the free energy function and stress tensor are divided into two parts related to the matrix and fiber aggregation, respectively. The nonlinear biomechanical responses of the tissues are described by the piecewise linearization theory with hypo-elastic relations of fiber aggregation. The evolution equations of viscoplasticity are formulated from the dissipation inequalities by the co-directionality hypotheses. The anisotropy is considered in the hypo-elastic relations and viscoplastic flow rules by introducing some material parameters dependent on the loading direction. Then the capability of the proposed model to describe the nonlinear time-dependent deformation of soft biological tissues is verified by comparing the predictions with the corresponding experimental results of three tissues. It is seen that the predicted monotonic stress–strain responses, stress relaxation, creep and ratchetting of soft biological tissues are in good agreement with the corresponding experimental ones.

A damage-based elastic-viscoplastic constitutive model for amorphous glassy polycarbonate polymers

This paper presents a new damage-based elastic-viscoplastic constitutive model for amorphous glassy polycarbonate (PC) within the framework of irreversible thermodynamics and continuum damage mechanics (CDM). To this end, experimental investigation, theoretical formulation and numerical implementation are performed. In the experiment part, noticeable strain rate and temperature dependent mechanical responses were observed in uniaxial compression tests over a wide range of strain rates and temperatures. Moreover, damage evolution associated with the decreasing elastic modulus was highlighted in cyclic loading-unloading tests. Based on the experimental data, an elastic-viscoplastic model coupled with damage formulation is developed. Constitutive equations, specifically the strain rate and temperature dependent yield criteria, the viscoplastic flow rule and the damage evolution law, are derived from the Helmholtz free energy and the Clausius-Duhem entropy inequality. Introducing an elastic-damage predictor/viscoplastic corrector scheme, a time-discrete frame of the constitutive equations is presented. The nonlinear time-discrete constitutive system is further simplified into a single-scalar Newton-Raphson scheme in view of computational efficiency. The model is then implemented into the finite element program LS-DYNA, by using a user-defined material subroutine (UMAT). The good correlation between model predictions and experimental data demonstrates the capabilities of the proposed model capturing mechanical behavior and damage evolution of PC over a wide range of strain rates and temperatures.

Viscoplastic flow rule for dislocation-mediated plasticity from systematic coarse-graining

The plastic response of metals is determined by the collective, coarse-grained dynamics of dislocations, rather than by the dynamics of individual dislocations. The evolution equations at both levels are quite different, for example considering their dependence on the applied mechanical load. On the one hand, the relation between the configurational force and dislocation velocity for individual dislocations is linear. On the other hand, in phenomenological crystal plasticity models, the relation between load and plastic slip is highly non-linear and often taken of power-law form. In this work, it is shown that this difference is justified and a consequence of emergent effects. Previously, an expression for the macroscopic dislocation flux was derived by systematic coarse graining (Kooiman et al., 2015). This expression has been evaluated numerically in this paper. The resulting relation between dislocation flux and applied mechanical load is found to be of power-law form with an exponent 3.7, while the underlying Discrete Dislocation Dynamics has a linear flux–load relation.

Micromechanical modeling of hardening mechanisms in commercially pure α-titanium in tensile condition

Tensile tests on commercially pure α-titanium show a three-stage behavior giving rise to a well on the strain dependence of the work hardening. An opposite strain rate effect on the well depth is found whether specimens are elongated along the rolling or the transverse direction. Slip lines analysis reveals an initial predominance of prismatic slip, particularly pronounced in specimens strained along the rolling direction. The relative activity of prismatic slip is then observed to decrease with the samples deformation. These results provide grounds for elaboration of an elasto-viscoplastic self-consistent model based on the translated field method and an affine linearization of the viscoplastic flow rule, and capable of explaining such peculiar work hardening behavior. The model considers crystal plasticity and deals separately with mobile dislocation density and dislocation velocity. It assumes lower strain rate sensitivity as well as higher dislocation multiplication rate for prismatic systems. Based on these assumptions, the model reproduces correctly the stress–strain curves and gives sound estimates of Lankford coefficients, prismatic slip activity and textures evolution. Most importantly, the opposite effect of strain rate on the well depth with regard to the orientation of the tensile axis is qualitatively retrieved, which allows putting forward an explanation of the observed phenomena.

Two-temperature continuum thermomechanics of deforming amorphous solids

There is an ever-growing need for predictive models for the elasto-viscoplastic deformation of solids. Our goal in this paper is to incorporate recently developed out-of-equilibrium statistical concepts into a thermodynamically consistent, finite-deformation, continuum framework for deforming amorphous solids. The basic premise is that the configurational degrees of freedom of the material – the part of the internal energy/entropy that corresponds to mechanically stable microscopic configurations – are characterized by a configurational temperature that might differ from that of the vibrational degrees of freedom, which equilibrate rapidly with an external heat bath. This results in an approximate internal energy decomposition into weakly interacting configurational and vibrational subsystems, which exchange energy following a Fourier-like law, leading to a thermomechanical framework permitting two well-defined temperatures. In this framework, internal variables, that carry information about the state of the material equilibrate with the configurational subsystem, are explicitly associated with energy and entropy of their own, and couple to a viscoplastic flow rule. The coefficients that determine the rate of flow of entropy and heat between different internal systems are proposed to explicitly depend on the rate of irreversible deformation. As an application of this framework, we discuss two constitutive models for the response of glassy materials, a simple phenomenological model and a model related to the concept of Shear-Transformation-Zones as the basis for internal variables. The models account for several salient features of glassy deformation phenomenology. Directions for future investigation are briefly discussed.