Equations Of Plastic Flow Research Articles

Flow Stress Behavior and Governing Equation of Plastic Flow of Low-Carbon Steel at Elevated Temperatures

Flow Stress Behavior and Governing Equation of Plastic Flow of Low-Carbon Steel at Elevated Temperatures

MICROSCOPIC DAMAGE MODEL CONSIDERING THE RESOLVED NORMAL STRESS ON CRYSTAL SLIP PLANE

Studies show that the macroscopic failure of structure results from the microscopic damage within materials, such as cleavage and slip plane decohesion. Therefore, it is helpful to understand the deformation and failure process of materials by studying the damage evolution at micro-scale. Based on the crystal plasticity theory, the microscopic damage in material is studied by analyzing the stress and deformation of slip system, and the microscopic damage model is proposed to consider the resolved normal stress on crystal slip plane. This study provides a new approach for the analysis of cleavage fracture of crystalline materials. First, the gradient tensor of damage deformation is introduced in addition to the crystal elastic-plastic deformation configuration. The constitutive equation with damage energy dissipation is established from the deformation kinematics analysis, and the plastic flow equation and the damage evolution equation are derived. Second, the numerical method is established including the updating algorithm of stress and state variables and the derivation of Jacobian matrix. After that, the single crystal copper with $[100]$ orientation is studied as an example. Through comparing the results obtained by finite element computation and by experimental test, the 11 material microscopic parameters are calibrated using the particle swarm optimization algorithm. Finally, the proposed microscopic damage model is applied to the simulation of RVE under uniaxial tension. The curve of stress versus strain considering the damage effect is obtained, and the development of plastic flow and damage evolution are analyzed. The results show that the proposed model is able to compute the damage accumulation of materials and reasonably reflect the microscopic damage mechanism of crystalline materials.

Mathematical modeling of single action pressing of powder materials under dry friction conditions

The paper presents a theoretical analysis of the single action pressing of powder materials featuring plasticity and compressibility. It takes into account dry external friction between the die material and side walls, which determines the strong nonlinearity of the problem considered. This problem has a number of features that complicate its numerical solution: the presence of external friction, the elastic-plastic law of material behavior description, as well as the calculation of large displacements and, as a consequence, strong geometric nonlinearity. To consider these features, a combination of Fleck–Kuhn–McMeeking and Gurson– Tvergaard–Needleman models was used to consider a wide range of changes in the porosity of materials. The numerical solution of the problem was carried out using finite element analysis with isoparametric elements. The increment of plastic deformations at each step was determined from nonlinear equations of plastic flow. Stresses at the Gaussian points were updated according to the specified increments of deformations to calculate the material behavior during deformation. Unknown density and strain values as functions of coordinate and time were calculated. The influence of the different height-to-diameter ratio of the blank and the value of external friction of the material stress-strain state and compaction kinetics were considered. The distribution of equivalent stresses and the value of volumetric plastic deformations in the material, as well as the nonuniformity of relative density at the end of the pressing period were studied. The theoretical analysis made it possible to study the basic compaction kinetics laws for powder materials with nonuniform density under conditions of dry friction on side walls. The results obtained are relevant for predicting possible negative changes in the blank geometry when implementing the single action pressing scheme for powder materials.

Analysis on pre-peak deformation and energy dissipation characteristics of sandstone under triaxial cyclic loading

Based on conventional triaxial cyclic loading–unloading compression tests, the axial and lateral elastic–plastic strains of sandstone under confining pressures of 5, 10, 20 and 30 MPa were analyzed. The results showed that the lateral stress–strain plastic hysteresis loop is more obvious than the axial direction under cyclic loading. The axial and lateral elastic strains of the specimens are nonlinearly related to the axial deviatoric stress, and the axial and lateral plastic strains of the specimens show obvious non-linear plastic hardening characteristics. The Duncan–Chang model is used to analyze the elastic stress–strain characteristics of sandstone during loading. Without considering the initial stage of fracture compaction, the tangent elastic modulus of the sandstone specimens decreases continuously with the increase of the axial elastic strain and the Poisson’s ratio increases continuously with the increase of lateral elastic strain under the same confining pressure. Based on the plastic strain characteristics of sandstone specimens, the corresponding plastic flow equation is established, which can preferably describe the axial and lateral strain characteristics of the specimen. The plastic flow rate of the specimens increases nonlinearly with the increase of the axial deviatoric stress, which approaches the ideal plastic flow state near the peak strength.

Alternative approach in distortion gradient plasticity theory

This study presents an alternative thermodynamically consistent distortion gradient plasticity theory for polycrystalline materials. The formulation employs the principle of virtual power, laws of thermodynamics, and codirectionality hypothesis for the constructions of macroscopic and microscopic balance laws governing the elastic and plastic responses of the materials. Expressing the plastic distortion tensor in terms of accumulated plastic strain and plastic rotation vector shows that the two quantities are governed by independent plastic flow equations. In addition, it is obtained that the flow equation which governs the accumulated plastic strain interacts with the elastic response of the materials, while the flow equation governing the plastic rotation vector is independent of the elastic response. Finally and among other things, it is shown that the present model is computationally less expensive in the sense that it requires lesser numbers of equations for the prediction of plastic behavior of the materials.

Plastic flow equations for the local strain approach in the multiaxial case

This paper presents a system of plastic flow equations which uses and generalizes to the multiaxial case a number of concepts commonly employed in the so-called Local Strain Approach to low cycle fatigue. Everything is built upon the idea of distance between stress points. It is believed that this will ease the generalization to the multiaxial case of the intuitive methods used in low cycle fatigue calculations, based on hysteresis loops, Ramberg?Osgood equations, Neuber or ESED rule, etc. It is proposed that the stress space is endowed with a quadratic metric whose structure is embedded in the yield criterion. Considerations of initial isotropy of the material and of the null influence of the hydrostatic stress upon yielding leads to the realization of the simplest metric, which is associated with the von Mises yield criterion. The use of the strain?hardening hypothesis leads in natural way to a normal flow rule and this establishes a linear relationship between the plastic strain increment and the stress increment.

On the significance to use dislocation-density-related constitutive equations to correlate strain hardening with microstructure of metallic alloys: The case of conventional and austempered ductile irons

In order to highlight the importance of selecting the proper constitutive equation of plastic flow in metallic alloys, the strain hardening behaviours of conventional (DI) and austempered ductile irons (ADI) with similar hardness were investigated as study-case. The empirical Hollomon equation and the dislocation-density-related Voce equation were compared. Hollomon equation showed strong limitations in describing the different strain hardening behaviours of DIs and ADIs. Conversely, Voce equation approximated properly all the flow curves, and Voce parameters could describe successfully the different strain hardening behaviours of the two groups of alloys. Voce parameters, rationalised by the Kocks-Mecking-Estrin physical concepts, could also give an insight to the micro-mechanisms that underlie strain hardening, like dislocation density multiplication, dynamic recovery and microstructure features that affect these micro-mechanisms during straining. Through the analysis of Voce parameters it could be highlighted that the DIs strain hardening behaviour was mainly caused by the fine pearlitic structure, consisting of ferritic lamellae with sub-micrometric widths. The improvement of austempered alloys in strength and ductility was related to a significantly lower propensity of ADIs to recover dynamically, which was mainly attributed to the dual phase structure of these alloys. Furthermore, the strain hardening analysis through Voce formalism could identify in ADIs a critical condition of strain hardening rate to be investigated to improve further their ductility, which could not be found with Hollomon equation. The dislocation-density-related Voce equation describes properly the strain hardening behaviour of metallic alloys and give certain correlations between strain hardening behaviour and microstructure.

Implicit Damping Iterative Algorithm to Solve Elastoplastic Static and Dynamic Equations

This paper presents an implicit damping iterative algorithm to simultaneously solve equilibrium equations, yield function, and plastic flow equations, without requiring an explicit expression of elastoplastic stiffness matrices and local iteration for “return mapping” stresses to the yield surface. In addition, a damping factor is introduced to improve the stiffness matrix conformation in the nonlinear iterative process. The incremental iterative scheme and whole amount iterative scheme are derived to solve the dynamical and static and dynamical elastoplastic problems. To validate the proposed algorithms, computation procedures are designed and the numerical tests are implemented. The computational results verify the correctness and reliability of the proposed implicit iteration algorithms.

An inelastic finite strip method for thin-walled compression members

Over the years the finite strip method (FSM) has proved to be an invaluable tool in the study of buckling modes in thin-walled structural members. This paper presents a formulation of the FSM, which is able to predict the buckling stresses of initially perfectly straight thin-walled inelastic members under uniform compression. Plasticity is accounted for by means of plastic flow equations. Previous short-comings of plastic flow theory with respect to the modeling of buckling problems are overcome by deriving an expression for the inelastic shear stiffness from second order considerations.

Novel Multi-step Inverse Finite Element Method Based on Constitutive Equations of Plastic Flow in Sheet Metal Forming

Because of the shortcoming of plastic deformation theory of the traditional inverse finite element method of sheet metal forming:No consideration of the effect of deformation history,a novel multi-step inverse finite element method based on the principle of virtue work is proposed.Elasto-plastic material model is adopted to improve calculation accuracy.DKT12 shell element is introduced to consider the effect of deformation history such as bending-unbending around die entrance.In order to improve the stress calculation,a novel constitutive equation based on plastic flow theory is proposed,which can well reflect the actual forming condition such as elastic/plastic deformation or loading/unloading condition.The notion of the equivalent stress is adopted for stress and strain updating,which in turn can avoid numerous iterations to ensure the numerical stability and maintain the calculation efficiency at the same time.Above mentioned improvements are implemented in our in-house inverse analysis software InverStamp/Multi-step module.The presented algorithms are applied to S-rail deep drawing example.The numerical results compared with explicit dynamic solver LS-DYNA confirm its validity in prediction of the thickness distribution of formed part.

МОДЕЛИРОВАНИЕ БОЛЬШИХ УПРУГОПЛАСТИЧЕСКИХ ДЕФОРМАЦИЙ ОБОЛОЧЕК. ТЕОРЕТИЧЕСКИЕ ОСНОВЫ КОНЕЧНО-ЭЛЕМЕНТНЫХ МОДЕЛЕЙ

The present article presents theoretical basis for modeling large deformations of shells using isoparametric finite element approximations. A physical model of an elastoplastic material is based on multiplicative decomposition of the deformation gradient into elastic and plastic components. The deformation process is described in curvilinear coordinates using Lagrange description. Constitutive equations and the law of plastic flow are obtained from thermodynamic equations for Almansi tensor and Kirchhoff tensor. The tensor equations are reduced to a scalar form which includes free energy gradients, gradients of the equation of ultimate state and a special tensor characterizing the plastic deformation level. An explicit and implicit integration schemes for the equation of plastic flow are discussed. The issues of applying the constructed relations to analyzing shells are tackled. Initially, the use of the stepped loading method (integration in time) is discussed, hence some relations for incrementing the basic vector and tensor values are presented. Key words: shells, large elastoplastic deformations, finite elements, isoparametric approximation, stepped loading method. &nbsp

Analysis of the stressed state of a circular thin-walled cylinder under complex loading and with nonlinear hardening of the material

The generalized differential equations of plastic flow for a material with nonlinear hardening are derived using the Prager kinematic model. An example of numerical analysis for stress variation under elastoplastic deformation of a thin-walled cylinder of a structural carbon steel is given for different elastoplastic material models.

Solution of the plastic flow equations under axisymmetric deformation

In [1–3], equations describing the deformation of an ideal rigidly plastic medium in curvilinear coordinates are given. In [4], it is shown that, in the case of plane strain, these equations have simple solutions, and the stresses depend only on the metric coordinates.

An implicit finite element method for simulating inhomogeneous deformation and shear bands of amorphous alloys based on the free-volume model

Inhomogeneous deformation of amorphous alloys is caused by the initiation, multiplication and interaction of shear bands (i.e. narrow bands with large plastic deformation). Based on the free-volume model under the generalized multiaxial stress state, this work develops a finite element scheme to model the individual processes of shear bands that contribute to the macroscopic plasticity behaviour. In this model, the stress-driven increase in the free volume reduces the viscosity and thus leads to the strain localization in the shear band. Using the small-strain and rate-dependent plasticity framework, the plastic strain is assumed to be proportional to the deviatoric stress, and the flow stress is a function of the free volume, while the temporal change in the free volume is also coupled with the stress state. Nonlinear equations from the incremental finite element formulation are solved by the Newton–Raphson method, in which the corresponding material tangent is obtained by simultaneously and implicitly integrating the plastic flow equation and the evolution equation of the free-volume field. This micromechanical model allows us to study the interaction between individual shear bands and between the shear bands and the background stress fields. To illustrate its capabilities, the method is used to solve representative boundary value problems.

A constitutive model for elastoplastic deformation under variable amplitude multiaxial cyclic loading

This paper discusses the applicability of recently reported plastic flow equations (based on the concept of distance in the stress space) to variable amplitude multiaxial cyclic loading. The main aim of these equations is to provide the means to generalize to the multiaxial case some of the techniques commonly employed nowadays by fatigue designers which are known collectively as the Local Strain Approach to (low cycle) fatigue. The stress space is assumed to have a non-Euclidean metric where hydrostatic directions are null displacements of zero length. The structure of the metric is reflected in the form of the yield surfaces of the material and it can thus be inferred from the yield criterion. Consideration of unloading processes leads to the analysis of equivalent paths and to the definition of a generalized separation, which provides a new representation of kinematic hardening. The generalization of the well known memory effect to the multiaxial case arises naturally from the formalism presented.

Effect of Wall Roughness on Laminar Flow of Bingham Plastic Fluids through Microtubes

Miniaturization of traditional devices is in high demand formicro-electro-mechanical systems~MEMS!. Examples includebut are not limited to optics, communication and information sys-tems, fluidics, biotechnology, medicine, automotive, and aero-space. An area of interest in several engineering fields is the con-trol of fluid flow within microchannels and microtubes. Theapplications include, but are not limited to, fields such as vibrationcontrol of structures and systems using small devices, reactors formodification and separation of biological cells, energy systems asa mobile power supply, heat exchangers for micro and macro de-vices, and propulsion engines @1–3#.In recent years, there has been growing attention given to theliquid flow in microchannels in parallel with the development ofminiaturized devices and systems. The understanding of flowcharacteristics, such as velocity distribution and pressure loss isnecessary in design and process control of microfluidic devices.Peng, Peterson, and Wang @4# experimentally studied the flowcharacteristics of water flowing through rectangular microchan-nels having hydraulic diameters of 0.133 to 0.367 mm and heightto width ratios of 0.333 to 1. Their results indicated that the lami-nar flow transition occurred for the range of the Re number be-tween 200 and 700. They also claimed that friction behavior forboth laminar and turbulent flow depart from classical thermofluidcorrelations, and the friction factor is proportional to Re

A general framework for the numerical solution of problems in finite elasto-plasticity

This article discusses a general framework for the analysis of initial/boundary-value problems of rate-independent finite elasto-plasticity based on the theory of Green and Naghdi. A constitutive model is developed within the context of the above theory employing generalized measures of Lagrangian strain and work-conjugate measures of stress. Computational implications of the proposed formulation are discussed in conjunction with an implicit time integrator for the differential/algebraic equations of plastic flow. Representative numerical simulations demonstrate the applicability and predictive capacity of the model in the presence of large plastic deformations.

The flow characteristics of hot-pressing of beta-Al 2O 3

The Murray's plastic flow equation of hot-pressing was solved using the methods of progressive approximation and direct integration. In the solution, ln [1/(1 − 0)] (0 = relative density) is expressed as a function of the hot-pressing time ( t), in an approximate form of power series and in a precise form of an exponential function or an infinite power series. The solution can be used to describe the hotpressing behavior of Beta-Al 2O 3. Within the temperature and pressure range used, the powder parameters, such as surface tension γ, yield stress f, and viscosity η, were calculated or estimated and used to rationally explain the effect of the chemical composition on the hot-pressing behavior.

A remark on the time-integration of plastic flow and creep

The evolution equations of plastic flow and creep often amount to a degenerate problem, and nearly always they represent a singular differential equation problem as will be illustrated by discussing an, ordinary differential equation model. The singular character will be removed by substituting the primary problem by a regularized one, to which a control term must be added for numerical reasons.

Plastic deformation analysis of strain-rate sensitive materials under plane strain conditions

The two-dimensional plane strain equation of plastic flow in accordance with the Levy-Mises constitutive relation is expressed in terms of stream functions of complex variables. Expressions for the stress, strain-rate and velocity are derived, assuming the stream function in the forms of both the summation and product of conjugate flow functions, for plastic flow in a nonlinear viscous (strain-rate sensitive) medium. The plastic states are also derived using a mixed mode solution expressed in terms of non-separable, independent conjugate complex variables. Application of the summation form solution is illustrated through the block indentation problem. Calculations are made on the effect of variation of the strain-rate sensitivity exponent on the contact stress. The predicted behavior of the contact stress suggests the possibility of the development of a specially instrumented plane strain block indentation test for the rapid determination of the strain-rate sensitivity of real materials. By reducing the results of the indentation of a perfectly plastic material it is found that the contact stress is uniform and the external load is constant. The stress on the contact surface obtained using the present analysis is identical to that available from a slip line solution to the problem.