Plastic Constitutive Relations Research Articles

Elastic–Plastic Constitutive Relationship of Polymer Fiber–Reinforced Clay Considering the Effect of Anisotropic Distribution

Elastic–Plastic Constitutive Relationship of Polymer Fiber–Reinforced Clay Considering the Effect of Anisotropic Distribution

Study on Plastic Constitutive Relation and Ductile Fracture Criterion of AM60B Magnesium Alloy.

It is currently a challenge to accurately predict the deformation and fracture behavior of metal parts in automobile crashes. Many studies have shown that the deformation and fracture behavior of materials are significantly affected by the stress state during automobile crashes with complex stress state characteristics. In order to further promote the application of die-cast magnesium alloys in automobiles, it is particularly important to study the material deformation and fracture behavior of die-cast magnesium alloys. In this paper, the mechanical properties of the AM60B die-cast magnesium alloy sheet under four stress states (shear, tension, R10 notch tension, and cupping) were designed and tested. Based on the von Mises isotropic constitutive model and Swift weighted Hockett-Sherby hardening model, the plastic constitutive model of die-cast magnesium alloy was established. Based on the plastic model and the fracture model (JC, MMC, and DIEM) considering the influence of three stress states, the deformation and fracture behavior of the AM60B die-cast magnesium alloy front-end members in three-point bending were predicted by experiments and finite element simulation. The experimental results show that the deformation mode and loading-displacement curve trend of the AM60B die-cast magnesium alloy front members are the same, the crack initiation point and crack initiation time are the same, and the crack shape is similar. The results show that the complex stress state constitutive model parameters and the DIEM fracture model obtained in this paper can accurately predict the deformation and fracture failure behavior of the AM60B die-cast magnesium alloy sheet.

A Perspective on Plasticity, Dissipation and the Second Law of Thermodynamics

Abstract The requirement of a non-negative dissipation rate for all possible deformation histories is generally imposed on plastic constitutive relations. This is a constraint analogous to the Coleman–Noll [Coleman, B. D., and Noll, W., 1964, “The Thermodynamics of Elastic Materials With Heat Conduction and Viscosity,” Arch. Ration. Mech. Anal., 13, pp. 167–178. 10.1007/BF01262690] postulate that the Clausius–Duhem inequality needs to be satisfied for all possible deformation histories. The physical basis for the Clausius–Duhem inequality is as a statistical limit for a large number of discrete events for a long time and is not a fundamental physical requirement for small systems for a short time. The relation between the requirement of a non-negative dissipation rate and the Clausius–Duhem inequality is considered. The consequences of imposing a non-negative dissipation rate for all possible deformation histories are illustrated for: (i) a single crystal plasticity framework that accounts for elastic lattice curvature changes as well as elastic lattice straining and (ii) for discrete defect theories of plasticity, with attention specifically on discrete dislocation plasticity for crystalline solids and discrete shear transformation zone (STZ) plasticity for amorphous solids. Possible less restrictive conditions on the evolution of dissipation in plasticity formulations are considered as are implications for stability. The focus is on open questions and issues.

Modeling the viscoplastic transient dynamic process

A hollow viscoelastic right circular cylinder subjected to a time-dependent load is considered. The cylinder is with finite dimensions. A transient load acts upon the inner surface of the cylinder, whereas no loads are applied to outer surfaces. The end surfaces of the cylinder are restrained to move along the cylinder axis. The viscoelastic behaviour of the material is modelled using Boltzmann – Volterra equations. A yield criterion is implemented in the numerical analyses. By assumption, upon reaching the yield surface, the linear viscoelastic model switches to a plastic constitutive relation. The transient dynamic behaviour of the material is simulated for different intensities of the applied load and model parameters.

Crystal plasticity-based analysis and modelling of grain size and strain path dependent micro-scaled deformation mechanisms of ultra-thin sheet metals

Multi-stage microforming is increasingly used in manufacturing of the miniaturized parts with complex structures and geometries. Optimization design of this type of manufacturing process, however, is still challenging due to its unknown intrinsic nature of the process. In the multi-stage deformation and microforming, the size effect (SE), the strain path change (SPC), and the intragranularly misoriented grain boundary (IMGB) generated in previous forming stage, are the main factors governing the deformation behaviors of ultra-thin sheets. To gain thorough insights into their influences on deformation and elucidate the associated micro-scaled mechanisms, SS 316L ultra-thin sheets with the thickness of 0.1 mm and different mean grain sizes (d¯) were used for two-stage tensile tests under the conditions with distinct pre-strain εpre and intersection angle θSPC (the angle between the previous loading direction and subsequent one). Mechanical tests reveal that increasing εpre and θSPC reduces the yield stress and hardening rate in SPC tension, but this reduction gets smaller after increasing d¯. This is because more IMGBs with complex pattern are formed in relatively large grains, raising the permanent deformation resistance inside grains in subsequent tension. To model the IMGB-induced hardening, the SPC-induced softening, and the SE-induced concurrent facilitations to the hardening and softening, an enhanced crystal plasticity constitutive relation was established. The physical essence that IMGB obstructs dislocation movement, is used to model the IMGB-induced hardening based on the interactions between the main slip planes and IMGBs. The hardening facilitation caused by increasing grain size is implicitly incorporated in determining the saturated slip resistance. Modelling of the SPC-induced softening is based on the high Schmid factor grain fraction and its influences are governed by pre-strain and the defined SPC parameter. The softening facilitation caused by coarsening grains is modelled in establishing the initial slip resistance in SPC deformation. The proposed simulation procedure and constitutive relationship help with accurately predicting the mechanical response and microstructure evolution in micro-scaled SPC deformation, and also provide a basis for modelling and design of the multi-stage microforming of complex miniaturized parts.

Constitutive modeling of non-ideal isotropic materials based on a novel theoretical framework

If a plastic constitutive relation is used to simulate a metal forming process, the main concern should be its ability to predict the plastic strain parameters. At present, constitutive modelling basically relies on the properties of stress parameters, and emphasizes on how to improve the prediction accuracy of initial yield and subsequent hardening stresses. Although, corresponding to the anisotropic materials, the experimental data of plastic strain ratios are also used, the experimental plastic strains data obtained from all related experiments, in addition to the one as the equivalent plastic strain, are not employed at all. The lack of the most important experimental data in the constitutive model will inevitably weaken its ability to predict the plastic strain parameters which will determine the final shape of formed parts.To make a constructed constitutive relation that can fully reflect the experimental characteristics of materials, a new theoretical framework of constitutive model is proposed, and the application of experimental plastic strain data can be achieved with the introduction of a plastic strain-increment function. To illustrate the importance of the application of plastic strain-increment function to improve the predictive accuracy of plastic strain-increments, a new constitutive model is constructed and combined with a large amount of experimental data to verify it. Experimental verifications were carried out by some sheet metals with different anisotropic properties under uniaxial and equi-biaxial tension stress states, and also deal with multiple sets of experimental data obtained under complex tension-tension and tension-compression stress states. Discussions also include the results predicted by the models constructed based on the classical theory. Calculation examples show, the stronger the anisotropic properties of material, the greater the prediction deviation of plastic strains generated by the classical-theory-based models. Moreover, such prediction deviations will gradually increase with the accumulation of predicted plastic strain-increments. Predictive ability of the new constitutive relation is completely unimpacted by the increased degree of anisotropic features of materials. Predicted results are in good agreement with all experimental data.

Finite–Discrete Element Method Prediction of Advanced Fractures in Extra-Thick Coal Seams Based on a Constitutive Model of Rock Deformation–Fragmentation Failure Process

Roof fall is a frequent and destructive disaster in the working face of extra-thick coal seams. The important technology for disaster elimination is roof grouting, and the key to its success is to accurately predict the distance of the advanced fractures based on a reasonable rock constitutive relationship. In this paper, the constitutive relationship reflecting the progressive failure process of rock was established, including the elastic–plastic constitutive relation of intact rock, the fracture constitutive relation of non-penetrating fracture, and the shear friction constitutive relation of penetrating fracture. On this basis, the finite–discrete element method (FDEM) numerical calculation method was developed. Taking Yushupo Coal mine with a 16-m-thick coal seam as an example, the numerical results showed that the fractures in the roof appear 15~35 m ahead of the working face, and the maximum value of advance bearing pressure is between 16 and 30 MPa. Meanwhile the laboratory test results showed that the compressive strength of the grouted coal is 14.91 MPa after solidification for 7d. The above data mean that the grouting slurry can solidify the broken roof into a whole without roof fall disaster. At the same time, the rock pressure of the extra-thick coal seam can effectively crush the top coal, which is conducive to the top-coal caving operation. The in situ test shows that when the pre-grouting is carried out in the range of 20~30 m in front of the working face, the roof fall disaster can be effectively avoided, which is consistent with the numerical simulation results. It shows the rationality of the FDEM numerical method and the constitutive model of rock deformation–fragmentation failure process.

Smoothed particle hydrodynamics for cohesive dense granular media

• An elastic–viscoplastic constitutive model for cohesive dense granular media is built. • Rheological model considering cohesion for liquid-like phase is put forward. • The solid-like phase and liquid-like phase are both modelled and simulated. • Smoothed particle hydrodynamics was first used for cohesive dense granular media. • Cohesive force plays an important role in granular pile collapse, flow and depositing. Dense granular media with cohesion exist widely in nature. A constitutive model that considers cohesion in terms of solid-like and liquid-like phases in dense granular media was designed in this study. In this new model, a linear elastic–plastic constitutive relationship based on the Drucker–Prager yield criterion affected by the parameters of cohesion was used to describe the solid-like phase. A viscoplastic constitutive model based on rheology, which also considers cohesion, was obtained through formula derivation for the liquid-like phase after plastic yielding. The conversion and transitional relationships from elastic to elastic-micro viscoplastic and then to complete elastic–viscoplastic were built. Smoothed particle hydrodynamics was adopted to solve the new model discretely, and the relationship between smoothed and actual particles and an approach employed to avoid particle aggregation caused by cohesion were illustrated. Several typical cases including the collapse of cohesive granular accumulation, collapse of dry and wet granular column, and flow of cohesive powder in a rotating drum were used for verification. A comparison between cohesive and dry granular flows and among numerical results, experimental results, and those of other numerical methods proved that the proposed model and algorithm are effective for simulating dense granular flow with cohesion; further, they provide a valid approach to describe issues of granular flow with veritable cohesion that exist in nature.

Modeling the non-Schmid crystallographic slip in MAX phases

We present a crystal plasticity constitutive relation for the description of experimentally observed non-Schmid crystallographic slip in a class of ternary carbides and nitrides commonly referred to as MAX phases. In the constitutive relation, we assume that the evolution of the slip system strength in MAX phases has two components – a classical component that depends on the Taylor cumulative shear strain and a non-Schmid component that depends on the stress normal to the slip plane. The non-Schmid crystal plasticity constitutive relation is then used to carry out finite element simulations of micropillar compression of single crystals of two MAX phases, Ti2AlC and Ti3AlC2. The finite element simulations not only quantitatively predict the stress – strain response of a wide range of crystallographic orientations of the micropillars but also rationalize the non-uniform deformation and the deformed shape of the micropillars observed in the experiments for the two materials. Parametric studies are also carried out to quantify the role of the non-Schmid effect and understand the effects of key experimental parameters on the stress – strain response of the micropillars of the two MAX phases.

Research on dynamic mechanical properties and plastic constitutive relation of Ti3Al intermetallic compounds under mechanical-thermal coupling

Ti3Al intermetallic compounds is a lightweight, high-temperature resistant rare aerospace materials, whose dynamic mechanical properties concern the accuracy of simulation analysis and further development in material manufacturing. To obtain the plastic constitutive relation of Ti3Al under the action of mechanical-thermal coupling, material rheological behavior was investigated by using the separation Hopkinson pressure bar (SHPB) system with a high-temperature synchronous assembly. Meanwhile, the Johnson-Cook plastic constitutive model was calculated and modified by introducing strain compensation and adiabatic temperature rise at the range of strain rate of 0.001–6500 s−1 and deformation temperature of 293–1173 K. Furthermore, the microstructure evolution of Ti3Al fracture surface morphology was discussed. Experimental results showed that the flow stress had a weak positive strain rate enhancement effect, significant negative temperature thermal softening effect and strain rate plasticizing effect. Adiabatic temperature rise was accompanied by the high strain rate, and its value increased exponentially with the strain rate. The adiabatic deformed shear band (ADSB) appeared at an edge of 45° along the loading direction at the strain rate 4500 s−1 and the deformation temperature 273 K, respectively. ADSB tended to expand internally along with the strain rate and evolved into an adiabatic transformed shear band (ATSB) when the temperature rose to 973 K. The maximum relative errors in the modified J-C model at the high strain rate and the room temperature, the high strain rate and the high temperature were 3.11% and 14.91%, respectively. Consequently, the modified J-C model described in dynamic mechanical properties of Ti3Al was more accurate than the conventional J-C model.

Mesh size effects on fracture locus of high strength bolts: A mesoscale critical equivalent plastic strain (MCEPS) approach

With the improvement of computational capability, finite element simulation is an increasingly practical method to accurately predict the ultimate capacity and ductile fracture behavior of high-strength bolts. However, the mesh size affects the results of FE simulations but related research on mesh size effects is relatively limited. In the present contribution, the mesoscale critical equivalent plastic strain (MCEPS) is used as a failure index for calibrating the parameters of ductile fracture locus of high-strength bolts with different mesh sizes. The identified fracture locus is compared with a large bulk of experimental data taken from the previously published literature. The results showed that mesh size can have high effects on the calibrated parameters of the plastic constitutive relationship after necking and ductile fracture locus of high-strength bolts.

Multi-scale modelling of rafting behaviour under complex stress states for Ni3Al superalloys

Directional coarsening behaviour, also known as rafting, largely determines the mechanical properties of Ni3Al superalloys. However, owing to a vague understanding of the micro-scale mechanisms of the superalloys, modelling their rafting behaviour is still challenging, especially under complex stress states. The present work aims to predict rafting behaviours under complex stress states through macro-scale thermodynamic approaches assisted by micro-scale molecular dynamics. First, an atomistic representative cell of the γ'/γ structure in the Ni3Al superalloy was constructed to simulate the dislocation motion and elastic and dislocation energy distribution in the γ matrix channels under different stress states at the initial rafting stage. Accordingly, an energy-based orientation criterion was proposed for the rafting behaviour under complex stress states. The criterion was then applied to a thermodynamic system of the γ'/γ structure to model the rafting behaviour on the basis of energy dissipation theory coupled with crystal plasticity constitutive relations. The new rafting model was verified by the experimental observations in the literature and was used to evaluate the effects of temperature and stress states on rafting. This research offers a basic understanding of the directional coarsening and provides a prediction tool for the direction and variation law of the rafting process.

Thermodynamic Constitutive Model for Sands with Consideration of Distributed Fabric

A micromechanics-inspired fabric-dependent thermodynamic model is developed to simulate the critical-state and phase-transformation behavior of sands. A generalized fabric tensor that considers the statistical distribution of granular fabric is proposed. The fabric tensor is the sum of three families of a transversely isotropic tensor, which is assumed to be characterized by the probability density function of the normal distribution. The new tensor variable is incorporated into the reversible and irreversible thermodynamic models, and the fabric-dependent hyperelastic and plastic constitutive relations are then derived. The hyperelastic instability leads to a yield criterion that depends on fabric distribution and its evolution. Plastic strain development is linked to fabric distribution by a concept of accumulative yielding probability transitions, which in turn provides smooth transition from macroscopic elasticity to plasticity and finally to the critical state upon shearing. Using this theoretical framework, the physical mechanism underlying the phase-transformation and critical-state behavior of sands are interpreted from the evolution of fabric distribution. The performance of the model is verified by simulating various shear tests of Toyoura sand and a discrete element method (DEM)-modeled sand. The effects of b-value and principal stress direction on the stress-strain behavior are related to the evolution of fabric distribution. The noncoaxiality among stress, strain, and fabric is also captured by the model.

A thermo-mechanical model for pile-soil interface behavior

This study proposes a constitutive model of the pile-soil interface based on a thermodynamic approach. Firstly, the thermo-elastic coupled nonlinear hyperelastic relation of the pile-soil interface is derived by defining the interface potential energy function of the pile-soil interface subjected to normal and tangential forces. The plastic interface constitutive relations are then developed using non-equilibrium thermodynamics, based on the energy dissipations caused by the pile-soil interface’s shear-induced slipping behavior. The stress- and temperature-dependent pile-soil interface behavior is described using the concept of interface density and the normal elastic thermal expansion coefficient.

New equivalent damping for the ATC nonlinear static procedure

The Nonlinear Static Procedure, also known in the literature as Pushover Analysis procedure or Capacity Spectrum Method (CSM), is mainly developed in the report of Applied Technology Council entitled “Seismic evaluation and retrofit of concrete buildings” ATC (1996). This is a widely practical engineering method to calculate the seismically induced ductility demand of a System, Structure or Component (SSC) without performing nonlinear transient analyses. Many authors have improved this method by introducing different equivalent linear concepts. However, discrepancy between predictive results by CSM and references obtained by transient nonlinear analyses remains quite important. In this paper, we present a new equivalent linear concept, in which the conventional equivalent frequency corresponding to the secant stiffness is adopted, and new equivalent damping value is calibrated so that the maximum displacement of equivalent linear oscillator is the same as the one of nonlinear oscillator. The new proposed damping values are derived from systematic analysis of the response of a series of nonlinear single degree of freedom (SDOF) oscillators to seismic type input motions. Oscillators have a bilinear elastic–plastic constitutive relationship with a hardening slope equal to 10% or 20% of the initial stiffness. Their natural frequency (f0) is such that f0/fc varies between 0.1 and 2.0, where fc is the central frequency of the input signal and their damping in elastic regime remained at 5%. Input motion is a set of one thousand synthetic filtered white noise signals. The novelty of the proposed approach is that new equivalent damping values exhibit strong frequency dependence on the f0/fc ratio, which has not been discussed so far. In order to validate the proposed equivalent linear concept, predictive results for nonlinear SDOF oscillators undergoing one hundred natural signals are performed by CSM with the new equivalent damping values and compared to reference results obtained from nonlinear transient analyses. It is concluded that the CSM using the new proposed equivalent damping values can determine the seismically induced ductility demand of SDOF oscillators with a better accuracy than methods used at the moment as a routine.

Nonlinear analysis of constructions from different materials based on unified plastic constitutive relations

In the paper the plastic constitutive relations based on the hypotheses of the plastic flow theory, unified for different materials, such as soil ground, concrete, metal etc. are proposed. These plastic constitutive relations are based on the hypotheses of the Mises-Schleicher-Botkin theory of failure. The strength characteristics of materials, determined by standard methods, are redefined for octahedral sites, they are invariants for. The problem is solved by the finite element method (FEM). The calculation model MFOE made up of singular finite elements (tetrahedral, triangular plates, rods). This fact increases the accuracy of the description of loading paths in finite elements and ensures a unique correspondence of the constitutive equations to elementary volumes of the construction. The use of singular finite elements is associated with a large amount of access memory for storing the stiffness matrix of the system. So, the range of practically solvable problems limits significantly. To eliminate this contradiction, the Newton’s iteration method – SOR method is developed for equilibrium equations solving in the design of finite elements structures. An iterative algorithm is obtained. It does not require assembly of the system stiffness matrix for its implementation. The volume of operational information is proportional to the number of finite elements in the system. Due to the traditional approach, which requires the assembly of the system stiffness matrix, the amount of operational information is proportional to the square of the degree of kinematic indeterminacy of the system. While using the iterative algorithm, the size of the stiffness matrix and the time of solving are reduced. The results of a nonlinear analysis of soil massifs and concrete structures are represented.

Characterization of plastically compressible solids via spherical indentation

The response in spherical indentation is used to identify material properties for plastically compressible materials with presumed known elastic properties. The materials are assumed to be characterized by a Deshpande–Fleck constitutive relation (Deshpande and Fleck, 2000). The indentation force versus indentation depth responses and the residual surface profiles (the surface profiles after unloading) are calculated for two sets of material parameters. These are regarded as the “experimental” input materials. The constitutive parameters for these two materials are identified using the Bayesian-type statistical approach of Zhang et al. (2019) both for noise-free and for noise-contaminated data. The uniaxial stress–strain responses obtained from the spherical indentation responses are good approximations of those of the “experimental” input materials, particularly if surface profile data is used, but the quality of the approximation decreases with increasing noise amplitude. Plastic compressibility is found to have a relatively small effect on the correction factor β in the Oliver–Pharr relation (Oliver and Pharr, 2004) between the unloading slope and the effective (or reduced) elastic modulus. The indentation response of the plastically compressible materials can be well-represented by a nearly incompressible plastic constitutive relation but the inferred uniaxial stress–strain response is a poor representation of the “experimental” material uniaxial stress–strain response. The predicted residual surface profile is less dependent on the assumed elastic constant values than is the indentation force versus indentation depth response. The indentation force versus indentation depth responses in spherical indentation for three materials with very different uniaxial stress–strain curves are found to be indistinguishable if the indentation depth is sufficiently small but are distinguishable if the indentation depth is sufficiently large.

A thermodynamics-based hyperelastic-plastic coupled model unified for unbonded and bonded soils

A hyperelastic-plastic coupled constitutive model unified for bonded and unbonded soils is developed in this paper based on thermodynamics. An elastic potential function applicable for different kinds of soils is proposed to derive a hyperelastic model accounting for the pressure- and -density dependency, the stress-induced anisotropy and the bonding effects as well as their couplings with plasticity. From the perspective of elastic stability, state boundary and failure surfaces of different soils can be naturally predicted by the hyperelasticity without any additional definitions and parameters. Based on the classical nonequilibrium thermodynamics, novel plastic constitutive relations are derived and naturally coupled with the hyperelasticity. As a result, elasto-plastic coupling features such as the dissipative history effect on elastic stiffness, the cyclic shear behavior, the degradation of shear modulus under small strain conditions, the stress-induced anisotropy of plastic behavior and the cohesion degradation can be reproduced. The model is well validated by predicting the undrained/drained monotonic and cyclic shear behavior of unbonded and bonded sands, providing useful insights into their critical state behavior, irreversible shear-dilation/contraction and effects of bonding and cohesion degradation. It is also shown that the cohesion degradation in different shearing stages to a large extent determines both the monotonic and cyclic behavior of bonded soils.

Numerical analysis of seismic performance of inclined piles in liquefiable sands

Previous post-earthquake damage investigations have indicated that inclined piles have either performed satisfactorily or poorly during seismic events; hence their effectiveness is still questionable. This paper aims to clarify the performance of inclined piles installed in the liquefiable ground during earthquakes. To this end, a three-dimensional finite element model has been developed using the OpenSees program. The effects of soil-pile interaction have been evaluated by applying interface elements between the soil and the pile. Subsequently, the behavior of sand is described with the help of a multi-yield-surface plasticity constitutive relation. By changing the pile configurations and the amplitude of loading patterns, a series of parametric analyses are conducted, and the system responses (i.e., soil, cap, and pile responses) are compared. The results clearly illustrate that the presence of inclined piles has a beneficial effect on the dynamic response of the soil-pile-cap system in non-liquefied soils and of the cap response in liquefied soils. However, highly detrimental effects of inclined piles on the soil and pile responses have been observed as the soil liquefies. Based on the results, the conditions wherein inclined piles may perform satisfactorily or poorly are highlighted.

Numerical assessment on fatigue damage evolution of materials at crack tip of CT specimen based on CPFEM

Fatigue failure of metal parts is common in engineering, and the failure mechanisms remain to be studied. The crystal plastic finite element method (CPFEM) was used to study the damage evolution of materials at the crack tip of precracked compact tension (CT) specimen based on the established crystal plasticity constitutive relation which considered the grain size effect and cyclic hardening. The parameters used in polycrystal model were obtained through reverse optimization method combined with genetic algorithm, which are performed through the combination of commercial software ABAQUS and MATLAB. The stress–strain curve obtained by 2D polycrystal model finite element simulation is closed to the experimental curve of 304 stainless steel. Then, the 3D Voronoi-like polycrystalline model was established to investigate the effects of stress state, grain size and inclusion on the fatigue properties of materials at crack tip of precracked 304SS CT specimen by evaluating the evolution of plastic strain energy (PSE) and Von Mises stress under cyclic load. These factors will affect the fatigue damage evolution of materials from different aspects which were analyzed in detail. The proposed method and model are helpful to study the fatigue failure of metal materials at mesoscopic scale.