Analysis of localized shear deformation of ductile metal based on gradient-dependent plasticity

Shear localization in linear strain softening heterogeneous material under simple shear was investigated analytically. The closed-form solutions obtained based on gradient plasticity theory considering interactions and interplaying among microstructures due to heterogeneity of metal material show that in the normal direction of shear band, elastic shear displacement is linear; while plastic and total shear displacement are non-linear. Elastic shear strain in the band is uniform and the non-uniformity of total shear displacement stems from localized plastic shear displacement. In the center of the band, plastic and total shear displacement all reach their maximum values. In strain-softening process, elastic displacement decreases as flow shear stress decreases. Contrarily, plastic and total shear displacement increase and manifest shear localization occurs progressively. Under the same shear stress level, plastic and total shear displacement increase as strain softening modulus and elastic shear modulus decrease. The present analytical solutions were compared with many experimental results and the agreement is good.

Strain localization of Mohr-Coulomb soils with non-associated plasticity based on micropolar continuum theory

To address the problems of strain localization, the exact Mohr-Coulomb (MC) model is used based on second-order cone programming (mpcFEM-SOCP) in the framework of micropolar continuum finite element method. Using the uniaxial compression test, we focused on the earth pressure problem of rigid wall segment involving non-associated plasticity. The numerical results reveal that when mpcFEM-SOCP is applied, the problems of mesh dependency can be effectively addressed. For geotechnical strain localization analysis involving non-associated MC plasticity, mpcFEM-SOCP in conjunction with the pseudo-time discrete scheme can improve the numerical stability and avoid the unreasonable softening issue in the pressure-displacement curves, which may be encountered in the conventional FEM. It also shows that the pressure-displacement responses calculated by mpcFEM-SOCP with the pseudo-time discrete scheme are higher than those calculated by mpcFEM-SOCP with the Davis scheme. The inclination angle of shear band predicted by mpcFEM-SOCP with the pseudo-time discrete scheme agrees well with the theoretical solution of non-associated MC plasticity.

Numerical simulation of shear band localization in geotechnical materials based on a nonlocal plasticity model

A new nonlocal plasticity model,which is based on the integral-type nonlocal model and the cubic representative volumetric element（RVE）,is proposed to simulate shear band localization in geotechnical materials such as soils and rocks.An algorithm is developed to solve the resulting nonlinear system of equations.In this algorithm,the nonlocal averaging of plastic strain over the RVE is evaluated using C0 elements instead of using C1 elements to solve the second-order gradient of plastic strains.To obtain the average plastic strain,a set of special elements,called the nonlocal elements,are constructed to approximate the RVE.The updating of average stresses of the local element is based on the nonlocal plastic strain of the corresponding nonlocal elements.Numerical examples show that meshindependent results can be achieved using the proposed model and the algorithm,and the thickness of the shear band is insensitive to the mesh refinement.

A Numerical Study of Localized Plastic Deformationin Polycrystals

A finite element formulation which derives constitutive response from crystal plasticity theory was used to examine localized deformation in fee polycrystals. The polycrystal model was an idealized planar array of 22 hexagonal grains. The constitutive description used is based on a finite strain kinematical theory that accounts for lattice rotations. Formation of shear bands was successfully modeled in both single crystal and polycrystals. Stress and strain distribution around triple junctions was also analyzed. Results show the distributions of stresses and strains are distinctly inhomogeneous. Stress and strain fields across grain boundaries are highly discontinuous. However, this discontinuity will be restrained when shear bands are fully developed.

Dynamic analysis of fault rockburst based on gradient-dependent plasticity and energy criterion

Fault rockburst is treated as a strain localization problem under dynamicloading condition considering strain gradient and strain rate. As a kind of dynamic fracturephenomena, rockburst has characteristics of strain localization, which is considered as aone-dimensional shear problem subjected to normal compressive stress and tangential shear stress.The constitutive relation of rock material is bilinear (elastic and strain softening) and sensitiveto shear strain rate. The solutions proposed based on gradient-dependent plasticity show thatintense plastic strain is concentrated in fault band and the thickness of the band depends on thecharacteristic length of rock material. The post-peak stiffness of the fault band was determinedaccording to the constitutive parameters of rock material and shear strain rate. Fault bandundergoing strain softening and elastic rock mass outside the band constitute a system and theinstability criterion of the system was proposed based on energy theory. The criterion depends onthe constitutive relation of rock material, the structural size and the strain rate. The staticresult regardless of the strain rate is the special case of the present analytical solution. Highstrain rate can lead to instability of the system.

THE VOID-SIZE EFFECT ON PLASTIC FLOW LOCALIZATION IN THE GURSON MODEL

Recent studies have shown that the size of microvoids has a significant effect on the void growth rate.The purpose of this paper is to explore whether the void size effect can influence the plastic flow localization in ductile materials.We have used the extended Gurson's dilatational plasticity theory,which accounts for the void size effect,to study the plastic flow localization in porous solids with long cylindrical voids.The localization model of Rice is adopted,in which the material inside the band may display a different response from that outside the band at the incipient plastic flow localization.The present study shows that it has little effect on the shear band angle.

Corrosion-Facilitating Local Plastic Deformation for α-Ti in Methanol

The effect of corrosion process on facilitating local plastic deformation for α-Ti in methanol has been investigated with the micro-multiplicative moire interferometer technique. The size of plastic zone and the plastic strain ahead of loaded notch have been measured. The results show that the anodic dissolution, or corrosion process itself can enhance the plastic zone and the plastic strain ahead of a loaded notch during stress corrosion of α-Ti in methanol solution

Calculation of temperature distribution in adiabatic shear band based on gradient-dependent plasticity

A method for calculation of temperature distribution in adiabatic shear band is proposed in terms of gradient-dependent plasticity where the characteristic length describes the interactions and interplaying among microstructures. First, the increment of the plastic shear strain distribution in adiabatic shear band is obtained based on gradient-dependent plasticity. Then, the plastic work distribution is derived according to the current flow shear stress and the obtained increment of plastic shear strain distribution. In the light of the well-known assumption that 90% of plastic work is converted into the heat resulting in increase in temperature in adiabatic shear band, the increment of the temperature distribution is presented. Next, the average temperature increment in the shear band is calculated to compute the change in flow shear stress due to the thermal softening effect. After the actual flow shear stress considering the thermal softening effect is obtained according to the Johnson-Cook constitutive relation, the increment of the plastic shear strain distribution, the plastic work and the temperature in the next time step are recalculated until the total time is consumed. Summing the temperature distribution leads to rise in the total temperature distribution. The present calculated maximum temperature in adiabatic shear band in titanium agrees with the experimental observations. Moreover, the temperature profiles for different flow shear stresses are qualitatively consistent with experimental and numerical results. Effects of some related parameters on the temperature distribution are also predicted.

ANALYSIS OF SIZE EFFECT,SHEAR DEFORMATION AND DILATION IN DIRECT SHEAR TEST BASED ON GRADIENT- DEPENDENT PLASTICITY

The paper concerns the issue of size law,localized deformation and dilation or compaction due to shear localization. It is assumed that the shear localization initiates at the peak shear stress in the form of single shear band,and based on gradient-dependent plasticity,an analytical solution on size effect or snap-back is obtained. The results show that the post peak response becomes steeper and even exhibits snap-back with increasing of length. For small specimen,the relative shear displacement when specimen failure occurs is lower than that of larger specimen and the shear stress-relative displacement curve becomes steeper. The theoretical solution on non-uniformity of strains in shear band is obtained and evolution of the relative shear displacement is represented. By resorting to the linear relation between local plastic shear strain and local plastic volumetric strain,the dilation and compaction within shear band are analyzed. Relation between apparent shear strain and apparent normal strain and relation between shear displacement and vertical displacement are established.

Localized shear deformation during shear band propagation in titanium considering interactions among microstructures

Closed-form analytical solutions of plastic shear strain and relative plastic shear displacement during shear band propagation are proposed under dynamic loadings based on gradient-dependent plasticity considering the effect of microstructures due to heterogeneous texture of Ti. According to the differences in shear stress levels, Ti specimen is divided into three regions: residual region, strain-softening region and elastic region. Well-developed shear band is formed in the residual region and the relative plastic shear displacement no longer increases. In the normal and tangential directions, the plastic strain and the displacement are nonuniform in the strain-softening region. At the tip of shear band, the shear stress acting on the band is increased to shear strength from the elastic state and the shear localization just occurs. Prior to the tip, Ti remains elastic. At higher strain rates, the extent of plastic strain concentration is greater than that under static loading. Higher strain rate increases the relative plastic shear displacement. The present analytical solution for evolution or propagation of shear localization under nonuniform shear stress can better reproduce the observed localized characteristics for many kinds of ductile metals.

INVESTIGATION ON ELASTO-PLASTIC CONSTITUTIVE MODEL COUPLED WITH DAMAGE FOR LOCALIZATION PHENOMENA

On the basis of existing plasticity-based damage model for plasticity coupled with damage for localization analysis, constitutive parameter identification was carried out through a series of numerical tests at local level.And then improvements were made on the expressions of the evolution laws of damage. Strain localization phenomena were simulated with a typical double-notched specimen under tensions. Numerical results indicate the validity of the proposed theory.

STUDY OF TRANSFORMATION PLASTICITY IN TETRAGONAL ZIRCONIA POLYCRYSTALS BY MOIRE INTERFEROMETRY

Transformation plasticity in ceria-stabilized tetragonal zirconia poly- crystals due to the stress-induced tetragonal-to-monoclinic martensitic transforma- tion under tension and bending is studied by moire interferometry. The whole fringe patterns including u fields and v fields are acquired. According to these patterns, the distributions of transformation plasticity in transformation zones are obtained, and the phenomenon of plastic flow localization for transformation is revealed. The above work provides a significant experimental foundation for establishing transformation constitutive relations

The Expression Plasticity of Hypoxia Related Genes in High-Altitude and Plains Nanorana parkeri Populations

For species that have a broad geographic distribution, adaptive variation may be attributable to gene expression plasticity. Nanorana parkeri is an anuran endemic to the southern Tibetan Plateau where it has an extensive altitudinal range（2850 to 5100 m asl）. Low oxygen concentration is one of the main environmental characteristics of the Tibetan Plateau. Hypoxia-inducible factor α subunits（HIF-1α and HIF-2α, encoded by Endothelial PAS domain protein 1（EPAS1）） and associated genes（e.g., vascular endothelial growth factor（VEGF） and Erythropoietin（EPO）） play crucial roles in maintaining oxygen homeostasis. In this study, we compared the expression of HIF-1A, VEGF, EPAS1 and EPO mRNA between two populations of N. parkeri： one population inhabiting the native high altitudes, and the second living in, and being acclimated to, the lower plains（70 m asl）. The expression of HIF-1A, VEGF and EPAS1 mRNA in the high altitude population were significantly higher than in the acclimated population, whereas there was no significant difference for EPO between two groups. Our results indicated that gene expression plasticity may make significant contributions to local adaptation of species that have broad altitudinal distributions. In addition, we deepen our understanding of the adaptive potential of this species by evaluating the experiments in the scope of its evolutionary history.

Plastic Flow Macrolocalization: Autowave and Quasi-Particle

A new approach is proposed to describe the autowave processes responsible for plastic deformation localiza-tion in metals and alloys. The existence of a quasi-particle, which corresponds to a localized plastic flow autowave, is postulated and its characteristics are determined. The above postulate leads to a number of cor-ollaries and quantitative assessments that are considered herein. The deformation processes occurring on the macro- and micro-scale levels are found to be directly related.

Deformation of plastically compressible hardening-softening-hardening solids

Motivated by a model of the response of vertically aligned carbon nanotube （VACNT） pillars in uniaxial compression, we consider the deformation of a class of compressible elastic-viscoplastic solids with a hardeningsoftening-hardening variation of flow strength with plastic strain. In previous work （Hutchens et al. 2011） a constitutive relation was presented and used to model the response of VACNT pillars in axisymmetric compression. Subsequently, it was found that due to a programming error the constitutive relation presented in the paper （Hutchens et al. 2011） was not the one actually implemented. In particular, the plastic flow rule actually used did not satisfy plastic normality. Here, we present the constitutive formulation actually implemented in the previous work （Hutchens et al. 2011）. Dynamic, finite deformation, finite element calculations are carried out for uniaxial compression, uniaxial tension and for indentation of a ＂half-space＂ by a conical indenter tip. A sequential buckling-like deformation mode is found in compression when there is plastic non-normality and hardening- softening-hardening. The same material characterization gives rise to a Lfiders band-like deformation mode in ten- sion. When there is a deformation mode with a sharp front along mesh boundaries, the overall stress-strain response contains high frequency oscillations that are a mesh artifact. The responses of non-softening solids are also analyzed and their overall stress-strain behavior and deformation modes are compared with those of hardening-softening- hardening solids. We find that indentation with a sharp indenter tip gives a qualitatively equivalent response for hardening and hardening-softening-hardening solids.

EXPERIMENTAL INVESTIGATION INTO PLASTIC DAMAGE OF MICROVOIDS WITH INTERACTION

A mechanism of plastic how localization in ductile matter near microvoids is studied. The voids with the size-scale of micromillimeter exist in sheet specimens under tensile loading, and the plastic strain held around voids is obtained by digital image processing of deformed grids. The size growth of the microvoids, the spacing change of the neighboring voids, and the development of shear bands in the ligament between the voids, are presented by experimental results accompanied with the plastic strain distribution, that gives good interpretation to the process of void growth and coalescence with the how localization in the ligaments.

Revealing the role of local shear strain partition of transformable particles in a TRIP-reinforced bulk metallic glass composite via digital image correlation

The coupling effects of the metastable austenitic phase and the amorphous matrix in a transformation-induced plasticity(TRIP)-reinforced bulk metallic glass(BMG)composite under compressive loading were investigated by employing the digital image correlation(DIC)technique.The evolution of local strain field in the crystalline phase and the amorphous matrix was directly monitored,and the contribution from the phase transformation of the metastable austenitic phase was revealed.Local shear strain was found to be effectively consumed by the displacive phase transformation of the metastable austenitic phase,which relaxed the local strain/stress concentration at the interface and thus greatly enhanced the plasticity of the TRIP-reinforced BMG composites.Our current study sheds light on in-depth understanding of the underlying deformation mechanism and the interplay between the amorphous matrix and the metastable crystalline phase during deformation,which is helpful for design of advanced BMG composites with further improved properties.

Non-Local Analysis of Forming Limits of Ductile Material Considering Void Growth

The current study performed a finite element analysis of the strain localization behavior of a voided ductile material using a non-local plasticity formulation in which the yield strength depends on both an equivalent plastic strain measurement (hardening parameter) and Laplacian equivalent. The introduction of gradient terms to the yield function was found to play an important role in simulating the strain localization behavior of the voided ductile material. The effect of the mesh size and characteristic length on the strain localization were also investigated. An FEM simulation based on the proposed non-local plasticity revealed that the load-strain curves of the voided ductile material subjected to plane strain tension converged to one curve, regardless of the mesh size. In addition, the results using non-local plasticity also exhibited that the dependence of the deformation behavior of the material on the mesh size was much less sensitive than that with classical local plasticity and could be successfully eliminated through the introduction of a large value for the characteristic length.

STUDY OF TEXTURE EFFECT ON STRAIN LOCALIZATION OF BCC STEEL SHEETS

Using elastic crystalline viscoplastic finite element (FE) annlysis, the formability of BCC steel sheets was assessed. An orientation probability assignment method in the FE modeling procedure, which can be categorized as an inhomogenized material modeling, was newly proposed. In the study, the crystal orientations of three materials, mild steel, dual phase steel and the high strength steel, were obtained by Xray diffraction and orientation distribution function (ODF) analyses. The measured ODF results have revealed clearly different textures in the sheets, featured by orientation fibers, skeleton lines and selected orientations in Euler angle space, which are closely related to the plastic anisotropy. Then, the crystal orientations were assigned to FE integration points by using this ODF data, individually. The FE analyses of the standard limiting dome height (LDH) test show how the fiber textures affect the extent of strain localization in the forming processes. It was confirmed by comparison with experimental results that this FE code could predict the extreme strain localization and assess the sheet formability.

Quantitative calculation of local shear deformation in adiabatic shear band for Ti-6Al-4V

JOHNSON-COOK(J-C) model was used to calculate flow shear stress—shear strain curve for Ti-6Al-4V in dynamic torsion test. The predicted curve was compared with experimental result. Gradient-dependent plasticity(GDP) was introduced into J-C model and GDP was involved in the measured flow shear stress—shear strain curve, respectively, to calculate the distribution of local total shear deformation(LTSD) in adiabatic shear band(ASB). The predicted LTSDs at different flow shear stresses were compared with experimental measurements. J-C model can well predict the flow shear stress—shear strain curve in strain-hardening stage and in strain-softening stage where flow shear stress slowly decreases. Beyond the occurrence of ASB, with a decrease of flow shear stress, the increase of local plastic shear deformation in ASB is faster than the decrease of elastic shear deformation, leading to more and more apparent shear localization. According to the measured flow shear stress—shear strain curve and GDP, the calculated LTSDs in ASB are lower than experimental results. At earlier stage of ASB, though J-C model overestimates the flow shear stress at the same shear strain, the model can reasonably assess the LTSDs in ASB. According to the measured flow shear stress—shear strain curve and GDP, the calculated local plastic shear strains in ASB agree with experimental results except for the vicinity of shear fracture surface. In the strain-softening stage where flow shear stress sharply decreases, J-C model cannot be used. When flow shear stress decreases to a certain value, shear fracture takes place so that GDP cannot be used.