Effects of constitutive parameters on adiabatic shear localization for ductile metal based on JOHNSON-COOK and gradient plasticity models

By using the widely used JOHNSON-COOK model and the gradient-dependent plasticity to consider microstructural effect beyond the occurrence of shear strain localization,the distributions of local plastic shear strain and deformation in adiabatic shear band(ASB)were analyzed.The peak local plastic shear strain is proportional to the average plastic shear strain,while it is inversely proportional to the critical plastic shear strain corresponding to the peak flow shear stress.The relative plastic shear deformation between the top and base of ASB depends on the thickness of ASB and the average plastic shear strain.A parametric study was carried out to study the influence of constitutive parameters on shear strain localization.Higher values of static shear strength and work to heat conversion factor lead to lower critical plastic shear strain so that the shear localization is more apparent at the same average plastic shear strain.Higher values of strain-hardening exponent,strain rate sensitive coefficient,melting point,thermal capacity and mass density result in higher critical plastic shear strain,leading to less apparent shear localization at the same average plastic shear strain.The strain rate sensitive coefficient has a minor influence on the critical plastic shear strain,the distributions of local plastic shear strain and deformation in ASB.The effect of strain-hardening modulus on the critical plastic shear strain is not monotonous.When the maximum critical plastic shear strain is reached,the least apparent shear localization occurs.

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 damage localization for ductile metal in process of shear band propagation

Distribution of localized damage in shear band can’ t be predicted theoretically based on classical elastoplastic theory. The average damage variable in shear band was considered to be a non-local variable. Based on non-local theory, an analytical expression for the localized damage in strain-softening region of shear band in the process of shear band propagation was presented using boundary condition and symmetry of local damage variable, etc. The results show that dynamic shear softening modulus, dynamic shear strength and shear elastic modulus influence the distribution of the localized damage in shear band. Internal length of ductile metal only governs the thickness of shear band. In the strain-softening region of shear band, the local damage variable along shear band’s tangential and normal directions is non-linear and highly non-uniform. The non-uniformities in the normal and tangential directions of shear band stem from the interactions and interplaying among microstructures and the non-uniform distribution of shear stress, respectively. At the tail of the strain-softening region, the maximum value of local damage variable reaches 1. This means that material at this position fractures completely. At the tip of shear band and upper as well as lower boundaries, no damage occurs. Local damage variable increases as dynamic shear softening modulus decreases or shear elastic modulus increases, leading to difficulty in identification or detection of damage for less ductile metal material at higher strain rates.

Analysis of progressive failure of pillar and instabilitycriterion based on gradient-dependent plasticity

A mechanical model for strain softening pillar is proposed considering the characteristics of progressive shear failure and strain localization. The pillar undergoes elastic, strain softening and slabbing stages. In the elastic stage, vertical compressive stress and deformation at upper end of pillar are uniform, while in the strain softening stage there appears nonuniform due to occurrence of shear bands, leading to the decrease of load-carrying capacity. In addition, the size of failure zone increases in the strain softening stage and reaches its maximum value when slabbing begins. In the latter two stages, the size of elastic core always decreases. In the slabbing stage, the size of failure zone remains a constant and the pillar becomes thinner. Total deformation of the pillar is derived by linearly elastic Hookes law and gradient-dependent plasticity where thickness of localization band is determined according to the characteristic length. Post-peak stiffness is proposed according to analytical solution of averaged compressive stress-average deformation curve. Instability criterion of the pillar and roof strata system is proposed analytically (using) instability condition given by Salamon. It is found that the constitutive parameters of material of pillar, the geometrical size of pillar and the number of shear bands influence the stability of the system; stress gradient controls the starting time of slabbing, however it has no influence on the post-peak stiffness of the pillar.

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.

Entire deformational characteristics and strain localization of jointed rock specimen in plane strain compression

Shear band (SB), axial, lateral and volumetric strains as well as Poisson’s ratio of anisotropic jointed rock specimen (JRS) were modeled by Fast Lagrangian Analysis of Continua (FLAC). Failure criterion of rock was a composited Mohr-Coulomb criterion with tension cut-off. An inclined joint was treated as square elements of ideal plastic material beyond the peak strength. Several FISH functions were written to automatically find the addresses of elements in the joint and to calculate the entire deformational characteristics of plane strain JRS. The results show that for moderate joint inclination (JI), strain is only concentrated into the joint governing the behavior of JRS, leading to ideal plastic responses in axial and lateral directions. For higher JI, the post-peak stress-axial and lateral strain curves become steeper as JI increases owing to the increase of new SB’s length. Lateral expansion and precursor to the unstable failure are the most apparent, resulting in the highest Poisson’s ratio and even negative volumetric strain. For lower JI, the entire post-peak deformational characteristics are independent of JI. The lowest lateral expansion occurs, leading to the lowest Poisson’s ratio and positive volumetric strain all along. The present prediction on anisotropic strength in plane strain compression qualitatively agrees with the results in triaxial tests of rocks. The JI calculated by Jaeger’s formula overestimates that related to the minimum strength. Advantages of the present numerical model over the Jaeger’s model are pointed out.

Numerical simulation of influence of shear dilatancy on deformation characteristics of shear band-elastic body system

The paper was numerically focused on investigation of deformation, failure and instability of shear band-surrounding elastic rock system in plane strain direct shear test considering shear dilatancy according to fast lagrangian analysis of continua (FLAC). The adopted failure criterion was a composite Mohr-Coulomb criterion with tension cut-off and post-peak constitutive relation of rock, i.e. linear strain-softening. Numerical results show that dilation angle affects the responses of elements, the number and the position of yielded elements. Increasing dilation angle results in higher load-carrying capability of elements, higher deformation or strain corresponding to peak stress, less brittle post-peak stress-deformation curve. Strain-hardening behavior can occur if dilation angle is high. Therefore, shear band-elastic rock body system tends to be stable and rock burst does not occur as dilation angle is increased. Moreover, the number of yielded elements is in- creased with dilation angle increase and two parallel plastic zones initially generated in the middle of two loading ends of specimen no longer remain collinear, reflecting increase in deformation resistant of the system. Evolution of volumetric strain rate was investigated based on three-dimensional curved surface diagrams. Approximately, volumetric strain rate concentration regions coincide with plastic zones. Volumetric strain rate in yielded elements is very evident. However, in elastic zones volumetric strain has not been nearly changed throughout the numerical tests.

Shear stress distribution and characteristics of deformation for shear band-elastic body system at pre-peak and post-peak

The distributed shear stress and the displacement across shear band, the evolution of plastic zones, and the load-carrying capacity of rock specimen were investigated in plane strain direct shear test according to Fast Lagrangian Analysis of Continua （FLAC）. And then the shear displacement distribution in normal direction of system composed of localized shear band and elastic rock was analyzed based on gradient-dependent plasticity. The adopted failure criterion was a composite of Mohr-Coulomb criterion, that is, the relation between tension cut-off and postpeak constitutive of rock was linear strain-softening. Numerical results show that shear stress field approximately undergoes three different stages. At first, shear stress is only concentrated in the middle of top and base of specimen. Next, shear stress in the middle of specimen tends to increase, owing to superposition of shear stresses. Interestingly, two peaks of shear stress appear far from the loading ends of specimen, and the peaks approach with the increase in timestep until elements at the center of specimen yield. Finally, relatively lower shear stress level is reached in large part of specimen except in the regions near the two ends. As flow stress decreases, the analytical shear displacement distribution in shear band based on gradient-dependent plasticity becomes steeps outside the band, it is linear and its slope tends to decrease. These theoretical results qualitatively agree with that of the present numerical predicted results. Main advantage of the analytical solution over the numerical results according to FLAC is that it is continuous, smooth and non-linear （except at elastic stage）.

Effect of height of rock specimen on strain localization, precursor to failure and entire deformational characteristics

Patterns of shear band, precursors to shear failure occurring in strain-softening stage, axial, lateral and volumetric strains as well as Poisson＇s ratio of plane strain rock specimens in compression for different heights were investigated by use of Fast Lagrangian Analysis of Continua（FLAC）. A material imperfection closer to the lower-left corner of the specimen was prescribed. For finer mesh, the imperfection was modeled by four null elements, while it was modeled by a null element for coarser mesh. FISH functions were written to calculate the entire deformational characteristics of the specimen. In elastic stage, the adopted constitutive relation was linear elastic; in strain-softening stage, a composite Mohr-Coulomb criterion with tension cut-off and a post-peak linear constitutive relation were adopted. Height of rock specimen does not influence shear band＇s pattern （including the thickness and inclination angle of shear band）. The slopes of the post-peak stress-axial strain curve, stress-lateral strain curve, lateral strain-axial strain curve, Poisson＇s ratio-axial strain curve and volumetric strain-axial strain curve depend on the height. Hence, the slopes of these curves cannot be considered as material properties. Nonlinear deformation prior to the peak stress is a kind of precursors to shear failure, which is less apparent for shorter specimen. For the same axial strain, lower lateral expansion is reached for shorter specimen, leading to lower Poisson＇s ratio and higher volumetric strain. The maximum volumetric strain of longer specimen is less than that of shorter specimen. The conclusions drawn from numerical results using finer mesh qualitatively agree with those using coarser mesh.

Effect of confining pressure on deformation and failure of rock at higher strain rate

Influence of confining pressure from 0 to 28 MPa, which acts on the two lateral edges of rock specimen in plane strain compression, on the shear failure processes and patterns as well as on the macroscopically mechanical responses were numerically modeled by use of FLAC. A material imperfection with lower strength in comparison with the intact rock, which is close to the lower-left corner of the specimen, was prescribed. In elastic stage, the adopted constitutive relation of rock was linear elastic; in strain-softening stage, a composite Mohr-Coulomb criterion with tension cut-off and a post-peak linear constitutive relation were adopted. The numerical results show that with an increase of confining pressure the peak strength of axial stress-axial strain curve and the corresponding axial strain linearly increase; the residual strength and the stress drop from the peak strength to the residual strength increase; the failure modes of rock transform form the multiple shear bands close to the loading end of the specimen （confining pressure=0-0.1 MPa）, to the conjugate shear bands （0.5-2.0 MPa）, and then to the single shear band （4-28 MPa）. Once the tip of the band reaches the loading end of the specimen, the direction of the band changes so that the reflection of the band occurs. At higher confining pressure, the new-formed shear band does not intersect the imperfection, bringing extreme difficulties in prediction of the failure of rock structure, such as rock burst. The present results enhance the understanding of the shear failure processes and patterns of rock specimen in higher confining pressure and higher loading strain rate.

SLIPPAGE SOLUTION OF GAS PRESSURE DISTRIBUTION IN PROCESS OF LANDFILL GAS SEEPAGE

A mathematical model of landfill gas migration was established under presumption of the effect of gas slippage. The slippage solutions to the nonlinear mathematical model were accomplished by the perturbation and integral transformation method. The distribution law of gas pressure in landfill site was presented under the conditions of considering and neglecting slippage effect. Sensitivity of the model input parameters was analyzed. The model solutions were compared to observation values. Results show that gas slippage effect has a large impact on gas pressure distribution. Landfill gas pressure and pressure gradient considering slippage effect is lower than that neglecting slippage effect, with reasonable agreement between model solution and measured data. It makes clear that the difference between considering and neglecting slippage effect is obvious and the effects of coupling cannot be ignored. The theoretical basis is provided for engineering design of security control and decision making of gas exploitation in landfill site. The solutions give scientific foundation to analyzing well test data in the process of low-permeability oil gas reservoir exploitation.

Phenomenon of transformed adiabatic shear band surrounded by deformed adiabatic shear band of ductile metal

The coexistent phenomenon of deformed and transformed adiabatic shear bands(ASBs) of ductile metal was analyzed using the JOHNSON-COOK model and gradient-dependent plasticity(GDP).The effects of melting point,density,heat capacity and work to heat conversion factor were investigated.Higher work to heat conversion factor,lower density,lower heat capacity and higher melting point lead to wider transformed ASB and higher local plastic shear deformation between deformed and transformed ASBs.Higher work to heat conversion factor,lower density,lower heat capacity and lower melting point cause higher local plastic shear deformation in the deformed ASB.Three reasons for the scatter in experimental data on the ASB width were pointed out and the advantages of the work were discussed.If the transformed ASB width is used to back-calculate the internal length parameter in the GDP,undoubtedly,the parameter will be extremely underestimated.

Effect of pore pressure on deformation and unstable snap-back for shear band and elastic rock system

Fast Lagrangian analysis of continua(FLAC) was used to study the influence of pore pressure on the mechanical behavior of rock specimen in plane strain direct shear, the distribution of yielded elements, the distribution of displacement and velocity across shear band as well as the snap-back (elastic rebound) instability. The effective stress law was used to represent the weakening of rock containing pore fluid under pressure. Numerical results show that rock specimen becomes soft (lower strength and hardening modulus) as pore pressure increases, leading to higher displacement skip across shear band. Higher pore pressure results in larger area of plastic zone, higher concentration of shear strain, more apparent precursor to snap-back (unstable failure) and slower snap-back. For higher pore pressure, the formation of shear band-elastic body system and the snap-back are earlier; the distance of snap-back decreases; the capacity of snap-back decreases, leading to lower elastic strain energy liberated beyond the instability and lower earthquake or rockburst magnitude. In the process of snap-back, the velocity skip across shear band is lower for rock specimen at higher pore pressure, showing the slower velocity of snap-back.

Analyses of axial,lateral and circumferential deformations of rock specimen in triaxial compression

The axial,lateral and circumferential strains were analyzed for a rock specimen subjected to shear failure in the form of a shear band bisecting the specimen in triaxial compression.Plastic deformation of the specimen stemmed from shear strain localization initiated at the peak shear stress.Beyond the onset of strain localization,the axial,lateral and circumferential strains were decomposed into two parts,respectively.One is the elas- tic strain described by general Hooke's law.The other is attributable to the plastic shear slips along shear band with a certain thickness dependent on the internal length of rock. The post-peak circumferential strain-axial strain curve of longer specimen is steeper than that of shorter specimen,as is consistent with the previous experiments.In elastic stage, the circumferential strain-axial strain curve exhibits nonlinear characteristic,as is in agreement with the previous experiment since confining pressure is loaded progressively until a certain value is reached.When the confining pressure is loaded completely,the circumferential strain-axial strain curve is linear in elastic and strain-softening stages.The predicted circumferential strain-axial strain curve in elastic and strain-softening stages agrees with the previous experiment.

Temperature distribution in adiabatic shear band for ductile metal based on JOHNSON-COOK and gradient plasticity models

Gradient-dependent plasticity considering interactions and interplay among microstructures was included into JOHNSON-COOK model to calculate the temperature distribution in adiabatic shear band(ASB), the peak and average temperatures as well as their evolutions. The differential local plastic shear strain was derived to calculate the differential local plastic work and the temperature rise due to the microstructural effect. The total temperature in ASB is the sum of initial temperature, temperature rise at strain-hardening stage and non-uniform temperature due to the microstructural effect beyond the peak shear stress. The flow shear stress—average plastic shear strain curve, the temperature distribution, the peak and average temperatures in ASB are computed for Ti-6Al-4V. When the imposed shear strain is less than 2 and the shear strain rate is 1 000 s?1, the dynamic recovery and recrystalliza-tion processes occur. However, without the microstructural effect, the processes might have not occurred since heat diffusion decreases the temperature in ASB. The calculated maximum temperature approaches 1 500 K so that phase transformation might take place. The present predictions support the previously experimental results showing that the transformed and deformed ASBs are observed in Ti-6Al-4V. Higher shear strain rate enhances the possibility of dynamic recrystallization and phase transformation.

Volume Change of Heterogeneous Quasi-brittle Materials in Uniaxial Compression

The volumetric strain was categorized into elastic and plastic parts. The farmer camposed of axial and lateral strains is uniform and determined by Hooke＇s law ; however, the latter consisting of axial and lateral strains is a fuaction af thickness af shear band determined by grndieat-dependeat plasticity by cansidering the heterngeneity of quasi- brittle materials. The non- uniform lateral strain due to the fact that shear band was farmed in the middle of specimen was averaged within specimen to precisely assess the volumetric strain. Then, the analytical expression for volumetric strain was verified by comparison with two earlier experimental results for concrete and rack. Finally, a detailed parametric study was carried out to investigate effects of constitutive parameters （ shear band thickness, elastic and softening rnoduli ） and geometrical size of specimen（ height and width of specimen ） on the volume dilatancy.

Unified analytical stressstrain curve for quasibrittle geomaterial in uniaxial tension, direct shear and uniaxial compression

Considering strain localization in the form of a narrow band initiated just at peak stress, three analytical expressions for stressstrain curves of quasibrittle geomaterial (such as rock and concrete) in uniaxial tension, direct shear and uniaxial compression were presented, respectively. The three derived stressstrain curves were generalized as a unified formula. Beyond the onset of strain localization, a linear strain-softening constitutive relation for localized band was assigned. The size of the band was controlled by internal or characteristic length according to gradient-dependent plasticity. Elastic strain within the entire specimen was assumed to be uniform and decreased with the increase of plastic strain in localized band. Total strain of the specimen was decomposed into elastic and plastic parts. Plastic strain of the specimen was the average value of plastic strains in localized band over the entire specimen. For different heights, the predicted softening branches of the relative stressstrain curves in uniaxial compression are consistent with the previously experimental results for normal concrete specimens. The present expressions for the post-peak stressdeformation curves in uniaxial tension and direct shear agree with the previously numerical results based on gradient-dependent plasticity.

Adiabatic shear localization evolution for steel based on the Johnson-Cook model and gradient-dependent plasticity

Gradient-dependent plasticity is introduced into the phenomenological Johnson-Cook model to study the effects of strainhardening, strain rate sensitivity, thermal-softening, and microstructure. The microstructural effect （interactions and interplay among microstructures） due to heterogeneity of texture plays an important role in the process of development or evolution of an adiabatic shear band with a certain thickness depending on the grain diameter. The distributed plastic shear strain and deformation in the shear band are derived and depend on the critical plastic shear strain corresponding to the peak flow shear stress, the coordinate or position, the internal length parameter, and the average plastic shear strain or the flow shear stress. The critical plastic shear strain, the distributed plastic shear strain, and deformation in the shear band are numerically predicted for a kind of steel deformed at a constant shear strain rate. Beyond the peak shear stress, the local plastic shear strain in the shear band is highly nonuniform and the local plastic shear deformation in the band is highly nonlinear. Shear localization is more apparent with the increase of the average plastic shear strain. The calculated distributions of the local plastic shear strain and deformation agree with the previous numerical and experimental results.

Adiabatic Shear Localization for Steels Based on Johnson-Cook Model and Second-and Fourth-Order Gradient Plasticity Models

To consider the effects of the interactions and interplay among microstructures, gradient-dependent models of second- and fourth-order are included in the widely used phenomenological Johnson-Cook model where the effects of strain-hardening, strain rate sensitivity, and thermal-softening are successfully described. The various parameters for 1006 steel, 4340 steel and S-7 tool steel are assigned. The distributions and evolutions of the local plastic shear strain and deformation in adiabatic shear band （ASB） are predicted. The calculated results of the second- and fourth- order gradient plasticity models are compared. S-7 tool steel possesses the steepest profile of local plastic shear strain in ASB, whereas 1006 steel has the least profile. The peak local plastic shear strain in ASB for S-7 tool steel is slightly higher than that for 4340 steel and is higher than that for 1006 steel. The extent of the nonlinear distribution of the local plastic shear deformation in ASB is more apparent for the S-7 tool steel, whereas it is the least apparent for 1006 steel. In fourth-order gradient plasticity model, the profile of the local plastic shear strain in the middle of ASB has a pronounced plateau whose width decreases with increasing average plastic shear strain, leading to a shrink of the portion of linear distribution of the profile of the local plastic shear deformation. When compared with the sec- ond-order gradient plasticity model, the fourth-order gradient plasticity model shows a lower peak local plastic shear strain in ASB and a higher magnitude of plastic shear deformation at the top or base of ASB, which is due to wider ASB. The present numerical results of the second- and fourth-order gradient plasticity models are consistent with the previous numerical and experimental results at least qualitatively.

ANALYSE OF PULSE WAVE PROPAGATION IN ARTERIES

Based upon the blood vessel of being regarded as the elasticity tube, and that the tissue restricts the blood vessel wall, the rule of pulse wave propagation in blood vessel was studied. The viscosity of blood, the elastic modulus of blood vessel, the radius of tube that influenced the pulse wave propagation were analyzed. Comparing the result that considered the viscosity of blood with another result that did not consider the viscosity of blood, we finally discover that the viscosity of blood that influences the pulse wave propagation can not be neglected; and with the accretion of the elastic modulus the speed of propagation augments and the press value of blood stream heightens; when diameter of blood vessel reduces, the press of blood stream also heightens and the speed of pulse wave also augments. These results will contribute to making use of the information of pulse wave to analyse and auxiliarily diagnose some causes of human disease.