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...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.展开更多
The tensile deformation localization and the shear band fracture behaviors of sheet metals with strong anisotropy are numerically simulated by using Updating Lagrange finite element method, Quasi-how plastic constitut...The tensile deformation localization and the shear band fracture behaviors of sheet metals with strong anisotropy are numerically simulated by using Updating Lagrange finite element method, Quasi-how plastic constitutive theory([1]) and B-L planar anisotropy yield criterion([2]). Simulated results are compared with experimental ones. Very good consistence is obtained between numerical and experimental results. The relationship between the anisotropy coefficient R and the shear band angle theta is found.展开更多
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) ...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.展开更多
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 ...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.展开更多
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 ...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.展开更多
α-titanium and its alloys with a dual-phase structure (α+β) were deformed dynamically under strain rate of about 10^4 s^-1. The formation and microstructural evolution of the localized shear bands were character...α-titanium and its alloys with a dual-phase structure (α+β) were deformed dynamically under strain rate of about 10^4 s^-1. The formation and microstructural evolution of the localized shear bands were characterized by scanning electron microscopy (SEM) and transmission electron microscopy (TEM). The results reveal that both the strain and strain rate should be considered simultaneously as the mechanical conditions for shear band formation, and twinning is an important mode of deformation. Both experimental and calculation show that the materials within the bands underwent a superhigh strain rate (9×10^5 S^-1) deformation, which is two magnitudes of that of average strain rate required for shear band formation; the dislocations in the bands can be constricted and developed into cell structures; the phase transformation from α to α2 within the bands was observed, and the transformation products (α2) had a certain crystallographic orientation relationship with their parent; the equiaxed grains with an average size of 10 μm in diameter observed within the bands are proposed to be the results of recrystallization.展开更多
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 an...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.展开更多
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 a...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.展开更多
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 micr...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.展开更多
Flow localization, which is an importantmode of deformation in engineering materi-als, has been the interesting subject ofa number of experimental observationsand theoretical investigations in recentyears. However, th...Flow localization, which is an importantmode of deformation in engineering materi-als, has been the interesting subject ofa number of experimental observationsand theoretical investigations in recentyears. However, the basic mechanism ofthe phenomena is not well understood atpresent. In a tensile test, the initiationand growth of a shear band are often simul-taneous. Therefore, it is rather difficult展开更多
In contrast to the traditional interpretation of shear bands in sand as a bifurcation problem in continuum mechanics,shear bands in sand are considered as high-strain phase(plastic phase) of sand and the materials out...In contrast to the traditional interpretation of shear bands in sand as a bifurcation problem in continuum mechanics,shear bands in sand are considered as high-strain phase(plastic phase) of sand and the materials outside the bands are still in low-strain phase(elastic phase),namely,the two phases of sand can coexist under certain condition.As a one-dimensional example,the results show that,for materials with strain-softening behavior,the two-phase solution is a stable branch of solutions,but the method to find two-phase solutions is very different from the one for bifurcation analysis.The theory of multi-phase equilibrium and the slow plastic flow model are applied to predict the formation and patterns of shear bands in sand specimens,discontinuity of deformation gradient and stress across interfaces between shear bands and other regions is considered,the continuity of displacements and traction across interfaces is imposed,and the Maxwell relation is satisfied.The governing equations are deduced.The critical stress for the formation of a shear band,both the stresses and strains inside the band and outside the band,and the inclination angle of the band can all be predicted.The predicted results are consistent with experimental measurements.展开更多
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 s...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.展开更多
文摘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.
基金The project supported by the National Natural Science Foundation of China and the Excellent Youth Teacher Foundation of the State Education Commission of China
文摘The tensile deformation localization and the shear band fracture behaviors of sheet metals with strong anisotropy are numerically simulated by using Updating Lagrange finite element method, Quasi-how plastic constitutive theory([1]) and B-L planar anisotropy yield criterion([2]). Simulated results are compared with experimental ones. Very good consistence is obtained between numerical and experimental results. The relationship between the anisotropy coefficient R and the shear band angle theta is found.
基金Project(2004F052) supported by the Educational Department of Liaoning Province, China
文摘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.
基金Project(2004F052) supported by the Educational Department of Liaoning Province, China
文摘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.
基金support for this work provided by Louisiana Board of Regents (RCS and Pfund)high performance grid computing resources (HPC) provided by LONI (Louisiana Optical Network Initiative)
文摘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.
基金This research was supported by the National Nature Science Foundation of China(No.50071064).
文摘α-titanium and its alloys with a dual-phase structure (α+β) were deformed dynamically under strain rate of about 10^4 s^-1. The formation and microstructural evolution of the localized shear bands were characterized by scanning electron microscopy (SEM) and transmission electron microscopy (TEM). The results reveal that both the strain and strain rate should be considered simultaneously as the mechanical conditions for shear band formation, and twinning is an important mode of deformation. Both experimental and calculation show that the materials within the bands underwent a superhigh strain rate (9×10^5 S^-1) deformation, which is two magnitudes of that of average strain rate required for shear band formation; the dislocations in the bands can be constricted and developed into cell structures; the phase transformation from α to α2 within the bands was observed, and the transformation products (α2) had a certain crystallographic orientation relationship with their parent; the equiaxed grains with an average size of 10 μm in diameter observed within the bands are proposed to be the results of recrystallization.
文摘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.
基金Project(2004F052) supported by the Education Department of Liaoning Province,China
文摘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.
文摘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.
文摘Flow localization, which is an importantmode of deformation in engineering materi-als, has been the interesting subject ofa number of experimental observationsand theoretical investigations in recentyears. However, the basic mechanism ofthe phenomena is not well understood atpresent. In a tensile test, the initiationand growth of a shear band are often simul-taneous. Therefore, it is rather difficult
基金Project(2007CB714001) supported by the National Basic Research Program of China (973 Program)
文摘In contrast to the traditional interpretation of shear bands in sand as a bifurcation problem in continuum mechanics,shear bands in sand are considered as high-strain phase(plastic phase) of sand and the materials outside the bands are still in low-strain phase(elastic phase),namely,the two phases of sand can coexist under certain condition.As a one-dimensional example,the results show that,for materials with strain-softening behavior,the two-phase solution is a stable branch of solutions,but the method to find two-phase solutions is very different from the one for bifurcation analysis.The theory of multi-phase equilibrium and the slow plastic flow model are applied to predict the formation and patterns of shear bands in sand specimens,discontinuity of deformation gradient and stress across interfaces between shear bands and other regions is considered,the continuity of displacements and traction across interfaces is imposed,and the Maxwell relation is satisfied.The governing equations are deduced.The critical stress for the formation of a shear band,both the stresses and strains inside the band and outside the band,and the inclination angle of the band can all be predicted.The predicted results are consistent with experimental measurements.
基金Supported by the National Natural Science Foundation of China(50309004)。
文摘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.