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Jan Vorel - One of the best experts on this subject based on the ideXlab platform.
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energy conservation error due to use of green naghdi objective stress Rate in commercial finite element codes and its compensation
Journal of Applied Mechanics, 2014Co-Authors: Zdeněk P Bažant, Jan VorelAbstract:The objective stress Rates used in most commercial finite element programs are the Jaumann Rate of Kirchhoff stress, Jaumann Rates of Cauchy stress, or Green–Naghdi Rate. The last two were long ago shown not to be associated by work with any finite strain tensor, and the first has often been combined with tangential moduli not associated by work. The error in energy conservation was thought to be negligible, but recently, several papers presented examples of structures with high volume compressibility or a high degree of orthotropy in which the use of commercial software with the Jaumann Rate of Cauchy or Kirchhoff stress leads to major errors in energy conservation, on the order of 25–100%. The present paper focuses on the Green–Naghdi Rate, which is used in the explicit nonlinear algorithms of commercial software, e.g., in subroutine VUMAT of ABAQUS. This Rate can also lead to major violations of energy conservation (or work conjugacy)—not only because of high compressibility or pronounced orthotropy but also because of large material rotations. This fact is first demonstRated analytically. Then an example of a notched steel cylinder made of steel and undergoing compression with the formation of a plastic shear band is simulated numerically by subroutine VUMAT in ABAQUS. It is found that the energy conservation error of the Green–Naghdi Rate exceeds 5% or 30% when the specimen shortens by 26% or 38%, respectively. Revisions in commercial software are needed but, even in their absence, correct results can be obtained with the existing software. To this end, the appropriate transformation of tangential moduli, to be implemented in the user's material subroutine, is derived.
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elastic soft core sandwich plates critical loads and energy errors in commercial codes due to choice of objective stress Rate
Journal of Applied Mechanics, 2013Co-Authors: Jan Vorel, Zdeněk P Bažant, Mahendra GattuAbstract:Most commercial finite element codes, such as ABAQUS, LS-DYNA, ANSYS and NASTRAN, use as the objective stress Rate the Jaumann Rate of Cauchy (or true) stress, which has two flaws: It does not conserve energy since is not work-conjugate to any finite strain tensor and, as previously shown for the case of sandwich columns, does not give a correct expression for the work of in-plane forces during buckling. This causes no appreciable errors when the skins and the core are subdivided by several layers of finite elements. However, in spite of a linear elastic behavior of the core and skins, the errors are found to be large when either the sandwich plate theory with the normals of the core remaining straight or the classical equivalent homogenization as an orthotropic plate with the normals remaining straight is used. Numerical analysis of a plate intended for the cladding of the hull of a light long ship shows errors up to 40%. It is shown that a previously derived stress-dependent transformation of the tangential moduli eliminates the energy error caused by Jaumann Rate of Cauchy stress and yields the correct critical buckling load. This load corresponds to the the Truesdell objective stress Rate, which is work-conjugate to the GreenLagrangian finite strain tensor. The commercial codes should switch to this Rate. The classical differential equations for buckling of elastic soft-core sandwich plates with a constant shear modulus of the core are shown to have a form that corresponds to the Truesdell Rate and Green-Lagrangian tensor. The critical in-plane load is solved analytically from these differential equations with typical boundary conditions, and is found to agree perfectly with the finite element solution based on the Truesdell Rate. Comparisons of the errors of various approaches are tabulated. 1 Motivation and Nature of Problem This paper is motivated by the design of large foam-core sandwich panels, intended for the cladding of a ribbed hull of light long ships with superior maneuverability and fuel-efficiency. In a laboratory test, one such panel failed at one third of the axial compressive load predicted by a standard commercial finite element program. Although the main cause of this gross underestimation of strength is probably the neglect of size effect due to cohesive delamination fracture, which was previously identified for cylindrical buckling of sandwich plates [7] and is the subject of a sepaRate study, an important additional cause to be studied here appears to lie in two long-ignored flaws [2, 5] in the handling of finite strain by standard commercial codes [1, 15, 16, e.g.]: 1) One flaw of these code is the use of objective stress Rates that are not work-conjugate to any finite strain tensor [2]. In the implicit updated Lagrangian analysis, it is the Jaumann Rate Visiting Scholar, Northwestern University; Assistant Professor on leave from Czech Technical University in Prague. McCormick Institute Professor and W.P. Murphy Professor of Civil Engineering and Materials Science, Northwestern University, 2145 Sheridan Road, CEE/A135, Evanston, Illinois 60208; z-bazant@northwestern.edu (corresponding author) Graduate Research Assistant, Northwestern University. of Cauchy stress and, in the explicit analysis, the use of the Green-Naghdi Rate. 2) Another flaw is that the bifurcation analysis uses the Jaumann Rate of Kirchhoff stress. Although this Rate is work-conjugate to the Hencky (or logarithmic) strain tensor, it cannot correctly capture the work of initial in-plane stresses in soft-in-shear highly orthotropic structures compressed in the strong direction. This work can be captured correctly only by the Truesdell objective stress Rate [3, 4]. The former flaw can lead to major errors in volume changes of polymeric, ceramic and metallic foams, fiber-reinforced foams and other highly compressible porous materials such as loess, silt, tuff, snow, under-consolidated granular materials, light wood, honeycomb, osteoporotic bones and various biological tissues. But it is unimportant for elastic buckling of a sandwich, which is the only case to be studied here. The reason is that even though the foam core is highly compressible, its hydrostatic stress is negligible (except for inelastic buckling with delamination, or for indentation [6]). This study will focus on the latter flaw. Its seriousness has already been demonstRated for sandwich columns [3, 4] and for highly orthotropic columns [12, 13], but not for sandwich plates. The energetically correct form of the differential equations of equilibrium of sandwich plates will also be identified, and their solution will be compared to finite element analysis of two kinds–a two-dimensional analysis with sandwich-type elements having a linear strain profile across the core, and three-dimensional analysis in which the core thickness is subdivided into several elements. In contrast to finite elements for the entire cross section of sandwich column, this subdivision has already been shown to yield correct results regardless of the choice of objective stress Rate; see [4] (and for elastomeric bearings see [28]). It will be verified whether this is also true for sandwich plates. A salient characteristic of sandwich plates is that the shear strain in a soft core is important for buckling. The shear buckling is a problem with a hundred-year controversial history. It requires using the stability criteria for a three-dimensional continuum, which were for half a century a subject of polemics. Although the polemics were resolved four decades ago, some authors still dispute various aspects. All the historical controversies can be traced to the arbitrariness in choosing the finite strain measure and to inattention to the work-conjugacy requirement, which means that the (doubly contracted) product of the incremental objective stress tensor with the incremental finite strain tensor must give a correct expression for the second-order work [5, ch.11]. How does the choice of strain measure affect the differential equations of equilibrium and the eigenvalues of compressed sandwich plates? And which choice is correct? These questions will be addressed first analytically, and then in the context of finite element analysis, with and without subdividing the core thickness into several layers of elements. The aim is to appraise the magnitude of errors and choose the best practical approach. 2 Review of Objective Stress Rates and Their Energy-Variational Basis A broad class of equally admissible finite strain measures which comprises virtually all of those ever used is represented by the Doyle-Ericksen tensors (m) = (U − I) /m, where m is a real parameter, I = unit tensor and U = right-stretch tensor [5]. The second-order approximation of these tensors is (m) ij = eij + 1 2 uk,iuk,j − αekiekj, eki = 12 (uk,i + ui,k) , α = 1− 1 2 m (1)
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work conjugacy error in commercial finite element codes its magnitude and how to compensate for it
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2012Co-Authors: Zdeněk P Bažant, Mahendra Gattu, Jan VorelAbstract:Most commercial finite-element programs use the Jaumann (or co-rotational) Rate of Cauchy stress in their incremental (Riks) updated Lagrangian loading procedure. This Rate was shown long ago not to be work-conjugate with the Hencky (logarithmic) finite strain tensor used in these programs, nor with any other finite strain tensor. The lack of work-conjugacy has been either overlooked or believed to cause only negligible errors. Presented are examples of indentation of a naval-type sandwich plate with a polymeric foam core, in which the error can reach 28.8 per cent in the load and 15.3 per cent in the work of load (relative to uncorrected results). Generally, similar errors must be expected for all highly compressible materials, such as metallic and ceramic foams, honeycomb, loess, silt, organic soils, pumice, tuff, osteoporotic bone, light wood, carton and various biological tissues. It is shown that a previously derived equation relating the tangential moduli tensors associated with the Jaumann Rates of Cauchy and Kirchhoff stresses can be used in the user’s material subroutine of a black-box commercial program to cancel the error due to the lack of work-conjugacy and make the program perform exactly as if the Jaumann Rate of Kirchhoff stress, which is work-conjugate, were used.
Zdeněk P Bažant - One of the best experts on this subject based on the ideXlab platform.
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energy conservation error due to use of green naghdi objective stress Rate in commercial finite element codes and its compensation
Journal of Applied Mechanics, 2014Co-Authors: Zdeněk P Bažant, Jan VorelAbstract:The objective stress Rates used in most commercial finite element programs are the Jaumann Rate of Kirchhoff stress, Jaumann Rates of Cauchy stress, or Green–Naghdi Rate. The last two were long ago shown not to be associated by work with any finite strain tensor, and the first has often been combined with tangential moduli not associated by work. The error in energy conservation was thought to be negligible, but recently, several papers presented examples of structures with high volume compressibility or a high degree of orthotropy in which the use of commercial software with the Jaumann Rate of Cauchy or Kirchhoff stress leads to major errors in energy conservation, on the order of 25–100%. The present paper focuses on the Green–Naghdi Rate, which is used in the explicit nonlinear algorithms of commercial software, e.g., in subroutine VUMAT of ABAQUS. This Rate can also lead to major violations of energy conservation (or work conjugacy)—not only because of high compressibility or pronounced orthotropy but also because of large material rotations. This fact is first demonstRated analytically. Then an example of a notched steel cylinder made of steel and undergoing compression with the formation of a plastic shear band is simulated numerically by subroutine VUMAT in ABAQUS. It is found that the energy conservation error of the Green–Naghdi Rate exceeds 5% or 30% when the specimen shortens by 26% or 38%, respectively. Revisions in commercial software are needed but, even in their absence, correct results can be obtained with the existing software. To this end, the appropriate transformation of tangential moduli, to be implemented in the user's material subroutine, is derived.
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elastic soft core sandwich plates critical loads and energy errors in commercial codes due to choice of objective stress Rate
Journal of Applied Mechanics, 2013Co-Authors: Jan Vorel, Zdeněk P Bažant, Mahendra GattuAbstract:Most commercial finite element codes, such as ABAQUS, LS-DYNA, ANSYS and NASTRAN, use as the objective stress Rate the Jaumann Rate of Cauchy (or true) stress, which has two flaws: It does not conserve energy since is not work-conjugate to any finite strain tensor and, as previously shown for the case of sandwich columns, does not give a correct expression for the work of in-plane forces during buckling. This causes no appreciable errors when the skins and the core are subdivided by several layers of finite elements. However, in spite of a linear elastic behavior of the core and skins, the errors are found to be large when either the sandwich plate theory with the normals of the core remaining straight or the classical equivalent homogenization as an orthotropic plate with the normals remaining straight is used. Numerical analysis of a plate intended for the cladding of the hull of a light long ship shows errors up to 40%. It is shown that a previously derived stress-dependent transformation of the tangential moduli eliminates the energy error caused by Jaumann Rate of Cauchy stress and yields the correct critical buckling load. This load corresponds to the the Truesdell objective stress Rate, which is work-conjugate to the GreenLagrangian finite strain tensor. The commercial codes should switch to this Rate. The classical differential equations for buckling of elastic soft-core sandwich plates with a constant shear modulus of the core are shown to have a form that corresponds to the Truesdell Rate and Green-Lagrangian tensor. The critical in-plane load is solved analytically from these differential equations with typical boundary conditions, and is found to agree perfectly with the finite element solution based on the Truesdell Rate. Comparisons of the errors of various approaches are tabulated. 1 Motivation and Nature of Problem This paper is motivated by the design of large foam-core sandwich panels, intended for the cladding of a ribbed hull of light long ships with superior maneuverability and fuel-efficiency. In a laboratory test, one such panel failed at one third of the axial compressive load predicted by a standard commercial finite element program. Although the main cause of this gross underestimation of strength is probably the neglect of size effect due to cohesive delamination fracture, which was previously identified for cylindrical buckling of sandwich plates [7] and is the subject of a sepaRate study, an important additional cause to be studied here appears to lie in two long-ignored flaws [2, 5] in the handling of finite strain by standard commercial codes [1, 15, 16, e.g.]: 1) One flaw of these code is the use of objective stress Rates that are not work-conjugate to any finite strain tensor [2]. In the implicit updated Lagrangian analysis, it is the Jaumann Rate Visiting Scholar, Northwestern University; Assistant Professor on leave from Czech Technical University in Prague. McCormick Institute Professor and W.P. Murphy Professor of Civil Engineering and Materials Science, Northwestern University, 2145 Sheridan Road, CEE/A135, Evanston, Illinois 60208; z-bazant@northwestern.edu (corresponding author) Graduate Research Assistant, Northwestern University. of Cauchy stress and, in the explicit analysis, the use of the Green-Naghdi Rate. 2) Another flaw is that the bifurcation analysis uses the Jaumann Rate of Kirchhoff stress. Although this Rate is work-conjugate to the Hencky (or logarithmic) strain tensor, it cannot correctly capture the work of initial in-plane stresses in soft-in-shear highly orthotropic structures compressed in the strong direction. This work can be captured correctly only by the Truesdell objective stress Rate [3, 4]. The former flaw can lead to major errors in volume changes of polymeric, ceramic and metallic foams, fiber-reinforced foams and other highly compressible porous materials such as loess, silt, tuff, snow, under-consolidated granular materials, light wood, honeycomb, osteoporotic bones and various biological tissues. But it is unimportant for elastic buckling of a sandwich, which is the only case to be studied here. The reason is that even though the foam core is highly compressible, its hydrostatic stress is negligible (except for inelastic buckling with delamination, or for indentation [6]). This study will focus on the latter flaw. Its seriousness has already been demonstRated for sandwich columns [3, 4] and for highly orthotropic columns [12, 13], but not for sandwich plates. The energetically correct form of the differential equations of equilibrium of sandwich plates will also be identified, and their solution will be compared to finite element analysis of two kinds–a two-dimensional analysis with sandwich-type elements having a linear strain profile across the core, and three-dimensional analysis in which the core thickness is subdivided into several elements. In contrast to finite elements for the entire cross section of sandwich column, this subdivision has already been shown to yield correct results regardless of the choice of objective stress Rate; see [4] (and for elastomeric bearings see [28]). It will be verified whether this is also true for sandwich plates. A salient characteristic of sandwich plates is that the shear strain in a soft core is important for buckling. The shear buckling is a problem with a hundred-year controversial history. It requires using the stability criteria for a three-dimensional continuum, which were for half a century a subject of polemics. Although the polemics were resolved four decades ago, some authors still dispute various aspects. All the historical controversies can be traced to the arbitrariness in choosing the finite strain measure and to inattention to the work-conjugacy requirement, which means that the (doubly contracted) product of the incremental objective stress tensor with the incremental finite strain tensor must give a correct expression for the second-order work [5, ch.11]. How does the choice of strain measure affect the differential equations of equilibrium and the eigenvalues of compressed sandwich plates? And which choice is correct? These questions will be addressed first analytically, and then in the context of finite element analysis, with and without subdividing the core thickness into several layers of elements. The aim is to appraise the magnitude of errors and choose the best practical approach. 2 Review of Objective Stress Rates and Their Energy-Variational Basis A broad class of equally admissible finite strain measures which comprises virtually all of those ever used is represented by the Doyle-Ericksen tensors (m) = (U − I) /m, where m is a real parameter, I = unit tensor and U = right-stretch tensor [5]. The second-order approximation of these tensors is (m) ij = eij + 1 2 uk,iuk,j − αekiekj, eki = 12 (uk,i + ui,k) , α = 1− 1 2 m (1)
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work conjugacy error in commercial finite element codes its magnitude and how to compensate for it
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2012Co-Authors: Zdeněk P Bažant, Mahendra Gattu, Jan VorelAbstract:Most commercial finite-element programs use the Jaumann (or co-rotational) Rate of Cauchy stress in their incremental (Riks) updated Lagrangian loading procedure. This Rate was shown long ago not to be work-conjugate with the Hencky (logarithmic) finite strain tensor used in these programs, nor with any other finite strain tensor. The lack of work-conjugacy has been either overlooked or believed to cause only negligible errors. Presented are examples of indentation of a naval-type sandwich plate with a polymeric foam core, in which the error can reach 28.8 per cent in the load and 15.3 per cent in the work of load (relative to uncorrected results). Generally, similar errors must be expected for all highly compressible materials, such as metallic and ceramic foams, honeycomb, loess, silt, organic soils, pumice, tuff, osteoporotic bone, light wood, carton and various biological tissues. It is shown that a previously derived equation relating the tangential moduli tensors associated with the Jaumann Rates of Cauchy and Kirchhoff stresses can be used in the user’s material subroutine of a black-box commercial program to cancel the error due to the lack of work-conjugacy and make the program perform exactly as if the Jaumann Rate of Kirchhoff stress, which is work-conjugate, were used.
Mahendra Gattu - One of the best experts on this subject based on the ideXlab platform.
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elastic soft core sandwich plates critical loads and energy errors in commercial codes due to choice of objective stress Rate
Journal of Applied Mechanics, 2013Co-Authors: Jan Vorel, Zdeněk P Bažant, Mahendra GattuAbstract:Most commercial finite element codes, such as ABAQUS, LS-DYNA, ANSYS and NASTRAN, use as the objective stress Rate the Jaumann Rate of Cauchy (or true) stress, which has two flaws: It does not conserve energy since is not work-conjugate to any finite strain tensor and, as previously shown for the case of sandwich columns, does not give a correct expression for the work of in-plane forces during buckling. This causes no appreciable errors when the skins and the core are subdivided by several layers of finite elements. However, in spite of a linear elastic behavior of the core and skins, the errors are found to be large when either the sandwich plate theory with the normals of the core remaining straight or the classical equivalent homogenization as an orthotropic plate with the normals remaining straight is used. Numerical analysis of a plate intended for the cladding of the hull of a light long ship shows errors up to 40%. It is shown that a previously derived stress-dependent transformation of the tangential moduli eliminates the energy error caused by Jaumann Rate of Cauchy stress and yields the correct critical buckling load. This load corresponds to the the Truesdell objective stress Rate, which is work-conjugate to the GreenLagrangian finite strain tensor. The commercial codes should switch to this Rate. The classical differential equations for buckling of elastic soft-core sandwich plates with a constant shear modulus of the core are shown to have a form that corresponds to the Truesdell Rate and Green-Lagrangian tensor. The critical in-plane load is solved analytically from these differential equations with typical boundary conditions, and is found to agree perfectly with the finite element solution based on the Truesdell Rate. Comparisons of the errors of various approaches are tabulated. 1 Motivation and Nature of Problem This paper is motivated by the design of large foam-core sandwich panels, intended for the cladding of a ribbed hull of light long ships with superior maneuverability and fuel-efficiency. In a laboratory test, one such panel failed at one third of the axial compressive load predicted by a standard commercial finite element program. Although the main cause of this gross underestimation of strength is probably the neglect of size effect due to cohesive delamination fracture, which was previously identified for cylindrical buckling of sandwich plates [7] and is the subject of a sepaRate study, an important additional cause to be studied here appears to lie in two long-ignored flaws [2, 5] in the handling of finite strain by standard commercial codes [1, 15, 16, e.g.]: 1) One flaw of these code is the use of objective stress Rates that are not work-conjugate to any finite strain tensor [2]. In the implicit updated Lagrangian analysis, it is the Jaumann Rate Visiting Scholar, Northwestern University; Assistant Professor on leave from Czech Technical University in Prague. McCormick Institute Professor and W.P. Murphy Professor of Civil Engineering and Materials Science, Northwestern University, 2145 Sheridan Road, CEE/A135, Evanston, Illinois 60208; z-bazant@northwestern.edu (corresponding author) Graduate Research Assistant, Northwestern University. of Cauchy stress and, in the explicit analysis, the use of the Green-Naghdi Rate. 2) Another flaw is that the bifurcation analysis uses the Jaumann Rate of Kirchhoff stress. Although this Rate is work-conjugate to the Hencky (or logarithmic) strain tensor, it cannot correctly capture the work of initial in-plane stresses in soft-in-shear highly orthotropic structures compressed in the strong direction. This work can be captured correctly only by the Truesdell objective stress Rate [3, 4]. The former flaw can lead to major errors in volume changes of polymeric, ceramic and metallic foams, fiber-reinforced foams and other highly compressible porous materials such as loess, silt, tuff, snow, under-consolidated granular materials, light wood, honeycomb, osteoporotic bones and various biological tissues. But it is unimportant for elastic buckling of a sandwich, which is the only case to be studied here. The reason is that even though the foam core is highly compressible, its hydrostatic stress is negligible (except for inelastic buckling with delamination, or for indentation [6]). This study will focus on the latter flaw. Its seriousness has already been demonstRated for sandwich columns [3, 4] and for highly orthotropic columns [12, 13], but not for sandwich plates. The energetically correct form of the differential equations of equilibrium of sandwich plates will also be identified, and their solution will be compared to finite element analysis of two kinds–a two-dimensional analysis with sandwich-type elements having a linear strain profile across the core, and three-dimensional analysis in which the core thickness is subdivided into several elements. In contrast to finite elements for the entire cross section of sandwich column, this subdivision has already been shown to yield correct results regardless of the choice of objective stress Rate; see [4] (and for elastomeric bearings see [28]). It will be verified whether this is also true for sandwich plates. A salient characteristic of sandwich plates is that the shear strain in a soft core is important for buckling. The shear buckling is a problem with a hundred-year controversial history. It requires using the stability criteria for a three-dimensional continuum, which were for half a century a subject of polemics. Although the polemics were resolved four decades ago, some authors still dispute various aspects. All the historical controversies can be traced to the arbitrariness in choosing the finite strain measure and to inattention to the work-conjugacy requirement, which means that the (doubly contracted) product of the incremental objective stress tensor with the incremental finite strain tensor must give a correct expression for the second-order work [5, ch.11]. How does the choice of strain measure affect the differential equations of equilibrium and the eigenvalues of compressed sandwich plates? And which choice is correct? These questions will be addressed first analytically, and then in the context of finite element analysis, with and without subdividing the core thickness into several layers of elements. The aim is to appraise the magnitude of errors and choose the best practical approach. 2 Review of Objective Stress Rates and Their Energy-Variational Basis A broad class of equally admissible finite strain measures which comprises virtually all of those ever used is represented by the Doyle-Ericksen tensors (m) = (U − I) /m, where m is a real parameter, I = unit tensor and U = right-stretch tensor [5]. The second-order approximation of these tensors is (m) ij = eij + 1 2 uk,iuk,j − αekiekj, eki = 12 (uk,i + ui,k) , α = 1− 1 2 m (1)
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work conjugacy error in commercial finite element codes its magnitude and how to compensate for it
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2012Co-Authors: Zdeněk P Bažant, Mahendra Gattu, Jan VorelAbstract:Most commercial finite-element programs use the Jaumann (or co-rotational) Rate of Cauchy stress in their incremental (Riks) updated Lagrangian loading procedure. This Rate was shown long ago not to be work-conjugate with the Hencky (logarithmic) finite strain tensor used in these programs, nor with any other finite strain tensor. The lack of work-conjugacy has been either overlooked or believed to cause only negligible errors. Presented are examples of indentation of a naval-type sandwich plate with a polymeric foam core, in which the error can reach 28.8 per cent in the load and 15.3 per cent in the work of load (relative to uncorrected results). Generally, similar errors must be expected for all highly compressible materials, such as metallic and ceramic foams, honeycomb, loess, silt, organic soils, pumice, tuff, osteoporotic bone, light wood, carton and various biological tissues. It is shown that a previously derived equation relating the tangential moduli tensors associated with the Jaumann Rates of Cauchy and Kirchhoff stresses can be used in the user’s material subroutine of a black-box commercial program to cancel the error due to the lack of work-conjugacy and make the program perform exactly as if the Jaumann Rate of Kirchhoff stress, which is work-conjugate, were used.
Ahmed K Noor - One of the best experts on this subject based on the ideXlab platform.
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sensitivity analysis of the non linear dynamic viscoplastic response of 2 d structures with respect to material parameters
International Journal for Numerical Methods in Engineering, 1995Co-Authors: Makarand Kulkarni, Ahmed K NoorAbstract:A computational procedure is presented for evaluating the sensitivity coefficients of the viscoplastic response of structures subjected to dynamic loading. A state of plane stress is assumed to exist in the structure, a velocity strain-Cauchy stress formulation is used, and the geometric non-linearities arising from large strains are incorpoRated. The Jaumann Rate is used as a frame indifferent stress Rate. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. The equations of motion emanating from a finite element semi-discretization are integRated using an explicit central difference scheme with an implicit stress update. The sensitivity coefficients are evaluated using a direct differentiation approach. Since the domain of integration is the current configuration, the sensitivity coefficients of the spatial derivatives of the shape functions must be included. Numerical results are presented for a thin plate with a central circular cutout subjected to an in-plane compressive loading. The sensitivity coefficients are geneRated by evaluating the derivatives of the response quantities with respect to Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients, and spatial distributions at selected times are presented.
Hamid Jahed - One of the best experts on this subject based on the ideXlab platform.
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eulerian framework for inelasticity based on the Jaumann Rate and a hyperelastic constitutive relation part ii finite strain elastoplasticity
Journal of Applied Mechanics, 2013Co-Authors: Amin Eshraghi, Hamid Jahed, Katerina D PapouliaAbstract:An Eulerian Rate formulation of finite strain elastoplasticity is developed based on a fully integrable Rate form of hyperelasticity proposed in Part I of this work. A flow rule is proposed in the Eulerian framework, based on the principle of maximum plastic dissipation in six-dimensional stress space for the case of J2 isotropic plasticity. The proposed flow rule bypasses the need for additional evolution laws and/or simplifying assumptions for the skew-symmetric part of the plastic velocity gradient, known as the material plastic spin. Kinematic hardening is modeled with an evolution equation for the backstress tensor considering Prager’s yielding-stationarity criterion. Nonlinear evolution equations for the backstress and flow stress are proposed for an extension of the model to mixed nonlinear hardening. Furthermore, exact deviatoric/volumetric decoupled forms for kinematic and kinetic variables are obtained. The proposed model is implemented with the Zaremba–Jaumann Rate and is used to solve the problem of rectilinear shear for a perfectly plastic and for a linear kinematic hardening material. Neither solution produces oscillatory stress or backstress components. The model is then used to predict the nonlinear hardening behavior of SUS 304 stainless steel under fixed-end finite torsion. Results obtained are in good agreement with reported experimental data. The Swift effect under finite torsion is well predicted by the proposed model.
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eulerian framework for inelasticity based on the Jaumann Rate and a hyperelastic constitutive relation part i Rate form hyperelasticity
Journal of Applied Mechanics, 2013Co-Authors: Amin Eshraghi, Katerina D Papoulia, Hamid JahedAbstract:An integrable Eulerian Rate formulation of finite deformation elasticity is developed, which relates the Jaumann or other objective corotational Rate of the Kirchhoff stress with material spin to the same Rate of the left Cauchy–Green deformation measure through a deformation dependent constitutive tensor. The proposed constitutive relationship can be written in terms of the Rate of deformation tensor in the form of a hypoelastic material model. Integrability conditions, under which the proposed formulation yields (a) a Cauchy elastic and (b) a Green elastic material model are derived for the isotropic case. These determine the deformation dependent instantaneous elasticity tensor of the material. In particular, when the Cauchy integrability criterion is applied to the stress-strain relationship of a hyperelastic material model, an Eulerian Rate formulation of hyperelasticity is obtained. This formulation proves crucial for the Eulerian finite strain elastoplastic model developed in part II of this work. The proposed model is formulated and integRated in the fixed background and extends the notion of an integrable hypoelastic model to arbitrary corotational objective Rates and coordinates. Integrability was previously shown for the grade-zero hypoelastic model with use of the logarithmic (D) Rate, the spin of which is formulated in principal coordinates. Uniform deformation examples of rectilinear shear, closed path four-step loading, and cyclic elliptical loading are presented. Contrary to classical grade-zero hypoelasticity, no shear oscillation, elastic dissipation, or ratcheting under cyclic load is observed when the simple Zaremba–Jaumann Rate of stress is employed.
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Eulerian framework for inelasticity based on the Jaumann Rate and a hyperelastic constitutive relation–part II: finite strain elastoplasticity
2013Co-Authors: Amin Eshraghi, Katerina D Papoulia, Hamid JahedAbstract:An integrable Eulerian Rate formulation of finite deformation elasticity is developed, which relates the Jaumann or other objective corotational Rate of the Kirchhoff stress with material spin to the same Rate of the left Cauchy-Green deformation measure through a deformation dependent constitutive tensor. The proposed constitutive relationship can be written in terms of the Rate of deformation tensor in the form of a hypoelastic material model. Integrability conditions, under which the proposed formulation yields (a) a Cauchy elastic and (b) a Green elastic material model are derived for the isotropic case. These determine the deformation dependent instantaneous elasticity tensor of the material. In particular, when the Cauchy integrability criterion is applied to the stressstrain relationship of a hyperelastic material model, an Eulerian Rate formulation of hyperelasticity is obtained. This formulation proves crucial for the Eulerian finite strain elastoplastic model developed in part II of this work. The proposed model is formulated and integRated in the fixed background and extends the notion of an integrable hypoelastic model to arbitrary corotational objective Rates and coordinates. Integrability was previously shown for the grade-zero hypoelastic model with use of the logarithmic (D) Rate, the spin of which is formulated in principal coordinates. Uniform deformation examples of rectilinear shear, closed path four-step loading, and cyclic elliptical loading are presented. Contrary to classical grade-zero hypoelasticity, no shear oscillation, elastic dissipation, or ratcheting under cyclic load is observed when the simple Zaremba-Jaumann Rate of stress is employed
