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The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 22 Feb 2024 14:37:11 GMT2024-02-22T14:37:11ZNumerical integration of rate-independent BCC single crystal plasticity models: comparative study of two classes of numerical algorithms
http://hdl.handle.net/10985/13535
Numerical integration of rate-independent BCC single crystal plasticity models: comparative study of two classes of numerical algorithms
AKPAMA, Holanyo K.; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
In an incremental formulation suitable to numerical implementation, the use of rate-independent theory of crystal plasticity essentially leads to four fundamental problems. The first is to determine the set of potentially active slip systems over a time increment. The second is to select the active slip systems among the potentially active ones. The third is to compute the slip rates (or the slip increments) for the active slip systems. And the last problem is the possible non-uniqueness of slip rates. The purpose of this paper is to propose satisfactory responses to the above-mentioned first three issues by presenting and comparing two novel numerical algorithms. The first algorithm is based on the usual return-mapping integration scheme, while the second follows the so-called ultimate scheme. The latter is shown to be more relevant and efficient than the former. These comparative performances are illustrated through various numerical simulations of the mechanical behavior of single crystals and polycrystalline aggregates subjected to monotonic and complex loadings. Although these algorithms are applied in this paper to Body-Centered-Cubic (BCC) crystal structures, they are quite general and suitable for integrating the constitutive equations for other crystal structures (e.g., FCC and HCP).
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/135352016-01-01T00:00:00ZAKPAMA, Holanyo K.BEN BETTAIEB, MohamedABED-MERAIM, FaridIn an incremental formulation suitable to numerical implementation, the use of rate-independent theory of crystal plasticity essentially leads to four fundamental problems. The first is to determine the set of potentially active slip systems over a time increment. The second is to select the active slip systems among the potentially active ones. The third is to compute the slip rates (or the slip increments) for the active slip systems. And the last problem is the possible non-uniqueness of slip rates. The purpose of this paper is to propose satisfactory responses to the above-mentioned first three issues by presenting and comparing two novel numerical algorithms. The first algorithm is based on the usual return-mapping integration scheme, while the second follows the so-called ultimate scheme. The latter is shown to be more relevant and efficient than the former. These comparative performances are illustrated through various numerical simulations of the mechanical behavior of single crystals and polycrystalline aggregates subjected to monotonic and complex loadings. Although these algorithms are applied in this paper to Body-Centered-Cubic (BCC) crystal structures, they are quite general and suitable for integrating the constitutive equations for other crystal structures (e.g., FCC and HCP).Prediction of Localized Necking Based on Crystal Plasticity: Comparison of Bifurcation and Imperfection Approaches
http://hdl.handle.net/10985/11858
Prediction of Localized Necking Based on Crystal Plasticity: Comparison of Bifurcation and Imperfection Approaches
AKPAMA, Holanyo K.; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
In the present work, a powerful modeling tool is developed to predict and analyze the onset of strain localization in polycrystalline aggregates. The predictions of localized necking are based on two plastic instability criteria, namely the bifurcation theory and the initial imperfection approach. In this tool, a micromechanical model, based on the self-consistent scale-transition scheme, is used to accurately derive the mechanical behavior of polycrystalline aggregates from that of their microscopic constituents (the single crystals). The mechanical behavior of the single crystals is developed within a large strain rate-independent constitutive framework. This micromechanical constitutive modeling takes into account the essential microstructure-related features that are relevant at the microscale. These microstructural aspects include key physical mechanisms, such as initial and induced crystallographic textures, morphological anisotropy and interactions between the grains and their surrounding medium. The developed tool is used to predict sheet metal formability through the concept of forming limit diagrams (FLDs). The results obtained by the self-consistent averaging scheme, in terms of predicted FLDs, are compared with those given by the more classical full-constraint Taylor model. Moreover, the predictions obtained by the imperfection approach are systematically compared with those given by the bifurcation analysis, and it is demonstrated that the former tend to the latter in the limit of a vanishing size for the initial imperfection.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/118582016-01-01T00:00:00ZAKPAMA, Holanyo K.BEN BETTAIEB, MohamedABED-MERAIM, FaridIn the present work, a powerful modeling tool is developed to predict and analyze the onset of strain localization in polycrystalline aggregates. The predictions of localized necking are based on two plastic instability criteria, namely the bifurcation theory and the initial imperfection approach. In this tool, a micromechanical model, based on the self-consistent scale-transition scheme, is used to accurately derive the mechanical behavior of polycrystalline aggregates from that of their microscopic constituents (the single crystals). The mechanical behavior of the single crystals is developed within a large strain rate-independent constitutive framework. This micromechanical constitutive modeling takes into account the essential microstructure-related features that are relevant at the microscale. These microstructural aspects include key physical mechanisms, such as initial and induced crystallographic textures, morphological anisotropy and interactions between the grains and their surrounding medium. The developed tool is used to predict sheet metal formability through the concept of forming limit diagrams (FLDs). The results obtained by the self-consistent averaging scheme, in terms of predicted FLDs, are compared with those given by the more classical full-constraint Taylor model. Moreover, the predictions obtained by the imperfection approach are systematically compared with those given by the bifurcation analysis, and it is demonstrated that the former tend to the latter in the limit of a vanishing size for the initial imperfection.Ductility prediction of substrate-supported metal layers based on rate-independent crystal plasticity theory
http://hdl.handle.net/10985/19115
Ductility prediction of substrate-supported metal layers based on rate-independent crystal plasticity theory
AKPAMA, Holanyo K.; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
In this paper, both the bifurcation theory and the initial imperfection approach are used to predict localized necking in substrate-supported metal layers. The self-consistent scale-transition scheme is used to derive the mechanical behavior of a representative volume element of the metal layer from the behavior of its microscopic constituents (the single crystals). The mechanical behavior of the elastomer substrate follows the neo-Hookean hyperelastic model. The adherence between the two layers is assumed to be perfect. Through numerical results, it is shown that the limit strains predicted by the initial imperfection approach tend towards the bifurcation predictions when the size of the geometric imperfection in the metal layer vanishes. Also, it is shown that the addition of an elastomer layer to a metal layer enhances ductility.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/191152016-01-01T00:00:00ZAKPAMA, Holanyo K.BEN BETTAIEB, MohamedABED-MERAIM, FaridIn this paper, both the bifurcation theory and the initial imperfection approach are used to predict localized necking in substrate-supported metal layers. The self-consistent scale-transition scheme is used to derive the mechanical behavior of a representative volume element of the metal layer from the behavior of its microscopic constituents (the single crystals). The mechanical behavior of the elastomer substrate follows the neo-Hookean hyperelastic model. The adherence between the two layers is assumed to be perfect. Through numerical results, it is shown that the limit strains predicted by the initial imperfection approach tend towards the bifurcation predictions when the size of the geometric imperfection in the metal layer vanishes. Also, it is shown that the addition of an elastomer layer to a metal layer enhances ductility.Prediction of the ductility limit of sheet metals during forming processes using the loss of ellipticity approach
http://hdl.handle.net/10985/19681
Prediction of the ductility limit of sheet metals during forming processes using the loss of ellipticity approach
AKPAMA, Holanyo K.; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
The prediction of the ductility limit of sheet metals during forming processes represents nowadays and ambitious challenge. To reach this goal, a new numerical approach, based on the loss of ellipticity criterion [1], is proposed in this work. A polycrystalline model is implemented as a user-material (UMAT) into ABAQUS FE code. The polycrystalline aggregate is associated with each integration point of the FE mesh. To derive the mechanical behavior of this polycrystalline aggregate from the behavior of its microscopic constituents, the self-consistent model is used [2]. The mechanical behavior of the single crystals is described by a finite strain rate-independent constitutive framework, where the Schmid law is used to model the plastic flow. The condition of loss of ellipticity at the macroscale, where the macroscopic behavior is derived by using the self-consistent scheme, is used as ductility criterion in the FE modeling. This numerical approach, which couples the FE method with the self-consistent scheme, is used to simulate some forming processes (deep drawing process, Nakazima testâ€¦), and the above criterion is used to predict the ductility limit of the studied sheets during these operations.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/196812016-01-01T00:00:00ZAKPAMA, Holanyo K.BEN BETTAIEB, MohamedABED-MERAIM, FaridThe prediction of the ductility limit of sheet metals during forming processes represents nowadays and ambitious challenge. To reach this goal, a new numerical approach, based on the loss of ellipticity criterion [1], is proposed in this work. A polycrystalline model is implemented as a user-material (UMAT) into ABAQUS FE code. The polycrystalline aggregate is associated with each integration point of the FE mesh. To derive the mechanical behavior of this polycrystalline aggregate from the behavior of its microscopic constituents, the self-consistent model is used [2]. The mechanical behavior of the single crystals is described by a finite strain rate-independent constitutive framework, where the Schmid law is used to model the plastic flow. The condition of loss of ellipticity at the macroscale, where the macroscopic behavior is derived by using the self-consistent scheme, is used as ductility criterion in the FE modeling. This numerical approach, which couples the FE method with the self-consistent scheme, is used to simulate some forming processes (deep drawing process, Nakazima testâ€¦), and the above criterion is used to predict the ductility limit of the studied sheets during these operations.Prediction of localized necking based on crystal plasticity: Comparison of bifurcation and imperfection approaches
http://hdl.handle.net/10985/19658
Prediction of localized necking based on crystal plasticity: Comparison of bifurcation and imperfection approaches
AKPAMA, Holanyo K.; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
In the present work, a powerful modeling tool is developed to predict and analyze the onset of strain localization in polycrystalline aggregates. The predictions of localized necking are based on two plastic instability criteria, namely the bifurcation theory and the initial imperfection approach. In this tool, a micromechanical model, based on the self-consistent scale-transition scheme, is used to accurately derive the mechanical behavior of polycrystalline aggregates from that of their microscopic constituents (the single crystals). The mechanical behavior of the single crystals is developed within a large strain rate-independent constitutive framework. This micromechanical constitutive modeling takes into account the essential microstructure-related features that are relevant at the microscale. These microstructural aspects include key physical mechanisms, such as initial and induced crystallographic textures, morphological anisotropy and interactions between the grains and their surrounding medium. The developed tool is used to predict sheet metal formability through the concept of forming limit diagrams (FLDs). The results obtained by the self-consistent averaging scheme, in terms of predicted FLDs, are compared with those given by the more classical full-constraint Taylor model. Moreover, the predictions obtained by the imperfection approach are systematically compared with those given by the bifurcation analysis, and it is demonstrated that the former tend to the latter in the limit of a vanishing size for the initial imperfection.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/196582016-01-01T00:00:00ZAKPAMA, Holanyo K.BEN BETTAIEB, MohamedABED-MERAIM, FaridIn the present work, a powerful modeling tool is developed to predict and analyze the onset of strain localization in polycrystalline aggregates. The predictions of localized necking are based on two plastic instability criteria, namely the bifurcation theory and the initial imperfection approach. In this tool, a micromechanical model, based on the self-consistent scale-transition scheme, is used to accurately derive the mechanical behavior of polycrystalline aggregates from that of their microscopic constituents (the single crystals). The mechanical behavior of the single crystals is developed within a large strain rate-independent constitutive framework. This micromechanical constitutive modeling takes into account the essential microstructure-related features that are relevant at the microscale. These microstructural aspects include key physical mechanisms, such as initial and induced crystallographic textures, morphological anisotropy and interactions between the grains and their surrounding medium. The developed tool is used to predict sheet metal formability through the concept of forming limit diagrams (FLDs). The results obtained by the self-consistent averaging scheme, in terms of predicted FLDs, are compared with those given by the more classical full-constraint Taylor model. Moreover, the predictions obtained by the imperfection approach are systematically compared with those given by the bifurcation analysis, and it is demonstrated that the former tend to the latter in the limit of a vanishing size for the initial imperfection.Influence of the Yield Surface Curvature on the Forming Limit Diagrams Predicted by Crystal Plasticity Theory
http://hdl.handle.net/10985/11189
Influence of the Yield Surface Curvature on the Forming Limit Diagrams Predicted by Crystal Plasticity Theory
AKPAMA, Holanyo K.; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
The aim of this paper is to investigate the impact of the microscopic yield surface (i.e., at the single crystal scale) on the forming limit diagrams (FLDs) of face centred cubic (FCC) materials. To predict these FLDs, the bifurcation approach is used within the framework of rate-independent crystal plasticity theory. For this purpose, two micromechanical models are developed and implemented. The first one uses the classical Schmid law, which results in the formation of vertices (or corners) at the yield surface, while the second is based on regularization of the Schmid law, which induces rounded corners at the yield surface. In both cases, the overall macroscopic behavior is derived from the behavior of the microscopic constituents (the single crystals) by using two different scale-transition schemes: the selfconsistent approach and the Taylor model. The simulation results show that the use of the classical Schmid law allows predicting localized necking at realistic strain levels for the whole range of strain paths that span the FLD. However, the application of a regularized Schmid law results in much higher limit strains in the range of negative strain paths. Moreover, rounding the yield surface vertices through regularization of the Schmid law leads to unrealistically high limit strains in the range of positive strain paths.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/111892016-01-01T00:00:00ZAKPAMA, Holanyo K.BEN BETTAIEB, MohamedABED-MERAIM, FaridThe aim of this paper is to investigate the impact of the microscopic yield surface (i.e., at the single crystal scale) on the forming limit diagrams (FLDs) of face centred cubic (FCC) materials. To predict these FLDs, the bifurcation approach is used within the framework of rate-independent crystal plasticity theory. For this purpose, two micromechanical models are developed and implemented. The first one uses the classical Schmid law, which results in the formation of vertices (or corners) at the yield surface, while the second is based on regularization of the Schmid law, which induces rounded corners at the yield surface. In both cases, the overall macroscopic behavior is derived from the behavior of the microscopic constituents (the single crystals) by using two different scale-transition schemes: the selfconsistent approach and the Taylor model. The simulation results show that the use of the classical Schmid law allows predicting localized necking at realistic strain levels for the whole range of strain paths that span the FLD. However, the application of a regularized Schmid law results in much higher limit strains in the range of negative strain paths. Moreover, rounding the yield surface vertices through regularization of the Schmid law leads to unrealistically high limit strains in the range of positive strain paths.Localized necking predictions based on rate-independent self-consistent polycrystal plasticity: Bifurcation analysis versus imperfection approach
http://hdl.handle.net/10985/11856
Localized necking predictions based on rate-independent self-consistent polycrystal plasticity: Bifurcation analysis versus imperfection approach
AKPAMA, Holanyo K.; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
The present study focuses on the development of a relevant numerical tool for predicting the onset of localized necking in polycrystalline aggregates. The latter are assumed to be representative of thin metal sheets. In this tool, a micromechanical model, based on the rate-independent self-consistent multi-scale scheme, is developed to accurately describe the mechanical behavior of polycrystalline aggregates from that of their single crystal constituents. In the current paper, the constitutive framework at the single crystal scale follows a finite strain formulation of the rate-independent theory of crystal elastoplasticity. To predict the occurrence of localized necking in polycrystalline aggregates, this micromechanical modeling is combined with two main strain localization approaches: the bifurcation analysis and the initial imperfection method. The formulation of both strain localization indicators takes into consideration the plane stress conditions to which thin metal sheets are subjected during deformation. From a numerical point of view, strain localization analysis with this crystal plasticity approach can be viewed as a strongly nonlinear problem. Hence, several numerical algorithms and techniques are developed and implemented in the aim of efficiently solving this non-linear problem. Various simulation results obtained by the application of the developed numerical tool are presented and extensively discussed. It is demonstrated from these results that the predictions obtained with the MarciniakeKuczynski procedure tend towards those yielded by the bifurcation theory, when the initial imperfection ratio tends towards zero. Furthermore, the above result is shown to be valid for both scale-transition schemes, namely the full-constraint Taylor model and self-consistent scheme.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/118562017-01-01T00:00:00ZAKPAMA, Holanyo K.BEN BETTAIEB, MohamedABED-MERAIM, FaridThe present study focuses on the development of a relevant numerical tool for predicting the onset of localized necking in polycrystalline aggregates. The latter are assumed to be representative of thin metal sheets. In this tool, a micromechanical model, based on the rate-independent self-consistent multi-scale scheme, is developed to accurately describe the mechanical behavior of polycrystalline aggregates from that of their single crystal constituents. In the current paper, the constitutive framework at the single crystal scale follows a finite strain formulation of the rate-independent theory of crystal elastoplasticity. To predict the occurrence of localized necking in polycrystalline aggregates, this micromechanical modeling is combined with two main strain localization approaches: the bifurcation analysis and the initial imperfection method. The formulation of both strain localization indicators takes into consideration the plane stress conditions to which thin metal sheets are subjected during deformation. From a numerical point of view, strain localization analysis with this crystal plasticity approach can be viewed as a strongly nonlinear problem. Hence, several numerical algorithms and techniques are developed and implemented in the aim of efficiently solving this non-linear problem. Various simulation results obtained by the application of the developed numerical tool are presented and extensively discussed. It is demonstrated from these results that the predictions obtained with the MarciniakeKuczynski procedure tend towards those yielded by the bifurcation theory, when the initial imperfection ratio tends towards zero. Furthermore, the above result is shown to be valid for both scale-transition schemes, namely the full-constraint Taylor model and self-consistent scheme.A comparative study of Forming Limit Diagrams predicted by two different plasticity theories involving vertex effects
http://hdl.handle.net/10985/10048
A comparative study of Forming Limit Diagrams predicted by two different plasticity theories involving vertex effects
AKPAMA, Holanyo K.; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
The main objective of this contribution is to compare the Forming Limit Diagrams (FLDs) predicted by the use of two different vertex theories. The first theory is micromechanical and is based on the use of the Schmid law, within the framework of crystal plasticity coupled with the Taylor scale-transition scheme. The second theory is phenomenological and is based on the deformation theory of plasticity. For both theories, the mechanical behavior is formulated in the finite strain framework and is assumed to be isotropic and rate-independent. The theoretical framework of these approaches will be presented in details. In the micro-macro modeling, the isotropy is ensured by considering an isotropic initial texture. In the phenomenological modeling, the material parameters are identified on the basis of micro-macro simulations of tensile tests.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10985/100482015-01-01T00:00:00ZAKPAMA, Holanyo K.BEN BETTAIEB, MohamedABED-MERAIM, FaridThe main objective of this contribution is to compare the Forming Limit Diagrams (FLDs) predicted by the use of two different vertex theories. The first theory is micromechanical and is based on the use of the Schmid law, within the framework of crystal plasticity coupled with the Taylor scale-transition scheme. The second theory is phenomenological and is based on the deformation theory of plasticity. For both theories, the mechanical behavior is formulated in the finite strain framework and is assumed to be isotropic and rate-independent. The theoretical framework of these approaches will be presented in details. In the micro-macro modeling, the isotropy is ensured by considering an isotropic initial texture. In the phenomenological modeling, the material parameters are identified on the basis of micro-macro simulations of tensile tests.Prediction of Plastic Instability in Sheet Metals During Forming Processes Using the Loss of Ellipticity Approach
http://hdl.handle.net/10985/11915
Prediction of Plastic Instability in Sheet Metals During Forming Processes Using the Loss of Ellipticity Approach
AKPAMA, Holanyo K.; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
The prediction of plastic instability in sheet metals during forming processes represents nowadays an ambitious challenge. To reach this goal, a new numerical approach, based on the loss of ellipticity criterion, is proposed in the present contribution. A polycrystalline model is implemented as a user-material subroutine into the ABAQUS/Implicit finite element (FE) code. The polycrystalline constitutive model is assigned to each integration point of the FE mesh. To derive the mechanical behavior of this polycrystalline aggregate from the behavior of its microscopic constituents, the multiscale self-consistent scheme is used. The mechanical behavior of the single crystals is described by a finite strain rateindependent constitutive framework, where the Schmid law is used to model the plastic flow. The condition of loss of ellipticity at the macroscale is used as plastic instability criterion in the FE modeling. This numerical approach, which couples the FE method with the self-consistent scheme, is used to simulate a deep drawing process, and the above criterion is used to predict the formability limit of the studied sheets during this operation.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/119152017-01-01T00:00:00ZAKPAMA, Holanyo K.BEN BETTAIEB, MohamedABED-MERAIM, FaridThe prediction of plastic instability in sheet metals during forming processes represents nowadays an ambitious challenge. To reach this goal, a new numerical approach, based on the loss of ellipticity criterion, is proposed in the present contribution. A polycrystalline model is implemented as a user-material subroutine into the ABAQUS/Implicit finite element (FE) code. The polycrystalline constitutive model is assigned to each integration point of the FE mesh. To derive the mechanical behavior of this polycrystalline aggregate from the behavior of its microscopic constituents, the multiscale self-consistent scheme is used. The mechanical behavior of the single crystals is described by a finite strain rateindependent constitutive framework, where the Schmid law is used to model the plastic flow. The condition of loss of ellipticity at the macroscale is used as plastic instability criterion in the FE modeling. This numerical approach, which couples the FE method with the self-consistent scheme, is used to simulate a deep drawing process, and the above criterion is used to predict the formability limit of the studied sheets during this operation.Numerical integration of rate-independent BCC single crystal plasticity models: comparative study of two classes of numerical algorithms
http://hdl.handle.net/10985/10654
Numerical integration of rate-independent BCC single crystal plasticity models: comparative study of two classes of numerical algorithms
AKPAMA, Holanyo K.; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
In an incremental formulation suitable to numerical implementation, the use of rate-independent theory of crystal plasticity essentially leads to four fundamental problems. The first is to determine the set of potentially active slip systems over a time increment. The second is to select the active slip systems among the potentially active ones. The third is to compute the slip rates (or the slip increments) for the active slip systems. And the last problem is the possible non-uniqueness of slip rates. The purpose of this paper is to propose satisfactory responses to the above-mentioned first three issues by presenting and comparing two novel numerical algorithms. The first algorithm is based on the usual return-mapping integration scheme, while the second follows the so-called ultimate scheme. The latter is shown to be more relevant and efficient than the former. These comparative performances are illustrated through various numerical simulations of the mechanical behavior of single crystals and polycrystalline aggregates subjected to monotonic and complex loadings. Although these algorithms are applied in this paper to Body-Centered-Cubic (BCC) crystal structures, they are quite general and suitable for integrating the constitutive equations for other crystal structures (e.g., FCC and HCP).
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/106542016-01-01T00:00:00ZAKPAMA, Holanyo K.BEN BETTAIEB, MohamedABED-MERAIM, FaridIn an incremental formulation suitable to numerical implementation, the use of rate-independent theory of crystal plasticity essentially leads to four fundamental problems. The first is to determine the set of potentially active slip systems over a time increment. The second is to select the active slip systems among the potentially active ones. The third is to compute the slip rates (or the slip increments) for the active slip systems. And the last problem is the possible non-uniqueness of slip rates. The purpose of this paper is to propose satisfactory responses to the above-mentioned first three issues by presenting and comparing two novel numerical algorithms. The first algorithm is based on the usual return-mapping integration scheme, while the second follows the so-called ultimate scheme. The latter is shown to be more relevant and efficient than the former. These comparative performances are illustrated through various numerical simulations of the mechanical behavior of single crystals and polycrystalline aggregates subjected to monotonic and complex loadings. Although these algorithms are applied in this paper to Body-Centered-Cubic (BCC) crystal structures, they are quite general and suitable for integrating the constitutive equations for other crystal structures (e.g., FCC and HCP).