SAM
https://sam.ensam.eu:443
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Fri, 15 Oct 2021 20:14:55 GMT2021-10-15T20:14:55ZComparative study of three techniques for the computation of the macroscopic tangent moduli by periodic homogenization scheme
http://hdl.handle.net/10985/19176
Comparative study of three techniques for the computation of the macroscopic tangent moduli by periodic homogenization scheme
ZHU, Jianchang; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
In numerical strategies developed for determining the efective macroscopic properties of heterogeneous media, the efcient and robust computation of macroscopic tangent moduli represents an essential step to achieve. Indeed, these tangent moduli are usually required in several numerical applications, such as the FE2 method and the prediction of the onset of material and structural instabilities in heterogeneous media by loss of ellipticity approaches. This paper presents a comparative study of three numerical techniques for the computation of such tangent moduli in the context of periodic homogenization: the perturbation technique, the condensation technique and the fuctuation technique. The practical implementations of these techniques within ABAQUS/Standard fnite element (FE) code are especially underlined. These implementations are based on the development of a set of Python scripts, which are connected to the fnite element computations to handle the computa‑ tion of the tangent moduli. The extension of these techniques to mechanical problems exhibiting symmetry properties is also detailed in this contribution. The reliability, accuracy and ease of implementation of these techniques are evaluated through some typical numerical examples. It is shown from this numerical and technical study that the condensation method reveals to be the most reliable and efcient. Also, this paper provides valuable reference guidelines to ABAQUS/Standard users for the determination of the homogenized tangent moduli of linear or nonlinear heterogeneous materials, such as composites, polycrystalline aggregates and porous solids.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10985/191762020-01-01T00:00:00ZZHU, JianchangBEN BETTAIEB, MohamedABED-MERAIM, FaridIn numerical strategies developed for determining the efective macroscopic properties of heterogeneous media, the efcient and robust computation of macroscopic tangent moduli represents an essential step to achieve. Indeed, these tangent moduli are usually required in several numerical applications, such as the FE2 method and the prediction of the onset of material and structural instabilities in heterogeneous media by loss of ellipticity approaches. This paper presents a comparative study of three numerical techniques for the computation of such tangent moduli in the context of periodic homogenization: the perturbation technique, the condensation technique and the fuctuation technique. The practical implementations of these techniques within ABAQUS/Standard fnite element (FE) code are especially underlined. These implementations are based on the development of a set of Python scripts, which are connected to the fnite element computations to handle the computa‑ tion of the tangent moduli. The extension of these techniques to mechanical problems exhibiting symmetry properties is also detailed in this contribution. The reliability, accuracy and ease of implementation of these techniques are evaluated through some typical numerical examples. It is shown from this numerical and technical study that the condensation method reveals to be the most reliable and efcient. Also, this paper provides valuable reference guidelines to ABAQUS/Standard users for the determination of the homogenized tangent moduli of linear or nonlinear heterogeneous materials, such as composites, polycrystalline aggregates and porous solids.Numerical investigation of necking in perforated sheets using the periodic homogenization approach
http://hdl.handle.net/10985/17463
Numerical investigation of necking in perforated sheets using the periodic homogenization approach
ZHU, Jianchang; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
Due to their attractive properties, perforated sheets are increasingly used in a number of industrial applications, such as automotive, architecture, pollution control, etc. Consequently, the accurate modeling of the mechanical behavior of this kind of sheets still remains a valuable goal to reach. This paper aims to contribute to this effort by developing reliable numerical tools capable of predicting the occurrence of necking in perforated sheets. These tools are based on the coupling between the periodic homogenization technique and three plastic instability criteria. The periodic homogenization technique is used to derive equivalent macroscopic mechanical behavior for a representative volume element of these sheets. On the other hand, the prediction of plastic instability is based on three necking criteria: the maximum force criterion (diffuse necking), the general bifurcation criterion (diffuse necking), and the loss of ellipticity criterion (localized necking). The predictions obtained by applying the three instability criteria are thoroughly analyzed and compared. A sensitivity study is also conducted to numerically investigate the influence on the prediction of necking of the design parameters (dimension, aspect-ratio, orientation, and shape of the holes), the macroscopic boundary conditions and the metal matrix material parameters (plastic anisotropy, hardening).
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10985/174632020-01-01T00:00:00ZZHU, JianchangBEN BETTAIEB, MohamedABED-MERAIM, FaridDue to their attractive properties, perforated sheets are increasingly used in a number of industrial applications, such as automotive, architecture, pollution control, etc. Consequently, the accurate modeling of the mechanical behavior of this kind of sheets still remains a valuable goal to reach. This paper aims to contribute to this effort by developing reliable numerical tools capable of predicting the occurrence of necking in perforated sheets. These tools are based on the coupling between the periodic homogenization technique and three plastic instability criteria. The periodic homogenization technique is used to derive equivalent macroscopic mechanical behavior for a representative volume element of these sheets. On the other hand, the prediction of plastic instability is based on three necking criteria: the maximum force criterion (diffuse necking), the general bifurcation criterion (diffuse necking), and the loss of ellipticity criterion (localized necking). The predictions obtained by applying the three instability criteria are thoroughly analyzed and compared. A sensitivity study is also conducted to numerically investigate the influence on the prediction of necking of the design parameters (dimension, aspect-ratio, orientation, and shape of the holes), the macroscopic boundary conditions and the metal matrix material parameters (plastic anisotropy, hardening).Numerical investigation of the ductility limit of perforated sheets
http://hdl.handle.net/10985/19680
Numerical investigation of the ductility limit of perforated sheets
ZHU, Jianchang; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
Perforated sheets are widely used in automotive, architecture, pollution control, and other fields. Because perforated sheets are lightweight and aesthetically attractive, and also allow specific elements such as water, air, and light to pass through them, press-formed products with high added-value from these advantages are in great demand. For these reasons, the accurate modeling of the mechanical behavior of this family of sheets becomes a major scientific and industrial challenge. The main objective of the current contribution is to numerically predict the ductility limit of this kind of sheet metals. To achieve this objective, the periodic homogenization technique ([1], [2]) will be used. This technique allows us to determine the homogenized macroscopic behavior (macroscopic stress and macroscopic tangent modulus) of an RVE (representative volume element) of the perforated sheet. To predict the ductility limit, represented as forming limit diagram, the periodic homogenization technique is coupled with two necking criteria: the macroscopic maximum force criterion ([3]) and the bifurcation criterion ([4]). The various numerical techniques are implemented into the finite element (FE) code Abaqus. The forming limit diagrams obtained by both necking criteria will be carefully analyzed and compared. A sensitivity study will also be conducted to numerically investigate the influence of the mechanical behavior of the sheets as well as several other design parameters: shape of the holes, inter distance between the holes...
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10985/196802018-01-01T00:00:00ZZHU, JianchangBEN BETTAIEB, MohamedABED-MERAIM, FaridPerforated sheets are widely used in automotive, architecture, pollution control, and other fields. Because perforated sheets are lightweight and aesthetically attractive, and also allow specific elements such as water, air, and light to pass through them, press-formed products with high added-value from these advantages are in great demand. For these reasons, the accurate modeling of the mechanical behavior of this family of sheets becomes a major scientific and industrial challenge. The main objective of the current contribution is to numerically predict the ductility limit of this kind of sheet metals. To achieve this objective, the periodic homogenization technique ([1], [2]) will be used. This technique allows us to determine the homogenized macroscopic behavior (macroscopic stress and macroscopic tangent modulus) of an RVE (representative volume element) of the perforated sheet. To predict the ductility limit, represented as forming limit diagram, the periodic homogenization technique is coupled with two necking criteria: the macroscopic maximum force criterion ([3]) and the bifurcation criterion ([4]). The various numerical techniques are implemented into the finite element (FE) code Abaqus. The forming limit diagrams obtained by both necking criteria will be carefully analyzed and compared. A sensitivity study will also be conducted to numerically investigate the influence of the mechanical behavior of the sheets as well as several other design parameters: shape of the holes, inter distance between the holes...A comparative study of three techniques for the computation of the macroscopic tangent moduli by periodic homogenization
http://hdl.handle.net/10985/18995
A comparative study of three techniques for the computation of the macroscopic tangent moduli by periodic homogenization
ZHU, Jianchang; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
The robust and efficient computation of the macroscopic tangent moduli represents a challenging numerical task in the process of the determination of the effective macroscopic properties of heterogeneous media. The aim of the present contribution is to compare the performances of three numerical techniques for the computation of the tangent moduli via the periodic homogenization multiscale scheme: the condensation technique, the fluctuation technique and the perturbation technique. A total Lagrangian approach is adopted in the formulation of the equations governing the periodic homogenization scheme as well as in the derivation of the macroscopic tangent moduli. Through a comparative study, the condensation technique is shown to have better performance as compared to the two other techniques.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10985/189952019-01-01T00:00:00ZZHU, JianchangBEN BETTAIEB, MohamedABED-MERAIM, FaridThe robust and efficient computation of the macroscopic tangent moduli represents a challenging numerical task in the process of the determination of the effective macroscopic properties of heterogeneous media. The aim of the present contribution is to compare the performances of three numerical techniques for the computation of the tangent moduli via the periodic homogenization multiscale scheme: the condensation technique, the fluctuation technique and the perturbation technique. A total Lagrangian approach is adopted in the formulation of the equations governing the periodic homogenization scheme as well as in the derivation of the macroscopic tangent moduli. Through a comparative study, the condensation technique is shown to have better performance as compared to the two other techniques.Investigation of the competition between void coalescence and macroscopic strain localization using the periodic homogenization multiscale scheme
http://hdl.handle.net/10985/19117
Investigation of the competition between void coalescence and macroscopic strain localization using the periodic homogenization multiscale scheme
ZHU, Jianchang; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
In most voided metallic materials, the failure process is often driven by the competition between the phenomena of void coalescence and plastic strain localization. This paper proposes a new numerical approach that allows an accurate description of such a competition. Within this strategy, the ductile solid is assumed to be made of an arrangement of periodic voided unit cells. Each unit cell, assumed to be representative of the voided material, may be regarded as a heterogeneous medium composed of two main phases: a central primary void surrounded by a metal matrix, which can itself be assumed to be voided. The mechanical behavior of the unit cell is then modeled by the periodic homogenization multiscale scheme. To predict the occurrence of void coalescence and macroscopic strain localization, the above multiscale scheme is coupled with several relevant criteria and indicators (among which the bifurcation approach and an energy-based coalescence criterion). The proposed approach is used for examining the occurrence of failure under two loading configurations: loadings under proportional stressing (classically used in unit cell computations to study the effect of stress state on void growth and coalescence), and loadings under proportional in-plane strain paths (traditionally used for predicting forming limit diagrams). It turns out from these numerical investigations that macroscopic strain localization acts as precursor to void coalescence when the unit cell is proportionally stressed. However, for loadings under proportional in-plane strain paths, only macroscopic strain localization may occur, while void coalescence is not possible. Meanwhile, the relations between the two configurations of loading are carefully explained within these two failure mechanisms. An interesting feature of the proposed numerical strategy is that it is flexible enough to be applied for a wide range of void shapes, void distributions, and matrix mechanical behavior. To illustrate the broad applicability potential of the approach, the effect of secondary voids on the occurrence of macroscopic strain localization is investigated. The results of this analysis reveal that the presence of secondary voids promotes the occurrence of macroscopic strain localization, especially for positive strain-path ratios.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10985/191172020-01-01T00:00:00ZZHU, JianchangBEN BETTAIEB, MohamedABED-MERAIM, FaridIn most voided metallic materials, the failure process is often driven by the competition between the phenomena of void coalescence and plastic strain localization. This paper proposes a new numerical approach that allows an accurate description of such a competition. Within this strategy, the ductile solid is assumed to be made of an arrangement of periodic voided unit cells. Each unit cell, assumed to be representative of the voided material, may be regarded as a heterogeneous medium composed of two main phases: a central primary void surrounded by a metal matrix, which can itself be assumed to be voided. The mechanical behavior of the unit cell is then modeled by the periodic homogenization multiscale scheme. To predict the occurrence of void coalescence and macroscopic strain localization, the above multiscale scheme is coupled with several relevant criteria and indicators (among which the bifurcation approach and an energy-based coalescence criterion). The proposed approach is used for examining the occurrence of failure under two loading configurations: loadings under proportional stressing (classically used in unit cell computations to study the effect of stress state on void growth and coalescence), and loadings under proportional in-plane strain paths (traditionally used for predicting forming limit diagrams). It turns out from these numerical investigations that macroscopic strain localization acts as precursor to void coalescence when the unit cell is proportionally stressed. However, for loadings under proportional in-plane strain paths, only macroscopic strain localization may occur, while void coalescence is not possible. Meanwhile, the relations between the two configurations of loading are carefully explained within these two failure mechanisms. An interesting feature of the proposed numerical strategy is that it is flexible enough to be applied for a wide range of void shapes, void distributions, and matrix mechanical behavior. To illustrate the broad applicability potential of the approach, the effect of secondary voids on the occurrence of macroscopic strain localization is investigated. The results of this analysis reveal that the presence of secondary voids promotes the occurrence of macroscopic strain localization, especially for positive strain-path ratios.Prediction of Localized Necking in Polycrystalline Aggregates Based on Periodic Homogenization
http://hdl.handle.net/10985/20262
Prediction of Localized Necking in Polycrystalline Aggregates Based on Periodic Homogenization
ZHU, Jianchang; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
Ductile failure is the main mechanism that limits the formability of thin metal sheets during forming processes. In the current contribution, ductile failure is assumed to be solely induced by the occurrence of localized necking within the sheet metal. Hence, other failure scenarios such as damage evolution are excluded in this study. It is well known from a number of previously reported works that the onset of localized necking in thin sheets is strongly dependent on the microstructure-related parameters, such as initial and induced textures, grain morphology and crystallographic structure. To accurately understand and analyze this dependency, several micromechanical constitutive frameworks have been coupled in the literature with various localized necking criteria ([1]). The aim of the current investigation is to contribute to this effort by developing a powerful and efficient numerical tool to predict the onset of localized necking in polycrystalline aggregates, which are assumed to be representative of thin metal sheets. In this tool, the periodic homogenization multiscale scheme is used to accurately derive the mechanical behavior of polycrystalline aggregates from that of their microscopic constituents (the single crystals). Compared to other multiscale schemes, such as the Taylor model or the self-consistent approach, the periodic homogenization technique allows us to acurately consider some important aspects in the modeling of the polycristalline behavior (realistic description of the morphology of the grains, boundary conditions, grain boundaries...). The constitutive framework at the single crystal scale follows a finite strain formulation of rate-independent crystal plasticity. At this scale, the Schmid law is used to model the plastic flow and the hardening is defined by a non-linear power law relating the evolution of the critical shear stresses to the slip rates of the crystallographic slip systems. From a numerical point of view, the periodic homogenization problem is solved by using the finite element method. In the present study, we have used ABAQUS/Implicit finite element code, where the polycrystalline aggregate is discretized by 3D finite elements. Each grain is assigned to one finite element. The periodic boundary conditions as well as the macroscopic loading are applied on the polycrystalline aggregate by using the Homtools developed by Lejeunes and Bourgeois ([2]). As the developed model is used to study the ductility limit of thin sheets, the macroscopic plane stress condition (in the thickness direction of the sheet) is legitimately adopted ([3]). The single crystal constitutive equations are integrated by using a powerful and robust numerical implicit scheme belonging to the family of ultimate algorithms ([4]). This numerical scheme is implemented via a user material subroutine (UMAT) into Abaqus. To predict the incipience of localized necking in polycrystalline aggregates, the developed periodic homogenization scheme is coupled with the bifurcation approach ([5]). With this approach, the localization phenomenon is viewed as a consequence of instability in the constitutive description of uniform deformation. In other words, the occurrence of strain localization is a result of jump in the macroscopic velocity gradient of the deformed solid. It is noteworthy that, besides its sound theoretical foundations, the bifurcation approach does not need the calibration of any additional parameter, such as the geometric imperfection factor required when the M–K analysis is used ([6]). The use of the Schmid law at the single crystal scale allows predicting localized necking at realistic strain levels. To apply the bifurcation approach for the prediction of strain localization, the macroscopic tangent modulus should be determined. In the current work, this tangent modulus is derived by a condensation of the global stiffness matrix ([7]). This global stiffness matrix is determined by coupling a 3D user element (UEL), used to compute the element stiffness matrices, with a Python Script developed to assembly these matrices. We have also developed a set of Python Scripts to condense the global stiffness matrix and then to obtain the macroscopic tangent modulus. Compared to other numerical techniques, such as the fluctuation technique ([8]) and the perturbation technique ([9]), the condensation technique seems to be the most appropriate and the most robust to derive the macroscopic tangent modulus. The performance and accuracy of the proposed computational methods will be demonstrated for representative numerical problems dealing with polycrystals with microstructures of FCC single crystals. The evolution of the macroscopic tangent modulus during the loading will be particularly analyzed and commented. A sensitivity study will be conducted in order to investigate the effect of some microstructure parameters (initial crystallogrpahic texture, initial morphology of the grains...) on the prediction of localized necking. In order to investigate the effect of the multiscale scheme on the predictions of strain localization, the results obtained with the current model will be compared with the results obtained with other multiscale schemes (the Taylor model and the self-consistent approach).
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10985/202622018-01-01T00:00:00ZZHU, JianchangBEN BETTAIEB, MohamedABED-MERAIM, FaridDuctile failure is the main mechanism that limits the formability of thin metal sheets during forming processes. In the current contribution, ductile failure is assumed to be solely induced by the occurrence of localized necking within the sheet metal. Hence, other failure scenarios such as damage evolution are excluded in this study. It is well known from a number of previously reported works that the onset of localized necking in thin sheets is strongly dependent on the microstructure-related parameters, such as initial and induced textures, grain morphology and crystallographic structure. To accurately understand and analyze this dependency, several micromechanical constitutive frameworks have been coupled in the literature with various localized necking criteria ([1]). The aim of the current investigation is to contribute to this effort by developing a powerful and efficient numerical tool to predict the onset of localized necking in polycrystalline aggregates, which are assumed to be representative of thin metal sheets. In this tool, the periodic homogenization multiscale scheme is used to accurately derive the mechanical behavior of polycrystalline aggregates from that of their microscopic constituents (the single crystals). Compared to other multiscale schemes, such as the Taylor model or the self-consistent approach, the periodic homogenization technique allows us to acurately consider some important aspects in the modeling of the polycristalline behavior (realistic description of the morphology of the grains, boundary conditions, grain boundaries...). The constitutive framework at the single crystal scale follows a finite strain formulation of rate-independent crystal plasticity. At this scale, the Schmid law is used to model the plastic flow and the hardening is defined by a non-linear power law relating the evolution of the critical shear stresses to the slip rates of the crystallographic slip systems. From a numerical point of view, the periodic homogenization problem is solved by using the finite element method. In the present study, we have used ABAQUS/Implicit finite element code, where the polycrystalline aggregate is discretized by 3D finite elements. Each grain is assigned to one finite element. The periodic boundary conditions as well as the macroscopic loading are applied on the polycrystalline aggregate by using the Homtools developed by Lejeunes and Bourgeois ([2]). As the developed model is used to study the ductility limit of thin sheets, the macroscopic plane stress condition (in the thickness direction of the sheet) is legitimately adopted ([3]). The single crystal constitutive equations are integrated by using a powerful and robust numerical implicit scheme belonging to the family of ultimate algorithms ([4]). This numerical scheme is implemented via a user material subroutine (UMAT) into Abaqus. To predict the incipience of localized necking in polycrystalline aggregates, the developed periodic homogenization scheme is coupled with the bifurcation approach ([5]). With this approach, the localization phenomenon is viewed as a consequence of instability in the constitutive description of uniform deformation. In other words, the occurrence of strain localization is a result of jump in the macroscopic velocity gradient of the deformed solid. It is noteworthy that, besides its sound theoretical foundations, the bifurcation approach does not need the calibration of any additional parameter, such as the geometric imperfection factor required when the M–K analysis is used ([6]). The use of the Schmid law at the single crystal scale allows predicting localized necking at realistic strain levels. To apply the bifurcation approach for the prediction of strain localization, the macroscopic tangent modulus should be determined. In the current work, this tangent modulus is derived by a condensation of the global stiffness matrix ([7]). This global stiffness matrix is determined by coupling a 3D user element (UEL), used to compute the element stiffness matrices, with a Python Script developed to assembly these matrices. We have also developed a set of Python Scripts to condense the global stiffness matrix and then to obtain the macroscopic tangent modulus. Compared to other numerical techniques, such as the fluctuation technique ([8]) and the perturbation technique ([9]), the condensation technique seems to be the most appropriate and the most robust to derive the macroscopic tangent modulus. The performance and accuracy of the proposed computational methods will be demonstrated for representative numerical problems dealing with polycrystals with microstructures of FCC single crystals. The evolution of the macroscopic tangent modulus during the loading will be particularly analyzed and commented. A sensitivity study will be conducted in order to investigate the effect of some microstructure parameters (initial crystallogrpahic texture, initial morphology of the grains...) on the prediction of localized necking. In order to investigate the effect of the multiscale scheme on the predictions of strain localization, the results obtained with the current model will be compared with the results obtained with other multiscale schemes (the Taylor model and the self-consistent approach).