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Discussion of a numerical study of localized deformation in bi-crystals
Mechanics of Materials 4 (1985) 437-438 North-Holland
437
DISCUSSION OF A NUMERICAL STUDY OF LOCALIZED D E F O R M A T I O N IN BI-CRYSTALS by J. LeMonds, R.J. Asaro and A. Needleman Thomas J.R. H U G H E S Division of Applied Mechanics, Stanford University,Stanford, CA 94305, U.S.A. Received 22 October
Dr. G.L. Goudreau of Lawrence Livermore National Laboratory was originally selected by the organizing committee to be the discussor of this paper. Unfortunately, he was taken ill at the time of the meeting and was unable to attend. It was the intention of the organizing committee that Dr. Goudreau would be able to provide the point of view of a computational mechanics team leader at a national laboratory and would contrast his concerns with those of the authors. I filled in for Dr. Goudreau at the meeting and I will attempt to incorporate into my discussion Dr. Goudreau's perspective as I am able to perceive it. The papers at the Workshop were divided into the categories of theory, experiment and computation. The paper under discussion was classified within the computational group. The subject addressed in this paper, namely the large-deformation inelastic response of bi-crystals, is sufficiently complex that very little can be done using nonnumerical analytical techniques. The authors employ a finite element method in order to simulate shear-band development in single crystals and bicrystals. The results of the author's calculations provide insight regarding the effect of the presence of a grain boundary, and enable comparisons between shear-band development in single crystals and bi-crystals, and between rate-dependent and rate-independent constitutive theories. Numerical methods in solid mechanics are most commonly viewed as tools for engineering analysis. The authors, on the other hand, employ numerical methods in an experimental mode to learn more about physical phenomena. This role of computa-
tional mechanics is extremely important and has perhaps been underutilized by the solid mechanics community so far. The authors are to be commended for demonstrating the value of this approach. The large-deformation inelastic response of metals and localization in particular are subjects of essential interest to the national laboratory community. If one just looks at the impressive pictorial results of the calculations in this paper, one might get the impression that the computation of shear-band development in metals in routine. This is not the case. The inadequacy of classical Jz-flow theory with respect to localization is welldocumented and is reiterated in this paper. Nevertheless, J2-flow models remain the state-of-the-art in the national laboratory community for lack of simple and viable alternatives. Various improvements, such as Christoffersen-Hutchinson corner theory, have been proposed to better represent localization and, at the same time, remain consistent with micromechanical considerations, but it appears that none has yet evolved into a complete constitutive theory which may be subjected to arbitrary strain-history inputs, and which is suitable for incorporation in general-purpose computer programs. It is thus apparent that there is a significant gap between the accomplishments of this paper and the needs of the engineering community. It seems clear to the writer that considerable research is needed to bridge this gap. The results of the numerical calculations, rather than the numerical methods themselves, are the focal point of this paper. Nevertheless, it would be
0167-6636/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
438
72J.R. Hughes / Discussion of paper t~v LeMonds, Asaro, and Needleman.
useful if the authors would demonstrate that the numerical procedures employed do represent precisely the physics inherent in the constitutive theory. It is the writer's opinion that the care used in developing and presenting the numerical algorithms should be no less than that used in developing and presenting the physical ideas. Unfortunately, the author's presentation of their methodology is extremely brief and consequently it is impossible to attain a detailed appreciation of this aspect of their work. However, what is mentioned raises the following points: (i) The authors employ a 'tangent method' without equilibrium iterations. The last out-of-balance residual is taken to the right-hand side to provide some correction to equilibrium although it is clear that this procedure never produces a completely equilibrated stress field. In most computational mechanics circles this is viewed as physically unacceptable. It may be argued that if small enough steps are taken, the violation of equilibrium will be tolerable. The fact is that it typically takes an inordinate number of steps to achieve this. Furthermore, much greater efficiency and precise equilibrium can be simply achieved if an equilibrium iteration loop is imbedded in the algorithm. Historically, some investigators argued against iteration in conjunction with path-dependent materials. However, in a correctly formulated constitutive algorithm, the path of deformation is not influenced by the iterative phase and, therefore, iteration can be effectively used in this context.
(ii) The authors do not present their algorithm for integrating the constitutive equation. The constitutive theory presented in Section 2 is sufficiently complicated that a careful step-by-step delineation of how strain increments, etc., are converted into stresses is definitely required. It is suggested from the description of the overall algorithm, which employes an explicit Euler time integration, that a similar technique would be employed in the constitutive integration. Such approaches are particularly inaccurate and may lead to anomalies such as rigid rotations creating nonzero stresses. (iii) The authors advocate the use of quadrilateral macro-elements composed of crossed constant strain triangles. It is pointed out that this element can accomodate the isochoric deformations associated with plastic flow and that crossed triangles can accurately represent an oblique shear band. Nevertheless, my own experience with this element and experience at Lawrence Livermore National Laboratory seem to indicate that the commonly-used mean-dilatation quadrilateral is a better element. It would seem worthwhile for someone to publish the results of a careful comparative study to settle this issue once and for all. Questions of methodology notwithstanding, this paper is an excellent contribution in which concrete steps are taken in developing an understanding of the large deformation inelastic behavior of metals.
437
DISCUSSION OF A NUMERICAL STUDY OF LOCALIZED D E F O R M A T I O N IN BI-CRYSTALS by J. LeMonds, R.J. Asaro and A. Needleman Thomas J.R. H U G H E S Division of Applied Mechanics, Stanford University,Stanford, CA 94305, U.S.A. Received 22 October
Dr. G.L. Goudreau of Lawrence Livermore National Laboratory was originally selected by the organizing committee to be the discussor of this paper. Unfortunately, he was taken ill at the time of the meeting and was unable to attend. It was the intention of the organizing committee that Dr. Goudreau would be able to provide the point of view of a computational mechanics team leader at a national laboratory and would contrast his concerns with those of the authors. I filled in for Dr. Goudreau at the meeting and I will attempt to incorporate into my discussion Dr. Goudreau's perspective as I am able to perceive it. The papers at the Workshop were divided into the categories of theory, experiment and computation. The paper under discussion was classified within the computational group. The subject addressed in this paper, namely the large-deformation inelastic response of bi-crystals, is sufficiently complex that very little can be done using nonnumerical analytical techniques. The authors employ a finite element method in order to simulate shear-band development in single crystals and bicrystals. The results of the author's calculations provide insight regarding the effect of the presence of a grain boundary, and enable comparisons between shear-band development in single crystals and bi-crystals, and between rate-dependent and rate-independent constitutive theories. Numerical methods in solid mechanics are most commonly viewed as tools for engineering analysis. The authors, on the other hand, employ numerical methods in an experimental mode to learn more about physical phenomena. This role of computa-
tional mechanics is extremely important and has perhaps been underutilized by the solid mechanics community so far. The authors are to be commended for demonstrating the value of this approach. The large-deformation inelastic response of metals and localization in particular are subjects of essential interest to the national laboratory community. If one just looks at the impressive pictorial results of the calculations in this paper, one might get the impression that the computation of shear-band development in metals in routine. This is not the case. The inadequacy of classical Jz-flow theory with respect to localization is welldocumented and is reiterated in this paper. Nevertheless, J2-flow models remain the state-of-the-art in the national laboratory community for lack of simple and viable alternatives. Various improvements, such as Christoffersen-Hutchinson corner theory, have been proposed to better represent localization and, at the same time, remain consistent with micromechanical considerations, but it appears that none has yet evolved into a complete constitutive theory which may be subjected to arbitrary strain-history inputs, and which is suitable for incorporation in general-purpose computer programs. It is thus apparent that there is a significant gap between the accomplishments of this paper and the needs of the engineering community. It seems clear to the writer that considerable research is needed to bridge this gap. The results of the numerical calculations, rather than the numerical methods themselves, are the focal point of this paper. Nevertheless, it would be
0167-6636/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
438
72J.R. Hughes / Discussion of paper t~v LeMonds, Asaro, and Needleman.
useful if the authors would demonstrate that the numerical procedures employed do represent precisely the physics inherent in the constitutive theory. It is the writer's opinion that the care used in developing and presenting the numerical algorithms should be no less than that used in developing and presenting the physical ideas. Unfortunately, the author's presentation of their methodology is extremely brief and consequently it is impossible to attain a detailed appreciation of this aspect of their work. However, what is mentioned raises the following points: (i) The authors employ a 'tangent method' without equilibrium iterations. The last out-of-balance residual is taken to the right-hand side to provide some correction to equilibrium although it is clear that this procedure never produces a completely equilibrated stress field. In most computational mechanics circles this is viewed as physically unacceptable. It may be argued that if small enough steps are taken, the violation of equilibrium will be tolerable. The fact is that it typically takes an inordinate number of steps to achieve this. Furthermore, much greater efficiency and precise equilibrium can be simply achieved if an equilibrium iteration loop is imbedded in the algorithm. Historically, some investigators argued against iteration in conjunction with path-dependent materials. However, in a correctly formulated constitutive algorithm, the path of deformation is not influenced by the iterative phase and, therefore, iteration can be effectively used in this context.
(ii) The authors do not present their algorithm for integrating the constitutive equation. The constitutive theory presented in Section 2 is sufficiently complicated that a careful step-by-step delineation of how strain increments, etc., are converted into stresses is definitely required. It is suggested from the description of the overall algorithm, which employes an explicit Euler time integration, that a similar technique would be employed in the constitutive integration. Such approaches are particularly inaccurate and may lead to anomalies such as rigid rotations creating nonzero stresses. (iii) The authors advocate the use of quadrilateral macro-elements composed of crossed constant strain triangles. It is pointed out that this element can accomodate the isochoric deformations associated with plastic flow and that crossed triangles can accurately represent an oblique shear band. Nevertheless, my own experience with this element and experience at Lawrence Livermore National Laboratory seem to indicate that the commonly-used mean-dilatation quadrilateral is a better element. It would seem worthwhile for someone to publish the results of a careful comparative study to settle this issue once and for all. Questions of methodology notwithstanding, this paper is an excellent contribution in which concrete steps are taken in developing an understanding of the large deformation inelastic behavior of metals.