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Recursive bifurcation of tensile steel specimens
International Journal of Engineering Science 39 (2001) 1913±1934
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Recursive bifurcation of tensile steel specimens Kiyohiro Ikeda a,*, Shigenobu Okazawa a, Kenjiro Terada a, Hirohisa Noguchi b, Tsutomu Usami c b
a Department of Civil Engineering, Tohoku University, Aoba, Sendai 980-8579, Japan Department of System Design Engineering, Keio University, Yokohama 223-8522, Japan c Department of Civil Engineering, Nagoya University, Nagoya 464-8603, Japan
Received 20 October 2000; accepted 31 January 2001
Abstract Failure modes of steel specimens subjected to uniaxial tension are investigated. These modes are well known to display complex geometrical characteristics of deformation accompanied by the plastic instability behavior. As an underlying mechanism of such complexity, we here focus on the recursive occurrence of bifurcations. In the theory, the rule of recursive bifurcation of a rectangular parallelepiped domain is obtained by the group-theoretic bifurcation theory so as to exhaust all the mathematically possible courses of bifurcation. In the experiment, we examine the representative failure modes with reference to the rules to identify actual courses of recursive bifurcation. Three-dimensional ®nite element analysis of a thin specimen is conducted to observe the recursive bifurcation, in which diuse necking is formed by the direct bifurcation and the single shear band by the secondary bifurcation. The recursive bifurcation has thus been identi®ed as the mechanism to create the complex failure modes. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Recursive bifurcation; Group-theoretic bifurcation theory; Steel specimens
1. Introduction The failure mechanism of a steel specimen subjected to uniform tensile loading still receives much attention under the name of plastic instability. On the basis of Hill's [1] theory of uniqueness and stability of solutions of elastic±plastic solids, Hill and Hutchinson [2] developed the theory on the plastic bifurcation behavior observed in tension tests. As for geometrical instability, this theory has been applied to the bifurcation analyses for necking behavior in steel specimens (e.g. [3,4]). On *
Corresponding author. Tel.: +81-22-217-7416; fax: +81-22-217-7418. E-mail address: [email protected] (K. Ikeda).
0020-7225/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 1 ) 0 0 0 4 0 - 4
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the other hand, the implication of the theory has been realized in a considerable number of theoretical and computational studies on the constitutive modeling within the framework of material instability. In particular, various phenomenological models for localized deformations have been introduced; to name a few, the ®nite-strain version of deformation theories [5], the constitutive models with microvoids [6] or with thermal-softening capability [7]. A great deal of related theories and computational techniques have also been developed to simulate the shear band formation leading to the ultimate failure (e.g. [8±11]). In most of these studies, an emphasis was placed on the direct bifurcation that creates necking or diagonal shear bands. However, the direct bifurcation modes of uniform domains are harmonic and cannot, in general, account for the complexity of actual deformation patterns observed in experiments. For such complexity, the recursive occurrence of bifurcation that entails complex deformation patterns can be advanced as the underlying mechanism (see, for theoretical developments [12,13]). In the bifurcation theory, a mathematical tool to describe recursive bifurcation is readily applicable (e.g. [14,15]). With reference to the symmetry of the system under consideration, we can enumerate the symmetries of possible deformation patterns with reduced symmetries. In experimental studies, however, only few attempts have so far been made at the visual observation of deformation pattern change for a steel specimen. In this respect, the theory of recursive bifurcation eminently deserves a pertinent means to investigate the complex pattern change of deformation of a steel specimen. The aim of this paper is to explain deformation characteristics of steel specimens subjected to uniaxial tension through the general standpoint of ``recursive bifurcation''. The geometrical pattern change of the deformed specimens observed in both an experiment and a numerical analysis can be systematically and successfully explained by the bifurcation rule developed in this paper. This paper is outlined as follows. In Section 2, the rule of recursive bifurcation of a rectangular parallelepiped domain is brie¯y described by the group-theoretic bifurcation theory, while the details are worked out in Appendix A. Section 3 is devoted to the experimental study, as well as numerical one, on the recursive bifurcation phenomena of steel members with an emphasis on the deformation pattern change. The representative failure modes are studied with reference to the bifurcation rule presented in Section 2. 2. Theory of recursive bifurcation In this section, we shall explain how the symmetry of a rectangular parallelepiped domain is labeled by a group made up of a series of geometrical actions, and introduce the group-theoretic rule of recursive bifurcation of this domain. As bifurcation phenomena are accompanied by an observable loss of symmetry of the system in question, it is convenient to express the bifurcation rule as a set of admissible transitions among groups that label symmetries and partial symmetries of the system under consideration. Namely, we can ®nd a group G that labels its full symmetry, and a hierarchy of subgroups G ! G1 ! G2 !
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1915
Fig. 1. Rectangular parallelepiped domain.
that characterizes the hierarchical transition of symmetries. Here ! corresponds to the loss of symmetry caused by a bifurcation, and Gi (i 1; 2; . . .) stand for the subgroups of G that label the partial symmetries of bifurcating branches. 2.1. Symmetry of a rectangular parallelepiped domain We consider a rectangular parallelepiped domain shown in Fig. 1, that is, X
x; y; z 2 R 3
Lx Lx 6x6 ; 2 2
Ly Ly 6y6 ; 2 2
Lz Lz 6z6 : 2 2
The geometric symmetry of this domain is expressed by the following eight elements of geometric operations: E;
rx ;
ry ;
rz ;
rx ry ;
ry rz ;
rz rx ;
rx ry rz ;
where E is the identity element; rx , ry and rz , respectively denote re¯ections in yz-, zx- and xyplanes; rx ry , ry rz and rz rx , respectively denote p-rotations about z-, x- and y-axes; rx ry rz is the inversion with respect to the origin; rx ry ry rx ; ry rz rz ry ; rz rx rx rz and r2x r2y r2z E. Fig. 2 illustrates the actions of these operations. The group of these operations is de®ned in the Schoen¯ies notation 1 as D2h hrx ; ry ; rz i fE; rx ; ry ; rz ; rx ry ; ry rz ; rz rx ; rx ry rz g; where hi denotes the group generated by the elements therein and fg denotes the group consisting of the elements therein. Partial symmetries emanating from a D2h -symmetric state are labeled by the proper subgroups of D2h , including: 1
See, e.g. [16] for this notation and [17] for relevant mathematical backgrounds.
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Fig. 2. Geometric operations.
hrx ; ry i fE; rx ; ry ; rx ry g; hry ; rz i fE; ry ; rz ; ry rz g; hrz ; rx i fE; rz ; rx ; rz rx g; hrx ; ry rz i fE; rx ; ry rz ; rx ry rz g;
2
hry ; rz rx i fE; ry ; rz rx ; rx ry rz g; hrz ; rx ry i fE; rz ; rx ry ; rx ry rz g; hrx ry ; ry rz i fE; rx ry ; ry rz ; rz rx g; hrx i fE; rx g; hry i fE; ry g; hrz i fE; rz g; hrx ry i fE; rx ry g;
3
hry rz i fE; ry rz g; hrz rx i fE; rz rx g; hrx ry rz i fE; rx ry rz g; hEi fEg:
4
The symmetries expressed by D2h and its subgroups are illustrated in Fig. 3. With reference to Remark 1, we assume an additional symmetry of the x-directional translation t l at a distance of l 0 6 l < Lx de®ned by t l : x 7! x l:
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
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Fig. 3. Symmetries expressed by D2h and its subgroups.
One may interpret l as the wavelength in the longitudinal direction. The rectangular parallelepiped domain consequently has the symmetry labeled by G hrx ; ry ; rz ; t li:
5
This group also expresses the symmetry of uniform deformation prior to bifurcation. Remark 1. We note that rectangular parallelepiped steel specimens under consideration are long and uniform in the longitudinal direction. In order to express the local uniformity in this direction, we assume the periodic symmetry in the x-direction. Such assumption of periodic symmetry is often employed in the investigation of bifurcation behavior (e.g. [12,15,18]). 2.2. Recursive bifurcation rule The rule of recursive bifurcation of a system with a symmetry labeled by the group G hrx ; ry ; rz ; t li is brie¯y introduced here whereas the details are worked out in Appendix A.
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In view of actual deformation pattern change of tensile steel specimens (see, e.g. Sections 3.1 and 3.2), among a series of mathematically possible symmetries for the deformation patterns, we restrict ourselves in this paper to the bifurcation process G hrx ; ry ; rz ; t li ! D2h hrx ; ry ; rz i: As shown in Fig. 4, this bifurcation process is associated with the occurrence of necking around the center of the rectangular parallelepiped specimen. As described in Section A.2, the direct bifurcation from D2h -symmetric state has the symmetry labeled by one of the seven subgroups in (2). The secondary bifurcation from this state leads to the symmetry labeled by one of the seven subgroups in (3). Further bifurcation will lead to completely asymmetric states labeled by hEi. To sum up, we focus in this paper on the recursive bifurcation expressed by a hierarchy of subgroups shown in Fig. 5 that exhausts all mathematical possibilities for the recursive bifurcation below D2h . It should be remarked that the group-theoretic study presents all mathematically possible bifurcation modes, or, to be precise, the symmetries of these modes. Actual bifurcation modes are to be restricted by the boundary conditions to be employed in experiments or numerical analyses (cf. [19]). 3. Steel specimens under uniform tension We focus our attention to complex geometrical pattern change of a steel specimen under uniform tension, which can commonly be observed in an experiment and a numerical analysis.
Fig. 4. Necking associated with the bifurcation process G hrx ; ry ; rz ; t li ! D2h hrx ; ry ; rz i.
Fig. 5. The rule of bifurcation of G-symmetric system expressed in terms of a hierarchy of subgroups.
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
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The group-theoretic method presented in Section 2 is employed here to untangle this complexity and clarify the geometrical aspects of ultimate failures. 3.1. Experimental behavior We employ various steel specimens with dierent dimensions listed in Table 1, all of which have the same material composition for metal. Specimen M is for calibration, while Specimens A, B, C and D of four dierent width±thickness ratios are for the visual observation of deformed geometry. A typical geometry of these steel specimens is depicted in Fig. 6; we put L Lx , W Ly and t Lz in Fig. 1 in the remainder of this paper. A tensile load is applied to each specimen statically at constant and moderate temperature by a testing machine with a sucient capacity. Fig. 7 shows the load versus elongation curve obtained in a single test on Specimen M, from which representative Young's modulus is determined as 204 GPa. After a relatively small elastic deformation, the plastic yielding occurs uniformly in the rectangular parallelepiped region of the specimen at the initial yield stress (429 MPa). As the specimen is further elongated, the load increases to reach the maximum load (591 MPa), thereafter a necking deformation occurs around the center of the specimen and is concentrated in the narrower region. The ultimate failure is caused by the inherent straining limit of the metal. The other Specimens A, B, C and D have exhibited similar behavior up to the maximum load, while the post-peak behavior is dependent on the cross-sectional shapes as shown in Figs. 8±11 taken during the tests. The states shown in (a), (b) and (c) in each ®gure, respectively, correspond to the beginning of the non-uniform deformation, the appearance of prominent strain localization and the failure mode. We carefully describe the geometrical aspects of each of the states in order. Table 1 Dimensions of steel specimens Specimen
Width, W (mm)
Thickness, t (mm)
Width±thickness ratio, W =t
Length, L (mm)
M A B C D
39.7 39.9 40.0 39.9 27.8
9.79 3.83 9.86 26.0 28.2
4.05 10.4 4.06 1.54 0.984
100 220 220 220 220
Fig. 6. Geometry of a steel specimen.
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Fig. 7. Typical load versus elongation curve obtained in a tension test for Specimen M for calibration (: maximum load).
Fig. 8. Transient deformation change of Specimen A (W =t 10:4).
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Fig. 9. Transient deformation change of Specimen B (W =t 4:06).
First, we observe the thinnest Specimen A. As shown in Fig. 8(a), diuse necking appears around the center of this specimen. As a result of strain localization triggered by the non-uniform deformation, two diagonal lines, sometimes called shear bands, emerge in the middle part of the necking; see Fig. 8(b). Then the intense straining concentrates on one of the shear bands, while the other becomes less discernible. In association with the growth of this single distinct shear band, the specimen undergoes ductile failure as shown in Fig. 8(c). Next, we investigate the deformation patterns of the thicker Specimens B, C and D. As shown in Figs. 9(a), 10(a) and 11(a), diuse necking appears around the center, while the shear bands are not observed even in the state of fracturing; see Figs. 9(b), 10(b) and 11(b). Instead, dimples are formed on all the side surfaces. The ®nal states shown in Figs. 9(c), 10(c) and 11(c) display the states after the brittle failures. The complexity of failure modes notwithstanding, they seem to present the distinctive features in view of the reduction of symmetries. As we have seen, the width±thickness ratio of the specimens is in¯uential on the resulting instability phenomena especially at the ®nal state. The transient pattern change leading to this state,
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Fig. 10. Transient deformation change of Specimen C (W =t 1:54).
which progresses rapidly and hence is unclear, will be discussed in Section 3.2 based on the theory of recursive bifurcation presented in Section 2. 3.2. Representative numerical simulation We perform a three-dimensional numerical analysis for Case A in order to demonstrate clearly the occurrence of pattern change due to recursive bifurcation, as the occurrence of bifurcation in the experiment may seem controversial for most of the readers. Since speci®c material behavior is not the central issue in our study, we employ a standard constitutive model for plastic deformations, which is presented in Appendix B together with the values of material constants. In the sequel, only the major result for recursive bifurcation is presented. The load±elongation curves are shown in Fig. 12. The primary solution path exhibits one limit point, which is marked by . The direct bifurcation point marked by is found immediately after the limit point. We are here concerned with only the physically meaningful bifurcations though other bifurcation points were found on the primary path after the bifurcation point marked by .
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
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Fig. 11. Transient deformation change of Specimen D (W =t 0:984).
Fig. 12. Numerical load versus elongation curves (: limit point, : bifurcation point; P: applied load, A: initial crosssection; rY : yields stress; u: axial displacement; L: member length).
At the direct bifurcation point, the critical eigenvector of the tangent stiness matrix is related to the lowest harmonic mode, which leads to diuse necking, and is given in Fig. 13(a). In addition, the two eigenvectors with eigenvalues close to zero are presented in Figs. 13(b) and (c). As
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Fig. 13. Eigenvectors near the direct bifurcation point.
can be seen, the last two eigenvectors are higher harmonic modes than the critical one. The critical eigenvector is associated with D2h -symmetry whose course is given in (6), while other ones with other symmetries labeled by the groups in (A.7). It is noteworthy that the formation of an oblique (single) shear band is not a direct bifurcation mode. Fig. 14(b) shows the deformed con®guration with the contour plot of eective plastic strains at the load level P = ArY 0:80. The loaded region is localized to shape a cross band while the unloaded one spreads from the edge surface to the center part of the specimen. Furthermore, the numerical analysis here presents the secondary bifurcation behavior, whose solution branch is depicted by dotted lines in Fig. 12. The corresponding deformed con®guration given in Fig. 14(c) agrees with the experimental result in Section 3.1 (cf. Fig. 8). 3.3. Theoretical interpretation of recursive bifurcation phenomena The recursive bifurcation observed in Sections 3.1 and 3.2 is examined with reference to the theory presented in Section 2. For all the specimens, the formation of the necking from the uniform state results from the direct bifurcation associated with G hrx ; ry ; rz ; t li ! D2h hrx ; ry ; rz i;
6
in which the symmetry of the uniform state is labeled by G and the necking by the group D2h .
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
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Fig. 14. Progress of deformation expressed by the contour of eective plastic strain of FE model of a rectangular parallelepiped specimen (6750 elements for Case A).
Let us start with the observation of the thinnest Specimen A. As shown in Fig. 8 and also in Fig. 14, the diagonal shear bands are formed after the diuse necking due to the intense localized straining. Such formation is characteristic from a physical standpoint, but is not associated with bifurcation in that both the state of necking and that of the diagonal shear bands share the same symmetry labeled by the group D2h . The secondary bifurcation takes place at the onset of the formation of a single distinct shear band, which is one of the diagonal shear bands. This secondary bifurcation is associated with a further reduction of symmetry associated with a bifurcation process D2h ! hrz ; rx ry i;
7
in which D2h denotes the symmetry of the diagonal shear bands and hrz ; rx ry i indicates that of the single distinct shear band. The ®nal failure state, the symmetry of which is labeled by the same group hrz ; rx ry i, is no longer caused by bifurcation. The deformation pattern change and the loss
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of symmetry associated with the direct and secondary bifurcations (6) and (7) are illustrated in Fig. 15. Next, we deal with the thicker Specimens B, C and D. These specimens undergo the direct bifurcation (6) to reach the state of diuse necking. Then, as shown in Figs. 9±11, dimples are formed by the localized straining; such formation is a non-bifurcation process. By brittle and premature failure, the specimens arrive at the ®nal states as shown in the last of each ®gure. With reference to these states, we can advance the following possible secondary bifurcation processes: D2h ! hry ; rz i & hry ; rz rx i
for B and C;
8
for D:
These processes, which take place just prior to their ®nal failure, presumably cause the failure. The deformation pattern change and the loss of symmetry associated with the direct and secondary bifurcations (6) and (8) are illustrated in Figs. 16 and 17. To sum up, from (6)±(8), we can arrive at a hierarchy of subgroups % hrz ; rx ry i for A; G hrx ; ry ; rz ; t li ! D2h ! hry ; rz i
for B and C;
9
& hry ; rz rx i for D; which is nothing but a part of the whole hierarchy in Fig. 5. Of course, the three courses of recursive bifurcation we have found here may not exhaust all the physically possible courses. More detailed observation based on the knowledge of the recursive bifurcation rule will be a future topic.
Fig. 15. Illustration of the hierarchical deformation pattern change due to recursive bifurcation for Specimen A (W =t 10:4).
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Fig. 16. Illustration of the hierarchical deformation pattern change due to recursive bifurcation for Specimens B (W =t 4:06) and C (W =t 1:54).
Fig. 17. Illustration of the hierarchical deformation pattern change due to recursive bifurcation for Specimen D (W =t 0:984).
4. Conclusions The geometrical pattern change of rectangular parallelepiped steel specimens subjected to uniaxial tension has been investigated with reference to the theory of recursive bifurcation. The single, distinct shear band does not emanate from the direct bifurcation but branches from the
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secondary one whereas the direct bifurcation creates diuse necking. The recursive bifurcation has thus emerged as a source of the instability behavior. While the group-theoretic bifurcation theory presents a plethora of mathematically possible courses of recursive bifurcation of rectangular parallelepiped specimens, the experiment and the numerical analysis on steel specimens demonstrate physically feasible ones among those possible courses of bifurcation. In the experiment, a transient state of deformations, which is usually overlooked, has been interpolated on the basis of the theoretical results. In the numerical analysis, the recursive bifurcation is clearly observed for a thin specimen, in which diuse necking is formed by the direct bifurcation and the distinct, single shear band by the secondary bifurcation. As a result of these, the failure modes are understood as a consequence of the recursive bifurcation. The study of the geometrical pattern change, the tail of which has been caught in this paper, is expected to shed light on the plastic instability behavior. Appendix A. Derivation of bifurcation rules The recursive bifurcation rule for a system equivariant to G hrx ; ry ; rz ; t li is derived on the basis of the standard mathematical technique called the group-theoretic bifurcation theory. In this derivation, we refer to the rules for systems with smaller symmetries, such as D2h and D1h (see, e.g. [16,20]). A.1. Group-theoretic bifurcation theory A very brief account of the main ideas of the group-theoretic bifurcation analysis is oered (see, e.g. [14,15,18]). The deformed state of the material in question is assumed to be described by a (vector-valued) function u x, where x is the coordinate of a point in the domain, and that u x satis®es a system of governing equations F u; f 0;
A:1
where f is a loading parameter and F is a nonlinear function satisfying pertinent regularity conditions, such as the dierentiability. It should be noted the governing equation (A.1) already implements material properties, such as the constitutive laws. Let G be a group of transformations acting on x. Denote by g x and g u the transformation caused by an element g of G. If u is a scalar-valued function, for example, this transformation is expressed as g u x u g
1
x:
Eq. (A.1) is called equivariant with respect to G if g F u; f F g u; f 8g 2 G is satis®ed. A solution u is called invariant under G (or G-invariant) if
A:2
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
guu
1929
8g 2 G
is satis®ed. The solutions u; f are assumed to be lying on the main path and retain G-invariance, whereas those on a bifurcation path have a reduced symmetry. Such a reduced symmetry can be labeled by a subgroup G0 of G. If uc ; fc is a critical point on the main path, at which the derivative (or the Frechet derivative) of F with respect to u: F 0 u; f
oF u; f ou
has a zero eigenvalue. It should be noted that F 0 u; f reduces to the tangential stiness matrix or the Jacobian matrix if the system (A.1) is ®nite dimensional. Usually a critical point is either a limit point of loading parameter f or a bifurcation point. The bifurcation paths, branching from the main path at a bifurcation point, are usually made up of solutions u; f of reduced symmetries labeled by subgroups G0 of G. The multiplicity M of a critical point uc ; fc is de®ned as the number of zero eigenvalues of F 0 uc ; fc . The kernel space X of the operator F 0 uc ; fc is easily veri®ed to be invariant under G, i.e., gu2X
for u 2 X ;
if uc is G-invariant. Then the critical point is divided into two types, group-theoretic or parametric, according to whether X is G-irreducible or not. Here X is said to be G-irreducible if there exists no non-zero proper subspace in X that is invariant under G. The group-theoretic critical point occurs due to the symmetry of the system while the parametric critical point occurs by the accidental coincidence of two or more critical points. Since the latter is rare, only the former type is considered. Let uc ; fc be a group-theoretic critical point. Then, by de®nition, the kernel space X is associated with one of the irreducible representations, which we denote by l. The irreducible representation, in turn, is associated with a subgroup Gl of G. To be more speci®c, let T l g g 2 G be the representation matrices of the irreducible representation l. Then Gl fg 2 G j T l g Ig; where I is the identity matrix. This subgroup is called the symmetry group of the kernel space. For example, a limit point of the loading parameter f is associated with the unit (one-dimensional) representation l for which T l g 1 for all g 2 G, and Gl G. A bifurcation point is usually associated with an irreducible representation l for which T l g 6 I for some g 2 G, and hence Gl is a proper subgroup of G. Such a bifurcation point is called a symmetry-breaking bifurcation point. The degree of the irreducible representation is equal to the multiplicity M of the critical point. The symmetry of the system on a bifurcation path can be determined by solving the bifurcation equation, which inherits the equivariance condition (A.2) (see, e.g. [12,14]). It should be noted that this symmetry is often higher than the symmetry of Gl .
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A.2. Bifurcation from D2h -invariant state The bifurcation structure of a system equivariant to the group D2h hrx ; ry ; rz i fE; rx ; ry ; rz ; rx ry ; ry rz ; rz rx ; rx ry rz g is described. We index the family of non-equivalent irreducible representations of D2h by R D2h f m1 ; m2 ; m3 D2h j m1 ; m2 ; m3 ; g:
A:3
Here ; ; D2h corresponds to the unit (one-dimensional) irreducible representation and the others to one-dimensional irreducible representations, being de®ned by the one-dimensional representation matrices given in Table 2. The unit representation ; ; D2h corresponds to the limit point of loading parameter f with D2h symmetry. The remaining representations correspond to simple bifurcation points and the symmetries of their bifurcation paths are labeled by the proper subgroups of D2h in (2), that is, hrx ; ry i;
hry ; rz i;
hrz ; rx i;
hrx ; ry rz i;
hry ; rz rx i;
hrz ; rx ry i;
hrx ry ; ry rz i:
A:4
By investigating further bifurcations from these bifurcation paths with symmetries labeled by the groups in (A.4), we can arrive at the bifurcation rule below the group D2h ; see Fig. 5. A.3. Bifurcation from G-invariant state The recursive bifurcation rule for a system equivariant to the group G hrx ; ry ; rz ; tx lx i hrx ; tx lx i hry i hrz i in (5) is derived. Note that the group G is isomorphic to the group O 2 Z2 Z2 , and the group hrx ; t li hry i is isomorphic to the group D1h in the Schoen¯ies notation. See, e.g. [16] for this notation and [17,20] for the studies on a D1h -equivariant system. Table 2 Representation matrices and symmetry groups of D2h -invariant system Irreducible representation l
T l g rx
ry
rz
; ; D2h ; ; D2h ; ; D2h ; ; D2h
1 )1 1 )1
1 1 )1 )1
1 1 1 1
; ; ; ; ; ; ; ;
1 )1 1 )1
1 1 )1 )1
)1 )1 )1 )1
D2h D2h D2h D2h
Symmetry groups Gl for X
Bifurcation paths D2h hrx ; ry ; rz i hry ; rz i hrz ; rx i hrz ; rx ry i hrx ; ry i hry ; rz rx i hrx ; ry rz i hrx ry ; ry rz i
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
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With reference to these studies, we can index the family of non-equivalent irreducible representations of G hrx ; ry ; rz ; t li by R G f m1 ; m2 ; m3 ; n; m2 ; m3 j m1 ; m2 ; m3 ; ; n 1; 2; . . .g:
A:5
Here ; ; corresponds to the unit representation, which is associated with the limit point of bifurcation parameter. Other one-dimensional ones ; ; ;
; ; ;
; ; ;
; ; ;
; ; ;
; ; ;
; ; ;
which are de®ned by the one-dimensional representation matrices T l given in Table 3, are associated with simple bifurcation points. The remaining irreducible representations in (A.5) are of degree two, and their actions are de®ned by the representation matrices listed in Table 3, where I2
1 0 ; 0 1
P
0 ; 1
1 0
R
cos nu sin nu
sin nu cos nu
A:6
with the correspondence of u 2p
lx ; Lx
0 6 u < 2p; 0 6 lx < Lx :
Though the analysis on the bifurcation equation is omitted here, the symmetries of the bifurcating solutions associated with these two-dimensional irreducible representations are obtained as Table 3 Representation matrices of G-equivariant system G hrx ; ry ; rz ; tx lx ; r rx tx lx for some lx i M
l
T l g
Symmetry groups
rx
ry
rz
tx lx
Gl
Bifurcation paths
1 1 )1 )1 1 1 )1 )1
1 1 1 1 )1 )1 )1 )1
1 1 1 1 1 1 1 1
G hry ; rz ; tx lx i hrx ; rz ; tx lx i hrz ; rx ry ; tx lx i hrx ; ry ; tx lx i hry ; rz rx ; tx lx i hrx ; ry rz ; tx lx i hrx ry ; ry rz ; tx lx i
No bifurcation hry ; rz ; tx lx i hrx ; rz ; tx lx i hrz ; rx ry ; tx lx i hrx ; ry ; tx lx i hry ; rz rx ; tx lx i hrx ; ry rz ; tx lx i hrx ry ; ry rz ; tx lx i
1
; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;
1 )1 1 )1 1 )1 1 )1
2
n; ;
P
I2
I2
R
hry ; rz ; tx Lnx i
hr; ry ; rz ; tx Lnx i
n; ;
P
I2
I2
R
Lx hrz ; ry tx 2n i
Lx hr; rz ; ry tx 2n i
n; ;
P
I2
I2
R
Lx hry ; rz tx 2n i
Lx hr; ry ; rz tx 2n i
n; ;
P
I2
I2
R
Lx Lx hry tx 2n ; rz tx 2n i
Lx Lx hr; ry tx 2n ; rz tx 2n i
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K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
Lx Lx r; ry ; rz ; tx ; r; rz ; ry tx ; n 2n Lx Lx ; rz tx ; n 1; 2; . . . ; r; ry tx 2n 2n
Lx r; ry ; rz tx ; 2n A:7
where r rx tx lx for some lx . Note that D2h (studied in Section A.2) is obtained by setting n 1 and r rx in hr; ry ; rz ; tx Lx =ni in (A.7) and, therefore, is associated with 1; ; . Appendix B. Description of the numerical model In this section, we present the formulation, procedure and analysis conditions for the numerical analysis in Section 3.2. Our large-scale computations rely on the standard formulation of the ®nite element method for classical ®nite-strain elastoplasticity, which is brie¯y summarized below. B.1. Formulation of ®nite-strain elastoplasticity We here describe the boundary value problem for the classical rate-independent plasticity model. By using the updated-Lagrangian formulation, which is widely used in practical computations, we employ the rate form of the momentum balance equation in the context of ®nite element (FE) analyses. For elaborate discussions for this type of formulations, one refers to, e.g. [21,22]. Let B Rndim (ndim 1; 2 or 3) be the reference con®guration of an elastic±plastic solid with material particles denoted by X 2 B and be subjected to a deformation u : B 7! Rndim , with J : det rX u > 0. We let oB be the boundary of B and assume that the deformation is prescribed ^ is prescribed on ot B oB, with the ^ whereas the nominal traction vector T on ou B oB as u u ^ We here consider the quasi-static nominal stress tensor P and the unit normal N, as PN T. equilibrium problem with the given body force B in B. In the current con®guration u B, the Kircho stress tensor s and the spatial velocity ®eld v : u_ u 1 are used to describe the equilibrium state. In terms of the admissible spatial velocity ®eld g in an appropriate function space V, the rate form of the variational equation of this problem is given by Z Z Z dv dv ^t_ g ds; g 2 V; rg : rvs Lv s b_ g B:1 J J u B u B ot u B ^ respectively, with the Cauchy stress where b and ^t rn are the spatial representations of B and T, r and the unit normal n on ot u. Here, Lv s is the Lie derivative of s and related to its Jaumann rate r s as r
Lv s s
ds
sd : a : d;
where d is the spatial rate-deformation tensor and a are the symmetric moduli.
B:2
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1933
In this study, we simply assume that the material reveals isotropy and that the elastic strains are small compared with plastic ones. Also, by assuming that the plastic deformation is incompressible, we neglect the volumetric change of this metal by J 1 so that the classical J2 ¯ow theory can be extended to ®nite-strain range. These assumptions are valid for the metal specimens under consideration whose constitutive equation is given in terms of the rates as r
s c : d;
B:3
in which the symmetric moduli c take the same constant values ce as those in isotropic linear elasticity for elastic deformation and, for plastic ¯ow, 9l2 1 e r0 r0 : B:4 cc 0 2 3l H r Here, l is the shear modulus, H 0 the plastic modulus, r the equivalent stress and r0 the deviatoric stress tensor. B.2. Procedure and analysis conditions In order to produce a uniform loading state, we consider a rectangular parallelepiped region where the deformation is dominated in actual experiments (cf. Figs. 1 and 6). In order to evaluate the critical eigenvector that plays a role of a branch-switching predictor, we utilize the method based on a scaled corrector, which has been proposed by Noguchi and Hisada [23]. The secondary bifurcation path(s) can also be computed essentially in the same manner. The detailed discussion about the algorithm for the branch-switching procedure is found elsewhere (e.g. [24]). A tensile force is applied on the edge surface located at x L=2 while all the other surfaces are free from stress. Note that our actual computation must be performed on the whole rectangular domain to recover all the possible bifurcation patterns. The material properties used in the analyses are chosen as follows: Young's modulus E 200 GPa, Poisson's ratio m 0:3333, and initial yield stress rY 400 MPa. For plastic hardening, the following power law is assumed so that its derivative leads to the parameter H 0 in (B.4): r rY
ep 1 eY
!0:0625 > 0;
B:5
where eY rY =E 1=500 and ep is the eective plastic strain. It is remarked that the nonlinear hardening law (B.5) excludes softening responses of the material, since material instability plays no role in this paper. References [1] R. Hill, A general theory of uniqueness and stability in elastic±plastic solids, J. Mech. Phys. Solids 16 (1958) 236± 240.
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[2] R. Hill, J.W. Hutchinson, Bifurcation phenomena in the plane tension test, J. Mech. Phys. Solids 23 (1975) 239± 264. [3] D.M. Norris Jr., B. Moran, J.K. Scudder, D.F. Quinones, A computer simulation of the tension test, J. Mech. Phys. Solids 26 (1977) 1±19. [4] M.A. Burke, W.D. Nix, A numerical study of necking in the plane tension test, Int. J. Solids Struct. 15 (1979) 379± 393. [5] J. Christoersen, J.W. Hutchinson, A class of phenomenological corner theories of plasticity, J. Mech. Phys. Solids 25 (1979) 465±487. [6] A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I ± Yield criteria and ¯ow rules porous ductile media, J. Eng. Mater. Tech. 99 (1977) 2±15. [7] A. Needleman, Dynamic shear band development in plane strain, J. Appl. Mech. 56 (1989) 1±9. [8] V. Tvergaard, A. Needleman, K.K. Lo, Flow localization in the plane strain tensile test, J. Mech. Phys. Solids 29 (2) (1981) 115±142. [9] V. Tvergaard, A. Needleman, Analysis of the cup±cone fracture in a round tensile bar, Acta Metall. 32 (1) (1984) 157±169. [10] S. Nemat-Nasser, Phenomenological theories of elastoplasticity and strain localization at high strain rates, Part 2, Appl. Mech. Rev. 45 (3) (1992) S19±S45. [11] H. Petryk, K. Thermann, On discretized plasticity problems with bifurcations, Int. J. Solids Struct. 29 (6) (1992) 745±765. [12] K. Ikeda, K. Murota, M. Nakano, Echelon modes in uniform materials, Int. J. Solids Struct. 31 (1994) 2709±2733. [13] K. Ikeda, K. Murota, Y. Yamakawa, E. Yanagisawa, Mode switching and recursive bifurcation in granular materials, J. Mech. Phys. Solids 45 (11±12) (1997) 1929±1953. [14] D.H. Sattinger, Group Theoretic Methods in Bifurcation Theory, Lecture Notes in Mathematics, vol. 762, Springer, Berlin, 1979. [15] M. Golubitsky, I. Stewart, D.G. Schaeer, Singularities and Groups in Bifurcation Theory 2, Springer, New York, 1988. [16] S.F.A. Kettle, Symmetry and Structure, second ed., Wiley, Chichester, 1995. [17] S. Dinkevich, Finite symmetric systems and their analysis, Int. J. Solids Struct. 27 (10) (1991) 215±253. [18] K. Murota, K. Ikeda, K. Terada, Bifurcation mechanism underlying echelon mode formation, Comput. Meth. Appl. Mech. Eng. 170 (3±4) (1999) 423±448. [19] K. Ikeda, M. Nakazawa, Bifurcation hierarchy of a rectangular plate, Int. J. Solids Struct. 35 (1998) 593±617. [20] K. Ikeda, K. Murota, Recursive bifurcation as sources of complexity in soil shearing behavior, Soils Foundations 37 (3) (1997) 17±29. [21] K.J. Bathe, Finite Element Procedures, Prentice-Hall, Englewood Clis, NJ, 1996. [22] J.C. Simo, T.J.R. Hughes, Computational Inelasticity, Springer, New York, 1998. [23] H. Noguchi, T. Hisada, Determination of a new branch-switching algorithm in nonlinear FEM using scaled corrector, JSME Int. J. 37 (1994) 225±263. [24] S. Okazawa, T. Usami, H. Noguchi, F. Fujii, 3D necking bifurcation in tensile steel specimens, 2001 (preprint).
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Recursive bifurcation of tensile steel specimens Kiyohiro Ikeda a,*, Shigenobu Okazawa a, Kenjiro Terada a, Hirohisa Noguchi b, Tsutomu Usami c b
a Department of Civil Engineering, Tohoku University, Aoba, Sendai 980-8579, Japan Department of System Design Engineering, Keio University, Yokohama 223-8522, Japan c Department of Civil Engineering, Nagoya University, Nagoya 464-8603, Japan
Received 20 October 2000; accepted 31 January 2001
Abstract Failure modes of steel specimens subjected to uniaxial tension are investigated. These modes are well known to display complex geometrical characteristics of deformation accompanied by the plastic instability behavior. As an underlying mechanism of such complexity, we here focus on the recursive occurrence of bifurcations. In the theory, the rule of recursive bifurcation of a rectangular parallelepiped domain is obtained by the group-theoretic bifurcation theory so as to exhaust all the mathematically possible courses of bifurcation. In the experiment, we examine the representative failure modes with reference to the rules to identify actual courses of recursive bifurcation. Three-dimensional ®nite element analysis of a thin specimen is conducted to observe the recursive bifurcation, in which diuse necking is formed by the direct bifurcation and the single shear band by the secondary bifurcation. The recursive bifurcation has thus been identi®ed as the mechanism to create the complex failure modes. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Recursive bifurcation; Group-theoretic bifurcation theory; Steel specimens
1. Introduction The failure mechanism of a steel specimen subjected to uniform tensile loading still receives much attention under the name of plastic instability. On the basis of Hill's [1] theory of uniqueness and stability of solutions of elastic±plastic solids, Hill and Hutchinson [2] developed the theory on the plastic bifurcation behavior observed in tension tests. As for geometrical instability, this theory has been applied to the bifurcation analyses for necking behavior in steel specimens (e.g. [3,4]). On *
Corresponding author. Tel.: +81-22-217-7416; fax: +81-22-217-7418. E-mail address: [email protected] (K. Ikeda).
0020-7225/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 1 ) 0 0 0 4 0 - 4
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the other hand, the implication of the theory has been realized in a considerable number of theoretical and computational studies on the constitutive modeling within the framework of material instability. In particular, various phenomenological models for localized deformations have been introduced; to name a few, the ®nite-strain version of deformation theories [5], the constitutive models with microvoids [6] or with thermal-softening capability [7]. A great deal of related theories and computational techniques have also been developed to simulate the shear band formation leading to the ultimate failure (e.g. [8±11]). In most of these studies, an emphasis was placed on the direct bifurcation that creates necking or diagonal shear bands. However, the direct bifurcation modes of uniform domains are harmonic and cannot, in general, account for the complexity of actual deformation patterns observed in experiments. For such complexity, the recursive occurrence of bifurcation that entails complex deformation patterns can be advanced as the underlying mechanism (see, for theoretical developments [12,13]). In the bifurcation theory, a mathematical tool to describe recursive bifurcation is readily applicable (e.g. [14,15]). With reference to the symmetry of the system under consideration, we can enumerate the symmetries of possible deformation patterns with reduced symmetries. In experimental studies, however, only few attempts have so far been made at the visual observation of deformation pattern change for a steel specimen. In this respect, the theory of recursive bifurcation eminently deserves a pertinent means to investigate the complex pattern change of deformation of a steel specimen. The aim of this paper is to explain deformation characteristics of steel specimens subjected to uniaxial tension through the general standpoint of ``recursive bifurcation''. The geometrical pattern change of the deformed specimens observed in both an experiment and a numerical analysis can be systematically and successfully explained by the bifurcation rule developed in this paper. This paper is outlined as follows. In Section 2, the rule of recursive bifurcation of a rectangular parallelepiped domain is brie¯y described by the group-theoretic bifurcation theory, while the details are worked out in Appendix A. Section 3 is devoted to the experimental study, as well as numerical one, on the recursive bifurcation phenomena of steel members with an emphasis on the deformation pattern change. The representative failure modes are studied with reference to the bifurcation rule presented in Section 2. 2. Theory of recursive bifurcation In this section, we shall explain how the symmetry of a rectangular parallelepiped domain is labeled by a group made up of a series of geometrical actions, and introduce the group-theoretic rule of recursive bifurcation of this domain. As bifurcation phenomena are accompanied by an observable loss of symmetry of the system in question, it is convenient to express the bifurcation rule as a set of admissible transitions among groups that label symmetries and partial symmetries of the system under consideration. Namely, we can ®nd a group G that labels its full symmetry, and a hierarchy of subgroups G ! G1 ! G2 !
1
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Fig. 1. Rectangular parallelepiped domain.
that characterizes the hierarchical transition of symmetries. Here ! corresponds to the loss of symmetry caused by a bifurcation, and Gi (i 1; 2; . . .) stand for the subgroups of G that label the partial symmetries of bifurcating branches. 2.1. Symmetry of a rectangular parallelepiped domain We consider a rectangular parallelepiped domain shown in Fig. 1, that is, X
x; y; z 2 R 3
Lx Lx 6x6 ; 2 2
Ly Ly 6y6 ; 2 2
Lz Lz 6z6 : 2 2
The geometric symmetry of this domain is expressed by the following eight elements of geometric operations: E;
rx ;
ry ;
rz ;
rx ry ;
ry rz ;
rz rx ;
rx ry rz ;
where E is the identity element; rx , ry and rz , respectively denote re¯ections in yz-, zx- and xyplanes; rx ry , ry rz and rz rx , respectively denote p-rotations about z-, x- and y-axes; rx ry rz is the inversion with respect to the origin; rx ry ry rx ; ry rz rz ry ; rz rx rx rz and r2x r2y r2z E. Fig. 2 illustrates the actions of these operations. The group of these operations is de®ned in the Schoen¯ies notation 1 as D2h hrx ; ry ; rz i fE; rx ; ry ; rz ; rx ry ; ry rz ; rz rx ; rx ry rz g; where hi denotes the group generated by the elements therein and fg denotes the group consisting of the elements therein. Partial symmetries emanating from a D2h -symmetric state are labeled by the proper subgroups of D2h , including: 1
See, e.g. [16] for this notation and [17] for relevant mathematical backgrounds.
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Fig. 2. Geometric operations.
hrx ; ry i fE; rx ; ry ; rx ry g; hry ; rz i fE; ry ; rz ; ry rz g; hrz ; rx i fE; rz ; rx ; rz rx g; hrx ; ry rz i fE; rx ; ry rz ; rx ry rz g;
2
hry ; rz rx i fE; ry ; rz rx ; rx ry rz g; hrz ; rx ry i fE; rz ; rx ry ; rx ry rz g; hrx ry ; ry rz i fE; rx ry ; ry rz ; rz rx g; hrx i fE; rx g; hry i fE; ry g; hrz i fE; rz g; hrx ry i fE; rx ry g;
3
hry rz i fE; ry rz g; hrz rx i fE; rz rx g; hrx ry rz i fE; rx ry rz g; hEi fEg:
4
The symmetries expressed by D2h and its subgroups are illustrated in Fig. 3. With reference to Remark 1, we assume an additional symmetry of the x-directional translation t l at a distance of l 0 6 l < Lx de®ned by t l : x 7! x l:
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
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Fig. 3. Symmetries expressed by D2h and its subgroups.
One may interpret l as the wavelength in the longitudinal direction. The rectangular parallelepiped domain consequently has the symmetry labeled by G hrx ; ry ; rz ; t li:
5
This group also expresses the symmetry of uniform deformation prior to bifurcation. Remark 1. We note that rectangular parallelepiped steel specimens under consideration are long and uniform in the longitudinal direction. In order to express the local uniformity in this direction, we assume the periodic symmetry in the x-direction. Such assumption of periodic symmetry is often employed in the investigation of bifurcation behavior (e.g. [12,15,18]). 2.2. Recursive bifurcation rule The rule of recursive bifurcation of a system with a symmetry labeled by the group G hrx ; ry ; rz ; t li is brie¯y introduced here whereas the details are worked out in Appendix A.
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In view of actual deformation pattern change of tensile steel specimens (see, e.g. Sections 3.1 and 3.2), among a series of mathematically possible symmetries for the deformation patterns, we restrict ourselves in this paper to the bifurcation process G hrx ; ry ; rz ; t li ! D2h hrx ; ry ; rz i: As shown in Fig. 4, this bifurcation process is associated with the occurrence of necking around the center of the rectangular parallelepiped specimen. As described in Section A.2, the direct bifurcation from D2h -symmetric state has the symmetry labeled by one of the seven subgroups in (2). The secondary bifurcation from this state leads to the symmetry labeled by one of the seven subgroups in (3). Further bifurcation will lead to completely asymmetric states labeled by hEi. To sum up, we focus in this paper on the recursive bifurcation expressed by a hierarchy of subgroups shown in Fig. 5 that exhausts all mathematical possibilities for the recursive bifurcation below D2h . It should be remarked that the group-theoretic study presents all mathematically possible bifurcation modes, or, to be precise, the symmetries of these modes. Actual bifurcation modes are to be restricted by the boundary conditions to be employed in experiments or numerical analyses (cf. [19]). 3. Steel specimens under uniform tension We focus our attention to complex geometrical pattern change of a steel specimen under uniform tension, which can commonly be observed in an experiment and a numerical analysis.
Fig. 4. Necking associated with the bifurcation process G hrx ; ry ; rz ; t li ! D2h hrx ; ry ; rz i.
Fig. 5. The rule of bifurcation of G-symmetric system expressed in terms of a hierarchy of subgroups.
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The group-theoretic method presented in Section 2 is employed here to untangle this complexity and clarify the geometrical aspects of ultimate failures. 3.1. Experimental behavior We employ various steel specimens with dierent dimensions listed in Table 1, all of which have the same material composition for metal. Specimen M is for calibration, while Specimens A, B, C and D of four dierent width±thickness ratios are for the visual observation of deformed geometry. A typical geometry of these steel specimens is depicted in Fig. 6; we put L Lx , W Ly and t Lz in Fig. 1 in the remainder of this paper. A tensile load is applied to each specimen statically at constant and moderate temperature by a testing machine with a sucient capacity. Fig. 7 shows the load versus elongation curve obtained in a single test on Specimen M, from which representative Young's modulus is determined as 204 GPa. After a relatively small elastic deformation, the plastic yielding occurs uniformly in the rectangular parallelepiped region of the specimen at the initial yield stress (429 MPa). As the specimen is further elongated, the load increases to reach the maximum load (591 MPa), thereafter a necking deformation occurs around the center of the specimen and is concentrated in the narrower region. The ultimate failure is caused by the inherent straining limit of the metal. The other Specimens A, B, C and D have exhibited similar behavior up to the maximum load, while the post-peak behavior is dependent on the cross-sectional shapes as shown in Figs. 8±11 taken during the tests. The states shown in (a), (b) and (c) in each ®gure, respectively, correspond to the beginning of the non-uniform deformation, the appearance of prominent strain localization and the failure mode. We carefully describe the geometrical aspects of each of the states in order. Table 1 Dimensions of steel specimens Specimen
Width, W (mm)
Thickness, t (mm)
Width±thickness ratio, W =t
Length, L (mm)
M A B C D
39.7 39.9 40.0 39.9 27.8
9.79 3.83 9.86 26.0 28.2
4.05 10.4 4.06 1.54 0.984
100 220 220 220 220
Fig. 6. Geometry of a steel specimen.
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Fig. 7. Typical load versus elongation curve obtained in a tension test for Specimen M for calibration (: maximum load).
Fig. 8. Transient deformation change of Specimen A (W =t 10:4).
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Fig. 9. Transient deformation change of Specimen B (W =t 4:06).
First, we observe the thinnest Specimen A. As shown in Fig. 8(a), diuse necking appears around the center of this specimen. As a result of strain localization triggered by the non-uniform deformation, two diagonal lines, sometimes called shear bands, emerge in the middle part of the necking; see Fig. 8(b). Then the intense straining concentrates on one of the shear bands, while the other becomes less discernible. In association with the growth of this single distinct shear band, the specimen undergoes ductile failure as shown in Fig. 8(c). Next, we investigate the deformation patterns of the thicker Specimens B, C and D. As shown in Figs. 9(a), 10(a) and 11(a), diuse necking appears around the center, while the shear bands are not observed even in the state of fracturing; see Figs. 9(b), 10(b) and 11(b). Instead, dimples are formed on all the side surfaces. The ®nal states shown in Figs. 9(c), 10(c) and 11(c) display the states after the brittle failures. The complexity of failure modes notwithstanding, they seem to present the distinctive features in view of the reduction of symmetries. As we have seen, the width±thickness ratio of the specimens is in¯uential on the resulting instability phenomena especially at the ®nal state. The transient pattern change leading to this state,
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Fig. 10. Transient deformation change of Specimen C (W =t 1:54).
which progresses rapidly and hence is unclear, will be discussed in Section 3.2 based on the theory of recursive bifurcation presented in Section 2. 3.2. Representative numerical simulation We perform a three-dimensional numerical analysis for Case A in order to demonstrate clearly the occurrence of pattern change due to recursive bifurcation, as the occurrence of bifurcation in the experiment may seem controversial for most of the readers. Since speci®c material behavior is not the central issue in our study, we employ a standard constitutive model for plastic deformations, which is presented in Appendix B together with the values of material constants. In the sequel, only the major result for recursive bifurcation is presented. The load±elongation curves are shown in Fig. 12. The primary solution path exhibits one limit point, which is marked by . The direct bifurcation point marked by is found immediately after the limit point. We are here concerned with only the physically meaningful bifurcations though other bifurcation points were found on the primary path after the bifurcation point marked by .
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
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Fig. 11. Transient deformation change of Specimen D (W =t 0:984).
Fig. 12. Numerical load versus elongation curves (: limit point, : bifurcation point; P: applied load, A: initial crosssection; rY : yields stress; u: axial displacement; L: member length).
At the direct bifurcation point, the critical eigenvector of the tangent stiness matrix is related to the lowest harmonic mode, which leads to diuse necking, and is given in Fig. 13(a). In addition, the two eigenvectors with eigenvalues close to zero are presented in Figs. 13(b) and (c). As
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Fig. 13. Eigenvectors near the direct bifurcation point.
can be seen, the last two eigenvectors are higher harmonic modes than the critical one. The critical eigenvector is associated with D2h -symmetry whose course is given in (6), while other ones with other symmetries labeled by the groups in (A.7). It is noteworthy that the formation of an oblique (single) shear band is not a direct bifurcation mode. Fig. 14(b) shows the deformed con®guration with the contour plot of eective plastic strains at the load level P = ArY 0:80. The loaded region is localized to shape a cross band while the unloaded one spreads from the edge surface to the center part of the specimen. Furthermore, the numerical analysis here presents the secondary bifurcation behavior, whose solution branch is depicted by dotted lines in Fig. 12. The corresponding deformed con®guration given in Fig. 14(c) agrees with the experimental result in Section 3.1 (cf. Fig. 8). 3.3. Theoretical interpretation of recursive bifurcation phenomena The recursive bifurcation observed in Sections 3.1 and 3.2 is examined with reference to the theory presented in Section 2. For all the specimens, the formation of the necking from the uniform state results from the direct bifurcation associated with G hrx ; ry ; rz ; t li ! D2h hrx ; ry ; rz i;
6
in which the symmetry of the uniform state is labeled by G and the necking by the group D2h .
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
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Fig. 14. Progress of deformation expressed by the contour of eective plastic strain of FE model of a rectangular parallelepiped specimen (6750 elements for Case A).
Let us start with the observation of the thinnest Specimen A. As shown in Fig. 8 and also in Fig. 14, the diagonal shear bands are formed after the diuse necking due to the intense localized straining. Such formation is characteristic from a physical standpoint, but is not associated with bifurcation in that both the state of necking and that of the diagonal shear bands share the same symmetry labeled by the group D2h . The secondary bifurcation takes place at the onset of the formation of a single distinct shear band, which is one of the diagonal shear bands. This secondary bifurcation is associated with a further reduction of symmetry associated with a bifurcation process D2h ! hrz ; rx ry i;
7
in which D2h denotes the symmetry of the diagonal shear bands and hrz ; rx ry i indicates that of the single distinct shear band. The ®nal failure state, the symmetry of which is labeled by the same group hrz ; rx ry i, is no longer caused by bifurcation. The deformation pattern change and the loss
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of symmetry associated with the direct and secondary bifurcations (6) and (7) are illustrated in Fig. 15. Next, we deal with the thicker Specimens B, C and D. These specimens undergo the direct bifurcation (6) to reach the state of diuse necking. Then, as shown in Figs. 9±11, dimples are formed by the localized straining; such formation is a non-bifurcation process. By brittle and premature failure, the specimens arrive at the ®nal states as shown in the last of each ®gure. With reference to these states, we can advance the following possible secondary bifurcation processes: D2h ! hry ; rz i & hry ; rz rx i
for B and C;
8
for D:
These processes, which take place just prior to their ®nal failure, presumably cause the failure. The deformation pattern change and the loss of symmetry associated with the direct and secondary bifurcations (6) and (8) are illustrated in Figs. 16 and 17. To sum up, from (6)±(8), we can arrive at a hierarchy of subgroups % hrz ; rx ry i for A; G hrx ; ry ; rz ; t li ! D2h ! hry ; rz i
for B and C;
9
& hry ; rz rx i for D; which is nothing but a part of the whole hierarchy in Fig. 5. Of course, the three courses of recursive bifurcation we have found here may not exhaust all the physically possible courses. More detailed observation based on the knowledge of the recursive bifurcation rule will be a future topic.
Fig. 15. Illustration of the hierarchical deformation pattern change due to recursive bifurcation for Specimen A (W =t 10:4).
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
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Fig. 16. Illustration of the hierarchical deformation pattern change due to recursive bifurcation for Specimens B (W =t 4:06) and C (W =t 1:54).
Fig. 17. Illustration of the hierarchical deformation pattern change due to recursive bifurcation for Specimen D (W =t 0:984).
4. Conclusions The geometrical pattern change of rectangular parallelepiped steel specimens subjected to uniaxial tension has been investigated with reference to the theory of recursive bifurcation. The single, distinct shear band does not emanate from the direct bifurcation but branches from the
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secondary one whereas the direct bifurcation creates diuse necking. The recursive bifurcation has thus emerged as a source of the instability behavior. While the group-theoretic bifurcation theory presents a plethora of mathematically possible courses of recursive bifurcation of rectangular parallelepiped specimens, the experiment and the numerical analysis on steel specimens demonstrate physically feasible ones among those possible courses of bifurcation. In the experiment, a transient state of deformations, which is usually overlooked, has been interpolated on the basis of the theoretical results. In the numerical analysis, the recursive bifurcation is clearly observed for a thin specimen, in which diuse necking is formed by the direct bifurcation and the distinct, single shear band by the secondary bifurcation. As a result of these, the failure modes are understood as a consequence of the recursive bifurcation. The study of the geometrical pattern change, the tail of which has been caught in this paper, is expected to shed light on the plastic instability behavior. Appendix A. Derivation of bifurcation rules The recursive bifurcation rule for a system equivariant to G hrx ; ry ; rz ; t li is derived on the basis of the standard mathematical technique called the group-theoretic bifurcation theory. In this derivation, we refer to the rules for systems with smaller symmetries, such as D2h and D1h (see, e.g. [16,20]). A.1. Group-theoretic bifurcation theory A very brief account of the main ideas of the group-theoretic bifurcation analysis is oered (see, e.g. [14,15,18]). The deformed state of the material in question is assumed to be described by a (vector-valued) function u x, where x is the coordinate of a point in the domain, and that u x satis®es a system of governing equations F u; f 0;
A:1
where f is a loading parameter and F is a nonlinear function satisfying pertinent regularity conditions, such as the dierentiability. It should be noted the governing equation (A.1) already implements material properties, such as the constitutive laws. Let G be a group of transformations acting on x. Denote by g x and g u the transformation caused by an element g of G. If u is a scalar-valued function, for example, this transformation is expressed as g u x u g
1
x:
Eq. (A.1) is called equivariant with respect to G if g F u; f F g u; f 8g 2 G is satis®ed. A solution u is called invariant under G (or G-invariant) if
A:2
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
guu
1929
8g 2 G
is satis®ed. The solutions u; f are assumed to be lying on the main path and retain G-invariance, whereas those on a bifurcation path have a reduced symmetry. Such a reduced symmetry can be labeled by a subgroup G0 of G. If uc ; fc is a critical point on the main path, at which the derivative (or the Frechet derivative) of F with respect to u: F 0 u; f
oF u; f ou
has a zero eigenvalue. It should be noted that F 0 u; f reduces to the tangential stiness matrix or the Jacobian matrix if the system (A.1) is ®nite dimensional. Usually a critical point is either a limit point of loading parameter f or a bifurcation point. The bifurcation paths, branching from the main path at a bifurcation point, are usually made up of solutions u; f of reduced symmetries labeled by subgroups G0 of G. The multiplicity M of a critical point uc ; fc is de®ned as the number of zero eigenvalues of F 0 uc ; fc . The kernel space X of the operator F 0 uc ; fc is easily veri®ed to be invariant under G, i.e., gu2X
for u 2 X ;
if uc is G-invariant. Then the critical point is divided into two types, group-theoretic or parametric, according to whether X is G-irreducible or not. Here X is said to be G-irreducible if there exists no non-zero proper subspace in X that is invariant under G. The group-theoretic critical point occurs due to the symmetry of the system while the parametric critical point occurs by the accidental coincidence of two or more critical points. Since the latter is rare, only the former type is considered. Let uc ; fc be a group-theoretic critical point. Then, by de®nition, the kernel space X is associated with one of the irreducible representations, which we denote by l. The irreducible representation, in turn, is associated with a subgroup Gl of G. To be more speci®c, let T l g g 2 G be the representation matrices of the irreducible representation l. Then Gl fg 2 G j T l g Ig; where I is the identity matrix. This subgroup is called the symmetry group of the kernel space. For example, a limit point of the loading parameter f is associated with the unit (one-dimensional) representation l for which T l g 1 for all g 2 G, and Gl G. A bifurcation point is usually associated with an irreducible representation l for which T l g 6 I for some g 2 G, and hence Gl is a proper subgroup of G. Such a bifurcation point is called a symmetry-breaking bifurcation point. The degree of the irreducible representation is equal to the multiplicity M of the critical point. The symmetry of the system on a bifurcation path can be determined by solving the bifurcation equation, which inherits the equivariance condition (A.2) (see, e.g. [12,14]). It should be noted that this symmetry is often higher than the symmetry of Gl .
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A.2. Bifurcation from D2h -invariant state The bifurcation structure of a system equivariant to the group D2h hrx ; ry ; rz i fE; rx ; ry ; rz ; rx ry ; ry rz ; rz rx ; rx ry rz g is described. We index the family of non-equivalent irreducible representations of D2h by R D2h f m1 ; m2 ; m3 D2h j m1 ; m2 ; m3 ; g:
A:3
Here ; ; D2h corresponds to the unit (one-dimensional) irreducible representation and the others to one-dimensional irreducible representations, being de®ned by the one-dimensional representation matrices given in Table 2. The unit representation ; ; D2h corresponds to the limit point of loading parameter f with D2h symmetry. The remaining representations correspond to simple bifurcation points and the symmetries of their bifurcation paths are labeled by the proper subgroups of D2h in (2), that is, hrx ; ry i;
hry ; rz i;
hrz ; rx i;
hrx ; ry rz i;
hry ; rz rx i;
hrz ; rx ry i;
hrx ry ; ry rz i:
A:4
By investigating further bifurcations from these bifurcation paths with symmetries labeled by the groups in (A.4), we can arrive at the bifurcation rule below the group D2h ; see Fig. 5. A.3. Bifurcation from G-invariant state The recursive bifurcation rule for a system equivariant to the group G hrx ; ry ; rz ; tx lx i hrx ; tx lx i hry i hrz i in (5) is derived. Note that the group G is isomorphic to the group O 2 Z2 Z2 , and the group hrx ; t li hry i is isomorphic to the group D1h in the Schoen¯ies notation. See, e.g. [16] for this notation and [17,20] for the studies on a D1h -equivariant system. Table 2 Representation matrices and symmetry groups of D2h -invariant system Irreducible representation l
T l g rx
ry
rz
; ; D2h ; ; D2h ; ; D2h ; ; D2h
1 )1 1 )1
1 1 )1 )1
1 1 1 1
; ; ; ; ; ; ; ;
1 )1 1 )1
1 1 )1 )1
)1 )1 )1 )1
D2h D2h D2h D2h
Symmetry groups Gl for X
Bifurcation paths D2h hrx ; ry ; rz i hry ; rz i hrz ; rx i hrz ; rx ry i hrx ; ry i hry ; rz rx i hrx ; ry rz i hrx ry ; ry rz i
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
1931
With reference to these studies, we can index the family of non-equivalent irreducible representations of G hrx ; ry ; rz ; t li by R G f m1 ; m2 ; m3 ; n; m2 ; m3 j m1 ; m2 ; m3 ; ; n 1; 2; . . .g:
A:5
Here ; ; corresponds to the unit representation, which is associated with the limit point of bifurcation parameter. Other one-dimensional ones ; ; ;
; ; ;
; ; ;
; ; ;
; ; ;
; ; ;
; ; ;
which are de®ned by the one-dimensional representation matrices T l given in Table 3, are associated with simple bifurcation points. The remaining irreducible representations in (A.5) are of degree two, and their actions are de®ned by the representation matrices listed in Table 3, where I2
1 0 ; 0 1
P
0 ; 1
1 0
R
cos nu sin nu
sin nu cos nu
A:6
with the correspondence of u 2p
lx ; Lx
0 6 u < 2p; 0 6 lx < Lx :
Though the analysis on the bifurcation equation is omitted here, the symmetries of the bifurcating solutions associated with these two-dimensional irreducible representations are obtained as Table 3 Representation matrices of G-equivariant system G hrx ; ry ; rz ; tx lx ; r rx tx lx for some lx i M
l
T l g
Symmetry groups
rx
ry
rz
tx lx
Gl
Bifurcation paths
1 1 )1 )1 1 1 )1 )1
1 1 1 1 )1 )1 )1 )1
1 1 1 1 1 1 1 1
G hry ; rz ; tx lx i hrx ; rz ; tx lx i hrz ; rx ry ; tx lx i hrx ; ry ; tx lx i hry ; rz rx ; tx lx i hrx ; ry rz ; tx lx i hrx ry ; ry rz ; tx lx i
No bifurcation hry ; rz ; tx lx i hrx ; rz ; tx lx i hrz ; rx ry ; tx lx i hrx ; ry ; tx lx i hry ; rz rx ; tx lx i hrx ; ry rz ; tx lx i hrx ry ; ry rz ; tx lx i
1
; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;
1 )1 1 )1 1 )1 1 )1
2
n; ;
P
I2
I2
R
hry ; rz ; tx Lnx i
hr; ry ; rz ; tx Lnx i
n; ;
P
I2
I2
R
Lx hrz ; ry tx 2n i
Lx hr; rz ; ry tx 2n i
n; ;
P
I2
I2
R
Lx hry ; rz tx 2n i
Lx hr; ry ; rz tx 2n i
n; ;
P
I2
I2
R
Lx Lx hry tx 2n ; rz tx 2n i
Lx Lx hr; ry tx 2n ; rz tx 2n i
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K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
Lx Lx r; ry ; rz ; tx ; r; rz ; ry tx ; n 2n Lx Lx ; rz tx ; n 1; 2; . . . ; r; ry tx 2n 2n
Lx r; ry ; rz tx ; 2n A:7
where r rx tx lx for some lx . Note that D2h (studied in Section A.2) is obtained by setting n 1 and r rx in hr; ry ; rz ; tx Lx =ni in (A.7) and, therefore, is associated with 1; ; . Appendix B. Description of the numerical model In this section, we present the formulation, procedure and analysis conditions for the numerical analysis in Section 3.2. Our large-scale computations rely on the standard formulation of the ®nite element method for classical ®nite-strain elastoplasticity, which is brie¯y summarized below. B.1. Formulation of ®nite-strain elastoplasticity We here describe the boundary value problem for the classical rate-independent plasticity model. By using the updated-Lagrangian formulation, which is widely used in practical computations, we employ the rate form of the momentum balance equation in the context of ®nite element (FE) analyses. For elaborate discussions for this type of formulations, one refers to, e.g. [21,22]. Let B Rndim (ndim 1; 2 or 3) be the reference con®guration of an elastic±plastic solid with material particles denoted by X 2 B and be subjected to a deformation u : B 7! Rndim , with J : det rX u > 0. We let oB be the boundary of B and assume that the deformation is prescribed ^ is prescribed on ot B oB, with the ^ whereas the nominal traction vector T on ou B oB as u u ^ We here consider the quasi-static nominal stress tensor P and the unit normal N, as PN T. equilibrium problem with the given body force B in B. In the current con®guration u B, the Kircho stress tensor s and the spatial velocity ®eld v : u_ u 1 are used to describe the equilibrium state. In terms of the admissible spatial velocity ®eld g in an appropriate function space V, the rate form of the variational equation of this problem is given by Z Z Z dv dv ^t_ g ds; g 2 V; rg : rvs Lv s b_ g B:1 J J u B u B ot u B ^ respectively, with the Cauchy stress where b and ^t rn are the spatial representations of B and T, r and the unit normal n on ot u. Here, Lv s is the Lie derivative of s and related to its Jaumann rate r s as r
Lv s s
ds
sd : a : d;
where d is the spatial rate-deformation tensor and a are the symmetric moduli.
B:2
K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
1933
In this study, we simply assume that the material reveals isotropy and that the elastic strains are small compared with plastic ones. Also, by assuming that the plastic deformation is incompressible, we neglect the volumetric change of this metal by J 1 so that the classical J2 ¯ow theory can be extended to ®nite-strain range. These assumptions are valid for the metal specimens under consideration whose constitutive equation is given in terms of the rates as r
s c : d;
B:3
in which the symmetric moduli c take the same constant values ce as those in isotropic linear elasticity for elastic deformation and, for plastic ¯ow, 9l2 1 e r0 r0 : B:4 cc 0 2 3l H r Here, l is the shear modulus, H 0 the plastic modulus, r the equivalent stress and r0 the deviatoric stress tensor. B.2. Procedure and analysis conditions In order to produce a uniform loading state, we consider a rectangular parallelepiped region where the deformation is dominated in actual experiments (cf. Figs. 1 and 6). In order to evaluate the critical eigenvector that plays a role of a branch-switching predictor, we utilize the method based on a scaled corrector, which has been proposed by Noguchi and Hisada [23]. The secondary bifurcation path(s) can also be computed essentially in the same manner. The detailed discussion about the algorithm for the branch-switching procedure is found elsewhere (e.g. [24]). A tensile force is applied on the edge surface located at x L=2 while all the other surfaces are free from stress. Note that our actual computation must be performed on the whole rectangular domain to recover all the possible bifurcation patterns. The material properties used in the analyses are chosen as follows: Young's modulus E 200 GPa, Poisson's ratio m 0:3333, and initial yield stress rY 400 MPa. For plastic hardening, the following power law is assumed so that its derivative leads to the parameter H 0 in (B.4): r rY
ep 1 eY
!0:0625 > 0;
B:5
where eY rY =E 1=500 and ep is the eective plastic strain. It is remarked that the nonlinear hardening law (B.5) excludes softening responses of the material, since material instability plays no role in this paper. References [1] R. Hill, A general theory of uniqueness and stability in elastic±plastic solids, J. Mech. Phys. Solids 16 (1958) 236± 240.
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K. Ikeda et al. / International Journal of Engineering Science 39 (2001) 1913±1934
[2] R. Hill, J.W. Hutchinson, Bifurcation phenomena in the plane tension test, J. Mech. Phys. Solids 23 (1975) 239± 264. [3] D.M. Norris Jr., B. Moran, J.K. Scudder, D.F. Quinones, A computer simulation of the tension test, J. Mech. Phys. Solids 26 (1977) 1±19. [4] M.A. Burke, W.D. Nix, A numerical study of necking in the plane tension test, Int. J. Solids Struct. 15 (1979) 379± 393. [5] J. Christoersen, J.W. Hutchinson, A class of phenomenological corner theories of plasticity, J. Mech. Phys. Solids 25 (1979) 465±487. [6] A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I ± Yield criteria and ¯ow rules porous ductile media, J. Eng. Mater. Tech. 99 (1977) 2±15. [7] A. Needleman, Dynamic shear band development in plane strain, J. Appl. Mech. 56 (1989) 1±9. [8] V. Tvergaard, A. Needleman, K.K. Lo, Flow localization in the plane strain tensile test, J. Mech. Phys. Solids 29 (2) (1981) 115±142. [9] V. Tvergaard, A. Needleman, Analysis of the cup±cone fracture in a round tensile bar, Acta Metall. 32 (1) (1984) 157±169. [10] S. Nemat-Nasser, Phenomenological theories of elastoplasticity and strain localization at high strain rates, Part 2, Appl. Mech. Rev. 45 (3) (1992) S19±S45. [11] H. Petryk, K. Thermann, On discretized plasticity problems with bifurcations, Int. J. Solids Struct. 29 (6) (1992) 745±765. [12] K. Ikeda, K. Murota, M. Nakano, Echelon modes in uniform materials, Int. J. Solids Struct. 31 (1994) 2709±2733. [13] K. Ikeda, K. Murota, Y. Yamakawa, E. Yanagisawa, Mode switching and recursive bifurcation in granular materials, J. Mech. Phys. Solids 45 (11±12) (1997) 1929±1953. [14] D.H. Sattinger, Group Theoretic Methods in Bifurcation Theory, Lecture Notes in Mathematics, vol. 762, Springer, Berlin, 1979. [15] M. Golubitsky, I. Stewart, D.G. Schaeer, Singularities and Groups in Bifurcation Theory 2, Springer, New York, 1988. [16] S.F.A. Kettle, Symmetry and Structure, second ed., Wiley, Chichester, 1995. [17] S. Dinkevich, Finite symmetric systems and their analysis, Int. J. Solids Struct. 27 (10) (1991) 215±253. [18] K. Murota, K. Ikeda, K. Terada, Bifurcation mechanism underlying echelon mode formation, Comput. Meth. Appl. Mech. Eng. 170 (3±4) (1999) 423±448. [19] K. Ikeda, M. Nakazawa, Bifurcation hierarchy of a rectangular plate, Int. J. Solids Struct. 35 (1998) 593±617. [20] K. Ikeda, K. Murota, Recursive bifurcation as sources of complexity in soil shearing behavior, Soils Foundations 37 (3) (1997) 17±29. [21] K.J. Bathe, Finite Element Procedures, Prentice-Hall, Englewood Clis, NJ, 1996. [22] J.C. Simo, T.J.R. Hughes, Computational Inelasticity, Springer, New York, 1998. [23] H. Noguchi, T. Hisada, Determination of a new branch-switching algorithm in nonlinear FEM using scaled corrector, JSME Int. J. 37 (1994) 225±263. [24] S. Okazawa, T. Usami, H. Noguchi, F. Fujii, 3D necking bifurcation in tensile steel specimens, 2001 (preprint).