International Journal of Plasticity 42 (2013) 141–167
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Direct evaluation of the limit states of engineering structures exhibiting limited, nonlinear kinematical hardening J.-W. Simon ⇑ Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe-Str. 1, 52074 Aachen, Germany
a r t i c l e
i n f o
Article history: Received 24 May 2012 Received in final revised form 12 October 2012 Available online 24 October 2012 Keywords: Limit states Shakedown analysis Lower bound approach Limited kinematical hardening Nonlinear hardening
a b s t r a c t The lower bound shakedown analysis is a most convenient tool to determine the load bearing capacity of engineering structures subjected to thermo-mechanical loadings. In order to achieve realistic results, limited nonlinear kinematical hardening needs to be taken into account. Although there exist different formulations incorporating limited kinematical hardening in the literature, it is still not conclusively clarified, whether or not these are applicable to generally-nonlinear hardening laws as well. Thus, the aim of this paper is to propose a method to determine the shakedown limit loads accounting for limited, generally-nonlinear kinematical hardening, and to close the discussion about the effect of the nonlinearity of the hardening law. The proposed method is based on an extension of the statical shakedown theorem by Melan using a two-surface model, which captures both incremental collapse and alternating plasticity. Furthermore, it is implemented into an interior-point algorithm, which is tailored to shakedown analysis and thus capable of handling large-scale problems. The algorithm allows for an arbitrary number of thermomechanical loadings. To illustrate the method’s potential, numerical results are shown for several examples from the field of power plant engineering. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction One of the most important tasks for construction engineers is the determination of the load bearing capacity of the engineering structure under consideration. This is particular challenging, if the applied thermo-mechanical loadings vary with time beyond the elastic limit. Then, the computation of the so-called shakedown loading factor aSD is necessary, which is the maximum loading factor a such that the system does neither fail due to spontaneous or incremental collapse – often referred to as ratcheting in the case of cyclic loadings – nor due to alternating plasticity. Using the classical step-by-step method for such calculations can be a complex and extensive challenge in experiments as well as numerical simulations (see e.g., Chaboche, 1991; Bari and Hassan, 2000, 2001, 2002; Zhang and Jiang, 2005; Taleb et al., 2006; Khan et al., 2007; Chaboche, 2008; Jiang and Zhang, 2008; Rahman et al., 2008; Hassan et al., 2008; Taleb and Hauet, 2009; Krishna et al., 2009; Abdel-Karim, 2009, 2010; Abdel-Karim and Khan, 2010; Shamsaei et al., 2010; Sai, 2011; Dafalias and Feigenbaum, 2011; Guo et al., 2011; Taleb and Cailletaud, 2011; Yu et al., 2012) to name only a few. Moreover, the loading history has to be given deterministically, which is not realistic in many technical applications. If the exact stress–strain distribution is not necessary to be determined but only the limit state, these problems can be overcome by use of direct methods, namely limit and shakedown analysis (see e.g., Mroz et al., 1995; Weichert and Maier, 2000; Maier et al., 2003; Weichert and Ponter, 2009). These do not require the exact knowledge of the loading history but
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only its bounding envelope, as shown by König (1987). Therefore, in this paper shakedown analysis is applied to determine the maximum loading factor aSD such that the system can be considered ‘safe’. Generally, there exist two different approaches to shakedown analysis: (1) the lower bound approach by Melan (1938a,b), which is formulated in statical quantities; (2) the upper bound approach of Koiter (1960), which is formulated in kinematical quantities. In this work, the static approach of Melan (1938a,b) is adopted, because it leads to conservative solutions in principle, which is desired from engineering point of view. Even more, Melan’s theorem is suited for the extension to kinematical hardening, because of the formulation in stresses. Since in many engineering applications the materials exhibit kinematical hardening, the latter needs to be incorporated into the procedure in order to obtain realistic results. The statical shakedown theorem in its original formulation is valid for elastic-perfectly plastic continua as well as for unlimited kinematical hardening ones. Notably, accounting for only unlimited kinematical hardening does not cover incremental collapse but solely alternating plasticity (e.g., Ponter, 1975; Zarka and Casier, 1981; König, 1987; König and Siemaszko, 1988). Thus, accounting for limited (or bounded) kinematical hardening is inevitable, and consequently has been addressed by several authors in the field of shakedown analysis (e.g., Mandel, 1976; Polizzotto, 1986; Weichert and Groß-Weege, 1988; Groß-Weege and Weichert, 1992; Stein et al., 1990, 1992, 1993; Corigliano et al., 1995; Pycko and Maier, 1995; Fuschi, 1999; Pham and Weichert, 2001; Staat and Heitzer, 2002; Nguyen, 2003; Pham, 2007; Pham, 2008; Pham et al., 2010; Polizzotto, 2010). The first explicit formulation for limited kinematical hardening materials has been given by Weichert and Groß-Weege (1988), who introduced a two-surface model. Almost at the same time, Stein et al. (1990, 1992, 1993) proposed another approach based on an overlay model, which leaded to an equivalent formulation. Later, Heitzer (1999) showed how to transfer these approaches one to the other. Even though the limited kinematical hardening has been intensively investigated in the context of shakedown analysis (see the references above), there is still an open discussion about whether or not the above mentioned formulations are restricted to specific hardening rules. Particularly, in some of these works only the limited linear kinematical hardening is considered, in accordance to the proof given by Weichert and Groß-Weege (1988), which is based on the Generalized Standard Material Model proposed by Halphen and Nguyen (1975). By contrast, some others claim, that the same formulation of the shakedown theorem can also be obtained for limited nonlinear kinematical hardening in accordance to the proof for the generally-nonlinear case given by Pham (2008). Hence, the shakedown theorem seems to be restricted to linear hardening on the one hand side, but has been proven to hold in the nonlinear case as well on the other hand side. In consequence, several publications state that the type of hardening has no influence on the shakedown limit but only the initial yield state and the ultimate state. Surprisingly, independently of each other, Staat and Heitzer (2002) and Bouby et al. (2006, 2009) have presented results with significant differences in the shakedown limit load between the limited linear hardening and the limited nonlinear hardening. To date, this seeming paradox remains unresolved. Thus, one aim of the present paper is to clarify the effect of the type of different hardening rules in lower bound shakedown analysis. However, the statical shakedown theorem leads to nonlinear convex optimization problems, which are typically characterized by large numbers of unknowns and constraints when problems of practical relevance are considered. In this work, the interior-point method (see e.g., Potra and Wright, 2000; Forsgren et al., 2002; Wright, 2004) is used to solve these. Based on this method, there exist several powerful algorithms, such as IPOPT (Wächter and Biegler, 2005, 2006), KNITRO (Byrd et al., 2000; Waltz et al., 2006) and LOQO (Vanderbei, 1999; Griva et al., 2008). These are, however, designed to solve a wide variety of problems, which may result in less efficient performances compared to problem-tailored codes. One possibility to overcome too extensive computations, following Christiansen and Andersen (1999), second order cone programming (SOCP) has come into the picture of direct methods in recent years. In particular, the software package MOSEK (Andersen et al., 2003, 2009) has been used e.g. in Trillat and Pastor (2005), Bisbos et al. (2005), Makrodimopoulos (2006), Krabbenhøft et al. (2007a), Pastor et al. (2008) and Skordeli and Bisbos (2010), as well as the codes SEDUMI (Sturm, 1999) and SDPT3 (Tütüncü et al., 2003) in Munoz et al. (2009). Nevertheless, in order to increase the numerical efficiency – and since not all problems can be formulated as SOCP – a number of alternative algorithms have been presented for both limit analysis (e.g. Lyamin and Sloan, 2002; Krabbenhøft and Damkilde, 2003; Krabbenhøft et al., 2007b; Pastor et al., 2008; Pastor et al., 2009; Pastor and Loute, 2010), and shakedown analysis (e.g. Zouain et al., 2002; Vu et al., 2004; Hachemi et al., 2005; Liu et al., 2005; Akoa et al., 2007; Vu and Staat, 2007). It should be mentioned, that a variety of alternative methods have been developed in recent years. For example, Zhang and Raad (2002) have proposed an eigen-mode method, whereas the so-called Linear Matching Method (LMM) has been suggested by Ponter (2002) and Ponter and Chen (2005). Further, a bipotential approach has been invented by Bousshine et al. (2003) and Bouby et al. (2009), and a homogenized method has been examined e.g. in Magoariec et al. (2004). In addition, Ngo and Tin-Loi (2007) and Ardito et al. (2008) propose a piece-wise linearization of the yield surface, while a strain-driven strategy is presented by Garcea and Leonetti (2011). More recently, a new direct method has been suggested by Spiliopoulos and Panagiotou (2012), in which the cyclic nature of the expected residual stress distribution at the steady cycle is investigated. However, in the present paper, the convex optimization problem resulting from the statical shakedown theorem is solved via the interior-point algorithm IPSA recently developed by the authors, which is especially tailored to shakedown analysis problems for von Mises-type materials. Founded on a previous interior-point algorithm IPDCA (Akoa et al., 2007; Hachemi et al., 2005; Hachemi et al., 2009), which has been developed for elastic-perfectly plastic engineering problems with either one or two varying loads, the new algorithm is distinguished by a particularly problem-oriented solution strategy (Simon
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and Weichert, 2011; Simon et al., 2012). Moreover, IPSA is capable to solve shakedown problems with multidimensional loading spaces (Simon and Weichert, 2012), such that the examined structures can be subjected to arbitrary numbers of loadings. In this work, the generally-nonlinear limited kinematical hardening is incorporated into the method by using a two-surface model. The extended procedure is applied to several numerical examples from the field of power plant engineering. Among these, the computation of the shakedown domain for a flanged pipe subjected to three independently varying thermo-mechanical loads should be highlighted, because this is the first presentation of a three-dimensional shakedown domain at all, in which hardening is taken into account. 2. Lower bound shakedown analysis accounting for nonlinear limited kinematical hardening In the following, an elastic-perfectly plastic body K with volume V and surface A is considered, which is subjected to: temperature loads Tðx; tÞ in V, body forces f V ðx; tÞ in V, surface loads f A ðx; tÞ on Af # A, and prescribed displacements uðx; tÞ on Au # A, such that A ¼ Af \ Au and Af [ Au ¼ ;. Only time- and temperature-independent material behavior is taken into account, and material damage as well as geometrical nonlinearity are neglected. The existence of a convex yield function fY ½rðx; tÞ; rY ðxÞ and the validity of the normality rule are assumed. Then, the elastic limit is described by a yield surface in stress space S as closure of the convex domain CY # S of admissible states of stress with the strict interior CiY :
CiY ¼ fr 2 S j f Y ½rðx; tÞ; rY ðxÞ < 0; 8x 2 V; 8t g
ð1Þ
2.1. Melan’s statical shakedown theorem for elastic-perfectly plastic materials The current formulation is based on the statical shakedown theorem by Melan (1938a,b), which provides a lower bound to the shakedown loading factor. For this, the total stress rðx; tÞ in a point x 2 V at time t is decomposed into an elastic reference stress rE ðx; tÞ and a residual stress qðx; tÞ induced by the evolution of plastic strains:
rðx; tÞ ¼ rE ðx; tÞ þ qðx; tÞ
ð2Þ E
E
Here, r ðx; tÞ denotes the stress state, which would occur in a fictitious purely elastic reference body K under the same conditions as the original one. Both the elastic reference stresses and the residual stresses satisfy the equilibrium constraints as well as the statical boundary conditions (bc):
equilibrium : r rE ¼ f V statical bc : n rE ¼ fA
r q ¼ 0 in V n q ¼ 0 on Af
ð3Þ ð4Þ
Then, Melan’s shakedown theorem for elastic-perfectly plastic materials can be formulated as follows: ðxÞ, such that the yield condition fY 6 0 is If there exist a loading factor a > 1 and a time-independent residual stress field q satisfied for any loading path within the considered loading domain X at any time t and in any point x of the structure, then the system will shake down:
ðxÞ; rY ðxÞ 6 0; fY arE ðx; tÞ þ q
8x 2 V; 8t
ð5Þ
Noteworthy, the numerical procedure allows for computing values a < 1 as long as a is positive. However, from the proof by Melan it is clear, that shakedown only can be guaranteed if a > 1 holds, because only then the plastic dissipative energy is guaranteed to be bounded. 2.2. Extended shakedown theorem accounting for limited kinematical hardening To extend the statical shakedown theorem in order to take into account limited kinematical hardening, a two-surface model is used, which initially has been proposed by Weichert and Groß-Weege (1988). Therein, kinematical hardening is considered as a translational motion of the yield surface – described by fY ¼ 0 – in stress space without change of orientation, form or size. The limitation of this kinematical hardening is captured through introducing a bounding surface, fH ¼ 0, which corresponds to the ultimate stress rH . The translational motion is defined by the six-dimensional vector of back-stresses p representing the movement of the yield surface’s center, Fig. 1. Thereby, the total stresses rðx; tÞ are decomposed into the back stresses p and the reduced stresses t, which are responsible for the occurrence of plastic strains:
rðx; tÞ ¼ pðx; tÞ þ tðx; tÞ
ð6Þ
Since the split of stresses into the elastic reference and the residual part (2) still holds, the reduced stresses tðx; tÞ can be expressed as follows:
tðx; tÞ ¼ rðx; tÞ p ðxÞ ¼ rE ðx; tÞ þ q ðxÞ p ðxÞ
ð7Þ
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Fig. 1. Kinematic hardening considered as translation of the yield surface in stress space.
ðxÞ, since the Noticeably, in (7) the fact that the back-stresses are time-independent has already been incorporated, p ¼ p bounding surface is fixed in stress space. Finally, the extended shakedown theorem taking into account limited kinematical hardening reads: and a time-independent field of If there exist a loading factor a > 1, a time-independent self-equilibrated (residual) stress field q , such that the yield condition fY 6 0 and the bounding condition fH 6 0 are satisfied for any loading path within back-stresses p the considered loading domain X at any time t and in any point x of the structure, then the system will shake down:
ðxÞ p ðxÞ; rY ðxÞ 6 0 fY arE ðx; tÞ þ q ðxÞ; rH ðxÞ 6 0 fH arE ðx; tÞ þ q
ð8Þ ð9Þ
This theorem has been proven initially by Weichert and Groß-Weege (1988) for linear limited kinematical hardening. In particular, the proof is given within the framework of the Generalized Standard Material Model (Halphen and Nguyen, 1975), which is based on a linear kinematical hardening rule. It should be mentioned, that the consideration of specific hardening . These restrictions have to be dealt with rules – such as the linear one – implies restrictions on the field of back-stresses p carefully, as will be shown in the closure (Section 5). However, the same formulation can be derived for generally-nonlinear hardening laws, as shown by Pham (2008). The only restriction on the considered hardening is the positive hysteresis postulate, which states that for any closed cycle of plastic deformations (t 2 ½0; h) the following condition has to hold:
I
p : depp ¼
Z 0
h
p : epp dt P 0
epp ð0Þ ¼ epp ðhÞ
ð10Þ
where p denotes the back-stresses, and epp denotes the corresponding plastic deformation. In the case of a simple loading– unloading closed plastic cycle, this restriction implicates that the hysteresis loop is followed in clockwise direction, but not anti-clockwise. Although this postulate seems to be self-evident, it could not be proven yet. However, in the present paper, it is assumed that the positive hysteresis postulate (10) is always satisfied. 2.3. Description of the loading domain In the following, only loading histories Hðx; tÞ are considered, which can be described as superposition of finite numbers NL of different loading sets P ‘ ðx; tÞ. Then, load multipliers l‘ ðtÞ can be introduced for any loading case ‘ reflecting the timedependence of the loading. Here, all loads are normalized by the unity load P0 ðxÞ:
Hðx; tÞ ¼
NL NL X X P‘ ðx; tÞ ¼ l‘ ðtÞ P0 ðxÞ ‘¼1
ð11Þ
‘¼1
As shown by König (1987), it is sufficient to consider only the convex hull of the loading history, which – in consequence of (11) – is polyhedral with NC ¼ 2NL corners. To define these corners in the loading space, the bounding values lþ ‘ and l‘ of each multiplier l‘ are introduced. Thereby, the set U of all possible combinations of loading sets within these bounds can be defined through merging all loading multipliers to the vector l ¼ l‘ e‘ :
U ¼ l 2 RNL j l‘ 6 l‘ 6 lþ‘ ; 8‘ 2 ½1; NL
ð12Þ
J.-W. Simon / International Journal of Plasticity 42 (2013) 141–167
145
Then, the loading domain X is described as set of all possible loading histories contained within U:
(
X¼
Hðx; tÞ j Hðx; tÞ ¼
NL X
)
l‘ ðtÞ P0 ðxÞ; 8l 2 U
ð13Þ
‘¼1
Consequently, the elastic reference stresses are split in analogy to (11):
rE ðx; tÞ ¼
NL X
l‘ ðtÞ rE‘ ðxÞ
ð14Þ
‘¼1
2.4. Discretization Using the finite element method (FEM), the stresses are approximately represented by their values in the Gaussian points, which will be referred to by the index r 2 ½1; NG, where NG is the total number of Gaussian points in the system. Then, the fictitious elastic stresses rEr;‘ can be computed for any loading case ‘ by purely elastic analysis:
rEr ðtÞ ¼
NL X
l‘ ðtÞ rEr;‘
ð15Þ
‘¼1
In order to ensure shakedown for all possible loading paths inside of X, it is sufficient to examine only its corners. Thus, the time-dependence of rEr can be expressed through the stress states in the corners j 2 ½1; NC of the loading domain. To do so, the matrix U NL 2 RNCNL with entries U j‘ is introduced, where j 2 ½1; NC and ‘ 2 ½1; NL:
rE;j r ¼
NL X
U j‘ rEr;‘
ð16Þ
‘¼1
Each row of these matrices U NL represents the coordinates of one corner of the loading domain in the NL-dimensional þ loading space, which are defined by the factors l ‘ and l‘ as introduced in (12). According to Simon and Weichert (2012), the U NL can be defined for arbitrary numbers of loading cases NL simply by arranging the corners of the loading domain in a specific order. This is illustrated by the simple case of three independent loads, NL ¼ 3. The according domain in the three-dimensional loading space is illustrated in Fig. 2 with the associated matrix U 3 (17):
ð17Þ The third column consists of two divisions of length 4, where l is the value of all entries in the first part, and in the second part all entries have the value l 3 . Such a substructure is denoted by block in the following. Then, the second column can be divided into two blocks having two entries lþ 2 and l2 each, whereas the first column comprises four blocks with one entry þ l1 and l1 each. This ordering scheme can be generalized for the case of arbitrary finite numbers of loadings NL. The last column of the associated matrix U NL 2 RNCNL consists of one block with NC=2 entries lþ NL and lNL . The penultimate column þ is composed of two blocks with NC=4 entries lNL1 and lNL1 , and so on. Finally, the first column can be divided into NC=2 NCNL blocks, where each of the blocks consists of one pair lþ can be constructed 1 and l1 only. Thereby, the matrices U NL 2 R column-wise in the following way: þ 3
Fig. 2. Loading domain in a three-dimensional loading space.
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J.-W. Simon / International Journal of Plasticity 42 (2013) 141–167
For l ¼ 1; 2; . . . ; NL do the following: in the considered column l write 2NLl blocks one below the other, where each block consists of 2l1 entries with the maximum value lþ followed by 2l1 entries with the minimum value l l l Thereby, the elastic reference stresses are appropriately represented in the discretized formulation. For the discretization of the residual stresses, the equilibrium condition (3) is recalled. Since the elastic reference stress field rE is in equilibrium has to be self-equilibrated. This can be expressed using the principle of with the external loading, the residual stress field q virtual work (Groß-Weege, 1997), where de denotes any virtual strain field which satisfies the kinematical boundary conditions:
Z
dV ¼ 0 de : q
ð18Þ
V
Using the FEM with isoparametric elements, the displacements u are approximated by appropriate shape functions and nodal displacements uK . Introducing the differentiation matrix BðxÞ, the strain field e can be expressed by uK as well:
1 2
e ¼ ð$u þ u$Þ ¼ BðxÞ uK
ð19Þ
Then, the principle of virtual work (18) reads as follows:
Z
dV ¼ duK de : q
Z
V
dV ¼ 0 BðxÞ : q
V
)
Z
dV ¼ 0 BðxÞ : q
ð20Þ
V
This integration is carried out numerically using Gaussian points GP with Cartesian GP x and natural coordinates GP n, respectively, and weighting factors wi . The transformation between the coordinate systems is given by the Jacobian J ¼ @x=@n:
Z
dV ¼ 00 BðxÞ : q
V
NE X
NGE X 00
GP xji wi det J j GP nji B GP xji q
ð21Þ
j¼1 i¼1
P Here, 00 00 represents the transition from the element-level to the system-level. Thereby (18) is approximated by a system r in the Gaussian points: of linear equations for the residual stresses q
Z
dV ¼: BðxÞ : q
V
NG X r ¼ 0 Cr q
ð22Þ
r¼1
The equilibrium matrices Cr 2 RmE 6 depend only on the geometry of the system and the applied element type and take into account the kinematical boundary conditions. Their dimension is mE ¼ 3 NK NBC, where NK is the total number of nodes and NBC the number of kinematical boundary conditions. 2.5. Resulting nonlinear optimization problem Using (16) and (22), Melan’s theorem accounting for limited kinematical hardening can be given in terms of an optimization problem for the loading factor a > 1:
ðP H Þ aSD ¼ max a q ; p
NG X r ¼ 0 Cr q
ð23aÞ
r¼1
8j 2 ½1; NC; 8r 2 ½1; NG f Y arE;j r þ qr pr ; rY;r 6 0; f H arE;j þ q ; r 8 j 2 ½1; NC; 8r 2 ½1; NG 6 0; r H;r r
ð23bÞ ð23cÞ
It is worth to mention, that due to the discretization the bounding properties of the computed shakedown factor may not hold anymore in the strict sense. 2.6. Tailoring to von Mises criterion In this work, both the yield surface f Y ðt; rY Þ as well as the bounding surface f H ðr; rH Þ are described by the von Mises criterion, which can be written as follows, where d and r are used as placeholders:
h i f ðd; rÞ ¼ ðd1 d2 Þ2 þ ðd2 d3 Þ2 þ ðd3 d1 Þ2 þ 6 ðd4 Þ2 þ ðd5 Þ2 þ ðd6 Þ2 2 r2 As proposed by Akoa et al. (2007), the following constant transformation matrices are introduced:
ð24Þ
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0 pffiffiffi pffiffiffi pffiffiffi 6 6 6 p ffiffiffi pffiffiffi pffiffiffi B pffiffiffi B 6 6 6 C B B pffiffiffi pffiffiffi 3 C B1 B 6 pffiffiffi pffiffiffi C 1 B 1 6 6 C and T ¼ pffiffiffi B L ¼ pffiffiffi B 2 B C p ffiffiffi B 2 6 2B 2 C B B 2 A @ B pffiffiffi 2 @ 2 0
1
1
2
C C C C C C C C C A
ð25Þ
2
2 R65 the matrix T without its third column T 3 2 R6 , and by T 1 2 R56 the matrix T 1 without its Further, denoting by T 1 6 third row T 3 2 R , the criterion (24) can be reformulated as follows: 2
f ðd; rÞ ¼ kdk2 2 r2
where : d ¼ LT T 1 d
ð26Þ
Note, that this mathematical transformation from the six-dimensional vector d to the five-dimensional one d can be justified from physical point of view as well, because through this transformation one of the components is extracted, which corresponds to the hydrostatic pressure and thus has no influence on the von Mises criterion. The according formulations for the yield and the bounding condition are achieved by substitution of rjr and tjr for the placeholder d:
2 f ujr ; rH ¼ ujr 2 2 r2H where : ujr ¼ LT T 1 rjr 2 f m jr ; rY ¼ m jr 2 2 r2Y where : m jr ¼ LT T 1 tjr
ð27Þ ð28Þ
Then, the condition (22) for the residual stresses has to be transformed in the same manner: NG NG X X r ¼ Cr q Cr rjr arrE;j ¼ 0 r¼1
)
~ u1 þ B ~ v ab ¼ 0 A
ð29Þ
r¼1
Here, the extracted sixth component of the transformed stress vector in the first corner of the loading domain, j ¼ 1, is denoted by v :
0
v
1 T 1 3 r1
B 1 1 B T 3 r2 B ¼B .. B @ .
1 T 1 3 rNG
1
C C C C 2 RNG C A
ð30Þ
Furthermore, the following abbreviations have been used. For details we refer to Simon and Weichert (2011):
T u1 ¼ u11 ; . . . ; u1r ; . . . ; u1NG
ð31Þ
NG X b¼ Cr rE;1 r
ð32Þ
r¼1
h
~ ¼ C1 T LT j . . . jCr T LT j . . . jCNG T LT A
i
ð33Þ
~ ¼ ½C1 T 3 j . . . jCr T 3 j . . . jCNG T 3 B
ð34Þ
in (29), the information that the residual stresses are time-independent gets lost and has to be reinBy substituting for q implies an independence of the considered troduced to the problem as an additional constraint. The time-independence of q corner j of the loading domain.
q r ¼ rjr arE;j r ¼ constðjÞ
ð35Þ
This condition has to hold for all j 2 ½1; NC. Thus, it can be used to link the stresses of different corners of the loading domain, e.g. the corners j and j þ 1. Doing so, the following relation is achieved:
r ¼ rjr arE;j rjþ1 arE;jþ1 ¼q r r r E;jþ1 ! rjþ1 ¼ rjr a rE;j r r rr
ð36aÞ ð36bÞ
Recalling the transformation (27), this leads to an additional constraint for the variables
ujþ1 ¼ ujr acjr r
E;jþ1 where : cjr ¼ LT T 1 rE;j r rr
ujr :
ð37Þ j r
In the same manner, an additional constraint for the variables m can be given representing the fact that the back stresses
p r are time-independent, too,
mjþ1 ¼ m jr acjr r
ð38Þ
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Finally, the optimization problem resulting from the statical shakedown theorem for limited kinematical hardening von Mises materials is formulated as follows:
ðP Hv M Þ aSD ¼ max a u; m; v
~ u1 þ B ~ v ab ¼ 0 A
ð39aÞ
ujþ1 r
ð39bÞ
ujr
j r;
¼ ac 8j 2 ½1; NC 1; 8r 2 ½1; NG j j mjþ1 ¼ m ac ; 8 j 2 ½1; NC 1; 8r 2 ½1; NG r r r j 2 u 2 r2 6 0; r 2 H;r j 2 m 2 r2 6 0; r 2 Y;r
ð39cÞ
8j 2 ½1; NC; 8r 2 ½1; NG
ð39dÞ
8j 2 ½1; NC; 8r 2 ½1; NG
ð39eÞ
3. Solving the optimization problem by use of the interior-point method For the purpose of a clear presentation, the problem is rewritten in the following concise form:
ðP HIP Þ
min f ðxÞ ¼ a AH x ¼ 0
ð40aÞ 2 H;r
c H ðxÞ ¼ 2 r
2 Y;r
c Y ðxÞ ¼ 2 r x2R
2 ujr 2 P 0 2 m j P 0
ð40bÞ ð40cÞ
r 2
n
ð40dÞ
where the variables of the problem are merged to the solution vector x of dimension n:
T 1 1 j NC x ¼ u11 ; u12 ; . . . ; ujr ; . . . ; uNC 2 Rn NG ; m 1 ; m 2 ; . . . ; m r ; . . . ; m NG ; v ; a
ð41Þ
ðP HIP Þ
The problem consists of mE equality constraints (39a)–(39c), represented by the system of Eq. (40a), and 2 mI nonlinear inequality constraints (40b) and (40c). Since the objective function is linear, the system of equality constraints is affine linear, and the inequality constraints are concave, the optimization problem is strictly convex. The dimensions of the problem are given by:
n ¼ 10 NC NG þ NG þ 1
ð42Þ
mE ¼ mE þ 10NG ðNC 1Þ
ð43Þ
mI ¼ NC NG
ð44Þ mI
The inequality constraints are converted into equality constraints by introducing slack variables wH 2 R and wY 2 RmI . Moreover, split variables y 2 Rn and z 2 Rn are utilized in order to avoid numerical instabilities due to the unboundedness of the solution vector (40d). In addition, as a key-idea of the interior-point method, the objective function is perturbed by logarithmic barrier terms. The latter shall penalize those directions, which would lead outside of the feasible region. Thereby, the barrier parameter l is introduced, which is a sequence tending to zero during the iteration:
f l ðx; y; z; wH ; wY Þ ¼ f ðxÞ l
" n X
logðyi Þ þ
i¼1
n X i¼1
logðzi Þ þ
mI X
logðwH;j Þ þ
j¼1
mI X
# logðwY;j Þ
ð45Þ
j¼1
The resulting optimization problem can then be expressed as follows.
ðP Hl Þ
min f l ðx; y; z; wH ; wY Þ AH x ¼ 0
ð46aÞ
c H ðxÞ wH ¼ 0 c Y ðxÞ wY ¼ 0
ð46bÞ ð46cÞ
xyþz ¼0
ð46dÞ
wH > 0; wY > 0; y > 0; z > 0
ð46eÞ
This optimization problem is convex, as the original one, and hence the Karush–Kuhn–Tucker condition is both necessary and sufficient. This condition states that the solution is optimal, if and only if the Lagrangian of the problem possesses a saddle point. In particular, the Lagrangian LH of ðP Hl Þ can be expressed as follows:
LH ¼ f l kE ðAH xÞ kH ðc H ðxÞ wH Þ kY ðc Y ðxÞ wY Þ s ðx y þ zÞ; mE
where kE 2 R ; kH 2 evaluated as
I Rm þ ; kY
2
I Rm þ
and s 2
Rnþ
ð47Þ
are appropriate Lagrange multipliers. The saddle point condition can then be
J.-W. Simon / International Journal of Plasticity 42 (2013) 141–167
rP LH ðPÞ ¼ 0
149
ð48Þ
where P ¼ ½x; y; z; wH ; wY ; kE ; kH ; kY ; sT denotes the vector of all variables included in this problem. Eq. (48) constitutes a system of nonlinear equations, which is solved approximately by use of the Newton’s method. The variables Pkþ1 of the subsequent iteration step k þ 1 are computed from the variables Pk of the previous one k and the step values DPk as follows:
Pkþ1 ¼ Pk þ !k DPk
ð49Þ
where !k denotes a diagonal matrix of damping factors, which is introduced for numerical reasons. The step values DPk are determined from the following linearized system of equations:
JðPk Þ DPk ¼ rP LH ðPk Þ
ð50Þ
where : JðPk Þ ¼ rP LH ðPÞ rP jP¼Pk In each iteration step, the condensed linearized system of Eq. (50) is solved for the step values DPk . Then, an inner loop is used to ensure that this solution is sufficiently accurate for the original nonlinear system. In case of negative components in the slack or slip variables or in the Lagrange multipliers of the inequality constraints, the computed step is damped to satisfy the non-negativity conditions. Further damping may be necessary, which is done by a linesearch procedure using the ‘2 -merit function. Once the damped step values are computed, the new variables can be easily determined from (49). With these, the break condition is checked based on appropriate convergence criteria. If the solution is not yet converged, the barrier parameter l is decreased and the next iteration step is entered. For further descriptions of the numerical procedure, the reader is referred to Simon and Weichert (2011). 4. Numerical examples The described method was applied to the following three examples from the field of power plant engineering: (1) a plate subjected to tension and a Bree-type temperature distribution, (2) a thin pipe subjected to internal pressure and temperature, (3) a flanged pipe subjected to internal pressure and an axial force. In each of these examples, two-dimensional loading spaces were considered, such that the loads varied independently in the following ranges:
0 6 P1 6 lþ1 P 0
ð51aÞ
0 6 P2 6 lþ2 P 0
ð51bÞ
þ þ þ The larger one of the two multipliers lþ 1 and l2 was chosen equal to 1, and the other one was scaled to the ratio l1 =l2 or þ l l1 , respectively. To illustrate the potential of solving problems with n-dimensional loading spaces, the flanged pipe was finally considered in a three-dimensional loading situation by adding an internal temperature load. For the three systems, the considered material was assumed to be homogeneous isotropic equipped with the material parameters of steel X6CrNiNb 18–10, see Table 1. These parameters were assumed to be temperature-independent. Furthermore, only steady-state processes were considered, and no transient thermal effects were taken into account. In addition, creep due to high temperature was not incorporated. In all calculations, the FEM-analyses were carried out with the software package ANSYS using isoparametric solid elements with 8 nodes. Since these elements are known to show relatively poor performance in thin shell-like applications, several elements were used over the thickness (three elements in the first example Section 4.1, five elements in the second one Section 4.2, and a varying number up to six elements in the third one Section 4.3). For all three meshes, convergence studies showed that the change of the maximum value of equivalent stresses is less than 1% when using more elements over the thickness. Worth mentioning, shell elements could be used alternatively. The presented method allows for use of a number of different element types, but then the computation of the equilibrium matrices Cr needs to be adapted to the element formulation. These were computed by a user-defined subroutine UEL in ANSYS. For the temperature loading cases, the FEM-analyses was carried out in two steps: (1) using the hexahedral thermal element solid70, the body temperature distribution was computed resulting from prescribed temperature bounding conditions þ 2=
Table 1 Thermal and mechanical characteristics. Young’s modulus [MPa] Yield stress [MPa] Poisson’s ratio Density [kg/m3] Thermal conductivity [W/(m K)] Specific heat capacity [J/(kg K)] Coefficient of thermal expansion [1/K]
2:0 105 205 0.3 7:9 103 15 500 1:6 105
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at the inner and the outer surface, respectively; (2) based on the body temperature distribution, nodal temperature loadings were defined for the structural analysis with element solid185, leading to the respective equivalent elastic stress distribution. For all computations a Dell Precision T7500 with Xeon E5620-processor with 2400 MHz and 12 GB RAM was used. 4.1. Plate under thermo-mechanical loading In the first example, a plate was subjected to a normal traction p in x-direction and a linear temperature distribution across its width, see Fig. 3. The illustrated distribution of equivalent elastic stresses resulting from the temperature load was calculated with the arbitrary value DT ¼ T 1 T 0 ¼ 100 K. Taking into account the system’s symmetry, the mesh consisted of 676 nodes and 432 elements, where three elements over the thickness were used. For the normal traction, the arbitrary value p = 100 MPa was chosen. This leaded to a homogeneous elastic stress field with rx ¼ p and ry ¼ m p, whereas the remaining stress components were zero, which resulted in an equivalent elastic stress of 88.882 MPa. As result of the shakedown analysis, Fig. 4 presents: (i) the elastic limit domain (dotted line with ); (ii) the shakedown domains without consideration of hardening for the yield stress rY ¼ 205 MPa (solid line with }) and for some multiples of the yield stress rY;1 ¼ 1:25 rY ; rY;2 ¼ 1:5 rY and rY;3 ¼ 1:75 rY (dash-dot lines with ; , and H, respectively); (iii) the shakedown domains including hardening for different values of ultimate stresses rH;1 ¼ 1:25 rY ; rH;2 ¼ 1:5 rY and rH;3 ¼ 1:75 rY (solid lines with ; , and H, respectively); (iv) the shakedown domain for unlimited kinematical hardening (solid line with ). The plot’s axes are scaled to the according value p0 and DT 0 , respectively, for perfectly plastic material behavior. To validate these results, they were compared to those given by Heitzer et al. (2000), Schwabe (2000) and Mouhtamid (2007), see Fig. 5. Whilst all of these works were based on the static approach as well, they differed in the chosen solution strategies: Heitzer et al. (2000) applied the basis reduction technique, whereas Schwabe (2000) used the program LANCELOT
Fig. 3. System and equivalent elastic stresses due to temperature loading.
Fig. 4. Results of shakedown analysis of the plate.
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(Conn et al., 1992) based on the augmented Lagrangian method, and Mouhtamid (2007) used the interior-point algorithm IPDCA (Akoa et al., 2007), which can be considered as the precursor of the one presented herein. Noteworthy, in these reference solutions only unlimited hardening as well as limited hardening for the case rH ¼ 1:5 rY were investigated.
4.2. Thin pipe subjected to thermo-mechanical loading As a second example, the proposed method was applied to a thin pipe subjected to an internal pressure p and a temperature load DT ¼ T 1 T 0 , which varied independently of each other, Fig. 6. The pipe was assumed to be long, open-ended and thin with a ratio of radius to thickness R=h ¼ 10. Using five elements over the thickness and taking into account the system’s symmetry, the mesh consisted of 984 nodes and 600 elements, see Fig. 7(a). The resulting distributions of equivalent elastic stresses are presented in Fig. 7, where the arbitrarily chosen values p ¼ 10 MPa and DT ¼ 100 K have been used. As in the previous example, the following results of the shakedown analysis are presented in Fig. 8: (i) the elastic limit domain and twice the elastic limit domain (dotted lines with and , respectively); (ii) the shakedown domains without consideration of hardening for the yield stress rY ¼ 205 MPa (solid line with }) and for some multiples of the yield stress rY;1 ¼ 1:2 rY ; rY;2 ¼ 1:35 rY and rY;3 ¼ 1:5 rY (dash-dot lines with j; , and H, respectively); (iii) the shakedown domains including hardening for different values of ultimate stresses rH;1 ¼ 1:2 rY ; rH;2 ¼ 1:35 rY and rH;3 ¼ 1:5 rY (solid lines with ; , and H, respectively); (iv) the shakedown domain for unlimited kinematical hardening (solid line with ). The plot’s axes are scaled to the according value p0 and DT 0 , respectively, for perfectly plastic material behavior.
Fig. 5. Validation of numerical results for the plate.
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Fig. 6. System of the open-ended pipe.
Fig. 7. Model and elastic equivalent stresses of the open-ended pipe.
Fig. 8. Results of shakedown analysis of the open-ended pipe.
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153
For validation purposes, these results were compared to those from the literature, see Fig. 9: The perfectly plastic and unlimited hardening case were investigated by Mouhtamid (2007) using the interior-point algorithm IPDCA (Akoa et al., 2007). Furthermore, limited hardening for rH ¼ 1:2 rY was considered by Hachemi (2005), who applied the BFGS algorithm (Morris, 1982), and for rH ¼ 1:35 rY by Heitzer et al. (2000) utilizing the basis reduction technique. 4.3. Flanged pipe under internal pressure, axial force and temperature loading The third example was a flanged pipe with three different outer radii, see Fig. 10(a), already considered by Mouhtamid (2007), Weichert et al. (2008) and Weichert and Hachemi (2010). The dimensions were adopted from Mouhtamid (2007), see Table 2. Taking advantage of the system’s rotational symmetry – as shown in Fig. 10(a), – the applied mesh consisted of 265 elements and 678 nodes, where one element across the thickness was used. 4.3.1. Two-dimensional loading space At first, the system was investigated in a two-dimensional loading space. In particular, the pipe was subjected to an internal pressure p and an axial force Q, which varied independently from each other in the ranges p 2 ½0; pmax and Q 2 ½0; Q max , respectively. In order to compute the elastic stresses presented in Fig. 10(c) and (d), the arbitrary values p ¼ 10 MPa and Q ¼ 113:097 kN were applied, respectively. Again, the following results were obtained from the shakedown analysis, see Fig. 11: (i) the elastic limit domain and twice the elastic limit domain (dotted lines with and , respectively); (ii) the shakedown domains without consideration of hard-
Fig. 9. Validation of numerical results for the open-ended pipe.
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Fig. 10. System, model and equivalent elastic stresses for the flanged pipe.
Table 2 Dimensions of flanged pipe. Length L in [mm] Inner radius Ri in [mm] Outer radius Ra;1 in [mm] Outer radius Ra;2 in [mm] Outer radius Ra;3 in [mm]
386.9 60.0 68.1 77.8 90.5
ening for the yield stress rY ¼ 205 MPa (solid line with }) and for some multiples of the yield stress rY;1 ¼ 1:25 rY and rY;2 ¼ 1:5 rY (dash-dot lines with and , respectively); (iii) the shakedown domains including hardening for different values of ultimate stresses rH;1 ¼ 1:25 rY and rH;2 ¼ 1:5 rY (solid lines with and , respectively); (iv) the shakedown domain for unlimited kinematical hardening (solid line with H). The plot’s axes are scaled to the according value p0 and Q 0 , respectively, for perfectly plastic material behavior. 4.3.2. Three-dimensional loading space To illustrate the influence of hardening in a three-dimensional loading space, a temperature load DT 2 ½0; DT max was applied additionally. In particular, the body temperature distribution was computed resulting from prescribed temperature bounding conditions of T i ¼ 100 K and T 0 ¼ 20 K at the inner and the outer surface of the pipe, respectively, see Fig. 13(a), resulting in the equivalent elastic stress distribution shown in Fig. 13(b). Applying the proposed algorithm, the three-dimensional shakedown domain was computed for elastic-perfectly plastic material, see Fig. 14. Further, the influence of limited kinematical hardening was investigated by calculations with different ultimate stresses: rH ¼ 1:1 rY (Fig. 15 and blue dash-dot line1), rH ¼ 1:25 rY (Fig. 16 and blue dashed line), and rH ¼ 1:5 rY (Fig. 17 and black dash-dot line). Subsequently, the domain was determined for unlimited hardening (Fig. 18 and black solid line). 4.4. Discussion of the results It can be summarized that the presented results are in agreement with the reference solutions. In particular, for the plate, the only difference can be observed for the perfectly plastic shakedown domain, where IPSA captures the transition from one mechanism to the other in a sharper way than any of the reference solutions, see Fig. 5. Concerning the pipe, the results particularly agree with those of Heitzer et al. (2000) – which are slightly lower in the case of hardening, see Fig. 9(c) – and Mouhtamid (2007), where the only discrepancy occurs in the regime of predominating pressure, see Fig. 9(a). By contrast, the computed shakedown domains of Hachemi (2005) are above the ones computed by the presented method, Fig. 9(b). Consequently, the shakedown loads are overestimated in both cases with and without considering hardening. Nevertheless, the curves are qualitatively similar as well as the inclinations at the intercept points. 1
For interpretation of color in Figs. 14–18 and 22, the reader is referred to the web version of this article.
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155
Fig. 11. Results of shakedown analysis of the flanged pipe.
Fig. 12. Comparison with results from Mouhtamid (2007).
For the flanged pipe, matching of the results is satisfying, especially for limited kinematical hardening with rH ¼ 1:5 rY , see Fig. 12. However, slight differences exist resulting from different elastic solutions, which can be explained by the use of different meshes. In particular, in the current calculation the mesh refinement around the stress concentration point has been improved, leading to a maximum equivalent stress of 100.143 MPa under axial force, whereas the mesh used in (Mouhtamid, 2007) yields 106.465 MPa. This example illustrates the importance of an accurate elastic solution, because the effect on the shakedown load is clearly observable, especially in the case of unlimited hardening, see Fig. 12. Another important issue is the type of the failure mechanism. Generally, the shakedown domain is defined by either the incremental collapse criterion or the alternating plasticity criterion (instantaneous collapse is then automatically included). In principle, in a two-dimensional loading space, this allows for three possible situations: (i) the incremental collapse crite-
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Fig. 13. Body temperature distribution and resulting equivalent elastic stresses.
Fig. 14. Three-dimensional shakedown domain without hardening.
rion is decisive in the whole loading domain; (ii) the alternating plasticity criterion is decisive in the whole loading domain; (iii) there exist two separate regions, one with incremental collapse and the other with alternating plasticity. To achieve the third situation in the presented examples, the loadings have been particularly chosen, such that one of them enforces failure due to incremental collapse, while the other one enforces failure due to alternating plasticity. Note, that the simplest way to enforce alternating plasticity is to choose a thermal loading. Consequently, one can clearly distinguish the two mechanisms of alternating plasticity and incremental collapse in all presented examples, in both the perfectly plastic and the hardening case. In particular, in the two-dimensional loading situations, alternating plasticity occurs in the regime of predominating temperature for the plate and the pipe, and in the regime of predominating axial force for the flanged pipe (see Figs. 4, 8, and 11). Here, no influence of hardening can be observed. Hence, all shakedown curves – with and without hardening – coincide with the one for unlimited hardening. On the other hand, the respective second loading case leads to incremental collapse. The limited kinematical hardening influences the shakedown curves, such that the according domains increase in direct proportion with the ratio rH =rY . Thus, the limited hardening curves (solid lines) coincide with the ones without hardening but with premultiplied yield stress
J.-W. Simon / International Journal of Plasticity 42 (2013) 141–167
Fig. 15. Three-dimensional shakedown domain with hardening
rH ¼ 1:1 rY .
Fig. 16. Three-dimensional shakedown domain with hardening
rH ¼ 1:25 rY .
157
(dash-dot lines) in this range. Noteworthy, in all cases, the two curves representing the two different mechanisms pass into each other seamlessly. These results are in perfect agreement with the ‘‘simplified theorems’’ formulated by Pham (2007), where the alternating plasticity problem and the incremental collapse problem are separated:
aSD ¼ min faAP ; aIC g aAP ¼ sup a j a q0 þ rE 2 CY ; 8rE 2 L q0
E
aIC ¼ sup a j a q þ r 2 CH ; 8rE 2 L
ð52aÞ ð52bÞ ð52cÞ
q2R
In the above theorems, CY and CH denote the domains described by the yield surface and by the bounding surface, respectively, and L denotes the bounded loading domain. Note, that in (52b) q0 represents any time-independent stress field, whereas in (52c) the stress fields q are required to be time-independent as well as self-equilibrated, q 2 R. Clearly, shake-
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Fig. 17. Three-dimensional shakedown domain with hardening
rH ¼ 1:5 rY .
Fig. 18. Three-dimensional shakedown domain with unlimited hardening.
down is due to either alternating plasticity or incremental collapse, where the smaller loading factor is decisive. The theorem for incremental collapse (52c) is equivalent to the static theorem for the elastic-perfectly plastic material but with the yield stress rH . In this regard, the two mechanisms – again – can be clearly distinguished in the three-dimensional case as well. To highlight this, all computed points leading to alternating plasticity are marked by red circles in Figs. 14–18. In the regime of predominating temperature, all shakedown domains coincide, which implicates alternating plasticity to be decisive. Here, hardening does not affect the solution. By contrast, an influence of hardening can be observed in the regime of predominating axial force. While incremental collapse leads to failure in the elastic-perfectly plastic case as well as when considering limited hardening with rH ¼ 1:1 rY , further increasing the ultimate stress has no impact, because alternating plasticity occurs starting from rH ¼ 1:2 rY . Finally, when the internal pressure is superior, hardening enlarges the shakedown domain in direct proportion with the ratio rH =rY in all calculations with limited hardening. Only for unlimited hardening, its effect is restricted and alternating plasticity appears. Furthermore, it is worth to mention, that the unlimited hardening curve does only accord partly with double the elastic domain. Even so, it is frequently stated in the literature, that these curves have to accord in the whole domain, which simply
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J.-W. Simon / International Journal of Plasticity 42 (2013) 141–167 Table 3 Influence of hardening on numerical details for the flanged pipe. 2 independent loads
n mE mI £ Iterations £ CPU-time [s]a a
3 independent loads
Perfectly plastic
Hardening
Perfectly plastic
Hardening
44,521 33,834 8480
86,921 65,634 16,960
86,921 76,234 16,960
171,721 150,434 33,920
400 48
481 57
2318 295
3 017 837
Dell Precision T7500 with Xeon E5620-processor with 2400 MHz and 12 GB RAM.
is wrong. In fact, they have to coincide at the axis intercepts, because the origin of the loading space is one corner of the loading domain in the considered examples. Nevertheless, in the remaining domain, they may – but do not must – be the same. For illustration, the curve describing double the elastic domain is plotted additionally in Figs. 8 and 11 (dotted line with ). As one can see, this curve coincides with the alternating plasticity curve only at the axis intercepts for the pipe (Fig. 8), while for the flanged pipe two separate regions exist, in the one of which they do coincide and in the other they do not (Fig. 11). Finally, for the plate Fig. 4 the alternating plasticity curve and double the elastic domain match in the whole loading domain, which is the reason why the latter is not plotted at all. Closing, the characteristic numerical details for the flanged pipe are reported in Table 3. Noticeably, the number of iterations is not as much affected as the running time. Moreover, in the considered example, the number of loadings has a larger impact than the hardening, even though the numbers of variables and constraints are comparable. 5. Closure: effect of the nonlinearity In this section, the effect of the underlying limited kinematical hardening model on the shakedown domain is investigated. Noteworthy, it is frequently stated in literature that shakedown limits only depend on the initial yield stress rY and the ultimate stress rH , and not at all on the hardening behavior in between (see e.g. Weichert and Groß-Weege, 1988; Stein et al., 1992; Heitzer et al., 2000; Pham and Weichert, 2001; Abdel-Karim, 2005; Pham, 2007, 2008). However, this is widely accepted in limited linear kinematical hardening based on the associative hardening law. Nevertheless, as shown by Pycko and Maier (1995), in case of non-associative hardening models it is necessary to consider the plastic potentials additionally, leading to different formulations for the associative and the non-associative hardening model, respectively. In the following, the influence of the hardening model is addressed in more detail. It is worth to mention, that although the classical model of Armstrong and Frederick (1966) is used exemplarily, the arguments presented below hold in a similar manner for more advanced models, such as the ones presented by Ohno and Wang (1993), Chaboche (2008) or Abdel-Karim (2009). Even more, since the hardening law needs to be addressed explicitly even for the simple nonlinear model it is clear that this holds the more for more advanced ones. 5.1. Effect of the hardening model on the resulting optimization problem As has been shown in Section 2.2, the proposed method is based on the extended statical shakedown theorem, which is valid for any kind of kinematical hardening – both linear and nonlinear – satisfying the positive hysteresis postulate (10). Denoting by R the set of all time-independent and self-equilibrated stress fields and by B the set of all time-independent stress fields, which satisfy the positive hysteresis postulate, this theorem can be rewritten in a more formal way: and p , such that the following conditions hold, then the system will shake down: If there exist a scalar a > 1 and fields q
p 2 B
ð53aÞ
q 2 R
ð53bÞ
fY arE;j 8j 2 ½1; NC; 8r 2 ½1; NG r þ qr pr ; rY;r 6 0; E;j fH arr þ qr ; rH;r 6 0; 8j 2 ½1; NC; 8r 2 ½1; NG Comparison of this theorem with the optimization problem formulated in Section 2.5
ð53cÞ ð53dÞ
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ðP H Þ aSD ¼ max a q ; p
NG X r ¼ 0 Cr q r¼1
ð54aÞ
r p r ; rY;r 6 0; 8j 2 ½1; NC; 8r 2 ½1; NG f Y arE;j þq r f H arE;j þ q ; r 6 0; 8j 2 ½1; NC; 8r 2 ½1; NG r H;r r
ð54bÞ ð54cÞ
shows, that not all of the conditions (53a)–(53d) are reflected. Obviously, the yield and the bounding conditions (53c) and is self-equilibrated (53b). On the (53d) are represented by (54b) and (54c), respectively, while (54a) mirrors the fact that q contrary, the condition (53a) is not incorporated anymore. In fact, as long as the generally-nonlinear hardening case is con is not restricted by any specific kind of hardening law but only by the positive hysteresis possidered, and the evolution of p tulate, the optimization problem ðP H Þ is equivalent to (53) without loss of generality. Then the back-stresses can be , the solution of the optimiconsidered as unrestricted variables of the optimization problem. Since a is maximized over p , which leads to the maximum value of a. The according kinematical zation problem ðP H Þ involves the one particular field p to evolve under the given loading domain, is the most advantageous one, hardening rule, which allows the back-stresses p which is called unrestricted hardening in the following. As could be shown in Section 4.4, only the yield and bounding surface need to be defined by rY and rH , respectively, when the unrestricted hardening is considered using the optimization problem ðP H Þ, which is in accordance with the above mentioned references. Nevertheless, the shakedown limit may depend on the hardening behavior in between the initial yield state and the ultimate state, even though no hardening law is explicitly defined. However, if a specific hardening law is 2 B and B B. This restriction has to be to be considered, then the feasible set of back-stresses is restricted, such that p R B . Thus, included to the optimization problem additionally, because otherwise its solution might not be feasible, p the optimization problem needs to be modified:
ðP H Þ aSD ¼ max a q ; p
p 2 B
ð55aÞ
NG X r ¼ 0 Cr q
ð55bÞ
r¼1
8j 2 ½1; NC; 8r 2 ½1; NG f Y arE;j r þ qr pr ; rY;r 6 0; f H arE;j þ q ; r 8 j 2 ½1; NC; 8r 2 ½1; NG 6 0; r H;r r
ð55cÞ ð55dÞ
For example, if the special case of limited linear kinematical hardening shall be investigated, the set of feasible back can evolve under the given loading domain following stresses B has to be formulated such that the resulting solution p p _ _ the linear hardening law, p ¼ C e , where C is a material parameter. To include such additional restrictions to the optimization problem entails difficulties, because of the introduction of kinematic variables (e.g. plastic strains ep ) into the statical theorem, which is formulated in stresses. In any case, the generally-nonlinear hardening can be directly incorporated into the procedure as shown in the previous Sections. Noticeably, in all of the presented numerical results the back-stresses are not restricted and thus, the unrestricted hardening law is applied automatically. This is in perfect accordance with the seminal idea of the original shakedown theorems by Melan (1938a,b) and Koiter (1960). Following their spirit of direct approaches, neither the specific loading path nor the underlying hardening law need to be given. On the other hand, it cannot be expected to keep this beauty and simplicity of the theorems while considering more specialized cases at the same time. 5.2. Illustrative example: sample under constant tension and alternating torsion In order to illustrate the correlation between different hardening laws, a specimen is considered, which is subjected to a > 0 and alternating torsion s with zero mean shear stress, such that smin ¼ smax . The according loading constant tension r ; smax Þ and ðr ; smin Þ. This system has been previously examined by e.g. Portier et al. domain consists of only two points ðr (2000), where ratcheting has been investigated experimentally as well as numerically. Furthermore, numerical and analytical results of shakedown analysis taking into account different types of kinematical hardening are presented in Lemaitre and Chaboche (1990), De Saxcé et al. (2000), Staat and Heitzer (2002) and Heitzer et al. (2003) for the plain stress state, while the plain strain state has been investigated in Bouby et al. (2006, 2009). In the following, the plain stress state is considered:
0
1
0
1
X Y 0 r s 0 B B C r ¼ @ s 0 0C A and p ¼ @ Y 0 0 A 0
0 0
0
0
ð56Þ
0
are not As in this simple problem both the stress and the strain field are uniformly distributed, the residual stresses q considered. In fact, the back-stresses p play the role of residual stresses here. Then, the von Mises yield condition reads:
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X Þ2 þ 3ðs Y Þ2 r2Y ¼ 0 fY ðr p; rY Þ ¼ ðr
ð57Þ
; smax Þ and ðr ; smax Þ shall be located on the yield surface (57), one gets the following two equations: Since both maxima ðr
X Þ2 þ 3ðsmax Y Þ2 r2Y ¼ 0 ðr X Þ2 þ 3ðsmax Y Þ2 r2Y ¼ 0 ðr
ð58aÞ ð58bÞ
The difference between these equations, (58a) and (58b), gives:
ðsmax Y Þ2 ðsmax Y Þ2 ¼ 4 smax Y ¼ 0
ð59Þ
A non-trivial solution, smax – 0, can therefore only be obtained if Y ¼ 0. Thus, to reach the shakedown state, the yield surface is moved in stress space only in the direction of r. The yield condition simplifies to:
X Þ2 þ 3 s2 r2Y ¼ 0 fY ðr p; rY Þ ¼ ðr
ð60Þ
The positive solution of (60) is:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi XÞ2 3 s ¼ r2Y ðr
ð61Þ
Furthermore, the bounding condition can be written independently of the back-stresses, where
2
2
2 H
fH ðr; rH Þ ¼ r þ 3 s r ¼ 0
rH 6 2 rY is assumed: ð62Þ
This leads to the following maximum value of shear stresses:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 3 s ¼ r2H r
ð63Þ
As can be seen from (61), the shakedown load s can be dependent on the according back-stresses X. These, however, are restricted by the applied hardening law. To illustrate the influence of different kinematical hardening rules, the following types of plastic behavior are incorporated: 1. Perfectly plastic behavior: No hardening occurs and thus no back-stress evolves, X ¼ 0, leading to the following maximum value of admissible shear stress:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 3 s0 ¼ r2Y r
ðdashed line in Fig:22Þ
¼ 0, whereas the remaining shakedown domain represents incremental Alternating plasticity only occurs in pure shear, r collapse. 2. Limit load for proportional loading path: As described above, the shakedown domain is the same as in the perfectly plastic case substituting rH for rY . It is defined by (62), leading to the following maximum value of admissible shear stress:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 3 sL ¼ r2H r
ðblack solid line with þ in Fig:22Þ
3. Unlimited kinematical hardening: The limit state is defined solely by alternating plasticity. It can be obtained by any hardening rule setting rH ! 1. The : X¼r , leading to the following maximum value evolution of back-stresses is not restricted at all, and consequently 8r of admissible shear stress:
pffiffiffi 3 su ¼ rY
ðblack solid line with in Fig:22Þ
4. Limited unrestricted kinematical hardening: In accordance with the discussion in Section 4.4, the unrestricted kinematical hardening – as a combination of the the unlimited hardening case and the perfectly plastic one multiplied by the ratio rH =rY , which is the limit load in this particular example – consists of two different regions. The back-stress X is not restricted (besides the positive hysteresis qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : X¼r . This leads to alternating plasticity in case of r 6r ¼ r2H r2Y . However, if assumption), and thus 8r
r > r , the bounding condition enforces incremental collapse following (62) independently of the back-stresses, see Fig. 19. This leads to the following maximum value of admissible shear stress:
8 pffiffiffi < 3 su ¼ rY 6r if r pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 sadv ¼ pffiffiffi : 3 sL ¼ r2H r >r 2 if r
ðblue solid line with in Fig: 22Þ
It should be noticed, that the yield surface is allowed to partly move outside of the bounding surface, as long as the con ; smax Þ and ðr ; smin Þ stay inside. sidered stress points on the yield surface ðr
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Fig. 19. Kinematic hardening for constant tension and alternating torsion: limited unrestricted hardening.
5. Limited linear kinematical hardening: For the limited linear kinematical hardening, the hardening rule of Prager (1959) is applied: p_ ¼ C e_ p , where C denotes the kinematical hardening modulus and e_ p denotes the plastic strain rate. As shown by Stein et al. (1993) and Heitzer et al. (2003), the back-stresses are restricted by f ðp; rH rY Þ ¼ 0:
f ðp; rH rY Þ ¼ X 2 ðrH rY Þ2 ¼ 0
ð64Þ
Hence, the back-stress X 6 r cannot exceed the value r ¼ rH rY . In consequence, alternating plasticity can only occur 6r , because then X ¼ r is possible, see Fig. 20(a). On the contrary, for r >r , the restriction of the back-stresses if r , which enforces incremental collapse. This is illustrated in Fig. 20(b), where r < r
8 pffiffiffi < 3 su ¼ rY 6r if r pffiffiffi 3 sP ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 : r2 ð r r Þ if r >r Y
ðdotted line in Fig: 22Þ
It is worth to mention, that this is in agreement with the solution presented by Bodovillé and De Saxcé (2001) for a specific nonlinear Prager’s rule: p_ ¼ C e_ p ðc=CÞ2 X 2eq e_ p . The back-stress corresponding to the stabilized cycle has only to be ¼ C=c. replaced by r 6. Limited nonlinear kinematical hardening: Finally, for the limited nonlinear kinematical hardening the hardening rule of Armstrong and Frederick (1966) is used:
Fig. 20. Kinematic hardening for constant tension and alternating torsion: limited linear hardening.
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2 3
p_ ¼ C e_ p C
p _ p; where p_ ¼
X1
163
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p p e_ : e_ and X 1 ¼ r ¼ rH rY 3
For this hardening law, the considered example has been intensively investigated by several authors. The first solution has been proposed by Lemaitre and Chaboche (1990), followed by De Saxcé et al. (2000), where an analytical solution is derived and verified by an alternative theoretical calculation on the basis of the bipotential approach. In Bouby et al. (2006, 2009) an analytical solution is presented for the plain strain state, which can be transferred to the plain stress state considered here simply by setting m ¼ 0. Moreover, a numerical implementation is given, which is in perfect agreement with the analytical solution. Finally, Staat and Heitzer (2002) obtained equivalent results using a step-by-step finite element computation. In all these references, it turns out that the back-stresses are restricted even more than in the linear hardening case:
X¼r
rH rY rH
ð65Þ
: X
rY , this leads to 8r ¼ 0, as shown in Fig. 21. Consequently, a signifdomain, and alternating plasticity only can occur in case of pure shear, r icant influence of the restriction on the back-stresses can be obtained, which contradicts some statements presented in literature (see e.g. Abdel-Karim, 2005). The according maximum value of admissible shear stress reads:
pffiffiffi 3 sAF ¼
ffi rY qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2H r 2 rH
ðdash-dot line in Fig: 22Þ
The results for all of the above mentioned cases are shown in Fig. 22 for an arbitrarily chosen value rH =rY ¼ 3=2, where both axes of the plot are scaled to the according shakedown value in the perfectly plastic case, r0 and s0 , respectively. Noteworthy, the limited linear hardening model (Prager, 1959) predicts a higher shakedown load than the limited nonlinear one (Armstrong and Frederick, 1966). Furthermore, the nonlinear model only predicts failure due to incremental collapse, whereas in the linear case two different regions exist, one of which represents the incremental collapse and the other one represents alternating plasticity. However, both models give a lower value for the shakedown load than the unrestricted one resulting from the optimization problem ðP H Þ, as expected according to the discussion above. 5.3. Interpretation of the different results In the presented results for the sample under constant tension and alternating torsion, one can observe significant differences between the linear (associative) model, the nonlinear (non-associative) model and the unrestricted hardening one being obtained as result of the optimization problem ðP H Þ. In particular, the differences are not only quantitatively but even qualitatively, because different failure mechanisms are predicted by the different models. Concerning the difference between the associative and the non-associative hardening laws, an explanation has already been given by Pycko and Maier (1995). There, a lower bound loading factor lL and an upper bound loading factor lU are determined by the sufficient shakedown condition – introducing plastic potentials Ua in the subsidiary conditions – and the necessary shakedown condition – using the yield function ua in the subsidiary conditions, as in the classical formulation of Melan. Thereby, the shakedown factor is bracketed: lL 6 lSD 6 lU . Only in the case of associative hardening ðUa ¼ ua Þ these values coincide, lL ¼ lSD ¼ lU . Using this notion, the current formulation only accounts for the necessary shakedown condition. Hence, an upper bound to the shakedown factor is obtained. However, as shown in the above example, this upper bound is not necessarily defined
Fig. 21. Kinematic hardening for constant tension and alternating torsion: limited nonlinear hardening.
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Fig. 22. Shakedown domains for a specimen under constant tension and alternating torsion with different hardening rules.
by the associative (linear) case. The respective result in Pycko and Maier (1995) uses the restriction (64), X 6 rH rY , which is only true for the linear model, but which is not in accordance with the considered optimization problem. The true upper bound is the one shown above for the unrestricted hardening case. Nevertheless, the unrestricted and the linear hardening rules correspond to each other if r is variable as well. Then, the yield surface cannot leave the bounding surface at all for neither hardening law, and the result using the unrestricted hardening coincides with the one obtained by the linear model. In both cases, the solution is simply given by (61). Concluding, based on the shakedown theorem by Pham (2007, 2008) shakedown limits of complex structures incorporating generally-nonlinear hardening can be determined without knowing the exact loading history deterministically nor the hardening rule explicitly. Even though, one needs to be aware, that the unrestricted hardening law is thereby applied automatically. Nonetheless, specific hardening laws can be incorporated as indicated above by formulating additional constraints in the optimization problem. This is still covered by the shakedown theorem and its proof. In that sense, the general character of the theorem itself is not curtailed by the discussion above. As a final remark, the choice of material parameters rY and rH is essential for obtaining realistic results. Since this aspect is out of scope of the present paper, the reader is kindly referred to Pham (2010), where a profound discussion on that issue is given. In particular, it is suggested therein to take the initial yield stress as small as the fatigue limit. Further, the ultimate yield stress shall be the lowest limit observed in multicycle loading experiments. Since in addition it needs to be ensured that the small deformation assumption framework is not violated, Pham (2010) suggests to take the yield stress corresponding to some allowable small amount of plastic deformation (i.e. 0:2%) from the standard monotonic loading experiment using the Ramberg–Osgood empirical formula. 6. Conclusions The statical shakedown theorem by Melan (1938a,b) has been extended to limited kinematical hardening with generallynonlinear hardening laws using a two-surface model based on the works of Pham (2007, 2008). Since neither the specific loading path nor the underlying hardening rule need to be given, the power of the original theorem is perfectly captured. Even though the effect of the nonlinearity of the considered hardening law on the shakedown loads has been addressed in literature several times during the last decades, it could not be conclusively clarified therein. The present paper closes this discussion by showing that the method allows for incorporating the generally-nonlinear case on the basis of the unrestricted hardening rule, while the application to specific hardening laws – e.g. the linear model by Prager (1959) – demands the formulation of additional restrictions for the back-stresses. The proposed method has been implemented into the interior-point algorithm IPSA recently developed by the authors, which is especially tailored to shakedown analysis of large-scale engineering problems. Several results have been presented for thermo-mechanical problems from the field of power plant engineering illustrating the method’s potential. Among these, the presentation of the shakedown domain for a flanged pipe subjected to three independently varying thermo-mechanical loadings should be highlighted, because this is the first three-dimensional presentation of a shakedown domain
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incorporating kinematical hardening. Finally, the influence of limited kinematical hardening on the shakedown domain – in particular on the alternating plasticity and incremental collapse regimes – has been discussed intensively. Acknowledgments First of all, the encouragement and support from Prof. Dieter Weichert is appreciated. Further, I thank Prof. Manfred Staat for the interesting discussion, which gave rise to my investigations concerning the influence of nonlinearity. Finally, I thank the reviewers for the suggestions and remarks, which helped to significantly improve this paper. References Abdel-Karim, M., 2005. Shakedown of complex structures according to various hardening rules. Int. J. Press. Vessels Piping 82, 427–458. Abdel-Karim, M., 2009. Modified kinematic hardening rules for simulations of ratcheting. Int. J. Plast. 25, 1560–1587. Abdel-Karim, M., 2010. An evaluation for several kinematic hardening rules on prediction of multiaxial stress-controlled ratcheting. Int. 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