Shakedown static and kinematic theorems for elastic-plastic limited linear kinematic-hardening solids

European Journal of Mechanics A/Solids 24 (2005) 35–45

Shakedown static and kinematic theorems for elastic-plastic limited linear kinematic-hardening solids Pham Duc Chinh Vietnamese Academy of Science and Technology, Vien Co hoc, 264 Doi Can, Hanoi, Viet Nam Received 20 August 2004; accepted 8 November 2004 Available online 19 December 2004

Abstract Shakedown static and kinematic theorems are constructed for variable-loaded elastic-plastic hardening bodies in classical spirit. Prager’s linear kinematic hardening is limited by the initial and ultimate yield surfaces. The theorems indicate that the shakedown safety of a structure does not depend on the plastic modulus, but on the initial and ultimate yield stresses. While the ultimate yield strength determines the unbounded incremental collapse pattern, the initial yield stress is responsible for the bounded cyclic plasticity collapse modes. Though the usual yield criteria do not bound the hydrostatic part of the stresses, the restrictions on the hydrostatic stresses are included in a specific way that is appropriate for structures’ safety assessment and for completeness of general shakedown analysis.  2004 Elsevier SAS. All rights reserved. Keywords: Shakedown; Elastic-plastic material; Residual stress; Strain compatibility; Limited linear kinematic hardening

1. Introduction An elastic plastic body under variable external loads would shake down if the overall response of the body should converge to some purely elastic state due to a residual stress field developed inside the body and caused by respective incompatible plastic deformations with the total amount of plastic work bounded. Otherwise the structure should fail as the plastic deformations would accumulate unrestrictively or be bounded but vary cyclically and endlessly. Classical shakedown theory for elasticperfectly plastic bodies has been formulated in the classical works of Melan (1938), Koiter (1963), and revisited extensively in Ho (1972), Corradi and Maier (1974), Debordes and Nayroles (1976), Ceradini (1980), König (1987), Pham (1992, 1996, 2001, 2003b), Kamenjarzh (1995). The essential contents of the classical shakedown theory are its dual static and kinematic theorems, which are path-independent: both theorems determine the shakedown boundary in the loading space under which a loaded structure should be safe regardless of the loading history, while it should fail if the boundary is allowed to be violated unlimitedly. The shakedown theorems involve the respective plastic limit theorems as a simpler limiting case, and for many practical structures under quasistatic loads the safer shakedown limits often coincide with or are lower but not differ much from the plastic limits. However extended shakedown theorems apply to the much larger class of dynamic loading processes, which lie outside the framework of plastic limit theorems. Usual yield criteria, including the Mises and Tresca ones, do not restrict the hydrostatic part of the stresses leading to some singularity in constructing shakedown theorems. For completeness of shakedown theorems in the general setting, restrictions on the hydrostatic stresses (but not by a real yield condition!), which are quite natural requirements, should be incorporated appropriately. Shakedown theory has been developed for elastic plastic hardening solids (Melan, 1938; Maier, 1972; Ponter, 1975; Mandel, 1976; Zarka and Casier, 1981; König and Siemaszko, 1988; Weichert and Gross-Weege, 1988; Corigliano et al., 1995a; 0997-7538/$ – see front matter  2004 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2004.11.001

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Pham Duc Chinh / European Journal of Mechanics A/Solids 24 (2005) 35–45

Fuschi, 1999; Bodoville and de Saxce, 2001; Pham and Weichert, 2001; Nguyen and Pham, 2001; Nguyen, 2003). Prager’s (1959) kinematic hardening appears the simplest law that captures the main features from general behaviour of many practical elastic-plastic materials while keeping the fine structure of classical mathematical plasticity theory: elasticity, plasticity, hardening, Bauschinger effect, translated yield surface, normality yield rule. It has been observed that the assumption of unlimited kinematic hardening is unrealistic and makes it impossible to predict failure due to incremental collapse, but only the bounded cyclic plasticity mode. Hence a saturation ultimate yield surface of perfect plasticity is postulated to limit the hardening and assumes the role of the initial translated yield surface at the saturation. Thus, the concept of limited linear kinematic hardening has been incorporated into shakedown analysis (Mandel, 1976; Weichert and Gross-Weege, 1988; Polizzotto et al., 1991; Corigliano et al., 1995a; Pham and Weichert, 2001; Nguyen, 2003). The initial yield stress, which appears to affect the alternating plasticity collapse mode, should not be taken as the convenient one specified at the amount of plastic deformations of 0.2%, since the respective failure limit would vary and depend on particular loading processes – from the low cycle to high cycle ones. Shakedown analysis has been extended further to include large deformations, nonstandard plasticity, viscoplasticity, damage mechanics, poroplasticity (Polizzotto et al., 1991; Corigliano et al., 1995b; Bodoville and de Saxce, 2001; Maier, 2001; Weichert and Maier, 2002, etc.), which leads to specific applications. However the extensions should be made often at the expense of losing certain features and generality of the classical plasticity theory and shakedown theorems. Without the theorems in Melan– Koiter sense, which are valid only within certain restrictions, generally in practice one has to check for shakedown of a structure under specific loading histories implementing numerical incremental analysis. In this work we derive shakedown static and kinematic theorems for elastic-plastic limited kinematic hardening solids, strictly following the original approach and spirit of Koiter (1963), Corradi and Maier (1974), Pham and Weichert (2001), Pham (2003b) with necessary modifications and new constructions. The proof of the static theorem is more refined than that given in Pham and Weichert (2001), while the complete kinematic theorem is new and constructed in a mathematically-consistent manner as that of Koiter. In Section 2 we state the mathematical essentials of the limited kinematic hardening plasticity model necessary for subsequent uses. The shakedown static theorem is presented in Section 3, and the kinematic theorem is constructed in Section 4. Section 5 discusses specific collapse modes. Last section resumes the main results.

2. Limited linear kinematic hardening plasticity Let σ e (x, t) denote the (quasistatic or dynamic) stress response of the body V (x ∈ V , t ∈ [0, T ]) to external agencies over a period of time in assumption of its perfectly elastic behaviour. We call it a loading process, as the actions of the external agencies upon V can be expressed explicitly through σ e . At every point x ∈ V , the elastic stress response σ e (x, t) is confined to a bounded time-independent domain with prescribed limits in the stress space called a local loading domain Lx . As a field over V , σ e (x, t) belongs to the global loading domain L: 
(1) L = σ e | σ e (x, t) ∈ Lx , x ∈ V , t ∈ [0, T ] . In the spirit of classical shakedown analysis, the bounded loading domain L, not a particular loading process σ e (x, t), is given a priori. A body said to shake down in L means it shakes down for all possible loading processes σ e (x, t) ∈ L. For a detailed discussion on the sense of the dynamic stress response taken here, see Pham (1996; 2001), and also the final part of Section 4. v, ε, e, σ , and εp , ep , ε r , er , σ r denote respectively the actual velocity, strain, strain rate, stress, and plastic strain, plastic strain rate, residual strain, residual strain rate, residual stress fields. ve , εe , ee are the elastic velocity, strain, strain rate fields corresponding to the elastic stress field σ e . We have e(x, t) = ee + ep + er , σ (x, t) = σ e + σ r ,

ε(x, t) = εe + ε p + ε r , εr (x, t) = C : σ r ,

(2)

where C is the elastic compliance tensor. As both σ and σ e satisfy generally dynamic equilibrium equations of the problem, the difference σ − σ e satisfies the respective dynamic homogeneous equilibrium equations on V : ∇ · (σ − σ e ) − m(˙v − v˙ e ) = 0

(3)

and homogeneous boundary conditions; Here m is the mass density, the dot over a variable means time derivative. p Prager’s (1959) linear kinematic hardening rule is assumed, which relates the back stress α to the plastic deformation ε α by p

α = H εα ,

H > 0,

(4)

p where H is the plastic modulus (the current yield stress in simple tension of a bar with longitudinal plastic strain ε α is σYi + 3 H εp ). The yield surface Γ , which envelopes the elasticity domain Y centered at α in the stress space, e.g. for Mises α α α 2

material, is described by the equation

(σ¯ − α) : (σ¯ − α) = 2/3(σYi )2 ,

(5)

Pham Duc Chinh / European Journal of Mechanics A/Solids 24 (2005) 35–45

37

where σYi is the initial yield stress, σ¯ denotes the deviatoric part of the stress tensor σ . The dissipation function corresponding to (5) is defined as  p p p p (6) Dα (eα ) = (σ − α) : eα = 2/3 σYi (eα : eα )1/2 . Γα and Yα with the back stress α in the center are translated in the stress space without changing sizes and forms. As the hardening is limited, the set of all allowable Γα in the stress space is enveloped by the fixed yield ultimate yield surface Γu , which encompasses the domain Yu (Yα ⊂ Yu ) and is described by the respective equation σ¯ : σ¯ = 2/3(σYu )2 ,

(7)

where σYu is the ultimate yield stress. The dissipation function corresponding to (7) is  p p p p Du (eu ) = σ : eu = 2/3 σYu (eu : eu )1/2 .

(8)

As Γα is bounded by Γu , the back stress is bounded by  α = (α : α)1/2
αˆ = 2/3(σYu − σYi ).

(9)

Hence, according to (4) p

p

p

p

ˆ ε α  = (ε α : ε α )1/2
εˆ α = α/H.

(10)

The usual normality yield rule is assumed on both yield surfaces Γα and Γu . The total plastic strain rate ep and plastic strain ε p are composed of those two components: p

p

p

ep = eα + eu ,

p

εp = εα + εu .

(11)

p A peculiar feature of this two-surface model is that if the yield stress is strictly under Γu , then eα is the only possible regime p p p p p (eα = 0, α˙ = 0, eu = 0). If the yield stress is fixed on Γu , then only eu is active (eu = 0, eα = α˙ = 0). If the yield stress is p p moving on Γu then both regimes eu and eα may be possible, but they remain separable following their separate normality rules according to their own yield surfaces Γu and Γα .

3. Static theorem The set R of admissible residual stress fields is defined as a set of bounded time-independent stress fields ρ(x), which satisfy static homogeneous equilibrium equations on V including those on the boundary. B denotes the set of all time-independent back stresses α that are bounded by (9); hence Yα ⊂ Yu , ∀α ∈ B. A stress distribution σ  is called safe in Yu if the stress in each point is inside the domain (bounded by Γu ), then from Drucker postulate (σ − σ  ) : eu > 0, p

p

eu = 0,

σ on Γu ,

p

eu at σ ,

(12)

while σ  is allowable if it is nowhere outside the domain. Also a stress distribution σ  is called safe in Yα with center in α if it falls inside the referential domain in the sense (σ − α − σ  ) : eα > 0, p

p

eα = 0,

σ on Γα ,

p

eα at σ ,

(13)

while σ  is allowable if it is nowhere outside the domain. Because B is bounded by (9), then once β ∈ B and σ  − β is safe (or allowable) in Yα then σ  is also safe (or allowable) in Yu , which contains all Yα with α moving in B. Theorem. If (time-independent) fields ρ ∈ R and β ∈ B can be found so that ρ + σ e − β is safe in Yα for all σ e ∈ L, then the body shakes down in L. On the other hand, shakedown in L is impossible if no ρ ∈ R and β ∈ B can be found so that ρ + σ e − β is allowable. The second statement is self-evident. To prove the first statement, let us consider a positive functional:    1 m 1 w= (σ − σ e − ρ) : C : (σ − σ e − ρ) dV + (v − ve ) · (v − ve ) dV + (α − β) : (α − β) dV . 2 2 2H V

Take the derivative of w with respect to time:

V

V

(14)

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Pham Duc Chinh / European Journal of Mechanics A/Solids 24 (2005) 35–45



 (σ − σ e − ρ) : C : (σ˙ − σ˙ e ) dV +

w˙ = V

 m(˙v − v˙ e ) · (v − ve ) dV +

V

V





(σ − σ e − ρ) : (e − ep − ee ) dV +

= V

1 (α − β) : α˙ dV H



m(˙v − v˙ e ) · (v − ve ) dV + V

p

(α − β) : eα dV .

(15)

V

Sice ρ ∈ R, and e − ee is compatible and satisfies the homogeneous boundary conditions, then it is evident that  ρ : (e − ee ) dV = 0,

(16)

V

thus:



 (σ − σ e ) : (e − ee ) dV −

w˙ = V



=−

p

p



(σ − σ e − ρ) : (eα + eu ) dV + V

  ∇ · (σ − σ e ) − m(˙v − v˙ e ) · (v − ve ) dV −

V

 m(˙v − v˙ e ) · (v − ve ) dV +

V



V p

(σ − σ e − ρ − α + β) : eα dV V



p

(σ − σ e − ρ) : eu dV .

−

p

(α − β) : eα dV

(17)

V

Taking into account Eq. (3), finally we have    p  p   σ − α − (σ e + ρ − β) : eα dV − σ − (σ e + ρ) : eu dV . w˙ = − V

(18)

V

Because σ e +ρ −β is safe in Yα – hence σ e +ρ is also safe in Yu , relations (12), (13) and (18) indicate that w˙ is always negative whenever non-vanishing plastic strain rates occur in the actual loading program σ e (x, t). As the positive-definite quadratic form w can never become negative, plastic flow cannot continue indefinitely (in the sense that the total dissipation should be finite) and the body must shake down ultimately to some distribution of residual stresses. In particular, the boundedness of the plastic dissipation T wp (T ) =

 dt

0

V

p Dα (eα ) dV +

T



0

p

Du (eu ) dV ,

dt

(19)

V

which is a measure of total plastic deformations over V , can be proved if the structure has a safety factor k > 1, i.e. k(ρ +σ e −β) is an allowable stress:   p   p σ − α − k(ρ + σ e − β) : eα + σ − k(ρ + σ e ) : eu  0. (20) In fact, from (18)–(20), we have      p p p p (σ − α) : eα dV + σ : eu dV w˙ = (σ e + ρ − β) : eα dV + (σ e + ρ) : eu dV − V






V

V



   1 k−1 p p (σ − α) : eα dV + σ : eu dV = −w˙ p −1 , k k V

V

(21)

V

and after integration with respect to time from 0 to T   k k
w(0) , wp (T )
w(0) − w(T ) k−1 k−1

(22)

for all T . Eq. (22) means the total plastic deformations over V are bounded for the structure shakedown under the conditions of the theorem. Define the shakedown safety factor ks as the largest multiplier of σ e such that the body shakes down, then the static theorem can be restated as: 
ks = sup (23) k | k(ρ + σ e − β) ∈ Yα , ∀σ e ∈ L ρ∈R, β∈B

Pham Duc Chinh / European Journal of Mechanics A/Solids 24 (2005) 35–45

39

(in other words, at ks > 1 the body will shake down, while it will not at ks < 1, and ks = 1 defines the boundary of the shakedown domain).  The proof procedure (14)–(22) of the static theorem requires implicitly the initial w(0), and hence also the quadratic form V ρ : C : ρ dV , be bounded. This requirement may be violated if the hydrostatic part of the residual stress ρ becomes infinity, which is allowed by the Mises or Tresca yield criteria. This singularity becomes more apparent in Koiter’s original procedure for the proof of the kinematic theorem, which will be dealt with in the next section. Though the usual yield stresses do not restrict the hydrostatic part of the stresses, practical materials cannot be safe against arbitrarily high hydrostatic stresses. In fact they often tolerate relatively high hydrostatic tension stresses (compared to the yield limits, which put restrictions on the shear stresses) but fracture after certain limits. They can resist even higher hydrostatic compression stresses, but these too could not be unlimited. Hence for completeness of shakedown safety assessment, the stress ρ + σ e from the optimal solution of the static theorem (23) should be checked against those particular restrictions on the hydrostatic part of the stresses. Whenever a limit is violated, the structure should be considered as failed before it could reach the shakedown boundary, and the safety factor should be counted from this point, not from the shakedown boundary. For many practical loaded structures, in particular – the thin-walled ones, the hydrostatic stresses often do not build up very high because there are no kinematic constraints on the structures’ thickness directions, so this hydrostatic-stress fact may not deserve special consideration. However for some three-dimensional geometries, or some quasi-brittle materials having hydrostatic tension strength comparable to the yield one, it should be considered as an indispensable factor. For porous and cellular materials, hydrostatic compression limit also should not be disregarded.

4. Kinematic theorem Koiter’s (1963) procedure for the proof of the shakedown kinematic theorem (in particular, the second part) requires the yield surface be bounded in the stress space, so that all plastic strain rate directions are possible and any possible plastic strain rate cycle could be completed to be an admissible kinematic field. However usual Mises and Tresca yield criteria do not restrict hydrostatic part of the stresses. Still, in many practical thin-walled structures, such as bars, beams, plates and shells, there are no kinematic constraints in the structures’ thickness directions for the hydrostatic stresses to build up arbitrarily high, hence the stresses are bounded definitely. For them Koiter’s assumption is secured. For general three-dimensional structures, Koiter’s procedure needs some modifications (Pham, 2003b). Here we develop further the approach to construct the kinematic theorem for the plastic hardening model. Consider the same loaded body V , but made of a fictitious material. This fictitious material has the same elastic and plastic properties as those of the real material with the only added property that it could yield in bulk tension (corresponding hydrostatic yield stress σF+ ) and in bulk compression (corresponding hydrostatic yield stress σF− ). Actually we add into the ultimate yield surface Γu (previously hydrostatically-unrestricted) two new planar faces:

1 σ = σY+ , and σ = −σY− σ = σii (24) 3 u ⊂ Yu ; Here σ +  0, σ −  0, σij are components of the to form a combined yield surface Γ u , which bounds the domain Y Y Y stress tensor σ , conventional assumption on repeating indices from 1 to 3 is assumed. Hence, not only the deviatoric part of the stresses, but also the hydrostatic part should be bounded by the ultimate yield surface for our fictitious material. The normality rule is also assumed. It is clear that the plastic deformations are not restricted to be deviatoric anymore. We will keep almost the same notations as those for the real material presented above, and add only the tilde sign ( ) overhead the respective variables p when it is necessary to emphasize the difference. The plastic strain rate e˜ u of the fictitious material can be decomposed into deviatoric and hydrostatic parts: 1 p p p p p (25) e˜ u = eu + e˜u e, e˜u = (e˜u )ii , 3 p p u (˜ep where e is the unit tensor, and eu now becomes the deviatoric part of e˜ u . The respective dissipation function D u ) has the form

p + p p p p u (˜eu ) = σ : e˜ u = Du (eu ) + σF |e˜u |, σF = σF , e˜up > 0, D (26) σF− , e˜u < 0, where Du has been defined for deviatoric plastic strain rate earlier, e.g. by (8) with Mises material. There are no changes on p the part of Γα and eα . The procedure of the previous section is valid also for the fictitious material. Similar to (23), via the shakedown safety factor k˜s , the static theorem for the fictitious material can be stated as 
u , ∀σ e ∈ L . ˜ + σ e) ∈ Y ˜ + σ e − β) ∈ Yα , k(ρ sup (27) k˜ | k(ρ k˜s = ρ∈R, β∈B

40

Pham Duc Chinh / European Journal of Mechanics A/Solids 24 (2005) 35–45

Let E denote the set of second order symmetric tensor fields over V ; E is a subset of E involving the fields that are deviatoric; C is a subset of E , a subset gathering the fields that satisfy compatibility conditions for deformations on V including the kinematic p p constraints on the boundary. Define the set of admissible plastic strain rate cycles ep = eα + e˜ u over the period 0
t
T :   T  T  p  p p p p p A = eα ∈ E, e˜ u ∈ E  eα dt = 0, ε α 
εˆ α , e˜ u dt ∈ C , 0

(28)

0

p where εˆ α is defined in (10), (9). p

p

Theorem. The body will not shake down if a cycle ep = eα + e˜ u ∈ A and a loading program σ e ∈ L can be found such that T



T

p

u (˜ep D u ) dV

dt

V

0



σ e : e˜ u dV >

dt

(29)

V

0

or 

T

V

0



T

p

σ e : eα dV >

dt

p

Dα (eα ) dV ;

dt

(30)

V

0

On the other hand, the shakedown occurs if a number k > 1 can be found that for all ep ∈ A and all σ e ∈ L we have 

T k

dt 0

V

T



p σ e : e˜ u dV




T dt

u (˜ep D u ) dV

(31)

V

0

and k

dt

p σ e : eα dV


V

0

T



p

Dα (eα ) dV .

dt

(32)

V

0

 p  p p p p p Proof. Without losing generality, presume ε˜ u (0) = εα (0) = 0, hence ε˜ u (t) = 0t e˜ u dt , ε α (t) = 0t eα dt . If shakedown would indeed occur in spite of (29) and (30), then it would be possible by the static theorem to find time u and σ e + ρ − β is allowable referring to Yα , that is independent fields ρ ∈ R and β ∈ B such that σ e + ρ is allowable in Y p p for any plastic fields e˜ u , eα and corresponding yield stresses T



p

(σ − σ e − ρ) : e˜ u dV  0

dt

(33)

V

0

and 

T

p

(σ − α − σ e − ρ + β) : eα dV  0.

dt

(34)

V

0

With ep ∈ A and time-independent fields ρ ∈ R, β, we have: T



0

V

T





V

p

p

ρ : ε˜ u dV = 0 (because ε˜ u ∈ C),

(35)

V p



T

ρ : eα dV =

dt 0

p

ρ : e˜ u dV =

dt

0

p

T

β : eα dV = 0 (because

dt V

p

eα dt = 0). 0

(36)

Pham Duc Chinh / European Journal of Mechanics A/Solids 24 (2005) 35–45

41

(33)–(36) yield 

T dt

p σ e : e˜ u dV


V

0

V

u (˜ep D u ) dV ,

(37)

V



T

p

σ e : eα dV


dt 0

dt 0



T



T

p

Dα (eα ) dV .

dt

(38)

V

0

(37) and (38) are contrary to (29) and (30), therefore the contradicting assumption is false and the first part of the theorem has been proved. The second part is proved by showing that the total amount of plastic work in any process is bounded. Consider an arbitrary p p process ep = e˜ u + eα over the time interval 0
t
θ . The actual plastic strain rate ep needs not constitute an admissible cycle  p  p p in A by itself, i.e. ε˜ u (θ) = 0θ e˜ u dt , which leads to the residual strain ε˜ ru (θ) = 0θ e˜ ru dt , needs not belong to C; and ε α (θ) = θ p 0 eα dt may not equal 0. However it can be completed on θ
t
T to be an admissible cycle over 0
t
T (T > θ) by p assigning (note that though the residual strain is arbitrary, the kinematically admissible e˜ u is also not restricted to be deviatoric, p but eu is so!)  

1 ε˜ r (θ), θ
t
T  , θ −T u  0, T 
t
T,  θ
t
T ,  0, p eα (t) = 1 p  εα (θ), T 
t
T , T −T p

e˜ u (t) =

(39)

(40)

 p  p p p where T  is between θ and T – hence ε˜ u (T ) = 0T e˜ u dt ∈ C (because ε˜ ru (T ) = 0), and ε α (T ) = 0T eα dt = 0 as required. The virtual work equation at any moment t of the actual loading program 0
t
θ can be given as: 

 σ e : (e − ee ) dV =

V

 σ : (e − ee ) dV +

V

 V

σ e : ep dV =

⇔ V



σ : ep dV + V





V



σ e : (ep + er ) dV =

⇔

m(˙v − v˙ e ) · (v − ve ) dV

V



σ : ep dV + V



 m d (v − ve ) · (v − ve ) dV 2 dt

(σ e + σ r ) : er dV + 

σ r : er dV + V

V

 m d (v − ve ) · (v − ve ) dV . 2 dt

(41)

V

With the last equality (over 0
t
θ ) and (39), (40) (over θ
t
T ) we make an integration over 0
t
T 

T dt 0



θ σ e : ep dV =

V

dt θ

= 0

V

θ

 σ : ep dV +

dt V

1 +  T −θ

σ r : er dV + V





σ e : ε˜ ru (θ) dV +

dt θ

θ

 dt

0 T

σ e : ep dV

dt θ

V

0



T σ e : ep dV +

V



 m d (v − ve ) · (v − ve ) dV 2 dt

dt V

0

1 T −T



T T

p

σ e : ε α (θ) dV .

dt V

(42)

42

Pham Duc Chinh / European Journal of Mechanics A/Solids 24 (2005) 35–45

Denote: t w p (t) =

 dt

0

 wr (t) =

u (˜ep D u ) dV +

t

V



p

Dα (eα ) dV ,

dt V

0

1 r σ : C : σ r dV , 2

V



wv (t) =

(43) m (v − ve ) · (v − ve ) dV , 2

V



wα (t) =

1 α : α dV , 2H

V

and notice that θ  θ  θ  p p p dt σ : e dV = dt σ : e˜ u dV + dt σ : eα dV V

0

0

V

0

V

θ



θ



=

dt

p σ : e˜ u dV +

V

0

dt 0

p (σ − α) : eα dV +



θ

V

dt

1 α : α˙ dV H

V

0

=w p (θ) + wα (θ) − wα (0),

(44)

then (42) can be given as T

 σ e : ep dV = w p (θ) + wα (θ) − wα (0) + wr (θ) − wr (0) + wv (θ) − wv (0)

dt 0

V

1 +  T −θ

T



 σ e : ε˜ ru (θ) dV +

dt θ

V

1 T −T

T



T

p

σ e : ε α (θ) dV .

dt

(45)

V

On the other hand, taking into account (39), (40), from (31), (32) one get   T  T  T    1 1 p p  p  e p u (˜ep dt σ : e dV
dt Du (˜eu ) + Dα (eα ) dV = D w p (θ) + dt u ) + Dα (eα ) dV k k 0

V

0

V

      r   1 u ε˜ u (θ) + Dα ε p w p (θ) + = D α (θ) dV . k

θ

V

(46)

V

Combination of (45) and (46) yields the following inequality:

1 w p (θ)
wα (0) − wα (θ) + wr (0) − wr (θ) + wv (0) − wv (θ) 1− k 



 T  T  1 1 p e r e − dt σ : ε˜ u (θ) dV + dt σ : ε α (θ) dV T−θ T −T θ V V T   p  1   r  + Du ε˜ u (θ) + Dα ε α (θ) dV . k

(47)

V

u are bounded (though Yu is not!), σ e and ε˜ ru = C : (σ − σ e ) (where σ is restricted by Γ u ) are also bounded. Because L and Y p In addition, the deviatoric ε α is bounded (see (9), (10)). Hence the last two terms of (47) are bounded. Furthermore, wα (θ), p (θ) is bounded and the proof of the theorem is completed. 2 wr (θ) and wv (θ) are positive quadratic forms. Thus w The shakedown kinematic theorem for the fictitious material can be restated as k˜s−1 = Max{U, B},

(48)

Pham Duc Chinh / European Journal of Mechanics A/Solids 24 (2005) 35–45

43

where  e p 0 dt V σ : e˜ u dV , T  p u (˜ep e˜ u ∈A; σ e ∈L 0 dt V D u ) dV T  e p 0 dt V σ : eα dV B= sup . T  p p eα ∈A; σ e ∈L 0 dt V Dα (eα ) dV T

U=

sup

(49)

(50)

One feels intuitively that with the limits σF+ and σF− increasing to infinity, the solution of the problem (48)–(50) for the fictitious material should approach that for usual Mises (or Tresca) model. In fact, if at sufficiently large values of σF+ and σF− , p p the optimal solution e˜ u of the problem (49) should have vanishing hydrostatic part (i.e. e˜u ≡ 0 – see (25)) to become deviatoric: p p p p e˜ u = eu , then Du (˜eu ) = Du (eu ) and the fictitious material behaves identically as the real material without yielding hydrostatically. Thus, from (48)–(50) for the fictitious material we get the solution for our real material. Practical materials cannot be safe against arbitrary high values of σF+ and σF− , but certain finite ones. They cannot yield in bulk tension and compression, but p p p fracture under those limiting stresses. Hence if e˜ u (from the solution {˜eu , eα } of k˜s = 1 in (48)–(49)) should have nonvanishing + − hydrostatic part (under given limitations σF and σF ), then the real material structure should be considered as failed because of fracture in bulk tension or compression before it could reach the shakedown boundary. The theorems are stated in the general dynamic setting. In the statements of both static and kinematic theorems there, only the elastic stress field σ e is explicitly presented, but not the boundary and initial conditions causing it. In the spirit of classical shakedown analysis only important is that the boundary of the loading stress domain L that contains all possible σ e is given, not any particular point σ e inside it or which loading path in L leading to the point. That is how the theorems are path-independent dynamically or quasistatically. Otherwise in applications when one deals with corresponding convenient loading domain in the space of external loads then initial conditions also have effects and should be dealt with appropriately in particular cases. The elastic stress domain L should be bounded for the subsequent shakedown safety factor ks not to decrease to zero. Allowances for possible stress concentrations as at the cracks or sharp corners in a shakedown problem should be treated separately. p Looking at (48)–(50) we see that B involves σYi and determines the bounded cyclic plastic collapse mode ε α , while U p involves σYu and causes unrestricted incremental collapse mode ε˜ u . Both static and kinematic theorems do not contain the plastic modulus H . The fact agrees with the observations of many practitioners in the field when they study shakedown of various loaded structures applying numerical incremental techniques: The shakedown of structures appears not to depend on the plastic modulus, but just on the initial and ultimate yield stresses (consult also Corigliano et al., 1995b; Pham and Weichert, 2001). While the ultimate yield strength is well defined physically, the initial yield stress is not so: for most practical materials it is not easy to specify this point. Plastic deformations often develop from microscopic to macroscopic levels gradually without abrupt boundary. So in engineering practice one usually takes the stress corresponding to plastic deformations at the amount of 0.2% as the standard initial yield stress σYi . This convenient assumption does not affect practically the accuracy of the usual incremental elastic plastic analysis as long as the stress-strain curve does not decline significantly from linear elasticity up to this yield point. However the specification of the initial yield stress becomes of primary importance for shakedown analysis, where the safety limits are to be determined, especially for the alternating plasticity collapse criterion, which is governed by the initial yield stress σYi . We know that at high cycles the yield strength for many materials can decrease considerably, f

down to the fatigue limit σY at ca. 106 –107 cycles (Yokobori, 1965). Relatively, note that the fatigue limit interpreted as lack of shakedown at microscale has been studied in Dang Van (1999). At the opposite extreme when a structure is subjected to just a few cycles, the alternating plasticity collapse criterion based on the convenient initial yield stress corresponding to the amount 0.2% of plastic deformations may underestimate the load-bearing capacity of a structure, which may be high up to the ultimate yield strength σYu . In the loading path-independence spirit of classical shakedown theorems, one should take the lowest f

(fatigue) limit σY as the initial yield stress for the alternating plasticity collapse limit. Otherwise σYi may be taken according to particular loading processes considered, which could range from the low-cycle to high-cycle ones, hence σYi may take all f

possible values from σY up to σYu .

5. Simplified kinematic deductions Application of the kinematic approach to shakedown analysis of particular structures has been developed in a number of works (Gokhfeld and Cherniavski, 1980; König, 1987; Pham, 1992, 1993, 2000, 2003a; Pham and Stumpf, 1994; Pham and Weichert, 2001, etc.).

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Pham Duc Chinh / European Journal of Mechanics A/Solids 24 (2005) 35–45

In the specific case of perfect plasticity, (50) falls out from (48)–(50), and we get the same result of (Pham, 2003b) and the following simplified estimation (k˜perf designates the respective safety factor in the perfect plasticity case) −1 = U  Max{I, R}, k˜perf

where

(51)



I=

p e V sup0
tx
T σ (x, tx ) : ε˜ u (x) dV ,  εp p u ) dV σ e ∈L; ε˜ u ∈C V Du (˜

sup

 e p0 0 dt V σ : e˜ u dV R= sup , T  p0 p0 σ e ∈L; e˜ u ∈E0 0 dt V Du (˜eu ) dV

(52)

T

(53)

 p0 p0 E0 = {˜eu ∈ E | 0T e˜ u dt = 0}; (51) represents the perfect incremental collapse mode; (53) is a bounded cyclic plasticity mode −1 = U = Max{I, R}? We still referred to in (Pham, 2003b) as the rotating plasticity collapse one. The question is whether k˜perf cannot prove it yet. We cannot either find a counter-example where U > Max{I, R}, which means that the structure may fail by some mixed mode of collapse not described by (52) or (53). Coming back to the general hardening case and compare the modes (50) and (53) one sees that both are bounded cyclic plasticity collapse modes, but the first one determined by σYi appears more imperative compared to the second one determined by the larger σYu , hence one can deduce the following simple estimation k˜s−1  Max{I, B}, where = B

(54)

 e p0 0 dt V σ : e˜ u dV sup , T  p0 p0 e˜ u ∈E0 ; σ e ∈L 0 dt V Dα (˜eu ) dV T

(55)

u in (26) with the only difference in that σ i takes the place of σ u . In fact (55) is the upper α has the same form as that of D D Y Y envelope of (50) and (53). Following the approach of Pham and Stumpf (1994) we can deduce even a simpler estimation k˜s−1  Max{I, A},

(56)

where [σ e (x, t) − σ e (x, t  )] : e˜ u (x) α (˜ep 2D u) p

A=

sup p

e˜ u ∈E ; x,t,t 

(57)

referred to as the perfect alternating plasticity collapse mode. Moreover k˜s−1 = Max{I, A} for those simple loading processes where the components of the plastic deformation tensor at every point x ∈ V should change proportionally (but may alternate the directions) during the cycles. Our previous results for particular structures made of perfectly plastic materials can be extended to the hardening ones as well. The only modification is that the yield stress for the mode A should be taken as σYi , while p that for the mode I should be taken as σYu . Note that the kinematically admissible plastic deformation field ε˜ u in (52) is not restricted to be deviatoric, hence can be constructed from any trial displacement field on V . Then, the solution is checked to p be acceptable for the real material if the optimal solution ε˜ u should have vanishing hydrostatic part, or otherwise it indicates premature hydrostatic collapse for the material.

6. Conclusion Classical approach for shakedown analysis of elastic perfectly plastic bodies has been modified and extended to construct static and kinematic theorems for elastic plastic linear kinematic hardening solids. The hardening is started at the initial yield surface and is bounded by the ultimate yield limit. The positive constant plastic hardening modulus does not enter the final statements of the theorems, but the initial and ultimate yield stresses do. The kinematic theorem indicates that the initial yield stress σYi determines the bounded cyclic plasticity collapse modes, while the ultimate yield stress σYu is responsible for the unbounded incremental collapse ones. Though the kinematic theorem is constructed for fictitious materials that can yield in the pure tension-compression states, it gives the results for the real materials if the plastic deformation solution of the problem k˜s = 1 should have vanishing hydrostatic part. Otherwise the real materials should be considered as failed because of hydrostatic fractures before they could reach the shakedown boundary.

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Acknowledgements This work is supported by the Program of Basic Research in Natural Science.

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