Plasticity with non-linear kinematic hardening: modelling and shakedown analysis by the bipotential approach

Eur. J. Mech. A/Solids 20 (2001) 99–112  2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0997-7538(00)01109-8/FLA

Plasticity with non-linear kinematic hardening: modelling and shakedown analysis by the bipotential approach Guillaume Bodovillé a,1 , Géry de Saxcé b a Laboratoire de Mécanique de Lille, URA CNRS 1441, École centrale de Lille, Cité scientifique, BP48, 59651 Villeneuve d’Ascq

cedex, France (Now at ONÉRA, BP72, 29, avenue de la division Leclere, 92322 Châtillon cedex, France) b Laboratoire de Mécanique de Lille, URA CNRS 1441, Université des sciences et technologies de Lille, UFR de Mathématiques

pures et appliquées, Cité scientifique, Bâtiment M3, 59655 Villeneuve d’Ascq cedex, France (Received 1 February 2000; revised and accepted 14 June 2000) Abstract – The class of generalised standard materials is not relevant to model the non-associative constitutive equations. The possible generalisation of Fenchel’s inequality proposed by de Saxcé allows the recovery of flow rule normality for non-associative behaviours. The normality rule is written in the weak form of an implicit relation. This leads to the introduction of the class of implicit standard materials. This formulation is applied to constitutive equations involving non-linear kinematic hardening, indispensable to describe accurately and realistically the cyclic plasticity of metallic materials. For these plastic flow rules shakedown bound theorems can be extended; an analytical example of the shakedown of a thin-walled tube under constant traction and alternate cyclic torsion is considered and the obtained solution is proved to be exact.  2001 Éditions scientifiques et médicales Elsevier SAS non-associative plasticity / non-linear kinematic hardening / shakedown / implicit standard materials

1. Introduction Nowadays, the concept of convex superpotential of dissipation is a powerful tool customarily used to model the constitutive dissipative laws. This concept leads to the generalised standard material theory (Germain et al., 1983), very suitable to model multivalued normality rules. Unfortunately, it is not relevant to model all the dissipative laws. In particular, it fails when applied to non-associative flow rules. Because the key-concepts of convexity and normality are very convenient tools, the implicit standard material theory was proposed by de Saxce (de Saxcé, 1992; de Saxcé and Feng, 1998) in order to extend in a natural way the interesting properties of the generalised standard materials to the non-associative flow rules. The cornerstone of this theory is to consider that the normality rule is written in the weak form of an implicit relation. One of the original results of the implicit standard material theory is to prove that many non-standard dissipative laws can be simply represented by a suitable pseudo-potential depending on the dual variables, internal variable rates and associated variables. The properties of the so-called bipotential are based on an extension of Fenchel’s inequality and allow the generalisation of, in the framework of convex analysis, Moreau’s superpotential. The constitutive equations involving non-linear kinematic hardening are indispensable to describe accurately and realistically the cyclic plasticity of metallic materials (Chaboche, 1991, 1994; Chaboche and Nouailhas, 1989; Chaboche et al., 1991), possibly giving rise to ratchetting effects, that is the progressive plastic strain 1 E-mail: [email protected]

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accumulation, cycle by cycle. The main drawback is the non-associative nature of the non-linear kinematic hardening rules, that is, the lack of normality to the yield surface of the internal variable rate of kinematic hardening, source of many theoretical and numerical troubles; the constitutive equations with non-linear kinematic hardening do not come with the scope of generalised normality rule. The bipotential approach is applied to constitutive equations involving the Armstrong–Frederick non-linear kinematic hardening rule with a threshold (the case of the non-linear Prager’s rule is presented in the Appendix). This approach enables us to write these constitutive equations in a compact form and to recover an implicit multivalued normality rule structure. It is known (König and Maier, 1981; Polizzotto, 1993; Zarka et al., 1990) that an elastic-plastic metallic material subjected to a cyclic loading may reach after a certain finite number of cycles, − an elastic shakedown or adaptation behaviour: plastic flow no longer occurs, the plastic strain rate and all internal variable rates vanish and thus the structural response to loads becomes purely elastic; − a plastic shakedown or accommodation behaviour: periodic plastic strain response; or − a ratchetting behaviour: incremental increase of the plastic strain which will rapidly cause the failure. During elastic shakedown the material can endure a very large number of cycles (high-cycle fatigue) (Dang Van, 1999) but during plastic shakedown low-cycle fatigue will occur. Unfortunately, for implicit standard materials no generalisation of Melan’s elastic shakedown theorem (Koiter, 1960) has been rigorously proved up to now. In spite of this, shakedown bound theorems can be extended (de Saxcé et al., 1998). They provide the safety factor, that is, the critical value of the load factor beyond which elastic shakedown does not occur. The analytical example of the shakedown of a thin-walled tube under constant traction and alternate cyclic torsion, proposed by de Saxcé, Tritsch and Hjiaj (1998), is applied to the constitutive equations involving the Armstrong-Frederick non-linear kinematic hardening rule with a threshold (and to the non-linear Prager’s rule in the Appendix). In particular, it is proved that the obtained solution is exact.

2. Constitutive law p

p

The set of internal variable rates of the model is κ˙ = (˙εij , −α˙ ij , −p), ˙ where ε˙ ij is the incompressible plastic p strain rate (˙εkk = 0), −α˙ ij and −p˙ are, respectively, the kinematic and isotropic hardening variable rates. The set of associated variables is π = (σij , Xij , R) where σij is the Cauchy stress, Xij represents the back-stress which is introduced as thePcentre of the elastic domain in the deviatoric stress space (several back-stresses can be super-imposed: Xij = k Xij k ) and R the size increase of the elastic domain. The convex elastic domain is defined by the von Mises yield function f : f (σij − Xij , R) = (σij − Xij )eq − R − σy 6 0, s

with (σij − Xij )eq =

3 1 (sij − Xij )(sij − Xij ), sij = σij − σkk δij , 2 3

σy being the yield stress of the material. The constitutive equations with isotropic and non-linear kinematic hardenings, introduced by Chaboche et al. (Chaboche, 1991, 1994; Chaboche et al., 1991), are given by: p

(˙εij , −p) ˙ ∈ ∂I f (σij − Xij , R),

(1)

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101

s

Xij p 3γ p α˙ ij = ε˙ ij − ε˙ , hXeq − Xl im 2c Xeq eq

with Xeq =

3 Xij Xij , 2

(2)

p If being the indicator function of f , and ε˙ eq the accumulated plastic strain rate:

s

2 p p ε˙ ε˙ , 3 ij ij

p ε˙ eq =

c > 0, γ > 0, m > 1 and Xl > 0 are material coefficients. h i is the McCauley bracket defined by: hui = u if u > 0,

hui = 0 if u 6 0.

Indeed, if f (σij − Xij , R) = 0, (1) means that there exists a plastic multiplier λ > 0 such that the plastic p strain rate ε˙ ij and the internal variable rate of isotropic hardening – p˙ are given by the normality rule: p

∂f 3 sij − Xij =λ , ∂σij 2 (σij − Xij )eq ∂f p − p˙ ∈ ∂I f (σij − Xij , R) ⇔ −p˙ = λ . → λ = p˙ = ε˙ eq ∂R p

ε˙ ij ∈ ∂I f (σij − Xij , R) ⇔ ε˙ ij = λ

The plastic multiplier λ is determined by the consistency condition f = f˙ = 0 during plastic flow: 



∂f ∂f ∂f ˙ ∂f ˙ 1 f˙ = Xij + σ˙ ij + σ˙ ij , R=0→λ= ζ ∂σij ∂Xij ∂R h ∂σij

with ζ = 0 if f < 0, ζ = 1 if f = 0.

With the state laws of the material (Chaboche et al., 1991; Lemaitre and Chaboche, 1988): 2 Xij = cαij 3

and

R = Q(1 − e−bp ) → R˙ = b(Q − R)p, ˙

the hardening modulus h is given by: 3 Xij sij − Xij h = c − γ hXeq − Xl im + b(Q − R), 2 Xeq (σij − Xij )eq Q > 0 and b > 0 being material coefficients. The materials generally present an accommodation limit, under which ratchetting does not occur. The modification (2) of the Armstrong–Frederick rule, proposed by Chaboche et al., consists of introducing a threshold Xl in the recall term, that plays the role of an accommodation limit. It allows an accurate description of a large number of aspects of cyclic plasticity of metallic materials and are indispensable to model accurately the Bauschinger effect, together with the ratchetting or mean-stress relaxation effects. The particular case where Xl = 0 leads to Henshall’s rule (Chaboche, 1991); m = 1 and Xl = 0 to the Armstrong–Frederick rule; and with γ = 0 the Prager’s linear kinematic hardening rule is recovered.

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3. Bipotential and implicit standard material Let b(κ; ˙ π ) be a function with values in [−∞, +∞], lower semi-continuous and bi-convex with respect to κ˙ and π (that is, convex with respect to κ, ˙ when π is fixed, and convex with respect to π , when κ˙ is fixed). This function is called bipotential if it satisfies the following fundamental inequality generalising Fenchel’s one (de Saxcé, 1992): ∀(κ˙ 0 ; π 0 ),

b(κ˙ 0 ; π 0 ) > κ˙ 0 π 0 .

(3)

A set (κ; ˙ π ) of couples of dual variables related by the dissipative law of the material, is said to be extremal if the equality is reached in the previous relation for this couple: b(κ; ˙ π ) = κ.π. ˙

(4)

Then, any extremal couple satisfies the following relations: ∀ κ˙ 0 ,

b(κ˙ 0 ; π ) − b(κ; ˙ π ) > π(κ˙ 0 − κ), ˙

∀ π 0,

b(κ; ˙ π 0 ) − b(κ; ˙ π ) > κ(π ˙ 0 − π ).

(5)

Therefore, κ˙ and π are related by the following differential inclusions: κ˙ ∈ ∂π b(κ; ˙ π ),

(6)

representing an implicit multivalued normality law, and: π ∈ ∂κ˙ b(κ; ˙ π ),

(7)

that can be interpreted as the inverse law of (6). An implicit standard material is a material for which the physical behaviours correspond to the extremal couples of a given bipotential. In other words, the relations (6) and (7) define the (multivalued) constitutive law of this material. Remark: Let ϕ(˙εij , −p) ˙ be a convex superpotential, and ϕ ∗ (σij − Xij , R) be its convex conjugate function by Legendre–Fenchel transform: p

p




ϕ ε˙ ij , −p˙ =

sup

σij −Xij ,R p







(σij − Xij )˙εij − R p˙ − ϕ ∗ , p




p

ϕ ∗ (σij − Xij , R) = I f (σij − Xij , R),

p ϕ ε˙ ij , −p˙ = σy ε˙ eq + I f ∗ ε˙ ij , −p˙ ,

with:

p

ε˙ ij ,−p˙




p



ϕ ∗ (σij − Xij , R) = sup (σij − Xij )˙εij − R p˙ − ϕ ,




p f ∗ ε˙ ij , −p˙ = ε˙ eq − p˙ 6 0. p

The corner-stone inequation (3) holds and degenerates to Fenchel’s inequality: p0

∀ (˙εij , −p˙ 0 ; σij0 − Xij0 , R 0 ), p0

p0

p0

b(˙εij , −p˙ 0 ; σij0 − Xij0 , R 0 ) = ϕ(˙εij , −p˙ 0 ) + ϕ ∗ (σij0 − Xij0 , R 0 ) > (σij0 − Xij0 )˙εij − R 0 p˙ 0 .

(8)

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Then, the sum of ϕ and ϕ ∗ is a separated bipotential of a generalised standard material of which the extremal couples of dual variables related by the dissipative law of the material verify the differential inclusions:


ε˙ ij , −p˙ ∈ ∂ϕ ∗ (σij − Xij , R), p

representing a multivalued normality law, and: p

(σij − Xij , R) ∈ ∂ϕ(˙εij , −p), ˙ that can be interpreted as the inverse dissipative law. These relationships define the constitutive law of a generalised standard material (Germain et al., 1983), with an explicit dependence between the internal variable rates and their associated variables. On the other hand, in (6) and (7), the dependence is implicit. This explains the choice of the name for the new class of materials. The bipotential concept sheds a new light on some non-associative laws (Bodovillé, 1999; de Saxcé, 1992; de Saxcé and Bousshine, 1998); to illustrate the interest of the previous general definitions, a particular application of this concept is presented, devoted to the non-linear kinematic hardening rule with a threshold (2) proposed by Chaboche et al. for cyclic plasticity of metallic materials. 4. The constitutive law admits a bipotential 4.1. Bipotential of dissipation Unfortunately, the non-linear kinematic hardening rule (2) does not come with the scope of generalised normality rule. Its main drawback is its non-associative nature resulting from the fact that: 

−α˙ ij ∈ / ∂Xij I f (σij − Xij , R) ⇔ α˙ ij 6= |

{z

}

p ε˙ ij ;

∂f λ ∂Xij



Xij p 3γ − α˙ ij + ε˙ ∈ ∂Xij I f (σij − Xij , R) . hXeq − Xl im 2c Xeq eq | {z } ∂f λ ∂Xij

Nevertheless, the constitutive law (1) and (2) admits a bipotential of dissipation b: p

p b(κ; ˙ π ) = σy ε˙ eq + I f ∗ (˙εij , −p) ˙ + I f (σij − Xij , R)



+I{0}



p α˙ ij − ε˙ ij

Xij p 3γ γ p + ε˙ eq + hXeq − Xl im Xeq ε˙ eq . hXeq − Xl im 2c Xeq c

(9)

b is bi-convex with respect to κ˙ and π ; b stands for the dissipation, and thus, is supposed to be positive to satisfy the second principle of thermodynamics. Moreover, two propositions can be demonstrated: 4.2. PROPOSITION 1. - The function (9) is a bipotential. Proof. – It is sufficient to verify that for any set (κ; ˙ π ) of dual variables satisfying (2), the inequality (3) holds. Taking account of the kinematic hardening rule (2) and Fenchel’s inequality (8), the inequality (3) holds, and this achieves the proof. 2

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4.3. PROPOSITION 2. - The constitutive laws are implicit multivalued normality laws. Proof. – A set (κ; ˙ π ) of dual variables satisfying (2) is considered and the satisfaction of (6) has to be demonstrated, that is equivalent to the satisfaction of (4). For any π 0 satisfying: p

α˙ ij = ε˙ ij −

Xij0 p 3γ 0 , hXeq − Xl im 0 ε˙ eq 2c Xeq

(10)

the following relation holds: I f (σij0 − Xij0 , R 0 ) − I f (σij − Xij , R) +


p γ 0 0 hXeq − Xl im Xeq − hXeq − Xl im Xeq ε˙ eq > κ.(π ˙ 0 − π ). c

Taking account of (2) and (10), the following inequality is satisfied: 



I f (σij0 − Xij0 , R 0 ) − I f (σij − Xij , R) > (σij0 − Xij0 ) − (σij − Xij ) ε˙ ij − (R 0 − R)p. ˙ p

Thus, it results that: p

ε˙ ij ∈ ∂I f (σij − Xij , R),

−p˙ ∈ ∂I f (σij − Xij , R),

that achieves the proof (the satisfaction of (2) is trivially fulfilled).

2

4.4. Flow rules and Lagrange’s multipliers Any set (κ; ˙ π ) of extremal couples of dual variables related by the dissipative law of the material satisfies: inf(b − κπ ˙ ) = inf(b − κ.π ˙ ) = 0. κ˙

π

As proposed in (Bodovillé, 1999), let λ, η and ν be three Lagrange multipliers and L be the functional: 



γ p p L(κ; ˙ π, λ, η, ν) = σy + hXeq − Xl im Xeq ε˙ eq − σij ε˙ ij + Xij α˙ ij + R p˙ c 



p

p + λ[(σij − Xij )eq − R − σy ] + η(˙εeq − p) ˙ + ν α˙ ij − ε˙ ij +

Xij p 3γ ε˙ , hXeq − Xl im 2c Xeq eq

subjected to f 6 0, f ∗ 6 0 and: p

α˙ ij = ε˙ ij −

Xij p 3γ ε˙ . hXeq − Xl im 2c Xeq eq

Therefore, any set (κ; ˙ π ) of extremal couples satisfies: sup inf L = sup inf L = 0. λ>0, η>0, ν6=0

κ˙

λ>0, π η>0, ν6=0

The Kuhn–Tucker conditions then lead to: ∂L = 0; ∂ κ˙

∂L = 0; ∂π

∂L 6 0, ∂λ

∂L 6 0, ∂η

∂L = 0; and L = 0 when plastic flow occurs. ∂ν

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The plastic flow laws are obtained as (2) and: p

ε˙ ij = λ

3 sij − Xij , 2 (σij − Xij )eq

p λ = p˙ = ε˙ eq ⇔ f ∗ = 0;

η = (σij − Xij )eq − σy = R ⇔ f = 0,

ν = −Xij ; L = 0.

The implicit (multivalued) constitutive equations of the material are thus defined by: p

(˙εij , −α˙ ij , −p) ˙ ∈ ∂(σij ,Xij ,R) b,

(σij , Xij , R) ∈ ∂(˙εp ,−α˙ ij ,−p) ˙ b. ij

5. Bifunctional and variational formulation of shakedown problems for implicit standard materials Let  be an elastoplastic implicit standard material, subjected to variable periodic external actions varying between given limits controlled by a load factor µ. The following question arises: under which conditions does the material shake down? Unfortunately, no generalisation of Melan’s theorem (Koiter, 1960) to implicit standard materials has been rigorously proved up to now. In spite of this, let us admit the existence of a set of admissible associated variables π = (µσijel + ρ ij , X ij , R) in the sense that: (i) ρij is a residual stress field, (ii) ρ ij , X ij and R are time-independent and plastically admissible anywhere in  when adding to ρ ij the stress response µσijel in the fictitious elastic material: ∀ t,





f µσijel (t) + ρ ij − Xij , R 6 0.

On the other hand, let κ˙ be a set of admissible internal variable rates in the sense that: R p p (i) the increment of the plastic strain rate on the load cycle 1εij = cycle ε˙ ij dt is kinematically admissible with zero values of the corresponding displacement increment on the supports, R R p p (ii) ε˙ ij is plastically admissible:  cycle µσijel ε˙ ij dt d > 0. The admissible internal variable rates are normalised as: Z Z  cycle

p

σijel ε˙ ij dt d = 1.

(11)

By virtue of the virtual work principle, it holds: Z Z  cycle

p

ρ ij ε˙ ij dt d =

Z 

p

ρ ij 1εij d = 0.

(12)

A possible variational formulation of shakedown problems results from introducing the so-called bifunctional (de Saxcé et al., 1998): βs (κ; ˙ π) =

Z Z  cycle



p



b(κ; ˙ π ) − µσijel ε˙ ij + X ij α˙ ij + R p˙ dt d.

(13)

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A straightforward consequence of (3), (12) and (13) is that for any admissible set (κ˙ 0 ; π 0 ): βs (κ˙ 0 , π 0 ) > 0. In particular, for the exact solution (κ; ˙ π ), the constitutive law is exactly satisfied anywhere in  and at any time. As a consequence of (4), the bifunctional vanishes: βs (κ; ˙ π ) = 0.

(14)

The previous observation is crucial in the sequel. For the exact solution (κ; ˙ π ), condition (14) combined with the normalisation condition (11) allows us to calculate the value of the shakedown load factor: µ=

Z Z  cycle

[b(κ; ˙ π ) + X ij α˙ ij + R p] ˙ dtd.

Remark: A special event is the usual perfectly plastic, (generalised) standard material. The bifunctional splits up into two terms:


p




p




βs ε˙ ij ; ρ ij ; µσijel = 8 ε˙ ij ; µσijel + 5 µσijel + ρ ij , where: 8


p ε˙ ij ; µσijel

=

Z Z



 cycle

p

p

ϕ(˙εij ) − µσijel ε˙ ij dt d,

is the functional of Markov’s principle over a cycle and: 5

µσijel




+ ρ ij =

Z Z  cycle




ϕ ∗ µσijel + ρ ij dt d,

is the functional of Hill’s principle over a cycle, as introduced by de Saxcé (Save et al., 1989).

6. Application to the shakedown of a thin-walled tube under constant tension and alternate cyclic torsion This analytical example concerns the elastic shakedown of a thin-walled tube specimen submitted to a biaxial plane stress state due to a constant tension σ11 = σ and an alternate symmetrical cyclic torsion generating a shear stress state σ12 varying between given limits (±τa with τa > 0) and controlled by the load factor µ, as proposed by Picko and Maier (1995) and de Saxcé et al. (1998). The behaviour of the material is described by the constitutive equations formulated above considering only the stabilised cyclic behaviour of the material, without taking into account the isotropic hardening. The problem is to find the maximum load multiplier µ for which elastic shakedown occurs. el The unit value is taken as reference shear stress in a fictitious, purely elastic material (σ12 = 1), that allows the identification of the load factor to the maximum alternate shear stress: µ = τa .

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107

Therefore, considering the cyclic loading, the state will be such that, for the maximum of the cycle: (

el el σ11 = σ, σ12 = τa = µσ12 = µ (σ12 = 1), p p+ p p+ + + ε˙ 11 = ε˙ 11 , ε˙ 12 = ε˙ 12 , α˙ 11 = α˙ 11 , α˙ 12 = α˙ 12 ,

(15)

el el σ11 = σ, σ12 = −τa = µσ12 = −µ (σ12 = −1), p p− p p− − − ε˙ 11 = ε˙ 11 , ε˙ 12 = ε˙ 12 , α˙ 11 = α˙ 11 , α˙ 12 = α˙ 12 .

(16)

and for the minimum of the cycle: (

6.1. Calculation of the shakedown safety factor For the sake of simplicity, a unit representative volume element  is now considered, in order to avoid the volume integrals. It is assumed that the collapse can take place by ratchetting, that is the progressive plastic strain accumulation, cycle by cycle, only in the direction of the traction (Chaboche, 1994; Chaboche and Nouailhas, 1989; p Polizzotto, 1993). Then, the plastic strain increment, or ratchet strain, 1ε12 on the load cycle vanishes: Z

p

1ε12 =

p

cycle

ε˙ 12 dt = 0.

The non-vanishing contributions to the time integral are related to the extrema of the collapse cycle. At each extremum, considering the rates as constant during a unit time interval leads to: p+

p−

ε˙ 12 + ε˙ 12 = 0. The normalisation condition (11) gives:

Z

p

2 cycle

el σ12 ε˙ 12 dt = 1.

(17)

Together with (15) and (16), it entails that: 1 p+ p− ε˙ 12 = −˙ε12 = 4

and

p = ε˙ eq

v u u t



2 p p (˙ε11 )2 + √ ε˙ 12 3

s

2

=

p

(˙ε11 )2 +

1 . 12

(18)

The maximum and the minimum of the cycle being supposed to be located on the yield surface, X12 vanishes: X12 = 0, and the yield criterion becomes: (σ − X11 )2 + 3τa2 = σy2 ⇔

√

v u   u σ − X11 2 3 τa = σy t1 − .

σy

(19)

The goal now is thus to determine the value of X11 at collapse accounting for the plastic flow and hardening p p rules. Therefore, ε˙ ij and ε˙ eq are calculated, in order to get an explicit relation of X11 through the hardening

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G. Bodovillé, G. de Saxcé

rule. At the extrema of the cycle, the plastic flow rule (1) gives: p±

ε˙ 12 = ±λ±

3 τa . 2 σy

Combining with relation (18), the plastic multiplier is equal to: λ± =

σy , 6τa

so that: σ − X11 and 6τa It allows now to write the kinematic hardening rule (2) as: p±

ε˙ 11 =

± α˙ 11 =

p± ε˙ eq =

σy . 6τa



(20) 

1 γ σ − X11 − hX11 − Xl im σy . 6τa c

(21)

A straightforward consequence of the previous developments is: + − α˙ 11 = α˙ 11 .

(22)

The collapse by ratchetting occurs only for a load factor greater than the shakedown one. After a transient period during which plastic strain may occur, elastic shakedown takes place, that is, the response of the material becomes purely elastic, and the back-stress tends to a time-periodic solution. In other words, the back-stress increment over the collapse cycle vanishes: 1X11 =

Z cycle

X˙ 11 dt = c

Z




cycle

+ − α˙ 11 dt = c α˙ 11 − α˙ 11 = 0,

the back-stress being linearly dependent on the internal variable of kinematic hardening. Combining with (22) gives: + − α˙ 11 = α˙ 11 = 0. ± Consequently, the value of the back-stress X11 is deduced from expression (21) of α˙ 11 ; it is solution of:

γ hX11 − Xl im σy + X11 − σ = 0. c In the particular case where X11 6 Xl , X11 = σ and the Prager’s linear hardening rule is recovered. Prager’s linear rule does not describe any ratchetting effect. Under periodic cyclic loading, it always leads to elastic shakedown or plastic shakedown (Zarka et al., 1990). If X11 > Xl , introducing the new variable x as: x = X11 − Xl , the following relation has to be solved: g(x) =

γ m x σy + x − σ + Xl = 0. c

Plasticity with non-linear kinematic hardening: analysis by the bipotential approach

109

Unfortunately, it is not possible to obtain analytically closed-form solutions, that is, the positive roots of this equation, but the existence of solutions may be discussed. The derivative g 0 (x) of g(x) with respect to x is positive: γ g 0 (x) = m x m−1 σy + 1 > 0. c It entails that g(x) is a strictly increasing function. Besides, if σ > Xl , g(0) = Xl − σ < 0 and

g(σ − Xl ) =

γ (σ − Xl )m σy > 0. c

Then, the equation g(x) = 0 has a single, and only a single root in the interval ]0, σ − Xl [. This root may be determined numerically for a given value of m , using either the Newton’s method or a fixed-point algorithm. Initially (Chaboche, 1991), the non-linear kinematic hardening rule with a threshold was proposed with exponent m = 1, so that: X11 =

σ + γc Xl σy . 1 + γc σy

(23a)

In the particular case where m = 2: X11 =

−1 + 2 γc Xl σy +

q

1 + 4 γc σy (σ − Xl )

2 γc σy

.

(23b)

For the sake of simplicity and to obtain analytically closed form solutions, these values will be considered hereafter. Putting (23) into (19) leads to the following expression of the maximum load factor or safety factor for which elastic shakedown takes place: √

3 τa = σy

if X11 6 Xl ,

and, if X11 > Xl : √

√

1 3 τa = γ 2 c σy

s 

3 τa =

γ 2 σy2 c

2

c γ

σy + σy 

v u u c t

γ

2

+ σy

− (σ − Xl )2

γ − 2 σy (σ − Xl ) + 1 − c

r

for m = 1,

γ 1 + 4 σy (σ − Xl ) c

(24a)

2

for m = 2.

(24b)

In the particular case where m = 1 and Xl = 0, the maximum load factor for which elastic shakedown does occur for the classical Armstrong–Frederick non-linear kinematic hardening rule is recovered. Its expression has been given by Lemaitre and Chaboche (1988), Chaboche and Nouailhas (1989), Picko and Maier (1995) and demonstrated from the shakedown theory using the bipotential approach by de Saxcé, Tritsch and Hjiaj (1998).

110

G. Bodovillé, G. de Saxcé

6.2. PROPOSITION. - The previous solution (24) for the maximum load factor for which shakedown does occur, is the exact one. Proof. – The key-idea (de Saxcé et al., 1998) is to consider the possible variational formulation of the shakedown problem resulting from introducing the following bifunctional: βs =

Z

 cycle




p

p



p

b ε˙ ij , −α˙ ij ; σij , Xij − σ11 ε˙ 11 − 2σ12 ε˙ 12 + X11 α˙ 11 + 2X12 α˙ 12 dt, p

and to prove that the bifunctional vanishes. For the sets of dual variables (˙εij , −α˙ ij ; σij , Xij ) satisfying f 6 0 and the kinematic hardening rule (2), accounting for expression (9) of the bipotential and the normalisation condition (17), the bifunctional reduces to: βs =



Z cycle

σy +

 m q   γ q 2 p p 2 2 2 X11 + 3X12 − Xl X11 + 3X12 ε˙ eq − σ11 ε˙ 11 + X11 α˙ 11 + 2X12 α˙ 12 dt − τa . c

Moreover, as demonstrated in the previous calculations, X12 = 0. Accounting for the expressions (20) and (21) p p of ε˙ 11 , ε˙ eq and α˙ 11 , the bifunctional vanishes: βs =

 1  2 σy − (σ − X11 )2 − 3τa2 = 0, 3τa

the yield criterion (19) at the extrema of the collapse cycle being satisfied. Therefore, the previous analytical solution (24) for the maximum load factor for which shakedown does occur, is the exact one, and this achieves the proof. 2 6.3. Collapse by ratchetting Collapse by ratchetting takes place in traction when the load factor is greater than the shakedown one. Taking account of (1) and (23), if X11 > Xl the plastic ratchet strain per cycle is given by: 4 δε p = √ s 3

σ − Xl 2

c γ

+ σy

for m = 1,

− (σ − Xl )2 q

1 + 4 γc σy (σ − Xl ) 4 p δε = √ s 1ε12    2 2 q 3 2 γc σy2 − 2 γc σy (σ − Xl ) + 1 − 1 + 4 γc σy (σ − Xl ) p

2 γc σy (σ − Xl ) + 1 −

p

1ε12

for m = 2.

7. Conclusion The bipotential concept sheds a new light on some non-associative laws (Bodovillé, 1999; de Saxcé, 1992; de Saxcé and Bousshine, 1998). An advantage of the method is to design improved numerical algorithms (de Saxcé and Feng, 1998). A particular application of this concept has been presented, devoted to the modelling

Plasticity with non-linear kinematic hardening: analysis by the bipotential approach

111

of the Armstrong-Frederick with a threshold and Prager’s non-linear kinematic hardening rules proposed by Chaboche et al. for cyclic plasticity of metallic materials. Assuming the existence of so-called admissible internal variable rates and associated variables, a possible variational formulation of shakedown problems may be introduced using the bipotential approach. An analytical example of the shakedown of a thin-walled tube under constant traction and alternate cyclic torsion has been considered and it has been demonstrated that the obtained solution for the maximum load factor for which shakedown occurs, is exact.

Appendix. Non-linear Prager’s rule Chaboche and Nouailhas (1989) have introduced a modification of the Prager’s linear kinematic hardening rule:  2

α˙ ij =

p ε˙ ij

γ c

−

p

2 Xeq ε˙ ij .

Such a non-associative model describes an accommodation limit that improves greatly the description of ratchetting but gives an abnormal shape to the stress–strain cycles (non-convexity of the stress–strain loops) (Chaboche et al., 1991). The hardening modulus is now given by: h=c−

γ2 2 X + b(Q − R). c eq

The model admits the following bipotential: b(κ; ˙ π) =

p σy ε˙ eq

p + I f ∗ (˙εij , −p) ˙ + I f (σij



p − Xij , R) + I{0} α˙ ij − ε˙ ij

 2

γ + c

2 p Xeq ε˙ ij



 

+

and the propositions Sections 4.2 and 4.3 can be easily demonstrated. In the calculation of the shakedown safety factor Section 6.1, the relation (21) becomes: ± α˙ 11



 2

γ σ − X11 = 1− 6τa c

 2 X11

.

It entails that: X11 =

c , γ

so that the maximum load factor for which elastic shakedown occurs is now given by: v u

  u √ c 2 t 2 3 τa = σy − σ − , γ

and it can be proved, following proposition Section 6.2, that this solution is the exact one.

γ p 2 Xeq Xij ε˙ ij , c

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G. Bodovillé, G. de Saxcé

Collapse by ratchetting takes place in traction when the load factor is greater than the shakedown one. The plastic ratchet strain per cycle is now obtained as: σ − γc 4 p s √ δε = 1ε12 .   2 3 σy2 − σ − γc p

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