- Email: [email protected]
Elastoplasticity, second principle of thermodynamics and softening
Eur. J. Mech. A/Solids 19 (2000) 51–67 2000 Éditions scientifiques et médicales Elsevier SAS. All rights reserved
Elastoplasticity, second principle of thermodynamics and softening Nor-Edine Abriak, Thierry Fayet Ecole des Mines de Douai, 941, rue C. Bourseul, 59 500 Douai, France (Received 20 November 1998; accepted 16 September 1999) Abstract – This study deals with the elastoplastic materials that are modelled with a yield surface and an associated or non-associated flow rule. The hardening is assumed to be isotropic. The incremental behaviour equations are completed by the equations that characterize the evolution of the boundary conditions. The case of axisymmetrical conditions is sufficient to introduce the concepts of guiding and piloting, that are necessary to distinguish the passive aspects of the evolution of the boundary conditions from the active ones. This approach allows to take into account the more complex situations. This incremental problem enables the study of the existence and the uniqueness of the theoretical response and, moreover, to link this point to the respect of the second principle of thermodynamics, thanks to the Clausius–Duhem inequality and the plastic multiplier concept. It appears that even for a basic associated model it is possible to determine certain loading increments that violate the second principle. This could be interpreted as an instability of the behaviour. 2000 Éditions scientifiques et médicales Elsevier SAS elastoplasticity / softening / incremental problem / behaviour instability
1. Introduction From a general point of view, a (static) problem of mechanics of continuous solid media is defined by a partial differential equation (equilibrium), boundary conditions and material behaviour equations. The variational formulation enables to synthesize all those elements and leads to most problems being solved. However the proof of the existence and the uniqueness of a solution, if the behaviour is elastoplastic, is difficult to give for the general case. Besides, the validation stage imposes to carry out laboratory tests, for which the stress and strain states are, as far as possible, chosen and kept homogeneous. Moreover, in the general case, the assumed continuity of the stress and strain fields enables the consideration that, for any point, one can define a small enough size of elementary volume so that the material state can be found homogeneous. In those situations the problem comes down to a differential equation system, of which the numerical solving is relatively easy and above all that permits to clear up the question of the existence, in the general case, of a unique solution that enforces (or not) the second principle of thermodynamics. The study presented here deals with the particular case of axisymmetrical problems. The incremental elastoplastic relations have already been studied, notably by Maier and Hueckel (1977), Klisinsky et al. (1992), Darve et al. (1995), Imposimato and Nova (1998), under the form: dEε = C dE σ
or dE σ = D dEε ,
(1)
where C and D are the compliance and the stiffness matrices. They enable to deal with the questions of the uniqueness of the response (that is lost for det(C) = 0 or det(D) = 0) and of the theoretical stability of the material behaviour (using Drücker or Hill’s criteria). With this approach, it is necessary to study one matrix for each type of evolution of the boundary conditions (the compliance one when dE σ is given or the stiffness one for
52
N.-E. Abriak, T. Fayet
dEε , for example). One difficulty is to be sure that all the possibilities are taken into account. Moreover, all the state variables do not appear in the considered response: in particular, the plastic multiplier 3 has to be positive and, when the response exists, this point has to be checked, as done by Maier or Klisinsky. In this work the ‘incremental problem’ is defined first by all the constitutive relations, and second by the equations that characterize the evolution of the boundary conditions. The main hypothesis concerning those equations is that it is possible to determine a unique scalar variable, called the piloting variable, that defines the intensity of the process. Thus the problem can be presented under the form: P dE x = dE y,
(2)
where dE x is the vector constituted by the increments of all the state variables and by the plastic multiplier, corresponding to the response, dE y is the vector, the same size as dE x , constituted first by the piloting increment and then by nil components. P is a matrix that characterizes both the material behaviour and the evolution of the boudary conditions (with the exception of its intensity). This last form directly gives the plastic multiplier in the response and moreover enables to consider more general evolutions of the boundary conditions that can couple all the state variables that are concerned. It appears that it is always possible to find boundary conditions that lead to a negative plastic multiplier, and so that violates the second principle of thermodynamics (negative dissipation). Such a response cannot be admitted; physically, and following Schofield and Wroth (1968), it can be interpreted as an instability of the behaviour (Fayet, 1997). Nevertheless, if the physical interpretation of this result is admitted, it enables to conceive a negative hardening modulus, and thus to represent softening, if the sign of the plastic multiplier is checked. 2. Definition of the incremental problem Consider an elementary volume of the modelised material, in a given state of stress and strain, supposed homogeneous: ∂σ = 0 and ∂xi thus
∂ε = 0; ∂xi
− → div(σ ) = 0E
and the volume forces are necessarily null (the internal equilibrium has to be verified). The configurations considered here are restricted to the axisymmetrical geometry and loading. The used vectorial basis is a principal one, the symmetry axis being numbered by 1. 2.1. Behaviour equations 2.1.1. Description of the stress and strain states
σ1
0
σ = 0 0
σ3
0
0
1 0 0 = 3 tr(σ )1 + σ σ3
with
1
0
2 σ 0 = (σ1 − σ3 ) 0 −1/2 3 0 0
0 0 −1/2
.
(3)
Elastoplasticity, second principle of thermodynamics and softening
53
The invariants σ1 = tr(σ ) and σ50 = 12 tr(σ 02 ) are not sufficient to distinguish all the possible axisymmetrical: indeed, σ50 = 13 (σ1 − σ3 )2 does not depend on the sign of (σ1 − σ3 ), whereas this quantity is meaningful (σ1 minor or major principal stress; two distinct mean pressures correspond to the same shear intensity). The stress state is also characterized by the invariants: (
p = 13 tr(σ ) = 13 (σ1 + 2σ3 ), √ p q = − 3 · sin(3θ) · σ50 = (σ1 − σ3 ),
(4)
where θ is the Lode angle, that measures (−π/6) or (π/6) according to the case.
ε1
0
ε= 0
ε3
0
0
0
1 0 0 = 3 tr(ε)1 + ε ε3
1
2 ε0 = (ε1 − ε3 ) 0 3 0
with
0
0
−1/2
0
0
−1/2
.
(5)
The strain state is characterized by: (
εv = tr(ε) = ε1 + 2ε3 , √
εd = − 2 3 3 sin(3θ) ·
p
0 ε5 = 23 (ε1 − ε3 ).
(6)
Remark: σ : dε = p · dεv + q · dεd . Decomposition in elastic and plastic strains:
ε=ε +ε , e
p
εv = εve + εvp , i.e.: p εd = εde + εd
thus
dεv = dεve + dεvp , p dεd = dεde + dεd .
(7)
2.1.2. State laws Using the presentation of Lemaitre and Chaboche (1988), the specific potential of free energy, supposed to be strictly convex relative to all variables and consisting of an elastic potential and a plastic one (the hardening variables being noted α and pc ), not coupled to the previous one, is written: 9=
1 ψe εve , εde + ψp (α) ρ
(8)
with ∂ 2 ψe = ψ2v > 0, ∂εve 2
∂ 2 ψe = ψ2d > 0, ∂εde 2
and
∂ 2 ψp = ψ2α > 0. ∂α 2
However, the case where 9 = 9(εve , εde ), that concerns the plasticity without hardening, is considered as well. The state laws are then written: p=ρ
∂9 , ∂εve
q =ρ
∂9 , ∂εde
pc = ρ
∂9 ∂α
e dp = ψ2v · dεv ,
thus
dq = ψ2d · dεde , dpc = ψ2α · dα,
because of the assumption that the isotropic and deviatoric behaviours are not coupled.
(9)
54
N.-E. Abriak, T. Fayet
The strict convexity condition comes down to suppose that there cannot be any variation in some flux variable without variation in the corresponding force variable, those variations presenting the same sign. 2.1.3. Yielding potential; generalized flow rule In the case of an axisymmetrical problem and for an isotropic behaviour, the yielding potential can be noted g(p, q, pc ), and is supposed to be convex, containing the origin, and positive as soon as there is a plastic flow. The material respects the generalized flow rule: ∂g dα = − 3 = −gpc · 3 and ∂pc
dεvp = dε p d
=
∂g 3 = gp ∂p ∂g 3 = gq ∂q
· 3, · 3.
(10)
It has to be noticed that this rule introduces the plastic multiplier as a scalar that can be positive or negative. Those formulae show that gp and gq have to be of the same unity, the inverse of the one for 3; the ones of dα and gpc having to be coherent with the previous ones. Remark: If α = α(εp ), then dα =
∂α ∂εp
: dε p and the relation −
∂g ∂α ∂g = p: ∂pc ∂ε ∂σ
has to be respected. 2.1.4. Yielding surface; consistency condition; mechanical dissipation The yielding surface admits the expression: f (p, q, pc ) = 0. Consistency condition: if f = 0 and dpc 6= 0, then ∂f ∂f ∂f dpc = 0, dp + dq + ∂p ∂q ∂pc fp · dp + fq · dq + fpc · dpc = 0.
df = 0 ⇔
noted too
(11)
Besides, ∂ 29 dα = −ψ2α · fpc · gpc · 3 ∂α 2 thus, defining the hardening modulus (that is not necessarily homogeneous to a stress) by h = ψ2α · fpc · gpc , one can write: −−→ − → fp · dp + fp · dq − h · 3 = 0, that is noted too gradf · dσ = h · 3 (12) − → (dσ is the ‘non generalised’ stress increment, that means independent on dpc ). It is clear that if the material admits an associated flow rule (f = g), then h > 0; if moreover fpc 6= 0, then h > 0. A model shows a negative hardening modulus if, and only if, fpc and gpc are of opposite signs. At last, for 9 = 9(εve , εde ), then h = 0, whatever the state of the material may be (the variables α and pc are then no longer taken into account). The Clausius–Duhem inequality enables the definition of the (volumetric) mechanical dissipation: fpc · dpc = fpc · ρ
δqm = σ : dε p − pc · dα = (δwp − dψp ) > 0.
(13)
Elastoplasticity, second principle of thermodynamics and softening
55
The δ symbol means that the concerned increments are not exact differentials of the state variables. It appears that during a hardening phase a certain part of the plastic strain work contributes towards the increase of the internal energy of the material, while during a softening phase, the dissipation is greater than the only plastic strain energy (those two quantities are equal only if the hardening modulus is null). It seems that the term of free energy (from Helmoltz), that comes from the gas study, can hardly be generalized to such a material; it would be clearer to speak of ‘blocked’ energy, because the only elastic energy is free of recuperation under a mechanical form. 2.1.5. Behaviour equation checkup Taking into account the two strain decomposition equations, one has seven equations at one’s disposal. Transferring the first ones into the elastic behaviour ones, they come down to five: (1/ψ2v ) dp − dεv + dεvp = 0, p (1/ψ2d ) dq − dεd + dεd = 0,
dε p − gp · 3 = 0,
v p dε d − gq · 3 = 0,
(14)
fp · dp + fq · dq − h · 3 = 0.
This system holds seven unknown functions: two equations are missing to obtain an a priori soluble problem (this number obviously corresponds to the bidimensional aspect of the considered problem). 2.2. Equations characterizing the loading increment The assumption of the homogeneity of the material state leads the kinematic boundary conditions to be equivalent to imposed strains. The most general form of the equations that enables the description of the evolution of the boundary conditions is thus the one of a linear system of dp, dq, dεv , dεd , because the p unknown quantities dεvp , dεd and 3 are not directly fixed at the boundaries of the elementary volume. Thus they are written: rp · dp + rq · dq + rv · dεv + rd · dεd = dy1 ,
(15-a)
sp · dp + sq · dq + sv · dεv + sd · dεd = dy2 .
(15-b)
The quantities dy1 and dy2 cannot be simultaneously null: indeed in this case the response corresponds to the lack of any processing. Thus, one can suppose that for instance dy1 6= 0, in which case the previous equations come down to: rp · dp + rq · dq + rv · dεv + rd · dεd = dy1 ,
(16-a)
lp · dp + lq · dq + lv · dεv + ld · dεd = 0
(16-b0 )
with l• = s• − r• dy2 /dy1 . If dy1 and dy2 are all too different from zero, then the system (a0 ), (b) can obviously be used as well, with mp · dp + mq · dq + mv · dεv + md · dεd = 0, proportional to (b0 ) since m• = r• − s•
dy1 dy1 = −l• . dy2 dy2
(a0 )
56
N.-E. Abriak, T. Fayet
The equation (a0 ) or (b0 ) is called the guiding equation and the (b) or (a) one (respectively) the piloting one. Remark: Each pair of equations, that can be deducted from the previous one by multiplying them by any non null factor, defines the same loading increment, since the intensity of the dy values is not intrinsically meaningful (only their ratio is such); those equations therefore rigorously define loading increment classes. The simple compression and oedometric tests for example admit the following representations of their loading increments: (
dε1 = 13 dεv + dεd = dε 6= 0, dσ3 = dp − 13 dq = 0
(
and
dσ1 = dp + 23 dq = dσ 6= 0,
(17)
dε3 = 13 dεv − 12 dεd = 0;
it appears that guiding and piloting can be of different natures (stress or strain). Otherwise, for a sample (of volume Vs ), lying in a compressible fluid (of volume Vf and of compressibility modulus K), the whole being contained in a recipient that can be strained as wanted (of volume Vr ), one of the equations takes the form (when the soil mechanics convention is used, i.e. the compression is positive): dp − K
Vs dVr dεv = −K . Vf Vf
(18)
Thus guiding and piloting (for dVr = 0 and dVr 6= 0, respectively) can couple stresses and strains. 3. Elastic problem solving 3.1. Existence and uniqueness of the solution; well stated problem The elastic problem corresponds to the particular case where the plastic strains are null. It consists of a system of four equations for four unknowns, that can be written:
− → − → Pe · dX = dY ,
with
− → (dX)t = (dp, dq, dεv , dεd ),
and
− → (dY )t = (dy, 0, 0, 0), Thus,
rp
lp Pe = (1/ψ2v )
0
rq
rv
rd
lq
lv
ld
0
−1
(1/ψ2d )
0
. (19) 0
−1
1 1 1 1 rv lq + ld − rq + rd lp + lv . ψ2v ψ2d ψ2d ψ2v The problem is said to be well stated if the matrix Pe is inversible, i.e. for det(Pe ) 6= 0. For a well stated problem, if det(Pe ) is of small absolute value, then some unknown increments are big compared to dy. The study of the physical meaning of the badly stated problems leads to distinguish two classes of situations: on the one hand the ones where the determinant cancels itself independently of the elastic characteristics of the material (deficient boundary conditions) and on the other hand the ones where the behaviour equations are contradictory or redundant with the evolution of the boundary conditions. det(Pe ) = rp +
Remark: The loading increment
dε1 = dε 6= 0, dσ1 = dσ 6= 0
Elastoplasticity, second principle of thermodynamics and softening can be written
(1 3
57
dεv + dεd = dε 6= 0,
dp + 23 dq − 3dσdε dεv −
dσ dε
dεd = 0;
thus 2 1 − 6= 0 (a priori), 9ψ2v ψ2v whatever dε and dσ may be; the stress and strain increment can be fixed independently for one of the geometrical directions, if it is possible to operate as much as necessary on the increments of the complementary direction, even if they do not explicitly appear in the boundary condition equations. det(Pe ) =
3.1.1. Deficient boundary conditions Three different cases have to be taken into account: 1. lp = lq = lv = ld = 0 lack of equation. 2. rp = rq = rv = rd = 0 3.
(
(i) (
(iii)
rv = rd = lv = ld = 0, rp · lq − rq · lp = 0,
(ii) (
rv = rp = lv = lp = 0, rq · ld − rd · lq = 0
or (iv)
contradiction with dy1 6= 0. rp = rq = lp = lq = 0, rv · ld − rd · lv = 0, rd = rq = ld = lq = 0, rv · lp − rp · lv = 0
contradiction with dy1 6= 0.
Such evolutions of the boundary conditions are intrinsically either badly defined or contradictory: it can be considered that they are not physically meaningful; it is impossible to obtain them experimentally. 3.1.2. Incompatibility between the behaviour equations and the loading increment Besides the previous cases, the determinant does not cancel itself independently of the behaviour characteristics, and three other situations can be distinguished: 4. rv = rp = lv = lp = 0,
(i)
rd = rq = ld = lq = 0,
rq · ld − rd · lq 6= 0, lq 6= −ψ2v · lq
or
(ii)
rv · lp − rp · lv 6= 0, lv 6= −ψ2d · lq
incompatible guiding.
5.
6. (i)
rp + rq +
1 r ψ2v v 1 r ψ2v d
=0 =0
or (ii)
lv = −ψ2v · lp , ld = −ψ2d · lq .
rp + rq +
1 r ψ2v v 1 r ψ2v d
linked guiding.
6= 0 and
lp +
6= 0 and
lq +
1 l ψ2v v 1 l ψ2v d
6= 0, 6= 0
incompatible piloting.
58
N.-E. Abriak, T. Fayet
The boundary conditions are a priori totally independent of the material behaviour, thus those situations have to be considered. The case 4 shows a lack of equation for the missing invariant type and above all a guiding equation contradictory to one of the behaviour equations. The case 5 corresponds to the situation for which the first one is linked with the second ones, independently of any piloting. The case 6 corresponds to the situation where the piloting is incompatible with the pair guiding-behaviour. In other words, for a fixed guiding, the behaviour forbids a certain piloting type: for example, in the case of linear elasticity, the guiding dσ1 = 2ν · dσ3 imposes dε1 = 0, whatever the piloting may be, that can therefore not be conceived under the form dε1 6= 0 (see (5)). Finally, from a numerical point of view, those situations are essentially singularities, because the probability to numerically obtain the mentioned perfect equalities is very small. 3.2. Response form for a well stated problem 3.2.1. Justification of the guiding and piloting terms E For a well-stated problem, the total inversion of the matrix is not necessary: indeed, due to the form of the dY −1 vector, the first column of (Pe ) is sufficient and its elements are the ratios, on the one hand of the determinants associated with the elements of the first line of (Pe ), and on the other hand of det(Pe ):
dXi = Pe−1
i1 dy
=
det1i (Pe ) dy det(Pe )
with
detj k (P ) = (−1)(j +k) · det P (j,k) ,
(20)
where P (j,k) is the matrix obtained from P when the j th line and the kth column are suppressed. − → Now the det1i (Pe ) do not depend on the piloting, therefore the dX vector is completely fixed in direction by the guiding equation. In other words, for a given guiding, two pilotings µ and ν (that define well-stated problems) lead to two responses linked by: Eν = dX
det(Peµ ) · dy ν E µ dX . det(Peν ) · dy µ
(21)
They are parallel: the piloting operates only on the way and the rhythm (the intensity) of the processing. 3.2.2. Importance of the piloting The physical meaning of the concept of a well stated problem can be pointed out. Consider indeed some sensible (the l• not all null) and not linked guiding: it is always possible to define (at least) one piloting class corresponding to a well-stated problem. Then consider two distinct piloting classes, µ and ν: (
Assume that det(Peµ ) 6= 0.
µ: rp · dp + rq · dq + rv · dεv + rd · dεd = dy µ ; ν: sp · dp + sq · dq + sv · dεv + sd · dεd = dy ν .
(22)
Elastoplasticity, second principle of thermodynamics and softening
59
Note dν µ = sp · dp + sq · dq + sv · dεv + sd · dεd , where the increments dp, dq, dεv , dεd result from the problem defined by the µ piloting; thus: dνeµ =
[sp · det11 (Peµ ) + sq · det12 (Peµ ) + sv · det13 (Peµ ) + sd · det(Peµ )] µ det(Peν ) µ dy = dy , det(Peµ ) det(Peµ )
(23)
then det(Peν ) = 0 ⇔ dνeµ = 0. It appears that if the µ piloting leads to a response such as the (µ, ν µ ) diagram presents a tangent parallel to the µ axis, then the ν piloting leads to a badly stated problem for this stress state that presents a tangent orthogonal to the ν axis at the corresponding point of the (ν, µν ) diagram. 3.2.3. Integration of the differential system When the quantities ψ2v and ψ2d and the guiding and piloting equations are independent of the material state, and if the functions l• (y) and r• (y) are not too complicated, the analytical integration of the response is not problematic. 4. Elastoplastic problem solving If f < 0, the elastic problem is the only one to be considered; if f = 0, the stress state lying on the boundary of the elastic domain, the following criterion is used to judge if the response is purely elastic or not: −−→ − → • if (gradf · dσ ) 6 0, then the behaviour is purely elastic; −−→ − → • if (gradf · dσ ) > 0, then the behaviour is elastoplastic. This criterion is sufficient if the loading increment is exclusively expressed in stress (rv = rd = lv = ld = 0): the −−→ − → dp and dq increments are so given and the gradf · dσ calculus presents no ambiguity. − → Else, dσ has to be determined thanks to the behaviour equations and two distinct increments are obtained: − → − → dσ e if the elasticity equations are the only ones used and dσ p otherwise. Moreover the Clausius–Duhem inequality, expressing the second principle of thermodynamics, can be written: ∂g ∂g ∂g 3 p > 0 ⇔ 3 > 0, (24) +q + pc ∂p ∂q ∂pc because of the assumed properties of g (Lemaitre and Chaboche, 1988). So it is essential to state under which conditions the following involvement is verified: (for f = 0)
−−→
gradf · dE σe > 0 ⇒ 3 > 0 .
(25)
4.1. Existence and uniqueness of the elastoplastic response The elastoplastic problem comes down to: − → − → Pp · dx = dy,
( − →
with
p
(dx)t = (dp, dq, dεv , dεd , dεvp , dεd , 3), → − (dy)t = (dy, 0, 0, 0, 0, 0, 0)
(26)
60 and
N.-E. Abriak, T. Fayet
rp
lp (1/ψ2v ) Pp = 0 0 0
rq
rv
rd
0 0
0
lq
lv
ld
0 0
0
0
−1
0
1 0
0
(1/ψ2v )
0
−1
0 1
0
0
0
0
1 0 −gp
0
0
0
0 1 −gq
−fq
0
0
0 0
−fp
thus
det(Pp ) = h · det(Pe ) + fp · gp (rv · lq − rq · lv ) +
h
1 (rv · ld − rd · lv ) ψ2d
1 + fq · gq (rp · ld − rd · lp ) + (rv · ld − rd · lv ) ψ2v
+ fp · gq (rd · lq − rq · ld ) + fq · gp (rp · lv − rv · lp ) .
(27)
Again, the problem will be called well stated if det(Pp ) 6= 0, and it is also necessary to distinguish the cases where this condition is not verified independently of the behaviour equations or not. 4.1.1. Deficient boundary conditions The cases 1, 2 and 3 established thanks to the elastic problem study are valid again. However, in the particular case where h = 0, i.e. for plasticity without hardening, the 3(i) case is replaced with: 3(i0 )
rv = rd = lv = ld = 0.
This is a much less restrictive variant: if the loading increment is exclusively defined in stress, then the elastoplastic problem does not give any response, whatever the material state may be, since the consistency condition imposes a priori some yielding surface evolution, contradictory to the lack of hardening. When such a model is used, it is considered that, in this case, if the processing is neither elastic nor perfectly plastic, then the material presents an unstable behaviour (brittle failure). 4.1.2. Contradiction between the behaviour equations and the loading increment In the situations other than the previous ones and for which det(Pp ) = 0, the material characteristics (and possibly state) have to be taken into account. As in the case of elasticity, those situations seem like singularities. However, when the cases 1, 2 and 3 are separated and without exception: either det(Pp ) = 0 and
det(Pe ) 6= 0,
or det(Pp ) 6= 0
and
det(Pe ) = 0.
The reasoning held for the elastic problem is still accurate: for a given guiding, sensible and not linked, a piloting leading to a badly stated problem corresponds to a variable (combination of the stress and total strain invariants) that presents a horizontal tangent or plateau if it is considered as the response to a piloting leading to a well-stated problem (and defining the horizontal axis).
Elastoplasticity, second principle of thermodynamics and softening
61
4.2. Thermodynamical validity and form of the elastoplastic solution The analysis done in the elastic case is still right: for a given guiding, two piloting µ and ν (defining two well stated problems) lead to two parallel responses; however, contrary to the elastic case, one of the directions, associated with a negative plastic multiplier, reveals itself thermodynamically disqualified. Otherwise the concerned determinants are functions of the material state that evolves during the processing: Pp = Pp [y, p(y), q(y)] so the analytical resolution of the differential system is very uncertain, even if there is no problem for a numerical integration. 4.2.1. Thermodynamical validity: general case
−−→ If the problem is well stated, it is possible to express 3 versus (gradf · dE σe ) (this calculus is valid even if h = 0): 3=
det17 (Pp ) dy; det(Pp )
(28)
noting P 0 the matrix deduced from Pp , considering that gp = gq = 0 and h = 1, one can write that: det17 (P 0 ) det17 (Pp ) det(Pp ) −−→ gradf · dE σe = 30 = dy = dy = 3, 0 det(P ) det(Pe ) det(Pe )
(29)
so 3=
det(Pe ) −−→ gradf · dE σe . det(Pp )
(30)
Conclusion: −−→
gradf · dE σe > 0 ⇒ 3 > 0 ⇔
i.e.
det(Pp ) > 0, det(Pe )
1 1 h + fp · gp (rv · lq − rq · lv ) + (rv · ld − rd · lv ) + fq · gq (rp · ld − rd · lp ) + (rv · ld − rd · lv ) ψ2d ψ2v
+ fp · gq (rd · lq − rq · ld ) + fq · gp (rp · lv − rv · lp ) /det(Pe ) > 0.
(31)
When this condition is not verified, the model predicts either a purely elastic behaviour, or else an elastoplastic one, with negative dissipation, i.e. (for h 6= 0), with softening or hardening whether the hardening modulus is positive or not (respectively). It appears that this last expression permits to study the respect, by the response, of the second principle of thermodynamics, for all the possible evolutions of the boundary conditions. The absurd responses have obviously to be rejected. As for the models without hardening when the stress increment goes out from the elasticity domain, they can be interpreted as corresponding to an unstable behaviour. Moreover this interpretation is perfectly consistent with the tests performed in soil mechanics, as mentioned by Schofield and Wroth (1968). Otherwise, even if the hardening modulus is positive and big, certain loading increments leading to a very small det(Pe ) are likely to make the behaviour unstable. From a general point of view, the cases for which det(Pe ) = det(Pp ) = 0 being a priori non-existent (if the situations 1, 2 and 3 are rejected), there are
62
N.-E. Abriak, T. Fayet
whole domains in the space (rv , rp , rq , rd , lv , lp , lq , ld ) for which det(Pe ) · det(Pp ) < 0, and whose boundaries are properly the singularities where either det(Pe ) or det(Pp ) cancels itself and changes its sign: the second principle of thermodynamics widens their physical meaning. This physical meaning is linked to the peak concept. Consider indeed some given guiding, sensible and not linked, and two pilotings, µ and ν, defining elastic and elastoplastic well-stated problems, as in (22): Suppose for example [det(Peµ ) · det(Ppµ )] > 0 (thermodynamically acceptable response). Note dν µ = sp · dp + sq · dq + sv · dεv + sd · dεd , the increments dp, dq, dεv , dεd resulting from the problems defined by the µ piloting; thus dνpµ
=
det(Ppν ) det(Ppµ )
dy
µ
and
dνeµ
det(Peν ) µ = dy , det(Peµ )
thus
dνpµ
det(Ppν ) det(Peµ ) = · ; dνeµ det(Peν ) det(Ppµ )
(32)
outcome: det(Ppν ) det(Peν )
>0⇔
dνpµ det(Peµ ) µ · dν > 0 ⇔ µ > 0. p det(Peν ) · dy µ dνe
(33)
Conclusion: in the case where the sign of det(Peν )/det(Peµ ) does not depend on the stress state and where the considered loading is monotonic, if the µ piloting leads to a response such as the (µ, ν µ ) diagram presents a peak (non monotony of the ν µ variable versus µ, i.e. the sign of dν µ /dy µ changes), then the response to the ν piloting presents a peak in the (µν , ν) diagram, keeping merged with the previous response up to this peak, and then diverges, the two responses keeping (at least locally) parallel (dν µ /dy µ = dy ν /dµv ) but of opposite directions, one of which being necessarily inadmissible (see (5) and figure 1a). The cases where det(Pe ) = 0 or det(Pp ) = 0 simply correspond to the situations where the acceptable peak presents a horizontal tangent or where the peak ‘degenerates’ into a plateau (before or after the peak). 4.2.2. Particular case: loading increment defined in stress (rv = rd = lv = ld = 0) In the particular case when the loading increment is exclusively defined in stress, det(Pp ) > 0 ⇔ h > 0; det(Pe )
(34)
if the hardening modulus is strictly positive (hardening behaviour), the fundamental criterion is sufficient to distinguish the elastic and elastoplastic behaviour, and moreover to assure the positivity of the dissipation. In the case of negative hardening modulus (softening behaviour), then the incremental problem cannot give any admissible response, that is quite logical: indeed, for softening, it is impossible to increase the stress intensity without provoking the material failure; so the model gives an admissible response only for an elastic discharge. 4.2.3. Particular case: loading increment defined in strain (rp = rq = lp = lq = 0) In the particular case when the loading increment is exclusively defined in strain, det(Pp ) > 0 ⇔ (h + 2ψ2v · fp · gp + ψ2d · fq · gq ) > 0. det(Pe )
(35)
Elastoplasticity, second principle of thermodynamics and softening
(a)
(b)
(c)
(d)
63
Figure 1. Calculated responses for dσ1 = l · dσ3 guiding l1 = 0.599, l2 = 0.9, l3 = 0.994; thin lines: stable responses; thick lines: instable response (for l = l2 and dε1 > 0 piloting); dotted line: simple traction test.
It is important to underline that this condition is still independent of the boundary conditions; only the behaviour characteristics intervene. One remarks that in the case of a material with an associated flow rule (f = g), it is automatically verified. On the other hand, for a material with a non associated flow rule, a positive hardening modulus is not sufficient to guarantee the validity of the solution. Nevertheless, even if the hardening modulus is negative or null, a material with only an associated plastic rule, i.e. for which fp = gp and fq = gq but fpc 6= gpc , can lead to an admissible softening response for this increment type, whatever the material state may be, the smaller the hardening modulus (in absolute value), the more probable the stability: so it appears that this type of model can quite be conceived, because the problems for which the evolution of the boundary conditions is given in terms of strain are current. The Cam Clay model is a living example of such a model (Fayet, 1997). 4.2.4. Particular case: mixed loading increments (associated and pseudo-associated models) In the case of loading increments that couple stress and strain, the form of the reference equation does not allow to study the response corresponding to a material with only an associated plastic rule from a general point of view: it depends strongly on the different potentials; nevertheless it is clear that the instability possibility is not negligible.
64
N.-E. Abriak, T. Fayet
On the other hand, the study of the associated and pseudo-associated models permits to present some general conclusions. Simple mixed increments. The increments that decouple either the geometrical directions of the elementary volume or the invariants of the different tensors are called simple mixed increments. Therefore the reference inequality comes down to:
dεd 6= 0, dp 6= 0, for and dp = 0, dεd = 0 dεv 6= 0, dq 6= 0, for and dq = 0, dεv = 0
for
dσ1 6= 0, and dε3 = 0
dε3 6= 0, dσ1 = 0,
h + ψ2d · fq2 > 0;
(36)
h + ψ2v · fp2 > 0;
(37)
4 · ψ2v · ψ2d 3 h+ fp − fq 4 · ψ2d + 9 · ψ2v 2
2
> 0;
(38)
2 dσ3 6= 0, dε1 6= 0, 9 · ψ2v · ψ2d 1 for h+ > 0. (39) fp + fq and ψ2d + 9 · ψ2v 3 dε1 = 0 dσ3 = 0, Conclusions: for an associated model (with h > 0) the instability cannot occur here; for a pseudo-associated model, if the hardening modulus is negative (softening), then its absolute value has to be small enough to avoid instability whatever the material state may be (in particular a null hardening modulus ensures it); this value has sometimes, but not necessarily, to be smaller than in the case of a loading increment defined exclusively in strain. Otherwise it appears that the increments
dε1 6= 0, dσ3 = 0
and
dσ1 6= 0, dσ3 = 0,
for example, are clearly differentiated by the second principle of thermodynamics: if the guiding is enough to specify the direction of the processing, the piloting form is essential to forecast the stability of the behaviour. Examples of more problematic mixed loading increments. All the loading increments actually considered in the mentioned works on the incremental response are similar to the previous ones (stress increments, strain increments or simple mixed increments). However it is possible to define loading increments that lead to different conclusions. For example:
dσ1 6= 0, dεv 6= 0, dε3 6= 0, dq 6= 0, 3 for and h + ψ2v · fp fp − fq > 0; (40) 2 dεv = 0, dσ1 = 0, dq = 0 dε3 = 0, dσ3 6= 0, dεv 6= 0, dε1 6= 0, dq 6= 0, for and h + ψ2v · fp fp + 3fq > 0; (41) dεv = 0, dσ3 = 0, dq = 0 dε1 = 0, dσ1 6= 0, dεd 6= 0, dε3 6= 0, dp 6= 0, 2 for and h + ψ2d · fq fq − fp > 0; (42) 3 dεd = 0, dσ1 = 0, dp = 0 dε3 = 0, dσ3 6= 0, dεd 6= 0, dε1 6= 0, dp 6= 0, 1 for and h + ψ2d · fq fq + fp > 0. (43) 3 dεd = 0, dσ3 = 0, dp = 0 dε1 = 0, From an experimental point of view, most of those increments are difficult to implement. However they are not physically insignificant: the very first one exactly corresponds to the ‘triaxial’ test, classical in soil mechanics, with stress piloting, on a porous material, saturated but undrained (when the water compressibility and the one
Elastoplasticity, second principle of thermodynamics and softening
65
of the material of the grains is negligible); the very last one needs the same device and can be performed with non porous materials. So it is clear that, in those conditions, even an associated model can lead to an inadmissible response, unless the hardening modulus is positive and big enough. For a pseudo-associated model, the absolute value of the hardening modulus, when it is negative, does not operate: certain material states forbid any stable elastoplastic behaviour for certain types of loading.
5. Particular case: associated flow rule with plasticity independent of the mean pressure For this type of model, which is the reference in solid materials modeling, fq = gq and fp = gp = 0. The reference inequality thus gives:
h + fq2 ·
(rp · ld − rd · lp ) +
1 (r ψ2v v
· ld − rd · lv )
det(Pe )
> 0.
(44)
For all the types of loading increment previously seen, this inequality is always verified for h > 0 (unless f = 0 and only stresses are applied): such a model is thus intrinsically more stable; in practice, not any axisymmetrical test that can be easily performed in laboratory permits, theoretically, to obtain an unstable behaviour (for h > 0). Nevertheless, it is possible to imagine some virtual increments that lead to instability, like the following one (linear elasticity):
dε1 6= 0 dσ1 = l · dσ3
and
2ν < l <
2ν · h + E · fq2 h + E · fq2
.
(45)
The limits of the l values are respectively given by det(Pe ) = 0 and det(Pp ) = 0. The response for this particular guiding have been studied for a Prandtl–Reuss model with linear hardening. The elastic behaviour respects the Hooke’s law, defined by the Young’s modulus (E) and the Poisson’s ratio (ν). The Von Mises’s yield surface is given by the expression: f (p, q, pc ) = g(p, q, pc ) = |q| − pc = 0,
(46)
where pc is the elastic strength in simple traction (pc = F e). The plastic potential is defined by: 1 ψp (α) = hα 2 , 2 R
(47)
p
where α is the cumulated plastic deviatoric strain (α = |dεd |) and h is the constant hardening modulus. The particular material that is considered here is a standard steel for reinforced concrete, defined by the data: E = 2 · 105 MPa,
ν = 0.3,
h = E/69 and
F ei = 500 Mpa.
Three different values of l have been chosen: l1 = 0.599 (< 2n),
l2 = 0.9,
and
l3 = 0.995 >
2νh + Efq2 h + Efq2
.
(48)
66
N.-E. Abriak, T. Fayet
The responses corresponding to this guiding have been calculated first for the dσ1 > 0 piloting and second for the dε1 > 0 one. They are illustrated on the figure 1. The values l1 and l3 are admissible and merge for both of the pilotings (l1 −l10 lines in figure 1). They show the limits of the instability domain for the dε1 > 0 piloting (see figure 1c, corresponding to the yield surface space). It can be seen on figure 1a that l1 −l10 is almost horizontal in the elastic stage, and l3 −l30 is almost horizontal in the elastoplastic stage. The value l2 shows two different responses for each of the pilotings (l2 −l20 for dσ1 > 0 and l2 −l200 for dε1 > 0). In figure 1c it appears that the dε1 > 0 piloting leads to the softening of the material (diminution of the elastic domain), and the calculus of the dissipation gives a negative result. This theoritical response cannot be realistic. 6. Conclusions The incremental problem associated with an elastoplastic isostropic and homogeneous processing leads to introduce the guiding and piloting concepts: the first one, linked to the passive reactions at the boundaries, defines the direction of the processing (relative rates of the variations of the different state variables) and the second one, linked to the active loading on the boudaries, fixes the way and the rythm (the intensity). It allows to state that even a material with an associated flow rule (and thus a strictly positive hardening modulus) gives, for certain types of axisymmetrical loading, a response that violates the second principle of thermodynamics. This result can be interpreted as an instablility (brittle failure or at least homogeneity loss), which is quite convenient with the current approach of elastoplastic models without hardening (h = 0), when the (imposed) stress path crosses the elastic frontier. Otherwise, for a strictly hardening behaviour and if the loading increments are exclusively defined in stress or in strain, then such a response cannot occur. Morover it appears that taking into account the mean pressure for the plastic behaviour leads to models that predict more often the instability. For the material with a non associated flow rule, the more the gradients of the different potentials are distinct, the more the instability is probable; if the hardening modulus is strictly positive, then only the pure stress laoding increments can a priori guarantee the response stability. It appears at well that a softening behaviour can be reproduced with such a model, if its hardening modulus is negative; if it is small enough in absolute value, then the response, for a loading increment purely defined in strain, can be admissible whichever the material state may be; like for a model without hardening, the pure stress loading increments always lead to the rupture. At last the incremental problem concept can be generalized to the tridimensional cases. However it is necessary to take into account that the total strain and stress tensors have not then a priori the same principal directions, that can be distinct to those of their increments. The equations that characterize the loading increment are then more complex (the increments leading to a pure rotation of the stress tensor principal directions have notably to be distinguished from the one leading to a variation of its invariants). This study will be treated in another communication. References Darve F., Flavigny E., Meghachou M., 1995. Constitutive modelling and instabilities of soil behaviour. Computers and Geotechnics 17, 203–224. Fayet T., 1997. Modèles standards généralisés et respect du deuxième principe de la thermodynamique. In: 3ème Congrès de Mécanique du Maroc, Tétouan, tome Ia, pp. 158–163. Imposimato S., Nova R., 1998. An investigation on the uniqueness of the incremental response of elastoplastic model for virgin sand. Mechanics of Cohesice-Frictional Material 3, 65–87.
Elastoplasticity, second principle of thermodynamics and softening
67
Klisinski M., Mroz Z., Runesson K., 1992. Structure of constitutive equations in plasticity for different choices of state and control variables. Int. J. of Plasticity 8, 221–243. Lemaitre J., Chaboche J.-L., 1988. Mécanique des Matériaux Solides. Dunod, Paris. Maier G., Hueckel T., 1977. Non associated and coupled flow rules of elastoplasticity for geotechnical media. Special session “Constitutive relations for soils”, 9th ICSMFE, pp. 129–142. Schofield A., Wroth P., 1968. Critical State Soil Mechanics. McGraw-Hill, London.
Elastoplasticity, second principle of thermodynamics and softening Nor-Edine Abriak, Thierry Fayet Ecole des Mines de Douai, 941, rue C. Bourseul, 59 500 Douai, France (Received 20 November 1998; accepted 16 September 1999) Abstract – This study deals with the elastoplastic materials that are modelled with a yield surface and an associated or non-associated flow rule. The hardening is assumed to be isotropic. The incremental behaviour equations are completed by the equations that characterize the evolution of the boundary conditions. The case of axisymmetrical conditions is sufficient to introduce the concepts of guiding and piloting, that are necessary to distinguish the passive aspects of the evolution of the boundary conditions from the active ones. This approach allows to take into account the more complex situations. This incremental problem enables the study of the existence and the uniqueness of the theoretical response and, moreover, to link this point to the respect of the second principle of thermodynamics, thanks to the Clausius–Duhem inequality and the plastic multiplier concept. It appears that even for a basic associated model it is possible to determine certain loading increments that violate the second principle. This could be interpreted as an instability of the behaviour. 2000 Éditions scientifiques et médicales Elsevier SAS elastoplasticity / softening / incremental problem / behaviour instability
1. Introduction From a general point of view, a (static) problem of mechanics of continuous solid media is defined by a partial differential equation (equilibrium), boundary conditions and material behaviour equations. The variational formulation enables to synthesize all those elements and leads to most problems being solved. However the proof of the existence and the uniqueness of a solution, if the behaviour is elastoplastic, is difficult to give for the general case. Besides, the validation stage imposes to carry out laboratory tests, for which the stress and strain states are, as far as possible, chosen and kept homogeneous. Moreover, in the general case, the assumed continuity of the stress and strain fields enables the consideration that, for any point, one can define a small enough size of elementary volume so that the material state can be found homogeneous. In those situations the problem comes down to a differential equation system, of which the numerical solving is relatively easy and above all that permits to clear up the question of the existence, in the general case, of a unique solution that enforces (or not) the second principle of thermodynamics. The study presented here deals with the particular case of axisymmetrical problems. The incremental elastoplastic relations have already been studied, notably by Maier and Hueckel (1977), Klisinsky et al. (1992), Darve et al. (1995), Imposimato and Nova (1998), under the form: dEε = C dE σ
or dE σ = D dEε ,
(1)
where C and D are the compliance and the stiffness matrices. They enable to deal with the questions of the uniqueness of the response (that is lost for det(C) = 0 or det(D) = 0) and of the theoretical stability of the material behaviour (using Drücker or Hill’s criteria). With this approach, it is necessary to study one matrix for each type of evolution of the boundary conditions (the compliance one when dE σ is given or the stiffness one for
52
N.-E. Abriak, T. Fayet
dEε , for example). One difficulty is to be sure that all the possibilities are taken into account. Moreover, all the state variables do not appear in the considered response: in particular, the plastic multiplier 3 has to be positive and, when the response exists, this point has to be checked, as done by Maier or Klisinsky. In this work the ‘incremental problem’ is defined first by all the constitutive relations, and second by the equations that characterize the evolution of the boundary conditions. The main hypothesis concerning those equations is that it is possible to determine a unique scalar variable, called the piloting variable, that defines the intensity of the process. Thus the problem can be presented under the form: P dE x = dE y,
(2)
where dE x is the vector constituted by the increments of all the state variables and by the plastic multiplier, corresponding to the response, dE y is the vector, the same size as dE x , constituted first by the piloting increment and then by nil components. P is a matrix that characterizes both the material behaviour and the evolution of the boudary conditions (with the exception of its intensity). This last form directly gives the plastic multiplier in the response and moreover enables to consider more general evolutions of the boundary conditions that can couple all the state variables that are concerned. It appears that it is always possible to find boundary conditions that lead to a negative plastic multiplier, and so that violates the second principle of thermodynamics (negative dissipation). Such a response cannot be admitted; physically, and following Schofield and Wroth (1968), it can be interpreted as an instability of the behaviour (Fayet, 1997). Nevertheless, if the physical interpretation of this result is admitted, it enables to conceive a negative hardening modulus, and thus to represent softening, if the sign of the plastic multiplier is checked. 2. Definition of the incremental problem Consider an elementary volume of the modelised material, in a given state of stress and strain, supposed homogeneous: ∂σ = 0 and ∂xi thus
∂ε = 0; ∂xi
− → div(σ ) = 0E
and the volume forces are necessarily null (the internal equilibrium has to be verified). The configurations considered here are restricted to the axisymmetrical geometry and loading. The used vectorial basis is a principal one, the symmetry axis being numbered by 1. 2.1. Behaviour equations 2.1.1. Description of the stress and strain states
σ1
0
σ = 0 0
σ3
0
0
1 0 0 = 3 tr(σ )1 + σ σ3
with
1
0
2 σ 0 = (σ1 − σ3 ) 0 −1/2 3 0 0
0 0 −1/2
.
(3)
Elastoplasticity, second principle of thermodynamics and softening
53
The invariants σ1 = tr(σ ) and σ50 = 12 tr(σ 02 ) are not sufficient to distinguish all the possible axisymmetrical: indeed, σ50 = 13 (σ1 − σ3 )2 does not depend on the sign of (σ1 − σ3 ), whereas this quantity is meaningful (σ1 minor or major principal stress; two distinct mean pressures correspond to the same shear intensity). The stress state is also characterized by the invariants: (
p = 13 tr(σ ) = 13 (σ1 + 2σ3 ), √ p q = − 3 · sin(3θ) · σ50 = (σ1 − σ3 ),
(4)
where θ is the Lode angle, that measures (−π/6) or (π/6) according to the case.
ε1
0
ε= 0
ε3
0
0
0
1 0 0 = 3 tr(ε)1 + ε ε3
1
2 ε0 = (ε1 − ε3 ) 0 3 0
with
0
0
−1/2
0
0
−1/2
.
(5)
The strain state is characterized by: (
εv = tr(ε) = ε1 + 2ε3 , √
εd = − 2 3 3 sin(3θ) ·
p
0 ε5 = 23 (ε1 − ε3 ).
(6)
Remark: σ : dε = p · dεv + q · dεd . Decomposition in elastic and plastic strains:
ε=ε +ε , e
p
εv = εve + εvp , i.e.: p εd = εde + εd
thus
dεv = dεve + dεvp , p dεd = dεde + dεd .
(7)
2.1.2. State laws Using the presentation of Lemaitre and Chaboche (1988), the specific potential of free energy, supposed to be strictly convex relative to all variables and consisting of an elastic potential and a plastic one (the hardening variables being noted α and pc ), not coupled to the previous one, is written: 9=
1 ψe εve , εde + ψp (α) ρ
(8)
with ∂ 2 ψe = ψ2v > 0, ∂εve 2
∂ 2 ψe = ψ2d > 0, ∂εde 2
and
∂ 2 ψp = ψ2α > 0. ∂α 2
However, the case where 9 = 9(εve , εde ), that concerns the plasticity without hardening, is considered as well. The state laws are then written: p=ρ
∂9 , ∂εve
q =ρ
∂9 , ∂εde
pc = ρ
∂9 ∂α
e dp = ψ2v · dεv ,
thus
dq = ψ2d · dεde , dpc = ψ2α · dα,
because of the assumption that the isotropic and deviatoric behaviours are not coupled.
(9)
54
N.-E. Abriak, T. Fayet
The strict convexity condition comes down to suppose that there cannot be any variation in some flux variable without variation in the corresponding force variable, those variations presenting the same sign. 2.1.3. Yielding potential; generalized flow rule In the case of an axisymmetrical problem and for an isotropic behaviour, the yielding potential can be noted g(p, q, pc ), and is supposed to be convex, containing the origin, and positive as soon as there is a plastic flow. The material respects the generalized flow rule: ∂g dα = − 3 = −gpc · 3 and ∂pc
dεvp = dε p d
=
∂g 3 = gp ∂p ∂g 3 = gq ∂q
· 3, · 3.
(10)
It has to be noticed that this rule introduces the plastic multiplier as a scalar that can be positive or negative. Those formulae show that gp and gq have to be of the same unity, the inverse of the one for 3; the ones of dα and gpc having to be coherent with the previous ones. Remark: If α = α(εp ), then dα =
∂α ∂εp
: dε p and the relation −
∂g ∂α ∂g = p: ∂pc ∂ε ∂σ
has to be respected. 2.1.4. Yielding surface; consistency condition; mechanical dissipation The yielding surface admits the expression: f (p, q, pc ) = 0. Consistency condition: if f = 0 and dpc 6= 0, then ∂f ∂f ∂f dpc = 0, dp + dq + ∂p ∂q ∂pc fp · dp + fq · dq + fpc · dpc = 0.
df = 0 ⇔
noted too
(11)
Besides, ∂ 29 dα = −ψ2α · fpc · gpc · 3 ∂α 2 thus, defining the hardening modulus (that is not necessarily homogeneous to a stress) by h = ψ2α · fpc · gpc , one can write: −−→ − → fp · dp + fp · dq − h · 3 = 0, that is noted too gradf · dσ = h · 3 (12) − → (dσ is the ‘non generalised’ stress increment, that means independent on dpc ). It is clear that if the material admits an associated flow rule (f = g), then h > 0; if moreover fpc 6= 0, then h > 0. A model shows a negative hardening modulus if, and only if, fpc and gpc are of opposite signs. At last, for 9 = 9(εve , εde ), then h = 0, whatever the state of the material may be (the variables α and pc are then no longer taken into account). The Clausius–Duhem inequality enables the definition of the (volumetric) mechanical dissipation: fpc · dpc = fpc · ρ
δqm = σ : dε p − pc · dα = (δwp − dψp ) > 0.
(13)
Elastoplasticity, second principle of thermodynamics and softening
55
The δ symbol means that the concerned increments are not exact differentials of the state variables. It appears that during a hardening phase a certain part of the plastic strain work contributes towards the increase of the internal energy of the material, while during a softening phase, the dissipation is greater than the only plastic strain energy (those two quantities are equal only if the hardening modulus is null). It seems that the term of free energy (from Helmoltz), that comes from the gas study, can hardly be generalized to such a material; it would be clearer to speak of ‘blocked’ energy, because the only elastic energy is free of recuperation under a mechanical form. 2.1.5. Behaviour equation checkup Taking into account the two strain decomposition equations, one has seven equations at one’s disposal. Transferring the first ones into the elastic behaviour ones, they come down to five: (1/ψ2v ) dp − dεv + dεvp = 0, p (1/ψ2d ) dq − dεd + dεd = 0,
dε p − gp · 3 = 0,
v p dε d − gq · 3 = 0,
(14)
fp · dp + fq · dq − h · 3 = 0.
This system holds seven unknown functions: two equations are missing to obtain an a priori soluble problem (this number obviously corresponds to the bidimensional aspect of the considered problem). 2.2. Equations characterizing the loading increment The assumption of the homogeneity of the material state leads the kinematic boundary conditions to be equivalent to imposed strains. The most general form of the equations that enables the description of the evolution of the boundary conditions is thus the one of a linear system of dp, dq, dεv , dεd , because the p unknown quantities dεvp , dεd and 3 are not directly fixed at the boundaries of the elementary volume. Thus they are written: rp · dp + rq · dq + rv · dεv + rd · dεd = dy1 ,
(15-a)
sp · dp + sq · dq + sv · dεv + sd · dεd = dy2 .
(15-b)
The quantities dy1 and dy2 cannot be simultaneously null: indeed in this case the response corresponds to the lack of any processing. Thus, one can suppose that for instance dy1 6= 0, in which case the previous equations come down to: rp · dp + rq · dq + rv · dεv + rd · dεd = dy1 ,
(16-a)
lp · dp + lq · dq + lv · dεv + ld · dεd = 0
(16-b0 )
with l• = s• − r• dy2 /dy1 . If dy1 and dy2 are all too different from zero, then the system (a0 ), (b) can obviously be used as well, with mp · dp + mq · dq + mv · dεv + md · dεd = 0, proportional to (b0 ) since m• = r• − s•
dy1 dy1 = −l• . dy2 dy2
(a0 )
56
N.-E. Abriak, T. Fayet
The equation (a0 ) or (b0 ) is called the guiding equation and the (b) or (a) one (respectively) the piloting one. Remark: Each pair of equations, that can be deducted from the previous one by multiplying them by any non null factor, defines the same loading increment, since the intensity of the dy values is not intrinsically meaningful (only their ratio is such); those equations therefore rigorously define loading increment classes. The simple compression and oedometric tests for example admit the following representations of their loading increments: (
dε1 = 13 dεv + dεd = dε 6= 0, dσ3 = dp − 13 dq = 0
(
and
dσ1 = dp + 23 dq = dσ 6= 0,
(17)
dε3 = 13 dεv − 12 dεd = 0;
it appears that guiding and piloting can be of different natures (stress or strain). Otherwise, for a sample (of volume Vs ), lying in a compressible fluid (of volume Vf and of compressibility modulus K), the whole being contained in a recipient that can be strained as wanted (of volume Vr ), one of the equations takes the form (when the soil mechanics convention is used, i.e. the compression is positive): dp − K
Vs dVr dεv = −K . Vf Vf
(18)
Thus guiding and piloting (for dVr = 0 and dVr 6= 0, respectively) can couple stresses and strains. 3. Elastic problem solving 3.1. Existence and uniqueness of the solution; well stated problem The elastic problem corresponds to the particular case where the plastic strains are null. It consists of a system of four equations for four unknowns, that can be written:
− → − → Pe · dX = dY ,
with
− → (dX)t = (dp, dq, dεv , dεd ),
and
− → (dY )t = (dy, 0, 0, 0), Thus,
rp
lp Pe = (1/ψ2v )
0
rq
rv
rd
lq
lv
ld
0
−1
(1/ψ2d )
0
. (19) 0
−1
1 1 1 1 rv lq + ld − rq + rd lp + lv . ψ2v ψ2d ψ2d ψ2v The problem is said to be well stated if the matrix Pe is inversible, i.e. for det(Pe ) 6= 0. For a well stated problem, if det(Pe ) is of small absolute value, then some unknown increments are big compared to dy. The study of the physical meaning of the badly stated problems leads to distinguish two classes of situations: on the one hand the ones where the determinant cancels itself independently of the elastic characteristics of the material (deficient boundary conditions) and on the other hand the ones where the behaviour equations are contradictory or redundant with the evolution of the boundary conditions. det(Pe ) = rp +
Remark: The loading increment
dε1 = dε 6= 0, dσ1 = dσ 6= 0
Elastoplasticity, second principle of thermodynamics and softening can be written
(1 3
57
dεv + dεd = dε 6= 0,
dp + 23 dq − 3dσdε dεv −
dσ dε
dεd = 0;
thus 2 1 − 6= 0 (a priori), 9ψ2v ψ2v whatever dε and dσ may be; the stress and strain increment can be fixed independently for one of the geometrical directions, if it is possible to operate as much as necessary on the increments of the complementary direction, even if they do not explicitly appear in the boundary condition equations. det(Pe ) =
3.1.1. Deficient boundary conditions Three different cases have to be taken into account: 1. lp = lq = lv = ld = 0 lack of equation. 2. rp = rq = rv = rd = 0 3.
(
(i) (
(iii)
rv = rd = lv = ld = 0, rp · lq − rq · lp = 0,
(ii) (
rv = rp = lv = lp = 0, rq · ld − rd · lq = 0
or (iv)
contradiction with dy1 6= 0. rp = rq = lp = lq = 0, rv · ld − rd · lv = 0, rd = rq = ld = lq = 0, rv · lp − rp · lv = 0
contradiction with dy1 6= 0.
Such evolutions of the boundary conditions are intrinsically either badly defined or contradictory: it can be considered that they are not physically meaningful; it is impossible to obtain them experimentally. 3.1.2. Incompatibility between the behaviour equations and the loading increment Besides the previous cases, the determinant does not cancel itself independently of the behaviour characteristics, and three other situations can be distinguished: 4. rv = rp = lv = lp = 0,
(i)
rd = rq = ld = lq = 0,
rq · ld − rd · lq 6= 0, lq 6= −ψ2v · lq
or
(ii)
rv · lp − rp · lv 6= 0, lv 6= −ψ2d · lq
incompatible guiding.
5.
6. (i)
rp + rq +
1 r ψ2v v 1 r ψ2v d
=0 =0
or (ii)
lv = −ψ2v · lp , ld = −ψ2d · lq .
rp + rq +
1 r ψ2v v 1 r ψ2v d
linked guiding.
6= 0 and
lp +
6= 0 and
lq +
1 l ψ2v v 1 l ψ2v d
6= 0, 6= 0
incompatible piloting.
58
N.-E. Abriak, T. Fayet
The boundary conditions are a priori totally independent of the material behaviour, thus those situations have to be considered. The case 4 shows a lack of equation for the missing invariant type and above all a guiding equation contradictory to one of the behaviour equations. The case 5 corresponds to the situation for which the first one is linked with the second ones, independently of any piloting. The case 6 corresponds to the situation where the piloting is incompatible with the pair guiding-behaviour. In other words, for a fixed guiding, the behaviour forbids a certain piloting type: for example, in the case of linear elasticity, the guiding dσ1 = 2ν · dσ3 imposes dε1 = 0, whatever the piloting may be, that can therefore not be conceived under the form dε1 6= 0 (see (5)). Finally, from a numerical point of view, those situations are essentially singularities, because the probability to numerically obtain the mentioned perfect equalities is very small. 3.2. Response form for a well stated problem 3.2.1. Justification of the guiding and piloting terms E For a well-stated problem, the total inversion of the matrix is not necessary: indeed, due to the form of the dY −1 vector, the first column of (Pe ) is sufficient and its elements are the ratios, on the one hand of the determinants associated with the elements of the first line of (Pe ), and on the other hand of det(Pe ):
dXi = Pe−1
i1 dy
=
det1i (Pe ) dy det(Pe )
with
detj k (P ) = (−1)(j +k) · det P (j,k) ,
(20)
where P (j,k) is the matrix obtained from P when the j th line and the kth column are suppressed. − → Now the det1i (Pe ) do not depend on the piloting, therefore the dX vector is completely fixed in direction by the guiding equation. In other words, for a given guiding, two pilotings µ and ν (that define well-stated problems) lead to two responses linked by: Eν = dX
det(Peµ ) · dy ν E µ dX . det(Peν ) · dy µ
(21)
They are parallel: the piloting operates only on the way and the rhythm (the intensity) of the processing. 3.2.2. Importance of the piloting The physical meaning of the concept of a well stated problem can be pointed out. Consider indeed some sensible (the l• not all null) and not linked guiding: it is always possible to define (at least) one piloting class corresponding to a well-stated problem. Then consider two distinct piloting classes, µ and ν: (
Assume that det(Peµ ) 6= 0.
µ: rp · dp + rq · dq + rv · dεv + rd · dεd = dy µ ; ν: sp · dp + sq · dq + sv · dεv + sd · dεd = dy ν .
(22)
Elastoplasticity, second principle of thermodynamics and softening
59
Note dν µ = sp · dp + sq · dq + sv · dεv + sd · dεd , where the increments dp, dq, dεv , dεd result from the problem defined by the µ piloting; thus: dνeµ =
[sp · det11 (Peµ ) + sq · det12 (Peµ ) + sv · det13 (Peµ ) + sd · det(Peµ )] µ det(Peν ) µ dy = dy , det(Peµ ) det(Peµ )
(23)
then det(Peν ) = 0 ⇔ dνeµ = 0. It appears that if the µ piloting leads to a response such as the (µ, ν µ ) diagram presents a tangent parallel to the µ axis, then the ν piloting leads to a badly stated problem for this stress state that presents a tangent orthogonal to the ν axis at the corresponding point of the (ν, µν ) diagram. 3.2.3. Integration of the differential system When the quantities ψ2v and ψ2d and the guiding and piloting equations are independent of the material state, and if the functions l• (y) and r• (y) are not too complicated, the analytical integration of the response is not problematic. 4. Elastoplastic problem solving If f < 0, the elastic problem is the only one to be considered; if f = 0, the stress state lying on the boundary of the elastic domain, the following criterion is used to judge if the response is purely elastic or not: −−→ − → • if (gradf · dσ ) 6 0, then the behaviour is purely elastic; −−→ − → • if (gradf · dσ ) > 0, then the behaviour is elastoplastic. This criterion is sufficient if the loading increment is exclusively expressed in stress (rv = rd = lv = ld = 0): the −−→ − → dp and dq increments are so given and the gradf · dσ calculus presents no ambiguity. − → Else, dσ has to be determined thanks to the behaviour equations and two distinct increments are obtained: − → − → dσ e if the elasticity equations are the only ones used and dσ p otherwise. Moreover the Clausius–Duhem inequality, expressing the second principle of thermodynamics, can be written: ∂g ∂g ∂g 3 p > 0 ⇔ 3 > 0, (24) +q + pc ∂p ∂q ∂pc because of the assumed properties of g (Lemaitre and Chaboche, 1988). So it is essential to state under which conditions the following involvement is verified: (for f = 0)
−−→
gradf · dE σe > 0 ⇒ 3 > 0 .
(25)
4.1. Existence and uniqueness of the elastoplastic response The elastoplastic problem comes down to: − → − → Pp · dx = dy,
( − →
with
p
(dx)t = (dp, dq, dεv , dεd , dεvp , dεd , 3), → − (dy)t = (dy, 0, 0, 0, 0, 0, 0)
(26)
60 and
N.-E. Abriak, T. Fayet
rp
lp (1/ψ2v ) Pp = 0 0 0
rq
rv
rd
0 0
0
lq
lv
ld
0 0
0
0
−1
0
1 0
0
(1/ψ2v )
0
−1
0 1
0
0
0
0
1 0 −gp
0
0
0
0 1 −gq
−fq
0
0
0 0
−fp
thus
det(Pp ) = h · det(Pe ) + fp · gp (rv · lq − rq · lv ) +
h
1 (rv · ld − rd · lv ) ψ2d
1 + fq · gq (rp · ld − rd · lp ) + (rv · ld − rd · lv ) ψ2v
+ fp · gq (rd · lq − rq · ld ) + fq · gp (rp · lv − rv · lp ) .
(27)
Again, the problem will be called well stated if det(Pp ) 6= 0, and it is also necessary to distinguish the cases where this condition is not verified independently of the behaviour equations or not. 4.1.1. Deficient boundary conditions The cases 1, 2 and 3 established thanks to the elastic problem study are valid again. However, in the particular case where h = 0, i.e. for plasticity without hardening, the 3(i) case is replaced with: 3(i0 )
rv = rd = lv = ld = 0.
This is a much less restrictive variant: if the loading increment is exclusively defined in stress, then the elastoplastic problem does not give any response, whatever the material state may be, since the consistency condition imposes a priori some yielding surface evolution, contradictory to the lack of hardening. When such a model is used, it is considered that, in this case, if the processing is neither elastic nor perfectly plastic, then the material presents an unstable behaviour (brittle failure). 4.1.2. Contradiction between the behaviour equations and the loading increment In the situations other than the previous ones and for which det(Pp ) = 0, the material characteristics (and possibly state) have to be taken into account. As in the case of elasticity, those situations seem like singularities. However, when the cases 1, 2 and 3 are separated and without exception: either det(Pp ) = 0 and
det(Pe ) 6= 0,
or det(Pp ) 6= 0
and
det(Pe ) = 0.
The reasoning held for the elastic problem is still accurate: for a given guiding, sensible and not linked, a piloting leading to a badly stated problem corresponds to a variable (combination of the stress and total strain invariants) that presents a horizontal tangent or plateau if it is considered as the response to a piloting leading to a well-stated problem (and defining the horizontal axis).
Elastoplasticity, second principle of thermodynamics and softening
61
4.2. Thermodynamical validity and form of the elastoplastic solution The analysis done in the elastic case is still right: for a given guiding, two piloting µ and ν (defining two well stated problems) lead to two parallel responses; however, contrary to the elastic case, one of the directions, associated with a negative plastic multiplier, reveals itself thermodynamically disqualified. Otherwise the concerned determinants are functions of the material state that evolves during the processing: Pp = Pp [y, p(y), q(y)] so the analytical resolution of the differential system is very uncertain, even if there is no problem for a numerical integration. 4.2.1. Thermodynamical validity: general case
−−→ If the problem is well stated, it is possible to express 3 versus (gradf · dE σe ) (this calculus is valid even if h = 0): 3=
det17 (Pp ) dy; det(Pp )
(28)
noting P 0 the matrix deduced from Pp , considering that gp = gq = 0 and h = 1, one can write that: det17 (P 0 ) det17 (Pp ) det(Pp ) −−→ gradf · dE σe = 30 = dy = dy = 3, 0 det(P ) det(Pe ) det(Pe )
(29)
so 3=
det(Pe ) −−→ gradf · dE σe . det(Pp )
(30)
Conclusion: −−→
gradf · dE σe > 0 ⇒ 3 > 0 ⇔
i.e.
det(Pp ) > 0, det(Pe )
1 1 h + fp · gp (rv · lq − rq · lv ) + (rv · ld − rd · lv ) + fq · gq (rp · ld − rd · lp ) + (rv · ld − rd · lv ) ψ2d ψ2v
+ fp · gq (rd · lq − rq · ld ) + fq · gp (rp · lv − rv · lp ) /det(Pe ) > 0.
(31)
When this condition is not verified, the model predicts either a purely elastic behaviour, or else an elastoplastic one, with negative dissipation, i.e. (for h 6= 0), with softening or hardening whether the hardening modulus is positive or not (respectively). It appears that this last expression permits to study the respect, by the response, of the second principle of thermodynamics, for all the possible evolutions of the boundary conditions. The absurd responses have obviously to be rejected. As for the models without hardening when the stress increment goes out from the elasticity domain, they can be interpreted as corresponding to an unstable behaviour. Moreover this interpretation is perfectly consistent with the tests performed in soil mechanics, as mentioned by Schofield and Wroth (1968). Otherwise, even if the hardening modulus is positive and big, certain loading increments leading to a very small det(Pe ) are likely to make the behaviour unstable. From a general point of view, the cases for which det(Pe ) = det(Pp ) = 0 being a priori non-existent (if the situations 1, 2 and 3 are rejected), there are
62
N.-E. Abriak, T. Fayet
whole domains in the space (rv , rp , rq , rd , lv , lp , lq , ld ) for which det(Pe ) · det(Pp ) < 0, and whose boundaries are properly the singularities where either det(Pe ) or det(Pp ) cancels itself and changes its sign: the second principle of thermodynamics widens their physical meaning. This physical meaning is linked to the peak concept. Consider indeed some given guiding, sensible and not linked, and two pilotings, µ and ν, defining elastic and elastoplastic well-stated problems, as in (22): Suppose for example [det(Peµ ) · det(Ppµ )] > 0 (thermodynamically acceptable response). Note dν µ = sp · dp + sq · dq + sv · dεv + sd · dεd , the increments dp, dq, dεv , dεd resulting from the problems defined by the µ piloting; thus dνpµ
=
det(Ppν ) det(Ppµ )
dy
µ
and
dνeµ
det(Peν ) µ = dy , det(Peµ )
thus
dνpµ
det(Ppν ) det(Peµ ) = · ; dνeµ det(Peν ) det(Ppµ )
(32)
outcome: det(Ppν ) det(Peν )
>0⇔
dνpµ det(Peµ ) µ · dν > 0 ⇔ µ > 0. p det(Peν ) · dy µ dνe
(33)
Conclusion: in the case where the sign of det(Peν )/det(Peµ ) does not depend on the stress state and where the considered loading is monotonic, if the µ piloting leads to a response such as the (µ, ν µ ) diagram presents a peak (non monotony of the ν µ variable versus µ, i.e. the sign of dν µ /dy µ changes), then the response to the ν piloting presents a peak in the (µν , ν) diagram, keeping merged with the previous response up to this peak, and then diverges, the two responses keeping (at least locally) parallel (dν µ /dy µ = dy ν /dµv ) but of opposite directions, one of which being necessarily inadmissible (see (5) and figure 1a). The cases where det(Pe ) = 0 or det(Pp ) = 0 simply correspond to the situations where the acceptable peak presents a horizontal tangent or where the peak ‘degenerates’ into a plateau (before or after the peak). 4.2.2. Particular case: loading increment defined in stress (rv = rd = lv = ld = 0) In the particular case when the loading increment is exclusively defined in stress, det(Pp ) > 0 ⇔ h > 0; det(Pe )
(34)
if the hardening modulus is strictly positive (hardening behaviour), the fundamental criterion is sufficient to distinguish the elastic and elastoplastic behaviour, and moreover to assure the positivity of the dissipation. In the case of negative hardening modulus (softening behaviour), then the incremental problem cannot give any admissible response, that is quite logical: indeed, for softening, it is impossible to increase the stress intensity without provoking the material failure; so the model gives an admissible response only for an elastic discharge. 4.2.3. Particular case: loading increment defined in strain (rp = rq = lp = lq = 0) In the particular case when the loading increment is exclusively defined in strain, det(Pp ) > 0 ⇔ (h + 2ψ2v · fp · gp + ψ2d · fq · gq ) > 0. det(Pe )
(35)
Elastoplasticity, second principle of thermodynamics and softening
(a)
(b)
(c)
(d)
63
Figure 1. Calculated responses for dσ1 = l · dσ3 guiding l1 = 0.599, l2 = 0.9, l3 = 0.994; thin lines: stable responses; thick lines: instable response (for l = l2 and dε1 > 0 piloting); dotted line: simple traction test.
It is important to underline that this condition is still independent of the boundary conditions; only the behaviour characteristics intervene. One remarks that in the case of a material with an associated flow rule (f = g), it is automatically verified. On the other hand, for a material with a non associated flow rule, a positive hardening modulus is not sufficient to guarantee the validity of the solution. Nevertheless, even if the hardening modulus is negative or null, a material with only an associated plastic rule, i.e. for which fp = gp and fq = gq but fpc 6= gpc , can lead to an admissible softening response for this increment type, whatever the material state may be, the smaller the hardening modulus (in absolute value), the more probable the stability: so it appears that this type of model can quite be conceived, because the problems for which the evolution of the boundary conditions is given in terms of strain are current. The Cam Clay model is a living example of such a model (Fayet, 1997). 4.2.4. Particular case: mixed loading increments (associated and pseudo-associated models) In the case of loading increments that couple stress and strain, the form of the reference equation does not allow to study the response corresponding to a material with only an associated plastic rule from a general point of view: it depends strongly on the different potentials; nevertheless it is clear that the instability possibility is not negligible.
64
N.-E. Abriak, T. Fayet
On the other hand, the study of the associated and pseudo-associated models permits to present some general conclusions. Simple mixed increments. The increments that decouple either the geometrical directions of the elementary volume or the invariants of the different tensors are called simple mixed increments. Therefore the reference inequality comes down to:
dεd 6= 0, dp 6= 0, for and dp = 0, dεd = 0 dεv 6= 0, dq 6= 0, for and dq = 0, dεv = 0
for
dσ1 6= 0, and dε3 = 0
dε3 6= 0, dσ1 = 0,
h + ψ2d · fq2 > 0;
(36)
h + ψ2v · fp2 > 0;
(37)
4 · ψ2v · ψ2d 3 h+ fp − fq 4 · ψ2d + 9 · ψ2v 2
2
> 0;
(38)
2 dσ3 6= 0, dε1 6= 0, 9 · ψ2v · ψ2d 1 for h+ > 0. (39) fp + fq and ψ2d + 9 · ψ2v 3 dε1 = 0 dσ3 = 0, Conclusions: for an associated model (with h > 0) the instability cannot occur here; for a pseudo-associated model, if the hardening modulus is negative (softening), then its absolute value has to be small enough to avoid instability whatever the material state may be (in particular a null hardening modulus ensures it); this value has sometimes, but not necessarily, to be smaller than in the case of a loading increment defined exclusively in strain. Otherwise it appears that the increments
dε1 6= 0, dσ3 = 0
and
dσ1 6= 0, dσ3 = 0,
for example, are clearly differentiated by the second principle of thermodynamics: if the guiding is enough to specify the direction of the processing, the piloting form is essential to forecast the stability of the behaviour. Examples of more problematic mixed loading increments. All the loading increments actually considered in the mentioned works on the incremental response are similar to the previous ones (stress increments, strain increments or simple mixed increments). However it is possible to define loading increments that lead to different conclusions. For example:
dσ1 6= 0, dεv 6= 0, dε3 6= 0, dq 6= 0, 3 for and h + ψ2v · fp fp − fq > 0; (40) 2 dεv = 0, dσ1 = 0, dq = 0 dε3 = 0, dσ3 6= 0, dεv 6= 0, dε1 6= 0, dq 6= 0, for and h + ψ2v · fp fp + 3fq > 0; (41) dεv = 0, dσ3 = 0, dq = 0 dε1 = 0, dσ1 6= 0, dεd 6= 0, dε3 6= 0, dp 6= 0, 2 for and h + ψ2d · fq fq − fp > 0; (42) 3 dεd = 0, dσ1 = 0, dp = 0 dε3 = 0, dσ3 6= 0, dεd 6= 0, dε1 6= 0, dp 6= 0, 1 for and h + ψ2d · fq fq + fp > 0. (43) 3 dεd = 0, dσ3 = 0, dp = 0 dε1 = 0, From an experimental point of view, most of those increments are difficult to implement. However they are not physically insignificant: the very first one exactly corresponds to the ‘triaxial’ test, classical in soil mechanics, with stress piloting, on a porous material, saturated but undrained (when the water compressibility and the one
Elastoplasticity, second principle of thermodynamics and softening
65
of the material of the grains is negligible); the very last one needs the same device and can be performed with non porous materials. So it is clear that, in those conditions, even an associated model can lead to an inadmissible response, unless the hardening modulus is positive and big enough. For a pseudo-associated model, the absolute value of the hardening modulus, when it is negative, does not operate: certain material states forbid any stable elastoplastic behaviour for certain types of loading.
5. Particular case: associated flow rule with plasticity independent of the mean pressure For this type of model, which is the reference in solid materials modeling, fq = gq and fp = gp = 0. The reference inequality thus gives:
h + fq2 ·
(rp · ld − rd · lp ) +
1 (r ψ2v v
· ld − rd · lv )
det(Pe )
> 0.
(44)
For all the types of loading increment previously seen, this inequality is always verified for h > 0 (unless f = 0 and only stresses are applied): such a model is thus intrinsically more stable; in practice, not any axisymmetrical test that can be easily performed in laboratory permits, theoretically, to obtain an unstable behaviour (for h > 0). Nevertheless, it is possible to imagine some virtual increments that lead to instability, like the following one (linear elasticity):
dε1 6= 0 dσ1 = l · dσ3
and
2ν < l <
2ν · h + E · fq2 h + E · fq2
.
(45)
The limits of the l values are respectively given by det(Pe ) = 0 and det(Pp ) = 0. The response for this particular guiding have been studied for a Prandtl–Reuss model with linear hardening. The elastic behaviour respects the Hooke’s law, defined by the Young’s modulus (E) and the Poisson’s ratio (ν). The Von Mises’s yield surface is given by the expression: f (p, q, pc ) = g(p, q, pc ) = |q| − pc = 0,
(46)
where pc is the elastic strength in simple traction (pc = F e). The plastic potential is defined by: 1 ψp (α) = hα 2 , 2 R
(47)
p
where α is the cumulated plastic deviatoric strain (α = |dεd |) and h is the constant hardening modulus. The particular material that is considered here is a standard steel for reinforced concrete, defined by the data: E = 2 · 105 MPa,
ν = 0.3,
h = E/69 and
F ei = 500 Mpa.
Three different values of l have been chosen: l1 = 0.599 (< 2n),
l2 = 0.9,
and
l3 = 0.995 >
2νh + Efq2 h + Efq2
.
(48)
66
N.-E. Abriak, T. Fayet
The responses corresponding to this guiding have been calculated first for the dσ1 > 0 piloting and second for the dε1 > 0 one. They are illustrated on the figure 1. The values l1 and l3 are admissible and merge for both of the pilotings (l1 −l10 lines in figure 1). They show the limits of the instability domain for the dε1 > 0 piloting (see figure 1c, corresponding to the yield surface space). It can be seen on figure 1a that l1 −l10 is almost horizontal in the elastic stage, and l3 −l30 is almost horizontal in the elastoplastic stage. The value l2 shows two different responses for each of the pilotings (l2 −l20 for dσ1 > 0 and l2 −l200 for dε1 > 0). In figure 1c it appears that the dε1 > 0 piloting leads to the softening of the material (diminution of the elastic domain), and the calculus of the dissipation gives a negative result. This theoritical response cannot be realistic. 6. Conclusions The incremental problem associated with an elastoplastic isostropic and homogeneous processing leads to introduce the guiding and piloting concepts: the first one, linked to the passive reactions at the boundaries, defines the direction of the processing (relative rates of the variations of the different state variables) and the second one, linked to the active loading on the boudaries, fixes the way and the rythm (the intensity). It allows to state that even a material with an associated flow rule (and thus a strictly positive hardening modulus) gives, for certain types of axisymmetrical loading, a response that violates the second principle of thermodynamics. This result can be interpreted as an instablility (brittle failure or at least homogeneity loss), which is quite convenient with the current approach of elastoplastic models without hardening (h = 0), when the (imposed) stress path crosses the elastic frontier. Otherwise, for a strictly hardening behaviour and if the loading increments are exclusively defined in stress or in strain, then such a response cannot occur. Morover it appears that taking into account the mean pressure for the plastic behaviour leads to models that predict more often the instability. For the material with a non associated flow rule, the more the gradients of the different potentials are distinct, the more the instability is probable; if the hardening modulus is strictly positive, then only the pure stress laoding increments can a priori guarantee the response stability. It appears at well that a softening behaviour can be reproduced with such a model, if its hardening modulus is negative; if it is small enough in absolute value, then the response, for a loading increment purely defined in strain, can be admissible whichever the material state may be; like for a model without hardening, the pure stress loading increments always lead to the rupture. At last the incremental problem concept can be generalized to the tridimensional cases. However it is necessary to take into account that the total strain and stress tensors have not then a priori the same principal directions, that can be distinct to those of their increments. The equations that characterize the loading increment are then more complex (the increments leading to a pure rotation of the stress tensor principal directions have notably to be distinguished from the one leading to a variation of its invariants). This study will be treated in another communication. References Darve F., Flavigny E., Meghachou M., 1995. Constitutive modelling and instabilities of soil behaviour. Computers and Geotechnics 17, 203–224. Fayet T., 1997. Modèles standards généralisés et respect du deuxième principe de la thermodynamique. In: 3ème Congrès de Mécanique du Maroc, Tétouan, tome Ia, pp. 158–163. Imposimato S., Nova R., 1998. An investigation on the uniqueness of the incremental response of elastoplastic model for virgin sand. Mechanics of Cohesice-Frictional Material 3, 65–87.
Elastoplasticity, second principle of thermodynamics and softening
67
Klisinski M., Mroz Z., Runesson K., 1992. Structure of constitutive equations in plasticity for different choices of state and control variables. Int. J. of Plasticity 8, 221–243. Lemaitre J., Chaboche J.-L., 1988. Mécanique des Matériaux Solides. Dunod, Paris. Maier G., Hueckel T., 1977. Non associated and coupled flow rules of elastoplasticity for geotechnical media. Special session “Constitutive relations for soils”, 9th ICSMFE, pp. 129–142. Schofield A., Wroth P., 1968. Critical State Soil Mechanics. McGraw-Hill, London.