International Journal of Plasticity, Vol. 7, pp. 679-692, 1991 0749-6419/91 $3.00 + .00 Printed in the U.S.A. Copyright © 1991 Pergamon Press plc

D Y N A M I C S H A K E D O W N IN E L A S T O P L A S T I C S T R U C T U R E S

W I T H G E N E R A L I N T E R N A L V A R I A B L E C O N S T I T U T I V E L A W S

CLAUDIA COMI a n d ALBERTO CORIGLIANO

Technical University (Politecnico) of Milan

(Communicated by Cesare Davini, Universith degli studi di Udine)

Abs t r ac t - -The shakedown and nonshakedown theory in elastic-plastic dynamics is dealt with here on the following basis: for the material behaviour a quite general class of internal variables constitutive model is adopted allowing for general nonlinear hardening law; discrete structural models described in terms of generalized stresses and strains are referred to. Some earlier re- sults concerning shakedown are extended to the present more general elastic-plastic models; fol- lowing the "static" approach a necessary and sufficient condition for adaptation is established; the "dual ," kinematic, approach gives rise to a necessary and sufficient condition for inadap- tation; two extremum characterizations of the safety factor are presented.

I. I N T R O D U C T I O N

Elastic-plastic structures under external variable-repeated loads can attain a critical sit- uation (failure) either because plastic deformations increase unboundedly in time (ratch- etting) or because alternating plastic strains give rise to local failure (alternating plasticity or low cycle fatigue). The above phenomena, called "inadaptation" or nonshakedown, are characterized by the unbounded growth in time of the total dissipated energy. "Ad- aptation" or shakedown denotes the opposite occurrence, that is, a purely elastic struc- tural behaviour after a finite time period during which plastic yielding processes occur.

Surveys and comprehensive presentation of the shakedown literature can be found, for example, in KONIG and MAIER [1981], KONIG [1987], and POLIZZOTTO [1982]. The classical shakedown theory is based on pioneering works which established "static" shakedown conditions (MELA_~ [1938]), and "kinematic" nonshakedown conditions (SY- MO~rDS R, NEAL [1951]; KOXTER [1956]) for elastic-perfectly plastic continua, in the quasi- static range. Since those early studies, several extensions in different directions were obtained. Valuable contributions concerning shakedown theory for more general elas- tic-plastic constitutive laws with linear and nonlinear hardening are due to MAIER [1970,1987], MAIER and NOVATI [1987], and WEICHERT and GROSS-WEEGE [1988]. In other works the theory was extended to take into account "geometric effects" of defor- mations on equilibrium (see MAIER [1973] and WEICHERT [1986]) and thermal effects (KOmG [1969]). In the dynamic range the central "static" (shakedown) theorem was es- tablished by CERADIm [1969] (see also Ceradini [1980]) for perfectly plastic continua and recently extended by MAmR and NOVA_TX [1990] for structural models with piecewiselinear yield functions and a particular kind of nonlinear hardening. The "kinematic" theorem was established in dynamics by CORRADI and MAIER [1974,1973] for perfectly plastic continua and work hardening structures, respectively. Computational and other aspects of shakedown analysis were extensively studied and discussed, for example, by C~RADI~-I and GAVARINI [1969], COR_P, ADI and ZAVELAr;I [1974], PONTER [1983], MARTIN [1985], POUZZOTTO [1984], ZARKA and NAVIDI [1986] and other authors.

679

680 C. CoMI and A. CORIGLIANO

The aim of this paper is to generalize, as for the constitutive model, the dynamic shakedown theory, following both static and kinematic approaches, thus extending and unifying the above-mentioned earlier results. To this purpose, we assume herein the elas- tic-plastic time independent constitutive law proposed by HALPVlEN and NGUX'EN [1975] ("mat6riaux standard g~n6ralis~s", see also LE~aTRE a Cn_~OCI-IE [1985]). Although its thermodynamic soundness seems to be rather controversial (see ERICICSEN [1983] and DAVINI [1986]), this engineering-oriented description of phenomenological elastic-plas- tic behaviour is here adopted since it provides a convenient formal framework for the whole shakedown theory. The constitutive laws assumed are characterized by dual in- ternal variables, deriving from a thermodynamic potential; they can represent a broad class of nonlinear hardening behaviours. Assuming some constitutive restrictions (ba- sically, the validity of Drucker's stability postulate), and referring to discrete structural systems described in terms of generalized variables, shakedown and nonshakedown nec- essary and sufficient conditions are proved. Following both the static and kinematic ap- proach, extremum characterizations of the safety factor are presented.

I!. GOVERNING RELATIONS

II. 1. A class o f constitutive laws with internal variables

Consider a compatible finite element model of the body, encompassing m finite ele- ments run by index i. Let ~i be the volume of element i. The constitutive law is defined in terms of natural generalized variables. These are variables which govern a field through interpolations over the subdomain ~i such that: they can be interpreted as weighted integrals of the relevant field; they intervene in pairs of "conjugate" quanti- ties, whose scalar product is equal to the integral over ~'/i of the scalar product of the relevant fields. A detailed discussion about the generalized variables formulation can be found in CORRAOI [1978], and in CoMI et al. [in press].

The class of constitutive laws we assume here is defined, for the whole discretized structural model, by the following relations.

= e + p (1)

a w(,~) o = E e ; X - - - (2 )

O~t (a,X),~" / / - (a,X)A (3) p = ~ , ax

= ~ ' ( a , x ) -< 0; , i _> 0; ~ ' , ~ = 0 (4 )

b = a'l~ - x"~ -> 0. (5)

T h e s e r e l a t i o n s e x p r e s s t h e m a t e r i a l c o n s t i t u t i v e l a w in a n a v e r a g e s e n s e o v e r e a c h fi-

n i t e e l e m e n t ( n o t local ly) . T h e y p r e s e r v e t h e e s sen t i a l p r o p e r t i e s o f t h e e l a s t i c - p l a s t i c in-

~Bold face letters denote matrices or (column) vectors. A superposed dot means time derivative. Inequal- ities concerning vectors apply componentwise. In order to remove the ambiguity which may arise because pa- rentheses enclose both an argument and a factor, the latter case (multiplication) will be marked by a dot.

Dynamic shakedown in elastoplastic structures 681

ternal variables material constitutive law which allows for a general nonlinear hardening (HAI.PI-mN ~ N 6 t r ~ s [1975]; L E ~ r ~ & CRtmocrm [1985]).

All quantities involved in eqns (1)-(5) pertain to the whole structure: vectors contain the corresponding elemental vectors as subvectors (run by element index i); scalar en- ergy functions are sums of elemental contributions.

Equation (1) expresses the additivity of elastic, e, and plastic, p, strains. Equation (2a) defines stresses u through Hooke's law, eqn (2b) the statical internal variables X as gra- dient of a potential W, function of kinematic internal variables ~ (a and e, X and I/are conjugate variables in the above-specified sense). W(T/) represents the stored strain en- ergy due to the structural rearrangements at the microscale (which are described by T/ at the phenomenological macroscale level). Equations (3) govern the evolution of inelas- tic kinematic variables p and ~. The elastic domain is defined in eqn (4a) by m x y yield functions (y for each element). Each yield function depends nonlinearly on stresses and static internal variables, thus taking into account a general nonlinear hardening law. Equations (4b,c) express the loading-unloading criterion (Prager's consistency rule), which reflects the irreversible nature of material behaviour.

Equation (5a) defines the rate of dissipation as the difference between the rates of plas- tic work and stored strain energy; eqn (Sb) postulates the rate of dissipation positive- ness for thermodynamical reasons.

It is worth noting that each of the yield functions (4a) can always be expressed, with- out loss of generality, as the difference between a first-order positively homogeneous func t ionfk(a ,X) and a constant Yk(yield limit) (k -- 1 . . . . m x y ) , as shown by EvE et al. [1988]. Therefore, we can write, by Euler's theorem:

~(a,X) = f ( a , x ) - Y = 0f 0f 0a--- 7 (o,X)O + - - (a,X)X - Y. 0X t (6)

We assume in what follows some hypotheses which specialize the elastic-plastic consti- tutive law (1)-(4).

a. The yield functions and the plastic potentials are identical (associated flow rules), namely:

~bk(q,X) = ~bk(o,X) (k = 1 . . . . m x y) . (7)

Thus one of the requirements of Drucker 's stability postulate (normality) is satisfied.

b. The yield functions are differentiable and convex functions of stresses and static internal variables:

C.

a~bk 0~k ~ ' k ( a , X ) - - + + ( a ' , X ' ) > - - ( a ' , x ' ) . ( ~ - a ' ) + - - ( a ' , x ' ) • ( x - x ' )

- 0¢t OX t

for any ( a , X ) , ( a ; X ' ) . ( 8 )

As a consequence, relation (4a) defines a convex elastic domain according to an- other requirement of Drucker's postulate. W(T/) is a differentiable convex function. This implies, considering eqn (2b):

W(~) - W(T/') _> X 't. (T/-- T/') for any 7/,~1'. (9)

682 C. Corn and A. CORIGLIANO

Relation (9) rules out softening behaviour and satisfies Hill 's sufficient criterion for stability and a further requirement of Drucker 's stability postulate.

By virtue of hypotheses a and b, eqns (3), and (4b,c) furnish the following inequality:

(o - # ) t O - ( x - ~)'/~ -> 0

which holds for any (~r,~) such that:

0(# ,~) -< o.

(10)

(11)

Notice that relations (10), (11) imply the nonnegativeness o f / ) , provided that the ini- tial elastic domain (X - 0) contains the origin o = 0.

II.2. Elastic-plastic structural dynamic problem

Let us consider, as in section II. 1, a discretized structure subject to initial conditions and to a time history of equivalent nodal forces P ( t) (equivalent to the external forces in the virtual work sense of the compatibility finite element method).

Under the hypothesis that the geometry changes do not affect equilibrium equations ("small deformat ion" hypothesis), compatibili ty and equilibrium are expressed by the following linear equations:

~(t) = C u ( t ) (12)

C ' o ( t ) = P ( t ) - Mii( t ) - Vu( t ) . (13)

The initial conditions read:

u(0 ) = u0; u (0 ) = Up. (14)

Vector u collects the n nodal displacements (degrees of freedom); M and V denote the symmetric positive-definite (or at least semidefinite) inertia and viscous damping ma- trices, respectively; C is the compatibility matrix of rank n, that is, no rigid motions are possible in the structure.

The elastic-plastic structural dynamic response is governed by eqns (12)-(14) and con- stitutive relations 0) - (5) .

The stress response for the above problem can be interpreted as the sum of two quantities:

o( t ) = oe(t) + oP(t) = oe( t ) + Z p ( I ) . (15)

o P(t) is the elastic static response to current plastic deformations p ( t ) , conceived as distortions; matrix Z is symmetric negative semidefinite and depends on the elastic stiff- ness and structural geometry only (MATER [1970]):

Z = EC- ( C t E C ) - I C tE - E. (16)

The stress vector ae(t) , defined by eqn (15a), depends on the elastic dynamic stress re- sponse to external actions P ( t ) and initial conditions Up, u0, and on the dynamic effects

Dynamic shakedown in elastoplastic structures 683

of plastic strains p ( t ) . It is worth stressing that, when inertia effects are negligible, ae(t) is the elastic response of the structure to the given loads.

Equation (15), through Hooke's law (2a), defines a partition of elastic strains:

e( t ) = E - loe ( t ) + E - 1 Z p ( t ) . (17)

Since the stress rates #P are self-equilibrated and the total strains ¢ are compatible, the virtual work principle and eqns (1), (15b), and (17) provide the equations:

#P'( t )~( t ) = pt ( t )ZE-~ae( t ) + p t ( t )ZE- ~ Zp ( t ) + p t ( t )Zp ( t ) = 0. (18)

Noting that, by eqn (16), Z E - I Z = - Z , (18b) implies:

pt( t )ZE-~a~(t) = 0. (19)

1I!. SHAKEDOWN THEORY: "STATIC" A P P R O A C H

III. 1. Fictitious response and related inequality

Consider a fictitious stress response o*(t) defined as follows:

a*( t ) = a~' ( t ) + u p• = o e ' ( t ) + Zp* (20)

where ae*(t) is the fictitious elastic dynamic stress response of the structure to the ac- tual external actions P ( t ) and to arbitrary initial conditions u~, fit; aP" are the static self-equilibrated stresses arising in the structure supposed elastic as a consequence of con- stant distorsions p*.

Assuming that the constitutive restrictions (a)-(c) are fulfilled, through suitable com- parisons of real and fictitious responses, we can prove the following proposition.

Proposition 1. If a fictitious stress response a*(t) , a vector of constant static internal variables X*, an instant t* and a constant gap vector ~ can be found such that:

~* +/~ = #(ae*(t) + Zp*,x*) + ~/_< 0 Vt_> t*, (21)

then the following inequality holds for any t _> t*:

[~t,~(t) <_ ~t,~(t*) + 8(Ao~(t*)) + 3((All( t*)) -- ~ A p ( t * ) t Z A p ( t *)

+ W(~(t*)) - W(,I*) + X*'" ('1" - ,~(t*)). (22)

Here and henceforth A denotes the difference, at the same time, between quantities in the actual and in the fictitious process: 8 represents the elastic strain energy and 3( the kinetic energy:

8(Aae( t ) ) = ~A~e ' ( t )E-1dae( t ) ; 3 ( (Au( t ) ) = ~ A u t ( t ) M A u ( t ) . (23)

Proof." Multiplying inequality (21) by the nonnegative vector ,~, taking into account eqn (4c), one obtains:

~ t ) ( t ) <_ ( # ( t ) - # * ( t ) ) t ) ( t ) . (24)

684 C. Corn and A. CORIGLIANO

The right-hand side of (24) can be bounded f rom above using the convexity relation (8) and the flow rules (3). Considering also eqns (15) and (20) one can write:

# ' A < ( ~ - ~*)',~ _< ( f - f * ) ' p - ( x - x * ) ' / s

: ( re _ fe*)tp + (Zp -- Zp*)tp -- (X -- X*)t//. (25)

Note that zl~ = zle + Ap is compatible with Au and that Afe is in equilibrium with - M A i i - V A l i ; therefore the virtual work principle allows one to write the equality:

( f ie _ f e * ) t A p : _ ( f i e _ f e * ) t A ( ~ _ ( M A i i + V A u ) t A l l . (26)

The first addend in the right-hand side of eqn (26) can be easily t ransformed using eqns (17) and (19), and the virtual work principle:

( f l e - - f e * ) t A e = A f e ' E - I A 0 e + f i e lE - lZo -- f i e* 'E- lZp = Afie 'E-I A0 e. (27)

Substituting eqns (26)-(27) into relations (25), noting that d p = p (p* is constant) and

integrating over t ~ , one obtains:

f' f' f' f' B t ) dr <- - - A O e ' E - I A a e dr + - a f l t M A l l d r + - A C V A u dr

+ A p t Z A p dr + -- (X -- x*) t i l dr.

(28)

The right-hand side of inequality (28) can be bounded f rom above in view of the fol- lowing circumstances: the elastic and kinetic energies are nonnegative; the matrix V is positive semidefinite; the matrix Z is negative semidefinite; WOI) is a convex function. For each integral on the right-hand side of eqn (28) the following inequalities can be written:

ft t - A o e ' E - 1 A i T e d7 = 8 ( A o e ( t*)) -- g ( AOe( t )) <_ 8(Ao~(t*)) (29)

ftt t - - A u t M A ~ d ~ " = 3 ¢ (A i J ( t * ) ) -- 3( (Au( t ) ) _< 3( (Au( t*) ) (30)

f t - - A l l t V A i l d ' r <-- 0 (31)

ft t 1 1 1 A p t Z A p d r = ~ A p ( t ) t Z A p ( t ) -- ~ A p ( t * ) t Z A p ( t *) <_ - - - A p ( t * y Z Z l p ( t * )

• 2

(32)

ft t . - ( X - X*)t i~dr = W ( ~ ( t * ) ) - W ( ~ ( t ) ) + X*ql(t) - x*t~(t *)

<_ W ( ~ ( t * ) ) - W ( ~ * ) + x *t. (~* - ~(t*)). (33)

Dynamic shakedown in elastoplastic structures 685

Inequality (28), together with inequalities (29)-(33), completes the proof of Proposition 1 (relation (22)).

III.2. Shakedown theorem

In the present context of discrete structural model, shakedown (or "adaptation") can be identified with the following occurrence:

,1 = = l i m ,1 ( t ) < + o . . ( 3 4 ) l~Oo

As shown below, this condition implies boundedness in time of the total dissipated en- ergy, whose rate is defined by eqn (Sa). From eqns (3), (7), (6), and (4c), and assuming ,~ (0) = 0, it follows that:

fot fot fot D(t) = (otl~ - Xt//) dr = V(~, X),~ dr = y t~ dr = Yt,l ( t ) . (35)

Under the constitutive restrictions (a)-(c), a sufficient and a necessary condition for shakedown are proved in what follows.

Proposition 2 (generalization of the first dynamic shakedown theorem due to CERADn~X [1969]). Adaptation occurs if a fictitious stress response u*(t), a vector of constant static internal variables X* and a finite time instant t* can be found such that:

sup #(~re*(t) + Zp*,x*) < 0 vt _ t*. (36) t

Proof." Proposition 2 follows straightforwardly from proposition 1. First of all note that hypothesis (36) coincides with (21) if the components of the gap vector is are all strictly positive. Therefore, inequality (22) holds; by setting the time t to the limit oo, relation (22) furnishes a time-independent upper bound to ist,~o.. Being the components of ,~ and iS nonnegative and positive, respectively, the above circumstance implies relation (34) and, hence, shakedown.

Proposition 3 (necessary condition for shakedown). If adaptation occurs, then a ficti- tious stress response a*(t) , a vector of constant static internal variables X* and a finite time instant t* can be found such that:

~(oe*(t) + Zp*,x*) -< 0 vt _> t*. (37)

Proposit ion 3 can be easily proved by standard arguments.

III.3. Safety factor

A load factor a is defined as a nonnegative scalar which multiplies all given external actions P ( t ) and initial conditions u0, lio. The safety factor ors will denote the value of a such that shakedown in the sense of (34) occurs for o~ _ o~s, it does not for ot > oz~.

686 C. COMI a n d A. COR1GLIANO

Proposition 4. From propositions 2 and 3 it follows that o~s coincides with the optimal value of the constrained optimization problem:

c~, = max { c~ ] (38a) e~,p*,x*

t ,Uo,U 0

subject to:

t~(otoe*(t) + Zp*,x*) --< 0 Vt --> t*. (38b)

Remarks."

i. Optimization (38) is a nonalgebraic, difficult problem as t*, u~, ~i~ intervene through the solution of the differential equations governing the fictitious response a~*(t). The easier optimization with respect to o~, p*, X* only, furnishes a subop- timal value o~ for o~:

ii.

iii.

o~ = ~0(t ,u0,u0) = max {a} (39a) c~,p*,x*

subject to:

4~(~a~*(t) + Zp*,x*) -< 0 vt _> t* (39b)

c~s - c0. (40)

As a result of the constitutive hypothesis (b), the optimization problem (39) is convex. In order to recast problem (39) into a nonlinear programming problem, one has to eliminate the time dependence in the constraint inequalities. This is possible if one knows the minimal convex hull F containing, for any t, the elastic dynamic fictitious response oe*(t). Being the yield functions and F convex (by hypothe- sis (b) and by definition o f / " ), the fulfilment of the constraint (39b) for any value a e* on the boundary OF of F makes it satisfied for any t.

~(OtO "e* -[- Zp*,x*) -< 0 VO "e* E O f '

¢~ #(txae*(t) + Zp*,x*) -< 0 Vt _> t*. (41)

It is worth noticing, however, that to construct F one has to integrate the ficti- tious elastic dynamic problem and calculate ae*(t). In the quasistatic range the definition of a load domain enables one to avoid the time integration (PoLtZZOTTO [1982]), this no longer holds in the present dynamic context. Also, note that in relation (39b) ~ and a collect all corresponding elemental quantities. However, with particular suitable choices of generalized variables modelling (cp. CoMI et al. [in press]), each subset of inequalities concerning one element (say the ith) only contains the generalized stresses a i relative to the same element. Therefore, the minimum convex hull can be defined in the a / space for each element i in turn.

Dynamic shakedown in elastoplastic structures 687

iv.

V.

The actual initial conditions have no influence on the safety factor as; this is a self-evident consequence of the fact that u0 and Uo do not intervene in (38). Note, however, that changes in initial conditions, even if immaterial for shakedown, may clearly affect history-dependent quantities such as residual displacements. Consider periodic external actions of period T: P (t) = P (t + T). In the presence

. . . . which of damping effects it is always possible to find initial conditions u0,u0 make periodic of period T, from t = 0, the fictitious stress response ~e*(t) (that is, suppress the free motion). In this case the value of o~ corresponding to u o, Uo coincides with as; therefore, the optimization with respect to fictitious initial con- ditions can be avoided:

= o~(O, U o , ~ ) . O~ S (.~ --~ ~* (42)

The proof follows the same path of reasoning of CORRADI and MAIER [1973].

IV. S H A K E D O W N THEORY: KINEMATIC A P P R O A C H

IV. 1. Admissible yielding cycle

Extending Koiter's definition of "admissible plastic strain-rate cycle," we will call "ad- missible yielding cycle" any yielding process such that the plastic strain rate and the ki- nematic internal variable fields (p A,//.) satisfy in the time interval t ~ the following conditions:

f t t2 ~112 Ap ~ = p^(t) dt = Cdu ' ; A~/̂ = / f ( t ) dt = 0. (43)

1 1

Equation (43a) express the compatibility of plastic strain A pA developed during an ad- missible yielding cycle. As a consequence, the residual stress rates 0P^ (t) = zp^(t) due to the plastic strain rates satisfy the relation:

f t t2 f t2 O°"(t) dt = zpA(t) dt = 0. (44) 1 I

IV.2. Nonshakedown theorem

Assuming the constitutive restrictions (a)-(c), a sufficient and necessary condition for inadaptation (ratchetting or alternating plasticity), is formulated and justified as follows.

Proposition 5 (generalization of the "kinematic," dynamic shakedown theorem due to CORaADI and MAmR [1973,1974]). Inadaptation (lack of shakedown) will occur in the

* * t* structure if and only if for all u0, u0, and for some admissible cycle starting at time tl -> t* the following inequality holds:

( ' 2 a e " ( t ) p ' ( f ' 2 t) dt > / ) ( p ' , / f ) dt. (45) ~'/I I

Proof o f the sufficient condition: By contradiction with the thesis, let us assume that shakedown occurs. Then by virtue of proposition 3 (shakedown necessary condition)

688 C. Corn and A. CORIGLIANO

there exists a set of fictitious variables u~, ti~, t*, p*, X* satisfying inequality (37) for any t > t*. Relation (37) implies the validity of (10) with # = ae*(t) + Zp* and ~ = X*; therefore, for any admissible cycle with tl _>t*, taking into account (5a), one obtains:

/ )(pA,~^) = a A t p ^ _ x A t i f > o e * t p A + p*tZlT - X*tif . (46)

In eqn (46) and often henceforth, the time dependence is not explicitly expressed for brevity of notation. By integrating over the time interval tl t2 with t~ _ t , bemg p and X* constant, inequality (46) yields:

D(p A, i f ) dt >_ f f e * t p - dt + p*t Zp" dt - x*' i f dt. (47) 1 I t"'l I

The last two integrals in (47) vanish because of the very definition of admissible cycle; therefore, inequality (47) is in contradiction with hypothesis (45) and proposition 5 (suf- ficient part) is proved.

Proof o f the necessary condition: The proof is conducted as before by contradiction. Let us assume that a set fi~, fi~, [* can be found such that the following inequality holds for any admissible cycle with tl > / * :

t2 • f t ' 2 o e ' ( t )p '( t)dt<_. D(p ' , i l ' )d t . (48) t] 'l I

Under this condition, we prove that the structure shakes down, that is, that the safety factor ors is not less than one. To this purpose, consider the problem expressed by re- lations (39) in which the fictitious initial conditions and the time instant are fixed to the values fi~, fi~, i*. The Lagrangian functional for the problem (39) is:

f / *) P. L = - o t + . # " ( ~ ( o * , X + c ) dr t > _ (49)

In (49) o* = aoe*(t) + ZO*; t* is a vector of Lagrangian multipliers; vector c contains nonnegative components ci = y~, with the role of slack variables which transform con- straint (39b) into equality. A solution of problem (39) is characterized by vanishing first variation of L for each variation of or, p*, X*, #i, Y~ and by nonnegative second variation of L, in particular with respect to 3'~- These conditions lead to the Euler-Lagrange relations:

#(a*,X*) -< 0; t~ -> 0; t~t,(a*,X *) = 0 vt _> [* (50a,b,c)

~ t f f e ' t O¢~t • -~o ( o * , X * ) # d r = 1 (50d)

f / 00' . Z ~-a (a*,X*)#dr = 0 (50e)

f f 00' • -~X (a*,X*)~tdr = 0. (50f)

Dynamic shakedown in elastoplastic structures 689

Interpreting/z~ in eqns (50) as plastic multipliers, the vectors:

atkt 0¢t (o*, X*)/~ (51a,b) P" = ~ ( ~ * , x * ) ~ , ; ')^ = - a-~

I t * I define an admissible yielding cycle in the time interval tl = t2 = t. This fact can easily be proved noting that, by eqns (50e,f), they satisfy eqns (43b) and (44b), which characterize an admissible cycle. Inequality (48) holds in particular for the admissible yielding cycle defined by (51a,b). Taking into account eqns (5a) and (51) and the defi- nition of a*, inequality (48) can be rewritten as follows:

f/ ,°' :: f f e * t • -~u (o*,X*)#dr<- D ( p ' , i f ) d r

= q e * t * -~o (a*'X*)t~dr+p*t el* Z ~ (a*,X*)t~dr

-- X *t : t 0ht • ,r ~ (,,*,x*)~,d~. (52)

An optimal value co of problem (39) satisfies the Euler-Lagrange relations (50); substi- tuting (50d,e,f) into (52) one obtains:

I _< co. (53)

As stated, the optimal value co is a lower bound of the safety factor O~s for any assigned value of u~, li~, t*. Hence, from (53) one deduces that ors is not less than one, that is, shakedown occurs (in contrast to the hypothesis) and, therefore, the necessary condi- tion for inadaptation is proved.

IV.3. Alternative statements

The above sufficient and necessary condition for inadaptation can be reinterpreted as necessary and sufficient condition for shakedown. This can easily be seen, because proposition 5 has been proved by contradiction.

Proposition 6. Shakedown will occur in the structure if and only if vectors fi], -'* u 0, and a finite time instant i* can be found such that inequality (48) holds for all admissible cycles with t _> i*.

Note also that the first member in inequality (45) (and hence in (48)) can be suitably transformed by making use of the virtual work principle (for the proof see CORRAOI & MAIER [1973]), obtaining:

f t2 f t2 (P - Mfi* - V~l*)tflPAdt > / ) ( l~A,~ * ) d t .

1 I (54)

Proposition 5 is thus restated in a way similar to that of Koiter's theorem in the quasistatic range (KorrES [1956]).

690 c. COMI and A. CORmLtANO

IV.4. Safety factor

Let (~1 denotes the set of all admissible yielding cycles for which the following equal- ity holds (normalization condition):

ft t2 oe*t(t)p^(t) dt = 1.

I

(55)

Proposition 7. The safety factor as can be determined by solving the following saddle point problem which arises f rom the kinematic approach:

/r 1 0 = O(u~,h~,t*) = min / ) (p~ , / ( ) dt; (pA,/f) E 6tl (56a) k . ~ l I

as max I * "* t* . = 0(u0,uo, )1 (56b) u~,u~,t*

Proof." The proof of (56a,b) can be derived f rom proposit ion 6 as follows. The suffi- cient condition for shakedown of proposit ion 6 and the very definition of O (eqn (56a))

" * t * ensure that a = 0(Uo,U0, ) is a load factor corresponding to a safe situation for any fixed u~, U~, t*, namely:

0(Uo,U0, ) -<Us vu0 ,uo , t . (57)

Hence, the maximum value of 0 (56b) also satisfies the relation:

max 0 _< as. (58)

For a = as shakedown occurs; f rom the necessary part of proposition 6 a set Uo, u0, exists such that, for any admissible cycle, it results:

f t2D(oA, il^) dt cts -< (59)

f ,2 oe*t(t)O'(t) dt I

Therefore, taking into account (55) and (56a), we deduce:

and, a fo r t io r i ,

as -< 0(fi3,-* -* Uo, t ) (60)

as -< m a x O. (61)

Inequalities (58) and (61) prove proposit ion 7. As for the problem arising f rom the static approach (see Section III.3), the solution

of problem (56) is generally difficult or prohibitive; a much simpler problem is gener- ated, for instance, if one assumes fixed initial conditions u~, u] and a time instant t* and, thus, seeks a lower bound 0 on as.

Dynamic shakedown in elastoplastic structures 691

V. SPECIALIZATIONS A N D CLOSING REMARKS

The findings presented in this paper, in the formal context o f an internal variable con- stitutive law allowing for general nonl inear hardening and with reference to discrete structural systems, can easily be specialized to various available results. For the elastic perfectly plastic case, propos i t ion 2 reduces to a discrete version o f CERADrSI'S [1969] theorem; assuming piecewise-linear yield funct ions and isotropic nonl inear hardening, the same propos i t ion coincides with that proved by MAIER and NOVATI [1990]; for the linear hardening case the results conta ined in MATER [1970] are obtained. The present propos i t ion 5 reduces to CORI~.OI and MAmR'S [1974] theorem when a perfectly plas- tic con t inuum is considered; with piecewise-linear yield functions and linear hardening, propos i t ion 5 leads to the theorem presented in CORRAOI and MAma [1973].

In the quasi static range with reference to discretized structural models, the classical MELAN'S [1938] static theorem and KOITER'S [1956] kinematic theorem can be derived one f rom the other as shown by MArER [1969] using the duali ty theory in mathemat i - cal programming. In the dynamic range here considered, duality between kinematic and static approaches still holds at fixed fictitious initial condit ions. The inequality proved in propos i t ion 1 can be used to deduce some t ime-independent upper bounds to linear funct ions o f A, in part icular to the dissipation D ( t ) (see MATER R, NOVATI [1990]). In- equalities similar to (22) in propos i t ion 1 can be obtained starting f rom a per turbat ion o f the yield condi t ion more general than the gap vector/~ adopted in (21); for example, considering a modif ied stress vector (PoLtZZOTTO [1982]). With suitably chosen pertur- bations one can obtain upper bounds to other significant postshakedown values o f quan- tities such as plastic strains, plastic work and internal variables. These bounds may provide practically useful in format ion on the elastic-plastic response o f the structure. The present results concerning the determination o f the safety factor Ors with respect to inadaptat ion (lack o f shakedown) are far f rom complete: the computat ional and numer- ical aspects o f this problem in the present more general constitutive context require fur- ther studies.

Acknowledgements--Thanks are expressed to Professor G. Maier and Professor C. Davini for valuable ad- vice. The second author expresses his gratitude to Confalonieri Foundation. This paper was carried out in the framework of GNDT. Most of the present results were the object of two communications to Accademia Nazionale dei Lincei, Rome, February 10th and March 10th, 1990, which appeared in Italian in Rendiconti Accademia Nazionale dei Lincei.

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Department of Structural Engineering Technical University (Politecnico) of Milan Piazza L. da Vinci 32, 20133 Milano, Italy

(Received 28 April 1990; in final revised form 11 July 1990)