Basic Definitions and Results
The experimental tests show that many engineering structures subjected to variable repeated loads exhibit plastic strains. Above a critical value α a of the load factor, they collapse by ratchet or alternating plasticity. On the contrary, below α a , the
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Abstract. The experimental tests show that many engineering structures subjected to variable repeated loads exhibit plastic strains. Above a critical value a a of the load factor, they collapse by ratchet or alternating plasticity. On the contrary, below aa, the plastic deformations are stabilized and the dissipation is bounded in time. We say that the structure shakes down. Firstly, the standard plasticity model is briefly recalled. Next, we define the basic tools of the fictitious elastic and residual fields. The statical approach due to Melan allows to characterize the shakedown. The kinematical approach due to Halphen gives a description of the collapses. The end of the lecture is devoted to useful concepts of mathematical programming and non smooth mechanics.
1
Compatibility and Equilibrium
Before presenting the main ideas of the theory, some classical hypothesis concerning the structure, the loading and the material are recalled. We consider a body (or structure) Q occupying an open domain of the space R 3 , limited by the surface r. The part r 0 corresponds to the supports, while the remaining part is r 1 = r- r 0 . The mechanical fields are generally depending on the position vector x E Q and the time variable t > 0, namely the displacement field (x,t) ~ u(x,t), the strain field (x,t) ~ e(x,t) and the stress field (x,t) ~ a(x,t). The structure is subjected to time dependent actions : 1. body forces: (x,t) ~
J (x,t)
prescribed in Q x [O,+oo[,
r 1 X [O,+oo[, prescribed in ro X [ O,+oo[.
2. imposed surface forces: (x,t) ~ p(x,t) prescribed in 3. imposed displacements : (x, t) ~
u(x, t)
The strain field is related to the displacement one by the internal compatibility equations :
Using intrinsic notation, we write equivalently : e(u)
= gradsu
in Q.
D. Weichert et al. (eds.), Inelastic Behaviour of Structures under Variable Repeated Loads © Springer-Verlag Wien 2002
(1.1)
2
G. de Saxce The displacements are subjected to boundary kinematical conditions : (1.2)
A displacement field u is said to be kinematically admissible (K.A.) if it satisfies (1.2) and if there exists a strain field e associated to u by (1.1). On the other hand, the stress field has to fulfill the internal equilibrium equations :
or, in brief:
div (J + J = 0 in .Q . Let n be the unit normal vector to equilibrium equations :
(1.3)
r. The stress field is also subjected to boundary
In a more compact way, it holds : p(G) = G.n = p on
r 1•
(1.4)
A stress field G is said to be statically admissible (S.A.) if it satisfies (1.3) and (1.4). Obviously, e and G are both symmetric tensors of order two :
The 6-dimensional vector space of the strains (resp. of the stresses) is denoted E (resp. S). These two spaces are putted into duality by means of the following bilinear form (double contracted tensor product) :
representing the work by unit volume. Now, it is easy to prove that for any K.A. displacement field u and any S.A. stress fieldG, Green's formula holds:
J.Q G:e(u)dO.= J.Q ].udO.+ J,
~
p.udr+ J, p(G).udr, ~
where the do
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