Issue 18

G. Del Piero et alii, Frattura ed Integrità Strutturale, 18 (2011) 5-13; DOI: 10.3221/IGF-ESIS.18.01 7 with the inequality due to the presence of the dissipation inequality (4), which requires the restriction   ( x )  0 (6) on the perturbations  . The analysis of the first variation leads to the following characterization of the equilibrium configurations [DLM] :  The elastic deformation  ( x ) and the axial force    = w  (  ( x )) (7) are constant all over the bar,   The axial force cannot exceed a limit depending on the current inelastic deformation:     (  ( x )) . (8)  The equilibrium condition (8) is in fact a yield condition . It is remarkable that it has not been postulated, but deduced from the variational procedure. The energy minimization is insufficient to determine the evolution of the deformation under a loading process  =  ( t ). To do this, it is necessary to formulate an incremental equilibrium problem , in which at any time t the deformations  ( t ) and  ( x,t ) are supposed to be known, and the unknowns are the deformation increments ),( ), ( tx t     , produced by a given load increment )( t   . One considers the expansion ) ( )) ( ), (( )) ( ), (( )) ( ), (( )) ( ), (( 2 2 2 1              o t t E t t E t t E t t E         (9) and minimizes for  sufficiently small to legitimate the truncation at the second-order term. The term of order zero being known, a first-order approximation involves the minimization of the function  dx x t tx t t l t t E l ))( )) ( ),((' ( )( )( )) ( ), (( 0                (10) In it, everything is known except )( x   . Then E  is an affine function of   , and a proper use of the dissipation inequality leads to the Kuhn-Tucker conditions 0 )(  x   , 0 )) (('     x , 0 )( ) )) ((' (   x x      (11) as necessary conditions for a minimum. The first two conditions are (4) and (8), respectively. The third condition, the complementarity condition, states that when   is positive the force  must lie on the border of the yield surface, and that when  lies at the interior of the yield surface then   must be zero. Again, it is remarkable that these distinctive aspects of plastic response have not been postulated, but deduced from incremental energy minimization. Conditions (11) are still insufficient to determine the evolution of the deformation. This is done by minimizing the second-order term of (9) dx x t x tx t t l t t E l ))( )( )( )),(('' ( )( )( )) ( ), (( 0                    (12) which is a quadratic function of   . The new minimization provides a second set of Kuhn-Tucker conditions 0 )(  x   , 0 )( )) ((''        x x , 0 )( ) )( )) (('' (   x x x         ,  x (13) to be satisfied at the portion  of the bar at which (8) is satisfied as an equality. Indeed, we already know that   = 0 out of  . In (13), the first condition is again the dissipation inequality (4). The second condition provides a relation between the increments of the force and of the inelastic deformation. By the complementarity condition (13) 3 , inequality (13) 2 is satisfied as an equality when   > 0. This equality provides the flow rule for the inelastic strain rate in the regime of inelastic deformation. Condition (13) 3 is the consistency condition , which says that the inelastic deformation cannot increase when a