Issue 51

A. Gesualdo et alii, Frattura ed Integrità Strutturale, 51 (2020) 376-385; DOI: 10.3221/IGF-ESIS.51.27 378 N D    : boundary of the body  and N D     , N D   : free and constrained boundary of the body  . Given the body forces b in  , the tractions p on  N and the displacements u on  D , the solution of the boundary value problem is the triplet  { , } u λ b (displacement, anelastic strain, stress) fulfilling the following relations:  equilibrium and static boundary conditions     , , on N div T b 0 Tn p (1)  kinematic boundary conditions   , on D u u (2)  stress-strain law and constitutive restrictions on strain involving fractures      [ ] , tr 0 , det 0 T e λ T T (3)  normality law     tr 0 , det 0 , 0 λ λ T λ (4) where e is the infinitesimal strain and  is the elastic tensor. The inequalities (3) lead to the following partition of the domain  : 1 2 3 { : tr 0 , det 0} { : tr 0 , det 0} { : tr 0 , det 0}.                x T T x T T x T T (5) The domain  1 is that of biaxial compression and the material has the classical bilateral elastic behaviour. In the domain  2 the material is in uniaxial compression and can show fractures. In this case the compressive lines when  b 0 are straight lines. In the  3 domain the material is completely inert and any positive semidefinite fracture field is possible. Variational formulation It has been proved [23] the existence of a strain energy density for NT materials, so that a variational formulation of the problem can be derived, i.e. an equilibrium configuration corresponds to a minimum of the total Potential Energy: 1 ( ) . 2 N p ds         u E E p u  E (6) The equilibrium displacement solution may not be unique, due to the presence of the anelastic part. A dual formulation of the problem has been derived by [22], with the stress field 0 T as statically admissible solution minimizing the Complementary Energy functional: 1 1 ( ) . 2 D c ds          T T T Tn u  E (7) defined over the convex set of statically admissible stress fields.