PSI - Issue 35

S. YaŞayanlar et al. / Procedia Structural Integrity 35 (2022) 18– 24 Yas¸ayanlar et al. / Structural Integrity Procedia 00 (2021) 000–000

21

4

used here is Poh and Sun (2017)

(1 − R )exp( − η D ) + R − exp( − η ) 1 − exp( − η )

g =

(7)

where R and η are material parameters. To treat the incompressibility constraint that arises due to volume preserving nature of plastic deformations in case of von Mises plasticity, the displacement / pressure (u / p) mixed formulation is used, Bathe K.J. (2006), Xu et al. (2020). Hydrostatic pressure ( p ) is introduced as an independent field which is supposed to satisfy

p = K e

(8)

at every material point, where K is the bulk modulus and e is the volumetric strain defined in terms of normal compo nents of the total strain tensor as e = xx + yy + zz .

3. Three Field Finite Element Formulation

The presented elastoplasticity with non-local damage problem requires the solution of equations 1, 6 and 8. Finite element method is used to solve this set of equations and the corresponding weak forms are derived as, V p δ e + δ σ dV = Γ t δ u ¯ t d Γ (9) V δ ¯ eqv pl ¯ eqv pl dV + V g ( D ) l 2 c ∇ δ ¯ eqv pl · ∇ ¯ eqv pl dV − V δ ¯ pl eqv pl dV = 0 (10) V δ p ( p K − e ) dV = 0 (11) Equations 9 and 10 are coupled through the dependence of σ on damage (in turn on ¯ eqv pl ) and eqv pl (local equivalent plastic strain) that drives the evolution of ¯ eqv pl . The coupling betweeen equations 9 and 11 is due to p and the volumetric strain e whereas there is no coupling between equations 10 and 11. For temporal discretization, backward Euler integration is used to update the stress tensor leading to an implicit solution algorithm. Since non-local equivalent plastic strain is an independent field and the formulation is coupled, linearized incremental deviatoric stress d σ and linearized incremental plastic strain d ¯ eqv pl depend on both incremental strain and incremental non-local equivalent plastic strain and written as,

eqv pl eqv pl

d σ = C 1 : d + C 2 d ¯ d pl = C 3 : d + C 4 d ¯

(12) (13)

where C 1 , C 2 , C 3 and C 4 are implicitly defined and can be derived using the persistency condition d φ ( , ¯ eqv pl ) = 0 to resolve the relation between d and d ¯ eqv pl .

4. Numerical Examples

As mentioned before a two-field hexahedra element and a three-field tetrahedra element are implemented using localizing implicit gradient damage formulation. In this section, the performance of these elements are assessed by means of two numerical examples.