PSI - Issue 35

E.A. Dizman et al. / Procedia Structural Integrity 35 (2022) 91–97 Author name / Structural Integrity Procedia 00 (2021) 000–000

93 3

Fig. 2: Slip systems for uni-axially reinforced composite microstructure

in which F e and F p represent elastic and plastic parts of the deformation gradient, respectively. This kinematic split implies that there exists an intermediate configuration resulting from F p at each material point.

Table 1: Slip systems

System number (type)

s 0

m 0

(0 1 0)

(1 0 0) (1 0 0)

1 (longitudinal) 2 (longitudinal) 3 (longitudinal) 4 (transverse) 5 (transverse)

1 2

0

√ 3 2

2 1 2 1 2

0 1 2 0 −

√ 3

(1 0 0)

−

(0 − 1 0)

(0 0 − 1)

0 1 2 0 1 2

2

√ 3

√ 3 2

− √ 3

2

0

√ 3 2

6 (transverse)

−

Following crystal plasticity Hill and Rice (1972), the velocity gradient tensor L p associated with F p reads as,

6 η = 1

L p = ˙ F p F − 1

˙ γ η s ⊗ m

(2)

p =

where ˙ γ η is the slip rate on slip system η . The elastic response is considered with respect to the intermediate configura tion defined by F p . Since the material is uni-axially fiber reinforced, the elastic behaviour is assumed to be transversely isotropic. The second Piola-Kirchho ff stress tensor S (with respect to stress free intermediate configuration) is defined as S = D : E , where E and D are the Green-Lagrange strain tensor and the anisotropic elasticity tensor, respectively. The explicit form of D is obtained by inverting the compliance tensor H given in Voigt notation as,

    =     H     S 11 S 22 S 33 S 12 S 13 S 23     1 / E f − ν 21 / E m 2 − ν 31 / E m 3 − ν 12 / E f 1 / E m 2 − ν 32 / E m 3 − ν 13 / E f − ν 23 / E m 2 1 / E m 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 / (2 G 12 ) 0 1 / (2 G 13 ) 0 0 1 / (2 G 23 )

   

    E 11 E 22 E 33 E 12 E 13 E 23

(3)