Eng 2023 , 4
84
where the parameters A , B , C and D are calibrated from the experimental results at 0 ◦ (MD—machine direction) and 90 ◦ (CD—cross direction) without perforation obtained in a previous work [15], using the following Equations (5)–(12) [11]:
1 − 12 x 2
(5)
A =
B = 3 ( y − x ) C = 3 ( y + x )
(6)
(7)
n ( n + 1 )
K
12 √
(8)
D =
3
α 2 24 ( 3 α 2 + β 2 − 4 β + 4 )
β + 1 − 6 β − 3 α 2 − 3
(9)
x =
α 4 x −
(10)
A
y =
2 n ( n + 1 ) 33
2 n ( n + 1 ) 22
(11)
α = K
− K
2 n ( n + 1 ) 33
2 n ( n + 1 ) 22
β = K (12) The parameters K ii and n are related to the curve fit of the tensile test applying the Ramberg–Osgood methodology. For the MD tensile test (see Equation (13)): + K
n
σ 11 E 0
σ 11 E 11
(13)
ε 11 =
+
For the CD (see Equation (14)):
n
K kk E kk
E 0
σ kk E kk
, k = 2,3
(14)
ε kk =
+
Note that for Equation (13), the repeated indices do not mean the usual summation rule used in the indicial notation. Finally, the parameter K 12 is obtained using Equation (15).
n
K 12 σ 12
E 0
σ 12 G 12
(15)
γ 12 =
+
The Hooke’s law for plane stress, small strain, linear elastic orthotropic material is given using Equation (16). σ = C : ε e (16) Where σ is the second order Cauchy stress tensor, C is the four-order plane stress, linear elastic, orthotropic constitutive law tensor and ε e is the second order small strain elastic tensor using matrix notation. 2.3. Finite Element Model The implementation of this model follows the well-known J 2 flow theory for isotropic materials using the backward Euler algorithm [11]. The explicit solver was used to over- come convergence issues that are common when using the implicit solver for this type of simulation. On the other hand, the stable time increment is very small, which increases the computational costs. Simulations were performed using a workstation with two intel Xeon E5-2630 8 cores (16 cores total with 32 threads) with 256 Gb ram. The FE model dimensions, and boundary conditions are presented in Figure 3. The boundary conditions were imposed to represent a tensile test. Thus, all the displacement degrees of freedom are restricted (see Figure 3) in one side, and a prescribed displacement
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