6
LINDBERG AND KULACHENKO
FIGURE 5
Numerical tensile tests compared to the mean values of the tensile test curves for the respective direction
FIGURE 6
The Hill's surfaces for the two paperboards for zero shear stress. (A) Board A, (B) Board B and (C) Board A and B normalized in the
same figure for comparison of shapes
TABLE 2
Hill's parameters
2 þ F
2 þ 2 F
2 <1, ð 7 Þ
σ TW ¼ F 1 σ 11 þ F 2 σ 22 þ F 11 σ 11
12 σ 11 σ 22 þ F 66 σ 12
22 σ 22
R 11 [ ]
R 22 [ ]
R 33 [ ]
R 12 [ ]
R 23 [ ]
R 13 [ ]
where F i and F ij are defined as
Board A 2.3248 1.0 1.2198 1.69
1.0
1.0
Board B 2.576
1.0 1.0512 1.3
1.0
1.0
1
1
1
1
1
-
; F 2 ¼
-
; F 11 ¼
;
F 1 ¼
σ c
σ c
t σ c
σ t
σ t
σ 11
11
11
22
11
22
ð 8 Þ
1
1
2 ; F 12 ¼ k ffiffiffiffiffiffiffiffiffiffiffiffifffifi F 11 F 22 p ,
; F 66 ¼
F 22 ¼
t σ c
σ 22
σ t 12
22
study is presented. Note that the fit of R 33 is important even though plane stress is approximated, since R 33 influences the shape of the yield surface in the MD-CD plane.
and the indices ‘ t ’ and ‘ c ’ are for ultimate tensile stress and ultimate compressive stress respectively. In Equation 8, F 12 and the in-plane ultimate shear stress σ t 12 require some extra attention. These cannot be directly determined from tensile and compressive tests and require shear testing where the failure envelope is studied. For the current study, no such data is given for the two paperboards. Some estima- tions from the literature are required. For F 12 the constant k is chosen as k ¼ 0.5, which is suitable for most composites. 27,29 In Li et al., 27
2.5
Failure evaluation
|
Failure is not included in the numerical model but will be evaluated as a part of the post-processing of the final results using the Tsai – Wu stress failure criterion. 21 For plane stress, it reads 27,28
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