5
LINDBERG AND KULACHENKO
TABLE 1
Elastic material parameters
22 !
22 !
33 !
1 2
1 R 2 1 R 2
1 R 2
1 R 2
1 2
1 R 2
1 R 2
1 R 2
1 2
1 R 2
1 R 2
1 R 2
F ¼
; G ¼
; H ¼
;
12 þ 23 !
33
33 þ
11
11 þ
22
Shear Modulus G 12 [MPa]
Young's modulus E 1 , E 2 [MPa]
Poisson's ratio a ν
13 !
12 !
3 2
3 2
1 R 2
3 2
1 R 2
; M ¼
; N ¼
L ¼
:
12 [ ]
Board A Board B
7143, 3078
2258
0.446
ð 5 Þ
7501, 2948
1511
0.467
The Hill's parameters R ij in Equation 5 are defined as
y
y
y
y
p σ 12
σ 11
σ 22
σ 33
a ν 21 is determined from the above parameters due to the symmetry of the stiffness matrix.
; R 12 ¼ ffiffiffi 3
R 11 ¼
; R 22 ¼
; R 33 ¼
; R 23
σ y
σ y
σ y
σ y
y
y
p σ 23
p σ 13
¼ ffiffiffi 3
; R 13 ¼ ffiffiffi 3
,
ð 6 Þ
σ y
σ y
ffiffiffiffiffiffiffiffiffiffi E 1 E 2 p 2 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffi ν 12 ν 21 p
and ffiffiffiffiffiffiffiffiffiffiffiffiffi ν 12 ν 21 p
G 12 ¼
¼ 0 : 293 :
ð 3 Þ
and determine the shape of the yield surface, which initial size is determined by the initial yield stresses σ ij y and the isotropic yield stress σ y . The material parameters in Equation 6 are found with the previously mentioned fitting procedure. The stress – strain curve mea- sured in the CD is used as a master curve for the multilinear hardening and the rest of the parameters are fitted in Matlab using the fmincon - function to minimize the error between the measured and the calculated tensile test curves. The paperboard is modelled to yield at R p ¼ 0.0001, that is, the initial yield stress is in this study the stress giving a permanent deformation of 0.01% strain. Such a low value is required to get a good fit between the experimental and the numerical tensile test curves. The quality of the fit is shown in Figure 5. The cur- ves on the compressive side are only from the numerical tests since no data is given from experiments. In compression, the two paper- boards have a 30% reduction of the yield stress and a 50% reduction of the ultimate stress compared to the tensional side, which renders in the curves on the compressive side in Figure 5. The shape of the Hill yield surfaces for the two boards is shown in Figure 6, plotted for zero shear stress, τ 12 ¼ 0. As seen in Figure 6, the modelled difference in tension and compression for the paper- boards render in two yield surfaces per board, one for compression and one for tension. Which surface that applies for the current point is determined by the sign of the hydrostatic stress. With a positive sign, the hardening in the current point is evaluated towards the sur- face for tension, and a negative sign evaluates the evolution of the plastic strains towards the surface for compression. If a sign change would occur during the analysis, the point remains on the initially assigned surface which is shown with dotted lines in Figure 6. This approach of using two surfaces has an advantage in avoiding the diffi- culties in simultaneous fitting compressive and tensile behaviour with non-symmetric surfaces. The disadvantage is an abrupt change in the second and fourth quadrants. As the largest part of the paperboard appears in the first and third quadrant and the surface is not swapped during the simulations, this does not present a problem with conver- gence or thermodynamic inconsistency. The fitting procedure resulted in different yield surfaces for the two paperboards (see Figure 6) Board B has a yield surface close to circular and an earlier yield point, compared to Board A. In Table 2, the complete set of parameters for the Hill's plasticity used in this
Equation 3 along with the symmetry condition of the compliance matrix, giving ν 21 = E 2 ¼ ν 12 = E 1 , give the in-plane elastic material parameters which are listed in Table 1. The out-of-plane strain ε 33 can be derived as ε 33 ¼ ν 13 E 1 σ 11 ν 23 E 2 σ 22 as given by Equation 1 with σ zz ¼ 0, but requires estimations of the Poisson's ratios ν 31 and ν 23 .
2.4
Hardening model
|
Plasticity in paperboard has been modelled in many different studies. In Harrysson and Ristinmaa 20 a large strain orthotropic elasto-plastic model was developed with a yielding surface based on the Tsai – Wu failure surface, 21 which made it possible to directly introduce the dif- ference in tensional and compressive yield behaviour for paperboard. Several models based on the complex anisotropic yield surface intro- duced by Xia et al. 18 have been developed, such as the in-plane paper- board models established in Li et al. 22 and Tjahjanto et al. 23 The latter model is a viscoelastic-viscoplastic small strain approach developed to capture creep and relaxation for transient uniaxial loading. One of the latest publications on the subject is the one by Robertsson et al. 24 where the continuum model is based on previous models 15,25 using numerous sub-surfaces. In Robertsson et al., 24 results from simula- tions using solid continuum elements and shell elements are compared for some forming operations. They showed, amongst other things, that the shell elements had a better performance compared to the continuum elements. For the example simulating an actual forming operation from the industry, frictionless contacts, an explicit solver scheme and ideal plasticity were used. The evolution of the plastic strains in the current study is described using Hill's plasticity, 26 which is suitable for composites and a common way to model plasticity for orthotropic composite such as paperboard. Hill's plasticity is defined as
f σ , σ f ð Þ¼ F σ 22 σ 33 ð Þ 2
þ G σ 33 σ 11 ð Þ 2
þ H σ 11 σ 22 ð Þ 2
ð 4 Þ
þ 2 L σ 2
2 31 þ 2 N σ
2 12 σ y
2 ¼ 0
23 þ 2 M σ
where F , G , H , L , M and N are defined as
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