7
LINDBERG AND KULACHENKO
2.6
Finite element model
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for the interested reader, F 12 is analysed not only for closed failure surfaces but also for open surfaces. The ultimate shear stress σ t 12 in Equation 8 may be estimated by using the geometrical mean of the tensile strength values in MD and CD, as done by Fellers et al. 30 for evaluation of the compressive modes. This study utilizes the geometrical mean for the tensile modes as
The simulations are performed with the finite element solver Ansys 2019R1 in a quasi-static regime using an implicit time-integration method. It is common to use explicit solver schemes for models exhibiting large non-linearities to avoid convergence problems. The use of explicit solvers introduces a limitation of the time step, which is limited by the size of the smallest element in the model and often requires the use of increase rate, mass-scaling, and the use of reduced-integration elements to make the solution solvable. In addi- tion, to achieve the accuracy for the spring-back, damping or implicit solver should be used. The motivation for choosing the implicit solver was to avoid these limitations. The model consists of the paperboard blank, the blank holder, the punch and the die. The blank is modelled with shell element 181, here fully integrated and with five integration points through the thickness. The tools are modelled as rigid bodies. Due to the symmetry, a quarter model is simulated, as seen in Figure 8. Initially, the blank is located so that the MD is parallel with the global x-axis and CD with the global y-axis, compare Figure 2A. The blank is meshed with 0.5-mm quad elements over the area with the creases, and then up to 1 mm towards the centre of the model, as seen in Figure 9. In total, the model consists of about 50 300 elements. This is the finest mesh among those used to address similar problems. The creases are included in the geometry, in total 30 with a depth of 0.05 mm. The depth is based on the average depth measured on the actual blank. The depth is considerably lower than for creases used for folding corners of, for instance, boxes but it suffi- cient for this kind of converting operation. The thickness of the blank in the model is 0.5 mm. The blank holder is force controlled, with a force of 700N acting on it (a quarter of 2800 N due to the use of quarter symmetry).
σ t 12 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffifi σ t 11 σ t 22 q :
ð 9 Þ
In Figure 7 the failure surface is shown for the two boards for zero shear stress. The shape of the two surfaces is very similar, but Board A fails earlier compared to Board B. It is obvious that the combined stress state allows for considerably higher stress levels than can be concluded if only the uni-axial tensile tests are studied. Strain failure is also evaluated, here using maximum strain theory according to
ε 11 ε t 11
ε 22 ε t 22
ε 12 ε t 12
<1 :
ð 10 Þ
ε F ¼ max
;
; abs
For zero shear strain, the failure envelop is a rectangle limited by the uni-axial tensile and compressive strains. In Equation 10 the tensile shear strain must be estimated and is in this project estimated in the same way as the tensile shear stress as
ε t 12 ¼ ffiffiffiffiffiffiffiffiffiffiffiffifi ε t 11 ε t 22 q :
ð 11 Þ
In Table 3 the material parameters for the failure evaluation are listed, which are given by the expression in Equation 8 along with the end value of the MD and CD curves in Figure 5.
FIGURE 7
TheTsai – Wu failure envelops for
the two boards for zero shear stress
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