In the homogeneous anisotropic material the solution of the 2D regular- ized problem given by Ko and Jackson [4] coincides with the one given by Lekhnitskii et al. [19].
The regularized 3D model developed, when applied to the pure bending moment case, includes several important hypothesis regarding the forces and moments. It assumes that the forces and moments reach their regularized values, which are the inputs of the model, and those regularized values are determined by the free boundaries at the ends in the y direction and a bending moment applied uniformly at the ends of the θ direction (it is considered uniform only when it is regularized in the y direction, as it has actually different values at the free edge). This model is equivalent to a
4.3. Analysis of the three-dimensional effects over the unfolding failure 168
free torsion model, the beam can torsion freely while T0 = 0 due to the
boundary condition of null τθy at the free edges along the width. The axial
force is considered also null, N0 = 0, due to the boundary condition of null
σy at the free edges in the width as well as the applied in-plane shear force
is considered null also, P0 = 0.
A second model has been considered as a constrained model. In this model the torsion is constrained and an uniform bending moment is applied at the ends along θ (the condition of T0 = 0 used is substituted by γθy0 = 0,
which implies a null torsion). Hence, 3D deformations are constrained in this model. As a consequence, this model is not realistic for the ILTS four-point bending test [3].
Considering an orthotropic material with properties given in Table 4.1 and varying the orientation angle in a single-ply laminate, for a geometry determined by t = R = 1 mm and a load of M0 = −1 Nm/m, the evolution
of the maximum regularized radial stress as a function of the orientation angle is depicted in Figure 4.9.
Figure 4.9: Maximum radial stress for M0 = −1 Nm/m depending on the
orientation of the single ply laminate.
It can be seen that using a 2D model for the homogeneous material returns similar results to the 3D constrained model. However, in a more realistic model where the beam can freely torsion, the maximum radial stress is significantly higher than that given by the 2D model. Therefore, a 2D model is not accurate for this kind of configuration. The difference between the 3D model and the 2D model is maximum around 35o for the given
material properties, and it is null in the 0o and 90o single ply laminates.
Considering the case with a maximum difference, given by an orientation of 35o, all the stresses in the regularized beam calculated with both 3D
169 Three-dimensional models for evaluating interlaminar stresses
models are depicted in Figure 4.10.
(a) (b)
Figure 4.10: Stresses in a 35o single ply laminate under M
0 = −1 Nm/m
for (a) the free torsion model and (b) the constrained model.
The stresses calculated with the free torsion model present high differ- ences with the ones calculated with the constrained model (which are similar to those of the 2D models since the 3D deformations are constrained). Spe- cially, the differences are very high in the maximum values of the circumfer- ential and the radial stress, affecting significantly to the failure prediction.
4.3.2
Composite laminates
The 2D models for composite laminates are typically obtained by assum- ing a plane strain or plane stress state. Plane stress is not a realistic state because a null stress state in the y direction does not fulfil the compatibility equations, causing discontinuities in the displacements along the transversal direction. However, in the same way, a null strain state in the y direction, when the laminate has any ply with an orientation different to 0o and 90o,
causes discontinuities in the radial shear stress associated to the y direction, τry, and then it is neither physically possible.
Therefore, it is expected that a 3D model can return more accurate results than the 2D models, as a 3D model accomplish the continuity in the displacements and in the shear stress τry. For this reason, the constrained
3D model yields more accurate results than the 2D model in the laminate case.
If a typical laminate with plies oriented at 0o, ±45oand 90o is considered, it is expected to obtain higher errors in a 2D model when the number of
4.3. Analysis of the three-dimensional effects over the unfolding failure 170
±45o plies increases and when the asymmetry of the stacking sequence is
higher.
Although not shown here, it has been observed that, generally, the dif- ferences between the 2D model and the 3D model decrease when the number of plies increase, as the stacking sequence causes a more homogeneous ma- terial. Consequently, the 2D model is less accurate when it is applied to thin laminates or less quasi-isotropic laminates.
An interesting 3D effect is the difference between plies of 45o and -45o when both kind of plies are given in the same stacking sequence. In a 2D model the stiffnesses of a 45o ply are the same than the stiffnesses of
a -45o ply and, therefore, the stresses are not affected by the sign of the orientation. Consequently, when a ply of 45o is adjacent to a ply of -45o
the circumferential stresses are continuous. However, due to the torsion, in a 3D model when a ply of 45o is adjacent to a ply of -45o, circumferential stresses are not continuous across the interface. This effect can be seen in Figure 4.11, where a stacking sequence [45 -45 -45 45] has been used. In this Figure the free torsion model presents discontinuities in the circumferential stress, while the constrained model presents a continuous distribution.
(a) (b)
Figure 4.11: Differences between 45o plies and -45o plies when both are used in a laminate, showed in a [45 -45 -45 45] stacking sequence. (a) Free torsion model, (b) Constrained model.
4.3.3
Finite elements comparison
The solution of the model has been compared with FE results for the case of the bending moment. A curved beam with the material given in
171 Three-dimensional models for evaluating interlaminar stresses
Table 4.1 has been used with a single-ply configuration of orientation 45o,
1 mm thickness, a mean radius of 2 mm, a total angle of 300o and a width in the y direction of 10 mm. A −10 Nmm total bending moment has been applied. Considering the 10 mm width, a bending moment per width of M0 = −1 Nmm/mm is given. The solid has been meshed with 10 quadratic
elements in the thickness, so a high accuracy is obtained for the interlaminar stresses. By comparing the finite elements results obtained with the free torsion model and with the constrained model, Figure 4.12 is obtained.
Figure 4.12: Validation of the models (continuous and discontinuous lines) by using M0 = −1 N with finite elements results (asterisks) in the homoge-
neous case of 45o.
Small differences (about a 10%) are observed between the solution of the free torsion 3D model and the FE results in all the stress components. These differences are due to the finite width of the specimen in the FE model. The finite width in the y direction induces a stress state of plane stress in the
4.3. Analysis of the three-dimensional effects over the unfolding failure 172
boundary, causing the bending moment in the edges to be smaller than the bending moment in the center of the beam. The bending moment, M0 [N], has been calculated from the total moment, MT [Nmm], and from
the width, W [mm], as M0 = MT/W , supposing that it is homogeneously
distributed. Actually, due to the edge effect we have that the regularized bending moment is calculated as M0 = kWMT/W , where the parameter kW
depends mainly on the W/t ratio although it depends also on the stacking sequence, the material properties, the R/t ratio, etc. It can be deduced that kW −→ 1 when W/t −→ ∞.
Therefore, the edge effects causes that the maximum value of M0 is
higher than MT/W . Using a FE model for the present problem a value of
kW = 1.2 has been obtained. Therefore, by using a M0 = 1.2 N bending
moment, Figure 4.13 is obtained.
Figure 4.13: Validation of the models (continuous and discontinuous lines) by using M0 = −1.2 N and comparing with FE results (asterisks) in the
173 Three-dimensional models for evaluating interlaminar stresses
Considering the right value of the parameter kW and, therefore, the
right value of the bending moment per unit of length in the inner part of the sample M0, the differences between the finite element results and the
free torsion 3D model are considerably reduced. The free torsion model results are very accurate compared with the finite elements model, as no kind of displacement constraint has been imposed in the finite elements model. However, the constrained model (which has similar characteristics to a 2D model) is less accurate in this case due to the torsion induced by the orientation of the ply.
The importance of considering the effect of the finite width by the pa- rameter kW in the bending moment calculation has been showed. The ana-
lytical calculation of kW is necessary for an accurate calculation of the stress
components, the regularized 3D model not being satisfactory at that point. This effect is calculated by using the non-regularized model developed in the following section.