UD plies of composite laminates have different thermal expansion co- efficients in the fibre direction and the in-plane direction perpendicular to the fibre, due to the different expansion coefficient between the matrix and the fibre. When a laminate with different orientations of the plies is man- ufactured the free expansion of the plies is constrained by the plies with a different orientation, causing the appearance of in-plane residual stresses. This kind of residual stresses are well-known and correctly calculated for the flat laminates, which typically raise during the cooling after the curing pro- cess. However, when the laminate is curved the existence of axial stresses,
4.2. Analysis of the effect of residual stresses in the unfolding failure 162
σθ, causes the apparition of interlaminar stresses to fulfil the equilibrium
equation (4.8c).
In this way, a temperature change induces the apparition of interlaminar stresses in a curved laminate. These interlaminar stresses can be tensile or compressive, depending on the stacking sequence, respectively anticipating or delaying the unfolding failure. When the temperature change is higher the residual stresses are higher too. Therefore, residual stresses depend on the service temperature. When the service temperature is lower, residual stresses are higher.
4.2.1
Homogeneous anisotropic materials
At mesoscopic scale, composite materials are not homogeneous as each ply is oriented in a different direction. However, an individual ply, or a lam- inate with all the plies oriented in the same direction, can be approximated as homogeneous and orthotropic. In a homogeneous material there are no differences in the coefficients of thermal expansion between plies. Therefore, by using the previous explanation about the differences in these coefficients, the non-existence of the mentioned residual stresses becomes reasonable. However, the curvature may induce residual stresses in many cases, even in homogeneous materials.
An orthotropic material is considered in this section as an example of anisotropic material. When the material orthotropic directions are coinci- dent with the θ and y axes, it can shrink or expand freely in those directions without inducing residual stresses. However, when the orthotropic axes are oriented in other directions the curvature constrains the thermal expansion and residual stresses appear.
In this section the model is applied to a shell with a mean radius equal to the thickness: R = t = 1 mm. The stiffness properties and the coefficients of thermal expansion, expressed in the orthotropic axes, are given in Table 4.1.
Table 4.1: Material properties
E11 (MPa) 150 ν12 0.3 G12 (MPa) 4.8 α11 (K−1) -1·10−6
E22 (MPa) 10 ν13 0.3 G13 (MPa) 4.8 α22 (K−1) 3·10−5
E33 (MPa) 10 ν23 0.3 G23 (MPa) 4.8 α33 (K−1) 3·10−5
Considering that the 0o direction is defined by the direction 1 of the
material being coincident with the θ axis, a material with 45o of orienta-
tion under a temperature decrement of 160oC presents the residual stresses (calculated according to the regularized solution developed in the previous
163 Three-dimensional models for evaluating interlaminar stresses
section) depicted in Figure 4.3. These stresses expressed in the material axes are depicted in Figure 4.4.
(a) (b)
Figure 4.3: Residual stresses in a 45o single ply laminate with ∆T =
−160oC. (a) Circumferential, axial and shear stresses. (b) Interlaminar nor-
mal stress.
(a) (b)
Figure 4.4: Residual stresses in a 45osingle ply laminate with ∆T = −160oC expressed in the material axes. (a) Stress in the fibre direction. (b) Stresses in the orthotropic matrix directions and shear stress.
In Figure 4.4, it can be observed that the highest stresses are given in the stiffest direction of the material (direction 1). The depicted residual stresses represent the thermal stresses due to the curvature.
Considering a typical strength in the radial direction of the material of approximately 50 MPa, and considering that the unfolding failure is typi- cally given for radius smaller than the mean radius, the thermal stresses in the given configuration represent a 4% of the total failure load.
Figure 4.5 shows the maximum radial stress appearing in a homogeneous curved sample under a temperature decrement of 160oC, depending on the
4.2. Analysis of the effect of residual stresses in the unfolding failure 164
orientation of the material. It can be observed that the maximum radial stress for the present case is given for approximately 38o. Furthermore, directions 0oand 90o have null residual stresses, since they are able to deform
freely.
Figure 4.5: Maximum radial stress for ∆T = −160oC depending on the orientation of the single ply laminate.
4.2.2
Composite laminates
In the case of a composite laminate, the different thermal expansion coefficients in the same direction of each ply due to their different orienta- tions induces significant residual in-plane stresses, and according to (4.8c) interlaminar stresses too.
As an example of application of the model to calculate the regularized thermal stresses, let us consider the material properties of Table 4.1 and two different stacking sequences given by [45, 0, -45, 90]S and [45, 90, -45, 0]S.
In these cases, Figure 4.6 shows the radial stresses obtained when applying a temperature decrement ∆T = −160oC.
In a composite laminate, fibres have very low (or even negative) expan- sion coefficients, and therefore, the plies have very small expansion coeffi- cients in direction 1, the expansion coefficient being higher in direction 2. As a consequence, referring to the circumferential stresses, the 0o plies are compressed in the circumferential direction while the 90o plies are tensioned
in the same direction. By using equation (4.8c), the presence of compressed plies implies a decrease of the interlaminar stress when the radial coordi- nate increases, and, on the contrary, the presence of tensioned plies implies an increase of the interlaminar stress when the radial coordinate increases. Hence, when the 0o plies are outer, in the stacking sequence, than the 90o plies, the inner plies are compressed in the through-the-thickness direction
165 Three-dimensional models for evaluating interlaminar stresses
Figure 4.6: Radial stresses due to the temperature increment in different stacking sequences.
and the outer plies are tensioned, as can be seen in Figure 4.6. Inversely, when the 90o plies are outer than the 0o plies the inner plies are tensioned
in the through-the-thickness direction and the outer plies are compressed. The unfolding failure is given in these curved beams when the component is under a bending moment that induces an opening of the curvature. Under this load state the maximum of the radial stresses is located in the inner part. Therefore, from the residual stresses point of view, it is optimal to locate the 0oplies outer than the 90o plies in the stacking sequence. This configuration
is optimal also when considering the stresses distributions due to the bending moment, residual stresses reinforcing this optimization criterion.
Furthermore, considering an interlaminar strength with a value of ap- proximately 50 MPa with the selected material properties, the residual stresses in the case of the [45, 90, -45, 0]S stacking sequence anticipates
a 12% the unfolding failure, and in the case of the [45, 0, -45, 90]S stacking
sequence it improves a 12%, considering the given geometry with t = R. The interlaminar stresses are lower when the relation tl/R decreases,
tl being the thickness of a ply. This parameter can decrease in two cases,
when the number of plies increases maintaining the t/R relation, or when the mean radius increases maintaining the number of plies and the ply thickness. Figure 4.7 shows the decrement of the interlaminar normal stresses in both cases respect to the interlaminar normal stresses depicted in Figure 4.6.
Hence, the effect of the residual stresses is specially important in thin laminates (with a small number of plies) and in laminates with a small mean radius respect to the thickness.