4.4 Mesh dependency
4.5.1 Standard tests
The ability of the material model implemented in LS-Dyna [1] to reproduce the physics of shear and compression failure is here assessed. Standard experimental tests were carried with carbon-epoxy T300/913 (see Appendix A). The numerical models presented in this sub-section reproduce the geometry of specimens that were
(a) (b) (c) (d) (e) 0 1 2 0 0.05 0.1 0.15 Displ. (mm) Mesh (a) (b) (c) (d) Load (kN)
Figure 4.5: Models with different mesh densities; the failed elements are identified by a lighter colour; (a) shows the material axes; (b), (c) and (d) show the fracture planes; (e) load vs. displacement curves for different mesh refinement levels
actually tested, and use the material properties obtained. The objective is to assess the model’s capability to predict the main failure features observed experimentally, such as the inclined fracture for the transverse compression test, the ±45◦ failure for
the shear specimen and the kink-band formation for the longitudinal compression specimen. Note that none of these features is directly included in the model.
The elastic properties and strengths, obtained experimentally, are presented in Table 4.1. The experimental in-plane shear stress vs. strain curve was used as input for the numerical model, the Poisson’s ratio νba was obtained as 0.021 and the
fracture angle for pure transverse compression was measured as φo = 53◦. Regarding
the through-the-thickness (c) direction, the composite was assumed transversely isotropic, with νbc = 0.4. The shear modulus Gca was taken as being equal to
the (initial) in-plane shear modulus (Gab), and the Poisson’s ratio νca as νba. The
intralaminar toughness was measured experimentally using 4-point bending tests as Γb = 0.22 kJ/m2, and Γkink and Γa were obtained using CTS and CCS as Γkink =
79.9 kJ/m2 and Γ
a= 91.6 kJ/m2 respectively, see Chapters 5 and 6. The toughness
values ΓT and ΓL were taken as the mode II interlaminar fracture toughness for the
same material, ΓT = ΓL= 1.1 kJ/m2, see Appendix B.
In all the examples that follow, an element with slightly lower strength was used to trigger failure close to the middle of the specimen.
Table 4.1: Mechanical properties of T300/913 and HSC/913 Material Ea Eb Gab Xt Xc Yt Yc Sab
(GPa) (GPa) (GPa) (MPa) (MPa) (MPa) (MPa) (MPa) T300/913 132 8.8 4.6 2005 1355 68 198 150 HSC/913 130 9.2 4.6 1650 1100 60 200 100
4.5.1.1 Modelling shear failure of a (±45)8S test specimen
Shear tests were carried with (±45)8S specimens (angles relative to the loading direction), tested in tension, according to the appropriate ASTM standard [153], see Fig. 4.6(a). The shear stress vs. strain curve was nonlinear almost from the beginning of the test, but no strain localization was present until immediately before final failure, which happened at a shear strain of about 25%. The data reduction was done according to the ASTM standard [153], but, in order to obtain the full strain vs. stress curve, fibre scissoring and width reduction were taken into account. (Fibre scissoring was taken into account by considering the current orientation of the fibres in the data reduction (affected by the shear strain), rather than assuming that they remain at 45◦. The applied stress was calculated by dividing the load by
the current cross-sectional area, where the reduction in width was computed using the strains from the transverse strain gauge.)
An FE model of part of the specimen, containing the failed region, is presented in Fig. 4.6(b). The model has the same dimensions as the actual specimen—except for the length, which is smaller. The model has 16 solid elements across the thickness, in order to simulate each layer individually. The ±45◦ failure can be observed in
Fig. 4.6(b), and results from the damage variable affecting the local shear traction components in the predicted fracture plane, within each element. (In fact, when the shear stress is reduced at some angle β, it is also reduced at an angle β + 90◦. This
may give rise to unrealistic failure patterns, and is a feature typical to CDM-based models.) The numerical load vs. displacement curve is compared to the experimental in Fig. 4.6(c). The experimental displacement was computed by multiplying the strain in the longitudinal strain gauge by the length of the numerical specimen.
(a)
(b )
(c )
(±4 5 )
! " # $ % ! $Figure 4.6: (a) Shear specimen; (b) model of the shear specimen; (c) exper- imental and numerical load vs. displacement curves
For the good agreement obtained, the consideration of fibre scissoring and width reduction in the data-reduction were key factors, as well as the FE code’s capability to handle large rotations appropriately.
4.5.1.2 Modelling matrix compression failure
Pure transverse compression tests were carried out, and a typical fracture surface is shown in Fig. 4.1(b). Fracture occurred at an angle of 53◦ with the thickness
direction. A numerical model of this specimen was created, and is shown in Fig. 4.7. The fracture angle predicted by each failed element, available as a history variable, is 53◦. This correct prediction is a consequence of the matrix compression failure
criterion, expressed in Eq. 3.19, being maximized for this angle, when the material is subjected to pure transverse compression. The angle of the band of failed material (smeared fracture surface), which can be observed in Fig. 4.7, is about 50◦. In
this case, the correct prediction results from the shear traction components being degraded in a coordinate system aligned with the predicted fracture plane. If the failed elements in this example had been deleted immediately after they failed, the contact between the fracture surfaces would have not been properly modelled during the propagation of the fracture surface (failed band) across the specimen. The author has observed this to affect the angle of the failed band observed in the numerical
Figure 4.7: Model of the transverse compression test specimen specimen.
4.5.1.3 Modelling fibre compression failure
The formation of kink bands at a small angle β with the normal to the loading direction is predicted by the model, as a result of the damage variable acting on the shear stress in the misalignment frame. For T300/913, the author has observed it to be about 25 ± 5◦ for out-of-plane kinking in CCS (see Chapter 6), and 20 ± 5◦
for standard axial compression specimens with in-plane kinking, Fig. 4.8(a). Fig. 4.8(b) shows the FE mesh of an axial compression specimen, with the corresponding loading. The formation of the kink band can be observed in Figs. 4.8(c) to (e). The predicted kink band angle is about 15◦. After the kink band is formed, further
loading leads to kink-band broadening, as observed in Fig. 4.8(e).
Kink fronts have been reported to reorient themselves naturally, as they prop- agate, before stabilizing in a β direction [154] and the tip of kink bands to lie at different angles than the rest of the kink band [155]. These observations emphasize the role of damage propagation within the kink band, for the definition of its final orientation. Turning to the numerical model, the kink band orientation observed in Figs. 4.8(c) to (e) is never predicted explicitly, and is the result of the shear traction component being degraded in the misalignment coordinate system, whose orientation is updated during damage propagation. The author has observed that not updating the misalignment frame (where the traction components are degraded) results in a smaller angle predicted.
( a ) ( b ) ( c ) ( d ) ( e )
b
a
b
a
Figure 4.8: (a) Left half of a failed longitudinal compression test specimen; (b) model of the same specimen; (c) to (e) formation of a kink band and kink-band broadening