Table 5.3: Mechanical properties of carbon fibers
Property Symbol Value Unit
Longitudinal elastic modulus E1 240 GPa Transversal elastic modulus E2 14 GPa Longitudinal shear modulus G12 14 GPa Longitudinal Poisson’s ratio ν12 0.2 Transversal Poisson’s ratio ν23 0.25
general expression, able to catch very well non-linear behavior of materials, a Ramberg-Osgood [110] constitutive law has been used.
!!" !
� = 4206σ +4206σ ·�
|σ|
123
�3.96−1
Figure 5.10: Ramberg-Osgood regression of eopxy matrix tensile tests
Figure 5.10 shows the data taken from four experiments and the Ramberg-Osgood regression curve. The fitting data are listed in Table 5.4.
5.4 Micro-mechanical models
Here the models built to obtain the 13 parameters necessary to describe the complete behavior of the non-crimp fabric yarns are presented.
Using a model like in Figure 5.11, several analyses can be performed. In particular the following analysis is presented:
• tension along 1 in order to obtain EL and ν12
5.4. MICRO-MECHANICAL MODELS
Figure 5.11: Micro-mechanical model of fiber reinforced plastic composites
• shear along 2-3 in order to obtain GT, τTY, n6 and α23
• shear along 1-3 in order to obtain GL, τLY and n4
5.4.1 Tension along 1
It is possible to calculate the values of the elastic modulus ELand the Poisson’s ratio ν12from a micro-mechanical model simulating a tension along direction 1. The prevalent contribution of the fibers makes the behavior of the model linear, with an elastic modulus mainly dependent on the volume fraction of fibers. This is also predictable from the Chamis formula:
ELChamis= Vf· E11fibers+ (1 − Vf) · Ematrix (5.28) For a volume fraction of 70% and an elastic modulus of fibers and matrix of 240 GPa and 4.2 GPa respectively the longitudinal elastic modulus of the yarns will be 169.3 GPa. From the analysis, values of which are listed in Table 5.5, the value calculated by linear regression is 166.8 GPa (the difference is 1.5%).
Measuring the deformation along 2 due to the tension along 1 (transverse contraction) it is
Table 5.4: Mechanical properties of epoxy matrix
Property Symbol Value Unit
Tensile elastic modulus Em 4206 MPa
Yield stress σmY 123 MPa
Hardening exponent nm 3.96
5.4. MICRO-MECHANICAL MODELS
Figure 5.12: Micro-mechanical model σ11− �11
Table 5.5: Results from the micro-mechanical model “tension along 1”
frame �σ11� [MPa] ��11� [%] ��22� [%] ��33� [%]
0 0 0.00 0.00 0.00
1 620 0.37 -0.09 -0.09
2 1239 0.74 -0.18 -0.18
3 1858 1.11 -0.27 -0.28
4 2476 1.48 -0.36 -0.37
5 3093 1.85 -0.46 -0.47
6 3710 2.22 -0.55 -0.57
7 4327 2.59 -0.65 -0.68
8 4943 2.96 -0.75 -0.78
9 5559 3.33 -0.85 -0.88
10 6175 3.71 -0.94 -0.99
possible to calculate the Poisson’s ratio as follows:
ν12= −��22�
��11� (5.29)
In the same way it is possible to calculate ν13 but as the material is transversely isotropic must be ν13= ν12. From Table 5.5 and using again Chamis formula the following parameters can be calculated:
νChamis12 = Vf· ν12fibers+ (1 − Vf) · νmatrix= 0.315 (5.30) The value calculated from the finite element results is not so perfect due to numerical approxima-tions. In fact it is ν12= 0.252 and ν13= 0.263 with a mean value of ν = 0.258 with a difference of 22% respect to analytically achieved results.
5.4. MICRO-MECHANICAL MODELS
5.4.2 Tension along 2
It is possible to calculate the values of the elastic modulus ET, the Poisson’s ratio ν23, the yield stress σTY and the hardening coefficient n2 from a micro-mechanical model simulating a tension along direction 2 (or 3 as well). In fact along this direction the non-linear behavior of the matrix becomes important and also the yarns behave non-linearly. Thus a Ramberg-Osgood regression must be used.
The behavior of the micro-mechanical model loaded along 2 or along 3 is slightly different.
When loaded along direction 2 raises to the following regression parameters:
• E2= 8711 MPa
• σY2 = 304 MPa
Figure 5.13: Linear regression of �σ11�–��11� data
(a) Micro-mechanical model σ22− �22 (b) Micro-mechanical model σ33− �33
Figure 5.14: Micro-mechanical models along direction 2 and 3
5.4. MICRO-MECHANICAL MODELS
Table 5.6: Results from the micro-mechanical model “tension along 2 and 3”
(a) Tension along direction 2
frame �σ22� [MPa] ��22� [%]
0 0 0.00
1 17 0.20
2 34 0.39
3 50 0.59
4 66 0.78
5 81 0.98
6 96 1.17
7 109 1.36
8 122 1.56
9 134 1.75
10 145 1.95
(b) Tensione along direction 3
frame �σ33� [MPa] ��33� [%]
0 0 0.00
1 10 0.11
2 20 0.23
3 30 0.34
4 39 0.46
5 49 0.57
6 58 0.68
7 68 0.80
8 77 0.91
9 85 1.02
10 94 1.14
• n2= 3.37
while if a tension along direction 3 is performed the models gives the following results:
• E3= 8668 MPa
• σY3 = 285 MPa
• n3= 3.67
The small difference is even clearer in Figure 5.15 where both data from a tension along direction 2 and a tension along direction 3 have been plotted together. A mean behavior can be calculated using both set of data. The result is plotted in Figure 5.15. The parameters of the Ramberg-Osgood expression are:
• ET = 8703 MPa
• σY3 = 296 MPa
• n3= 3.46
Taking the advantage of the previous analyses, it is easily possible to calculate the Poisson’s ration ν23. In fact using the homogenized strain data along direction 2 and along direction 3 from an analysis performed loading along direction 2 it is possible to calculate ν23 and at the same way from an analysis performed loading along direction 3 is possible to calculate ν32 using the homogenized strain data along direction 3 and along direction 2. These two values must be theoretically the same. However because of the numerical approximations of the finite element approach the two values will be different but should be still very close to each other.
By linear regression of ��33� vs. ��22� the Poisson’s ratio is 0.361 and by linear regression of
��22� vs. ��33� the Poisson’s ratio is 0.374 with a difference of 4%. The used value will be the
5.4. MICRO-MECHANICAL MODELS
� = 8703σ +8703σ ·�
|σ|
296
�3.46−1
Figure 5.15: Ramberg-Osgood regression of �σ22�–��22� and �σ33�–��33� data
5.4.3 Shear along 2-3
The shear 2-3 is the one in the plane of symmetry, perpendicular to the direction of fibers.
This loading condition leads obviously to a non-linear behavior of the micro-mechanical model and so consequently of the yarns. Again a Ramberg-Osgood regression must be used and fitting parameters must be calculated.
Figure 5.16: Micro-mechanical model σ23− �23
From Table 5.7 it is possible to plot the data, showed in Figure 5.17. The Ramberg-Osgood regression leaded to the following results:
5.4. MICRO-MECHANICAL MODELS
Table 5.7: Results from the micro-mechanical model “shear along 2-3”
frame �τ23� [MPa] �γ23� [%]
0 0.00 0.00
1 7.27 0.23
2 14.51 0.45
3 21.65 0.68
4 28.60 0.91
5 35.28 1.14
6 41.61 1.36
7 47.59 1.59
8 53.21 1.82
9 58.49 2.05
10 63.46 2.28
• GT = 3205 MPa
• τTY = 135 MPa
• n6= 3.52
γ = 3205τ +8703τ ·�
|τ|
135
�3.52−1
Figure 5.17: Ramberg-Osgood regression of �τ23�–�γ23� data
5.4.4 Shear along 1-3
The shear along 1-3 is the one considering a plane that contains the axis of the fibers. The influence of the matrix anyway is introducing non-linear behavior in the shear response of the
5.4. MICRO-MECHANICAL MODELS
Table 5.8: Results from the micro-mechanical model “shear along 1-2 and 1-3”
(a) Shear along direction 1-2
frame �τ12� [MPa] �γ12� [%]
0 0 0.00
1 21 0.39
2 39 0.78
3 53 1.18
4 64 1.57
5 72 1.96
6 79 2.35
7 85 2.75
8 90 3.14
9 95 3.54
10 99 3.93
(b) Shear along direction 1-3
frame �τ13� [MPa] �γ13� [%]
0 0 0.00
1 13 0.23
2 25 0.46
3 35 0.68
4 44 0.91
5 52 1.14
6 58 1.37
7 63 1.60
8 68 1.83
9 72 2.05
10 75 2.28
Also in this case, considering the transversely isotropic nature of the material it is possible to define two numerical models, one with a loading condition along 1-2 and one with a loading condition along 1-3 (Figure 5.18). The comparison of the two models should give similar results, while the differences should remain related to numerical approximations due to the finite element approach. The analysis performed with the loading along direction 1 and 2 can be seen in Figure 5.18(a). The results, listed in Table 5.8(a), lead to the following regression parameters:
• E4= 5461 MPa
• σY4 = 94 MPa
• n4= 3.85
The analysis performed with the loading along direction 1 and 3 can be seen in Figure 5.18(b).
The results, listed in Table 5.8(b), lead to the following regression parameters:
(a) Micro-mechanical model τ12− γ12 (b) Micro-mechanical model τ13− γ13
Figure 5.18: Micro-mechanical models along direction 1-2 and 1-3