Basic strength parameters¶
Strength is a rather ambiguous term that most often needs a prefix or qualifier. For example, the yield strength and the tensile strength of most metals are very different quantities representing very different mechanisms. Due to the inherent material variability of composites, the strength must also be considered in a statistical framwork using qualifiers such as mean and characteristic strength.
Strength of unidirectional fiber composites is commonly given by the basic strength parameters corresponding to the different types of basic loading (tensile, compression and shear) along different principal directions (longitudinal and transverse) and on different material planes.
Although the basic strength parameters are well defined in terms of the modes of loading, it is not always trivial to quantify the values due to the progressive and nonlinear behavior.
The term basic strength parameters is used for orthotropic materials and includes:
$X_T$ : Tensile failure strength, 1-direction ($\sigma_1>0$)
$X_C$ : Compressive failure strength, 1-direction ($\sigma_1<0$)
$Y_T$ : Tensile failure strength, 2-direction ($\sigma_2>0$)
$Y_C$ : Compressive failure strength, 2-direction ($\sigma_2<0$)
$Z_T$ : Tensile failure strength, 3-direction ($\sigma_3>0$)
$Z_C$ : Compressive failure strength, 3-direction ($\sigma_3<0$)
$S_{12}$ : Shear failure strength, 1-2-plane ( $|\tau_{12}|$ )
$S_{13}$ : Shear failure strength, 1-3-plane ( $|\tau_{13}|$)
$S_{23}$ : Shear failure strength, 2-3-plane ( $|\tau_{23}|$)
There are no universal agreement on the nomenclature for the basic strength parameters. In the literature we find symbols all over the place, where every textbook and paper seems to have invented its own. The nomenclature in this compendium follows [1] with the exception of the shear strengths (those symbols were already occupied, and after all, it is a tradition to come up with variations).
This is the time to show up in the lecture, join the discussion on all the details and the nuances, and look at stress-strain curves and images of failed specimen. We will also briefly connect this topic to later topics on testing and material variability, and initially it can be useful to take a brief look at Material variability and characteristic strength
UD fiber composites¶
UD fiber composites are usually considered to be transversely isotropic and the number of independent strength parameters is reduced. When the 2-3 plane is the plane of isotropy,
$$ Z_T = Y_T \\ Z_C = Y_C \\ S_{13} = S_{12} $$Using absolute values¶
Note that the parameters for compressive strengths are implemented using the positive, or absolut value as in the following example:
m = {"name": "E-glass/Epoxy", "units": "MPa-mm-Mg", "type": "UD", "fiber": "E-glass",
"Vf": 0.55, "rho": 2000E-12,
"description": "Typical UD E-glass/Epoxy from TMM4175",
"E1": 40000, "E2": 10000, "E3": 10000,
"v12": 0.3, "v13": 0.3, "v23": 0.4,
"G12": 3800, "G13": 3800, "G23": 3400,
"a1": 7e-06, "a2": 2.2e-05, "a3": 2.2e-05,
"XT": 1000, "YT": 40, "ZT": 40,
"XC": 700, "YC": 120, "ZC": 120,
"S12": 70, "S13": 70, "S23": 40,
"f12":-0.5, "f13":-0.5, "f23":-0.5}
Obviously, when comparing a compressive stress to the corresponding material strength, we must take care of the sign during the calculation:
s1 = - 200 # normal stress in the 1-direction
if s1 > 0:
R = m['XT']/s1 # use tensile strength
else:
R = -m['XC']/s1 # use compressive strength with negative sign
print('Strength to load ratio is',R)
Alternative, simpler code:
R=max(m['XT']/s1, -m['XC']/s1)
print('Strength to load ratio is',R)
References and further readings¶
- Herakovich, Carl T. Mechanics of Fibrous Composites. New York: Wiley, 1998.
