## 2 Empirical Model for Calculation of Mechanical Properties
### Fundamental Strength of UID Composites  
The longitudinal tensile strength of unidirectional fiber-reinforced composite materials depends on the volume ratio and ultimate tensile strain of the fiber and the matrix. The longitudinal tensile strength is expressed in two cases. In the first case, the failure occurs when the matrix reaches its ultimate strain. In the second case, failure of the composite occurs when the fiber reaches its ultimate strain. The factor \(V_{min}\) plays an important role in handling both the cases independently. If \(V_{f}<V_{min}\), then the expression (9) will be used. However, if \(V_{f}\geq V_{min}\), then the expression of the second case, as expressed in Equation (10), is applied [22; 23]. Here, \(V_{min}\) and \(V_{f}\) are the minimum fiber volume fraction that ensures fiber-controlled composite failure, and the volume fraction of fiber, respectively. \[\sigma_{clu,t}=\sigma_{mu}(1-V_{f}) \tag{9}\]  
\[\sigma_{clu,t}=\sigma_{fu}V_{f}+(\sigma_{m})\epsilon_{f}^{*}(1-V_{f}) \tag{10}\]  
\[V_{min}=\frac{\sigma_{mu}-\sigma_{m}\epsilon_{f}^{*}}{\sigma_{fu}+\sigma_{mu} -\epsilon_{f}^{*}} \tag{11}\]  
The response of unidirectional fiber-reinforced composite to longitudinal compressive stress is quite different from the response to longitudinal tensile stress. This is because, under longitudinal compressive stress, the unidirectional fiber-reinforced composites exhibit various modes of failure. In the case of elastic behavior of the matrix, two different failure modes can occur. These are the extensional mode and the shear mode of micro-buckling of fibers. The extensional mode of failure occurs when the volume ratio of fibers is less than 0.2. For unidirectional fiber-reinforced composites having a higher fiber volume ratio, the shear mode of micro-buckling of fibers takes place, as depicted in Figure 3 [22; 23]. The expression of longitudinal compressive strength (\(\sigma_{clu,c}\)) for the extensional model is derived using the energy method [24]. The expression is as follows:  
\[\sigma_{clu,c}=2V_{f}\sqrt{\frac{V_{f}E_{m}E_{f}}{3(1-V_{f})}} \tag{12}\]  
The expression of longitudinal compressive strength (\(\sigma_{clu,c}\)) for the shear mode is as follows [25]:  
\[\sigma_{clu,c}=\frac{G_{m}}{1-V_{f}} \tag{13}\]  
Figure 3: Schematic representation of (**a**) extensional mode and (**b**) shear mode of micro-buckling of fibers (Figure adapted from [2]). The longitudinal compressive modulus of unidirectional fiber-reinforced composite is almost equal to the longitudinal tensile modulus of unidirectional fiber-reinforced composite and can be used as an alternate to each other [26]. The transverse tensile strength of the unidirectional reinforced composites is less than the transverse strength of the matrix. The unidirectional fibers in a composite result in stress concentration upon transverse loading. The stress concentration factor (SCF) is calculated to find the transverse tensile strength of the unidirectional fiber-reinforced composite. The transverse tensile strength of the unidirectional reinforced composites is controlled by the ultimate strength of the matrix and the stress concentration factor. The expression to predict the transverse tensile strength (\(\sigma_{ctu,t}\)) is as follows [2]:  
\[\sigma_{ctu,t}=\frac{\sigma_{mu}}{\text{SCF}} \tag{14}\]  
\[\text{SCF}=\frac{1-V_{f}[1-\frac{E_{m}}{E_{f}}]}{1-\frac{4V_{f}}{\pi}^{\frac{ 1}{2}}[1-\frac{E_{m}}{E_{f}}]} \tag{15}\]  
The micro-mechanical behavior of a unidirectional fiber-reinforced composite under transverse compressive loading is difficult to predict because, under transverse compressive loading, the iso-stress and the iso-strain approaches become invalid.