## 3 Verification and Comparison of Proposed MPFE  
### Experimental Results with the CFRP AS4/3501-6 under Various Strain Rates  
The subject of the investigation was a carbon/epoxy composite (AS4/3501-6) procured in the form of a prepreg, which was subsequently used for the fabrication of laminates using the standard autoclave process. To perform multi-axial experiments, unidirectional (UD) carbon/epoxy specimens were tested at diverse loading directions relative to the principal fiber reinforcement. These experiments primarily yielded stress states that combined transverse normal and in-plane shear stresses. For each set, between two and six specimens were tested. Experiments were conducted at three different strain rates, with quasi-static tests carried out on a servo-hydraulic testing machine at a strain rate of 0.0001 s\({}^{-1}\), as shown (1) in Figure 4. Intermediate rate tests were also conducted in the servo-hydraulic machine at an average strain rate of 1 s\({}^{-1}\). A unique fixture was devised featuring a gap between the specimen and the lower cross-head. This allowed for the crosshead to accelerate to almost uniform speed before transferring the load to the specimen, thus ensuring a more consistent strain rate during the test. High-strain-rate tests were conducted using a split  
Figure 3: Schematic diagram of the tensile–compressive failure envelope of UD CFRP. Hopkinson bar with strain rates ranging from 200 to 400, employing off-axis prismatic specimens, as shown (2) in Figure 4. The strain rate assigned to various tests represents an average value throughout the duration of the test. To maintain the most consistent strain rate possible, a rectangular pulse was applied to the incident bar. Experimental results derived from testing unidirectional (UD) CFRP specimens demonstrate a consistent correlation between mechanical properties and strain rate [6; 7; 8; 9; 10]. Therefore, it is understood that mechanical properties, such as elastic modulus and strength, are contingent on the particular loading conditions, with quasi-static loading representing a distinct scenario within this spectrum. In order to validate the proposed theoretical assumption, experimental data [7] from UD CFRP AS4/3501-6 subjected to both compression and tension at various angles relative to the reinforcement direction were used. These tests were conducted at different strain rates. The World-Wide Failure Exercise was set out to check the current capability of various methods for predicting the strength of composite lamina and laminates. Selected workers in the area of composite failure theories were invited to compare their theories with the experimental data provided by the organizers, from which we obtained \(\tau_{6}^{0}=125\) MPa [4]. Notably, for quasi-static loading, specific values of shear stress under compression at a 45\({}^{\circ}\) angle, as well as the tensile and compressive strength in the direction perpendicular to the reinforcement, were utilized as initial conditions. This can be represented symbolically as \(\sigma_{2}^{+}=65\) MPa, \(\sigma_{2}^{-}=-285\) MPa. With the experimental data from Table 1, we can derive the values of all coefficients for \(f(\sigma_{2},\tau_{6})\) under quasi-static loading: \(m=175\) MPa, \(\alpha=-110\) MPa, \(n=124.5\) MPa, and \(k=0.000972\) MPa\({}^{-1}\). By substituting these coefficients into Equation (5), the failure envelope \(f(\sigma_{2},\tau_{6})\) of the UD composite under quasi-static loading with a strain rate \(\dot{\epsilon}=10^{-4}\) s\({}^{-1}\) can be plotted. Similarly, failure envelopes for the same material under static (\(\dot{\epsilon}=1\) s\({}^{-1}\)) and dynamic loading (\(\dot{\epsilon}=400\) s\({}^{-1}\)) can be derived using the data in Table 2.