## 2 Influence of Material and Mechanical Parameters on Structural Response and Toughness  
Concerning structural design, different strategies may be adopted depending on the expected structural response and namely on the potential of the structure to redistribute internal forces. Two responses can, in general, be identified: those related to strain softening and those where the material can develop relatively large plastic deformations (or even harden while deforming plastically). The former (strain softening) is associated with brittle responses with potential strain localization, as the weakest section governs not only the strength but also the deformation capacity. In concrete, such strain localization develops at a single critical crack, whose progression leads to the failure of the element. Such strain localization is typically found in elements subjected to tensile or shear stresses and without reinforcement to control cracking [25]. However, even in cases where reinforcement is provided, strain localization may occur due to the geometry or in the presence of strong gradients of internal forces (as typically found for punching). Other than limiting the structural resistance by the capacity of the weakest element, it shall be noted that cases where brittle responses might be expected are also sensitive to imposed deformations, such as temperature, self-stresses, or support settlements. Concerning cases of ductile responses, plastic regions develop, allowing for large deformation capacities while the structural resistance is maintained (or even increased in case of strain hardening) [26]. This allows for the redistribution of internal forces between different elements of a system, and the structural resistance is eventually a result of the individual resistance of its components (and not governed by the strength of the weakest region). Also, such a response allows removing any potential effect of previous states of stresses (and thus of self-stresses or imposed strains) on the structural resistance. It shall be noted that in the case where the response of a member is not fully plastic, different levels of material toughness may be found. For instance, for concrete under compression, Figure 1a, compression softening occurs in a region whose length is approximately twice the cross-section dimension (for a typical value of the friction angle of the material equal to 37\({}^{\circ}\)[27; 28]). Despite the softening response and eventual localization of strains, the material has a certain level of toughness associated to the friction stresses developing at the failure surface. This allows for the application of plastic design methods [28], by suitably reducing the compression strength to an equivalent plastic strength. In so doing, the uniaxial compressive strength may be reduced as given below (as recently adopted by EN1992-1-1:2023 [29], following the works of [30]):  
\[f_{cp}=f_{\mathrm{c}}\cdot\left(\frac{f_{\mathrm{c0}}}{f_{\mathrm{c}}}\right) ^{\frac{1}{3}}\leq f_{\mathrm{c}} \tag{1}\]  
where a suitable value for \(f_{\mathrm{c0}}\) is typically 30 MPa. As it can be observed (see also Figure 1a), larger reductions apply for higher values of the uniaxial compressive resistance as the response of the material becomes more brittle. Concerning the tensile response of FRC, different responses can be expected depending on the properties of the fibers used and their dosage [31]. A detailed investigation of the structural response can be performed in a consistent manner by accounting for both the uncracked and cracked response of the material. The former is governed by an elastic response (Figure 1b), which allows for loading and unloading without any residual strain. As the level of load increases, one section may attain its tensile strength (\(f_{\mathrm{c}t}\)) at its weakest section and develop a crack. No other cracks form thereafter, as the stress in the element softens for increasing levels of deformation. This is the case shown in Figure 1c for a simplified linear softening relationship.