## 3 Materials and Methods
### Material Models to Describe the Behaviour of Materials  
Conducting experimental research on living organisms, including animals and humans, is not always possible for many reasons [31; 32; 33; 34]. Constitutive models represent a mathematical framework for mechanical behaviour of the material in any context. Depending on the complexity required for modelling, these models may depend on a varying number of parameters that usually represent certain mechanical properties of the material. For reliable simulations of numerical data, it is crucial to properly quantify these parameters (i.e., constants), included in the equations governing the adopted constitutive model [35]. In order to describe the strength limit of materials, constitutive strength models are being used. Among the most important ones currently used in equations representing the metallic strength material model, there is a model, Johnson-Cook, which is a special type of [PERSON] plasticity model, with analytical forms of the law of material hardening and softening under influence temperature. The J-C constitutive equation is suitable for computational purposes due to its simplicity and the use of parameters that are relatively easy to determine. It is commonly used in CAE programs such as ABAQUS, and LS-Dyna. Pattern takes into account both the influence of strain magnitude, strain rate, and temperature on the value of the plastic flow limit [36; 37; 38] (1). \[\sigma_{y}=\Big{(}A+B\xi^{p^{n}}\Big{)}\Big{(}1+Cln\dot{\varepsilon}^{*}\Big{)} \big{(}1-T^{*m}\big{)} \tag{1}\]  
\[\dot{\varepsilon}^{*}=\frac{\dot{\varepsilon}^{p}}{\dot{\varepsilon}_{0}} \tag{2}\]  
Figure 9: Numerical model in ABAQUS 6.13 software: (**a**) discretization; and (**b**) initial-boundary conditions. \[T^{*}=\frac{T-T_{room}}{T_{melt}-T_{room}} \tag{3}\]  
where: \(A\)--yield strength, \(B\)--strengthening constant, \(C\)--strain rate constant, \(n\)--strengthening exponent, \(n\)--thermal softening coefficient, \(\varepsilon^{p}\)--effective plastic strain, \(\dot{\varepsilon}\)--effective strain rate (dimensionless), \(\dot{\varepsilon}^{p}\)--strain rate, \(\dot{\varepsilon}_{0}\)--reference value for strain rate, \(T^{*}\)--homologated temperature (dimensionless), \(T_{room}\)--room temperature, \(T_{melt}\)--melting point, and \(T\)--current temperature. Material failure equations regulate material strength and/or material stiffness as a function of the damage variable. When the damage variable reaches a critical value, damage propagates (crack). One of the most frequently used constitutive models describing material destruction (especially the destruction of metallic materials) is the Johnson-Cook destruction model [36; 39; 40]. In this relationship, the deformation occurring at the moment of destruction is described by (4). \[\varepsilon_{f}=\Big{[}D_{1}+D_{2}e^{D_{3^{*}}}\Big{]}\Big{[}1+D_{4}ln\dot{ \varepsilon}^{*}\Big{]}\big{[}1+D_{5}T^{*}\big{]} \tag{4}\]  
\[\sigma^{*}=\frac{p}{\sigma_{eff}} \tag{5}\]  
where: \(D_{1}\)-\(D_{5}\)--failure parameters, \(\sigma^{*}\)--stress triaxiality coefficient, \(p\)--pressure, \(\sigma_{eff}\)--effective stress, \(T^{*}\)--approved temperature (dimensionless), and \(\dot{\varepsilon}^{*}\)--effective strain rate (dimensionless). To determine the material constants, it is necessary to carry out a series of experimental analyses. In the case of a material, as in this biological case, developing a failure model would require significant experimental effort, also for samples with complex shapes. For this purpose, the value of the limit strain followed by the conversion of the element into a free particle was adopted as the failure model for the bone (6). \[\varepsilon_{limit}\geq input\ value-destruction\ of\ material \tag{6}\]  
The material parameters used in the numerical simulations are presented in the table below (Table 1). These values are based on literature data, including a model based on the stress-strain curve for horse cortical bone (Figure 10).