# Investigating Bayesian Parameter Identification Using Non-Standard Laboratory Specimens
## 2 Modeling of Fracture Propagation
### Constitutive Law  
The material initially deforms elastically, following the principles of [PERSON]'s law. The linear elastic behavior is applied to the bulk region of the element, where the stress is defined as \(\sigma=C\overline{\epsilon}\). Here, \(C\) represents the constitutive matrix, determined by Young's modulus \(E\) and [PERSON]'s ratio \(\
u\), while \(\overline{\epsilon}\) denotes the strain in the bulk material. During the loading process, each finite element is evaluated for elastoplastic behavior with isotropic hardening. At every Gaussian integration point, the equivalent stress is calculated based on the [PERSON] yield criterion:  
\[\sigma_{eq}=\sqrt{\sigma_{xx}^{2}+\sigma_{yy}^{2}-\sigma_{xx}\sigma_{yy}+3 \tau_{xy}^{2}}. \tag{6}\]  
The corresponding yield function is defined as follows:  
\[f=\sigma_{eq}-\left(\sigma_{0}+H\cdot\varepsilon_{p}^{*}\right), \tag{7}\]  
where \(\sigma_{0}\) is the yield strength, \(H\) is the hardening modulus, and \(\varepsilon_{p}\) is the equivalent plastic strain for each Gauss integration point. If the yield function returns a positive value, elastoplastic isotropic hardening occurs at that integration point. The plastic strain is then updated using the radial return algorithm (see [1]), and the stress tensor is subsequently computed as follows: \(\sigma=C(\overline{\epsilon}-\epsilon_{p})\), where \(\epsilon_{p}\) represents the plastic strain tensor. As the stress continues to increase and surpasses the material strength, the first cracks begin to form. At this phase, the finite element is divided into the crack domain and the remaining bulk material. Crack behavior, including material degradation and energy dissipation, is governed by two uncoupled plasticity softening laws [1]. Each law corresponds to the separation modes, Mode I (opening) and Mode II (sliding), and is formulated within the framework of thermodynamics using Helmholtz free energy. These laws describe the dissipation of energy during the fracture process of material by irreversible accumulation of plastic deformations while preserving the stiffness of the material. Figure 2: Quadrilateral finite element with an embedded crack line, defined by point \(\mathbf{x}_{ED}^{*}\) and angle \(\alpha_{ED}^{*}\). Symbol \({}^{*}\) represents the local coordinate system. uncertain parameters are defined in Equation (11). These parameters are treated as random variables and are expressed in vector form as \(\mathbf{z}:\Omega\rightarrow\mathbb{R}^{n}\)  
\[\mathbf{z}(\omega):=\Big{[}E(\omega),\
u(\omega),\sigma_{0}(\omega),H(\omega), \sigma_{n}(\omega),G_{fn}(\omega)\Big{]}, \tag{13}\]  
where \(\omega\) represents one realization from the set of all possible realizations \(\Omega\). Each model parameter \(z_{i}\) within the vector \(\mathbf{z}\) is described by its probability density function \(\pi(z_{i})\). Assuming that the input parameters are mutually independent, their joint probability function \(\pi(\mathbf{z})\) can be expressed as [23; 50]:  
\[\pi(\mathbf{z})=\pi(z_{1},\dots,z_{N_{i}})=\prod_{i=1}^{N_{i}}\pi_{i}(z_{i}). \tag{14}\]  
The term \(\pi(\mathbf{z})\) is referred to as the prior distribution of the parameter values. The selection of the prior distribution is influenced by various factors, including prior experience, results from previous identification efforts, available measurement data, or even mere assumptions [23; 50]. The solution of the numerical forward model, \(\mathcal{G}\), with the \"true\" set of input parameters should ideally match the obtained measurements from the analyzed structure or specimen. However, in practice, this is not achievable due to inherent errors and imperfections in both the measurement and modeling processes, especially when dealing with nonlinear phenomena.