## 2 Method
### Preliminaries  
This section formalizes the structural mechanics problem, focusing on stress and strain prediction within materials under external loads. We begin with the foundational principles of continuum mechanics and gradually introduce the mathematical models that underpin our methodology. This formulation serves as the groundwork for our proposed model. The core equations governing structural mechanics are derived from conservation laws. The equilibrium equation ensures that the internal stresses balance the external forces:  
\[\mathbf{nabla}\cdot\mathbf{\sigma}+\mathbf{f}=0, \tag{1}\]  
where \(\mathbf{\sigma}\) is the Cauchy stress tensor, \(\mathbf{f}\) is the body force per unit volume, and \(\mathbf{nabla}\cdot\mathbf{\sigma}\) represents the divergence of the stress tensor. The strain tensor quantifies deformation in the material and is derived from the displacement field \(\mathbf{u}\):  
\[\mathbf{\varepsilon}=\frac{1}{2}\Big{(}\mathbf{nabla}+(\mathbf{nablau})^{\top}\Big{)}, \tag{2}\]where \(\epsilon\) is the strain tensor and \(\mathbf{nablau}\) is the gradient of the displacement field. Stress-strain relationships define how materials respond to external loads. For linear elasticity, [PERSON]'s law is given by  
\[\mathbf{\sigma}=\mathbf{C}:\epsilon, \tag{3}\]  
where \(\mathbf{C}\) is the fourth-order elasticity tensor, and : denotes a double contraction. For nonlinear materials, constitutive models depend on factors like plasticity or viscoelasticity, and the stress-strain relation is more complex:  
\[\mathbf{\sigma}=\mathcal{F}(\epsilon,\dot{\epsilon},\mathbf{p}), \tag{4}\]  
where \(\dot{\epsilon}\) is the strain rate and \(\mathbf{p}\) represents internal state variables. Solving the problem requires specifying boundary conditions:  
\[\mathbf{u}=\mathbf{u}_{0} \text{on}\ \Gamma_{u}, \tag{5}\] \[\mathbf{\sigma}\cdot\mathbf{n}=\mathbf{t}_{0} \text{on}\ \Gamma_{t}, \tag{6}\]  
where \(\Gamma_{u}\) and \(\Gamma_{t}\) are the boundaries with prescribed displacement \(\mathbf{u}_{0}\) and traction \(\mathbf{t}_{0}\), respectively, and \(\mathbf{n}\) is the outward normal. The weak form is essential for numerical methods:  
\[\int_{\Omega}(\mathbf{nabla}\cdot\mathbf{\sigma})\cdot\delta\mathbf{u}\,d\Omega+\int_ {\Omega}\mathbf{\mathrm{f}}\cdot\delta\mathbf{u}\,d\Omega=0, \tag{7}\]  
where \(\delta\mathbf{u}\) is the test function in the admissible space, and \(\Omega\) is the domain of interest. Finite element methods (FEM) are used to discretize the equations. The domain is divided into elements, and the displacement \(\mathbf{u}\) is approximated as  
\[\mathbf{u}\approx\sum_{i}N_{i}\mathbf{u}_{i}, \tag{8}\]  
where \(N_{i}\) are the shape functions, and \(\mathbf{u}_{i}\) are the nodal displacements. The discretized equations form a system of equations:  
\[\mathbf{K}\mathbf{u}=\mathbf{F}, \tag{9}\]  
where \(\mathbf{K}\) is the global stiffness matrix, \(\mathbf{u}\) is the nodal displacement vector, and \(\mathbf{F}\) is the force vector. The mathematical framework presented in this study is rooted in the fundamental principles of continuum mechanics and elasticity theory, as established in classical works such as Introduction to the Mechanics of a Continuous Medium [36] and Theory of elasticity [37]. The governing equations, including the equilibrium equation, strain-displacement relations, and constitutive models, are derived from conservation laws and thermodynamic principles. The equilibrium equation follows directly from [PERSON]'s second law applied to a differential volume element, ensuring force balance within the material. The strain tensor definition is derived from the kinematic relationship between displacements and deformations in a continuous medium. The constitutive equation represents [PERSON]'s law for linear elasticity and extends to nonlinear cases via phenomenological or physically based models, as described in Foundations of solid mechanics [38] and Nonlinear solid mechanics: a continuum approach for engineering science [39].