## 2 Numerical Formulation
### Strain-Displacement Relations  
The strain-displacement relations in the Cartesian-coordinate system are obtained using Green-Lagrange theory  
\[\epsilon_{ij}=\frac{1}{2}[u_{i,j}+u_{j,i}+u_{k,i}u_{k,j}] \tag{5}\]  
In which  
\[u_{i,j}=\frac{\partial u_{i}}{\partial x_{j}}[u_{1}=u,u_{2}=v,u_{3}=w,x_{1}=x,x_{2}=y,x_{3}=z] \tag{6}\]  
Figure 1: Schematic representation degenerated shell element showing different co-ordinate systems and degrees of freedom.  
Similarly, in local co-ordinate system \((x^{\prime},y^{\prime},z^{\prime})\), the strain displacement relations can be obtained by replacing \([u_{1}=u^{\prime},u_{2}=v^{\prime},u_{3}=w^{\prime},x_{1}=x^{\prime},x_{2}=y^{ \prime},x_{3}=z^{\prime}]\)  
\[e^{\prime}_{ij}=\frac{1}{2}[u^{\prime}_{i,j}+u^{\prime}_{j,i}+u^{\prime}_{k,j}u ^{\prime}_{k,j}] \tag{7}\]  
Now, local and global strains can be decomposed into linear and non-linear parts  
\[\epsilon^{\prime}_{ij,L}=\frac{1}{2}[u^{\prime}_{i,j}+u^{\prime}_{j,i}], \epsilon^{\prime}_{ij,NL}=\frac{1}{2}[u^{\prime}_{k,i}u^{\prime}_{k,j}] \tag{8}\]  
\[\epsilon_{ij,L}=\frac{1}{2}[u_{i,j}+u_{j,i}],\epsilon_{ij,NL}=\frac{1}{2}[u_{ k,i}u_{k,j}] \tag{9}\]  
Now, the linear and non-linear components of local strain tensor can be written in term of global strain components  
\[\epsilon^{\prime}_{L}=\epsilon_{ij,L}\mathbf{e}^{\prime}_{i}\otimes\mathbf{e }^{\prime}_{j},\mathbf{e}^{\prime}_{NL}=\epsilon_{ij,NL}\mathbf{e}^{\prime}_ {i}\otimes\mathbf{e}^{\prime}_{j},i,j\in\{1,2,3\} \tag{10}\]  
Here, \(\mathbf{e}^{\prime}_{1}\), \(\mathbf{e}^{\prime}_{2}\), and \(\mathbf{e}^{\prime}_{3}\) is the position vector of the local axes \(x^{\prime},y^{\prime},z^{\prime}\), respectively, with respect to global axes and they can be obtained from the direction cosine of the local axes  
\[\mathbf{e}^{\prime}_{1}=(l_{x},l_{y},l_{z}),\mathbf{e}^{\prime}_{2}=(m_{x},m_ {y},m_{z}),\mathbf{e}^{\prime}_{3}=(n_{x},n_{y},n_{z}) \tag{11}\]  
These direction cosines can be directly obtained from the Jacobian matrix, as \((x^{\prime},y^{\prime})\) can be expressed as \((\xi,\eta)\), which is curvilinear natural coordinate  
\[(l_{x},l_{y},l_{z})=(\frac{\partial x}{\partial\xi^{\prime}}, \frac{\partial y}{\partial\xi^{\prime}},\frac{\partial z}{\partial\xi^{\prime}}) \tag{12}\] \[(m_{x},m_{y},m_{z})=(\frac{\partial x}{\partial\eta^{\prime}}, \frac{\partial y}{\partial\eta^{\prime}},\frac{\partial z}{\partial\eta})\] (13) \[(n_{x},n_{y},n_{z})=(l_{x},l_{y},l_{z})\times(m_{x},m_{y},m_{z}) \tag{14}\]  
Now, utilizing Equations (1) and (2) one can write the linear and non-linear components of the local strain tensors with the help of Equation (10) in the following matrix form  
\[\{e^{\prime}_{L}\}=[B_{L}]\{d_{e}\} \tag{15}\] \[\{\epsilon^{\prime}_{NL}\}=[B_{NL}]\{d_{e}\} \tag{16}\]  
Here, \(\{\epsilon^{\prime}\}=[\epsilon^{\prime}_{xx^{\prime}},e^{\prime}_{yy^{\prime }},\epsilon^{\prime}_{zz^{\prime}},\epsilon^{\prime}_{xx^{\prime}},\epsilon^{ \prime}_{yz}]^{T}\) and \(\{d_{e}\}=[u_{1},v_{1},u_{1},\beta_{1},u_{2},v_{2},w_{2},a_{2},\beta_{2},\ \dots,\)\(u_{8},v_{8},w_{8},a_{8},\beta_{8}]^{T}\). \([B]\) is the strain-displacement matrix. However, here due plan strain assumption the the contribution from \(e^{\prime}_{zz}\) has been neglected. In practice for plate the the strain component \(\epsilon^{\prime}_{zz}\) is very less/almost equal to zero. Therefore we remove the row corresponding to \(e^{\prime}_{zz}\) from \([B_{L}]\) and \([B_{NL}]\) matrix for further calculation.