# A Posteriori Detection of Numerical Locking in \(hpq\)-Adaptive Finite Element Analysis
## 2 Preliminaries
### Locking Phenomena  
It was demonstrated in a numerous works (see [49; 50], for example) that the three-dimensional elasticity description of the strain energy \(U\equiv\frac{1}{2}B(u,u)\) of the thin-walled body \(V\) tends to the following limit value when the thickness of the body tends to zero, \(t\to 0\):  
\[ \begin{array}{rcl}U&=&\frac{1}{2}\int_{V}\,\sigma^{ij}\varepsilon_{ij}\,dV \equiv\frac{1}{2}\int_{V}\,[\sigma^{\kappa\beta}\varepsilon_{\alpha\beta}+ \sigma^{33}\varepsilon_{33}+2\sigma^{3\beta}\varepsilon_{3\beta}]\,dV\\ &\to&\frac{1}{2}t\int_{S_{M}}\,\frac{E}{1-\
u^{2}}[(1-\
u)\gamma_{\alpha \beta}\gamma_{\alpha\beta}+\
u\gamma_{\alpha\alpha}\gamma_{\beta\beta}]\,dS_ {M}+\frac{1}{2}t\int_{S_{M}}\,\frac{E\,k}{1+\
u}(\gamma_{\alpha 3}\gamma_{ \alpha 3}+\gamma_{3\beta}\gamma_{3\beta})\,dS_{M}\\ &+&\frac{1}{2}t^{3}\int_{S_{M}}\,\frac{E}{12(1-\
u^{2})}[(1-\
u) \kappa_{\alpha\beta}\kappa_{\alpha\beta}+\
u\kappa_{\alpha\alpha}\kappa_{ \beta\beta}]\,dS_{M}\\ &=&U_{m}+U_{s}+U_{b}\end{array}  \tag{8}\]  
Above, the three-dimensional components of the stress and strain tensors, \(\sigma^{ij}\) and \(\varepsilon_{ij}\), \(i,j=1,2,3\), of the three-dimensional theory of elasticity can be expressed in the thin limit with the two-dimensional longitudinal strain components of the mid-surface \(S_{M}\), \(\gamma_{\alpha\beta}\), \(\alpha,\beta=1,2\); transverse strain components, \(\gamma_{\alpha 3}\) and \(\gamma_{3\beta}\), \(\alpha,\beta=1,2\); and the components of the tensor of curvature variation, \(\kappa_{\alpha\beta}\), \(\alpha,\beta=1,2\). Additionally, in the thin limit, the three-dimensional isotropic elasticity constants can be replaced with the isotropic plane stress constants of the first-order shell theory, expressed by the [PERSON]'s modulus \(E\) and the [PERSON]'s ratio \(\
u\) and resulting from the plane stress assumption \(\sigma_{33}=0\). In addition, the kinematic condition of no elongation of the normals to the mid-surface, \(\gamma_{33}=0\), comes into play. The thin limit strain energy can then be approximated with the limit values of the membrane, shear, and bending parts, \(U_{m}\), \(U_{s}\), \(U_{b}\), of the strain energy \(U\). The quantity \(k\) is the shear strain correction factor, equal to \(5/6\) and resulting from the applied first-order model which leads to false (constant) transverse-shear stresses. Depending on the geometry, loading and kinematic boundary conditions, one may distinguish between the bending-dominated problems, when  
\[U_{b}\gg U_{m}+U_{s} \tag{9}\]  
and membrane-dominated or shear-membrane-dominated problems, where  
\[ \begin{array}{rcl}U_{m}\gg U_{b},&U_{s}\cong 0\\ U_{m}+U_{s}\gg U_{b}\end{array}  \tag{10}\]  
respectively. Then, it results from (9) that, in the analytical solution of the bending-dominated problems, the shear and membrane strains should be equal or very close to zero, i.e.,  
\[\gamma_{13}=\gamma_{31}\cong 0,\ \ \gamma_{23}=\gamma_{32}\cong 0 \tag{11}\]  
for the plate and shell structures, and  
\[\gamma_{11}\cong 0,\ \ \gamma_{12}\cong 0,\ \ \gamma_{22}\cong 0 \tag{12}\]  
for the shell structures where the coupling of the membrane and bending strains exists. Note that the decoupling of the membrane and bending strains shown by the limit expression of the relation (8) existsonly in the thin limit (\(t\to 0\)) for the first-order shells, while for the first-order plates it holds for any thickness \(t\). The shear and shear-membrane lockings are purely numerical phenomena which happen for poor discretizations and result from insufficiently accurate approximation of the analytical constraints (11) and/or (12). If such inaccurate approximation occurs then the shear and/or membrane strains are too large and the shear and membrane strain energies grow. These energies dominate over the bending strain energy, leading to numerical over-stiffening of the structure, which in turn gives too small (locked) values of displacements of the numerical solution.