Assessment of second Piola–Kirchhoff and Cauchy stress tensors in finite rotation sandwich and laminated shells under non-conservative pressure loads

Abstract

In this paper, the stress fields in nonlinear sandwich and laminated composite shells subjected to displacement-dependent loads are analyzed. To solve the problem, the geometrically exact (GeX) hybrid-mixed four-node solid-shell element is proposed. The term GeX means that the middle surface is described by analytical functions, i.e., the parametrization of the surface is known. The laminated solid-shell element formulation is based on the choice of an arbitrary number of sampling surfaces (SaS) parallel to the middle surface and located at the Chebyshev polynomial nodes within the layers, in order to introduce the displacements of these surfaces as unknown functions. The outer surfaces and interfaces are also included into a set of SaS. Due to analytical integration utilized to evaluate the tangent stiffness matrix, the developed GeX solid-shell element under non-conservative loading shows excellent performance in the case of coarse meshes and makes it possible to use only one load step in most of the considered benchmark problems. It is shown that the difference between the second Piola–Kirchhoff and Cauchy stress tensors in some non-conservative problems for sandwich and laminated shells undergoing arbitrarily large rotations can be significant.

Appendix A

The element stiffness matrix ${\mathbf{K}}_{{\text{F}}}$ of order $12N_{{{\text{SaS}}}} \times 12N_{{{\text{SaS}}}}$, introduced in Sect. 6 to take into account non-conservative loading, can be written in a closed form. The use of Eqs. (39), (40), (48), (53) and (55) leads to the following matrix:

$${\mathbf{K}}_{{\text{F}}} = - \frac{{\partial {\mathbf{F}}}}{{\partial {\mathbf{q}}}}({}^{t}{\mathbf{P}}\, + {\Delta }{\mathbf{P}},\,{}^{t}{\mathbf{q}}\, + {\Delta }{\mathbf{q}}^{[n]} {\mathbf{)}}\,{,}$$

(A1)

whose columns with nonzero elements are

$$\begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\left( {{\mathbf{K}}_{{\text{F}}} } \right)_{{{\mathbf{.}}\,\,i + 3N_{{{\text{SaS}}}} (r - 1)}} = \frac{1}{{4\overline{A}_{1} \overline{A}_{2} }}\sum\limits_{s} {N_{rs} \left( {{}^{t}\hat{p}_{s}^{ - } + \Delta \hat{p}_{s}^{ - } } \right)} \frac{{\partial \eta_{is}^{ - } }}{{\partial {\mathbf{q}}}}\left( {{}^{t}{\mathbf{q}}\, + {\Delta }{\mathbf{q}}^{[n]} } \right)\,, \hfill \\ \left( {{\mathbf{K}}_{{\text{F}}} } \right)_{{{\mathbf{.}}\,\,i + 3(N_{{{\text{SaS}}}} - 1) + 3N_{{{\text{SaS}}}} (r - 1)}} = - \frac{1}{{4\overline{A}_{1} \overline{A}_{2} }}\sum\limits_{s} {N_{rs} \left( {{}^{t}\hat{p}_{s}^{ + } + \Delta \hat{p}_{s}^{ + } } \right)} \frac{{\partial \eta_{is}^{ + } }}{{\partial {\mathbf{q}}}}\left( {{}^{t}{\mathbf{q}}\, + {\Delta }{\mathbf{q}}^{[n]} } \right)\,, \hfill \\ \end{gathered}$$

(A2)

where

$$\frac{{\partial \eta_{1s}^{ \pm } }}{{\partial {\mathbf{q}}}}\left( {{}^{t}{\mathbf{q}}\, + {\Delta }{\mathbf{q}}^{[n]} } \right) = A_{1s} A_{2s} \left[ {\left( {{{\varvec{\Gamma}}}_{1s}^{ \pm } + ({{\varvec{\Gamma}}}_{1s}^{ \pm } )^{{\text{T}}} } \right)\,\left( {{}^{t}{\mathbf{q}}\, + {\Delta }{\mathbf{q}}^{[n]} } \right) - c_{2s}^{ \pm } {{\varvec{\Xi}}}_{31s}^{ \pm } } \right]\,,$$

$$\frac{{\partial \eta_{2s}^{ \pm } }}{{\partial {\mathbf{q}}}}\left( {{}^{t}{\mathbf{q}}\, + {\Delta }{\mathbf{q}}^{[n]} } \right) = A_{1s} A_{2s} \left[ {\left( {{{\varvec{\Gamma}}}_{2s}^{ \pm } + ({{\varvec{\Gamma}}}_{2s}^{ \pm } )^{{\text{T}}} } \right)\,\left( {{}^{t}{\mathbf{q}}\, + {\Delta }{\mathbf{q}}^{[n]} } \right) - c_{1s}^{ \pm } {{\varvec{\Xi}}}_{32s}^{ \pm } } \right]\,,$$

(A3)

$$\frac{{\partial \eta_{3s}^{ \pm } }}{{\partial {\mathbf{q}}}}\left( {{}^{t}{\mathbf{q}}\, + {\Delta }{\mathbf{q}}^{[n]} } \right) = A_{1s} A_{2s} \left[ {\left( {{{\varvec{\Gamma}}}_{3s}^{ \pm } + ({{\varvec{\Gamma}}}_{3s}^{ \pm } )^{{\text{T}}} } \right)\,\left( {{}^{t}{\mathbf{q}}\, + {\Delta }{\mathbf{q}}^{[n]} } \right) + c_{1s}^{ \pm } {{\varvec{\Xi}}}_{22s}^{ \pm } + c_{2s}^{ \pm } {{\varvec{\Xi}}}_{11s}^{ \pm } } \right]\,,$$

where $A_{\alpha r}$ and $\overline{A}_{\alpha }$ are the coefficients of the first fundamental form at the nodes and the center of the element, and, as we remember, the nodal indices $r,\,s = 1,\,2,\,3,\,4$.