In order to derive a new axisymmetric shell model based on the three-field DM weak formulation (
10), a systematic dimensional reduction will be used for the trial and test functions
\(u_{k}\),
\(\phi ^{s}\),
\(\sigma ^{k\ell }\), and
\(\delta u_{k}\),
\(\delta \phi ^{s}\),
\(\delta \sigma ^{k\ell }\). As a starting step, the 3D translational equilibrium equation (
1) is written down in the mid-surface coordinate frame, which can be given for thin shells of revolution as
$$\begin{aligned} {\sigma }_{\,\,\,\,\,\mid \ell }^{\, k\ell }+{b}^{\, k}=0\qquad \;\mathrm {in}\;V \end{aligned}$$
(27)
where the 3D mid-surface covariant derivative
\(\sigma _{\,\,\,\,\,\mid \ell }^{\, k\ell }\) can be put in the form
$$\begin{aligned} {\sigma }_{\,\,\,\,\,\mid \ell }^{\, k\ell }={\sigma }_{\,\,\,\,\,\,\mid \alpha }^{\, k\alpha }+{\sigma }_{\,\,\,\,\,,3}^{\, k3} \end{aligned}$$
(28)
which can be separated into [
11,
36,
60]
$$\begin{aligned} {\sigma }_{\,\,\,\,\,\mid \ell }^{\,\kappa \ell }&= {\sigma }_{\,\,\,\,\,\,\parallel \alpha }^{\,\kappa \alpha }- b_{\,\alpha \,}^{\kappa }{\sigma }^{\,3\alpha }- b_{\,\alpha \,}^{\alpha }{\sigma }^{\,\kappa 3}+ {\sigma }_{\,\,\,\,\,,3}^{\,\kappa 3}, \end{aligned}$$
(29)
$$\begin{aligned} {\sigma }_{\,\,\,\,\,\,\mid \ell }^{\,3\ell }&= {\sigma }_{\,\,\,\,\,\,\parallel \alpha }^{\,3\alpha }+ b_{\kappa \alpha \,}{\sigma }^{\,\kappa \alpha }- b_{\,\alpha \,}^{\alpha }{\sigma }^{\,33}+ {\sigma }_{\,\,\,\,\,,3}^{\,33} \end{aligned}$$
(30)
where
$$\begin{aligned} {\sigma }_{\,\,\,\,\,\,\parallel \alpha }^{\,\kappa \alpha }= {\sigma }_{\,\,\,\,\,\,,\alpha }^{\,\kappa \alpha }+ {\varGamma }_{\beta \alpha \,}^{\,\kappa }{\sigma }^{\,\beta \alpha } +{\varGamma }_{\mu \alpha \,}^{\,\alpha }{\sigma }^{\,\kappa \mu }, \end{aligned}$$
(31)
and
$$\begin{aligned} {\sigma }_{\,\,\,\,\,\,\parallel \alpha }^{\,3\alpha }= {\sigma }_{\,\,\,\,\,\,,\alpha }^{\,3\alpha }+ {\varGamma }_{\beta \alpha \,}^{\,\alpha }{\sigma }^{\,3\beta }\,. \end{aligned}$$
(32)
The above five equations and the homogeneous form of Eq. (
27) are valid for the test function
\(\delta \sigma ^{k\ell }\) as well. As a next point of the derivation the trial and test functions, as well as the known functions occurring in Eq. (
10) are expanded into power series with respect to the thickness coordinate
\(\xi ^3\) on the mid-surface
\(S_0\):
$$\begin{aligned} \sigma ^{k\ell }(\xi ^{\alpha },\xi ^3)&= \sum _{i=0}^{\infty }\,_{i}\sigma ^{k\ell }\,(\xi ^{\alpha })\,\left( \xi ^3\right) ^i,\quad \delta \sigma ^{k\ell }(\xi ^{\alpha },\xi ^3)= \sum _{i=0}^{\infty }\delta _i\sigma ^{k\ell }\,(\xi ^{\alpha })\,\left( \xi ^3\right) ^i, \end{aligned}$$
(33)
$$\begin{aligned} u_k(\xi ^{\alpha },\xi ^3)&=\sum _{i=0}^{\infty }\,_iu_k\,(\xi ^{\alpha })\,\left( \xi ^3\right) ^i, \quad \delta u_k(\xi ^{\alpha },\xi ^3)=\sum _{i=0}^{\infty }\,\delta _iu_k\,(\xi ^{\alpha })\, \left( \xi ^3\right) ^i, \end{aligned}$$
(34)
$$\begin{aligned} \phi ^s(\xi ^{\alpha },\xi ^3)&=\sum _{i=0}^{\infty }\,_i\phi ^s\,(\xi ^{\alpha })\, \left( \xi ^3\right) ^i,\quad \delta \phi ^s(\xi ^{\alpha },\xi ^3)= \sum _{i=0}^{\infty }\,\delta _i\phi ^s\,(\xi ^{\alpha })\,\left( \xi ^3\right) ^i, \end{aligned}$$
(35)
as well as
$$\begin{aligned} b^k(\xi ^{\alpha },\xi ^3)&= \sum _{i=0}^{\infty }\,_ib^{k}\,(\xi ^{\alpha })\, \left( \xi ^3\right) ^i, \end{aligned}$$
(36)
$$\begin{aligned} {\tilde{u}}_k(\xi ^{\alpha },\xi ^3)&=\sum _{i=0}^{\infty } \,_i{\tilde{u}}_k\,(\xi ^{\alpha })\,\left( \xi ^3\right) ^i\,. \end{aligned}$$
(37)
Now all variables will be referred to unit mid-surface, covariant or contravariant base vectors, namely the physical components of the tensorial quantities, denoted by a hat over the variables, will be introduced as follows:
$$\begin{aligned} {\hat{\sigma }}^{k\ell }=\sqrt{a_{KK\,}a_{LL}}\,\sigma ^{k\ell }, \;{\hat{\varepsilon }}_{k\ell }=\sqrt{a^{KK\,}a^{LL}}\,\varepsilon _{k\ell }, \;{\hat{u}}_k=\sqrt{a^{KK}}\,u_k,\;{\hat{\phi }}^r=\sqrt{a_{RR}}\,\phi ^r, \end{aligned}$$
(38)
in which there is no summation for the uppercase indices. Furthermore, our investigations will be restricted to axisymmetric problems of shells of revolution, i.e., the variables will not depend on the polar coordinate
\(\xi ^2\). Thus, upon substitution of the curvature tensor components (
17)–(
18), the nonzero mid-surface Christoffel symbols of second kind (
19)–(
21) and the Taylor series expansions (
33) and (
36) into the translational equilibrium equation (
28) by the derivatives (
29)–(
32), expressing the results in terms of the physical components of tensorial variables (
38) and making separation the obtained equations with respect to the powers of the thickness coordinate
\(\xi ^3\), we find
$$\begin{aligned}&\frac{\,_{i}{\hat{\sigma }}_{\,\,\,,1}^{\,11}}{A^2}+ \frac{R'}{RA^2}\left( \,_{i}{\hat{\sigma }}^{\,11}-\,_{i}{\hat{\sigma }}^{22\,}\right) -\frac{x'R''-x''R'}{A^4}\left( \,_{i}{\hat{\sigma }}^{\,13}+ \,_{i}{\hat{\sigma }}^{\,31\,}\right) \nonumber \\&\quad +\frac{x'}{RA^2}\,_{i}{\hat{\sigma }}^{\,13\,} +\frac{i+1}{A}\,_{i+1}{\hat{\sigma }}^{13}+\frac{\,_{i}{\hat{b}}^{\,1}}{A}=0, \quad i=0,1,2,\ldots ,\infty , \end{aligned}$$
(39)
$$\begin{aligned}&\frac{\,_{i}{\hat{\sigma }}_{\,\,\,,1}^{\,21}}{AR}+ \frac{R'}{AR^2}\left( \,_{i}{\hat{\sigma }}^{\,12}+\,_{i}{\hat{\sigma }}^{\,21\,}\right) + \frac{x'}{AR^2}\left( \,_{i}{\hat{\sigma }}^{\,32}+\,_{i}{\hat{\sigma }}^{\,23\,}\right) + \frac{\,_{i}{\hat{b}}^{\,2}}{R}\nonumber \\&\quad -\frac{x'R''-x''R'}{RA^3}\,_{i}{\hat{\sigma }}^{\,23\,} +\frac{i+1}{R}\,_{i+1}{\hat{\sigma }}^{23}=0,\quad i=0,1,2,\ldots ,\infty , \end{aligned}$$
(40)
$$\begin{aligned}&\frac{\,_{i}{\hat{\sigma }}_{\,\,\,,1}^{\,31}}{A}+ \frac{x'R''-x''R'}{A^3}\left( \,_{i}{\hat{\sigma }}^{\,11}-\,_{i}{\hat{\sigma }}^{\,33\,}\right) +\frac{x'}{AR}\left( \,_{i}{\hat{\sigma }}^{\,33}-\,_{i}{\hat{\sigma }}^{\,22\,}\right) \nonumber \\&\quad +\frac{R'}{AR}\,_{i}{\hat{\sigma }}^{\,31}+(i+1)\,_{i+1}{\hat{\sigma }}^{33} +\,_{i}{\hat{b}}^{\,3}=0,\quad i=0,1,2,\ldots ,\infty , \end{aligned}$$
(41)
from which it follows that the stress components
\({\hat{\sigma }}^{k\lambda }\) have to be approximated along the thickness coordinate
\(\xi ^3\) by one degree lower than
\({\hat{\sigma }}^{k3}\) in order to satisfy any
ith scalar expanded translational equilibrium equations. This is admissible because the weak formulation (
10) allows the stress tensor to be treated as a priori non-symmetric. In this paper, the following linear and quadratic stress approximations are applied:
$$\begin{aligned} {\hat{\sigma }}^{k\lambda }(\xi ^1,\xi ^{3\,})&=\,_0{\hat{\sigma }}^{k\lambda } (\xi ^{1\,})+\,_1{\hat{\sigma }}^{k\lambda }(\xi ^{1\,})\,\xi ^3, \end{aligned}$$
(42)
$$\begin{aligned} {\hat{\sigma }}^{k3}(\xi ^1,\xi ^{3\,})&=\,_0{\hat{\sigma }}^{k3}(\xi ^{1\,}) +\,_1{\hat{\sigma }}^{k3}(\xi ^{1\,})\,\xi ^3+\,_2{\hat{\sigma }}^{k3}(\xi ^{1\,}) \,\left( \xi ^{3\,}\right) ^2, \end{aligned}$$
(43)
as well as
$$\begin{aligned} {\hat{b}}^k(\xi ^1,\xi ^{3\,})=\,_0{\hat{b}}^k(\xi ^{1\,})+ \,_1{\hat{b}}^k(\xi ^{1\,})\,\xi ^3 \end{aligned}$$
(44)
is valid for the components of the volumetric force density vector, namely the scalar, expanded translational equilibrium equations of
\(i=0\) and
\(i=1\) will be retained from Eqs. (
39)–(
41), while the remaining equations associated with
\(i=2,3,\ldots ,\infty \) are considered to be identically satisfied. Thereafter, the question comes up which approximations along the thickness coordinate have to be used for the components of the displacement- and rotation vectors. For this purpose, investigating the volumetric integrals in the second row of Eq. (
9) by means of Eqs. (
22), (
34)–(
35), (
38) and Eqs. (
39)–(
41) for
\(i=0\) and
\(i=1\), neglecting the detailed derivations, with regard to Eqs. (
42)–(
43), the displacement- and rotation components have to be approximated by polynomials of first degree in
\(\xi ^3\):
$$\begin{aligned} {\hat{u}}_k(\xi ^1,\xi ^{3\,})&=\,_{0}{\hat{u}}_k(\xi ^{1\,})+ \,_1{\hat{u}}_k(\xi ^{1\,})\,\xi ^3, \end{aligned}$$
(45)
$$\begin{aligned} {\hat{\phi }}^{s}(\xi ^1,\xi ^{3\,})&= \,_0{\hat{\phi }}^s\,(\xi ^{1\,})+\,_1{\hat{\phi }}^s\,(\xi ^{1\,})\,\xi ^3\,. \end{aligned}$$
(46)
As the final step in the dimensional reduction, the relating test functions
\(\delta {\hat{\sigma }}^{k\ell }\) and
\(\delta {\hat{u}}_k\),
\(\delta {\hat{\phi }}^{s}\) are approximated along
\(\xi ^3\), respectively, by the same degree polynomials as the trial functions
\({\hat{\sigma }}^{k\ell }\) and
\({\hat{u}}_k\),
\({\hat{\phi }}^{s}\):
$$\begin{aligned} \delta {\hat{\sigma }}^{k\lambda }(\xi ^1,\xi ^{3\,})&=\delta _0{\hat{\sigma }}^{k\lambda } (\xi ^{1\,})+\delta _1{\hat{\sigma }}^{k\lambda }(\xi ^{1\,})\,\xi ^3, \end{aligned}$$
(47)
$$\begin{aligned} \delta {\hat{\sigma }}^{k3}(\xi ^1,\xi ^{3\,})&=\delta _0{\hat{\sigma }}^{k3}(\xi ^{1\,}) +\delta _1{\hat{\sigma }}^{k3}(\xi ^{1\,})\,\xi ^3+\delta _2{\hat{\sigma }}^{k3}(\xi ^{1\,}) \,\left( \xi ^{3\,}\right) ^2 \end{aligned}$$
(48)
and
$$\begin{aligned} \delta {\hat{u}}_k(\xi ^1,\xi ^{3\,})&=\delta _{0}{\hat{u}}_k(\xi ^{1\,})+ \delta _1{\hat{u}}_k(\xi ^{1\,})\,\xi ^3, \end{aligned}$$
(49)
$$\begin{aligned} \delta {\hat{\phi }}^{s}(\xi ^1,\xi ^{3\,})&= \delta _0{\hat{\phi }}^s\,(\xi ^{1\,})+\delta _1{\hat{\phi }}^s\,(\xi ^{1\,})\,\xi ^3\,. \end{aligned}$$
(50)
According to Eqs. (
42), (
43) and (
45), (
46), the number of the unknown trial functions is 33 including the 21 stress coefficients
\(\,_0{\hat{\sigma }}^{k\ell }\),
\(\,_1{\hat{\sigma }}^{k\ell }\),
\(\,_2{\hat{\sigma }}^{k3}\) and the 6 displacement coefficients
\(\,_{0}{\hat{u}}_k\),
\(\,_{0}{\hat{u}}_k\), as well as the 6 rotation coefficients
\(\,_0{\hat{\phi }}^s\),
\(\,_1{\hat{\phi }}^s\), leading to a consistent shell model, see, for example, the consistency property of the displacement-based and stress-based shell theories, in [
24,
25,
31,
53,
54]. In the next Section, a systematic process will be presented for the reduction in the number of variables.