Thermodynamically consistent thermoelastic plate and shell formulations for small deformation and small strain using non-classical continuum mechanics incorporating internal rotations

Abstract

The work presented in a recent paper by the authors [35] for a thermodynamically consistent and kinematic assumption free plate and shell formulation for small deformation and small strain based on the conservation and balance laws of classical continuum mechanics (CCM) is extended here for non-classical continuum mechanics (NCCM). This formulation incorporates additional physics due to internal rotations that arise due to the deformation gradient tensor. This physics is neglected in CCM, hence is absent in the plate and shell formulation of reference [35]. Consideration of this new physics requires modifications of the current balance laws as well as consideration of a new balance law “balance of moment of moments” (BMM) [2, 3]. Cauchy stress tensor becomes non-symmetric. Cauchy moment tensor is conjugate to the symmetric part of the rotation gradient tensor which exists now due to new physics. Balance of angular momenta yields additional three differential equations as part of the mathematical model. The new balance law (BMM) establishes symmetry of the Cauchy moment tensor. The new physics considered here exists in all deforming solid continua as it is due to the deformation gradient tensor, but is ignored in CCM. The consequence of this new physics is additional stiffness, hence additional strain energy storage and change in the time history of displacements and stress field compared to formulations based on CCM. The basic mathematical model for the plate and shell deformation consists of conservation and balance laws in $\mathbb {R}^3$ based on NCCM incorporating internal rotations. The associated finite element formulations for obtaining the solution of the mathematical model consists of : (i) geometry of the plate or shell described by the flat or curved middle surface (as done conventionally) and nodal vectors locating the top and bottom faces of the plate/shell (ii) the displacement field approximation that is p-version hierarchical in the plane as well as in the transverse direction (iii) integral form is constructed using Galerkin Method with Weak Form (GM/WF) and the corresponding element equations. The formulation presented here remains valid and accurate for thin as well as thick plate/shell and naturally reduces to the formulation of reference [35] based on classical continuum mechanics. Model problem studies and comparisons with the studies based on CCM formulation [35] will be presented in a follow up paper.

A Appendix

Strains, rotations and their gradients, conjugate pairs and constitutive theories

In this appendix we present the conservation and balance laws of CCM as well as NCCM, some details of the strain tensor, rotation tensor, gradients of the rotation tensor and its decomposition, the conjugate pairs and linear constitutive theories for classical as well as non-classical continuum mechanics based on internal rotations and the non-classical continuum mechanics incorporating internal and the Cosserat rotations. The material presented here considers small deformation, small strain, linear as well as non linear reversible mechanical deformation.

1.1 A.1 Classical continuum mechanics (CCM)

A material point has only three translational degrees of freedom. The symmetric part of the displacement gradient tensor constitutes strain measures and the antisymmetric part of the displacement gradient tensor (a measure of internal rotations) is neglected in the derivation of the conservation and balance laws. We have conservation of mass, balance of linear momenta (BLM), first law of thermodynamics (FLT), the second law of thermodynamics (SLT) and considerations for constitutive theories [9].

$$\begin{aligned}&\rho _{_{_{0}}}(\pmb {\varvec{x }}) = |J|\rho (\pmb {\varvec{x }},t) \quad \quad \text { (CM)} \end{aligned}$$

(A1)

$$\begin{aligned}&\rho _{_{_{0}}}\frac{Dv_{i}}{Dt} - \rho _{_{_{0}}}F^{b}_i- \dfrac{\partial { {\sigma }_{ji}}}{\partial {x_{j}}} = 0 \quad \quad \text {(BLM) } \end{aligned}$$

(A2)

$$\begin{aligned}&\pmb {\varvec{\sigma }} = \pmb {\varvec{\sigma }} ^T \quad \quad \text { (BAM)}\end{aligned}$$

(A3)

$$\begin{aligned}&\rho _{_{_{0}}}\frac{De}{Dt} + \nabla .\pmb {\varvec{q }} - {\sigma }_{ji} \overset{\,\mathbf{. }}{\varepsilon } _{ij} = 0 \quad \quad \text { (FLT)}\end{aligned}$$

(A4)

$$\begin{aligned}&\rho _{_{_{0}}}\left( \frac{D\phi }{Dt}+\eta \frac{D\theta }{Dt}\right) + \frac{\pmb {\varvec{q.g }}}{\theta } - {\sigma }_{ji} \overset{\,\mathbf{. }}{\varepsilon } _{ij} \le 0 \quad \quad \text { (SLT)} \end{aligned}$$

(A5)

In which $\rho _{_{_{0}}}$ is the mass density in the reference configuration, $F^{b}_1$, $F^{b}_2$ and $F^{b}_3$ are body force per unit mass in $x_1$, $x_2$ and $x_3$ directions, $\sigma _{ij}$ is Cauchy stress tensor, $\varepsilon _{ij}$ is linear strain tensor, e is specific internal energy, $\pmb {\varvec{q }}$ is heat flux vector, $\pmb {\varvec{g }}$ is temperature gradient vector, $\phi$ is Helmholtz free energy density, $\eta$ is entropy density and $\theta$ is temperature.

From the conjugate pairs $\sigma _{ij}, \overset{\,\mathbf{. }}{\varepsilon } _{ij}$ and $\frac{\pmb {\varvec{q.g }}}{\theta }$, we can conclude that at the very least the following must hold (thermoelastic solid continua)

$$\begin{aligned} \pmb {\varvec{ {\sigma } }}= & {} \pmb {\varvec{ {\sigma } }}( \pmb {\varvec{\varepsilon }} , \theta ) \end{aligned}$$

(A6)

$$\begin{aligned} \pmb {\varvec{q }}= & {} \pmb {\varvec{q }}(\pmb {\varvec{g }}, \theta ) \end{aligned}$$

(A7)

Choices of the argument tensors for $\phi$ and $\eta$ are discussed in section 4 in conjunction with the derivation of the constitutive theories for $\pmb {\varvec{ {\sigma } }}$ and $\pmb {\varvec{q }}$.

1.2 A.2 Non-classical continuum theory incorporating internal rotations

Displacement gradient tensor $[{}^{\;d}J]$ and its decomposition into symmetric and antisymmetric tensors can be written as

$$\begin{aligned} \left[ \dfrac{\partial { \{u\} }}{\partial { \{x\} }} \right] = [{}^{\;d}J] = [{}_{s}^{\;d}J] + [{}_{a}^{\;d}J] = [\varepsilon ] + [ {}_{a}^{\,i}r ] \end{aligned}$$

(A8)

in which $[{}_{s}^{\;d}J]$ and $[{}_{a}^{\;d}J]$ are symmetric and antisymmetric tensor, thus $[\varepsilon ]$ and $[ {}_{a}^{\,i}r ]$ are strain and rotation tensors. $[ {}_{a}^{\,i}r ]$ contains rotations ${}_i\varTheta _{x_1}$, ${}_i\varTheta _{x_2}$ and ${}_i\varTheta _{x_3}$ about $x_1$, $x_2$ and $x_3$ axes. Alternatively

$$\begin{aligned} \pmb {\varvec{\nabla }}\times \pmb {\varvec{u }} = \pmb {\varvec{e }}_1({}_i\varTheta _{x_1}) + \pmb {\varvec{e }}_2({}_i\varTheta _{x_2}) + \pmb {\varvec{e }}_3({}_i\varTheta _{x_3}) \end{aligned}$$

(A9)

with this definitions of ${}_i\varTheta _{x_1}$, ${}_i\varTheta _{x_2}$ and ${}_i\varTheta _{x_3}$ in (A9), we have

$$\begin{aligned} \begin{aligned}&{}_{a}^{\,i}r _{12} = {}_i\varTheta _{x_3}\\ \qquad \text {and } \quad&{}_{a}^{\,i}r _{21} = - {}_{a}^{\,i}r _{12} \end{aligned} \begin{aligned}&\quad ; \quad \\&\quad ; \quad \end{aligned} \begin{aligned}&{}_{a}^{\,i}r _{13} = -{}_i\varTheta _{x_2}\\&{}_{a}^{\,i}r _{31} = - {}_{a}^{\,i}r _{13} \end{aligned} \begin{aligned}&\quad ; \quad \\&\quad ; \quad \end{aligned} \begin{aligned}&{}_{a}^{\,i}r _{23} = {}_i\varTheta _{x_1}\\&{}_{a}^{\,i}r _{32} = - {}_{a}^{\,i}r _{23} \end{aligned} \end{aligned}$$

(A10)

all others are zero. The rotations ${}_i\varTheta _{x_1}$, ${}_i\varTheta _{x_2}$ and ${}_i\varTheta _{x_3}$ are about the axes of the triad located at a material point. If we define the rotations as a vector $\{{}_i\varTheta \}$, $\{{}_i\varTheta \}^T = [{}_i\varTheta _{x_1}, {}_i\varTheta _{x_2}, {}_i\varTheta _{x_3}]$, the gradient of $\{{}_i\varTheta \}$ i.e. $[ {}^{{}_i \varTheta }J ]$ and its decomposition into symmetric and skew-symmetric tensor $[ {}_{\;s}^{{}_i \varTheta }J ]$ and $[ {}_{\;a}^{{}_i \varTheta }J ]$ can be written as

$$\begin{aligned} \left[ \dfrac{\partial {\{{}_i\varTheta \}}}{\partial {\{x\}}} \right] = [ {}^{{}_i \varTheta }J ] = [ {}_{\;s}^{{}_i \varTheta }J ] + [ {}_{\;a}^{{}_i \varTheta }J ] \end{aligned}$$

(A11)

We have the following conservation and balance laws [4, 5, 7]

$$\begin{aligned}&\rho _{_{_{0}}}(\pmb {\varvec{x }}) = |J|\rho (\pmb {\varvec{x }},t) \quad \quad \text { (CM)} \end{aligned}$$

(A12)

$$\begin{aligned}&\rho _{_{_{0}}}\frac{Dv_{i}}{Dt} - \rho _{_{_{0}}}F^{b}_i- \dfrac{\partial { {\sigma }_{ji}}}{\partial {x_{j}}} = 0 \quad \quad \text {(BLM) } \end{aligned}$$

(A13)

$$\begin{aligned}&m_{mk,m} + \epsilon _{ijk} \sigma _{ij} = 0 \quad \quad \text { (BAM)}\end{aligned}$$

(A14)

$$\begin{aligned}&\rho _{_{_{0}}}\frac{De}{Dt} + \nabla .\pmb {\varvec{q }} - {}_s {\sigma } _{ij} \overset{\,\mathbf{. }}{\varepsilon } _{ij} - m_{ij}( {}_{\;s}^{{}_i \varTheta }J _{ij}) = 0 \quad \quad \text { (FLT)}\end{aligned}$$

(A15)

$$\begin{aligned}&\rho _{_{_{0}}}\left( \frac{D\phi }{Dt}+\eta \frac{D\theta }{Dt}\right) + \frac{\pmb {\varvec{q.g }}}{\theta } - {}_s {\sigma } _{ji} \overset{\,\mathbf{. }}{\varepsilon } _{ij} - m_{ij}( {}_{\;s}^{{}_i \varTheta }J _{ij}) \le 0 \; \text { (SLT)} \end{aligned}$$

(A16)

Additionally we have used decomposition of Cauchy stress tensor $\pmb {\varvec{\sigma }}$ into symmetric tensor ${}_s \pmb {\varvec{\sigma }}$ and antisymmetric tensor ${}_a \pmb {\varvec{\sigma }}$

$$\begin{aligned} \pmb {\varvec{\sigma }} = {}_s \pmb {\varvec{\sigma }} + {}_a \pmb {\varvec{\sigma }} \end{aligned}$$

(A17)

The Cauchy moment tensor $\pmb {\varvec{m }}$ is symmetric due to balance of moment of moments balance law, an additional balance law needed in non-classical continuum theories [4, 5, 7] due to new physics associated with rotations.

From the conjugate pairs in the entropy inequality, at the very least, the following must hold (thermoelastic solid continua)

$$\begin{aligned} {}_s \pmb {\varvec{\sigma }}= & {} {}_s \pmb {\varvec{\sigma }} ( \pmb {\varvec{\varepsilon }} , \theta )\end{aligned}$$

(A18)

$$\begin{aligned} \pmb {\varvec{m }}= & {} \pmb {\varvec{m }} ( {}_{\;s}^{{}_i \varTheta }\pmb {\varvec{J }} , \theta )\end{aligned}$$

(A19)

$$\begin{aligned} \pmb {\varvec{q }}= & {} \pmb {\varvec{q }}(\pmb {\varvec{g }}, \theta ) \end{aligned}$$

(A20)

Choice of the argument tensors for $\phi$ and $\eta$ can be based on the principle of equipresence [9]. The derivation of the constitutive theory for ${}_s \pmb {\varvec{\sigma }}$ is same as for $\pmb {\varvec{\sigma }}$ in case of CCM (section). A linear constitutive theory for $\pmb {\varvec{m }}$ can be written as

(A21)

where is the material coefficient related to the constitutive theor for the Cauchy moment tensor.

1.3 A.3 Non-classical continuum theory incorporating internal rotations and Cosserat rotations

Let ${}_e\pmb {\varvec{\varTheta }}$ be external or Cosserat rotations (unknown) about the axes of the same triad at a material point about which internal rotations ${}_i\pmb {\varvec{\varTheta }}$ act, then the total rotations ${}_t\pmb {\varvec{\varTheta }}$ are given by

$$\begin{aligned} {}_t\pmb {\varvec{\varTheta }}= {}_i\pmb {\varvec{\varTheta }}+ {}_e\pmb {\varvec{\varTheta }}\end{aligned}$$

(A22)

and

$$\begin{aligned}{}[ {}_{a}^{\,t}r ] = [ {}_{a}^{\,i}r ] + [ {}_{a}^{\,e}r ] \end{aligned}$$

(A23)

Gradient of ${}_t\pmb {\varvec{\varTheta }}$, ${}^{{}_t\varTheta }\pmb {\varvec{J }}$ and its decomposition into symmetric and antisymmetric tensors gives

$$\begin{aligned}{}[ {}^{{}_t \varTheta }J ]= & {} \left[ \frac{ \partial \{{}_t\varTheta \}}{ \partial \{x\}} \right] = [ {}_{\;s}^{{}_t \varTheta }J ] + [ {}_{\;a}^{{}_t \varTheta }J ] = [ {}_{\;s}^{{}_t \varTheta }J ] + [ {}_{a}^{\,t}r ] \end{aligned}$$

(A24)

$$\begin{aligned}= & {} \frac{1}{2} \left( [ {}^{{}_t \varTheta }J ] + [ {}^{{}_t \varTheta }J ]^T \right) \end{aligned}$$

(A25)

$$\begin{aligned}= & {} \frac{1}{2} \left( [ {}^{{}_t \varTheta }J ] - [ {}^{{}_t \varTheta }J ]^T \right) \end{aligned}$$

(A26)

The CM, BLM, BAM and BMM balance laws in this case are same as in section A.2. The FLT and the SLT [8, 36] are given by

$$\begin{aligned}&\rho _{_{_{0}}}\frac{De}{Dt} + \nabla \cdot \pmb {\varvec{q }} - {}_s {\sigma } _{ij} \overset{\,\mathbf{. }}{\varepsilon } _{ij} - {{}_a {\sigma }}_{ij} ( {}_{a}^{\,t}\overset{\,\mathbf{. }}{r} _{ij}) \nonumber \\&\quad - m_{ij}( {}_{\;s}^{{}_i \varTheta }J _{ij}) = 0 \quad \quad \text { (FLT)} \end{aligned}$$

(A27)

$$\begin{aligned}&\rho _{_{_{0}}}\left( \frac{D\phi }{Dt}+\eta \frac{D\theta }{Dt}\right) + \frac{\pmb {\varvec{q\cdot g }}}{\theta } \nonumber \\&\quad - {}_s {\sigma } _{ji} \overset{\,\mathbf{. }}{\varepsilon } _{ij} - {{}_a {\sigma }}_{ij} ( {}_{a}^{\,t}\overset{\,\mathbf{. }}{r} _{ij})- m_{ij}( {}_{\;s}^{{}_i \varTheta }J _{ij}) \le 0 \quad \quad \text { (SLT)}\nonumber \\ \end{aligned}$$

(A28)

The decomposition of $\pmb {\varvec{\sigma }} = {}_s \pmb {\varvec{\sigma }} + {}_a \pmb {\varvec{\sigma }}$ (section A.2) is used here as well. The Cauchy moment tensor is symmetric in this case also (balance of moment of moments balance law [2, 3]). From the conjugate pairs in the entropy inequality (A28), at the very least the following must hold (thermoelastic solid continua)

$$\begin{aligned} {}_s \pmb {\varvec{\sigma }}= & {} {}_s \pmb {\varvec{\sigma }} ( \pmb {\varvec{\varepsilon }} , \theta ) \end{aligned}$$

(A29)

$$\begin{aligned} \pmb {\varvec{m }}= & {} \pmb {\varvec{m }} ( {}_{\;s}^{{}_t \varTheta }\pmb {\varvec{J }} , \theta )\end{aligned}$$

(A30)

$$\begin{aligned} {}_a \pmb {\varvec{\sigma }}= & {} {}_a \pmb {\varvec{\sigma }} ( {}_{a}^{\,t}\pmb {\varvec{r }} , \theta ) \end{aligned}$$

(A31)

Argument tensors for $\varPhi$ and $\eta$ at this stage can be established using principle of equipresence. Constitutive theories for $\pmb {\varvec{m }}$ and ${}_a \pmb {\varvec{\sigma }}$ in the absence of $\theta$ reduce to [8, 36]

(A32)

(A33)