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A Geometric Theory of Nonlinear Morphoelastic Shells

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Abstract

Many thin three-dimensional elastic bodies can be reduced to elastic shells: two-dimensional elastic bodies whose reference shape is not necessarily flat. More generally, morphoelastic shells are elastic shells that can remodel and grow in time. These idealized objects are suitable models for many physical, engineering, and biological systems. Here, we formulate a general geometric theory of nonlinear morphoelastic shells that describes both the evolution of the body shape, viewed as an orientable surface, as well as its intrinsic material properties such as its reference curvatures. In this geometric theory, bulk growth is modeled using an evolving referential configuration for the shell, the so-called material manifold. Geometric quantities attached to the surface, such as the first and second fundamental forms, are obtained from the metric of the three-dimensional body and its evolution. The governing dynamical equations for the body are obtained from variational consideration by assuming that both fundamental forms on the material manifold are dynamical variables in a Lagrangian field theory. In the case where growth can be modeled by a Rayleigh potential, we also obtain the governing equations for growth in the form of kinetic equations coupling the evolution of the first and the second fundamental forms with the state of stress of the shell. We apply these ideas to obtain stress-free growth fields of a planar sheet, the time evolution of a morphoelastic circular cylindrical shell subject to time-dependent internal pressure, and the residual stress of a morphoelastic planar circular shell.

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Notes

  1. Other examples of evolving material metrics in mechanics have been introduced in (Ozakin and Yavari 2010; Yavari and Goriely 2012a, b, 2013a, 2015, 2013b, 2014; Sadik and Yavari 2015).

  2. cf. (2.7) where the notation \({\bar{\varvec{G}}}_{{\mathcal {H}}}\) was introduced.

  3. The Lie derivative along the vector field \(\varvec{{\mathcal {V}}}\) is defined as \({\varvec{L}}_{\varvec{{\mathcal {V}}}}\varvec{{\mathcal {V}}}^\parallel = \left. \frac{\hbox {d}}{\hbox {d}t}\right| _{t=s} \left[ \left( \varphi _t \circ \varphi _s^{-1}\right) ^*\varvec{{\mathcal {V}}}^\parallel \right] \), where \(\varphi _t \circ \varphi _s^{-1}\) is the flow of \(\varvec{{\mathcal {V}}}\).

  4. Note that since the vector \({\varvec{U}}\) is tangent to \({\mathcal {H}}\) at \(\varphi (X,t)\), the vectors \(\left[ \varvec{{\mathcal {V}}}, {\varvec{U}} \right] ={\varvec{L}}_{\varvec{{\mathcal {V}}}}{\varvec{U}}\) and \(\tilde{\nabla }_{{\varvec{U}}} {\varvec{n}}\) are tangent to \({\mathcal {H}}\) as well.

  5. Note that (4.5) can also be obtained from (4.2) and (4.1) by writing

    $$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t} \int _{\varphi _t(\mathcal {U})}\varrho \hbox {d}s =\int _{\varphi _t(\mathcal {U})}s_m\hbox {d}s. \end{aligned}$$
  6. Since the Lagrangian density is a scalar, it depends on the metrics \({\varvec{G}}\) and \({\tilde{\varvec{g}}}\).

  7. For fixed X and t, we let \(\varphi _{\epsilon ,t}(X):=\varphi _{\epsilon }(X,t).\)

  8. We denote by \(F^{-A}{}_a\) the components of \({\varvec{F}}^{-1}\), the inverse of \({\varvec{F}}\). See Appendix 7 for the details of the derivation of (4.13) and (4.14) following (4.7).

  9. Recall the Piola identity \(\left( JF^{-A}{}_a\right) _{|A}=0\).

  10. We define the convected manifold to be the material manifold \({\mathcal {H}}\) equipped with the right Cauchy–Green deformation tensor \({\varvec{C}}\).

  11. The components of \({\varvec{C}}^{-1}\), the inverse of \({\varvec{C}}\), are denoted by \(C^{-AB}\).

  12. For details on the derivation of the Saint Venant–Kirchhoff shell model, see (Fox et al. 1993; Le Dret and Raoult 1993; Lods and Miara 1995; Miara 1998; Lods and Miara 1998; Friesecke et al. 2002a, b, 2003).

  13. Following (4.17b), there are three equilibrium equations

    figure c

    Because of the symmetry of the problem and the isotropy of the material, the stresses take the form (5.9). This implies that Eq. (5.10a) are trivially satisfied and the terms containing derivatives in (5.10b) vanish. Therefore, we are left with Eq. (5.11) as the only non-trivial equilibrium equation.

  14. When \(\omega =0\), the first fundamental form \({\varvec{G}}\) is not a dynamical variable anymore, and hence, the kinetic equation (5.20a) should be discarded.

  15. Recall that r depends on K as can be seen in (5.16), (5.17), or (5.18) depending on the value of the discriminant \(\Delta \).

  16. We let for example \(\omega _A=ZK_A(R,t)\) for \(A=R,\Theta \) in (5.24).

  17. Note that \(\left( \Sigma ^{AB}+C^{-AC}\Theta _{CD}\Lambda ^{DB}\right) _{||B} +C^{-AC}\Theta _{CD} \Lambda ^{DB}{}_{||B} =\left( \Sigma ^{AB}+2C^{-AC}\Theta _{CD}\Lambda ^{DB}\right) _{||B} -\left( C^{-AC}\Theta _{CD}\right) _{||B}\Lambda ^{DB}\). Also, we have \(\text {J} = \frac{r}{R}\). Therefore, the convected stress and couple-stress tensors read \({\varvec{\Sigma }}=\frac{R}{r}{\varvec{S}}\) and \({\varvec{\Lambda }}=\frac{R}{r}{\varvec{M}}\).

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Acknowledgments

SS was supported by a Fulbright Grant. AG is a Wolfson/Royal Society Merit Award Holder and acknowledges support from a Reintegration Grant under EC Framework VII. We thank M.F. Shojaei for his help with some of the numerical examples. This research was partially supported by AFOSR – Grant No. FA9550-12-1-0290 and NSF—Grant No. CMMI 1042559 and CMMI 1130856.

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Appendix: Derivation of the Euler–Lagrange Equations

Appendix: Derivation of the Euler–Lagrange Equations

In this appendix, we work out in detail the derivation of the Euler–Lagrange equations first assuming that \(\delta {\varvec{G}}=\delta {\varvec{B}}={\varvec{0}}\). We substitute (4.8), (4.9), (4.10), (4.11), and (4.12) into (4.7) to obtain

$$\begin{aligned}&\int _{t_0}^{t_1}\int _{{\mathcal {H}}}\bigg [ \frac{\partial {\mathcal {L}}}{\partial \varphi ^a}{\delta \varphi ^a} +\frac{\partial {\mathcal {L}}}{\partial \varphi ^n}{\delta \varphi ^n} -\left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _b\left( \delta \varphi ^a\beta ^b{}_a+\frac{\partial (\delta \varphi ^n)}{\partial x^a}g^{ab}\right) +\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }}.\frac{\hbox {D}\delta \varphi }{\hbox {d}t}\nonumber \\&\quad +\,\frac{\partial {\mathcal {L}}}{\partial C_{AB}}\left( 2F^b{}_B{g}_{bc}\delta \varphi ^c{}_{|A} -2\delta \varphi ^nF^a{}_AF^b{}_B{\beta }_{ab} \right) +\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}\bigg ( F^a{}_AF^b{}_B{\beta }_{ab|c}\delta \varphi ^c\nonumber \\&\quad +\,2F^a{}_A{\beta }_{ac}\delta \varphi ^c{}_{|B}-\delta \varphi ^nF^a{}_AF^b{}_B{\beta }_{ac}{\beta }_{bd}g^{cd} +F^b{}_A\left( \frac{\partial \delta \varphi ^n}{\partial x^b}\right) _{|B} \bigg ) \bigg ]\nonumber \\&\quad \sqrt{\det {\varvec{G}}}\hbox {d}X^I\hbox {d}t =0. \end{aligned}$$
(7.1)

Hence, we have

$$\begin{aligned} \begin{aligned}&\int _{t_0}^{t_1}\int _{{\mathcal {H}}}\bigg \{ \frac{\partial {\mathcal {L}}}{\partial \varphi ^a}{\delta \varphi ^a} +\frac{\partial {\mathcal {L}}}{\partial \varphi ^n}{\delta \varphi ^n} +\frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }}.\delta \varphi \right) \\&\quad -\,\frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }}\right) .\delta \varphi \\&\quad -\,\left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _b\beta ^b{}_a\delta \varphi ^a -\left( \left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _b\delta \varphi ^ng^{ab}F^{-A}{}_a\right) _{|A}\\&\quad +\,\left( \left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _bg^{ab}F^{-A}{}_a\right) _{|A}\delta \varphi ^n\\&\quad +\,2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_A{g}_{ac}\delta \varphi ^a\right) _{|B} -2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_A{g}_{ac}\right) _{|B}\delta \varphi ^a \\&\quad -\,2\frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^a{}_AF^b{}_B{\beta }_{ab}\delta \varphi ^n\\&\quad +\,\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_AF^b{}_B{\beta }_{bc|a}\delta \varphi ^a +2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac}\delta \varphi ^a\right) _{|B}\\&\quad -\,2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac}\right) _{|B}\delta \varphi ^a\\&\quad -\,\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}} F^a{}_AF^b{}_B{\beta }_{ac}{\beta }_{bd}g^{cd}\delta \varphi ^n +\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\frac{\partial \delta \varphi ^n}{\partial x^b}\right) _{|B} \\&\quad -\,\bigg [\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}\delta \varphi ^nF^{-D}{}_b\bigg ]_{|D}\\&\quad +\,\bigg [\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}F^{-D}{}_b\bigg ]_{|D}\delta \varphi ^n \bigg \}\sqrt{\det {\varvec{G}}}\hbox {d}X^I\hbox {d}t=0 . \end{aligned} \end{aligned}$$
(7.2)

We can rewrite (7.2) as

$$\begin{aligned}&\int _{t_0}^{t_1}\int _{{\mathcal {H}}}\bigg \{ \frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }}.\delta \varphi \right) +2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_A{g}_{ac}\delta \varphi ^a\right) _{|B}\nonumber \\&\quad +\,2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac}\delta \varphi ^a\right) _{|B}-\,\bigg [\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}\delta \varphi ^nF^{-D}{}_b\bigg ]_{|D}\nonumber \\&\quad -\left( \left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _b\delta \varphi ^ng^{ab}F^{-A}{}_a\right) _{|A} +\,\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}\frac{\partial \delta \varphi ^n}{\partial X^A}\right) _{|B}\nonumber \\&\quad +\,\bigg [ \frac{\partial {\mathcal {L}}}{\partial \varphi ^a} -\left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _b\beta ^b{}_a -\frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }^a}\right) -2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_A{g}_{ac}\right) _{|B}\nonumber \\&\quad +\,\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_AF^b{}_B{\beta }_{bc|a} -2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac}\right) _{|B} \bigg ]\delta \varphi ^a \nonumber \\&\quad +\,\bigg [ \frac{\partial {\mathcal {L}}}{\partial \varphi ^n} +\left( \left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _bg^{ab}F^{-A}{}_a\right) _{|A}-\,\frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }^n}\right) \nonumber \\&\quad -\,2\frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^a{}_AF^b{}_B{\beta }_{ab} -\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^a{}_AF^b{}_B{\beta }_{ac}{\beta }_{bd}g^{cd}\nonumber \\&\quad +\,\bigg [\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}F^{-D}{}_b\bigg ]_{|D} \bigg ]\delta \varphi ^n \bigg \}\hbox {d}S\hbox {d}t=0. \end{aligned}$$
(7.3)

At \(t=t_1\), we assume that \(\varphi _{\epsilon ,t_1}=\varphi _{t_1}\) so that \(\delta \varphi _{t_1}=0\). Therefore, by integrating the first term in (7.3) in the time domain, we obtain only one term at \(t=t_0\) giving the initial condition on the velocity at \(t=t_0\). By applying Stokes’ theorem to the following five terms, if we denote by \(\varvec{{\mathsf {T}}}\) the outward in-plane vector field normal to the boundary curve \(\partial {\mathcal {H}}\), we obtain

$$\begin{aligned}&-\left. \int _{{\mathcal {H}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }}.\delta \varphi \hbox {d}S\right| _{t=t_0}+ \int _{t_0}^{t_1}\int _{\partial {\mathcal {H}}}\bigg \{ \left( 2\frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_A{g}_{ac}+2\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac}\right) {\mathsf {T}}_{B}\delta \varphi ^a\nonumber \\&\quad -\,\bigg [\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AC}}F^b{}_A\right) _{|C}F^{-B}{}_b+\left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _bg^{ab}F^{-B}{}_a\bigg ]{\mathsf {T}}_{B}\delta \varphi ^n +\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}{\mathsf {T}}_{B}\frac{\partial \delta \varphi ^n}{\partial X^A}\bigg \}\hbox {d}L\hbox {d}t\nonumber \\&\quad \int _{t_0}^{t_1}\int _{{\mathcal {H}}}\bigg \{ \bigg [ \frac{\partial {\mathcal {L}}}{\partial \varphi ^a} -\left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _b\beta ^b{}_a -\frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }^a}\right) -2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_A{g}_{ac}\right) _{|B}\nonumber \\&\quad +\,\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_AF^b{}_B{\beta }_{bc|a} -2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac}\right) _{|B} \bigg ]\delta \varphi ^a \nonumber \\&\quad +\bigg [ \frac{\partial {\mathcal {L}}}{\partial \varphi ^n}+\,\left( \left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _bg^{ab}F^{-A}{}_a\right) _{|A} -\,\frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }^n}\right) \nonumber \\&\quad -2\frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_AF^b{}_B{\beta }_{bc} -\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^a{}_AF^b{}_B{\beta }_{ac}{\beta }_{bd}g^{cd}\nonumber \\&\quad +\,\bigg [\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}F^{-D}{}_b\bigg ]_{|D} \bigg ]\delta \varphi ^n \bigg \} \hbox {d}S\hbox {d}t =0. \end{aligned}$$
(7.4)

By arbitrariness of \(\delta \varphi ^\parallel \), \(\delta \varphi ^n\), and \(d(\delta \varphi ^n)\), the Euler–Lagrange equations for shells (4.13) together with the initial and boundary conditions (4.14) follow from (7.4). Note that following Codazzi’s equation (2.3), we have \({\beta }_{bc|a}={\beta }_{ac|b}.\) Therefore \(\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_AF^b{}_B{\beta }_{bc|a}=\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac|B}.\)

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Sadik, S., Angoshtari, A., Goriely, A. et al. A Geometric Theory of Nonlinear Morphoelastic Shells. J Nonlinear Sci 26, 929–978 (2016). https://doi.org/10.1007/s00332-016-9294-9

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