$\def\bP{\boldsymbol P}$ $\def\bF{\boldsymbol F}$ $\def\bN{\boldsymbol N}$ $\def\bG{\boldsymbol G}$ $\def\bM{\boldsymbol M}$

$\def\bK{\boldsymbol K}$

Geometry

We identify the shell with a surface $\mathcal S$, i.e. a two-dimensional submanifold of the Euclidean point space $\mathcal E$. We suppose $\mathcal S$ to be orientable, at least locally, and we shall denote by $\boldsymbol n(x)$ the orthogonal unit vector at $x$.

We denote by $V$ the linear space of vectors in three dimensions. Given $x\in\mathcal S$, we let the tangent space at $x$

We shall use the manifold $\mathcal S$ as the label set to identify the particles that comprise the shell. The label set should not be confused with the configuration of the shell, the latter being specified by a placement $\widehat y:\mathcal S\to\mathcal E$ and a director field $\boldsymbol d:\mathcal S\to \partial B(0,1)$. Note that the map $\widehat{y}$ defines a surface $\mathcal S'$ in the Euclidean space, but does not embody complete information on its configuration.

Equilibrium

We postulate ¹ the existence of pair of tensor fields $\boldsymbol S$ and $\boldsymbol M$ such that the resultant force and the resultant moment exerted on a part $\mathcal P$ are given by

In the above expressions, $\boldsymbol b(x)$ and $\boldsymbol d (x)\times \boldsymbol c(x)$ are, respectively, the applied force and the applied couple per unit reference area at $x$. Moreover, $\boldsymbol y(x)$ is the translation vector that takes the origin to the place $\widehat y(x)$. In the boundary integrals, $\boldsymbol\nu(x)\in T_x$ denotes the unit vector tangent to $\mathcal S$, orthogonal to the curve $\partial\mathcal P$, and pointing outside $\mathcal P$ at $x$. The terms $\boldsymbol S(x)\boldsymbol \nu(x)$ and $\boldsymbol d(x)\times\boldsymbol M(x)\boldsymbol \nu(x)$ represent, respectively, the linear densities of contact force and contact couple applied at the boundary of $\mathcal P$ at $x$.

Remarks:

• The fact that surface and line couples are orthogonal to $\boldsymbol d$ encodes the assumption that a shell cannot sustain and transmit couples parallel to the director.
• We take both $\boldsymbol c$ and $\boldsymbol M\boldsymbol\nu$ orthogonal to $\boldsymbol d$.
• While the tensor $\boldsymbol S(x)$ is a linear map from $T_x$ to $V$, $\boldsymbol M(x)$ is a linear map from $T_x$ to the orthogonal complement of the space spanned by $\boldsymbol d(x)$:

• The tensor fields $\boldsymbol S$ and $\boldsymbol M$ are the homologous of the Piola stress.
• As long as we are concerned with balance equations, it does not matter whether the surface $\mathcal S$ is flat or curved. In fact, the differential equations of equilibrium we write are independent on the choice of $\mathcal S$.
• All tensor fields we shall consider are superficial: their domain at $x$ is $T_x \mathcal S$.
• We shall call a tensor field tangential if its range is $T_x\mathcal S$.

Divergence theorems

We need a divergence theorem to transform integrals computed on $\partial\mathcal P$ into integrals over $\mathcal P$. To begin with, we record that if $\boldsymbol v$ is a tangential tensor field ​then the gradient and the tangential gradient of $\boldsymbol v$ are defined, respectively, by

with $\mathring{\boldsymbol P}=\mathbf I-\boldsymbol n\otimes\boldsymbol n$ the projector on the tangent space. We recall that the following divergence theorem holds:

Next, consider the tensor field $\mathbf A=\mathbf a\otimes\mathbf v$ where $\mathbf a$ is a constant vector and $\mathbf v$ is a tangential tensor field. Then

Let $\{\boldsymbol e_i,i=1,2,3\}$ be a basis for the space of translations of the physical space $\mathcal E$. Since every tensor field $\mathbf A$ can be written as $\boldsymbol A=\boldsymbol e_i\otimes\boldsymbol v^{(i)}$, for a suitable choice of a triplet of vector fields $\boldsymbol v^{(i)}$, the identity $\eqref{eq:divergence2}$ holds for whatever tensor field $\boldsymbol A$. Thus, we can conveniently take

as the definition of divergence of a tensor field $\boldsymbol T$.

Pointwise form of the equilibrium equations

We require that the resultant force $\boldsymbol r(\mathcal P)$ and the resultant moment $\boldsymbol m(\mathcal P)$ be null for every part $\mathcal P \subset \mathcal S$. It is not difficult, using the divergence theorem and a standard localization argument to deduce the equivalence:

A similar argument can be applied to the equilibrium of moments. However, the result of this procedure is more transparent if we use the identity

Then the the vanishing of the resultant moment for each part $\mathcal P$ is equivalent to

Then a procedure similar as above yields the following equivalence

On introducing the deformation gradient and the director gradient

on taking into account the equilibrium equation $\operatorname{div}\boldsymbol S+\boldsymbol b=\boldsymbol 0$, and on recalling that $\operatorname{skw}\boldsymbol A=-(\operatorname{skw}\boldsymbol A)^T=-\operatorname{skw}(\boldsymbol A^T)$ for every tensor $\boldsymbol A$, the last equation yields

Note that this equation coincides with the symmetry condition (5.18) found in ²

The decomposition

uniquely identifies the membrane tensor $\boldsymbol N$ satisfies and the shear vector $\boldsymbol q$. Note that on introducing the projector

we have

Note that $\boldsymbol q\cdot\mathring{\boldsymbol n}=0$ by definition, hence $\boldsymbol q$ is a tangential tensor field. We call this field the shear flux.

Thanks to the above decomposition we now can write:

Whence, on recalling the definition of $\boldsymbol q$ and $\boldsymbol N$:

From this equation we conclude:

The vector $\mathbf q=\boldsymbol S^T\boldsymbol d$ is the shear vector. Its interpretation is as follows: the scalar product $\boldsymbol q\cdot\boldsymbol\nu$ represents the component along $\boldsymbol d$ of the line contact force acting across a line element from the side of $\boldsymbol \nu$.

Virtual work

On using the divergence theorem

We next write

We define the shear strain:

We arrive at

Now we introduce the following strain measures

Note that the tensors $\mathbf C$ and $\mathbf K$ are tangential, in the sense that their range is $T_x\mathcal S$. Note in particular that they map $T_x\mathcal S$ into itself.

We can establish a connection with the standard strain measures by observing that, on denoting by $A_\alpha=\boldsymbol x_{,\alpha}$ and by $a_\alpha=\boldsymbol y_{,\alpha}$, respectively, the covariant basis in the reference and deformed configuration, then

Note also that if we regard $\boldsymbol F=a_\alpha\otimes A^\alpha$ as a map from $T_x \mathcal S$ to $T_y \mathcal S'$, then the inverse $\boldsymbol F^{-1}$ is well defined. We now introduce the referential stress measures

Then we can compute

In the last equation we have used the fact that $\mathbf N-\widetilde{\mathbf K}\mathbf M^T$ is a symmetric tensor. This follows from $\eqref{eq:0318151000}$. In fact, $\boldsymbol N\boldsymbol F^T+\boldsymbol M\boldsymbol G^T\in\operatorname{Sym}$ is equivalent to $\boldsymbol F^{-1}(\boldsymbol N\boldsymbol F^T+\boldsymbol M\boldsymbol G^T)\boldsymbol F^{-T}\in\operatorname{Sym}$, that is to say, $\mathbf N+\mathbf M\widetilde{\mathbf K}^T\in\operatorname{Sym}$. But this in turn is equivalent to to $\mathbf N-(\mathbf M\widetilde{\mathbf K}^T)^T\in\operatorname{Sym}$.

Here we have introduced the spatial strain measure

We introduce the Green-Lagrange strain

Then the above equation becomes

Remarks

• the book by Taylor, Zienkiewicz and Fox obtains the same expression of the internal power by starting from three-dimensional elasticity. We work out the expression of the stress working and our frame-indifference requirements without invoking a parent three-dimensional theory.
• The stress measures $\mathbf M$ and $\mathbf N$ are the homologous of the Cosserat (2nd P-K) stress in nonlinear elasticity
• As the notation suggests, $\boldsymbol C$ is the analogue of the right Green strain.
• Our derivation is completely component- and derivative-free. In fact, a definition of gradient of scalar and tensor field can be given, based on the notion of tangent space and Gateaux derivative can be given, which does not rely on coordinates (although in differential geometry the very notion of manifold requires the introduction of chart, and hence a coordinate system.)

Unshearable shells

From this point on, we shall restrict our attention to shell satisfying the constraint:

Note that in this case the shear strain vanishes: $\mathbf g=\mathbf 0$. In fact, the range of $\boldsymbol F$ is orthogonal to the space spanned by the normal $\boldsymbol n$ in the current configuration. Thus the equilibrium equations become

Quadratic strain energy

We for $h$ a thickness parameter, we assume that the strain energy is separable, and has the form

where $\mathring{\mathbf N}$ and $\mathring{\boldsymbol M}$ are the initial membrane stress and initial bending stress and $\mathbb C:T_x \mathcal S\to T_x\mathcal S$ is the fourth-order tensor defined by

where $\widetilde\mu$ and $\widetilde\lambda$ are the effective Lamé constants and $\mathbf I$ is the identity on $T_x\mathcal S$. We compute

The dissipation principle implies that for an hyperelastic shell the following constitutive equations hold:

Thus, the expressions of the Piola-like stresses $\boldsymbol N$ and $\boldsymbol M$ in terms of their duals $\boldsymbol F$ and $\boldsymbol G$ are:

Linearization

We are concerned with infinitesimal perturbations from an initial configuration.

We have at order 0

Small displacements

In the theory of linearly elastic shells the equations of equilibrium are written in the reference configuration, defined by $\boldsymbol y(\boldsymbol x)=\boldsymbol x$ and $\boldsymbol d=\boldsymbol n$. Then

where $\boldsymbol W$ is the Weingarten tensor. Furthermore $\boldsymbol S\boldsymbol P=\boldsymbol S$. We introduce the surface divergence $\operatorname{div}_s(\cdot)=\boldsymbol P\operatorname{div}(\cdot)$ and we recall that $\boldsymbol W$ is symmetric, then we recover the standard equilibrium equations used in the theory of linearly elastic shells:

Alternatively,

Remarks.

The geometrical structure of the shell emerges only at this stage, when we single out a stress-free configuration.

We note that both $\boldsymbol P$ and $\boldsymbol W$ are superficial tensor fields, hence:

Virtual work

In the linear theory the deformation of the shell is described by a displacement field $\boldsymbol u$ and a rotation field $\boldsymbol\varphi$. In the deformed configuration the typical point $x$ of the shell occupies the position $\boldsymbol y=\boldsymbol x+\boldsymbol u(x)$, and the associated director field is $\boldsymbol d=\boldsymbol n+\boldsymbol\omega\times\boldsymbol n=\boldsymbol n+\boldsymbol\varphi$. Then the work performed by the applied couples on a virtual rotation is:

The virtual work performed by forces and couples acting on a part $\mathcal P$ is:

Integration by parts by making use of the divergence theorem and of the equilibrium equations shows that

Introducing the decomposition

we can write

Note that the vector

is invariant under a infinitesimal change of observer. However, the tensors $\nabla_s\delta\boldsymbol u$ and $\nabla_s\delta\boldsymbol\varphi$ are not. Thus, we cannot take them as strain measures.

However, we can rewrite the internal work as

For a rigid motion $\delta\boldsymbol\varphi(x)=\boldsymbol\psi\times\boldsymbol n(x)$, hence $\nabla_s\delta\boldsymbol\varphi=-[\boldsymbol\psi\times]\boldsymbol W$, and $\nabla_s\delta\boldsymbol u=[\boldsymbol\psi\times]$. Thus, under a rigid motion the quantity $\nabla_s\delta\boldsymbol\varphi+\nabla_s\delta\boldsymbol u\boldsymbol W$ vanishes. Furthermore since $\boldsymbol N-\boldsymbol M\boldsymbol W$ is symmetric, then

Note that also $\operatorname{sym}\nabla_s\delta\boldsymbol u$ vanishes when $\delta\boldsymbol u$ is an infinitesimal rigid displacement. This and the previous results suggest that we adopt

as additional strain measures, so that the virtual work reads

Note that since $\boldsymbol W=-\nabla\boldsymbol n$ maps $T_x$ into itself, we have

where we have introduced the decomposition of the displacement into its tangential and normal components

Notice also that, since $\nabla\boldsymbol u=\nabla\boldsymbol v+\boldsymbol n\otimes\nabla w-w\boldsymbol W$, we have

Hence, given that $\boldsymbol W$ is symmetric,

###

Appendix

Gradient and divergence on a surface

The gradient of a vector field $\boldsymbol v$ on $\mathcal S$ is a tensor field whose value at $x$ is a map from $T_x$ to $V$, defined as follows: the image of a vector $\boldsymbol a$ is computed by taking any smooth curve $a:\mathbb R\to\mathcal S$ such that $a(0)=x$ and $a'(0)=\boldsymbol a$, and by letting

This definition is independent on the particular curve chosen, as can be seen by working on a chart. The covariant derivative of $\boldsymbol v$ is

Note that the covariant derivative is instead a linear map from $T_x$ to $T_x$ .

The divergence of a tangential vector field $\boldsymbol a$ is defined as $\operatorname{div}\boldsymbol a=\boldsymbol P\cdot\nabla\boldsymbol a=\boldsymbol P\nabla_s\boldsymbol a$, where $\boldsymbol P(x)$ is the projection of $V$ to $T_x$.

The best way to introduce the divergence of a tensorial quantity $\boldsymbol A$ mapping $T_x$ into some linear space is to consider first tensors of the form

where $\boldsymbol v$ is a constant vector.

The divergence of a tensor field $\boldsymbol A$ is defined by the identity:

Invariance under a change of observer

A rigid displacement from the reference configuration has the form

Then, on denoting by $[\delta\boldsymbol\psi\times]$ the skew-symmetric tensor whose axial vector id $\delta\boldsymbol\psi$, we can write

hence

Thus $\delta\boldsymbol\gamma$ is invariant under a change of observer.

The effective Lamé constants for a linearly elastic shell

For a three-dimensional isotropic, linearly elastic material, the stress-strain relation has the form

where $\mu$ and $\lambda$ are the Lamé constants. As discussed in Ciarlet (2005)](https://doi.org/10.1007/s10659-005-4738-8), Sec. 4.5, a rigorous argument based on asymtotic analysis shows that the mechanical response of a shell-like homogeneous linearly elastic body of thickness $h$, made of a material obeying the stress-strain relation $\eqref{eq:resp}$ is captured by a shell with Lamé constants (See the discussion about the convergence

These constants are obtained by considering the in-plane response under the assumption that $\mathbf S\boldsymbol n\cdot\boldsymbol n=0$, which leads to the system

Details to obtain the pointwise form of the balance equations

The divergence of a tensor field can be written as

In particular, $\boldsymbol v$ is a vector field and $\boldsymbol T$ is a tensor field, we have

We obtain

Hence,

On introducing the decomposition

we can write

Linear Koiter energy

A finite deformation of the shell is described by an one-to-one smooth map

The deformation gradient $\nabla\boldsymbol y(\boldsymbol x)$ is a linear map from $T_x$ to $T'_{\boldsymbol y(\boldsymbol x)}$, where $T'_{\boldsymbol x'}$ denotes the tangent space of $\mathcal S'$ at $\boldsymbol x'$. We define the strain as

$\boldsymbol D=\nabla\boldsymbol y^T\nabla\boldsymbol y/2$

Note if $\boldsymbol x:\mathbb R^2\to\mathcal S$ is a parametrization for $\mathcal S$, then the map $\boldsymbol x'=\boldsymbol y\circ\boldsymbol x$ is a parametrization for $\mathcal S'$.

We denote by $\boldsymbol n':\mathcal S'\to V$ the normal. The Weingarten tensor on $\mathcal S'$ is

For a Koiter shell, the strain energy of a part $\mathcal P$ is

The variation of $\boldsymbol D$ is \boldsymbold\boldsymbolsf D=\fr\boldsymbol(\boldsymbolsf P\nabla\boldsymbol u+\nabla\boldsymbolT\boldsymbolsf P)=\fr\boldsymbol(\boldsymbolsf P\nabla\boldsymbol v+\nabla\boldsymbolT\boldsymbols\boldsymbolw\boldsymbolsf W.

Hence, in we pre-multipl\boldsymbol$\boldsymbolsf P$,

For a Kirchhoff plate $d\boldsymbol n=\boldsymbol\varphi$, hence

Thus, \boldsymbold\boldsymbolsf W=-\b\boldsymbol-\boldsymbolsf W\nabla\boldsymbolw\boldsymbolsf W^2.

References:

[1] P. Podio-Guidugli: Lezioni sulla teoria lineare dei gusci elastici sottili, Masson, 1991

[2] C. Davini: Lezioni di Teoria dei Gusci (lecture notes in Italian).

[3] P.M. Naghdi: The Theory of Shells and Plates, Handbuch der Physik VIa/2 p.425-640, Springer-Verlag, 1972

[4] P.G. Ciarlet: An introduction to differential geometry with applications to elasticity, Springer, 2005.