Axisymmetric shell element allowing asymmetric loading

Let $\left(S,\theta \right)$ be coordinate functions parametrizing the reference surface of the shell and let $S_3\in \left[-h/2,h/2\right]$ be the coordinate function in the thickness direction, where h is the shell's initial thickness. (For a detailed account of the geometric description of the finite-strain shell formulation, see Finite-strain shell element formulation.) Then points in the reference or undeformed configuration are identified by the normal coordinates mapping

where $\mathbf{X}$ is the three-dimensional position of a material point, $\overline{\mathbf{X}}$ is the shell reference surface mapping, and ${\mathbf{T}}_3$ is the unit normal to the shell reference surface. The fact that ${\mathbf{T}}_3$ is a unit vector assumes that the reference configuration is (locally) of constant thickness. Owing to the axisymmetric reference configuration, $\overline{\mathbf{X}}$ can be given relative to a global Cartesian coordinate system as

where $r\left(S\right)$ is the radius, $z\left(S\right)$ is the axial position, and $\left(r,z,\theta \right)$ are the cylindrical coordinates. (Note that the usual convention for cylindrical coordinates $\left(r,\theta ,z\right)$ has been changed, which is consistent with the axisymmetric shell elements and the axisymmetric elements allowing nonlinear bending.) By definition the normal field to the shell reference surface is ${\mathbf{T}}_3={\overline{\mathbf{X}}}_{,S}\times {\overline{\mathbf{X}}}_{,\theta }/\parallel {\overline{\mathbf{X}}}_{,S}\times {\overline{\mathbf{X}}}_{,\theta }\parallel$, which by direct computation yields

where $r^{\prime }=\partial r/\partial S$ and $z^{\prime }=\partial z/\partial S$. Relative to the cylindrical coordinate system, ${\mathbf{T}}_3={\left[{\left(r^{\prime }\right)}^2+{\left(z^{\prime }\right)}^2\right]}^{-\frac{1}{2}}\left(-z^{\prime }{\mathbf{e}}_r+r^{\prime }{\mathbf{e}}_z\right)$.

The basic kinematic assumption is that for any deformed configuration, the position of a point in the body can be identified by

where $\mathbf{x}$ is the deformed position of the material point, $\overline{\mathbf{x}}$ is the deformed shell reference surface mapping, ${\mathbf{t}}_3$ is the deformed unit director field, and ${\overline{f}}_{33}$ is the thickness change parameter. Of critical importance for any shell formulation is the treatment of the rotation field; that is, the treatment of the director field ${\mathbf{t}}_3$. The geometric description and the incremental update procedure for the director field are given in detail below.

Under the kinematic assumption above, the deformed configuration of the shell is completely determined by the reference surface mapping $\overline{\mathbf{x}}$, the deformed director field ${\mathbf{t}}_3$ and the thickness parameter ${\overline{f}}_{33}$.

We define the following displacement quantities. Since $\overline{\mathbf{x}}$ is an element of a (linear) vector space, we can define the reference surface displacement vector $\mathbf{u}$ by the difference between the deformed reference surface and the undeformed reference surface; i.e.,

The director field, however, is a unit vector field that is not a member of a linear vector space. The orientation of the director field is defined in terms of a rotation vector $\mathit{\varphi }$ as

Here $\widehat{\mathit{\varphi }}$ is the skew-symmetric matrix with axial vector $\mathit{\varphi }$, defined by the properties

and $\exp \left[\widehat{\mathit{\varphi }}\right]$ is an orthogonal transformation given by the closed-form expression

Alternatively, quaternion algebra can be used to specify the orientation of the deformed director field ${\mathbf{t}}_3$. In this case the orthogonal matrix $\exp \left[\widehat{\mathit{\varphi }}\right]$ is replaced by the quaternion parameter $q=\left(q_0,\mathbf{q}\right)$, where

The orientation of the unit director field then follows as

Similarly, the orthogonal transformation $\exp \left[\widehat{\mathit{\varphi }}\right]$ can be extracted from the quaternion parameter as

Displacement and rotation components are given relative to the cylindrical coordinate system $\left(r,z,\theta \right)$ with orthonormal basis vectors $\left\{{\mathbf{e}}_r,{\mathbf{e}}_z,{\mathbf{e}}_{\theta }\right\}$ that are fixed in the reference or undeformed configuration. A general interpolation scheme for $\mathbf{u}=u_r{\mathbf{e}}_r+u_z{\mathbf{e}}_z+u_{\theta }{\mathbf{e}}_{\theta }$ and $\mathit{\varphi }={\varphi }_r{\mathbf{e}}_r+{\varphi }_z{\mathbf{e}}_z+{\varphi }_{\theta }{\mathbf{e}}_{\theta }$ using a Fourier expansion in the $\theta$ variable is

Here $H^m\left(S\right)$ are the polynomial interpolation functions along the generator lines of the axisymmetric reference configuration; $u_i^{m0}$, $u_{ic}^{mp}$, $u_{is}^{mp}$, ${\varphi }_i^{m0}$, ${\varphi }_{ic}^{mp}$, ${\varphi }_{is}^{mp}$ are the solution amplitude values (Fourier coefficients); M is the number of terms used in the interpolation along the generator lines; and P is the number of Fourier interpolation terms used around the circumference of the reference shell. Note that an axisymmetric deformation is obtained for the choice $P=0$.

The symmetry requirement in the r–z plane at $\theta =2n\pi$, $n=0,1,\mathrm{\dots }$, eliminates many of the above Fourier coefficients. For the displacement vector the only admissible terms are

For the rotation components, symmetry requirements switch the role of the r and z components with the $\theta$ components:

For practical reasons the values of $u_r$, $u_z$, and ${\varphi }_{\theta }$ are often required at specific locations around the circumference of the shell. Therefore, displacement and rotation components $u_r^{mp}$, $u_z^{mp}$, and ${\varphi }_{\theta }^{mp}$ are used instead of the Fourier coefficients $u_{rc}^{mp}$, $u_{zc}^{mp}$, and ${\varphi }_{\theta c}^{mp}$. Furthermore, a negative sign is introduced in the interpolation for ${\varphi }_{\theta }$ for the following reason: The Abaqus convention for axisymmetric shell elements is that the axial tangent direction is drawn between nodes in ascending node number (the shell local 1-direction). The normal to the shell is then obtained by a 90° counter-clockwise rotation of the tangent (the shell local 3-direction). However, a positive rotation of the normal field (about the shell local 2-direction) is counterclockwise. This convention implies a left-handed shell local coordinate system. For the axisymmetric shell bending elements, a right-handed shell local coordinate system is required at the integration points; thus, the direction of positive rotation is reversed there.

Rearranging the Fourier series expansions and making the substitution $u_{\theta s}^{mp}\mapsto u_{\theta }^{mp}$, the interpolations for the displacement components are

Similarly, replacing ${\varphi }_{rs}^{mp}$ and ${\varphi }_{zs}^{mp}$ with ${\varphi }_r^{mp}$ and ${\varphi }_z^{mp}$ respectively, the interpolation for the rotation components becomes

In the above interpolations, $u_r^{mp}$, $u_z^{mp}$, and ${\varphi }_{\theta }^{mp}$ are physical displacement and rotation components at ${\theta }^p=\pi \left(p-1\right)/P$ and $R^p\left(\theta \right)$ are trigonometric interpolation functions with the property that $R^p\left({\theta }^q\right)=\{\begin{array}{c}\hfill 1, p=q\hfill \\ \hfill 0, p\ne q\hfill \end{array}$ defined by:

$P=1$: ${\theta }^1=0$, ${\theta }^2=\pi$,

$P=2$: ${\theta }^1=0$, ${\theta }^2=\pi /2$, ${\theta }^3=\pi$,

$P=3$: ${\theta }^1=0$, ${\theta }^2=\pi /3$, ${\theta }^3=2\pi /3$, ${\theta }^4=\pi$,

$P=4$: ${\theta }^1=0$, ${\theta }^2=\pi /4$, ${\theta }^3=\pi /2$, ${\theta }^4=3\pi /4$, ${\theta }^5=\pi$,

As with the continuum axisymmetric bending element, $P=4$ is the highest-order interpolation offered with respect to $\theta$. The element becomes significantly more expensive as higher-order interpolations are used, and it is assumed that the general-purpose finite-strain shell is less expensive than using this element with $P>4$.

The virtual work expression from the three-dimensional theory is

where V is the current volume of the deformed body, ${\sigma }^{ij}$ are the curvilinear components of the Cauchy stress tensor, $E_{ij}$ are the components of the Lagrange strain tensor, and $\delta E_{ij}$ are the variational or linearized strain measure components. By definition the Lagrange strain tensor components are given by

Note that in the statement of virtual work, no choice has thus far been made regarding the curvilinear coordinate functions ${\xi }^i$$\left(i=1,2,3\right)$. Furthermore, the current volume measure $dV$ is given by the parametric relationship

We now introduce the kinematic assumption Equation 1 into the definition of $E_{ij}$ to find

where differentiation is now with respect to the parametric coordinates, so that $\alpha ,\beta \in \left\{S,\theta \right\}$ and $3=S_3$. Define the following shell strain measure components and kinematic relationships:

In the above, $b_{\alpha \beta }^0\stackrel{\mathrm{def}}{=}{\overline{\mathbf{X}}}_{,\alpha }\cdot {\mathbf{T}}_{3,\beta }=b_{\beta \alpha }^0$ are the components of the second fundamental form of the undeformed reference surface.

Substituting the above definitions into the virtual work expression, we find (after some manipulation) that the volume integral reduces to the following integral over the deformed reference surface

where $\delta \mu =\delta {\overline{f}}_{33}/{\overline{f}}_{33}$ and the current reference surface Jacobian determinant is $\overline{j}=\parallel {\overline{\mathbf{x}}}_{,1}\times {\overline{\mathbf{x}}}_{,2}\parallel$. In the above virtual work expression, the ${\left(S_3\right)}^2$ term in $E_{\alpha \beta }$ has been neglected. This term is $O\left(h^2/R^2\right)$—where h is the thickness and R is some characteristic radius of curvature—and is negligible in light of the kinematic assumption Equation 1. The shell stress resultant components are defined by the following integrals through the thickness of the shell:

For thin shells the Kirchhoff-Love approximation, which states that the deformed director field ${\mathbf{t}}_3$ is (approximately) the normal field to the deformed reference surface, is introduced along with the plane stress assumption ${\sigma }^{33}=0$. Consistent with these approximations, we neglect all $O\left(h/R\right)$ terms and terms proportional to the gradient of the thickness parameter. Accordingly, we set

We can now summarize the virtual work expression for thin (Kirchhoff-Love) shells:

where the shell resultant components are defined in terms of the Cauchy stress tensor components by the integrals

In the expression for $\delta \mathrm{\Pi }$, $Q^{\alpha }={\left({\overline{f}}_{33}\right)}^2{\int }_h{\sigma }^{3\alpha }d{S}_3$ is interpreted as a constraint stress that enforces that the director field remain normal to the reference surface. Two other contributions to the virtual work expression $M^{\alpha \beta }{\kappa }_{\alpha \beta }\delta \mu$ and $M^{3\alpha }{\left(\delta \mu \right)}_{,\alpha }$, where $M^{3\alpha }={\left({\overline{f}}_{33}\right)}^2{\int }_h{S}_3{\sigma }^{3\alpha }d{S}_3$ are $O\left(h/R\right)$ and, thus, neglected.

It is desirable to define stress resultant quantities relative to an orthonormal basis in the deformed configuration. To do this, we define a normal coordinate system $\left\{{\mathbf{n}}_1,{\mathbf{n}}_2,{\mathbf{n}}_3\right\}$, where ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$ are tangent to the deformed reference surface and ${\mathbf{n}}_3$ is the unit normal field.

Define the following notation. Let ${\overline{f}}_{\alpha }^a$$\left(a=1,2\right)$ be the components of ${\overline{\mathbf{x}}}_{,\alpha }$ relative to the basis $\left\{{\mathbf{n}}_1,{\mathbf{n}}_2\right\}$; that is

Furthermore, let the inverse of the matrix of components $\left[{\overline{f}}_{\alpha }^a\right]$ be given by $\left[{\overline{h}}_a^{\alpha }\right]$, such that

Note that the basis vectors ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$ induce distance measuring coordinates $s^1$ and $s^2$ such that

It follows from Equation 5 that the orthonormal tangent vectors are given by

For the material calculations, it is important to express both the strain and stress quantities relative to the local orthonormal frame $\left\{{\mathbf{n}}_1,{\mathbf{n}}_2\right\}$. Accordingly, let $N^{ab}=N^{ba}$ and $M^{ab}=M^{ba}$ be the membrane and bending stress resultant components relative to this local orthonormal basis. Thus, we can write

where we recall that ${\overline{h}}_a^{\alpha }\stackrel{\mathrm{def}}{=}\partial {\xi }^{\alpha }/\partial s^a$. Then the stress resultant contributions to the virtual work expression can be transformed as follows. First, the membrane contribution:

Recall, however, that by the definition of the coordinates $s^a$, $\partial \overline{\mathbf{x}}/\partial s^a={\mathbf{n}}_a$. Thus,

Similarly, the bending contribution is:

Let $e_a^c$ be the two-dimensional alternator, such that $e_1^1=e_2^2=0$ and $e_1^2=-e_2^1=1$. Then ${\mathbf{t}}_3\times {\mathbf{n}}_a=e_a^c{\mathbf{n}}_c$ and

Since the second term in the brackets is proportional to the bending curvature, we neglect this term relative to the first, yielding

We now write the virtual work expression Equation 4 in matrix operator notation. Define the following stress resultant component vectors:

Define the matrix strain displacement operators as follows:

where ${\mathbf{0}}_{1\times 3}$ is the column of zeros ${\left\{\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right\}}^T$. Then the virtual work expression Equation 4 is equivalently stated

Thus far ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$ are any two orthonormal vectors in the tangent plane to the reference surface. We can uniquely choose these two vectors by requiring the matrix components of the incremental reference surface deformation gradient $\left[\mathrm{\Delta }{\overline{f}}_b^a\right]$, where

to be symmetric; i.e., $\mathrm{\Delta }{\overline{f}}_2^1=\mathrm{\Delta }{\overline{f}}_1^2$. Note that by definition ${\overline{f}}_{\alpha }^{\left(t+\mathrm{\Delta }t\right)a}=\mathrm{\Delta }{\overline{f}}_b^a{\overline{f}}_{\alpha }^{\left(t\right)b}$. This symmetry condition defines a corotational orthonormal basis in the deformed configuration. This orthonormal basis is calculated as follows.

Obtain the Abaqus convention pair of orthonormal surface vectors ${\widehat{\mathbf{t}}}_a$. We then rotate these vectors in the tangent plane to the reference surface about the normal vector ${\mathbf{n}}_3^{t+\mathrm{\Delta }t}$ by the angle $\mathrm{\Delta }\psi$, where $\mathrm{\Delta }\psi$ is determined by the symmetry condition $\mathrm{\Delta }{\overline{f}}_2^1=\mathrm{\Delta }{\overline{f}}_1^2$. Thus, we define

It follows by definition that

Therefore, define the quantity

The symmetry condition requires that

From this it can be determined that

and $\left\{{\mathbf{n}}_1^{t+\mathrm{\Delta }t},{\mathbf{n}}_2^{t+\mathrm{\Delta }t}\right\}$ are then determined by Equation 6.

Having determined the updated vectors ${\mathbf{n}}_a^{t+\mathrm{\Delta }t}$, we can calculate the quantities required for coordinate transformation. First the incremental deformation gradient components and the Jacobian matrix components:

Note that due to the symmetry of the incremental deformation gradient components, there is no ambiguity in writing $\mathrm{\Delta }{\overline{f}}_b^a=\mathrm{\Delta }{\overline{f}}_{ab}$. We can now calculate the inverse components

The iterative solution procedure requires the calculation of the consistent tangent stiffness. This stiffness has two parts, one resulting from the material model and the other resulting from the changing geometry. We denote the second variational quantities with $\mathrm{d}$. Returning to Equation 4, the virtual work expression can be written as

where $J=j/j^0$, $j^0$ is the Jacobian determinant of the reference configuration parametrization given by $j^0=det\left[\nabla \mathbf{X}\right]$, and components are relative to the corotated frame with $a,b\in \left\{1,2\right\}$. We assume constitutive behavior such that

Thus, the variation of the virtual work expression yields