Membrane elements

Membrane elements are sheets in space that can carry membrane force but do not have any bending or transverse shear stiffness, so the only nonzero stress components in the membrane are those components parallel to the middle surface of the membrane: the membrane is in a state of plane stress.

The following topics are discussed:

Equilibrium
Jacobian
Thickness change
Total deformation

Equilibrium

The virtual work contribution from the internal forces in a membrane element is

\begin{matrix} (1) & δ W^{I} = \int_{V} σ : δ ε d V, \end{matrix}

where $σ$ is the Cauchy stress, $δ ε = sym (δ L)$ is the virtual rate of deformation ( $δ L = \partial δ u / \partial x$ , where $δ u$ is the virtual velocity field), and V is the current volume of the membrane.

We assume that only the membrane stress components in the surface of the membrane are nonzero: $σ_{3 i} = 0$ . Then Equation 1 simplifies to

\begin{matrix} δ W^{I} & = \int_{V} σ_{α β} δ ε_{α β} d V \\ = \int_{V} σ_{α β} δ L_{α β} d V (since σ is symmetric), \end{matrix}

where

δ L_{α β} = e_{α} \frac{\partial δ u}{\partial x} \cdot e_{β} = e_{α} \frac{\partial δ u}{\partial x_{β}}

and $d V = t d A$ , where t is the current thickness of the element and A is its current area.

Jacobian

The consistent Jacobian contribution from the element is

d δ W^{I} = \int_{A} t (δ ε : D : d ε + σ : (δ L^{T} \cdot d L - 2 δ ε \cdot d ε)) d A .

Since we assume that $σ_{3 i} = 0$ , the first term in the integrand is

δ ε_{α β} D_{α β γ δ} d ε_{γ δ} .

We also assume that there is no transverse shear strain of the element: $ε_{3 i} = 0$ , and, hence, $δ ε_{α i} d ε_{i β} = δ ε_{α γ} d ε_{γ β}$ . Thus, the second term in the integrand is

σ_{α β} (\frac{\partial u}{\partial x_{α}} \cdot \frac{\partial u}{\partial x_{β}} - \frac{1}{2} (e_{γ} \cdot \frac{\partial u}{\partial x_{α}} + e_{α} \cdot \frac{\partial u}{\partial x_{γ}}) (e_{γ} \cdot \frac{\partial u}{\partial x_{β}} + e_{β} \cdot \frac{\partial u}{\partial x_{γ}})) .

We can write this out as

\begin{matrix} σ_{11} \frac{\partial δ u}{\partial x_{1}} \cdot \frac{\partial d u}{\partial x_{1}} + σ_{22} \frac{\partial δ u}{\partial x_{2}} \cdot \frac{\partial d u}{\partial x_{2}} + σ_{12} (\frac{\partial δ u}{\partial x_{1}} \cdot \frac{\partial d u}{\partial x_{2}} + \frac{\partial δ u}{\partial x_{2}} \cdot \frac{\partial d u}{\partial x_{1}}) \\ - 2 σ_{11} e_{1} \cdot \frac{\partial δ u}{\partial x_{1}} e_{1} \cdot \frac{\partial d u}{\partial x_{1}} - 2 σ_{22} e_{2} \cdot \frac{\partial δ u}{\partial x_{2}} e_{2} \cdot \frac{\partial d u}{\partial x_{2}} \\ - \frac{1}{2} (σ_{11} + σ_{22}) (e_{2} \cdot \frac{\partial δ u}{\partial x_{1}} e_{2} \cdot \frac{\partial d u}{\partial x_{1}} + e_{1} \cdot \frac{\partial δ u}{\partial x_{2}} e_{1} \cdot \frac{\partial d u}{\partial x_{2}} + e_{1} \cdot \frac{\partial δ u}{\partial x_{2}} e_{2} \cdot \frac{\partial d u}{\partial x_{1}} + e_{2} \cdot \frac{\partial δ u}{\partial x_{1}} e_{1} \cdot \frac{\partial d u}{\partial x_{2}}) \\ - σ_{12} (e_{1} \cdot \frac{\partial δ u}{\partial x_{1}} e_{1} \cdot \frac{\partial d u}{\partial x_{1}} + e_{2} \cdot \frac{\partial δ u}{\partial x_{1}} e_{2} \cdot \frac{\partial d u}{\partial x_{2}} + e_{1} \cdot \frac{\partial δ u}{\partial x_{2}} e_{1} \cdot \frac{\partial d u}{\partial x_{1}} + e_{2} \cdot \frac{\partial δ u}{\partial x_{2}} e_{2} \cdot \frac{\partial d u}{\partial x_{1}} \\ + e_{1} \cdot \frac{\partial δ u}{\partial x_{2}} e_{2} \cdot \frac{\partial d u}{\partial x_{2}} + e_{2} \cdot \frac{\partial δ u}{\partial x_{1}} e_{1} \cdot \frac{\partial d u}{\partial x_{1}} + e_{1} \cdot \frac{\partial δ u}{\partial x_{1}} e_{2} \cdot \frac{\partial d u}{\partial x_{1}} + e_{2} \cdot \frac{\partial δ u}{\partial x_{2}} e_{1} \cdot \frac{\partial d u}{\partial x_{2}}) \end{matrix}

Thickness change

In geometrically nonlinear analyses the cross-section thickness changes as a function of the membrane strain with a user-defined “effective section Poisson's ratio,” $ν$ .

In plane stress $σ_{33} = 0$ ; linear elasticity gives

ϵ_{33} = - \frac{ν}{1 - ν} (ϵ_{11} + ϵ_{22}) .

Treating these as logarithmic strains,

\ln (\frac{t}{t_{0}}) = - \frac{ν}{1 - ν} (\ln (\frac{l_{1}}{l_{1}^{0}}) + \ln (\frac{l_{2}}{l_{2}^{0}})) = - \frac{ν}{1 - ν} \ln (\frac{A}{A^{0}}),

where A is the area on the membrane's reference surface. This nonlinear analogy with linear elasticity leads to the thickness change relationship:

\frac{t}{t_{0}} = {(\frac{A}{A_{0}})}^{- \frac{ν}{1 - ν}} .

For $ν = 0.5$ the material is incompressible; for $ν = 0$ the section thickness does not change.

Total deformation

The deformation gradient is $F = \partial x / \partial X$ . Since we take $e_{3}$ normal to the current membrane surface and assume no transverse shear of the membrane,

F_{3 α} = e_{3} \cdot \frac{\partial x}{\partial X_{α}} = 0 and F_{α 3} = e_{α} \cdot \frac{\partial x}{\partial X_{3}} = 0.

By the thickness change assumption above, the direct out-of-plane component of the deformation gradient is

F_{33} = \frac{1}{{(F_{11} F_{22} - F_{12} F_{21})}^{\frac{ν}{1 - ν}}} .

To calculate the deformation gradient at the end of the increment, first we calculate the two tangent vectors at the end of the increment defined by the derivative of the position with respect to the reference coordinates:

\frac{\partial x^{n + 1}}{\partial X_{β}} = \frac{\partial x^{n + 1}}{\partial ξ_{α}} \frac{\partial ξ_{α}}{\partial X_{β}},

where $\partial x^{n + 1} / \partial ξ_{α}$ is obtained by interpolation with the shape function derivatives from the nodal coordinates and the change of coordinate transformation $\partial ξ_{α} / \partial X_{β}$ is based on the reference geometry. The deformation gradient components are defined

F_{α β} = e_{α}^{n + 1} \cdot \frac{\partial x^{n + 1}}{\partial X_{β}} .

To choose the element basis directions $e_{α}^{n + 1}$ , we do the following. Find any pair of in-plane orthonormal vectors ${\hat{e}}_{α}^{n + 1}$ (by the standard Abaqus projection). Then find the angle $Δ ψ$ such that the element basis vectors $e_{α}^{n + 1}$ , defined

\begin{matrix} e_{1}^{n + 1} & = \cos (Δ ψ) {\hat{e}}_{1}^{n + 1} + \sin (Δ ψ) {\hat{e}}_{2}^{n + 1} \\ e_{2}^{n + 1} & = - \sin (Δ ψ) {\hat{e}}_{1}^{n + 1} + \cos (Δ ψ) {\hat{e}}_{2}^{n + 1}, \end{matrix}

satisfy the symmetry condition

F_{α β} = e_{α}^{n + 1} \cdot \frac{\partial x^{n + 1}}{\partial X_{β}} = F_{β α} .

Using the definitions of $e_{α}^{n + 1}$ in terms of ${\hat{e}}_{α}^{n + 1}$ in the above equation, the rotation angle $Δ ψ$ is found to be

Δ ψ = \tan^{- 1} [\frac{{\hat{F}}_{21} - {\hat{F}}_{12}}{{\hat{F}}_{11} + {\hat{F}}_{22}}],

where

{\hat{F}}_{α β} \overset{def}{=} {\hat{e}}_{α}^{n + 1} \cdot \frac{\partial x^{n + 1}}{\partial X_{β}} .

The deformation gradient then follows immediately.

For elastomers we work directly in terms of $B_{α β} = F_{α γ} F_{β γ}$ and $B_{33} = {(F_{33})}^{2}$ . For inelastic material models we need measures of incremental strain and average material rotation, which we compute from $Δ F$ defined by $F = Δ F \cdot F^{n}$ , where $F^{n}$ is the deformation gradient at the start of the current increment (at increment “n”):

{(F^{n})}_{α β} = {(e^{n})}_{α} \cdot \frac{\partial x^{n}}{\partial X_{β}} .

We can define the components of $Δ F^{- 1}$ by

{(Δ F^{- 1})}_{α β} = {(e^{n})}_{α} \cdot \frac{\partial x^{n}}{\partial x_{β}}

and, hence, define $Δ F_{α β}$ by inversion.

The incremental strain and rotation are then defined from the polar decomposition $Δ F = Δ V \cdot Δ R$ , where $Δ R$ is a rotation matrix and $Δ V$ is a pure stretch:

Δ V = \overset{3}{\sum_{I = 1}} Δ λ_{I} a^{I} a^{I}

(see Deformation). We find the $Δ λ_{I}$ and the corresponding eigenvectors $a_{I}$ by solving the eigenproblem for

Δ F \cdot Δ F^{T} = Δ V \cdot Δ V = \sum_{I = 1}^{3} {(Δ λ_{I})}^{2} a^{I} a^{I} .

Since we assume no transverse shear in the membrane, the normal direction (along $e_{3}$ ) is always a principal direction, so the eigenproblem is the $2 \times 2$ problem

Δ F_{α γ} F_{β γ} = \overset{2}{\sum_{I = 1}} {(Δ λ)}^{2} a_{α}^{I} a_{β}^{I} .

The logarithmic strain increment is then

Δ ε_{α β} = \overset{2}{\sum_{I = 1}} \ln (Δ λ_{I}) a_{α}^{I} a_{β}^{I},

and the average material rotation increment is defined from the polar decomposition of the increment:

Δ R_{α β} = \overset{2}{\sum_{I = 1}} \frac{1}{Δ λ_{I}} a_{α}^{I} a_{γ}^{I} Δ F_{γ β} .

Due to the choice of the element basis directions, we can assume that

Δ R_{α β} \approx δ_{α β} .