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and the approximation to the curvature
B _ {\alpha \beta} = - \frac {1}{2} \left(\frac {\partial \mathbf {N}}{\partial S ^ {\alpha}} \cdot \frac {\partial \mathbf {X}}{\partial S ^ {\beta}} + \frac {\partial \mathbf {N}}{\partial S ^ {\beta}} \cdot \frac {\partial \mathbf {X}}{\partial S ^ {\alpha}}\right).
The vectors \partial \mathbf { N } / \partial S ^ { \alpha } are defined from the derivatives of the interpolation functions and the "normals" at the nodes. These nodal normals are calculated as average values of the normals to the surfaces of all elements abutting the node. ABAQUS determines if the surface is intended to be smooth at the node (the criterion is that the angle between the normals at the node should be less than 20°). If the surface is not calculated as smooth, separate normals are set up in the different surface branches at the node. Thus, B _ { \alpha \beta } should be a reasonable approximation to the second fundamental form of the original reference surface.
Displacements
The nodal variables for shell elements are the displacements of the shell's reference surface, \mathbf { u } = \mathbf { x } - \mathbf { X } , and the normal direction, n. Since n is defined to be a unit vector, only two independent values are needed to define n, so that this type of shell element needs only five degrees of freedom per node. In ABAQUS this issue is addressed in two ways. At nodes in a smooth shell surface in those elements that naturally have five degrees of freedom per node, ABAQUS stores the values of the projections of the change in n projected onto two orthogonal directions in the shell surface at the start of the increment to define n. Otherwise, ABAQUS stores the usual rotation triplet, !, at the node. This latter method leaves a redundant degree of freedom if the node is on a smooth surface. A small stiffness is introduced locally at the node to constrain this extra degree of freedom to a measure of the same rotation of the shell's reference surface.
Interpolation
The same bipolynomial interpolation functions are used for all components of u, X, N, and n. The shear flexible shell elements in the library use bilinear interpolation (four nodes), fully biquadratic interpolation (nine nodes), and "serendipity" quadratics (eight nodes).
Strains
The reference surface membrane strains are
\epsilon_ {\alpha \beta} = \frac {1}{2} (g _ {\alpha \beta} - G _ {\alpha \beta}).
The curvature change is
\kappa_ {\alpha \beta} = B _ {\alpha \beta} - b _ {\alpha \beta} + \frac {1}{2} (B _ {\alpha} ^ {\gamma} \epsilon_ {\gamma \beta} + B _ {\beta} ^ {\gamma} \epsilon_ {\gamma \alpha}).
The transverse shears are
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\gamma_ {3 \alpha} = \mathbf {n} \cdot \mathbf {t} _ {\alpha},
where
\mathbf {t} _ {\alpha} = \frac {\partial \mathbf {x}}{\partial S ^ {\alpha}} \left/ \sqrt {\frac {\partial \mathbf {x}}{\partial S ^ {\alpha}} \cdot \frac {\partial \mathbf {x}}{\partial S ^ {\alpha}}} \left. \right. \mathrm{(nosumon} \alpha \mathrm{)}
is a unit vector, tangent to the d S ^ { \alpha } line in the current surface.
In addition to these strains, when six degrees of freedom are used at the nodes of the elements, the extra rotation degree of freedom is constrained with a penalty, as follows.
When such a node is the corner node of an element, define \mathbf { T } _ { 1 } , \mathbf { T } _ { 2 } , \mathbf { N } _ { 1 } , d S ^ { 1 } , and d S ^ { 2 } in the element as above. Notice that these will be different in each element at the node, since the interpolated surface is not generally continuous. Then the strain to be penalized is defined as
\gamma_ {S R C} = \frac {1}{2} (\mathbf {t} _ {1} \cdot \overline {{\mathbf {t}}} _ {2} - \mathbf {t} _ {2} \cdot \overline {{\mathbf {t}}} _ {1}),
where
\overline {{\mathbf {t}}} _ {\alpha} = \mathbf {C} \cdot \mathbf {T} _ {\alpha}
is the rotated tangent direction, as defined by the rotation values at the node, and
\mathbf {t} _ {\alpha} = \frac {\partial \mathbf {x}}{\partial S ^ {\alpha}} \left/ \sqrt {\frac {\partial \mathbf {x}}{\partial S ^ {\alpha}} \cdot \frac {\partial \mathbf {x}}{\partial S ^ {\alpha}}} \left. \right. \mathrm{(nosumon} \alpha \mathrm{)}
is the rotated tangent direction defined by the motion of the interpolated reference surface at the node.
At each midside node in the original configuration, define N as the average surface normal for the elements of this surface branch at the nodes (there will be at most two such elements) and T as the tangent to the edge. Then define
\overline {{\mathbf {t}}} = \mathbf {C} \cdot \mathbf {T} \quad \mathrm{and} \quad \mathbf {n} = \mathbf {C} \cdot \mathbf {N}
as rotated values of T and N, as defined by the rotation values at the node. The vector
\mathbf {p} = \overline {{\mathbf {t}}} \times \mathbf {n}
is then normal to n and to the edge.
The strain to be penalized at these midside nodes is then defined as
\gamma_ {S R M} = \mathbf {t} \cdot \mathbf {p},
where
\mathbf {t} = \frac {\partial \mathbf {x}}{\partial S} \left/ \sqrt {\frac {\partial \mathbf {x}}{\partial S} \cdot \frac {\partial \mathbf {x}}{\partial S}} \left. \right.
is the tangent to the edge of the element in the current position of the reference surface.
Penalties
The transverse shear strains are calculated at a set of reduced integration points and have the following stiffness associated with them:
K _ {3 \alpha} = G _ {3 \alpha} h \Delta (\mathrm{area}) / (1 + q \Delta (\mathrm{area}) / h ^ {2}),
where the G _ { 3 \alpha } are the elastic moduli associated with transverse shear. The G _ { 3 \alpha } are defined directly by the user or are computed from the elastic moduli given for the layers of the shell. h is the shell thickness; \Delta ( \mathrm { a r e a } ) is the value of reference surface area associated with this integration point in the numerical integration scheme; q is a numerical factor, currently set to 1 / 4 \times 1 0 ^ { - 4 } . (See Hughes et al., 1977, for a discussion of such factors.) Transverse shears are always treated elastically: nonlinear material calculations in shells are based on plane stress theory, using the membrane and bending strains to define the strain on the surface parallel to the shell's reference surface at each integration point through the shell's thickness.
When rotation constraints are required at nodes that use six degrees of freedom, the penalty used is
K = k G h \Delta (\mathrm{area}) / (1 + q \Delta (\mathrm{area}) / h ^ {2}).
This is the same as the transverse shear constraint, except that \Delta ( \mathrm { a r e a } ) is here an area "assigned" to the node and the factor k is introduced. This (small) factor has been chosen based on numerical experiments, to be large enough to avoid singularities yet small enough to avoid adding significantly to the stiffness of the model.
These strain measures, with the interpolation specified above, give zero strain for any general rigid body motion
\mathbf {x} = \mathbf {X} + \mathbf {u} _ {1} + \left(\mathbf {C} _ {1} - \mathbf {I}\right) \cdot \left(\mathbf {X} - \mathbf {X} _ {1}\right)
\mathbf {n} = \mathbf {C} _ {1} \cdot \mathbf {N},
where \mathbf { u } _ { 1 } , \mathbf { C } _ { 1 } , and { \bf X } _ { 1 } are constant.
First variations of strain
The first variations of the strains are
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\delta \epsilon_ {\alpha \beta} = \frac {1}{2} \delta g _ {\alpha \beta}
\delta \kappa_ {\alpha \beta} = - \delta b _ {\alpha \beta} + \frac {1}{2} (B _ {\alpha} ^ {\gamma} \delta \epsilon_ {\gamma \beta} + B _ {\beta} ^ {\gamma} \delta \epsilon_ {\gamma \alpha})
\delta \gamma_ {3 \alpha} = \delta \mathbf {n} \cdot \mathbf {t} _ {\alpha} + \mathbf {n} \cdot \delta \mathbf {t} _ {\alpha}
\delta \gamma_ {S R C} = \frac {1}{2} \big (\delta \mathbf {t} _ {1} \cdot \overline {{\mathbf {t}}} _ {2} - \delta \mathbf {t} _ {2} \cdot \overline {{\mathbf {t}}} _ {1} + \mathbf {t} _ {1} \cdot \delta \overline {{\mathbf {t}}} _ {2} - \mathbf {t} _ {2} \cdot \delta \overline {{\mathbf {t}}} _ {1} \big)
\delta \gamma_ {S R M} = \delta \overline {{\mathbf {t}}} _ {1} \cdot \mathbf {p} + \delta \mathbf {p} \cdot \overline {{\mathbf {t}}},
where
\delta g _ {\alpha \beta} = \frac {\partial \delta \mathbf {u}}{\partial S ^ {\alpha}} \cdot \frac {\partial \mathbf {x}}{\partial S ^ {\beta}} + \frac {\partial \mathbf {x}}{\partial S ^ {\alpha}} \cdot \frac {\partial \delta \mathbf {u}}{\partial S ^ {\beta}}
\delta b _ {\alpha \beta} = - \frac {1}{2} \left(\frac {\partial \delta \mathbf {n}}{\partial S ^ {\alpha}} \cdot \frac {\partial \mathbf {x}}{\partial S ^ {\beta}} + \frac {\partial \delta \mathbf {n}}{\partial S ^ {\beta}} \cdot \frac {\partial \mathbf {x}}{\partial S ^ {\alpha}} + \frac {\partial \mathbf {n}}{S ^ {\alpha}} \frac {\partial \delta \mathbf {u}}{\partial S ^ {\beta}} + \frac {\partial \mathbf {n}}{\partial S ^ {\beta}} \cdot \frac {\partial \delta \mathbf {u}}{\partial S ^ {\alpha}}\right)
\delta \mathbf {n} = \delta \boldsymbol {\omega} \times \mathbf {n}
\delta \overline {{\mathbf {t}}} _ {\alpha} = \delta \boldsymbol {\omega} \times \overline {{\mathbf {t}}} _ {\alpha}
\delta \mathbf {t} _ {\alpha} = \left(\frac {\partial \mathbf {x}}{\partial S ^ {\alpha}} \cdot \frac {\partial \mathbf {x}}{\partial S ^ {\alpha}}\right) ^ {- \frac {1}{2}} \left[ \mathbf {I} - \mathbf {t} _ {\alpha} \mathbf {t} _ {\alpha} \right] \cdot \frac {\partial \delta \mathbf {u}}{\partial S ^ {\alpha}} \quad \mathrm{(nosumon} \alpha \mathrm{)}
and at the midside nodes
\delta \mathbf {p} = \delta \boldsymbol {\omega} \times \mathbf {p}.
Second variations of strains
In forming the initial stress matrix we approximate by neglecting d±n, d±t, etc, to simplify the expressions and reduce the cost of forming the matrix. Numerical experiments have suggested that, at least for the problems tested, this does not significantly affect the convergence rate. With this approximation,
d \delta \epsilon_ {\alpha \beta} = \frac {1}{2} \left(\frac {\partial \delta \mathbf {u}}{\partial S ^ {\alpha}} \cdot \frac {\partial d \mathbf {u}}{\partial S ^ {\beta}} + \frac {\partial \delta \mathbf {u}}{\partial S ^ {\beta}} \cdot \frac {\partial d \mathbf {u}}{\partial S ^ {\alpha}}\right)
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\begin{array}{l} d \delta \kappa_ {\alpha \beta} = \frac {1}{2} \left(\frac {\partial \pmb {\omega}}{\partial S ^ {\alpha}} \times \mathbf {n} \cdot \frac {\partial d \mathbf {u}}{\partial S ^ {\beta}} + \delta \pmb {\omega} \times \frac {\partial \mathbf {n}}{\partial S ^ {\alpha}} \cdot \frac {\partial d \mathbf {u}}{\partial S ^ {\beta}} \right. \\ + \frac {\partial \pmb {\omega}}{\partial S ^ {\beta}} \times \mathbf {n} \cdot \frac {\partial d \mathbf {u}}{\partial S ^ {\alpha}} + \delta \pmb {\omega} \times \frac {\partial \mathbf {n}}{\partial S ^ {\beta}} \cdot \frac {\partial d \mathbf {u}}{\partial S ^ {\alpha}} \\ + \frac {\partial \pmb {\omega}}{\partial S ^ {\alpha}} \times \mathbf {n} \cdot \frac {\partial \delta \mathbf {u}}{\partial S ^ {\beta}} + d \pmb {\omega} \times \frac {\partial \mathbf {n}}{\partial S ^ {\alpha}} \cdot \frac {\partial \delta \mathbf {u}}{\partial S ^ {\beta}} \\ + \left(\frac {\partial \pmb {\omega}}{\partial S ^ {\beta}} \times \mathbf {n} \cdot \frac {\partial \delta \mathbf {u}}{\partial S ^ {\alpha}} + d \pmb {\omega} \times \frac {\partial \mathbf {n}}{\partial S ^ {\beta}} \cdot \frac {\partial \mathbf {u}}{\partial S ^ {\alpha}}\right) \\ + \frac {1}{2} (B _ {\alpha} ^ {\gamma} d \delta \epsilon_ {\gamma \beta} + B _ {\beta} ^ {\gamma} d \delta \epsilon_ {\gamma \alpha}) \\ \end{array}
d \gamma_ {3 \alpha} = g _ {\alpha} ^ {- \frac {1}{2}} \delta \pmb {\omega} \times \mathbf {n} \cdot (\mathbf {I} - \mathbf {t} _ {\alpha} \mathbf {t} _ {\alpha}) \cdot \frac {\partial d \mathbf {u}}{\partial S ^ {\alpha}}
+ g _ {\alpha} ^ {- \frac {1}{2}} d \pmb {\omega} \times \mathbf {n} \cdot (\mathbf {I} - \mathbf {t} _ {\alpha} \mathbf {t} _ {\alpha}) \cdot \frac {\partial \mathbf {u}}{\partial S ^ {\alpha}} \quad (\mathrm{nosumon} \alpha)
d \delta \gamma_ {S R C} = g _ {1} ^ {- \frac {1}{2}} \delta \pmb {\omega} \times \overline {{\pmb {t}}} _ {1} \cdot (\mathbf {I} - \mathbf {t} _ {2} \mathbf {t} _ {2}) \cdot \frac {\partial d \mathbf {u}}{\partial S ^ {2}}
- g _ {2} ^ {- \frac {1}{2}} \delta \pmb {\omega} \times \overline {{\mathbf {t}}} _ {2} \cdot (\mathbf {I} - \mathbf {t} _ {1} \mathbf {t} _ {1}) \cdot \frac {\partial d \mathbf {u}}{\partial S ^ {1}}
+ g _ {1} ^ {- \frac {1}{2}} d \pmb {\omega} \times \overline {{\mathbf {t}}} _ {1} \cdot (\mathbf {I} - \mathbf {t} _ {2} \mathbf {t} _ {2}) \cdot \frac {\partial \delta \mathbf {u}}{\partial S ^ {2}}
- g _ {2} ^ {- \frac {1}{2}} d \boldsymbol {\omega} \times \overline {{\mathbf {t}}} _ {2} \cdot (\mathbf {I} - \mathbf {t} _ {1} \mathbf {t} _ {1}) \cdot \frac {\partial \delta \mathbf {u}}{\partial S ^ {1}}
and
d \delta \gamma_ {S R M} = g ^ {- \frac {1}{2}} d \pmb {\omega} \times \mathbf {p} \cdot (\mathbf {I} - \mathbf {t t}) \cdot \frac {\partial \delta \mathbf {u}}{\partial S ^ {2}}
+ g ^ {- \frac {1}{2}} \delta \pmb {\omega} \times \mathbf {p} \cdot (\mathbf {I} - \mathbf {t t}) \cdot \frac {\partial d \mathbf {u}}{\partial S ^ {1}}.
Internal virtual work rate
For these shell elements the internal virtual work rate is assumed to be
\delta W ^ {I} = \int_ {A} \int_ {h} \sigma^ {\alpha \beta} \delta \varepsilon_ {\alpha \beta} ^ {f} d z d A
+ \sum_ {r} K _ {I} ^ {(r)} \gamma_ {3 \alpha} ^ {(r)} \delta \gamma_ {3 \alpha} ^ {(r)}
+ \sum_ {n ^ {c}} K _ {I I} ^ {(n)} \gamma_ {S R C} ^ {(n)} \delta \gamma_ {S R C} ^ {(n)}
+ \sum_ {n ^ {m}} K _ {I I I} ^ {(n)} \gamma_ {S R M} ^ {(n)} \delta \gamma_ {S R M} ^ {(n)},
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where K _ { I } ^ { ( r ) } , K _ { I I } ^ { ( n ) } (r) , K(n)II , and K _ { I I I } ^ { ( n ) } are the transverse shear stiffness and the penalties defined above and r indicates the integration points at which transverse shears are calculated, n ^ { c } indicates corner nodes at which six degrees of freedom are used, and n ^ { m } indicates midside nodes at which six degrees of freedom are used. Here \varepsilon _ { \alpha \beta } ^ { f } and \sigma ^ { \alpha \beta } are the strain and stress in the ( d S ^ { \alpha } , d S ^ { \beta } ) material directions in a surface offset by a distance z from the reference surface. The usual Kirchhoff assumption is adopted:
\varepsilon_ {\alpha \beta} ^ {f} = \epsilon_ {\alpha \beta} + z \kappa_ {\alpha \beta},
so that the first term above is
\int_ {A} \Bigl (\delta \epsilon_ {\alpha \beta} \int_ {h} \sigma^ {\alpha \beta} d z + \delta \kappa_ {\alpha \beta} \int_ {h} z \sigma^ {\alpha \beta} d z \Bigr) d A.
The thickness direction integrations are performed numerically in ABAQUS. The integration scheme is a Simpson's rule, of user-chosen order. The shell can also be considered layered, with different properties at each layer and a different integration scheme assigned (by the user) to each layer.
Pressure load stiffness
The load stiffness associated with pressure loading is often important in shells, especially in eigenvalue buckling estimates on elastic shells. In ABAQUS/Standard the pressure load stiffness is implemented as a symmetric form, thus assuming that the pressure magnitude is constant over the surface and neglecting free edge effects. See Hibbitt (1979) and Mang (1980) for details.
The load stiffness is obtained in such a form as follows. The external virtual work associated with pressure is
\delta W ^ {e} = \int_ {A} \mathbf {p} \cdot \delta \mathbf {u} d A,
where p is the pressure load per unit area, given in terms of the (externally prescribed) pressure magnitude, p , as
\mathbf {p} d A = p \frac {\partial \mathbf {x}}{\partial \theta^ {1}} \times \frac {\partial \mathbf {x}}{\partial \theta^ {2}} d \theta^ {1} d \theta^ {2}.
Thus,
\delta W ^ {e} = \int_ {A} p \left(\frac {\partial \mathbf {x}}{\partial \theta^ {1}} \times \frac {\partial \mathbf {x}}{\partial \theta^ {1}}\right) \cdot \delta \mathbf {u} d \theta^ {1} d \theta^ {2}.
The change in this term caused by change of displacement of the shell (the "load stiffness") is
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d \delta W ^ {e} = \int_ {A} p \delta \mathbf {u} \cdot \left(\frac {\partial \mathbf {x}}{\partial \theta^ {1}} \times \frac {\partial d \mathbf {u}}{\partial \theta^ {2}} - \frac {\partial \mathbf {x}}{\partial \theta^ {2}} \times \frac {\partial d \mathbf {u}}{\partial \theta^ {1}}\right) d \theta^ {1} d \theta^ {2},
since we assume that pressure magnitude, p , is externally prescribed and has no dependence on position, x. Neglecting free edge effects, and assuming the magnitude p is uniform, results in the symmetric form
\begin{array}{l} d \delta W ^ {e} = \int_ {A} \frac {1}{2} p \left[ \frac {\partial \mathbf {x}}{\partial \theta^ {1}} \cdot \left(\frac {\partial d \mathbf {u}}{\partial \theta^ {2}} \times \delta \mathbf {u} + \frac {\partial \delta \mathbf {u}}{\partial \theta^ {2}} \times d \mathbf {u}\right) \right. \\ \left. \right.\left. - \frac {\partial \mathbf {x}}{\partial \theta^ {2}} \cdot \left(\frac {\partial d \mathbf {u}}{\partial \theta^ {1}} \times \delta \mathbf {u} + \frac {\partial \delta \mathbf {u}}{\partial \theta^ {1}} \times d \mathbf {u}\right)\right] d \theta^ {1} d \theta^ {2}. \\ \end{array}
This is the pressure load stiffness provided in ABAQUS.
3.6.4 Triangular facet shell elements
Element type STRI3 in ABAQUS/Standard is a facet shell--a plate element used to approximate a shell. The element has three nodes, each with six degrees of freedom. The strains are based on thin plate theory, using a small-strain approximation. Arbitrary rigid body rotations are accounted for exactly by formulating the deformation of the element in a local coordinate system that rotates with the element. The element also satisfies the patch test, so that it will produce reliable results with appropriate meshes.
The bending of the element is based on a discrete Kirchhoff approach to plate bending, using Batoz's interpolation functions (Batoz et al., 1980). This formulation satisfies the Kirchhoff constraints all around the boundary of the triangle and provides linear variation of curvature throughout the element. However, the membrane strains are assumed constant within the element. In addition, a curved shell is approximated by this element as a set of facets formed by the planes defined by the three nodes of each element. For these reasons it is necessary to use a reasonably well refined mesh in most applications.
Kinematics
A local orthonormal basis system, \mathbf { T } ^ { 1 } and \mathbf { T } ^ { 2 } , is defined in the plane of each element in the reference configuration, using the standard ABAQUS convention. S ^ { 1 } and S ^ { 2 } measure distance along \mathbf { T } ^ { 1 } and \mathbf { T } ^ { 2 } in the reference configuration.
Figure 3.6.4-1 Triangular facet shell in the reference configuration.
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text_image
3 T² N T¹ 1 2 3 2 1
The membrane strains are then defined as
\varepsilon_ {\alpha \beta} = \frac {1}{2} (g _ {\alpha \beta} - G _ {\alpha \beta}), \quad \alpha \mathrm{and} \beta = 1, 2
where
g _ {\alpha \beta} = \frac {\partial \mathbf {x}}{\partial S ^ {\alpha}} \cdot \frac {\partial \mathbf {x}}{\partial S ^ {\beta}}
is the metric in the current configuration, and
G _ {\alpha \beta} = \frac {\partial \mathbf {X}}{\partial S ^ {\alpha}} \cdot \frac {\partial \mathbf {X}}{\partial S ^ {\beta}}
is the metric in the reference configuration.
Here x and X are the spatial coordinates of a point in the current and reference configurations, respectively. Curvature changes are defined incrementally. To account for large rigid body rotations we use a local coordinate system that rotates with the plane defined by the three nodes of the element. The basis vectors chosen for this local system are \mathbf { t } ^ { 1 } = \partial \mathbf { x } / \partial S ^ { 1 } and \mathbf { t } ^ { 2 } = \partial \mathbf { x } / \partial S ^ { 2 } . Since the membrane strains are assumed to be small, these vectors will be approximately orthonormal. The components of incremental rotation of the normal to the plate are defined as \Delta \omega _ { 1 } about \mathbf { t } ^ { 1 } and \Delta \omega _ { 2 } about \mathbf { t } ^ { 2 } . The incremental displacement of the reference surface of the plate along the normal to the plane of its nodes is defined as ¢w. (Note that \Delta w will be zero at the nodes at all times because the plane containing \mathbf { t } ^ { 1 } and \mathbf { t } ^ { 2 } always passes through the nodes.) The Kirchhoff constraints are, approximately,
- \Delta \omega_ {2} + \frac {\partial \Delta w}{\partial S ^ {1}} = 0
and
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\Delta \omega_ {1} + \frac {\partial \Delta w}{\partial S ^ {2}} = 0.
Batoz (1980) assumes that \Delta \omega _ { 1 } and \Delta \omega _ { 2 } vary quadratically over the element and that \Delta w is defined independently along each of the three sides of the element as a cubic function. The Kirchhoff constraints are then imposed at the corners and at the middle of each element edge along the direction of the edge to give
\Delta \omega_ {1} = H _ {1} ^ {P} \Delta \nu^ {P}
and
\Delta \omega_ {2} = H _ {2} ^ {P} \Delta \nu^ {P},
where \Delta \nu ^ { P } is the array
\left\lfloor \Delta \theta_ {1} ^ {1}, \Delta \theta_ {2} ^ {1}, \Delta \theta_ {1} ^ {2}, \Delta \theta_ {2} ^ {2}, \Delta \theta_ {1} ^ {3}, \Delta \theta_ {2} ^ {3} \right\rfloor .
In the above expressions H _ { 1 } ^ { P } ( g , h ) and H _ { 2 } ^ { P } ( g , h ) are interpolation functions that are defined by Batoz (1980), and the incremental rotation components at the nodes, \Delta \theta _ { \alpha } ^ { N } , are defined as
\Delta \theta_ {\alpha} ^ {N} = \mathbf {n} ^ {N} \cdot \mathbf {t} ^ {\alpha},
where
\mathbf {n} ^ {N} = \mathbf {C} (\Delta \phi_ {i} ^ {N}) \cdot \mathbf {N},
and \Delta \phi _ { i } ^ { N } are the increments of the rotational degrees of freedom at the node N , C is the rotation matrix defined by \Delta \phi _ { i } ^ { N } , and N is the normal to the plane of the element's nodes at the beginning of the increment. Finally, the incremental curvature change measures are defined as
\Delta k _ {1 1} = \frac {\partial \Delta \omega_ {2}}{\partial S ^ {1}}
\Delta k _ {2 2} = - \frac {\partial \Delta \omega_ {1}}{\partial S ^ {2}}
and
\Delta k _ {1 2} = - \frac {\partial \Delta \omega_ {1}}{\partial S ^ {1}} + \frac {\partial \Delta \omega_ {2}}{\partial S ^ {2}}.
The three membrane strains and three curvature strains complete the basic kinematic description of the element, except that the use of six degrees of freedom per node introduces a spurious rotation at each node (only two incremental rotations at each node appear in the above equations--the rotation about
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the normal to the plane of the element's nodes does not enter). To deal with this problem, we define a generalized strain to be penalized with a small stiffness at each node as
\gamma^ {i} = \mathbf {b} _ {1} \cdot \mathbf {a} _ {1} - \mathbf {b} _ {2} \cdot \mathbf {a} _ {2},
where
\mathbf {a} _ {1} = \mathbf {C} \cdot \mathbf {A} _ {1}
\mathbf {a} _ {2} = \mathbf {C} \cdot \mathbf {A} _ {2}
\mathbf {A} _ {1} = \mathbf {N} \times \overline {{\mathbf {A}}} _ {1}
\mathbf {A} _ {2} = \overline {{\mathbf {A}}} _ {2} \times \mathbf {N}
\overline {{\mathbf {A}}} _ {1} = (\mathbf {X} _ {j} - \mathbf {X} _ {i}) / | \mathbf {X} _ {j} - \mathbf {X} _ {i} |
\overline {{{\mathbf {A}}}} _ {2} = (\mathbf {X} _ {k} - \mathbf {X} _ {i}) / | \mathbf {X} _ {k} - \mathbf {X} _ {i} |
\mathbf {b _ {1}} = (\mathbf {x} _ {j} - \mathbf {x} _ {i}) / | \mathbf {x} _ {j} - \mathbf {x} _ {i} |
\mathbf {b _ {2}} = (\mathbf {x} _ {k} - \mathbf {x} _ {i}) / | \mathbf {x} _ {k} - \mathbf {x} _ {i} |
and j, k are the node numbers in cyclic order forming the two sides of the triangle at the node i.
First variations of strain
The first variations of strain are
\delta \varepsilon_ {\alpha \beta} = \frac {1}{2} \left(\frac {\partial \mathbf {x}}{\partial S ^ {\alpha}} \cdot \frac {\partial \delta \mathbf {x}}{\partial \mathbf {S} ^ {\beta}} + \frac {\partial \mathbf {x}}{\partial S ^ {\beta}} \cdot \frac {\partial \delta \mathbf {x}}{\partial S ^ {\alpha}}\right)
\delta K _ {1 1} = \frac {\partial (\delta \Delta \omega_ {2})}{\partial S ^ {1}}
\delta K _ {2 2} = - \frac {\partial (\delta \Delta \omega_ {1})}{\partial S ^ {2}}
\delta K _ {1 2} = \left(\frac {\partial (\delta \Delta \omega_ {2})}{\partial S ^ {2}} - \frac {\partial (\delta \Delta \omega_ {1})}{\partial S ^ {1}}\right),
where
\delta \Delta \omega_ {1} = H _ {1} ^ {P} \delta \Delta \nu^ {P}
\delta \Delta \omega_ {2} = H _ {2} ^ {P} \delta \Delta \nu^ {P},
and in \Delta \delta \nu ^ { P } ,
\delta \Delta \theta_ {\alpha} ^ {N} = \delta \pmb {\phi} ^ {N} \times \mathbf {n} ^ {N} \cdot \mathbf {t} ^ {\alpha} + \mathbf {n} ^ {N} \cdot \delta \mathbf {t} ^ {\alpha}.
Also, for the "strain" used to introduce the extra stiffness at the nodes to avoid singularity caused by the component of rotation about the normal,