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Elements
Then
\frac {4 \Delta \lambda_ {\alpha}}{(1 + \Delta \lambda_ {\alpha}) ^ {2}} = \frac {1 + e}{(1 + e / 2) ^ {2}} = (1 + e) (1 - e + \frac {3}{4} e ^ {2} - \frac {1}{2} e ^ {3} + \dots)
= (1 - \frac {1}{4} e ^ {2} + \frac {1}{4} e ^ {3} - \dots).
Again, if e ¼ 20 percent, this means that
\frac {4 \Delta \lambda_ {\alpha}}{(1 + \Delta \lambda_ {\alpha}) ^ {2}} \approx 1 -. 0 1 + 0 (e ^ {3}),
and so once again using the argument that practical applications will involve strain increments of no more than a few percent, we approximate
\frac {4 \Delta \lambda_ {\alpha}}{(1 + \Delta \lambda_ {\alpha}) ^ {2}} \approx 1.
This then gives
Equation 3.6.2-6
\Delta \varepsilon_ {\alpha \alpha} ^ {1} \approx \Delta \varepsilon_ {\alpha \alpha} + \eta \Delta k _ {\alpha \alpha}.
The stretch ratio in the thickness direction is assumed to be defined by an incompressibility condition on the reference surface:
\Delta \lambda_ {t} \Delta \lambda_ {1} \Delta \lambda_ {2} = 1 + \Delta D ^ {\mathrm{th}},
where \Delta { D ^ { \mathrm { t h } } } is the increase in volume caused by thermal strain and is approximated in ABAQUS as
\Delta D ^ {\mathrm{th}} = \frac {3}{2} (\Delta \varepsilon_ {1} ^ {\mathrm{th}} + \Delta \varepsilon_ {2} ^ {\mathrm{th}}),
where \Delta \varepsilon _ { 1 } ^ { \mathrm { t h } } , \Delta \varepsilon _ { 2 } ^ { \mathrm { t h } } are the thermal strain increments in the 1- and 2-directions on the reference surface.
From the definition of \Delta k _ { \alpha \alpha } ;
Equation 3.6.2-7
\Delta k _ {\alpha \alpha} = \lambda_ {t} ^ {o} \left(\frac {(1 + \Delta D ^ {\mathrm{th}})}{\Delta \lambda_ {1} \Delta \lambda_ {2}} \frac {b _ {\alpha \alpha}}{g _ {\alpha \alpha}} - \frac {B _ {\alpha \alpha}}{G _ {\alpha \alpha}}\right).
The transverse shear strains are written as
Equation 3.6.2-8
\Delta \gamma_ {\alpha} = \frac {\partial \mathbf {x}}{\partial \theta^ {\alpha}} \cdot \mathbf {n} - \frac {\partial \mathbf {X}}{\partial \theta^ {\alpha}} \cdot \mathbf {N}.
Elements
This simple form is used because these strains are always assumed to be small. This completes the statement of the incremental strain definitions, and so--together with a virtual work statement to represent equilibrium--a theory is available. However, it is necessary to satisfy the minimum requirement that the theory provide constant strain under appropriate motions. This is essential if the theory is to be suitable for many practical cases, most especially those involving thermal loading. Interestingly, the theory in Rodal and Witmer (1979) appears to violate this requirement. To achieve this, a modified incremental curvature change measure is defined as
\Delta \tilde {k} _ {\alpha \beta} = \Delta k _ {\alpha \beta} + \mathrm{sym} (\rho_ {\alpha} ^ {\gamma} \Delta \varepsilon_ {\gamma \beta}),
where
\rho _ { \alpha } ^ { \beta } is a tensor, de¯ned as follows.
We know that the radii of curvature of the ®-line at the end and at the beginning of an increment are given by
\frac { 1 } { r _ { \alpha } } = \frac { b _ { \alpha \alpha } } { g _ { \alpha \alpha } } r® = b®® at the end of an increment,
and
\frac { 1 } { R _ { \alpha } } = \frac { B _ { \alpha \alpha } } { G _ { \alpha \alpha } } G®® B®® at the beginning of the increment.
In these expressions, as in the following development, no summation is implied by a repeated index. If the ®-line is stretched uniformly by \Delta \lambda _ { \alpha } during the increment, we require that
r _ {\alpha} = \Delta \lambda_ {\alpha} R _ {\alpha};
and, further, such uniform stretch of the shell must give constant strain so that since we assume
\Delta \varepsilon_ {\alpha \alpha} ^ {1} = \Delta \varepsilon_ {\alpha \alpha} + \eta \Delta \tilde {k} _ {\alpha \alpha},
we need
\Delta \tilde {k} _ {\alpha \alpha} = 0
under such circumstances. In this motion
\begin{array}{l} \Delta k _ {\alpha \alpha} = \lambda_ {t} ^ {o} \left(\frac {1 + \Delta D ^ {\mathrm{th}}}{\Delta \lambda_ {\alpha} \Delta \lambda_ {\beta}} \frac {1}{r _ {\alpha}} - \frac {1}{R - \alpha}\right), \quad \beta \neq \alpha \\ = \frac {\lambda_ {t} ^ {o}}{R _ {\alpha}} \left(\frac {1 + \Delta D ^ {\mathrm{th}}}{\Delta \lambda_ {\alpha} \Delta \lambda_ {1} \Delta \lambda_ {2}} - 1\right). \\ \end{array}
Defining
Equation 3.6.2-9
\Delta \tilde {k} _ {\alpha \alpha} = \Delta k _ {\alpha \alpha} + \left(1 - \frac {1 + \Delta D ^ {\mathrm{th}}}{\Delta \lambda_ {\alpha} \Delta \lambda_ {1} \Delta \lambda_ {2}}\right) \lambda_ {t} ^ {o} \frac {B _ {\alpha \alpha}}{G _ {\alpha \alpha}}
and assuming
Equation 3.6.2-10
\Delta \varepsilon_ {\alpha \alpha} ^ {1} = \Delta \varepsilon_ {\alpha \alpha} + \eta \Delta \tilde {k} _ {\alpha \alpha}
satisfies the requirement. Equation 3.6.2-9 may be simplified by substituting in the definition of \Delta k _ { \alpha \alpha } in Equation 3.6.2-7 to give
\Delta \tilde {k} _ {\alpha \alpha} = \frac {\lambda_ {t} ^ {o} (1 + \Delta D ^ {\mathrm{th}})}{\Delta \lambda_ {1} \Delta \lambda_ {2}} \left(\frac {b _ {\alpha \alpha}}{g _ {\alpha \alpha}} - \frac {1}{\Delta \lambda_ {\alpha}} \frac {B _ {\alpha \alpha}}{G _ {\alpha \alpha}}\right),
and so
Equation 3.6.2-11
\Delta \tilde {k} _ {\alpha \alpha} = \frac {\lambda_ {t} ^ {o} (1 + \Delta D ^ {\mathrm{th}})}{g _ {\alpha \alpha} \Delta \lambda_ {1} \Delta \lambda_ {2}} (b _ {\alpha \alpha} - \Delta \lambda_ {\alpha} B _ {\alpha \alpha}).
The formulation is completed by the assumption that the virtual work equation can be written
Equation 3.6.2-12
\int_ {\theta^ {1}, \theta^ {2}} \left\{\lambda_ {t} ^ {o} \int_ {h} \sigma^ {\alpha \beta} \delta \varepsilon_ {\alpha \beta} ^ {1} d \eta + T ^ {\alpha} \delta \gamma_ {\alpha} \right\} (G _ {1 1} G _ {2 2}) ^ {\frac {1}{2}} d \theta^ {1} d \theta^ {2} = \delta W ^ {E},
where
\sigma^ {\alpha \beta}
are the Kirchhoff stresses at a point;
(\theta^ {1}, \theta^ {2})
in the shell, defined by plane stress theory using the summation of the strain increments in Equation 3.6.2-10 to define the strain at this point;
\delta \varepsilon_ {\alpha \beta} ^ {1}
are the variations of the strain increments in Equation 3.6.2-10;
T ^ {\alpha}
are the transverse shear forces per unit area, defined by T ^ { \alpha } = k G h \gamma _ { \alpha } ; where °®
Elements
are the transverse shear strains from Equation 3.6.2-8, h is the original thickness of the shell, kG is the elastic transverse shear stiffness (reduced according to the suggestions of Hughes et al. (1977) if the shell is too thin, to avoid numerical problems);
and \delta W^{E}
is the virtual external work rate.
This completes the statement of the formulation.
Implementation
In this section we summarize the basic equations of the formulation defined above. We take \theta ^ { 1 } = g _ { \mathrm { ; } } , an isoparametric coordinate along the reference surface in the meridional ( r-z) plane. In each element - 1 < g < 1 : We also take \theta ^ { 2 } = \theta , the angular position, measured in radians, in the circumferential direction. The metrics at the start and end of the increment are
G _ {1 1} = \frac {d \mathbf {X}}{d g} \cdot \frac {d \mathbf {X}}{d g},
G _ {2 2} = R ^ {2}
and
g _ {1 1} = \frac {d \mathbf {x}}{d g} \cdot \frac {d \mathbf {x}}{d g},
g _ {2 2} = r ^ {2}.
From these the incremental stretches of the reference surface are
\Delta \lambda_ {1} = (g _ {1 1} / G _ {1 1}) ^ {\frac {1}{2}},
\Delta \lambda_ {2} = \frac {r}{R}.
Curvature measures are
B _ {1 1} = \frac {d \mathbf {X}}{d g} \cdot \frac {d \mathbf {N}}{d g},
B _ {2 2} = R N _ {r}
Elements
and
b _ {1 1} = \frac {d \mathbf {x}}{d g} \cdot \frac {d \mathbf {n}}{d g},
b _ {2 2} = r n _ {r}.
First variations are then
\delta \Delta \lambda_ {1} = \frac {1}{\Delta \lambda_ {1} G _ {1 1}} \frac {d \mathbf {x}}{d g} \cdot \frac {d \delta \mathbf {x}}{d g},
\delta \Delta \lambda_ {2} = \frac {\delta r}{R},
\delta b _ {1 1} = \frac {d \mathbf {n}}{d g} \cdot \frac {d \delta \mathbf {x}}{d g} + \frac {d \mathbf {x}}{d g} \cdot \mathbf {t} \frac {d \delta \omega}{d g} + \frac {d \mathbf {x}}{d g} \cdot \frac {d \mathbf {t}}{d g} \delta \omega ,
where ! is the rotation of the thickness direction vector n and t is orthogonal to n, so that
\delta \mathbf {n} = \delta \omega \mathbf {t},
\delta b _ {2 2} = n _ {r} \delta r - n _ {z} r \delta \omega .
Second variations are
\begin{array}{l} d \delta b _ {1 1} = \frac {d \delta \mathbf {x}}{d g} \cdot \frac {d \mathbf {t}}{d g} d \omega + \delta \omega \frac {d \mathbf {t}}{d g} \cdot \frac {d d \mathbf {x}}{d g} \\ + \frac {d \delta \mathbf {x}}{d g} \cdot \mathbf {t} \frac {d d \omega}{d g} + \frac {d \delta \omega}{d g} \mathbf {t} \cdot \frac {d d \mathbf {x}}{d g} \\ - \mathbf {n} \cdot \frac {d \mathbf {x}}{d g} \left(\frac {d \delta \omega}{d g} d \omega - \delta \omega \frac {d d \omega}{d g}\right) \\ - \frac {d \mathbf {x}}{d g} \cdot \frac {d \mathbf {n}}{d g} \delta \omega d \omega \\ \end{array}
and
d \delta b _ {2 2} = \delta r t _ {r} d \omega + \delta \omega t _ {r} d r - \delta \omega d \omega t _ {r} r.
The incremental strains are
\Delta \varepsilon_ {\alpha \alpha} = \frac {2 (\Delta \lambda_ {\alpha} - 1)}{\Delta \lambda_ {\alpha} + 1}
\Delta \tilde {k} _ {\alpha \alpha} = \frac {\lambda_ {t} ^ {o} (1 + \Delta D ^ {\mathrm{th}})}{\Delta \lambda^ {2} \Delta \lambda_ {1} \Delta \lambda_ {2}} (b _ {\alpha \alpha} - \Delta \lambda_ {\alpha} B _ {\alpha \alpha}).
First variations of strains are
Elements
\delta \varepsilon_ {1 1} = \frac {4}{(1 + \Delta \lambda_ {1}) ^ {2} \Delta \lambda_ {1} G _ {1 1}} \frac {d \mathbf {x}}{d g} \cdot \frac {d \delta \mathbf {x}}{d g}
\delta \varepsilon_ {2 2} = \frac {4}{(1 + \Delta \lambda_ {2}) ^ {2} R} \delta r
\begin{array}{l} \delta \tilde {k} _ {1 1} = \frac {\lambda_ {t} ^ {o} (1 + \Delta D ^ {\mathrm{th}})}{\Delta \lambda_ {1} ^ {3} \Delta \lambda_ {2} G _ {1 1}} \left[ \frac {1}{\Delta \lambda_ {1} ^ {2} G _ {1 1}} (- 3 B _ {1 1} + 2 \Delta \lambda_ {1} B _ {1 1}) \frac {d \mathbf {x}}{d g} \cdot \frac {d \delta \mathbf {x}}{d g} \right. \\ - \left(b _ {1 1} - \Delta \lambda_ {1} B _ {1 1}\right) \frac {\delta r}{r} + \mathrm {\dot {n}} g \cdot \frac {d \delta \mathbf {x}}{d g} \\ \left. + \mathbf {t} \cdot \frac {d \mathbf {x}}{d g} \frac {d \delta \omega}{d g} + \frac {d \mathbf {t}}{d g} \cdot \dot {\mathrm{x}} g \delta \omega \right] \\ \end{array}
\delta \tilde {k} _ {2 2} = \frac {\lambda_ {t} ^ {o} (1 + \Delta D ^ {\mathrm{th}})}{\Delta \lambda_ {1} \Delta \lambda_ {2} ^ {3} G _ {2 2}} \left[ \frac {1}{\Delta \lambda_ {1} ^ {2} G _ {1 1}} (- b _ {2 2} + \Delta \lambda_ {2} B _ {2 2}) \frac {d \mathbf {x}}{d g} \cdot \frac {d \delta \mathbf {x}}{d g} \right.
\left. + \frac {1}{R} \left(\frac {- 3 b _ {2 2}}{\Delta \lambda_ {2 2}} + 2 B _ {2 2}\right) \delta r + n _ {r} \delta r - n _ {z} r \delta \omega \right].
Second variations of strains are
\begin{array}{l} d \delta \varepsilon_ {1 1} = \frac {- 4 (3 \Delta \lambda_ {1} + 1)}{G _ {1 1} ^ {2} \Delta \lambda_ {1} ^ {3} (1 + \Delta \lambda) ^ {3}} \frac {d \mathbf {x}}{d g} \cdot \frac {d \delta \mathbf {x}}{d g} \frac {d \mathbf {x}}{d g} \cdot \frac {d d \mathbf {x}}{d g} \\ + \frac {4}{(1 + \Delta \lambda_ {1}) ^ {2} \Delta \lambda_ {1} G _ {1 1}} \frac {d \delta \mathbf {x}}{d g} \cdot \frac {d d \mathbf {x}}{d g} \\ \end{array}
d \delta \varepsilon_ {2 2} = \frac {- 8}{R ^ {2} (1 + \Delta \lambda_ {2}) ^ {3}} \delta r d r
Elements
\begin{array}{l} d \delta \tilde {k} _ {1 1} = \frac {\lambda_ {t} ^ {o} (1 + \Delta D ^ {\mathrm{th}}}{G _ {1 1} \Delta \lambda_ {1} ^ {3} \Delta \lambda_ {2}} \left\{\left[ - \frac {1}{\Delta \lambda_ {1} ^ {2} G _ {1 1}} (3 b _ {1 1} - 2 \Delta \lambda_ {1} B _ {1 1}) \frac {d d \mathbf {x}}{d g} \right] \right. \\ + \left[ \frac {1}{\Delta \lambda_ {1} ^ {4} G _ {1 1} ^ {2}} \left\{1 5 b _ {1 1} - 8 \lambda_ {1} B _ {1 1} \right\} \right] \frac {d \delta \mathbf {x}}{d g} \cdot \frac {d \mathbf {x}}{d g} \frac {d \mathbf {x}}{d g} \cdot \frac {d d \mathbf {x}}{d g} \\ + \left[ \frac {- 3}{\Delta \lambda_ {1} ^ {2} r G _ {1 1}} \right] \left[ \frac {d \delta \mathbf {x}}{d g} \cdot \frac {d \mathbf {n}}{d g} \frac {d \mathbf {x}}{d g} \cdot \frac {d d \mathbf {x}}{d g} + \frac {d \delta \mathbf {x}}{d g} \cdot \mathbf {\dot {x}} g \mathbf {\dot {n}} g \cdot \frac {d d \mathbf {x}}{d g} \right] \\ + \left[ \frac {1}{\Delta \lambda_ {1} ^ {2} r G _ {1 1}} (3 b _ {1 1} - 2 \Delta \lambda_ {1} B _ {1 1}) \right] \left[ \frac {d \delta \mathbf {x}}{d g} \cdot \frac {d \mathbf {x}}{d g} d r + \delta r \frac {d \mathbf {x}}{d g} \cdot \frac {d d \mathbf {x}}{d g} \right] \\ + \left[ - \frac {1}{r} \right] \left[ \frac {d \delta \mathbf {x}}{d g} \cdot \frac {d \mathbf {n}}{d g} d r + \delta r \frac {d \mathbf {n}}{d g} \cdot \frac {d d \mathbf {x}}{d g} \right] \\ + \left[ \frac {2}{r ^ {2}} (b _ {1 1} - \Delta \lambda_ {1} B _ {1 1}) \right] \delta r d r \\ + \left[ - \frac {3}{\Delta \lambda_ {1} ^ {2} G _ {1 1}} \mathbf {t} \cdot \frac {d \mathbf {x}}{d g} \right] \left[ \frac {d \delta \mathbf {x}}{d g} \cdot \frac {d \mathbf {x}}{d g} d \omega + \delta \omega \frac {d \mathbf {x}}{d g} \cdot \frac {d d \mathbf {x}}{d g} \right] \\ + \frac {d \delta \mathbf {x}}{d g} \cdot \mathbf {t} \frac {d d \omega}{d g} + \frac {d \delta \omega}{d g} \mathbf {t} \frac {d d \mathbf {x}}{d g} + \frac {d \delta \mathbf {x}}{d g} \cdot \frac {d \mathbf {t}}{d g} d \omega + \delta \omega \frac {d \mathbf {t}}{d g} \cdot \frac {d d \mathbf {x}}{d g} \\ + \left[ - \frac {1}{r} \frac {d \mathbf {t}}{d g} \cdot \frac {d \mathbf {x}}{d g} \right] [ \delta \omega d r + \delta r d \omega ] + \left[ - \frac {1}{r} \mathbf {t} \cdot \frac {d \mathbf {x}}{d g} \right] \left[ \frac {d \delta \omega}{d g} d r + \delta r \frac {d d \omega}{d x} \right] \\ + \left\{\left[ - \mathbf {n} \cdot \frac {d \mathbf {x}}{d g} \frac {d \delta \omega}{d g} \right] \left[ d \omega + \delta \omega \frac {d d \omega}{d g} \right] + \delta \omega d \omega \left[ - \frac {d \mathbf {n}}{d g} \cdot \frac {d \mathbf {x}}{d g} \right] \right\} \\ \end{array}
d \delta \tilde {k} _ {2 2} = \frac {\lambda_ {t} ^ {o} (1 + \Delta D ^ {\mathrm{th}})}{G _ {2 2} \Delta \lambda_ {1} \Delta \lambda_ {2} ^ {3}} \Bigg \{- \frac {1}{\Delta \lambda_ {1} ^ {2} G _ {1 1}} (b _ {2 2} - \delta \lambda_ {2} B _ {2 2}) \frac {d \delta \mathbf {x}}{d g} \cdot \frac {d d \mathbf {x}}{d g}
+ \left[ \frac {3}{\Delta \lambda_ {1} ^ {4} G _ {1 1} ^ {2}} (b _ {2 2} - \Delta \lambda_ {2} B _ {2 2}) \right] \frac {d \delta \mathbf {x}}{d g} \cdot \frac {d \mathbf {x}}{d g} \frac {d \mathbf {x}}{d g} \cdot \frac {d d \mathbf {x}}{d g}
\left. \right. + \left[ \frac {1}{\Delta \lambda_ {1} ^ {2} G _ {1 1}} \left\{\frac {1}{R} \left(\frac {3 b _ {2 2}}{\Delta \lambda_ {2}}\right) - 2 B _ {2 2} - n _ {r} \right\}\right]\left[ \frac {d \delta \mathbf {x}}{d g} \cdot \frac {d \mathbf {x}}{d g} d r + \delta r \frac {d \mathbf {x}}{d g} \cdot \frac {d d \mathbf {x}}{d g} \right]
+ \left[ \frac {6}{r} \left\{\frac {1}{R} \left(\frac {b _ {2 2}}{\Delta \lambda_ {2}} - B _ {2 2}\right) - n _ {r} \right\} \right] \delta r d r
+ \frac {n _ {z} r}{\Delta \lambda_ {1} ^ {2} G _ {1 1}} \left[ \frac {d \delta \mathbf {x}}{d g} \cdot \frac {d \mathbf {x}}{d g} d \omega + \delta \omega \frac {d \mathbf {x}}{d g} \frac {d d \mathbf {x}}{d g} \right]
\left. + 2 n _ {z} \delta r d \omega + \delta \omega d r - n _ {r} r \delta \omega d \omega \right\}.
The transverse shear strain is written as
\Delta \gamma = g _ {1 1} ^ {- \frac {1}{2}} \frac {d \mathbf {x}}{d g} \cdot \mathbf {n} - G _ {1 1} ^ {- \frac {1}{2}} \frac {d \mathbf {X}}{d g} \cdot \mathbf {N}.
Elements
In the initial configuration \begin{array} { r } { \frac { d { \bf X } } { d g } \cdot { \bf N } = 0 } \end{array} . Ignoring terms involving \textstyle { \frac { d \mathbf { x } } { d g } } \cdot \mathbf { n } , the first variation is
\delta \gamma = g _ {1 1} ^ {- \frac {1}{2}} \left[ \frac {d \delta \mathbf {x}}{d g} \cdot \mathbf {n} + \frac {d \mathbf {x}}{d g} \cdot \mathbf {t} \delta \omega \right],
where \mathbf { t } ^ { T } = \{ - n _ { z } \ n _ { r } \} . The second variation is
d \delta \gamma = - g _ {1 1} ^ {- \frac {3}{2}} \frac {d \delta \mathbf {x}}{d g} \cdot \left[ \frac {d \mathbf {x}}{d g} \mathbf {n} + \mathbf {n} \frac {d \mathbf {x}}{d g} \right] \cdot \frac {d d \mathbf {x}}{d g},
where it has been assumed that d \gamma \approx 0 .
This completes the kinematic formulation. Two elements have been implemented: SAX1, which uses linear interpolation for x and ! and a single integration point along its length, and SAX2, which uses quadratic interpolation for x and ! and two integration points along its length. The integration through the thickness follows the usual numerical or exact scheme of ABAQUS.
3.6.3 Shear flexible small-strain shell elements
This section discusses the formulation of the small-strain shear flexible elements in ABAQUS/Standard, which are quadrilaterals (S4R5, S8R5, S9R5, and S8R), except for the 6-node triangle STRI65. The essential idea of these elements is that the position of a point in the shell reference surface--x--and the components of a vector n--which is approximately normal to the reference surface--are interpolated independently. The kinematics of the shell theory then consist of measuring membrane strain on the reference surface from the derivatives of x with respect to position on the surface and bending strain from the derivatives of n; the strain measures that are used for this purpose are approximations to Koiter-Sanders theory strains ( Budiansky and Sanders, 1963). The transverse shear strains are measured as the changes in the projections of n onto tangents to the shell's reference surface. For these element types the strain measures are suitable for large rotations but small strains, and the change in the shell's thickness caused by deformation is neglected.
Notation
A typical piece of shell surface is shown in Figure 3.6.3-1.
Figure 3.6.3-1 Shell reference surface.
text_image
n S² θ² S¹ θ¹ z y x
Let ( \theta ^ { 1 } , \theta ^ { 2 } ) be a set of Gaussian surface coordinates on the shell reference surface. Since these coordinates are only needed locally at an integration point, we use the element's isoparametric coordinates as these coordinates. \mathbf { x } ( \theta ^ { 1 } , \theta ^ { 2 } ) is the current position of a point on the interpolated reference surface, and \mathbf { X } ( \theta ^ { 1 } , \theta ^ { 2 } ) is the initial position of the same point. The unit vector
\mathbf {N} = \frac {\partial \mathbf {X}}{\partial \theta^ {1}} \times \frac {\partial \mathbf {X}}{\partial \theta^ {2}} \left/ \sqrt {\frac {\partial \mathbf {X}}{\partial \theta^ {1}} \times \frac {\partial \mathbf {X}}{\partial \theta^ {2}}} \left. \right.
is the unit normal to the interpolated reference surface in the initial configuration. This vector gives a "sidedness" to the surface--one surface of the shell is the "top" surface (in the positive direction along N from the shell's reference surface) and the other is the bottom surface. The vector corresponding to N in the current configuration, n, will be made approximately normal to the reference surface in the current configuration by imposing the Kirchhoff constraint discretely.
In the rest of this section Greek indices will be used to indicate values associated with the (two-dimensional) reference surface and so will sum over the range 1, 2 under the summation convention.
First, we establish convenient directions for stress and strain output. These will be local material directions, indistinguishable (to the order of approximation) from corotational directions, since we assume strains are small. The standard convention used throughout ABAQUS for such local directions on a surface is as follows.
It is most convenient to choose orthogonal directions. Define
\mathbf {T} _ {2} = \frac {\mathbf {N} \times \mathbf {i}}{\sqrt {\mathbf {N} \times \mathbf {i}}}
Elements
so long as N _ { 1 } < \cos 0 . 1 ^ { \circ } , where i is a unit vector in the global X-direction; otherwise,
\mathbf {T} _ {2} = \frac {\mathbf {N} \times \mathbf {k}}{\sqrt {\mathbf {N} \times \mathbf {k}}},
where k is a unit vector in the global Z-direction. Then define
\mathbf {T} _ {1} = \mathbf {T} _ {2} \times \mathbf {N}.
Let
d S ^ {\beta} = \mathbf {T} _ {\beta} \cdot \frac {\partial \mathbf {X}}{\partial \theta^ {\alpha}} d \theta^ {\alpha},
so that the d S ^ { \alpha } are locally defined distance measuring coordinates at each material point. The transformation
\frac {\partial}{\partial S ^ {\alpha}} = \frac {\partial \theta^ {\beta}}{\partial S ^ {\alpha}} \frac {\partial}{\partial \theta^ {\beta}}
transforms locally with respect to surface coordinates. Here
\frac {\partial \theta^ {\alpha}}{\partial S ^ {\beta}} = \left[ \begin{array}{c c} \mathbf {T} _ {1} \cdot \partial \mathbf {X} / \partial \theta^ {1} & \mathbf {T} _ {1} \cdot \partial \mathbf {X} / \partial \theta^ {2} \\ \mathbf {T} _ {2} \cdot \partial \mathbf {X} / \partial \theta^ {1} & \mathbf {T} _ {2} \cdot \partial \mathbf {X} / \partial \theta^ {2} \end{array} \right] ^ {- 1}.
Stress and strain components are formed in the ( d S ^ { 1 } , d S ^ { 2 } ) directions.
Surface measures
The following surface measures are defined. The metric of the deformed surface is
g _ {\alpha \beta} = \frac {\partial \mathbf {x}}{\partial S ^ {\alpha}} \cdot \frac {\partial \mathbf {x}}{\partial S ^ {\beta}},
and an approximation to the curvature tensor (the second fundamental form) is
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)
(this is only an approximation because n is not exactly normal to the surface in the current configuration).
The corresponding measures associated with the original reference surface are the metric
G _ {\alpha \beta} = \frac {\partial \mathbf {X}}{\partial S ^ {\alpha}} \cdot \frac {\partial \mathbf {X}}{\partial S ^ {\beta}}