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\delta \gamma^ {i} = \delta \mathbf {b} _ {1} \cdot \mathbf {a} _ {1} - \delta \mathbf {b} _ {2} \cdot \mathbf {a} _ {2} + \mathbf {b} _ {1} \cdot \delta \boldsymbol {\omega} \times \mathbf {a} _ {1} - \mathbf {b} _ {2} \cdot \delta \boldsymbol {\omega} \times \mathbf {a} _ {2}.
Second variations of strain
The second variations of strain are
d \delta \varepsilon_ {\alpha \beta} = \frac {1}{2} \left(\frac {\partial \delta \mathbf {x}}{\partial S ^ {\alpha}} \cdot \frac {\partial d \mathbf {x}}{\partial S ^ {\beta}} + \frac {\partial \delta \mathbf {x}}{\partial S ^ {\beta}} \cdot \frac {\partial d \mathbf {x}}{\partial S ^ {\alpha}}\right) \delta K _ {1 1}
= \frac {\partial d \delta \omega_ {2}}{\partial S ^ {1}}
d \delta K _ {2 2} = - \frac {\partial (d \delta \omega_ {1})}{\partial S ^ {2}}
d \delta K _ {1 2} = \left(\frac {\partial d \delta \omega_ {2}}{\partial S ^ {2}} - \frac {\partial d \delta \omega_ {1}}{\partial S ^ {1}}\right),
where
d \Delta \delta \theta_ {\alpha} ^ {N} = - \delta \pmb {\omega} ^ {N} \cdot d \pmb {\omega} ^ {N} \mathbf {n} ^ {N} \cdot \mathbf {t} ^ {\alpha} + \delta \pmb {\omega} ^ {N} \cdot \mathbf {n} ^ {N} d \pmb {\omega} ^ {N} \cdot \mathbf {t} ^ {\alpha}
+ \delta \pmb {\omega} ^ {N} \cdot \mathbf {n} ^ {N} \times \left(\frac {\partial g ^ {1}}{\partial S ^ {1}} (d \mathbf {x} ^ {2} - d \mathbf {x} ^ {1}) + \frac {\partial g ^ {2}}{\partial S ^ {1}} (d \mathbf {x} ^ {3} - d \mathbf {x} ^ {1})\right)
- \left(\frac {\partial g ^ {1}}{\partial S ^ {1}} (\delta \mathbf {x} ^ {2} - \delta \mathbf {x} ^ {1}) + \frac {\partial g ^ {2}}{\partial S ^ {1}} (\delta \mathbf {x} ^ {3} - \delta \mathbf {x} ^ {1})\right) \times \mathbf {n} ^ {N} \cdot d \pmb {\omega} ^ {N}
and
d \delta \gamma^ {i} = \delta \mathbf {b} _ {i} \cdot d \mathbf {a} _ {1} - \delta \mathbf {b} _ {2} \cdot d \mathbf {a} _ {2} + d \mathbf {b} _ {1} \cdot \delta \boldsymbol {\omega} \times \mathbf {a} _ {1} + \mathbf {b} _ {1} \cdot \delta \boldsymbol {\omega} \times d \mathbf {a} _ {1}
- d \mathbf {b} _ {2} \cdot \delta \pmb {\omega} \times \mathbf {a} _ {2} - b _ {2} \cdot \delta \pmb {\omega} \times d \mathbf {a} _ {2}.
Here g ^ { 1 } and g ^ { 2 } are coordinates in the plane of the element, normalized so that the nodes of the element are at (0,0), (1,0) and (0,1).
Internal virtual work rate
The internal virtual work rate is defined as
\delta W ^ {I} = \int_ {A} \int_ {h} \sigma^ {\alpha \beta} \delta \varepsilon_ {\alpha \beta} ^ {f} d z d A + \sum_ {\mathrm{nodes} (i)} K ^ {i} \gamma^ {i} \delta \gamma^ {i},
where \varepsilon _ { \alpha \beta } ^ { f } = \varepsilon _ { \alpha \beta } + z k _ { \alpha \beta } is the strain at a point, f , away from the reference surface; \sigma ^ { \alpha \beta } are the stress components at f ; h is the shell thickness; and K ^ { i } is the penalty stiffness used to constrain the spurious rotation.
The formulation now proceeds as for the shell elements described in ``Shear flexible small-strain shell elements,'' Section 3.6.3, using a 3-point integration scheme in the plane of the element.
3.6.5 Finite-strain shell element formulation
This section describes the formulation of the quadrilateral finite-membrane-strain element S4R, the triangular element S3R and S3 obtained through degeneration of S4R, and the fully integrated finite-membrane-strain element S4.
Geometric description
At a given stage in the deformation history of the shell, the position of a material point in the shell is defined by
\mathbf {x} (S _ {i}) = \overline {{\mathbf {x}}} (S _ {\alpha}) + \overline {{f}} _ {3 3} (S _ {\alpha}) \mathbf {t} _ {3} (S _ {\alpha}) S _ {3},
where the subscript i and other Roman subscripts range from 1 to 3. Subscripts ® and other lowercase Greek subscripts which describe the quantities in the reference surface of the shell range from 1 to 2. In the above equation \mathbf { t } _ { 3 } is the normal to the reference surface of the shell. The gradient of the position is
\frac {\partial \mathbf {x}}{\partial S _ {\beta}} = \frac {\partial \overline {{\mathbf {x}}}}{\partial S _ {\beta}} + \overline {{f}} _ {3 3} \frac {\partial \mathbf {t} _ {3}}{\partial S _ {\beta}} S _ {3}, \quad \frac {\partial \mathbf {x}}{\partial S _ {3}} = \overline {{f}} _ {3 3} \mathbf {t} _ {3},
where we have neglected derivatives of \overline { { f } } _ { 3 3 } with respect to S _ { \beta } . Note that in the above S _ { \alpha } are local surface coordinates that are assumed to be orthogonal and distance measuring in the reference state. S _ { 3 } is the coordinate in the thickness direction, distance measuring and orthogonal to S _ { \alpha } in the reference state. The thickness increase factor \overline { { f } } _ { 3 3 } is assumed to be independent of S _ { 3 } .
In the deformed state we define local, orthonormal shell directions \mathbf { t } _ { i } such that
\mathbf {t} _ {i} \cdot \mathbf {t} _ {j} = \delta_ {i j}, \quad \mathbf {t} _ {i} \mathbf {t} _ {i} = \mathbf {I},
where \delta _ { i j } is the Kronecker delta and I is the identity tensor of rank 2. Summation convention is used for repeated subscripts. The in-plane components of the gradient of the position are obtained as
f _ {\alpha \beta} = \mathbf {t} _ {\alpha} \cdot \frac {\partial \mathbf {x}}{\partial S _ {\beta}} = \overline {{f}} _ {\alpha \beta} + B _ {\alpha \beta} \overline {{f}} _ {3 3} S _ {3},
where we have introduced the reference surface deformation gradient
\overline {{f}} _ {\alpha \beta} \stackrel {\mathrm{def}} {=} \mathbf {t} _ {\alpha} \cdot \frac {\partial \mathbf {x}}{\partial S _ {\beta}} \bigg | _ {S _ {3} = 0} = \mathbf {t} _ {\alpha} \cdot \frac {\partial \overline {{\mathbf {x}}}}{\partial S _ {\beta}}
and the reference surface normal gradient
B _ {\alpha \beta} \stackrel {\mathrm{def}} {=} \mathbf {t} _ {\alpha} \cdot \frac {\partial \mathbf {t} _ {3}}{\partial S _ {\beta}}.
Elements
In the original (reference) configuration we denote the position by \mathbf { X } ( { \overline { { \mathbf { X } } } } for the reference surface) and the direction vectors by \mathbf { T } _ { i } , which yields
\mathbf {X} (S _ {i}) = \overline {{\mathbf {X}}} (S _ {\alpha}) + \mathbf {T} _ {3} (S _ {\alpha}) S _ {3}.
The gradient of the position is
\frac {\partial \mathbf {X}}{\partial S _ {\beta}} = \frac {\partial \overline {{\mathbf {X}}}}{\partial S _ {\beta}} + \frac {\partial \mathbf {T} _ {3}}{\partial S _ {\beta}} S _ {3}, \quad \frac {\partial \mathbf {X}}{\partial S _ {3}} = \mathbf {T} _ {3},
and the in-plane components of the gradient are obtained as
f _ {\alpha \beta} ^ {\circ} = \mathbf {T} _ {\alpha} \cdot \frac {\partial \mathbf {X}}{\partial S _ {\beta}} = \delta_ {\alpha \beta} + B _ {\alpha \beta} ^ {\circ} S _ {3},
where we have assumed that the in-plane direction vectors follow from the surface coordinates with
\mathbf {T} _ {\beta} = \frac {\partial \mathbf {X}}{\partial S _ {\beta}} \bigg | _ {S _ {3} = 0} = \frac {\partial \overline {{\mathbf {X}}}}{\partial S _ {\beta}}
and defined the original reference surface normal gradient,
B _ {\alpha \beta} ^ {\circ} \stackrel {\mathrm{def}} {=} \mathbf {T} _ {\alpha} \cdot \frac {\partial \mathbf {T} _ {3}}{\partial S _ {\beta}}.
The original reference surface normal gradient is obtained in the finite element formulation from the interpolation of the nodal normals with the shape functions. In the deformed configuration it is not derived from the nodal normals but is updated independently based on the gradient of the incremental rotations.
Parametric interpolation
The position of the points in the shell reference surface is described in terms of discrete nodal positions with parametric interpolation functions N ^ { I } ( \xi _ { \alpha } ) . The functions are C _ { \circ } continuous, and \xi _ { \alpha } are nonorthogonal, nondistance measuring parametric coordinates. For the reference surface positions one, thus, obtains
\overline {{\mathbf {x}}} (\xi_ {\alpha}) = N ^ {I} (\xi_ {\alpha}) \overline {{\mathbf {x}}} ^ {I}, \qquad \overline {{\mathbf {X}}} (\xi_ {\alpha}) = N ^ {I} (\xi_ {\alpha}) \overline {{\mathbf {X}}} ^ {I}.
The gradients of the position with respect to \xi _ { \beta } are
\frac {\partial \overline {{\mathbf {x}}}}{\partial \xi_ {\beta}} = \frac {\partial N ^ {I}}{\partial \xi_ {\beta}} \overline {{\mathbf {x}}} ^ {I}, \quad \frac {\partial \overline {{\mathbf {X}}}}{\partial \xi_ {\beta}} = \frac {\partial N ^ {I}}{\partial \xi_ {\beta}} \overline {{\mathbf {X}}} ^ {I}.
Elements
Note that uppercase Roman superscripts such as I denote nodes of an element and that repeated superscripts imply summation over all nodes of an element.
Now consider the original configuration. The unit normal to the shell reference surface is readily obtained as
\mathbf {T} _ {3} = \left(\frac {\partial \overline {{\mathbf {X}}}}{\partial \xi_ {1}} \times \frac {\partial \overline {{\mathbf {X}}}}{\partial \xi_ {2}}\right) \bigg / \left| \frac {\partial \overline {{\mathbf {X}}}}{\partial \xi_ {1}} \times \frac {\partial \overline {{\mathbf {X}}}}{\partial \xi_ {2}} \right|.
Subsequently, we define two orthonormal tangent vectors \mathbf { T } _ { \alpha } and distance measuring coordinates S _ { \alpha } along these vectors. The derivatives of these coordinates with respect to \xi _ { \beta } follow from
\frac {\partial S _ {\alpha}}{\partial \xi_ {\beta}} = \mathbf {T} _ {\alpha} \cdot \frac {\partial \overline {{\mathbf {X}}}}{\partial \xi_ {\beta}} = \mathbf {T} _ {\alpha} \cdot \overline {{\mathbf {X}}} ^ {I} \frac {\partial N ^ {I}}{\partial \xi_ {\beta}}.
The gradient of \xi _ { \alpha } with respect to S _ { \beta } is readily obtained by inversion:
\frac {\partial \xi_ {\alpha}}{\partial S _ {\beta}} = \left[ \frac {\partial S _ {\alpha}}{\partial \xi_ {\beta}} \right] ^ {- 1},
which makes it possible to obtain the gradient operator
\frac {\partial N ^ {I}}{\partial S _ {\beta}} = \frac {\partial N ^ {I}}{\partial \xi_ {\alpha}} \frac {\partial \xi_ {\alpha}}{\partial S _ {\beta}}.
The original reference surface normal gradient is obtained from the nodal normals \mathbf { T } _ { 3 } ^ { I } with
B _ {\alpha \beta} ^ {\circ} = \mathbf {T} _ {\alpha} \cdot \mathbf {T} _ {3} ^ {I} \frac {\partial N ^ {I}}{\partial S _ {\beta}}.
Since the original reference surface normal gradient is obtained by taking derivatives with respect to orthogonal distance measuring coordinates, we will call B _ { \alpha \beta } ^ { \circ } = b _ { \alpha \beta } ^ { \circ } the original curvature of the reference surface.
Membrane deformation and curvature
It is convenient to define the inverse of the reference surface deformation gradient
\overline {{h}} _ {\alpha \beta} = \left[ \overline {{f}} _ {\alpha \beta} \right] ^ {- 1}.
With this expression we can define the gradient operator in the current state:
\frac {\partial}{\partial s _ {\beta}} \stackrel {\mathrm{def}} {=} \overline {{h}} _ {\alpha \beta} \frac {\partial}{\partial S _ {\alpha}}, \quad \mathrm{orinverted} \quad \frac {\partial}{\partial S _ {\beta}} = \overline {{f}} _ {\alpha \beta} \frac {\partial}{\partial s _ {\alpha}}.
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The gradient operator in the current state can also be defined as the derivative with respect to distance measuring coordinates s _ { \alpha } along the base vectors \mathbf { t } _ { \alpha } , since
\mathbf {t} _ {\alpha} \cdot \frac {\partial \overline {{\mathbf {x}}}}{\partial s _ {\beta}} = \mathbf {t} _ {\alpha} \cdot \frac {\partial \overline {{\mathbf {x}}}}{\partial S _ {\gamma}} \overline {{h}} _ {\gamma \beta} = \overline {{f}} _ {\alpha \gamma} \overline {{h}} _ {\gamma \beta} = \delta_ {\alpha \beta}
and, hence,
\mathbf {t} _ {\alpha} = \frac {\partial \overline {{\mathbf {x}}}}{\partial s _ {\alpha}}.
Hence, it is possible to write for the \overline { { h } } _ { \alpha \beta }
\overline {{h}} _ {\alpha \beta} = \mathbf {T} _ {\alpha} \cdot \frac {\partial \overline {{\mathbf {X}}}}{\partial s _ {\beta}}
since
\overline {{f}} _ {\alpha \gamma} \overline {{h}} _ {\gamma \beta} = \mathbf {t} _ {\alpha} \cdot \frac {\partial \overline {{\mathbf {x}}}}{\partial \overline {{\mathbf {X}}}} \cdot \mathbf {T} _ {\gamma} \mathbf {T} _ {\gamma} \frac {\partial \overline {{\mathbf {X}}}}{\partial \overline {{\mathbf {x}}}} \cdot \mathbf {t} _ {\beta} = \mathbf {t} _ {\alpha} \cdot \frac {\partial \overline {{\mathbf {x}}}}{\partial \overline {{\mathbf {X}}}} \cdot \frac {\partial \overline {{\mathbf {X}}}}{\partial \overline {{\mathbf {x}}}} \cdot \mathbf {t} _ {\beta} = \mathbf {t} _ {\alpha} \cdot \mathbf {t} _ {\beta} = \delta_ {\alpha \beta}.
In an incremental analysis we can also define the incremental deformation tensor
\Delta \overline {{f}} _ {\alpha \beta} = \mathbf {t} _ {\alpha} ^ {t + \Delta t} \cdot \frac {\partial \overline {{\mathbf {x}}} ^ {t + \Delta t}}{\partial s _ {\beta} ^ {t}}
and its inverse
\Delta \overline {{h}} _ {\alpha \beta} = \mathbf {t} _ {\alpha} ^ {t} \cdot \frac {\partial \overline {{\mathbf {x}}} ^ {t}}{\partial s _ {\beta} ^ {t + \Delta t}}.
With a local coordinate system defined in the current state, the current gradient of the normal can be transformed into the curvature of the surface:
b _ {\alpha \beta} \stackrel {\mathrm{def}} {=} \mathbf {t} _ {\alpha} \cdot \frac {\partial \mathbf {t} _ {3}}{\partial s _ {\beta}} = B _ {\alpha \gamma} \overline {{h}} _ {\gamma \beta}.
Orientation update
The equations given in the earlier sections are valid for any local coordinate system defined in the current state. The \mathbf { t } _ { \alpha } vectors at the beginning of the analysis are determined following the standard ABAQUS conventions. In this section, we outline the way in which the in-plane coordinates are made corotational.
To obtain the updated version of \mathbf { t } _ { \alpha } , we follow a two-step approach. First, we construct orthogonal
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vectors \hat { \mathbf { t } } _ { \alpha } tangential to the surface (following ABAQUS conventions). Subsequently, we calculate
\hat {f} _ {\alpha \beta} = \hat {\mathbf {t}} _ {\alpha} \cdot \frac {\partial \overline {{\mathbf {x}}} ^ {t + \Delta t}}{\partial S _ {\beta}}.
We then apply an in-plane rotation \Delta R _ { \alpha \beta } to the vectors: \hat { \mathbf { t } } _ { \alpha } :
\Delta R _ {1 1} = \Delta R _ {2 2} = \cos \Delta \psi ,
\Delta R _ {2 1} = - \Delta R _ {1 2} = \sin \Delta \psi ,
where \Delta \psi is to be determined such that the resulting deformation tensor is symmetric, as
\bar {f} _ {\alpha \beta} = \Delta R _ {\alpha \gamma} \hat {f} _ {\gamma \beta} = \hat {f} _ {\gamma \alpha} \Delta R _ {\beta \gamma} = \bar {f} _ {\beta \alpha}.
From this follows
\tan \Delta \psi = \frac {(\hat {f} _ {1 2} - \hat {f} _ {2 1})}{\hat {f} _ {1 1} + \hat {f} _ {2 2}}.
Thus, we can calculate the updated local material directions as
\bar {\mathbf {t}} _ {\alpha} ^ {t + \Delta t} = \Delta R _ {\alpha \gamma} \hat {\mathbf {t}} _ {\gamma}.
Curvature change
We assume that the nodal spin will be interpolated with the interpolation functions N ^ { I } ( \xi _ { \alpha } ) . During an increment the nodal spin is assumed to be constant; consequently, the value of the spin at each material point will be constant. Hence, we can use the same interpolation functions for the incremental finite rotation vector \Delta \phi \mathrm { : }
\Delta \pmb {\phi} = \pmb {\omega} \Delta t = N ^ {I} (\xi_ {\alpha}) \pmb {\omega} ^ {I} \Delta t = N ^ {I} (\xi_ {\alpha}) \Delta \pmb {\phi} ^ {I}.
The finite rotation vector can be split in a rotation amplitude \Delta \phi and a rotation axis \mathbf { p } \mathrm { : }
\Delta \pmb {\phi} = \Delta \phi \mathbf {p}, \qquad \mathrm{with} \quad \Delta \phi \stackrel {\mathrm{def}} {=} | \Delta \pmb {\phi} | \quad \mathrm{and} \quad \mathbf {p} = \Delta \pmb {\phi} / \Delta \pmb {\phi}.
To rotate the shell normal, we use quaternion algebra. The incremental nodal rotation is represented by the rotation quaternion \Delta \pmb q . , which is defined by
\Delta \pmb {q} \stackrel {\mathrm{def}} {=} \left(\cos \frac {\Delta \phi}{2}, \sin \frac {\Delta \phi}{2} \mathbf {p}\right).
Elements
An updated shell normal is then obtained according to
\tilde {\mathbf {t}} _ {3} ^ {t + \Delta t} = \Delta \boldsymbol {q} \mathbf {t} _ {3} ^ {t} \Delta \boldsymbol {q} ^ {\dagger}.
This updated shell normal does not actually have to be calculated: it is used only for the derivation of the expression for the curvature change. It is not equal to the shell normal used at the start of the next increment \mathbf { t } _ { 3 } ^ { t + \Delta t } , which will again be chosen perpendicular to the reference surface. The updated normal used here will be approximately orthogonal to the reference surface, depending upon the amount of transverse shear deformation. The gradient of the updated shell normal can be obtained by differentiation:
\frac {\partial \tilde {\mathbf {t}} _ {3} ^ {t + \Delta t}}{\partial S _ {\beta}} = \frac {\partial \Delta \pmb {q}}{\partial S _ {\beta}} \mathbf {t} _ {3} ^ {t} \Delta \pmb {q} ^ {\dagger} + \Delta \pmb {q} \mathbf {t} _ {3} ^ {t} \frac {\partial \Delta \pmb {q} ^ {\dagger}}{\partial S _ {\beta}} + \Delta \pmb {q} \frac {\partial \mathbf {t} _ {3} ^ {t}}{\partial S _ {\beta}} \Delta \pmb {q} ^ {\dagger}.
The second term on the right-hand side can be written in the form
\Delta \pmb {q} \mathbf {t} _ {3} ^ {t} \frac {\partial \Delta \pmb {q} ^ {\dagger}}{\partial S _ {\beta}} = \left[ \frac {\partial \Delta \pmb {q}}{\partial S _ {\beta}} \left(\mathbf {t} _ {3} ^ {t}\right) ^ {\dagger} \Delta \pmb {q} ^ {\dagger} \right] ^ {\dagger} = - \left[ \frac {\partial \Delta \pmb {q}}{\partial S _ {\beta}} \mathbf {t} _ {3} ^ {t} \Delta \pmb {q} ^ {\dagger} \right] ^ {\dagger}.
Hence, the scalar parts of the first two terms cancel each other and the vector parts reinforce each other, leading to
\frac {\partial \tilde {\mathbf {t}} _ {3} ^ {t + \Delta t}}{\partial S _ {\beta}} = 2 \mathbf {V} \left(\frac {\partial \Delta \pmb {q}}{\partial S _ {\beta}} \mathbf {t} _ {3} ^ {t} \Delta \pmb {q} ^ {\dagger}\right) + \Delta \pmb {q} \frac {\partial \mathbf {t} _ {3} ^ {t}}{\partial S _ {\beta}} \Delta \pmb {q} ^ {\dagger}.
The inverse of a rotation quaternion such as \Delta \pmb q is equal to its conjugate ( \Delta \pmb q ^ { - 1 } = \Delta \pmb q ^ { \dag } ) . Hence, we can write
\begin{array}{l} \frac {\partial \tilde {\mathbf {t}} _ {3} ^ {t + \Delta t}}{\partial S _ {\beta}} = 2 \mathbf {V} \left(\frac {\partial \Delta \boldsymbol {q}}{\partial S _ {\beta}} \Delta \boldsymbol {q} ^ {\dagger} \Delta \boldsymbol {q} \mathbf {t} _ {3} ^ {t} \Delta \boldsymbol {q} ^ {\dagger}\right) + \Delta \boldsymbol {q} \frac {\partial \mathbf {t} _ {3} ^ {t}}{\partial S _ {\beta}} \Delta \boldsymbol {q} ^ {\dagger} \\ = \mathbf {V} \left(2 \frac {\partial \Delta \pmb {q}}{\partial S _ {\beta}} \Delta \pmb {q} ^ {\dagger} \tilde {\mathbf {t}} _ {3} ^ {t + \Delta t}\right) + \Delta \pmb {q} \frac {\partial \mathbf {t} _ {3} ^ {t}}{\partial S _ {\beta}} \Delta \pmb {q} ^ {\dagger} \\ = \Delta \mathbf {R} _ {\beta} \times \tilde {\mathbf {t}} _ {3} ^ {t + \Delta t} + \Delta \pmb {q} \frac {\partial \mathbf {t} _ {3} ^ {t}}{\partial S _ {\beta}} \Delta \pmb {q} ^ {\dagger}, \\ \end{array}
where we have formally defined the incremental gradient update vectors
\Delta \mathbf {R} _ {\beta} \stackrel {\mathrm{def}} {=} \mathbf {V} \left(2 \frac {\partial \Delta \pmb {q}}{\partial S _ {\beta}} \Delta \pmb {q} ^ {\dagger}\right),
which must be expressed in terms of the gradient of the incremental rotation. From the definition of the incremental quaternion \Delta \pmb q follows
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\frac {\partial \Delta \pmb {q}}{\partial S _ {\beta}} = \left(\frac {1}{2} \sin \frac {\Delta \phi}{2} \frac {\partial \Delta \phi}{\partial S _ {\beta}}, \frac {1}{2} \cos \frac {\Delta \phi}{2} \mathbf {p} \frac {\partial \Delta \phi}{\partial S _ {\beta}} + \sin \frac {\Delta \phi}{2} \frac {\partial \mathbf {p}}{\partial S _ {\beta}}\right);
thus, for \Delta \mathbf { R } _ { \beta } , again with use of the incremental quaternion definition
\Delta \mathbf {R} _ {\beta} = \mathbf {p} \frac {\partial \Delta \phi}{\partial S _ {\beta}} + \sin \Delta \phi \frac {\partial \mathbf {p}}{\partial S _ {\beta}} + (1 - \cos \Delta \phi) \mathbf {p} \times \frac {\partial \mathbf {p}}{\partial S _ {\beta}}.
From the definition of \Delta \phi and p follows
\frac {\partial \Delta \phi}{\partial S _ {\beta}} = \frac {1}{\Delta \phi} \Delta \phi \cdot \frac {\partial \Delta \phi}{\partial S _ {\beta}},
\frac {\partial \mathbf {p}}{\partial S _ {\beta}} = \frac {1}{\Delta \phi} \frac {\partial \Delta \pmb {\phi}}{\partial S _ {\beta}} - \frac {1}{\Delta \phi^ {3}} \Delta \pmb {\phi} \Delta \pmb {\phi} \cdot \frac {\partial \Delta \pmb {\phi}}{\partial S _ {\beta}} = \frac {1}{\Delta \phi} \left[ \frac {\partial \Delta \pmb {\phi}}{\partial S _ {\beta}} - \mathbf {p} \mathbf {p} \cdot \frac {\partial \Delta \pmb {\phi}}{\partial S _ {\beta}} \right].
After substitution in the expression for \Delta \mathbf { R } _ { \beta } and some algebra one obtains
\Delta \mathbf {R} _ {\beta} = \frac {\sin \Delta \phi}{\Delta \phi} \frac {\partial \Delta \pmb {\phi}}{\partial S _ {\beta}} + \frac {1 - \cos \Delta \phi}{\Delta \phi} \mathbf {p} \times \frac {\partial \Delta \pmb {\phi}}{\partial S _ {\beta}} + \left(1 - \frac {\sin \Delta \phi}{\Delta \phi}\right) \mathbf {p} \mathbf {p} \cdot \frac {\partial \Delta \pmb {\phi}}{\partial S _ {\beta}}.
Note that \Delta { \bf R } _ { \beta } \frac { \partial \Delta \phi } { \partial S _ { \beta } } @S¯ when \Delta \phi \to 0 .
For the gradient B _ { \alpha \beta } of the updated shell normal we obtain
\begin{array}{l} B _ {\alpha \beta} ^ {t + \Delta t} = \mathbf {t} _ {\alpha} ^ {t + \Delta t} \cdot \frac {\partial \tilde {\mathbf {t}} _ {3} ^ {t + \Delta t}}{\partial S _ {\beta}} = \mathbf {t} _ {\alpha} ^ {t + \Delta t} \left[ \Delta \mathbf {R} _ {\beta} \times \tilde {\mathbf {t}} _ {3} ^ {t + \Delta t} + \Delta \pmb {q} \mathbf {t} _ {\gamma} ^ {t} B _ {\alpha \beta} ^ {t} \Delta \pmb {q} ^ {\dagger} \right] \\ = \Delta \mathbf {R} _ {\beta} \cdot \left(\tilde {\mathbf {t}} _ {3} ^ {t + \Delta t} \times \mathbf {t} _ {\alpha} ^ {t + \Delta t}\right) + \mathbf {t} _ {\alpha} ^ {t + \Delta t} \cdot \mathbf {t} _ {\gamma} ^ {t + \Delta t} B _ {\gamma \beta} ^ {t} = \epsilon_ {\alpha} ^ {\gamma} \mathbf {t} _ {\gamma} ^ {t + \Delta t} \cdot \Delta \mathbf {R} _ {\beta} + B _ {\alpha \beta} ^ {t}, \\ \end{array}
where we have introduced the two-dimensional alternator \epsilon _ { \alpha } ^ { \beta } :
\epsilon_ {1} ^ {1} = \epsilon_ {2} ^ {2} = 0, \qquad \epsilon_ {1} ^ {2} = - \epsilon_ {2} ^ {1} = 1.
Note that the change in B _ { \alpha \beta } is independent of B _ { \alpha \beta } ^ { t }
Calculation of B _ { \alpha \beta } involves taking the gradient with respect to the reference configuration. It is more convenient to use the reference surface curvature tensor
b _ {\alpha \beta} \stackrel {\mathrm{def}} {=} B _ {\alpha \gamma} \overline {{h}} _ {\gamma \beta} = \mathbf {t} _ {\alpha} \cdot \frac {\partial \mathbf {t} _ {3}}{\partial s _ {\beta}}.
We then introduce the incremental curvature update vectors
\Delta \mathbf {r} _ {\beta} \stackrel {\mathrm{def}} {=} \Delta \mathbf {R} _ {\gamma} \overline {{h}} _ {\gamma \beta} = \frac {\sin \Delta \phi}{\Delta \phi} \frac {\partial \Delta \pmb {\phi}}{\partial s _ {\beta}} + \frac {1 - \cos \Delta \phi}{\Delta \phi} \mathbf {p} \times \frac {\partial \Delta \pmb {\phi}}{\partial s _ {\beta}} + \left(1 - \frac {\sin \Delta \phi}{\Delta \phi}\right) \mathbf {p} \cdot \frac {\partial \Delta \pmb {\phi}}{\partial s _ {\beta}},
which makes it possible to write the update equation as
b _ {\alpha \beta} ^ {t + \Delta t} = \epsilon_ {\alpha} ^ {\gamma} \mathbf {t} _ {\gamma} ^ {t + \Delta t} \cdot \Delta \mathbf {r} _ {\beta} + b _ {\alpha \gamma} ^ {t} \Delta \overline {{h}} _ {\gamma \beta}.
This expression makes it feasible to calculate the update in the reference surface curvature by taking gradients in the latest updated state only.
Deformation gradient
We already have obtained an expression for the deformation gradient in the reference surface, and we have assumed that the thickness change is constant:
\overline {{F}} _ {\alpha \beta} = \overline {{f}} _ {\alpha \beta}, \qquad \overline {{F}} _ {3 3} = \overline {{f}} _ {3 3}.
At other points in the shell we obtain for the in-plane component
F _ {\alpha \beta} = \mathbf {t} _ {\alpha} \cdot \frac {\partial \mathbf {x}}{\partial S _ {\gamma}} \left(\mathbf {T} _ {\beta} \cdot \frac {\partial \mathbf {X}}{\partial S _ {\gamma}}\right) ^ {- 1} = (\overline {{f}} _ {\alpha \gamma} + \overline {{f}} _ {3 3} S _ {3} B _ {\alpha \gamma}) (\delta_ {\gamma \beta} + S _ {3} B _ {\gamma \beta} ^ {\circ}) ^ {- 1}.
We neglect terms of order ( S _ { 3 } ) ^ { 2 } , which yields the simplified relation
F _ {\alpha \beta} = \overline {{F}} _ {\alpha \beta} + S _ {3} (\overline {{f}} _ {3 3} B _ {\alpha \beta} - \overline {{f}} _ {\alpha \gamma} B _ {\gamma \beta} ^ {\circ}).
We can write this as the product of a finite-membrane deformation and a bending perturbation:
\begin{array}{l} F _ {\alpha \beta} = \left[ \delta_ {\alpha \gamma} + S _ {3} (\overline {{f}} _ {3 3} B _ {\alpha \delta} \overline {{h}} _ {\delta \gamma} - \overline {{f}} _ {\alpha \delta} B _ {\delta \varepsilon} ^ {\circ} \overline {{h}} _ {\varepsilon \gamma}) \right] \overline {{f}} _ {\gamma \beta} \\ = \left[ \delta_ {\alpha \gamma} + S _ {3} (\overline {{f}} _ {3 3} b _ {\alpha \gamma} - \overline {{f}} _ {\alpha \delta} b _ {\delta \varepsilon} ^ {\circ} \overline {{h}} _ {\varepsilon \gamma}) \right] \overline {{f}} _ {\gamma \beta}. \\ \end{array}
It will be assumed that the deformation (strain and rotation) due to bending is small and, therefore,
S _ {3} (\overline {{f}} _ {3 3} b _ {\alpha \gamma} - \overline {{f}} _ {\alpha \delta} b _ {\delta \varepsilon} ^ {\circ} \overline {{h}} _ {\varepsilon \gamma}) \ll 1.
Membrane strain increment
The membrane strain increment follows from the incremental stretch tensor \Delta { \bf V } , whose components follow from the incremental deformation gradient \Delta \overline { { f } } _ { \alpha \beta } by the polar decomposition
\Delta \overline {{f}} _ {\alpha \beta} = \Delta \overline {{V}} _ {\alpha \gamma} \Delta \overline {{R}} _ {\gamma \beta}.
Let \overline { { f } } _ { \alpha \beta } ^ { t } and { \overline { { f } } } _ { \alpha \beta } ^ { t + \Delta t } be the deformation gradient at the beginning and the end of the increment, respectively. By definition \overline { { f } } _ { \alpha \beta } ^ { t + \Delta t } = \Delta \overline { { f } } _ { \alpha \delta } \overline { { f } } _ { \delta \beta } ^ { t } . The incremental deformation gradient follows as
\Delta \overline {{f}} _ {\alpha \beta} = \overline {{f}} _ {\alpha \delta} ^ {t + \Delta t} \left(\overline {{f}} ^ {t ^ {- 1}}\right) _ {\delta \beta}.
Elements
Since \Delta \overline { { R } } _ { \gamma \beta } are the components of an orthogonal matrix, the square of the incremental stretch tensor can be obtained by
\Delta \overline {{f}} _ {\alpha \gamma} \Delta \overline {{f}} _ {\beta \gamma} = \Delta \overline {{V}} _ {\alpha \gamma} \Delta \overline {{V}} _ {\beta \gamma} = \sum_ {I = 1} ^ {2} (\Delta \lambda_ {I}) ^ {2} a _ {\alpha} ^ {I} a _ {\beta} ^ {I}
(see ``Deformation,'' Section 1.4.1). The logarithmic strain increment is then
\Delta \epsilon_ {\alpha \beta} = \sum_ {I = 1} ^ {2} \ln (\Delta \lambda_ {I}) a _ {\alpha} ^ {I} a _ {\beta} ^ {I}
and the average material rotation increment is defined from the polar decomposition:
\Delta \overline {{R}} _ {\alpha \beta} = \sum_ {I = 1} ^ {2} \frac {1}{\Delta \lambda_ {I}} a _ {\alpha} ^ {I} a _ {\gamma} ^ {I} \Delta \overline {{f}} _ {\gamma \beta}.
Due to the choice of the element basis directions, it follows that
\Delta \overline {{R}} _ {\alpha \beta} \approx \delta_ {\alpha \beta}.
Curvature increment
Following Koiter-Sanders shell theory, and compensating for the rotation of the base vectors relative to the material, we define the physical curvature increment \Delta \kappa _ { \alpha \beta } as
\Delta \kappa_ {\alpha \beta} = \mathrm{sym} \left[ b _ {\alpha \beta} ^ {t + \Delta t} - b _ {\alpha \gamma} ^ {t} \Delta \overline {{R}} _ {\beta \gamma} + b _ {\alpha \gamma} ^ {t} \Delta \overline {{R}} _ {\delta \gamma} \Delta \overline {{\varepsilon}} _ {\delta \beta} \right] = \mathrm{sym} \left[ b _ {\alpha \beta} ^ {t + \Delta t} - b _ {\alpha \gamma} ^ {t} \Delta \overline {{R}} _ {\delta \gamma} (\delta_ {\delta \beta} - \Delta \overline {{\varepsilon}} _ {\delta \beta}) \right].
Neglecting terms of the order ( \Delta \overline { { \varepsilon } } _ { \alpha \beta } ) ^ { 2 } relative to \Delta \overline { { \mathcal { E } } } _ { \alpha \beta } , this expression can be rewritten as
\Delta \kappa_ {\alpha \beta} = \mathrm{sym} \left[ b _ {\alpha \beta} ^ {t + \Delta t} - b _ {\alpha \gamma} ^ {t} \Delta \overline {{R}} _ {\delta \gamma} \Delta \overline {{V}} _ {\delta \beta} ^ {- 1} \right] = \mathrm{sym} \left[ b _ {\alpha \beta} ^ {t + \Delta t} - b _ {\alpha \gamma} ^ {t} \Delta \overline {{h}} _ {\gamma \beta} \right] = \mathrm{sym} \left[ \epsilon_ {\alpha} ^ {\gamma} \mathbf {t} _ {\gamma} ^ {t + \Delta t} \cdot \Delta \mathbf {r} _ {\beta} \right],
where use was made of the curvature update formula. Observe that the curvature at the beginning of the increment, b _ { \alpha \beta } ^ { t } . , does not appear in this equation. Hence, there is no need to calculate the initial curvature b _ { \alpha \beta } ^ { \circ } , and we can assume b _ { \alpha \beta } ^ { \circ } = 0 . The deformation gradient can, hence, also be simplified to
F _ {\alpha \beta} = \overline {{f}} _ {\alpha \beta} + S _ {3} \overline {{f}} _ {3 3} b _ {\alpha \gamma} \overline {{f}} _ {\gamma \beta}.
For the material strain increment at a point through the shell thickness Koiter-Sanders theory thus yields