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Elements
r (R, Z) = R + u _ {r} (R, Z),
z (R, Z) = Z + u _ {z} (R, Z),
\theta (R, Z, \Theta) = \Theta + \phi (R, Z).
As the above description implies, the degrees of freedom u _ { r } , u _ { z } , , and \phi are independent of £ . Moreover, the reference cross-section of interest is at \Theta = 0 , but for the benefit of the mathematical analysis to follow it is important that £ be nonzero in the above expression for µ.
Parametric interpolation and integration
The following isoparametric interpolation scheme for the motion is used:
u _ {r} = N ^ {N} (g, h) \bar {u _ {r}} ^ {N},
u _ {z} = N ^ {N} (g, h) \bar {u _ {z}} ^ {N},
\phi = N ^ {N} (g, h) \bar {\phi} ^ {N},
where g, h are isoparametric coordinates in the reference r-z cross-section at \Theta = 0 ; , and \bar { u _ { r } } ^ { N } , \bar { u _ { z } } ^ { N } , \bar { \phi } ^ { N } are the nodal degrees of freedom. The interpolation functions N ^ { N } ( g , h ) are those described in ``Solid isoparametric quadrilaterals and hexahedra, '' Section 3.2.4, where the integration scheme of isoparametric solid elements is also discussed.
Deformation gradient
For a material point in space, the deformation gradient F is defined as the gradient of the current position x with respect to the original position X:
\mathbf {F} = \frac {\partial \mathbf {x}}{\partial \mathbf {X}}.
The current position x is given by Equation 3.2.8-1, and the gradient operator can be described in terms of partial derivatives with respect to the cylindrical coordinates:
\frac {\partial (\cdot)}{\partial \mathbf {X}} = \frac {\partial (\cdot)}{\partial R} \mathbf {e} _ {R} + \frac {\partial (\cdot)}{\partial Z} \mathbf {e} _ {Z} + \frac {1}{R} \frac {\partial (\cdot)}{\partial \Theta} \mathbf {e} _ {\Theta}.
Since the radial and circumferential base vectors depend on the original circumferential coordinate µ, the partial derivatives of these base vectors with respect to µ are nonvanishing:
\frac {\partial \mathbf {e} _ {r}}{\partial \theta} = \mathbf {e} _ {\theta}, \quad \frac {\partial \mathbf {e} _ {\theta}}{\partial \theta} = - \mathbf {e} _ {r}.
Thus, the chain rule allows us to write
\frac {\partial \mathbf {e} _ {r}}{\partial R} = \frac {\partial \mathbf {e} _ {r}}{\partial \theta} \frac {\partial \theta}{\partial R} = \frac {\partial \theta}{\partial R} \mathbf {e} _ {\theta} \quad \mathrm{and} \quad \frac {\partial \mathbf {e} _ {r}}{\partial Z} = \frac {\partial \mathbf {e} _ {r}}{\partial \theta} \frac {\partial \theta}{\partial Z} = \frac {\partial \theta}{\partial Z} \mathbf {e} _ {\theta}.
With these results, the deformation gradient is obtained as
Equation 3.2.8-3
\begin{array}{l} \mathbf {F} = \frac {\partial r}{\partial R} \mathbf {e} _ {r} \mathbf {e} _ {R} + \frac {\partial r}{\partial Z} \mathbf {e} _ {r} \mathbf {e} _ {Z} + \frac {\partial z}{\partial R} \mathbf {e} _ {z} \mathbf {e} _ {R} + \frac {\partial z}{\partial Z} \mathbf {e} _ {z} \mathbf {e} _ {Z} + \\ r \frac {\partial \theta}{\partial R} \mathbf {e} _ {\theta} \mathbf {e} _ {R} + r \frac {\partial \theta}{\partial Z} \mathbf {e} _ {\theta} \mathbf {e} _ {Z} + \frac {r}{R} \frac {\partial \theta}{\partial \Theta} \mathbf {e} _ {\theta} \mathbf {e} _ {\Theta}. \\ \end{array}
Alternatively, it can be written in matrix form as
\left[ F \right] = \left[ \begin{array}{c c c} 1 + \partial u _ {r} / \partial R & \partial u _ {r} / \partial Z & 0 \\ \partial u _ {z} / \partial R & 1 + \partial u _ {z} / \partial Z & 0 \\ r \partial \phi / \partial R & r \partial \phi / \partial Z & r / R \end{array} \right],
where the motion given by Equation 3.2.8-2 has been used explicitly.
Similarly, the inverse deformation gradient \mathbf { F } ^ { - 1 } is readily obtained as
\begin{array}{l} \mathbf {F} ^ {- 1} = \frac {\partial R}{\partial r} \mathbf {e} _ {R} \mathbf {e} _ {r} + \frac {\partial R}{\partial z} \mathbf {e} _ {R} \mathbf {e} _ {z} + \frac {\partial Z}{\partial r} \mathbf {e} _ {Z} \mathbf {e} _ {r} + \frac {\partial Z}{\partial z} \mathbf {e} _ {Z} \mathbf {e} _ {z} + \\ R \frac {\partial \Theta}{\partial r} \mathbf {e} _ {\Theta} \mathbf {e} _ {r} + R \frac {\partial \Theta}{\partial z} \mathbf {e} _ {\Theta} \mathbf {e} _ {z} + \frac {R}{r} \frac {\partial \Theta}{\partial \theta} \mathbf {e} _ {\Theta} \mathbf {e} _ {\theta}. \\ \end{array}
Virtual work
As discussed in ``Equilibrium and virtual work,'' Section 1.5.1, the formulation of equilibrium (virtual work) requires the virtual velocity gradient L, which is the variation in the gradient of the position with respect to the current state. This tensor is given by
Equation 3.2.8-4
\delta \mathbf {L} = \delta \mathbf {F} \cdot \mathbf {F} ^ {- 1}
where ±F is the linearized deformation gradient.
ABAQUS formulates the finite element equations in terms of a fixed spatial basis with respect to the axisymmetric twist degree of freedom. Therefore, the desired result for ±F in Equation 3.2.8-4 does not simply follow from the linearization of Equation 3.2.8-3. Namely, it is necessary to cancel out the contributions from the variations
\delta \mathbf {e} _ {r} = \delta \phi \mathbf {e} _ {\theta} \quad \mathrm{and} \quad \delta \mathbf {e} _ {\theta} = - \delta \phi \mathbf {e} _ {r}.
To this end F can be modified according to
\tilde {\mathbf {F}} = \mathbf {R} \cdot \mathbf {F},
where R = I instantaneously, but its variation is given by
\delta \mathbf {R} = \widehat {\delta \phi} \cdot \mathbf {R},
where \widehat { \delta \phi } is skew-symmetric with components
\left[ \widehat {\delta \phi} \right] = \left[ \begin{array}{c c c} 0 & \delta \phi & 0 \\ - \delta \phi & 0 & 0 \\ 0 & 0 & 0 \end{array} \right],
with respect to the basis \mathbf { e } _ { r } , \mathbf { e } _ { z } , and \mathbf { e } _ { \theta } at \Theta = 0
With this modification the corotational virtual deformation gradient is given by
\delta \tilde {\mathbf {F}} = \delta \mathbf {R} \cdot \mathbf {F} + \mathbf {R} \cdot \delta \mathbf {F} = \widehat {\delta \boldsymbol {\phi}} \cdot \mathbf {F} + \delta \mathbf {F},
and the corotational virtual velocity gradient by
\delta \tilde {\mathbf {L}} = \delta \tilde {\mathbf {F}} \cdot \tilde {\mathbf {F}} ^ {- 1} = \widehat {\delta \pmb {\phi}} + \delta \mathbf {L},
or
\delta \tilde {\mathbf {L}} = \frac {\partial \delta r}{\partial r} \mathbf {e} _ {r} \mathbf {e} _ {r} + \frac {\partial \delta r}{\partial z} \mathbf {e} _ {r} \mathbf {e} _ {z} + \frac {\partial \delta z}{\partial r} \mathbf {e} _ {z} \mathbf {e} _ {r} + \frac {\partial \delta z}{\partial z} \mathbf {e} _ {z} \mathbf {e} _ {z} +
r \frac {\partial \delta \theta}{\partial r} \mathbf {e} _ {\theta} \mathbf {e} _ {r} + r \frac {\partial \delta \theta}{\partial z} \mathbf {e} _ {\theta} \mathbf {e} _ {z} + \frac {\delta r}{r} \mathbf {e} _ {\theta} \mathbf {e} _ {\theta}.
Equation 3.2.8-5
The modified virtual rate of deformation tensor is simply
\delta \tilde {\mathbf {D}} = \frac {1}{2} \left(\delta \tilde {\mathbf {L}} + \delta \tilde {\mathbf {L}} ^ {T}\right).
Stiffness in the current state
As shown in ``Procedures: overview and basic equations, '' Section 2.1.1, the contribution of the internal work terms to the Jacobian of the Newton method that is used in ABAQUS/Standard for solid element formulations is
d \delta \Pi = \int_ {V} \left(d \pmb {\sigma}: \delta \tilde {\mathbf {D}} + \pmb {\sigma}: d \delta \tilde {\mathbf {D}}\right) d V.
The second variation in \delta \tilde { \mathbf { D } } is obtained as
d \delta \tilde {\mathbf {D}} = d \tilde {\mathbf {L}} ^ {T} \cdot \delta \tilde {\mathbf {L}} - 2 d \tilde {\mathbf {D}} \cdot \delta \tilde {\mathbf {D}} + d \delta \tilde {\mathbf {F}} \cdot \tilde {\mathbf {F}} ^ {- 1},
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where d \tilde { \mathbf { L } } has the same form as \delta \tilde { \mathbf { L } } in Equation 3.2.8-5. Moreover, in this formulation d \delta \tilde { \mathbf { F } } is nonzero, and it can be shown with the aid of ``Rotation variables,'' Section 1.3.1, that d \delta \tilde { \mathbf { F } } \cdot \tilde { \mathbf { F } } ^ { - 1 } has the form
d \delta \tilde {\mathbf {F}} \cdot \tilde {\mathbf {F}} ^ {- 1} = \frac {1}{2} (\widehat {\delta \pmb {\phi}} \cdot \widehat {d \pmb {\phi}} + \widehat {d \pmb {\phi}} \cdot \widehat {\delta \pmb {\phi}}) + \widehat {\delta \pmb {\phi}} \cdot d \tilde {\mathbf {L}} + \widehat {d \pmb {\phi}} \cdot \delta \tilde {\mathbf {L}} + d \delta \mathbf {F} \cdot \mathbf {F} ^ {- 1}.
In component form,
d \delta \tilde {\mathbf {F}} \cdot \tilde {\mathbf {F}} ^ {- 1} = \left(\frac {\partial d \theta}{\partial r} \delta r + d r \frac {\partial \delta \theta}{\partial r}\right) \mathbf {e} _ {\theta} \mathbf {e} _ {r} + \left(\frac {\partial d \theta}{\partial z} \delta r + d r \frac {\partial \delta \theta}{\partial z}\right) \mathbf {e} _ {\theta} \mathbf {e} _ {z}.
3.2.9 Axisymmetric elements allowing nonlinear bending
ABAQUS/Standard includes a library of solid elements whose geometry is initially axisymmetric and that allow for nonlinear analysis in which bending can occur about the plane \theta = \pi / 2 in the ( r , z , \theta ) cylindrical coordinate system of the model. The geometric model is defined in the r-z plane only. The displacements are the usual isoparametric interpolations with respect to r and z, augmented by Fourier expansions with respect to µ. Since the elements are written for bending about the plane \theta = \pi / 2 only, they cannot be used to model torsion of the structure about the original axis of symmetry. Because the elements are intended for nonlinear applications, the orthogonality properties associated with Fourier modes cannot be used to reduce the problem to a series of smaller, uncoupled, cases, since the stiffness before projection onto the Fourier modes is not necessarily constant. For this reason these elements are significantly more expensive to use than the corresponding axisymmetric elements intended for axisymmetric deformations.
Interpolation
The coordinate system used with these elements is the cylindrical system ( r , z , \theta ) , where r measures the distance of a point from the axis of the cylindrical system, z measures its position along this axis, and µ measures the angle between the plane containing the point and the axis of the coordinate system and some fixed reference plane that contains the coordinate system axis. The order in which the coordinates and displacements are taken in these elements is based on the convention used in ABAQUS for axisymmetric elements, so that z is the second coordinate. This allows these elements to be used in conjunction with other elements in the library that allow only axisymmetric deformation. This order is not the same as that used in three-dimensional elements in ABAQUS in which z is the third coordinate, nor is it the order ( r , \theta , z ) , usually taken in cylindrical systems.
The original geometry of the elements is assumed to be axisymmetric with respect to the axis of the coordinate system and, thus, independent of µ. Let \mathbf { e } _ { r } , \mathbf { e } _ { z } , and \mathbf { e } _ { \theta } be unit vectors in the radial, axial, and circumferential directions at a point in the undeformed state. The reference position X of the point can be represented in terms of the original radius R and the axial position Z:
\mathbf {X} = R \mathbf {e} _ {r} + Z \mathbf {e} _ {z}.
Similarly the displacement u of the point can be represented in terms of the components u _ { r } , u _ { z } , and u _ { \theta }
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with respect to these same vectors at the original position of the point:
\mathbf {u} = u _ {r} \mathbf {e} _ {r} + u _ {z} \mathbf {e} _ {z} + u _ {\theta} \mathbf {e} _ {\theta}.
For small radial and circumferential displacements the circumferential displacement is proportional to the change in circumferential angle ( u _ { \boldsymbol { \theta } } = R \Delta \boldsymbol { \theta } ) , but for large displacements this relation becomes nonlinear ( u _ { \theta } = \left( R + u _ { r } \right) \tan ( \Delta \theta ) ) , as shown in Figure 3.2.9-1. The distinction is of importance only in geometrically nonlinear analysis with radially applied concentrated loads and/or sliding radial boundary conditions.
Figure 3.2.9-1 Displacement and rotation in the r – \theta plane.
text_image
R θ Δθ x uθ u uᵣ x
This definition of the degrees of freedom is equivalent to applying transformations to the global ( x, y, z) degrees of freedom associated with a standard continuum element. Hence, the nonlinear equations associated with these elements have the same structure as the equations for standard continuum elements.
A general interpolation scheme for u using Fourier terms with respect to µ is used:
u _ {\alpha} = \sum_ {m = 1} ^ {M} H ^ {m} (g, h) \left(u _ {\alpha} ^ {m 0} + \sum_ {p = 1} ^ {P} \left(\cos p \theta u _ {\alpha c} ^ {m p} + \sin p \theta u _ {\alpha s} ^ {m p}\right)\right), \quad (\alpha = r, z, \theta)
where g , h are isoparametric coordinates in the original R { - } Z plane; H ^ { m } are polynomial interpolation functions; and u _ { \alpha } ^ { m 0 } , u _ { \alpha c } ^ { m p } , and u _ { \alpha s } ^ { m p } are solution amplitude values. M is the number of terms used for interpolation with respect to g , h ; and P is the number of terms used in the Fourier interpolation with respect to µ. Purely axisymmetric deformation results when P = 0 .
We reduce the number of variables in such an element by assuming that bending is allowed only about one plane, \theta = \pi / 2 , so that the plane \theta = 2 n \pi , n integer, is a plane of symmetry. The only terms that satisfy this condition are
Equation 3.2.9-1
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\left\{ \begin{array}{l} u _ {r} \\ u _ {z} \\ u _ {\theta} \end{array} \right\} = \sum_ {m = 1} ^ {M} H ^ {m} (g, h) \left(\left\{ \begin{array}{c} u _ {r} ^ {m 0} \\ u _ {z} ^ {m 0} \\ 0 \end{array} \right\} + \sum_ {p = 1} ^ {P} \cos p \theta \left\{ \begin{array}{c} u _ {r c} ^ {m p} \\ u _ {z c} ^ {m p} \\ 0 \end{array} \right\} + \sum_ {p = 1} ^ {P} \sin p \theta \left\{ \begin{array}{c} 0 \\ 0 \\ u _ {\theta s} ^ {m p} \end{array} \right\}\right).
For convenience we use the values of the u _ { r } and u _ { z } displacement components at specific locations around the model between \theta = 0 and \theta = \pi instead of the Fourier amplitudes u _ { r } ^ { m 0 } and u _ { r c } ^ { m p } . The main reason for this is to allow the elements to be used with interface elements, such as slide lines, for which physical displacement values are required. This is accurate only if it is assumed that the relative displacements in the µ-direction are small so that the interface conditions are considered with respect to u _ { r } and u _ { z } only; that is, in planes of constant µ. In addition, we omit the subscript s in the expression for the circumferential displacement: umpµs u _ { \theta s } ^ { m p } \to u _ { \theta } ^ { m p } ! uµ mp . Equation 3.2.9-1 is, therefore, rewritten
Equation 3.2.9-2
\left\{ \begin{array}{l} u _ {r} \\ u _ {z} \\ u _ {\theta} \end{array} \right\} = \sum_ {m = 1} ^ {M} H ^ {m} (g, h) \left(\sum_ {p = 1} ^ {P + 1} R ^ {p} (\theta) \left\{ \begin{array}{c} u _ {r} ^ {m p} \\ u _ {z} ^ {m p} \\ 0 \end{array} \right\} + \sum_ {p = 1} ^ {P} \sin p \theta \left\{ \begin{array}{c} 0 \\ 0 \\ u _ {\theta} ^ {m p} \end{array} \right\}\right),
where R ^ { p } ( \theta ) are trigonometric interpolation functions and u _ { r } ^ { m p } , u _ { z } ^ { m p } are physical radial and axial displacement components at \theta = \pi \left( p - 1 \right) / P .
The R ^ { p } interpolators at the associated positions \theta _ { p } are taken as
P = 1:
R ^ {1} = \frac {1}{2} (1 + \cos \theta)
R ^ {2} = \frac {1}{2} (1 - \cos \theta)
and
\theta^ {1} = 0
\theta^ {2} = \pi
P = 2:
R ^ {1} = \frac {1}{4} (1 + 2 \cos \theta + \cos 2 \theta)
R ^ {2} = \frac {1}{2} (1 - \cos 2 \theta)
R ^ {3} = \frac {1}{4} (1 - 2 \cos \theta + \cos 2 \theta)
and
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\theta^ {1} = 0
\theta^ {2} = \frac {1}{2} \pi
\theta^ {3} = \pi
P = 3:
R ^ {1} = \frac {1}{6} (1 + 2 \cos \theta + 2 \cos 2 \theta + \cos 3 \theta)
R ^ {2} = \frac {1}{3} (1 + \cos \theta - \cos 2 \theta - \cos 3 \theta)
R ^ {3} = \frac {1}{3} (1 - \cos \theta - \cos 2 \theta + \cos 3 \theta)
R ^ {4} = \frac {1}{6} (1 - 2 \cos \theta + 2 \cos 2 \theta - \cos 3 \theta)
and
\theta^ {1} = 0
\theta^ {2} = \frac {1}{3} \pi
\theta^ {3} = \frac {2}{3} \pi
\theta^ {4} = \pi
P = 4:
R ^ {1} = \frac {1}{8} (1 + 2 \cos \theta + 2 \cos 2 \theta + 2 \cos 3 \theta + \cos 4 \theta)
R ^ {2} = \frac {1}{4} (1 + \sqrt {2} \cos \theta - \sqrt {2} \cos 3 \theta - \cos 4 \theta)
R ^ {3} = \frac {1}{4} (1 - 2 \cos 2 \theta + \cos 4 \theta)
R ^ {4} = \frac {1}{4} (1 - \sqrt {2} \cos \theta + \sqrt {2} \cos 3 \theta - \cos 4 \theta)
R ^ {5} = \frac {1}{8} (1 - 2 \cos \theta + 2 \cos 2 \theta - 2 \cos 3 \theta + \cos 4 \theta)
and
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\begin{array}{l} \theta^ {1} = 0 \\ \theta^ {2} = \frac {1}{4} \pi \\ \theta^ {3} = \frac {1}{2} \pi \\ \theta^ {4} = \frac {3}{4} \pi \\ \theta^ {5} = \pi \\ \end{array}
P = 4 is the highest-order interpolation offered with respect to µ in these elements: the elements become significantly more expensive as higher-order interpolation is used; and it is assumed that, because of this, full three-dimensional modeling is less expensive than using these elements with P > 4 .
Integration
The integration scheme used in these elements is a product of integration with respect to element coordinates in surfaces that were originally in the R-Z plane and integration with respect to µ. For the former the same scheme is used as in the corresponding purely axisymmetric elements (for example, either full or reduced Gauss integration in the isoparametric quadrilaterals). For integration with respect to µ the trapezoidal rule is used, with the number of integration points set to 2 ( P + 1 ) .
Deformation gradient
For a material point in space the deformation gradient F is defined as the gradient of the current position x with respect to the original position X:
\mathbf {F} = \frac {\partial \mathbf {x}}{\partial \mathbf {X}}.
The current position x can be described in terms of the original position X and the displacement u;
\mathbf {x} = \mathbf {X} + \mathbf {u} = (R + u _ {r}) \mathbf {e} _ {r} + (Z + u _ {z}) \mathbf {e} _ {z} + u _ {\theta} \mathbf {e} _ {\theta},
and the gradient operator can be described in terms of partial derivatives with respect to the cylindrical coordinates:
\frac {\partial}{\partial \mathbf {X}} = \mathbf {e} _ {r} \frac {\partial}{\partial R} + \mathbf {e} _ {z} \frac {\partial}{\partial Z} + \mathbf {e} _ {\theta} \frac {\partial}{R \partial \theta}.
Since the radial and circumferential base vectors depend on the original circumferential coordinate µ: { \bf e } _ { r } = { \bf e } _ { r } ( \theta ) , { \bf e } _ { \theta } = { \bf e } _ { \theta } ( \theta ) , the partial derivatives of these base vectors with respect to µ are nonvanishing:
\frac {\partial \mathbf {e} _ {r}}{\partial \theta} = \mathbf {e} _ {\theta}, \quad \frac {\partial \mathbf {e} _ {\theta}}{\partial \theta} = - \mathbf {e} _ {r}.
With this result the deformation gradient is obtained as
\begin{array}{l} \mathbf {F} = \mathbf {e} _ {r} \mathbf {e} _ {r} \left(1 + \frac {\partial u _ {r}}{\partial R}\right) + \mathbf {e} _ {r} \mathbf {e} _ {z} \frac {\partial u _ {r}}{\partial Z} + \mathbf {e} _ {r} \mathbf {e} _ {\theta} \left(\frac {\partial u _ {r}}{R \partial \theta} - \frac {u _ {\theta}}{R}\right) \\ + \mathbf {e} _ {z} \mathbf {e} _ {r} \frac {\partial u _ {z}}{\partial R} + \mathbf {e} _ {z} \mathbf {e} _ {z} \left(1 + \frac {\partial u _ {z}}{\partial Z}\right) + \mathbf {e} _ {\theta} \mathbf {e} _ {z} \frac {\partial u _ {z}}{R \partial \theta} \\ + \mathbf {e} _ {\theta} \mathbf {e} _ {r} \frac {\partial u _ {\theta}}{\partial R} + \mathbf {e} _ {\theta} \mathbf {e} _ {z} \frac {\partial u _ {\theta}}{\partial Z} + \mathbf {e} _ {\theta} \mathbf {e} _ {\theta} \left(1 + \frac {u _ {r}}{R} + \frac {\partial u _ {\theta}}{R \partial \theta}\right). \\ \end{array}
Alternatively, this can be written in matrix form with components relative to the local reference basis \mathbf { e } _ { r } , \mathbf { e } _ { z } , \mathbf { e } _ { \theta } :
Equation 3.2.9-3
\left[ F \right] = \left[ \begin{array}{c c c} 1 + \partial u _ {r} / \partial R & \partial u _ {r} / \partial Z & \partial u _ {r} / R \partial \theta - u _ {\theta} / R \\ \partial u _ {z} / \partial R & 1 + \partial u _ {z} / \partial Z & \partial u _ {z} / R \partial \theta \\ \partial u _ {\theta} / \partial R & \partial u _ {\theta} / \partial Z & 1 + u _ {r} / R + \partial u _ {\theta} / R \partial \theta \end{array} \right].
To be able to analyze approximately incompressible material behavior, the volume change in the fully integrated 4-node quadrilaterals is assumed to be independent of g and h in an R-Z plane. Hence, \big [ F \big ] ¤ is modified according to
\left[ \overline {{F}} \right] = \left(\frac {J _ {0}}{J}\right) ^ {\frac {1}{3}} \left[ F \right],
where \boldsymbol { J } = \operatorname* { d e t } [ \boldsymbol { F } ] is the volume change at the integration point and J _ { 0 } is the average volume change over the R { - } Z plane of the element. In addition, the part of the axisymmetric hoop strain that does not depend on µ is made independent of g and h. Experience has shown this considerably improves the solution accuracy for axisymmetric problems. Thus, we use
\overline {{F}} _ {\theta \theta} = 1 + \overline {{u _ {r} / R}} + \partial u _ {\theta} / R \partial \theta ,
where
\overline {{u _ {r} / R}} = \left[ \sum_ {m = 1} ^ {M} \frac {H ^ {m} (0 , 0)}{R (0 , 0)} \sum_ {p = 1} ^ {P + 1} C ^ {p} + \sum_ {m = 1} ^ {M} \frac {H ^ {m} (g , h)}{R (g , h)} \sum_ {p = 1} ^ {P + 1} \left(R ^ {p} (\theta) - C ^ {p}\right) \right] u _ {r} ^ {m p},
with C ^ { p } the leading (constant) term in R ^ { p } ( \theta ) .
Strain and rotation increments
Strain and rotation increments are calculated from the integrated velocity gradient matrix, \Delta \mathbf { L } , defined as
Elements
\Delta \mathbf {L} = \frac {\partial \Delta \mathbf {u}}{\partial \mathbf {x} _ {t + \Delta t / 2}},
where \begin{array} { r } { \mathbf { x } _ { t + \Delta t / 2 } = \mathbf { x } _ { t } + \frac { 1 } { 2 } \Delta \mathbf { u } = \mathbf { x } _ { t + \Delta t } - \frac { 1 } { 2 } \Delta \mathbf { u } } \end{array} . This expression is not easily evaluated directly, since points that were in an R { - } Z plane in the undeformed shape will no longer be located in the same plane after deformation. Instead, we calculate the gradient of \Delta \mathbf { u } with respect to the reference state and obtain \Delta \mathbf { L } with the transformation
\Delta \mathbf {L} = \frac {\partial \Delta \mathbf {u}}{\partial \mathbf {X}} \cdot \frac {\partial \mathbf {X}}{\partial \mathbf {x} _ {t + \Delta t / 2}} = \frac {\partial \Delta \mathbf {u}}{\partial \mathbf {X}} \cdot \left(\frac {\partial \mathbf {x} _ {t + \Delta t / 2}}{\partial \mathbf {X}}\right) ^ {- 1} = \frac {\partial \Delta \mathbf {u}}{\partial \mathbf {X}} \cdot \left(\mathbf {F} - \frac {1}{2} \frac {\partial \Delta \mathbf {u}}{\partial \mathbf {X}}\right) ^ {- 1}.
In matrix form this can be written as
Equation 3.2.9-4
\left[ \Delta L \right] = \left[ \partial \Delta u / \partial X \right] \left[ \left[ F \right] - \frac {1}{2} \left[ \partial \Delta u / \partial X \right] \right] ^ {- 1},
with
Equation 3.2.9-5
\left[ \partial \Delta u / \partial X \right] = \left[ \begin{array}{c c c} \partial \Delta u _ {r} / \partial R & \partial \Delta u _ {r} / \partial Z & \partial \Delta u _ {r} / R \partial \theta - \Delta u _ {\theta} / R \\ \partial \Delta u _ {z} / \partial R & \partial \Delta u _ {z} / \partial Z & \partial \Delta u _ {z} / R \partial \theta \\ \partial \Delta u _ {\theta} / \partial R & \partial \Delta u _ {\theta} / \partial Z & \partial \Delta u _ {\theta} / R \partial \theta + \Delta u _ {r} / R \end{array} \right].
The strain increments are approximated as the symmetric part of \Delta \mathbf { L } :
\left[ \Delta \varepsilon \right] = \frac {1}{2} \left[ \left[ \Delta L \right] + \left[ \Delta L \right] ^ {T} \right].
As was the case for the deformation gradient, we modify the volume strain increment in the fully integrated 4-node quadrilaterals to be independent of g and h in an R-Z plane, which yields
\left[ \Delta \overline {{\varepsilon}} \right] = \frac {1}{2} \left[ \left[ \Delta \overline {{L}} \right] + \left[ \Delta \overline {{L}} \right] ^ {T} \right], \quad \mathrm{with} \quad \left[ \Delta \overline {{L}} \right] = \left[ \Delta L \right] + \frac {1}{3} \left(\mathrm{tr} (\Delta L ^ {0}) - \mathrm{tr} (\Delta L)\right) \left[ I \right],
where [ I ] is the unit matrix, tr ( \Delta L ) = \Delta L _ { r r } + \Delta L _ { z z } + \Delta L _ { \theta \theta } is the volume strain increment at the integration point, and \mathrm { t r } ( \Delta L ^ { 0 } ) is the average volume strain increment over the R { - } Z plane of the element. In addition we use the approximation
\overline {{\Delta u _ {r} / R}} = \left[ \sum_ {m = 1} ^ {M} \frac {H ^ {m} (0 , 0)}{R (0 , 0)} \sum_ {p = 1} ^ {P + 1} C ^ {p} + \sum_ {m = 1} ^ {M} \frac {H ^ {m} (g , h)}{R (g , h)} \sum_ {p = 1} ^ {P + 1} \left(R ^ {p} (\theta) - C ^ {p}\right) \right] \Delta u _ {r} ^ {m p}.
The spin increments, \Delta \overline { { \Omega } } , are approximated as the antisymmetric part of \Delta \mathbf { \overline { { L } } } , , which in matrix form becomes