MultiPhysicsVault/.raw/AbaqusTheoriesManual/AbaqusTheoriesManual_032.md

\left[ \Delta \overline {{\Omega}} \right] = \frac {1}{2} \left[ \left[ \Delta \overline {{L}} \right] - \left[ \Delta \overline {{L}} \right] ^ {T} \right].

Virtual work

The formulation of equilibrium (virtual work) requires linearization of the strain-displacement relation in the current state. For fully integrated 4-node elements the volume strain modification provides

\delta \overline {{\pmb {\varepsilon}}} = \frac {1}{2} \left(\delta \overline {{\mathbf {L}}} + \delta \overline {{\mathbf {L}}} ^ {T}\right),

where

\delta \overline {{\mathbf {L}}} = \delta \mathbf {L} + \frac {1}{3} \left[ \mathrm{tr} (\delta \mathbf {L} ^ {0}) - \mathrm{tr} (\delta \mathbf {L}) \right] \mathbf {I}

and

\delta \mathbf {L} = \frac {\partial \delta \mathbf {u}}{\partial \mathbf {x}}.

As was the case for the strain increments, the linearized strain-displacement relation involves taking derivatives in the deformed shape ( \mathbf { x } = \mathbf { x } _ { t + \Delta t } ) , which is troublesome since points that were in an R-Z plane in the undeformed shape will no longer be located in the same plane. Hence, ±L is computed in a similar manner to \Delta \mathbf { L } :

\delta \mathbf {L} = \frac {\partial \delta \mathbf {u}}{\partial \mathbf {X}} \cdot \frac {\partial \mathbf {X}}{\partial \mathbf {x}} = \frac {\partial \delta \mathbf {u}}{\partial \mathbf {X}} \cdot \left(\frac {\partial \mathbf {x}}{\partial \mathbf {X}}\right) ^ {- 1} = \frac {\partial \delta \mathbf {u}}{\partial \mathbf {X}} \cdot \mathbf {F} ^ {- 1}.

In matrix form this can be written as

Equation 3.2.9-6

\left[ \delta L \right] = \left[ \partial \delta u / \partial X \right] \left[ F \right] ^ {- 1},

where

\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].

Equation 3.2.9-7

For fully integrated, 4-node quadrilaterals we again use the approximation

Elements

\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 displacements and, hence, the displacement variations, are interpolated in terms of nodal displacement variations with Equation 3.2.9-2. The derivatives of the displacements with respect to R, Z , and µ are readily obtained from these expressions:

\begin{array}{l} \left\{ \begin{array}{l} \partial u _ {r} / \partial R \\ \partial u _ {z} / \partial R \\ \partial u _ {\theta} / \partial R \end{array} \right\} = \sum_ {m = 1} ^ {M} \frac {\partial H ^ {m}}{\partial R} \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), \\ \left\{ \begin{array}{l} \partial u _ {r} / \partial Z \\ \partial u _ {z} / \partial Z \\ \partial u _ {\theta} / \partial Z \end{array} \right\} = \sum_ {m = 1} ^ {M} \frac {\partial H ^ {m}}{\partial Z} \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), \\ \left\{ \begin{array}{l} \partial u _ {r} / \partial \theta \\ \partial u _ {z} / \partial \theta \\ \partial u _ {\theta} / \partial \theta \end{array} \right\} = \sum_ {m = 1} ^ {M} H ^ {m} \left(\sum_ {p = 1} ^ {P + 1} \frac {\partial R ^ {p}}{\partial \theta} \left\{ \begin{array}{c} u _ {r} ^ {m p} \\ u _ {z} ^ {m p} \\ 0 \end{array} \right\} + \sum_ {p = 1} ^ {P} p \cos p \theta \left\{ \begin{array}{c} 0 \\ 0 \\ u _ {\theta} ^ {m p} \end{array} \right\}\right). \\ \end{array}

Stiffness in the current state

Since the elements are formulated in terms of Cartesian components of displacements, the equations presented in ``Solid element formulation,'' Section 3.2.2, apply. For the 4-node quadrilaterals, we can adapt Equation 3.2.2-1 to the averaged volume change formulation, which yields

d \delta \Pi = \int_ {V} \left(d \pmb {\sigma}: \delta \overline {{\pmb {\varepsilon}}} + \pmb {\sigma}: d \delta \overline {{\pmb {\varepsilon}}}\right) d V.

The second variation in \overline { \varepsilon } is obtained with the standard procedure

\mathrm{d} \delta \overline {{\boldsymbol {\varepsilon}}} = \mathrm{d} \overline {{\mathbf {L}}} ^ {T} \cdot \delta \overline {{\mathbf {L}}} - 2 \mathrm{d} \overline {{\boldsymbol {\varepsilon}}} \cdot \delta \overline {{\boldsymbol {\varepsilon}}},

where d \mathbf { \overline { { L } } } has the same form as \delta \mathbf { \overline { { L } } } : This can be worked out in terms of nodal degrees of freedom with the expressions for ±L and ±" obtained in the previous paragraph on virtual work.

Hourglass control

In the 4-node reduced integration element the hourglass modes must be controlled. These modes are similar to the ones in regular axisymmetric elements but have some additional features.

1. The hourglass pattern can vary along the circumference, which requires application of an hourglass stiffness at multiple points around the circumference.
2. Hourglassing can also occur in the circumferential direction.

Hence, at each integration point around the circumference, we calculate the hourglass strains

Elements

\mathbf {q} (\theta) = \mathbf {x} _ {\gamma} (\theta) \cdot \mathbf {F} (\theta) - \mathbf {X} _ {\gamma}.

Here F is the deformation gradient as given by Equation 3.2.9-3 and \mathbf { x } _ { \gamma } and \mathbf { X } _ { \gamma } are the hourglass modes in the deformed and undeformed geometry respectively:

\mathbf {x} _ {\gamma} (\theta) = \sum_ {m = 1} ^ {M} \gamma^ {m} \mathbf {x} ^ {m} (\theta), \qquad \mathbf {X} _ {\gamma} = \sum_ {m = 1} ^ {M} \gamma^ {m} \mathbf {X} ^ {m},

where \gamma ^ { m } is the same hourglass operator as used for the 4-node axisymmetric continuum elements and \mathbf { x } ^ { m } ( \theta ) and \mathbf { X } ^ { m } are the nodal positions at angle µ in the deformed and undeformed states. Observe that since the initial geometry is axisymmetric, \mathbf { X } ^ { m } is independent of \theta \mathrm { : }

\mathbf {X} ^ {m} = \left\{ \begin{array}{c} R ^ {m} \\ Z ^ {m} \\ 0 \end{array} \right\}.

In the deformed state we write

\mathbf {x} ^ {m} (\theta) = \mathbf {X} ^ {m} + \mathbf {u} ^ {m} (\theta) = \left\{ \begin{array}{c} R ^ {m} + u _ {r} ^ {m} (\theta) \\ Z ^ {m} + u _ {z} ^ {m} (\theta) \\ u _ {\theta} ^ {m} (\theta) \end{array} \right\}.

With Equation 3.2.9-2 this becomes

\mathbf {x} ^ {m} = \left\{ \begin{array}{c} R ^ {m} \\ Z ^ {m} \\ 0 \end{array} \right\} + \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\}.

The hourglass "strain" transforms into an hourglass "force" with the hourglass stiffness c:

\mathbf {Q} = c \mathbf {q}.

This hourglass stiffness can be obtained with the same procedure as used for the regular axisymmetric elements, where the only difference is the scaling factor required to reflect the fact that each point reflects only part of the circumference. The first variation of \mathbf { q } is readily obtained as

\delta \mathbf {q} = \delta \mathbf {x} _ {\gamma} \cdot \mathbf {F} + \mathbf {x} _ {\gamma} \cdot \delta \mathbf {F} = \delta \mathbf {u} _ {\gamma} \cdot \mathbf {F} + \mathbf {x} _ {\gamma} \cdot \frac {\partial \delta \mathbf {u}}{\partial \mathbf {X}}.

Here \partial \delta \mathbf { u } / \partial \mathbf { X } follows from Equation 3.2.9-7, and for \delta \mathbf { u } _ { \gamma } we obtain

\delta \mathbf {u} _ {\gamma} = \sum_ {m = 1} ^ {M} \gamma^ {m} \delta \mathbf {u} ^ {m} = \sum_ {m = 1} ^ {M} \gamma^ {m} \left(\sum_ {p = 1} ^ {P + 1} R ^ {p} (\theta) \left\{ \begin{array}{c} \delta u _ {r} ^ {m p} \\ \delta u _ {z} ^ {m p} \\ 0 \end{array} \right\} + \sum_ {p = 1} ^ {P} \sin p \theta \left\{ \begin{array}{c} 0 \\ 0 \\ \delta u _ {\theta} ^ {m p} \end{array} \right\}\right).

Similarly, we obtain for the second variation

\mathrm{d} \delta \mathbf {q} = \delta \mathbf {x} _ {\gamma} \cdot \mathrm{d} \mathbf {F} + \mathrm{d} \mathbf {x} _ {\gamma} \cdot \delta \mathbf {F} = \delta \mathbf {u} _ {\gamma} \cdot \frac {\partial \mathrm{d} \mathbf {u}}{\partial \mathbf {X}} + \mathrm{d} \mathbf {u} _ {\gamma} \cdot \frac {\partial \delta \mathbf {u}}{\partial \mathbf {X}},

where \partial \mathrm { d } { \bf u } / \partial { \bf X } and { \bf { d u } } _ { \gamma } follow with the same expressions as used in the first variation.

Pressure loads and load stiffness

For geometrically linear problems equivalent nodal loads due to applied surface pressures and body forces are readily calculated since the geometry is axisymmetric. For geometrically nonlinear problems the treatment of body forces does not change because of the fixed direction of the forces and because the forces are proportional to the volume, which is assumed to change by a negligible amount. However, for surface pressures nonaxisymmetric deformations must be taken into consideration.

The equivalent nodal loads associated with surface pressure p can be obtained by considering the virtual work contribution

\int_ {A} p \mathbf {n} \cdot \delta \mathbf {u} d A = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} p \left(\frac {\partial \mathbf {x}}{\partial \xi} \times \frac {\partial \mathbf {x}}{\partial \theta}\right) \cdot \delta \mathbf {x} d \xi d \theta ,

where \xi is the parametric surface coordinate in the R-Z plane and

\mathbf {x} = r \mathbf {e} _ {r} + z \mathbf {e} _ {z} + u _ {\theta} \mathbf {e} _ {\theta},

with r = R + u _ { \ i } and z = Z + u _ { z } . Hence, the current position of a point can be expressed in terms of the surface interpolator h ^ { m } ( \boldsymbol { \xi } ) and the standard circumferential interpolators:

Equation 3.2.9-8

\left\{ \begin{array}{l} r \\ z \\ u _ {\theta} \end{array} \right\} = \sum_ {m = 1} ^ {M} h ^ {m} (\xi) \left(\sum_ {p = 1} ^ {P + 1} R ^ {p} (\theta) \left\{ \begin{array}{c} r ^ {m p} \\ 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).

The terms in Equation 3.2.9-8 can be worked out as follows:

\frac {\partial \mathbf {x}}{\partial \xi} = \frac {\partial r}{\partial \xi} \mathbf {e} _ {r} + \frac {\partial z}{\partial \xi} \mathbf {e} _ {z} + \frac {\partial u _ {\theta}}{\partial \xi} \mathbf {e} _ {\theta},

\frac {\partial \mathbf {x}}{\partial \theta} = \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) \mathbf {e _ {r}} + \frac {\partial z}{\partial \theta} \mathbf {e} _ {z} + \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) \mathbf {e} _ {\theta},

and, hence,

Elements

\frac {\partial \mathbf {x}}{\partial \xi} \times \frac {\partial \mathbf {x}}{\partial \theta} = \left[ - \frac {\partial z}{\partial \xi} \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) + \frac {\partial u _ {\theta}}{\partial \xi} \frac {\partial z}{\partial \theta} \right] \mathbf {e} _ {r} +

\left[ - \frac {\partial u _ {\theta}}{\partial \xi} \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) + \frac {\partial r}{\partial \xi} \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) \right] \mathbf {e} _ {z} +

\left[ - \frac {\partial r}{\partial \xi} \frac {\partial z}{\partial \theta} + \frac {\partial z}{\partial \xi} \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) \right] \mathbf {e} _ {\theta}.

Hence, we obtain the virtual work contribution

\int_ {A} p \mathbf {n} \cdot \delta \mathbf {u} d A = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} - p \left\{\left[ \frac {\partial z}{\partial \xi} \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) - \frac {\partial u _ {\theta}}{\partial \xi} \frac {\partial z}{\partial \theta} \right] \delta u _ {r} + \right.

\left[ \frac {\partial u _ {\theta}}{\partial \xi} \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) - \frac {\partial r}{\partial \xi} \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) \right] \delta u _ {z} +

\left. \left[ \frac {\partial r}{\partial \xi} \frac {\partial z}{\partial \theta} - \frac {\partial z}{\partial \xi} \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) \right] \delta u _ {\theta} \right\} d \xi d \theta .

With use of the interpolation functions we, thus, obtain the equivalent nodal forces:

F _ {r} ^ {m p} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} - p h ^ {m} (\xi) R ^ {p} (\theta) \left[ \frac {\partial z}{\partial \xi} \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) - \frac {\partial u _ {\theta}}{\partial \xi} \frac {\partial z}{\partial \theta} \right] d \xi d \theta

F _ {z} ^ {m p} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} - p h ^ {m} (\xi) R ^ {p} (\theta) \left[ \frac {\partial u _ {\theta}}{\partial \xi} \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) - \frac {\partial r}{\partial \xi} \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) \right] d \xi d \theta

F _ {\theta} ^ {m p} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} - p h ^ {m} (\xi) \sin p \theta \left[ \frac {\partial r}{\partial \xi} \frac {\partial z}{\partial \theta} - \frac {\partial z}{\partial \xi} \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) \right] d \xi d \theta .

For geometrically linear analysis this reduces to the standard axisymmetric equivalent nodal loads

F _ {r} ^ {m p} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} - p h ^ {m} (\xi) R ^ {p} (\theta) \frac {\partial Z}{\partial \xi} d \xi R d \theta

F _ {z} ^ {m p} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} p h ^ {m} (\xi) R ^ {p} (\theta) \frac {\partial R}{\partial \xi} d \xi R d \theta

F _ {\theta} ^ {m p} = 0.

The load stiffness matrix follows by linearization:

- \left\{ \begin{array}{l} \mathrm{d} F _ {r} ^ {m p} \\ \mathrm{d} F _ {z} ^ {m p} \\ \mathrm{d} F _ {\theta} ^ {m p} \end{array} \right\} = \left[ \begin{array}{l l l} K _ {r r} ^ {m p n q} & K _ {r z} ^ {m p n q} & K _ {r \theta} ^ {m p n q} \\ K _ {z r} ^ {m p n q} & K _ {z z} ^ {m p n q} & K _ {z \theta} ^ {m p n q} \\ K _ {\theta r} ^ {m p n q} & K _ {\theta z} ^ {m p n q} & K _ {\theta \theta} ^ {m p n q} \end{array} \right] \left\{ \begin{array}{l} \mathrm{d} u _ {r} ^ {n q} \\ \mathrm{d} u _ {z} ^ {n q} \\ \mathrm{d} u _ {\theta} ^ {n q} \end{array} \right\},

with

Elements

K _ {r r} ^ {m p n q} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} p h ^ {m} (\xi) R ^ {p} (\theta) \frac {\partial z}{\partial \xi} h ^ {n} (\xi) R ^ {q} (\theta) d \xi d \theta

\begin{array}{l} K _ {z r} ^ {m p n q} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} p h ^ {m} (\xi) R ^ {p} (\theta) \left[ - \frac {\partial r}{\partial \xi} h ^ {n} (\xi) R ^ {q} (\theta) - \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) \frac {d h ^ {n}}{d \xi} R ^ {q} (\theta) \right. \\ \left. + \frac {\partial u _ {\theta}}{\partial \xi} h ^ {n} (\xi) \frac {d R ^ {q}}{d \theta} \right] d \xi d \theta \\ \end{array}

K _ {\theta r} ^ {m p n q} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} p h ^ {m} (\xi) \sin p \theta \left[ \frac {\partial z}{\partial \theta} \frac {d h ^ {n}}{d \xi} R ^ {q} (\theta) - \frac {\partial z}{\partial \xi} h ^ {n} (\xi) \frac {d R ^ {q}}{d \theta} \right] d \xi d \theta

K _ {r z} ^ {m p n q} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} p h ^ {m} (\xi) R ^ {p} (\theta) \left[ \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) \frac {d h ^ {n}}{d \xi} R ^ {q} (\theta) - \frac {\partial u _ {\theta}}{\partial \xi} h ^ {n} (\xi) \frac {d R ^ {q}}{d \theta} \right] d \xi d \theta

K _ {z z} ^ {m p n q} = 0

K _ {\theta z} ^ {m p n q} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} p h ^ {m} (\xi) \sin p \theta \left[ \frac {\partial r}{\partial \xi} h ^ {n} (\xi) \frac {d R ^ {q}}{d \theta} - \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) \frac {d h ^ {n}}{d \xi} R ^ {q} (\theta) \right] d \xi d \theta

K _ {r \theta} ^ {m p n q} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} p h ^ {m} (\xi) R ^ {p} (\theta) \left[ q \frac {\partial z}{\partial \xi} h ^ {n} (\xi) \cos q \theta - \frac {\partial z}{\partial \theta} \frac {d h ^ {n}}{d \xi} \sin q \theta \right] d \xi d \theta

\begin{array}{l} K _ {z \theta} ^ {m p n q} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} p h ^ {m} (\xi) R ^ {p} (\theta) \left[ - \frac {\partial u _ {\theta}}{\partial \xi} h ^ {n} (\xi) \sin q \theta + \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) \frac {d h ^ {n}}{d \xi} \sin q \theta \right. \\ \left. - q \frac {\partial r}{\partial \xi} h ^ {n} (\xi) \cos q \theta \right] d \xi d \theta \\ \end{array}

K _ {\theta \theta} ^ {m p n q} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} p h ^ {m} (\xi) \sin p \theta \frac {\partial z}{\partial \xi} h ^ {n} (\xi) \sin q \theta d \xi d \theta .

In the case of hydrostatic pressure (p dependent on z) some additional terms appear. These terms are readily obtained from the expression

Elements

\int_ {A} \mathrm{d} p \mathbf {n} \cdot \delta \mathbf {u} d A = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} - \mathrm{d} u _ {z} \frac {d p}{d z} \left\{\left[ \frac {\partial z}{\partial \xi} \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) - \frac {\partial u _ {\theta}}{\partial \xi} \frac {\partial z}{\partial \theta} \right] \delta u _ {r} + \right.

\left[ \frac {\partial u _ {\theta}}{\partial \xi} \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) - \frac {\partial r}{\partial \xi} \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) \right] \delta u _ {z} +

\left. \left[ \frac {\partial r}{\partial \xi} \frac {\partial z}{\partial \theta} - \frac {\partial z}{\partial \xi} \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) \right] \delta u _ {\theta} \right\} d \xi d \theta .

With use of the interpolation functions we, thus, obtain the additional load stiffness contributions:

K _ {r z} ^ {m p n q} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} \frac {d p}{d z} h ^ {m} (\xi) R ^ {p} (\theta) \left[ \frac {\partial z}{\partial \xi} \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) - \frac {\partial u _ {\theta}}{\partial \xi} \frac {\partial z}{\partial \theta} \right] h ^ {n} (\xi) R ^ {q} (\theta) d \xi d \theta

K _ {z z} ^ {m p n q} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} \frac {d p}{d z} h ^ {m} (\xi) R ^ {p} (\theta) \left[ \frac {\partial u _ {\theta}}{\partial \xi} \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) - \frac {\partial r}{\partial \xi} \left(r + \frac {\partial u _ {\theta}}{\partial \theta}\right) \right] h ^ {n} (\xi) R ^ {q} (\theta) d \xi d \theta

K _ {\theta z} ^ {m p n q} = \int_ {0} ^ {2 \pi} \int_ {- 1} ^ {+ 1} \frac {d p}{d z} h ^ {m} (\xi) \sin p \theta \left[ \frac {\partial r}{\partial \xi} \frac {\partial z}{\partial \theta} - \frac {\partial z}{\partial \xi} \left(\frac {\partial r}{\partial \theta} - u _ {\theta}\right) \right] h ^ {n} (\xi) R ^ {q} (\theta) d \xi d \theta .

Mass matrix

At each material point the displacement components in the three directions (radial, axial, circumferential) are dependent only on the corresponding nodal displacement components. Hence, the mass matrix does not involve any coupling between the radial, axial, and circumferential degrees of freedom, and we can write the mass matrix in the form of three separate expressions:

M _ {r r} ^ {m n p q} = \int_ {V} \rho N _ {r} ^ {m p} N _ {r} ^ {n q} \mathrm{d} V

M _ {z z} ^ {m n p q} = \int_ {V} \rho N _ {z} ^ {m p} N _ {z} ^ {n q} \mathrm{d} V

M _ {\theta \theta} ^ {m n p q} = \int_ {V} \rho N _ {\theta} ^ {m p} N _ {\theta} ^ {n q} \mathrm{d} V.

Here the superscripts m and n refer to a particular node in the _ { r - z } plane, and the superscripts p and q refer to a particular position along the circumference. The interpolation functions N _ { r } ^ { m p } , N _ { z } ^ { m p } , and N _ { \theta } ^ { m p } are the product of interpolation functions H ^ { m } ( g , h ) in the _ { r - z } plane and interpolation functions in the µ-direction:

N _ {r} ^ {m p} = H ^ {m} (g, h) R ^ {p} (\theta) = N _ {z} ^ {m p}

N _ {\theta} ^ {m p} = H ^ {m} (g, h) \sin p \theta .

The volume integral used to form the mass matrix can be split into an integral over the \displaystyle r - z cross-section and an integral around the circumference. For the r { - } r component of the mass matrix this yields

Elements

M _ {r r} ^ {m n p q} = \int_ {A} \rho H ^ {m} (g, h) H ^ {n} (g, h) 2 \pi R \mathrm{d} A \frac {1}{2 \pi} \int_ {0} ^ {2 \pi} R ^ {p} (\theta) R ^ {q} (\theta) \mathrm{d} \theta .

This matrix can be written in a convenient form by defining the primitive mass matrix,

M _ {p r i m} ^ {m n} = \int_ {A} \rho H ^ {m} (g, h) H ^ {n} (g, h) 2 \pi R \mathrm{d} A.

This primitive mass matrix is the same mass matrix that is used for the regular axisymmetric elements. We can also define the circumferential distribution matrices

f _ {1} ^ {p q} = \frac {1}{2 \pi} \int_ {0} ^ {2 \pi} R ^ {p} (\theta) R ^ {q} (\theta) \mathrm{d} \theta \quad \mathrm{and}

f _ {2} ^ {p q} = \frac {1}{2 \pi} \int_ {0} ^ {2 \pi} \sin p \theta \sin q \theta \mathrm{d} \theta .

The radial, axial, and circumferential components of the mass matrix then take the form

M _ {r r} ^ {m n p q} = M _ {p r i m} ^ {m n} f _ {1} ^ {p q} = M _ {z z} ^ {m n p q}

M _ {\theta \theta} ^ {m n p q} = M _ {p r i m} ^ {m n} f _ {2} ^ {p q}.

The circumferential distribution matrices can be evaluated for various values of the number of terms P in the Fourier series. After some calculations the following results are obtained:

P = 1:

f _ {1} ^ {p q} = \frac {1}{8} \left( \begin{array}{c c} 3 & 1 \\ 1 & 3 \end{array} \right), \qquad f _ {2} ^ {p q} = \frac {1}{2} (1).

P = 2:

f _ {1} ^ {p q} = \frac {1}{3 2} \left( \begin{array}{c c c} 7 & 2 & - 1 \\ 2 & 1 2 & 2 \\ - 1 & 2 & 7 \end{array} \right), \qquad f _ {2} ^ {p q} = \frac {1}{2} \left( \begin{array}{c c} 1 & 0 \\ 0 & 1 \end{array} \right).

P = 3:

f _ {1} ^ {p q} = \frac {1}{7 2} \left( \begin{array}{c c c c} 1 1 & 2 & - 2 & 1 \\ 2 & 2 0 & 4 & - 2 \\ - 2 & 4 & 2 0 & 2 \\ 1 & - 2 & 2 & 1 1 \end{array} \right), \qquad f _ {2} ^ {p q} = \frac {1}{2} \left( \begin{array}{c c c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right).

P = 4:

f _ {1} ^ {p q} = \frac {1}{1 2 8} \left( \begin{array}{c c c c c} 1 5 & 2 & - 2 & 2 & - 1 \\ 2 & 2 8 & 4 & - 4 & 2 \\ - 2 & 4 & 2 8 & 4 & - 2 \\ 2 & - 4 & 4 & 2 8 & 2 \\ - 1 & 2 & - 2 & 2 & 1 5 \end{array} \right), \qquad f _ {2} ^ {p q} = \frac {1}{2} \left( \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right).

Hybrid and pore pressure elements

For hybrid and pore pressure elements additional degrees of freedom p are added. In the hybrid elements these degrees of freedom are internal to the element and represent the hydrostatic pressure in the material. In the pore pressure elements the degrees of freedom represent the hydrostatic pressure in the fluid as interpolated from the pressure variables at the external, user-defined nodes. Let the interpolation function for the (hydrostatic or pore) pressure in the r-z plane be denoted by G ^ { m } ( g , h ) . The interpolation functions are the same as for the regular axisymmetric hybrid and pore pressure elements, respectively. Along the circumference, we observe that in the geometrically linear formulation the volumetric strain Á only shows cosine dependence:

\phi = \frac {\partial u _ {r}}{\partial R} + \frac {\partial u _ {z}}{\partial Z} + \frac {u _ {r}}{R} + \frac {\partial u _ {\theta}}{R \partial \theta}.

Hence, we choose the hydrostatic/pore pressure to have cosine dependence only:

p (R, Z, \theta) = \sum_ {m = 1} ^ {M} G ^ {m} (g, h) \sum_ {p = 1} ^ {P = 1} R ^ {p} (\theta) p ^ {m p}.

In the nonlinear case Á will exhibit higher-order variations in µ. For approximately incompressible materials, these higher-order terms are likely to lead to "locking" of the finite element mesh for nonaxisymmetric deformations. In the hybrid formulation, however, the higher-order terms in Á are not used for calculation of hydrostatic pressure: only the cosine terms as used in the interpolation for p are used. Hence, the hybrid elements prevent locking in the nonaxisymmetric modes as well as in the axisymmetric modes.

Pore pressure gradient

For a material point in space in the pore pressure element, the pore pressure gradient calculation involves taking derivatives of the pore pressure with respect to the current position x. Again, we do not evaluate the gradient directly but calculate it with respect to the original position X, with the following transformation:

\frac {\partial p}{\partial \mathbf {x}} = \frac {\partial p}{\partial \mathbf {X}} \cdot \frac {\partial \mathbf {X}}{\partial \mathbf {x}} = \frac {\partial p}{\partial \mathbf {X}} \cdot \mathbf {F} ^ {- 1},

where F is the deformation gradient, and the cylindrical components of the scalar gradient of the pore pressure with respect to X are readily obtained from the following expressions:

\frac {\partial p}{\partial R} = \sum_ {m = 1} ^ {M} \frac {\partial G ^ {m}}{\partial R} \sum_ {p = 1} ^ {P + 1} R ^ {p} (\theta) p ^ {m p},

\frac {\partial p}{\partial Z} = \sum_ {m = 1} ^ {M} \frac {\partial G ^ {m}}{\partial Z} \sum_ {p = 1} ^ {P + 1} R ^ {p} (\theta) p ^ {m p},

\frac {\partial p}{\partial \theta} = \sum_ {m = 1} ^ {M} G ^ {m} \sum_ {p = 1} ^ {P + 1} \frac {\partial R ^ {p} (\theta)}{\partial \theta} p ^ {m p}.

3.2.10 Pressure load stiffness for continuum elements

In geometrically nonlinear analysis pressure loads are applied on the deformed structure. Hence, the equivalent nodal loads are dependent on the nodal displacements. This dependency leads to additional contributions to the Jacobian in the solution procedure used in ABAQUS/Standard. The external virtual work is

\delta W ^ {E} = \int_ {A} \delta \mathbf {u} \cdot p \mathbf {n} d A,

where A is the surface on which the pressure is applied; n is the normal to this surface, pointing into the material; ±u is the virtual displacement field; and p is the pressure magnitude.

Pressure load stiffness on a surface in three-dimensional space

The expression n dA can be rewritten as follows:

\mathbf {n} d A = \frac {\partial \mathbf {x}}{\partial g} \times \frac {\partial \mathbf {x}}{\partial h} d g d h,

where x are the current coordinates of a point on the surface, and the surface parametric coordinates (g, h) are chosen to give the correct sign to n through the cross product. The external virtual work is then given by

\delta W ^ {E} = \int_ {g, h} p \delta \mathbf {u} \cdot \frac {\partial \mathbf {x}}{\partial g} \times \frac {\partial \mathbf {x}}{\partial h} d g d h,

and the load stiffness matrix is obtained from

- d \delta W ^ {E} = - \int_ {g, h} p \delta \mathbf {u} \cdot \left(\frac {\partial d \mathbf {u}}{\partial g} \times \frac {\partial \mathbf {x}}{\partial h} + \frac {\partial \mathbf {x}}{\partial g} \times \frac {\partial d \mathbf {u}}{\partial h}\right) d g d h,

where, for a solid, dx = du.

Pressure load stiffness on a surface in two-dimensional space