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Geometrically nonlinear formulation
Since we want to use a formulation that can be used for any material model, we want to express the incompatible modes as a modification of the deformation gradient F. The most obvious approach is to add the incompatible modes to the deformation gradient:
Equation 3.2.5-3
\overline {{\mathbf {F}}} = \mathbf {F} + \tilde {\mathbf {F}} = \frac {\partial \mathbf {x}}{\partial \mathbf {X}} + \tilde {\mathbf {F}}.
This approach has been used successfully by Simo and Armero. Elements formulated on this basis satisfy the large-strain patch test; i.e., any patch of elements will be able to represent homogeneous deformations exactly. However, once the elements become distorted due to deformation, the patch test will no longer be satisfied in an incremental sense; that is, subsequent homogeneous deformations will not be represented exactly. This turns out to be a fatal flaw in the formulation for problems involving large distortions in compression.
Satisfaction of the instantaneous patch test requires the addition of an incompatible deformation rate tensor to the standard rate of deformation:
\dot {\bar {\varepsilon}} = \dot {\bar {\varepsilon}} + \dot {\tilde {\bar {\varepsilon}}}.
To obtain an approximate relation of this type, we write the total deformation gradient as the product of a series of incremental deformation gradients:
\overline {{\mathbf {F}}} ^ {(n + 1)} = \Delta \overline {{\mathbf {F}}} ^ {(n + 1)} \cdot \Delta \overline {{\mathbf {F}}} ^ {(n)} \cdot \ldots \cdot \Delta \overline {{\mathbf {F}}} ^ {(1)}.
Principal incompatible modes are then added to the incremental deformation gradient:
\Delta \overline {{\mathbf {F}}} _ {p} = \Delta \mathbf {F} + \Delta \tilde {\mathbf {F}} = \frac {\partial \mathbf {x} _ {t + \Delta t}}{\partial \mathbf {x} _ {t}} + \Delta \tilde {\mathbf {F}}.
Similar to Equation 3.2.5-2 the principal incompatible modes are described as a transformation of the parametric gradient field ¢g~(»):
Equation 3.2.5-4
\Delta \tilde {\mathbf {F}} (\pmb {\xi}) = \frac {j (0)}{j (\pmb {\xi})} \bigg | _ {t} \Delta \tilde {\mathbf {g}} (\pmb {\xi}) \cdot \mathbf {t} _ {t} ^ {- 1},
where \mathbf { t } _ { t } is the parametric transformation at the center of the element in the state at the start of the increment,
\mathbf {t} _ {t} = \left. \frac {\partial \mathbf {x} _ {t}}{\partial \pmb {\xi}} \right| _ {\pmb {\xi} = 0};
Elements
j ( \pmb \xi ) is the Jacobian of the parametric transformation at the location \pmb { \xi } ; and j ( 0 ) is the Jacobian at the centroid at the start of the increment. Note that j ( \pmb \xi ) is evaluated based on the deformation caused by the displacement degrees of freedom only and does not include the volume change due to the incompatible modes.
The incremental parametric gradient field \Delta \tilde { \bf g } ( \pmb { \xi } ) has exactly the same form as in the linear formulation,
\Delta \tilde {\mathbf {g}} (\pmb {\xi}) = \Delta \pmb {\alpha} ^ {i} \pmb {\xi} _ {i},
which yields the principal incremental incompatible deformation gradient
\Delta \tilde {\mathbf {F}} (\boldsymbol {\xi}) = \Delta \boldsymbol {\alpha} ^ {i} \frac {j (0)}{j (\boldsymbol {\xi})} \bigg | _ {t} \boldsymbol {\xi} _ {i} \cdot \mathbf {t} _ {t} ^ {- 1}.
Bilinear volumetric terms are added to the principal terms in a multiplicative way:
Equation 3.2.5-5
\Delta \overline {{\mathbf {F}}} = \Delta \overline {{\mathbf {F}}} _ {p} \Delta F _ {I} = (\Delta \mathbf {F} + \Delta \tilde {\mathbf {F}}) \exp \left(\Delta \beta^ {j} \frac {j (0)}{j (\pmb {\xi})} \Bigg | _ {t} \vartheta_ {j} (\pmb {\xi})\right).
The variation in the gradient of the position with respect to the current state is obtained from the fundamental relation
\delta \overline {{\mathbf {L}}} = \delta \overline {{\mathbf {F}}} \cdot \overline {{\mathbf {F}}} ^ {- 1} = \delta \Delta \overline {{\mathbf {F}}} \cdot \Delta \overline {{\mathbf {F}}} ^ {- 1} = \left[ (\delta \Delta \mathbf {F} + \delta \Delta \tilde {\mathbf {F}}) \Delta F _ {I} + \Delta \overline {{\mathbf {F}}} _ {p} \delta \Delta F _ {I} \right] \cdot \Delta \overline {{\mathbf {F}}} ^ {- 1}
= (\delta \Delta \mathbf {F} + \delta \Delta \tilde {\mathbf {F}}) \cdot \Delta \overline {{\mathbf {F}}} _ {p} ^ {- 1} + \delta \Delta F _ {I} \Delta F _ {I} ^ {- 1},
with
\delta \Delta \mathbf {F} = \frac {\partial \delta \mathbf {x}}{\partial \mathbf {x} _ {t}},
\delta \Delta \tilde {\mathbf {F}} = \frac {j (0)}{j (\boldsymbol {\xi})} \bigg | _ {t} \delta \tilde {\mathbf {g}} (\boldsymbol {\xi}) \cdot \mathbf {t} _ {t} ^ {- 1} = \delta \boldsymbol {\alpha} ^ {i} \frac {j (0)}{j (\boldsymbol {\xi})} \bigg | _ {t} \boldsymbol {\xi} _ {i} \cdot \mathbf {t} _ {t} ^ {- 1},
\delta \Delta F _ {I} = \delta \beta^ {j} \frac {j (0)}{j (\pmb {\xi})} \bigg | _ {t} \vartheta_ {j} (\pmb {\xi}) \Delta F _ {I}.
This allows us to write for \delta \mathbf { \overline { { L } } }
\delta \overline {{\mathbf {L}}} = \frac {\partial \delta \mathbf {x}}{\partial \mathbf {x} _ {t}} \cdot \Delta \overline {{\mathbf {F}}} _ {p} ^ {- 1} + \delta \boldsymbol {\alpha} ^ {i} \frac {j (0)}{j (\boldsymbol {\xi})} \bigg | _ {t} \boldsymbol {\xi} _ {i} \cdot \mathbf {t} _ {t} ^ {- 1} \cdot \Delta \overline {{\mathbf {F}}} _ {p} ^ {- 1} + \delta \beta^ {j} \frac {j (0)}{j (\boldsymbol {\xi})} \bigg | _ {t} \vartheta_ {j} (\boldsymbol {\xi}).
The integral of the principal incompatible modes over the element volume at the start of the increment is, hence, equal to
Elements
\begin{array}{l} \int_ {v _ {e l}} \delta \Delta \tilde {\mathbf {F}} \cdot \Delta \overline {{{\mathbf {F}}}} _ {p} ^ {- 1} d v _ {e l} = \int_ {v _ {e l}} \frac {j (0)}{j (\pmb {\xi})} \big | _ {t} \delta \tilde {\mathbf {g}} (\pmb {\xi}) \cdot \mathbf {t} _ {t} ^ {- 1} \cdot \Delta \overline {{{\mathbf {F}}}} _ {p} ^ {- 1} d v _ {e l} \\ = \int_ {V _ {p a r}} \left. j (0) \right| _ {t} \delta \widetilde {\mathbf {g}} (\pmb {\xi}) \cdot \mathbf {t} _ {t} ^ {- 1} \cdot \Delta \overline {{\mathbf {F}}} _ {p} ^ {- 1} d V _ {p a r}. \\ \end{array}
Note that the integral will vanish if the incremental deformation is homogeneous; that is, \Delta \overline { { \mathbf { F } } } _ { p } ( \pmb { \xi } ) = \Delta \overline { { \mathbf { F } } } _ { 0 } , since in that case the integral can be written as
\int_ {v _ {e l}} \delta \Delta \tilde {\mathbf {F}} \cdot \Delta \overline {{\mathbf {F}}} _ {p} ^ {- 1} d v _ {e l} = j (0) \big | _ {t} \left(\int_ {V _ {p a r}} \delta \tilde {\mathbf {g}} _ {p} (\pmb {\xi}) d V _ {p a r}\right) \cdot \mathbf {t} _ {t} ^ {- 1} \cdot \Delta \overline {{\mathbf {F}}} _ {0} ^ {- 1} = 0.
Hence, the incremental patch test will be satisfied. The rate of the gradient of the position with respect to the current state is obtained similar to the variation
\begin{array}{l} \mathrm{d} \overline {{\mathbf {L}}} = \mathrm{d} \overline {{\mathbf {F}}} \cdot \overline {{\mathbf {F}}} ^ {- 1} = \mathrm{d} \Delta \overline {{\mathbf {F}}} \cdot \Delta \overline {{\mathbf {F}}} ^ {- 1} = \left[ (\mathrm{d} \Delta \mathbf {F} + \mathrm{d} \Delta \tilde {\mathbf {F}}) \Delta F _ {I} + \Delta \overline {{\mathbf {F}}} _ {p} \mathrm{d} \Delta F _ {I} \right] \cdot \Delta \overline {{\mathbf {F}}} ^ {- 1} \\ = \left(\mathrm{d} \Delta \mathbf {F} + \mathrm{d} \Delta \tilde {\mathbf {F}}\right) \cdot \Delta \overline {{\mathbf {F}}} _ {p} ^ {- 1} + \mathrm{d} \Delta F _ {I} \Delta F _ {I} ^ {- 1}, \\ \end{array}
with
\begin{array}{l} \mathrm{d} \Delta \mathbf {F} = \frac {\partial \mathrm{d} \mathbf {x}}{\partial \mathbf {x} _ {t}}, \\ \mathrm{d} \Delta \tilde {\mathbf {F}} = \frac {j (0)}{j (\boldsymbol {\xi})} \bigg | _ {t} \mathrm{d} \tilde {\mathbf {g}} (\boldsymbol {\xi}) \cdot \mathbf {t} _ {t} ^ {- 1} = \mathrm{d} \boldsymbol {\alpha} ^ {i} \frac {j (0)}{j (\boldsymbol {\xi})} \bigg | _ {t} \boldsymbol {\xi} _ {i} \cdot \mathbf {t} _ {t} ^ {- 1}, \\ \mathrm{d} \Delta F _ {I} = \mathrm{d} \beta^ {j} \frac {j (0)}{j (\pmb {\xi})} \bigg | _ {t} \vartheta_ {j} (\pmb {\xi}) \Delta F _ {I}. \\ \end{array}
For a finite strain increment we use the midincrement approach proposed by Hughes and Winget. This yields
\Delta \overline {{\mathbf {L}}} \approx \Delta \overline {{\mathbf {F}}} \cdot \Delta \overline {{\mathbf {F}}} _ {t + \Delta t / 2} ^ {- 1}.
The second variation is obtained in the usual way. Since the regular deformation gradient and the principal incompatible modes are purely displacement based, the initial stress stiffness terms are readily obtained as
\int_ {v _ {e l}} \pmb {\sigma}: \left(\delta \overline {{\mathbf {L}}} _ {p} ^ {T} \cdot \mathrm{d} \overline {{\mathbf {L}}} _ {p} - 2 \delta \overline {{\mathbf {D}}} _ {p} \cdot \mathrm{d} \overline {{\mathbf {D}}} _ {p}\right) \mathrm{d} v _ {e l},
where \mathrm { d } \overline { { \mathbf { L } } } _ { p } = \mathrm { d } \overline { { \mathbf { F } } } _ { p } \cdot \Delta \overline { { \mathbf { F } } } _ { p } ^ { - 1 } is the gradient of the velocity and \delta \overline { { \mathbf { L } } } _ { p } = \delta \overline { { \mathbf { F } } } _ { p } \cdot \Delta \overline { { \mathbf { F } } } _ { p } ^ { - 1 } is the gradient of the displacement variation including the primary incompatible modes. Further, \begin{array} { r } { \mathrm { d } \overline { { \mathbf { D } } } _ { p } ^ { \bullet } = \frac { 1 } { 2 } ( \mathrm { d } \overline { { \mathbf { L } } } _ { p } + \mathrm { d } \overline { { \mathbf { L } } } _ { p } ^ { T } ) } \end{array} is the rate of deformation, and \begin{array} { r } { \delta \overline { { \mathbf { D } } } _ { p } = \frac { 1 } { 2 } ( \delta \overline { { \mathbf { L } } } _ { p } + \delta \overline { { \mathbf { L } } } _ { p } ^ { T } ) } \end{array} is the variation in the deformation.
The additional bilinear modes appear in the first variation only as variations (the values \beta ^ { i } themselves do not appear); hence, the contributions to the second variation can be neglected.
3.2.6 Triangular, tetrahedral, and wedge elements
The library of solid elements in ABAQUS includes first- and second-order triangles, tetrahedra, and wedge elements for planar, axisymmetric, and three-dimensional analysis.
Hybrid versions of these elements are provided for use with incompressible and nearly incompressible constitutive models (see ``Hybrid incompressible solid element formulation,'' Section 3.2.3, for a detailed discussion of the formulation used). However, these hybrid forms should be used only to fill in regions in meshes made of brick elements; otherwise, too many constraint variables may be introduced.
Second-order tetrahedra are not suitable for the analysis of contact problems: a constant pressure on an element face produces zero equivalent loads at the corner nodes. In contact problems this makes the contact condition at the corners indeterminate, with failure of the solution likely because of excessive gap chatter. The same argument holds true for contact on triangular faces of a wedge element.
Interpolation
The interpolation is defined in terms of the element coordinates g, h, and r shown in Figure 3.2.6-1. Since ABAQUS is a Lagrangian code for most applications, these are also material coordinates. They each span a range from 0 to 1 in an element but satisfy the constraint that g + h \leq 1 for triangles and wedges and g + h + r \leq 1 for tetrahedra. The node numbering convention used in ABAQUS for these elements is also shown in Figure 3.2.6-1. Corner nodes are numbered first, and then the midside nodes for second-order elements. The interpolation functions are as follows.
First-order triangle (3 nodes):
u = (1 - g - h) u _ {1} + g u _ {2} + h u _ {3}
Second-order triangle (6 nodes):
\begin{array}{l} u = 2 \left(\frac {1}{2} - g - h\right) (1 - g - h) u _ {1} + 2 g \left(g - \frac {1}{2}\right) u _ {2} + 2 h \left(h - \frac {1}{2}\right) u _ {3} \\ + 4 g (1 - g - h) u _ {4} + 4 g h u _ {5} + 4 h (1 - g - h) u _ {6} \\ \end{array}
First-order tetrahedron (4 nodes):
u = (1 - g - h - r) u _ {1} + g u _ {2} + h u _ {3} + r u _ {4}
Second-order tetrahedron (10 nodes):
Elements
\begin{array}{l} u = (2 (1 - g - h - r) - 1) (1 - g - h - r) u _ {1} + (2 g - 1) g u _ {2} + (2 h - 1) h u _ {3} \\ + (2 r - 1) r u _ {4} + 4 (1 - g - h - r) g u _ {5} + 4 g h u _ {6} + 4 (1 - g - h - r) h u _ {7} \\ + 4 (1 - g - h - r) r u _ {8} + 4 g r u _ {9} + 4 h r u _ {1 0} \\ \end{array}
Figure 3.2.6-1 Isoparametric master elements.
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| Point | g | h | |---|---|---| | 1 | 0 | 0 | | 2 | 2 | 0 | | 3 | 3 | 0 | | 4 | 4 | 0 | | 5 | 5 | 0 | | 6 | 6 | 0 |
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r 4 10 8 9 7 3 6 1 5 2 g h
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r 4 12 11 6 15 h 10 18 5 17 3 13 16 9 14 8 g 1 7 2
First-order wedge (6 nodes):
\begin{array}{l} u = \frac {1}{2} (1 - g - h) (1 - r) u _ {1} + \frac {1}{2} g (1 - r) u _ {2} + \frac {1}{2} h (1 - r) u _ {3} \\ + \frac {1}{2} (1 - g - h) (1 + r) u _ {4} + \frac {1}{2} g (1 + r) u _ {5} + \frac {1}{2} h (1 + r) u _ {6} \\ \end{array}
Elements
Second-order wedge (15 nodes):
\begin{array}{l} u = \frac {1}{2} ((1 - g - h) (2 (1 - g - h) - 1) (1 - r) - (1 - g - h) (1 - r ^ {2})) u _ {1} \\ + \frac {1}{2} (g (2 g - 1) (1 - r) - g (1 - r ^ {2})) u _ {2} + \frac {1}{2} (h (2 h - 1) (1 - r) - h (1 - r ^ {2})) u _ {3} \\ + \frac {1}{2} ((1 - g - h) (2 (1 - g - h) - 1) (1 + r) - (1 - g - h) (1 - r ^ {2})) u _ {4} \\ + \frac {1}{2} (g (2 g - 1) (1 + r) - g (1 - r ^ {2})) u _ {5} + \frac {1}{2} (h (2 h - 1) (1 + r) - h (1 - r ^ {2})) u _ {6} \\ + 2 (1 - g - h) g (1 - r) u _ {7} + 2 g h (1 - r) u _ {8} + 2 h (1 - g - h) (1 - r) u _ {9} \\ + 2 (1 - g - h) g (1 + r) u _ {1 0} + 2 g h (1 + r) u _ {1 1} + 2 h (1 - g - h) (1 + r) u _ {1 2} \\ + (1 - g - h) (1 - r ^ {2}) u _ {1 3} + g (1 - r ^ {2}) u _ {1 4} + h (1 - r ^ {2}) u _ {1 5} \\ \end{array}
Second-order variable 15-18 node wedge (assuming all 18 nodes are defined):
\begin{array}{l} u = (\frac {1}{2} ((1 - g - h) (2 (1 - g - h) - 1) (1 - r) - (1 - g - h) (1 - r ^ {2})) + \frac {1}{4} (N _ {1 6} + N _ {1 8})) u _ {1} \\ + (\frac {1}{2} (g (2 g - 1) (1 - r) - g (1 - r ^ {2})) + \frac {1}{4} (N _ {1 6} + N _ {1 7})) u _ {2} \\ + (\frac {1}{2} (h (2 h - 1) (1 - r) - h (1 - r ^ {2})) + \frac {1}{4} (N _ {1 7} + N _ {1 8})) u _ {3} \\ + (\frac {1}{2} ((1 - g - h) (2 (1 - g - h) - 1) (1 + r) - (1 - g - h) (1 - r ^ {2})) + \frac {1}{4} (N _ {1 6} + N _ {1 8})) u _ {4} \\ + (\frac {1}{2} (g (2 g - 1) (1 + r) - g (1 - r ^ {2})) + \frac {1}{4} (N _ {1 6} + N _ {1 7})) u _ {5} \\ + (\frac {1}{2} (h (2 h - 1) (1 + r) - h (1 - r ^ {2})) + \frac {1}{4} (N _ {1 7} + N _ {1 8})) u _ {6} \\ + (2 (1 - g - h) g (1 - r) - \frac {1}{2} N _ {1 6}) u _ {7} + (2 g h (1 - r) - \frac {1}{2} N _ {1 7}) u _ {8} \\ + (2 h (1 - g - h) (1 - r) - \frac {1}{2} N _ {1 8}) u _ {9} + (2 (1 - g - h) g (1 + r) + \frac {1}{2} N _ {1 6}) u _ {1 0} \\ + (2 g h (1 + r) + \frac {1}{2} N _ {1 7}) u _ {1 1} + (2 h (1 - g - h) (1 + r) + \frac {1}{2} N _ {1 8}) u _ {1 2} \\ + ((1 - g - h) (1 - r ^ {2}) - \frac {1}{2} (N _ {1 6} + N _ {1 8})) u _ {1 3} + (g (1 - r ^ {2}) - \frac {1}{2} (N _ {1 6} + N _ {1 7})) u _ {1 4} \\ + (h (1 - r ^ {2}) - \frac {1}{2} (N _ {1 7} + N _ {1 8})) u _ {1 5} + + N _ {1 6} u _ {1 6} + N _ {1 7} u _ {1 7} + N _ {1 8} u _ {1 8}, \\ \end{array}
where
N _ {1 6} = 4 g (1 - g - h) (1 - r ^ {2})
N _ {1 7} = 4 g h (1 - r ^ {2})
N _ {1 8} = 4 h (1 - g - h) (1 - r ^ {2}).
Integration
The first-order triangle and tetrahedron are constant stress elements and use a single integration point for the stiffness calculation when used in stress/displacement applications. A lumped mass matrix is used for both elements, with the total mass divided equally over the nodes. For heat transfer applications a three-point integration scheme is used for the conductivity and heat capacity matrices of the first-order triangle, with the integration points midway between the vertices and the centroid of the element; and a four-point integration scheme is used for the first-order tetrahedron. Distributed loads are integrated with two and three points for first-order triangles and tetrahedrons, respectively.
The three-point scheme is also used for the stiffness of the second-order triangle when it is used in stress/displacement applications. The mass matrix is integrated with a six-point scheme that integrates fourth-order polynomials exactly (Cowper, 1973). Distributed loads are integrated using three points. The heat transfer versions of the element use the six-point scheme for the conductivity and heat capacity matrices.
For stress/displacement applications the second-order tetrahedron uses 4 integration points for its stiffness matrix and 15 integration points for its consistent mass matrix. For heat transfer applications the conductivity and heat capacity matrices are integrated using 15 integration points. The first-order wedge uses 2 integration points for its stiffness matrix but 6 integration points for its lumped mass matrix. The second-order wedge uses 9 integration points for its stiffness matrix but 18 integration points for its consistent mass matrix. The integration schemes used for the second-order tetrahedra and wedge elements can be found in Stroud (1971).
3.2.7 Generalized plane strain elements
The generalized plane strain theory used in ABAQUS assumes that the model lies between two bounding planes, which may move as rigid bodies with respect to each other, thus causing strain of the "thickness direction" fibers of the model. It is assumed that the deformation of the model is independent of position with respect to this thickness direction, so the relative motion of the two planes causes a direct strain of the thickness direction fibers only. This strain and its first and second variations are defined as follows.
Let P _ { 0 } ( X _ { 0 } , Y _ { 0 } ) be a fixed point in one of the bounding planes. The length of the fiber between P _ { 0 } and its image in the other bounding plane is t _ { 0 } + \Delta u _ { z } , where t _ { 0 } is the length of this fiber in the initial configuration and \Delta u _ { z } is the change in length of this fiber. \Delta u _ { z } is the value of the degree of freedom at the first "generalized plane strain node" of the elements.
Figure 3.2.7-1 Generalized plane strain element.
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Arc of length t φ r P x y x a Bounding planes of the generalized plane strain model
The generalized plane strain nodes should be the same for all elements in any given connected region so that the bounding planes are the same for that region. Different regions may have different generalized plane strain nodes. Since the bounding planes are rigid, the length of a fiber at any other point ( x , y ) in the element is
t = t _ {0} + \Delta u _ {z} + (y - Y _ {0}) \Delta \phi_ {x} - (x - X _ {0}) \Delta \phi_ {y},
where
\Delta \phi_ {x} = (\Delta \phi_ {x}) | _ {0} + (\Delta \phi_ {x}) _ {1},
\Delta \phi_ {y} = (\Delta \phi_ {y}) | _ {0} + (\Delta \phi_ {y}) _ {1},
where ( \Delta \phi _ { i } ) | _ { 0 } ; \quad i = x , y ; are the initial values of \Delta \phi _ { i } , given on the *SOLID SECTION option associated with the elements; and ( \Delta \phi _ { i } ) | _ { 1 } are the degrees of freedom at the second "generalized plane strain node" of the elements.
The thickness direction logarithmic strain is
\varepsilon_ {z z} = \ln \left(\frac {t}{t _ {0}}\right).
The first variation of thickness direction strain is, therefore,
\delta \varepsilon_ {z z} = \frac {\delta t}{t},
where
\delta t = - \Delta \phi_ {y} \delta x + \Delta \phi_ {x} \delta y - x \delta \Delta \phi_ {y} + y \delta \Delta \phi_ {x} + \delta \Delta u _ {z}
and the second variation is
d \delta \varepsilon_ {z z} = \frac {d \delta t}{t} - \frac {\delta t d t}{t ^ {2}},
where
d \delta t = - \delta x d \Delta \phi_ {y} + \delta y d \Delta \phi_ {x} - d x \delta \Delta \phi_ {y} + d y \delta \Delta \phi_ {x}.
3.2.8 Axisymmetric elements
ABAQUS includes two libraries of solid elements, CAX and CGAX, whose geometry is axisymmetric (bodies of revolution) and which can be subjected to axially symmetric loading conditions. In addition, CGAX elements support torsion loading. As a result, CGAX elements will be referred to as generalized axisymmetric elements, and CAX elements as torsionless axisymmetric elements. In both cases, the body of revolution is generated by revolving a plane cross-section about an axis (the symmetry axis) and is readily described in cylindrical polar coordinates r , z , and µ. The radial and axial coordinates of a point on this cross-section are denoted by r and z, respectively. \mathbf { A } \mathbf { t } \theta = 0 , the radial and axial coordinates coincide with the global Cartesian X- and Y -coordinates.
If the loading consists of radial and axial components that are independent of µ and the material is either isotropic or orthotropic, with µ being a principal material direction, the displacement at any point will only have radial ( u _ { r } ) and axial ( u _ { z } ) components and the only stress components that will be nonzero are \sigma _ { r r } , \sigma _ { z z } , \sigma _ { \theta \theta } , and \sigma _ { r z } . Moreover, the deformation of any r-z plane completely defines the state of strain and stress in the body. Consequently, the geometric model is described by discretizing the reference cross-section at \theta = 0 .
If one allows for a circumferential component of loading (which is independent of µ) and for general material anisotropy, displacements and stress fields become three-dimensional, but the problem remains axisymmetric in the sense that the solution does not vary as a function of µ and the deformation of the reference r-z cross-section still characterizes the deformation in the entire body. The motion at any point will have, in addition to the aforementioned radial and axial displacements, a twist \phi (in radians) about the z-axis, which is independent of µ.
This section describes the formulation of the generalized axisymmetric elements. The formulation of the torsionless axisymmetric elements is a subset of this formulation.
Kinematic description
The coordinate system used with both families of 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 \theta measures the angle between the plane containing the point and the axis of the
Elements
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 that z is the second coordinate. 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.
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, as shown in Figure 3.2.8-1.
Figure 3.2.8-1 Cylindrical coordinate system and definition of position vectors.
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Y eθ eᵣ X eₒ eᵣ X θ φ θ e e_z, e_z X
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}.
Likewise, 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 deformed state. As shown in Figure 3.2.8-1, the radial and circumferential base vectors depend on the \theta coordinate: { \mathbf e } _ { r } = { \mathbf e } _ { r } ( \theta ) and \mathbf { e } _ { \theta } = \mathbf { e } _ { \theta } ( \theta ) . The current position x of the point can be represented in terms of the current radius r and the current axial position z:
Equation 3.2.8-1
\mathbf {x} = r \mathbf {e} _ {r} + z \mathbf {e} _ {z}.
The general axisymmetric motion at a point can be described by
Equation 3.2.8-2