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Loading and Constraints
\begin{array}{l} d x _ {j} ^ {m} (t _ {j} ^ {2} t _ {i} ^ {1} - t _ {j} ^ {1} t _ {i} ^ {2}) + d x _ {k} ^ {m} (t _ {k} ^ {2} t _ {i} ^ {1} - t _ {k} ^ {1} t _ {i} ^ {2}) + \\ - P d \mathbf {x} ^ {m} \cdot (\mathbf {t} ^ {2} t _ {i} ^ {1} - \mathbf {t} ^ {1} t _ {i} ^ {2}) - (1 - P) d \mathbf {x} ^ {1} \cdot (\mathbf {t} ^ {2} t _ {i} ^ {1} - \mathbf {t} ^ {1} t _ {i} ^ {2}) = 0. \\ \end{array}
In the two-dimensional case n lies in the x-y or r-z plane and { \bf t } ^ { 2 } = ( 0 , 0 , - 1 ) . This implies that the second constraint equation is satisfied automatically. The remaining constraint equation is
x _ {i} ^ {m} t _ {i} ^ {1} = \mathbf {x} ^ {1} \cdot \mathbf {t} ^ {1} - x _ {k} ^ {m} t _ {k} ^ {1},
and its derivative is
d x _ {i} ^ {m} t _ {i} ^ {1} + d x _ {k} ^ {m} t _ {k} ^ {1} - P d \mathbf {x} ^ {n} \cdot \mathbf {t} ^ {1} - (1 - P) d \mathbf {x} ^ {1} \cdot \mathbf {t} ^ {1} = 0.
6.4.2 Shell to solid constraint
The shell to solid constraints SS LINEAR, SS BILINEAR, and SSF BILINEAR are used to connect shell elements to a solid element mesh. These MPCs are used in conjunction with the sliding constraint SLIDER (see ``Sliding constraint,'' Section 6.4.1). The SLIDER MPC maintains consistency with standard shell theory by forcing initially straight lines through the thickness to remain straight despite rotation and displacement. Thus, the shell to solid MPC must enforce the remaining constraints:
- The displacement of the shell node at the interface must be equal to the displacement of the corresponding point on a line of nodes through the thickness of the solid;
- The rotation of the shell node at the interface must be compatible with the rotation of the corresponding line of nodes through the thickness.
The three MPC types impose essentially the same constraints but use different weighting factors to reflect the nature of the interpolations in the solid elements. SS LINEAR is used with first-order elements, SS BILINEAR is used at the edges of second-order elements, and SSF BILINEAR is used for the middle of second-order elements. The MPCs can be used in two-dimensional as well as three-dimensional models. The degrees of freedom will automatically adapt to the dimensionality of the problem. The shell to solid MPCs can be used with any number of points through the thickness of the solid. The weighting functions for the nodes in the solid will be chosen based on this number.
The displacement constraint for the shell node is obtained by setting the displacement of the shell node, \mathbf { u } _ { s } ; equal to the weighted average of the displacements of the nodes in the solid:
Equation 6.4.2-1
\mathbf {u} _ {s} = \sum_ {i = 1} ^ {n} w _ {i} \mathbf {u} _ {c} ^ {i},
where the MPC selects the appropriate weighting factors, w _ { i } ; based on the MPC type and the location of the nodes. If the SLIDER MPC is used to keep the nodes in the solid on a straight line, the choice of weighting factors does not influence the solution.
Loading and Constraints
For the formulation of the rotation constraint, we assume that the nodes on the solid remain on one line. Hence, this line of nodes can be represented by the normalized direction N in the undeformed configuration and n in the deformed configuration. Let the rotation of the shell node be given by the finite rotation vector, { \big . } \phi . Then N, n, and \phi are related by the equation
Equation 6.4.2-2
\mathbf {C} \cdot \mathbf {N} = \mathbf {n},
where C = \exp [ \hat { \phi } ] with \hat { \pmb { \phi } } the skew symmetric matrix form of the rotation vector \phi . See ``Rotation variables,'' Section 1.3.1, for a more detailed discussion of finite rotations.
This constraint equation does not completely define the rotation of the shell node: any solution to Equation 6.4.2-2 can be augmented by a rotation { \boldsymbol { \phi } } _ { f } = \mathbf { n } \phi _ { f } around the line of nodes in the solid, where \phi _ { f } can be chosen arbitrarily. Hence, Equation 6.4.2-2 only constrains the finite rotation vector \phi to within two components. The linearized form of the rotation constraint is
Equation 6.4.2-3
\delta \boldsymbol {\theta} \times \mathbf {n} = d \mathbf {n},
where \delta \pmb { \theta } is the linearized rotation. See ``Rotation variables,'' Section 1.3.1, for a more detailed discussion of the linearized rotation.
We now define two local directions, s and t, so that s, t, and n form a right-handed, orthonormal, local coordinate system. We then project Equation 6.4.2-3 onto s and t, which yields
(\delta \boldsymbol {\theta} \times \mathbf {n}) \cdot \mathbf {s} = d \mathbf {n} \cdot \mathbf {s}, \quad (\delta \boldsymbol {\theta} \times \mathbf {n}) \cdot \mathbf {t} = d \mathbf {n} \cdot \mathbf {t}.
With some standard vector algebra these equations can be transformed into the form
Equation 6.4.2-4
\delta \pmb {\theta} \cdot \mathbf {t} = d \mathbf {n} \cdot \mathbf {s}, \qquad \delta \pmb {\theta} \cdot \mathbf {s} = - d \mathbf {n} \cdot \mathbf {t}.
The change in the normal can be expressed in terms of the displacement difference between the two extreme nodes 1 and n in the continuum:
\delta \mathbf {n} = \frac {1}{h} \delta \Delta \mathbf {u} \cdot (\mathbf {I} - \mathbf {n n}),
where \Delta \mathbf { u } = \mathbf { u } _ { n } - \mathbf { u } _ { 1 } is the displacement difference and h = \mathbf { n } \cdot \left( \mathbf { x } _ { n } - \mathbf { x } _ { 1 } \right) is the distance between the nodes. Substitution in Equation 6.4.2-4 yields the constraint equations
Equation 6.4.2-5
\delta \pmb {\theta} \cdot \mathbf {t} = \frac {1}{h} \delta \Delta \mathbf {u} \cdot \mathbf {s}, \qquad \delta \pmb {\theta} \cdot \mathbf {s} = - \frac {1}{h} \delta \Delta \mathbf {u} \cdot \mathbf {t}.
Loading and Constraints
These constraint equations are formulated in terms of local components of \delta \pmb \theta . . To obtain the constraint in terms of global components of \delta \pmb \theta _ { ; } , we choose a basis ( \mathbf { e } _ { i } , \mathbf { e } _ { j } , \mathbf { e } _ { k } ) that is a cyclic permutation of the global basis \left( \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } , \mathbf { e } _ { 3 } \right) , such that n _ { k } \geq n _ { i } and n _ { k } \geq n _ { j } . The constraints can then be written in component form as follows:
\begin{array}{l} \delta \theta_ {i} t _ {i} + \delta \theta_ {j} t _ {j} + \delta \theta_ {k} t _ {k} = \frac {1}{h} \delta \Delta \mathbf {u} \cdot \mathbf {s}, \\ \delta \theta_ {i} s _ {i} + \delta \theta_ {j} s _ {j} + \delta \theta_ {k} s _ {k} = - \frac {1}{h} \delta \Delta \mathbf {u} \cdot \mathbf {t}. \\ \end{array}
From these two equations we solve for \delta \theta _ { i } and \delta \theta _ { j } :
Equation 6.4.2-6
\delta \theta_ {i} = \frac {n _ {i}}{n _ {k}} \delta \theta_ {k} - \frac {1}{h n _ {k}} (s _ {j} \delta \Delta \mathbf {u} \cdot \mathbf {s} + t _ {j} \delta \Delta \mathbf {u} \cdot \mathbf {t}),
\delta \theta_ {j} = \frac {n _ {j}}{n _ {k}} \delta \theta_ {k} + \frac {1}{h n _ {k}} (s _ {i} \delta \Delta \mathbf {u} \cdot \mathbf {s} + t _ {i} \delta \Delta \mathbf {u} \cdot \mathbf {t}).
The linearized constraints shown above are used directly in geometrically linear analysis, linear perturbations, and ABAQUS/Explicit. In geometrically nonlinear analysis in ABAQUS/Standard, the linearized constraints are used to solve for the general nonlinear constraint Equation 6.4.2-2 with a Newton method. When the rotation is large, the permutation i; j; k may change to maintain the conditions that n _ { k } \geq n _ { i } and n _ { k } \geq n _ { j } .
6.4.3 Revolute joint
A revolute joint is a joint between two nodes in which the rotations of the nodes differ by a relative rotation about an axis that is fixed in and, therefore, rotates with the joint. A simple example is a hinge.
A revolute joint is implemented in ABAQUS/Standard as a multi-point constraint, defining the total rotation of the constrained ("slave") node (the first node given on the *MPC data line), \mathbf { C } ( \boldsymbol { \phi } ^ { S } ) , as the total rotation of the "master node" (the second node given on the *MPC data line), { \bf C } ( \phi ^ { M } ) , followed by the relative rotation \phi ^ { J } , about the axis of the joint a:
\mathbf {C} (\pmb {\phi} ^ {S}) = \mathbf {C} (\phi^ {J} \mathbf {a}) \cdot \mathbf {C} (\pmb {\phi} ^ {M}).
The joint axis, a, also rotates with the rotation of the master node:
\mathbf {a} = \mathbf {C} (\boldsymbol {\phi} ^ {M}) \cdot \mathbf {a} \big | _ {0}.
The angular velocity of the slave node is
\dot {\boldsymbol {\phi}} ^ {S} = \dot {\boldsymbol {\phi}} ^ {M} + \dot {\boldsymbol {\phi}} ^ {J} \mathbf {a},
and the virtual variations of the rotations are, likewise,
Equation 6.4.3-1
\delta \pmb {\phi} ^ {S} = \delta \pmb {\phi} ^ {M} + \delta \phi^ {J} \mathbf {a}.
Thus, the joint imposes three constraints (each component of the angular velocity of the slave node is constrained) but introduces an additional degree of freedom in the form of the relative rotation \phi ^ { J } . This means the joint provides a total of two constraints to the model if \phi ^ { J } is not prescribed and three constraints if it is.
The virtual work contribution of the three nodes of the joint is
\mathbf {M} ^ {S} \cdot \delta \pmb {\phi} ^ {S} + \mathbf {M} ^ {M} \cdot \delta \pmb {\phi} ^ {M} + M ^ {J} \delta \phi^ {J} = 0,
where \mathbf { M } ^ { S } is the total moment at node S, \mathbf { M } ^ { M } is the total moment at node M , and M ^ { J } is the moment in the joint. Applying the constraints ( Equation 6.4.3-1), this is
(\mathbf {M} ^ {S} + \mathbf {M} ^ {M}) \cdot \delta \pmb {\phi} ^ {M} + (\mathbf {M} ^ {S} \cdot \mathbf {a} + M ^ {J}) \delta \phi^ {J} = 0.
If there are no further constraints associated with the nodes of the joint, \delta \phi ^ { M } and \delta \phi ^ { J } are independent variations, so the constrained virtual work equation implies that
\mathbf {M} ^ {S} = - \mathbf {M} ^ {M}
and that
M ^ {J} = - \mathbf {M} ^ {S} \cdot \mathbf {a} = \mathbf {M} ^ {M} \cdot \mathbf {a}.
Because the revolute is implemented in this manner, the relative rotation in the joint \phi ^ { J } appears as a degree of freedom in the model (degree of freedom 6 at the third node of the MPC). Thus, a moment, M ^ { J } , can be applied in the joint by giving its value in the *CLOAD option; \phi ^ { J } can have a prescribed variation in time by using the *BOUNDARY option; or stiffness and/or damping can be associated with relative rotation of the joint by attaching a spring and/or dashpot to ground to this degree of freedom (a spring or dashpot to ground is used because the variable is a relative rotation).
6.4.4 Universal joint
A universal joint is a joint between two nodes containing orthogonal hinges that provide two axes of relative rotation in the joint.
A universal joint is implemented in ABAQUS/Standard as a multi-point constraint, defining the total rotation of the constrained ("slave") node (the first node given on the *MPC data line), \mathbf { C } ( \boldsymbol { \phi } ^ { S } ) , as the total rotation of the "master node" (the second node given on the *MPC data line), { \bf C } ( \phi ^ { M } ) , followed by two relative rotations: \phi _ { 1 } about the first axis of the joint { \bf a } _ { 1 } , then \phi _ { 2 } about the second axis of the
Loading and Constraints
joint \mathbf { a } _ { 2 } (which is orthogonal to \mathbf { a } _ { 1 } ) :
\mathbf {C} (\pmb {\phi} ^ {S}) = \mathbf {C} (\phi_ {2} \mathbf {a} _ {2}) \cdot \mathbf {C} (\phi_ {1} \mathbf {a} _ {1}) \cdot \mathbf {C} (\pmb {\phi} ^ {M}).
The first joint axis, { \bf a } _ { 1 } , rotates with the rotation of the master node:
\mathbf {a} _ {1} = \mathbf {C} (\pmb {\phi} ^ {M}) \cdot \mathbf {a} _ {1} \big | _ {0}.
The second joint axis has this rotation plus the rotation about the first joint axis:
\mathbf {a} _ {2} = \mathbf {C} (\phi_ {1} \mathbf {a} _ {1}) \cdot \mathbf {C} (\boldsymbol {\phi} ^ {M}) \cdot \mathbf {a} _ {2} \big | _ {0}.
The angular velocity of the slave node is
\dot {\pmb {\phi}} ^ {S} = \dot {\pmb {\phi}} ^ {M} + \dot {\phi} _ {1} \mathbf {a} _ {1} + \dot {\phi} _ {2} \mathbf {a} _ {2};
and the virtual variations of the rotations are, likewise,
Equation 6.4.4-1
\delta \pmb {\phi} ^ {S} = \delta \pmb {\phi} ^ {M} + \delta \phi_ {1} \mathbf {a} _ {1} + \delta \phi_ {2} \mathbf {a} _ {2}.
Thus, the joint imposes three constraints (each component of the angular velocity of the slave node is constrained) but introduces two additional degrees of freedom in the form of the relative rotations \phi _ { 1 } and \phi _ { 2 } . This means the joint provides a total of one constraint to the model if \phi _ { 1 } and \phi _ { 2 } are not prescribed or up to three constraints if they are.
The virtual work contribution of the joint is
\mathbf {M} ^ {S} \cdot \delta \pmb {\phi} ^ {S} + \mathbf {M} ^ {M} \cdot \delta \pmb {\phi} ^ {M} + M _ {1} \delta \phi_ {1} + M _ {2} \delta \phi_ {2} = 0,
where \mathbf { M } ^ { S } is the total moment at node S, \mathbf { M } ^ { M } is the total moment at node M , , and M _ { 1 } and M _ { 2 } are the moments in the joint hinges. Applying the constraints ( Equation 6.4.4-1), this is
(\mathbf {M} ^ {S} + \mathbf {M} ^ {M}) \cdot \delta \boldsymbol {\phi} ^ {M} + (\mathbf {M} ^ {S} \cdot \mathbf {a} _ {1} + M _ {1}) \delta \phi_ {1} + (\mathbf {M} ^ {S} \cdot \mathbf {a} _ {2} + M _ {2}) \delta \phi_ {2} = 0.
If there are no further constraints associated with the nodes of the joint, \delta \phi ^ { M } , \delta \phi _ { 1 } and \delta \phi _ { 2 } are independent variations, so that the constrained virtual work equation implies that
\mathbf {M} ^ {S} = - \mathbf {M} ^ {M},
M _ {1} = - \mathbf {M} ^ {S} \cdot \mathbf {a} _ {1} = \mathbf {M} ^ {M} \cdot \mathbf {a} _ {1}
and
M _ {2} = - \mathbf {M} ^ {S} \cdot \mathbf {a} _ {2} = \mathbf {M} ^ {M} \cdot \mathbf {a} _ {2}.
Because the universal joint is implemented in this manner, the relative rotations in the joint, \phi _ { 1 } and \phi _ { 2 } , , appear as degrees of freedom in the model (degree of freedom 6 at the third and fourth nodes of the MPC). Moments M _ { 1 } and M _ { 2 } can, therefore, be applied in the joint by giving their values in the *CLOAD option; \phi _ { 1 } and \phi _ { 2 } can have prescribed variations in time by using the *BOUNDARY option; or stiffness and/or damping can be associated with relative rotations of the joint by attaching springs and/or dashpots to ground to these degrees of freedom (springs or dashpots to ground are used because the variables are relative rotations).
6.4.5 Local velocity constraint
The V LOCAL MPC in ABAQUS/Standard constrains the velocity components at the first MPC node to be equal to the velocity components at the third node along local, rotating, directions. These local directions rotate according to the rotation at the second node. In the initial configuration the first local direction is from the second to the third node of the MPC. The global z-axis is used if these nodes coincide. The velocity of the first node \dot { \mathbf { u } } ^ { 1 } , is as follows:
\dot {\mathbf {u}} ^ {1} = \sum_ {i = 1} ^ {3} \mathbf {e} _ {i} ^ {2} \dot {u} _ {i} ^ {3}.
The constraint is integrated approximately to define
\Delta \mathbf {u} ^ {1} = \sum_ {i = 1} ^ {3} \left. \mathbf {e} _ {i} ^ {2} \right| _ {t + \frac {1}{2} \Delta t} \Delta u _ {i} ^ {3},
where
\left. \mathbf {e} _ {i} ^ {2} \right| _ {t + \frac {1}{2} \Delta t} \stackrel {\mathrm{def}} {=} \left. \mathbf {C} \right| _ {t + \frac {1}{2} \Delta t} \cdot \left. \mathbf {e} _ {i} ^ {2} \right| _ {t},
where
\mathbf {C} \big | _ {t + \frac {1}{2} \Delta t} = \mathbf {C} (\frac {1}{2} \Delta \phi^ {2})
is the increment of rotation defined by half of the magnitude of the increment of rotation at the second node of the constraint, \Delta \phi ^ { 2 } , and \mathbf { e } _ { i } ^ { 2 } | _ { t } , i = 1 , 2 , 3 , are the local directions at the beginning of the increment.
6.4.6 Kinematic coupling
A kinematic coupling constrains a group of slave nodes to the translation and rotation of a master node in a customized manner; combinations of slave node degrees of freedom are selected to participate in the constraint. Since each slave node has a separate relationship with the master node, the kinematic coupling constraint can be considered as the combination of general master-slave constraints. These general constraints are described below.
Rotations
Each possible combination of selected constraints on a slave node rotation results in rotation relationships that are unique but analogous to existing MPC types: zero constrained rotational degrees of freedom results in a pin constraint; one, results in a universal constraint; two, results in a revolute constraint; and three, results in a beam constraint. Each of these combinations is treated according to the appropriate theory employed in ABAQUS/Standard. To implement these constraints, an additional node is created internally for each slave node. For example, for revolute and universal constraints this additional node is used in a similar manner to the nodes required in the specification of the analog MPC types.
Translations
The additional internal node, described above, also reflects the motion of the slave node, relative to that of a fully constrained slave node in the following manner. Let \mathbf { X } ^ { m } be the position of the master node and \mathbf { X } ^ { s } be the position of the slave node in the reference configuration. The reference configuration position of the slave node with respect to the master node is then
\mathbf {N} = \mathbf {X} ^ {s} - \mathbf {X} ^ {m}.
Then, in the current configuration
\hat {\mathbf {x}} ^ {s} = \mathbf {x} ^ {m} + \mathbf {n},
where \hat { \mathbf { x } } ^ { s } is the fully constrained slave node position and
\mathbf {n} = \boldsymbol {C} \left(\boldsymbol {\phi} ^ {m}\right) \cdot \mathbf {N},
where C ( \phi ^ { m } ) is the rotation matrix associated with the master node rotation, \phi ^ { m } . The selectively constrained slave node position can be described as
Equation 6.4.6-1
\mathbf {x} ^ {s} = \hat {\mathbf {x}} ^ {s} + y _ {i} \mathbf {e} _ {i},
where y _ { i } are the translation degrees of freedom at the additional node and \mathbf { e } _ { i } are the current configuration base vectors, which rotate from the reference global Cartesian base vectors \mathbf { e } _ { i } according to
\mathbf {e} _ {i} = \boldsymbol {C} \left(\boldsymbol {\phi} ^ {m}\right) \cdot \mathbf {e} _ {i}.
This base vector rotation is made regardless of the choice of rotation constraint at the slave node. Constraint or release of slave node translation degree of freedom i can now be described as the constraint ( y _ { i } = 0 ) or release of translation degrees of freedom on the additional node. With these constraints on y _ { i } , Equation 6.4.6-1 can be used to define the constraint equations.
Linearized form
The linearized form is readily obtained as
Equation 6.4.6-2
\begin{array}{l} \delta \mathbf {x} ^ {s} = \delta \mathbf {x} ^ {m} + \delta \mathbf {n} + \delta y _ {i} \mathbf {e} _ {i} + y _ {i} \delta \mathbf {e} _ {i} \\ = \delta \mathbf {x} ^ {m} + \delta \boldsymbol {\phi} ^ {m} \times \mathbf {r} + \delta y _ {i} \mathbf {e} _ {i}, \\ \end{array}
where \mathbf { r } = \mathbf { x } ^ { s } - \mathbf { x } ^ { m } .
Second-order form
The initial stress stiffness terms can be obtained from
Equation 6.4.6-3
\begin{array}{l} \mathrm{d} \delta \mathbf {x} ^ {s} = \mathrm{d} \delta \mathbf {n} + y _ {i} \mathrm{d} \delta \mathbf {e} _ {i} + \delta y _ {i} \mathrm{d} \mathbf {e} _ {i} + \mathrm{d} y _ {i} \delta \mathbf {e} _ {i} \\ = - (\delta \pmb {\phi} ^ {m} \cdot \mathrm{d} \pmb {\phi} ^ {m}) \mathbf {r} + \frac {1}{2} (\delta \pmb {\phi} ^ {m} \cdot \mathbf {r}) \mathrm{d} \pmb {\phi} ^ {m} + \frac {1}{2} (\mathrm{d} \pmb {\phi} ^ {m} \cdot \mathbf {r}) \delta \pmb {\phi} ^ {m} + \delta y _ {i} \mathrm{d} \pmb {\phi} ^ {m} \times \mathbf {e} _ {i} + \mathrm{d} y _ {i} \delta \pmb {\phi} ^ {m} \times \mathbf {e} _ {i}. \\ \end{array}
7. References
7.1 References
7.1.1 References
Agah-Tehrani, A., E. H. Lee, R. L. Mallett, and E. T. Onat, ``The Theory of Elastic-Plastic Deformation at Finite Strain with Induced Anisotropy Modeled as Combined Isotropic-Kinematic Hardening,'' Metal Forming Report, Rensselaer Polytechnic Institute, Troy, New York, 1986.
Al-Ani, A. M., and J. W. Hancock, ``J-Dominance of Short Cracks in Tension and Bending,'' Journal of the Mechanics and Physics of Solids, vol. 39, pp. 23-43, 1991.
Anagnastopoulos, S. A., ``Response Spectrum Techniques for Three Component Earthquake Design ,'' Earthquake Engineering and Structural Dynamics, vol. 9, pp. 459-476, 1981.
Aoki, S., K. Kishimoto, and M. Sakjata, ``Crack Tip Stress and Strain Singularity in Thermally Loaded Elastic-Plastic Material,'' Transactions of the ASME, Journal of Applied Mechanics , vol. 48, no. 2, pp. 428-429, 1981.
Aravas, N., ``On the Numerical Integration of a Class of Pressure-Dependent Plasticity Models ,'' International Journal for Numerical Methods in Engineering, vol. 24, pp. 1395-1416, 1987.
Arruda, E. M., and M. C. Boyce, ``A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials,'' Journal of the Mechanics and Physics of Solids , vol. 41, no. 2, pp. 389-412, 1993.
Ashwell, D. G., and R. H. Gallagher, Editors, Finite Elements for Thin Shells and Curved Members , John Wiley and Sons, London, 1976.
Atomic Energy Commission Regulatory Guide 1.60, "Design Response Spectra for Seismic Design of Nuclear Power Plants."
Atomic Energy Commission Regulatory Guide 1.92, "Combining Modal Responses."
Barlow, J., ``Optimal Stress Locations in Finite Element Models,'' International Journal for Numerical Methods in Engineering, vol. 10, pp. 243-251, 1976.
Barnett, D. M., and R. J. Asaro, ``The Fracture Mechanics of Slit-Like Cracks in Anisotropic Elastic Media,'' Journal of the Mechanics and Physics of Solids, vol. 20, pp. 353-366, 1972.
Bathe, K. J., and C. A. Almeida, ``A Simple and Effective Pipe Elbow Element--Linear Analysis,'' Transactions of the ASME, Journal of Applied Mechanics , vol. 47, no. 1, 1980.
Bathe, K. J., and E. N. Dvorkin, ``A Continuum Mechanics-Based Four-Node Shell Element for General Non-Linear Analysis,'' International Journal of Computer Aided Engineering Software, vol. 1, 1984.
Bathe, K. J., and E. L. Wilson, ``Large Eigenvalue Problems in Dynamic Analysis,'' Proceedings of the ASCE, EM6, 98, pp. 1471-1485, 1972.
References
Batoz, J. L., K. J. Bathe, and L. W. Ho, ``A Study of Three-Node Triangular Plate Bending Elements,'' International Journal for Numerical Methods in Engineering, vol. 15, pp. 1771-1821, 1980.
Bayliss, A., M. Gunzberger, and E. Turkel, ``Boundary Conditions for the Numerical Solution of Elliptic Equations in Exterior Regions ,'' SIAM Journal of Applied Mathematics., vol. 42, no. 2, pp. 430-451, 1982.
Bear, J., Dynamics of Fluids in Porous Media, American Elsevier Publishing Company, Dover, New York, 1972.
Belytschko, T., ``Survey of Numerical Methods and Computer Programs for Dynamic Structural Analysis,'' Nuclear Engineering and Design, vol. 37, pp. 23-34, 1976.
Belytschko, T., J. I. Lin, and C. S. Tsay, ``Explicit Algorithms for the Nonlinear Dynamics of Shells,'' Computer Methods in Applied Mechanics and Engineering, vol. 43, pp. 251-276, 1984.
Belytschko, T., B. L. Wong, and H. Y. Chiang, ``Advances in One-Point Quadrature Shell Elements,'' Computer Methods in Applied Mechanics and Engineering, vol. 96, pp. 93-107, 1992.
Bergan, P. B., G. Horrigmoe, B. Krakeland, and T. H. Soreide, ``Solution Techniques for Non-Linear Finite Element Problems,'' International Journal for Numerical Methods in Engineering, vol. 12, pp. 1677-1696, 1978.
Bergström, J. S., and M. C. Boyce, ``Constitutive Modeling of the Large Strain Time-Dependent Behavior of Elastomers,'' Journal of the Mechanics and Physics of Solids, vol. 46, pp. 931-954, 1998.
Betegón, C., and J. W. Hancock, ``Two-Parameter Characterization of Elastic-Plastic Crack-Tip Fields,'' Journal of Applied Mechanics, vol. 58, pp. 104-110, 1991.
Bilby, B. A., G. E. Goldthorpe, and I. C. Howard, "A Finite Element Investigation of the Effect of Specimen Geometry on the Fields of Stress and Strain at the Tip of Stationary Cracks, " Size Effects in Fracture, Institution of Mechanical Engineers, London, pp. 37-46, 1986.
Bilkhu, S., Private communication, 1987.
Budiansky, B., and J. L. Sanders, "On the `Best' First-Order Linear Shell Theory," Progress in Applied Mechanics, The Prager Anniversary Volume, Macmillan, London, pp. 129-140, 1963.
Calladine, C. R., ``A Microstructural View of the Mechanical Properties of Saturated Clay ,'' Geotechnique, vol. 21, no. 4, pp. 391-415.
Chen, W. F., and D. J. Han, Plasticity for Structural Engineers , Springer-Verlag, New York, 1988.
Chu, C. C., and A. Needleman, ``Void Nucleation Effects in Biaxially Stretched Sheets,'' Journal of Engineering Materials and Technology, vol. 102, pp. 249-256, 1980.
Clough, R. W., and J. Penzien, Dynamics of Structures, McGraw-Hill, New York, 1975.
Coffin, L. F., Jr., ``The Flow and Fracture of a Brittle Material,'' Journal of Applied Mechanics, vol. 72, pp. 233-248, 1950.
Cohen, M., and P. C. Jennings, Silent Boundary Methods for Transient Analysis (in Computational