Spring Foundation and Thin Elastic Layer
In this section, the equations for the spring type physics nodes are developed using boundaries, but the generalizations to geometrical objects of other dimensions are obvious. Also, for cases where rotational springs are present, the relations between moments and rotations are analogous to the relations between forces and displacements described below.
Spring Foundation
A spring gives a force that depends on the displacement and acts in the opposite direction. In the case of a force that is proportional to the displacement, this is called Hooke’s law. In a suitable coordinate system, a spring condition can be represented as
where fs is a force/unit area, u is the displacement deforming the spring, and K is a stiffness matrix. u0 is an optional deformation offset, which describes the stress-free state of the spring.
If the spring stiffness is not constant, then it is in general easier to directly describe the force as a function of the displacement, so that
In the same way, a viscous damping can be described as a force proportional to the velocity
where D is a matrix representing the viscosity.
Structural (“loss factor”) damping is only relevant for frequency domain analysis and is defined as
where η is the loss factor and i is the imaginary unit. It is also possible to give individual loss factors for each component in the stiffness matrix K.
If the elastic part of the spring definition is given as a force versus displacement relation, the stiffness K is taken as the stiffness at the linearization point at which the frequency response analysis is performed. Since the loss factor force is proportional to the elastic force, the equation can be written as
The contribution to the virtual work is
Thin Elastic Layer Between Two Parts
A spring or damper can also act between two boundaries of an identity pair. The spring force then depends on the difference in displacement between the two boundaries.
The uppercase indices refer to “source” and “destination”. When a force versus displacement description is used,
The viscous and structural damping forces have analogous properties,
or
The virtual work expression is formulated on the destination side of the pair as
Here the displacements from the source side are obtained using the src2dst operator of the identity pair. Select the side with the finer mesh as destination if there is a difference in mesh density on the two sides of the pair.
Thin Elastic Layer on Interior Boundaries
On an interior boundary, the Thin Elastic Layer decouples the displacements between two sides of the boundary. The two boundaries are then connected by elastic and viscous forces with equal size but opposite directions, proportional to the relative displacements and velocities.
If an interface which is active on boundaries (Shell or Membrane in 3D for example) is added on the same interior boundary as a Thin Elastic Layer, then the virtual slit between the two sides of the boundary may be closed again. This happens if the domain interface and the boundary interface share the same displacement degrees of freedom.
The spring force can be written as
or
The viscous force is
and the structural damping force is
or
The subscripts u and d denote the “upside” and “downside” of the interior boundary, respectively.
The virtual work expression is formulated as
Stiffness from Material Data
When the stiffness is given in terms of actual material data and layer thickness ds, the stiffness in the normal direction is computed based on a state of plane strain, so that
The assumption of plane strain conditions is relevant when the material in the elastic layer is softer than its surroundings, and this is normally the case.
The shear stiffness is isotropic in the tangential plane, having the value
Since the layer thickness is known in this case, it is also possible to compute a strain in the elastic layer. The strain tensor has the is stored in a variable with a name like <interface>.<feature>.etelij, for example solid.tel1.etelxx for the normal strain. The two shear strains are stored in the xy and xz components of the tensor. In 3D, the orientation of the two local directions y and z used for the two shear strain directions is obtained using the following scheme:
1
Choose an auxiliary direction. Unless the normal to the layer is very close to the global X direction, use . If the X direction cannot be used, the Y direction is instead used as the auxiliary direction, .
2
The local y direction is obtained from the part of the auxiliary direction which is orthogonal to the normal direction n:
3
The local z direction is orthogonal to the normal and the local y direction:
Springs and Dampers in the Structural Mechanics Modeling chapter.