Theory for the Wire Interface

Theory Background for the Wire Interface

The displacements in the Wire interface are represented by Lagrange shape functions.

The axial strain εn is calculated by expressing the global strains in tangential derivatives and projecting the global strains on the edge.

where t is the edge tangent vector and εgT is defined as

The strains is expressed as Green-Lagrange strains, allowing large displacements and rotations.

The axial strain written out becomes

Stress-Strain Relation

The constitutive relation for a wire is uniaxial. Since wires in most cases do not have a homogeneous cross section, it is not meaningful to compute a stress. Rather, a force vs. strain relation is used. The axial force in tension, N, is computed as

where

•

kA is the axial stiffness of the wire

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εn is the total axial strain

•

εn,el is the elastic axial strain

•

εinel is the sum of all inelastic strain contributions, such as:

Initial strain, ε0

Thermal strain, εth

Hygroscopic strain, εhs

•

Nex is the sum of all extra axial force contributions such as:

Initial axial force Ni

External axial force Next

In a geometrically nonlinear analysis, the strains are interpreted as Green-Lagrange strains. It should be noted that while Green-Lagrange strains are formally not additive, it assumed that even if displacements and rotations are large, the axial strains are small.

The ideal wire will not be able to sustain compressive forces. In practice, it may wrinkle in an unpredictable manner when tension is lost. In order to maintain numerical stability, a small stiffness is instead used when in compression. The following expression is used for the force in this case:

where β is a stiffness reduction factor, and

The reduced compressive stiffness is constant, βkA, below a certain compressive strain εn,c. Between the two domains of constant stiffness, there is a transition region in which the stiffness drops exponentially, so that the strain-force relation always has a continuous derivative.

Implementation

Using the principle of virtual work results in the following weak formulation

where the summation stands for summation over all points in the geometry. Here, stresses are not directly accessible. The volume integration must be replaced by a line integration.