Non-linear Material Force Term |
 
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Non-linear Material Force Term |
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The equilibrium equations outlined in the Finite Element Formulation section are developed for the case where all the generalised stress components are linearly related to their corresponding strain components. This linear stress-strain relationship is given by:
σ = D ε (1)
where σ and ε are generalised stress and strain components, respectively, and D is a component of the constitutive (stiffness) matrix.
Consider the more general case where the stress-strain relationship is nonlinear. Such a relationship is presented below.
Non-linear Material Model
This curve could represent any of the component stress-strain functions, such as axial force v axial strain, bending moment v curvature, or torque v twist relationships. The non-linear curve (shown dotted above) may be approximated as a series of straight-line segments. For a specified strain component, ε, the stress component, σ, is defined by the equation:
σ = σ0 + D(ε - ε0) (2)
where:
ε0 is the nearest strain value below ε where the line segment intersects with the stress-strain curve.
σ0 is the stress component corresponding to ε0.
D is the slope of the line segment.
Equation (2) may be rearranged to give:
σ = D ε - h0 (3)
where:
h0 = D ε0 - σ0 (4)
In Flexcom terminology, h0 is known as the non-linear material force.
Equation (4) is now in the same form as the linear relationship, with the only change being the subtraction of a new non-linear material force term from the right hand side of the equation. At any given iteration of the solution of the equilibrium equations, the non-linear material force is computed from Equation (4), and is accounted for in the Flexcom solution by putting it on the right hand side of the equilibrium equations.
•*MOMENT-CURVATURE is used to define moment-curvature curves for non-linear materials.
•*FORCE-STRAIN is used to define force-strain curves for non-linear materials.
•*TORQUE-TWIST is used to define torque-twist curves for non-linear materials.