Loss Factor Damping

Loss factor damping is only applicable in frequency domain. When using loss factor damping, a complex constitutive matrix is used. With an isotropic loss factor ηs, this means that

where D is the constitutive matrix computed from the material data, and Dc is the complex constitutive matrix used when computing the stresses. For a linear elastic material, this would be equivalent to multiplying Young’s modulus by the factor (1 + jηs). For a nonlinear elastic material, this applies to the tangential stiffness.

It is also possible to give individual loss factors for each entry in the constitutive matrix, so that

In the case of an orthotropic material, yet another option is available, where each individual component of Young’s modulus and shear modulus can be given an individual loss coefficient:

The complex moduli are then used to form the constitutive matrix.

For hyperelastic materials, the loss information appears as a contribution to the second Piola–Kirchhoff stress:

For loss factor damping, the following definition is used for the elastic part of the entropy:

note that Selast here denotes the entropy contribution and not any stress.

This is because the entropy is a function of state and thus independent of the strain rate, while the damping represents the rate-dependent effects in the material (for example, viscous or viscoelastic effects). The internal work of such inelastic forces averaged over the time period 2π/ω can be computed as:

Qh can be used as a heat source for modeling of the heat generation in vibrating structures, when coupled with the frequency-domain analysis for the stresses and strains.