Material damping

Material damping can be defined:

for direct-integration (nonlinear, implicit or explicit), subspace-based direct-integration, direct-solution steady-state, and subspace-based steady-state dynamic analysis; or
for mode-based (linear) dynamic analysis in Abaqus/Standard.

The following topics are discussed:

Rayleigh damping
Structural damping
Artificial damping in direct-integration dynamic analysis
Artificial damping in explicit dynamic analysis
Effects of damping on the stable time increment in Abaqus/Explicit
Damping in modal superposition procedures
Material options
Elements

Rayleigh damping

In direct-integration dynamic analysis you very often define energy dissipation mechanisms—dashpots, inelastic material behavior, etc.—as part of the basic model. In such cases there is usually no need to introduce additional damping: it is often unimportant compared to these other dissipative effects. However, some models do not have such dissipation sources (an example is a linear system with chattering contact, such as a pipeline in a seismic event). In such cases it is often desirable to introduce some general damping. Abaqus provides “Rayleigh” damping for this purpose. It provides a convenient abstraction to damp lower (mass-dependent) and higher (stiffness-dependent) frequency range behavior.

Rayleigh damping can also be used in direct-solution steady-state dynamic analyses and subspace-based steady-state dynamic analyses to get quantitatively accurate results, especially near natural frequencies.

To define material Rayleigh damping, you specify two Rayleigh damping factors: $α_{R}$ for mass proportional damping and $β_{R}$ for stiffness proportional damping. In general, damping is a material property specified as part of the material definition. For the cases of rotary inertia, point mass elements, and substructures, where there is no reference to a material definition, the damping can be defined in conjunction with the property references. Any mass proportional damping also applies to nonstructural features (see Nonstructural mass definition).

For a given mode i the fraction of critical damping, $ξ_{i}$ , can be expressed in terms of the damping factors $α_{R}$ and $β_{R}$ as:

ξ_{i} = \frac{α_{R}}{2 ω_{i}} + \frac{β_{R} ω_{i}}{2},

where $ω_{i}$ is the natural frequency at this mode. This equation implies that, generally speaking, the mass proportional Rayleigh damping, $α_{R}$ , damps the lower frequencies and the stiffness proportional Rayleigh damping, $β_{R}$ , damps the higher frequencies.

Mass proportional damping

The $α_{R}$ factor introduces damping forces caused by the absolute velocities of the model and so simulates the idea of the model moving through a viscous “ether” (a permeating, still fluid, so that any motion of any point in the model causes damping). This damping factor defines mass proportional damping, in the sense that it gives a damping contribution proportional to the mass matrix for an element. If the element contains more than one material in Abaqus/Standard, the volume average value of $α_{R}$ is used to multiply the element's mass matrix to define the damping contribution from this term. If the element contains more than one material in Abaqus/Explicit, the mass average value of $α_{R}$ is used to multiply the element's lumped mass matrix to define the damping contribution from this term. $α_{R}$ has units of (1/time).

Input File Usage

DAMPING, ALPHA= $α_{R}$

Abaqus/CAE Usage

Property module: material editor: MechanicalDamping: Alpha:  $α_{R}$

Defining variable mass proportional damping in Abaqus/Explicit

In Abaqus/Explicit you can define $α_{R}$ as a tabular function of temperature and/or field variables. Therefore, mass proportional damping can vary during an Abaqus/Explicit analysis.

Input File Usage

DAMPING, ALPHA=TABULAR

Stiffness proportional damping

The $β_{R}$ factor introduces damping proportional to the strain rate, which can be thought of as damping associated with the material itself. $β_{R}$ defines damping proportional to the elastic material stiffness. Since the model may have quite general nonlinear response, the concept of “stiffness proportional damping” must be generalized, since it is possible for the tangent stiffness matrix to have negative eigenvalues (which would imply negative damping). To overcome this problem, $β_{R}$ is interpreted as defining viscous material damping in Abaqus, which creates an additional “damping stress,” $σ_{d}$ , proportional to the total strain rate:

σ_{d} = β_{R} D^{e l} \dot{ε},

where $\dot{ε}$ is the strain rate. For hyperelastic (Hyperelastic behavior of rubberlike materials) and hyperfoam (Hyperelastic behavior in elastomeric foams) materials $D^{e l}$ is defined as the elastic stiffness in the strain-free state. For all other linear elastic materials in Abaqus/Standard and all other materials in Abaqus/Explicit, $D^{e l}$ is the material's current elastic stiffness. $D^{e l}$ will be calculated based on the current temperature during the analysis.

This damping stress is added to the stress caused by the constitutive response at the integration point when the dynamic equilibrium equations are formed, but it is not included in the stress output. As a result, damping can be introduced for any nonlinear case and provides standard Rayleigh damping for linear cases; for a linear case stiffness proportional damping is exactly the same as defining a damping matrix equal to $β_{R}$ times the (elastic) material stiffness matrix. Other contributions to the stiffness matrix (e.g., hourglass, transverse shear, and drill stiffnesses) are not included when computing stiffness proportional damping. $β_{R}$ has units of (time).

Input File Usage

DAMPING, BETA= $β_{R}$

Abaqus/CAE Usage

Property module: material editor: MechanicalDamping: Beta:  $β_{R}$

Defining variable stiffness proportional damping in Abaqus/Explicit

In Abaqus/Explicit you can define $β_{R}$ as a tabular function of temperature and/or field variables. Therefore, stiffness proportional damping can vary during an Abaqus/Explicit analysis.

Input File Usage

DAMPING, BETA=TABULAR

Structural damping

Structural damping assumes that the damping forces are proportional to the forces caused by stressing of the structure and are opposed to the velocity. Therefore, this form of damping can be used only when the displacement and velocity are exactly 90° out of phase. Structural damping is best suited for frequency domain dynamic procedures (see Damping in modal superposition procedures below). The damping forces are then

F_{D}^{N} = i s I^{N},

where $F_{D}^{N}$ are the damping forces, $i = \sqrt{- 1}$ , s is the user-defined structural damping factor, and $I^{N}$ are the forces caused by stressing of the structure. The damping forces due to structural damping are intended to represent frictional effects (as distinct from viscous effects). Thus, structural damping is suggested for models involving materials that exhibit frictional behavior or where local frictional effects are present throughout the model, such as dry rubbing of joints in a multi-link structure.

Structural damping can be added to the model as mechanical dampers such as connector damping or as a complex stiffness on spring elements.

Structural damping can be used in steady-state dynamic procedures that allow for nondiagonal damping.

Input File Usage

Use the following option to define structural damping:

DAMPING, STRUCTURAL= $s$

Abaqus/CAE Usage

Property module: material editor: MechanicalDamping: Structural:  $s$

Artificial damping in direct-integration dynamic analysis

In Abaqus/Standard the operators used for implicit direct time integration introduce some artificial damping in addition to Rayleigh damping. Damping associated with the Hilber-Hughes-Taylor and hybrid operators is usually controlled by the Hilber-Hughes-Taylor parameter $α$ , which is not the same as the $α_{R}$ parameter controlling the mass proportional part of Rayleigh damping. The $β$ and $γ$ parameters of the Hilber-Hughes-Taylor and hybrid operators also affect numerical damping. The $α$ , $β$ , and $γ$ parameters are not available for the backward Euler operator. See Implicit dynamic analysis using direct integration for more information about this other form of damping.

Artificial damping in explicit dynamic analysis

Rayleigh damping is meant to reflect physical damping in the actual material. In Abaqus/Explicit a small amount of numerical damping is introduced by default in the form of bulk viscosity to control high frequency oscillations; see Explicit dynamic analysis for more information about this other form of damping.

Effects of damping on the stable time increment in Abaqus/Explicit

As the fraction of critical damping for the highest mode ( $ξ_{\max}$ ) increases, the stable time increment for Abaqus/Explicit decreases according to the equation

Δ t \leq \frac{2}{ω_{\max}} (\sqrt{1 + ξ_{\max}^{2}} - ξ_{\max}),

where (by substituting $ω_{\max}$ , the frequency of the highest mode, into the equation for $ξ_{i}$ given previously)

ξ_{\max} = \frac{α_{R}}{2 ω_{\max}} + \frac{β_{R} ω_{\max}}{2} .

These equations indicate a tendency for stiffness proportional damping to have a greater effect on the stable time increment than mass proportional damping.

To illustrate the effect that damping has on the stable time increment, consider a cantilever in bending modeled with continuum elements. The lowest frequency is $ω_{\min} =$ 1 rad/sec, while for the particular mesh chosen, the highest frequency is $ω_{\max} =$ 1000 rad/sec. The lowest mode in this problem corresponds to the cantilever in bending, and the highest frequency is related to the dilation of a single element.

With no damping the stable time increment is

Δ t = \frac{2}{ω_{\max}} = 2 \times 10^{- 3} \sec .

If we use stiffness proportional damping to create 1% of critical damping in the lowest mode, the damping factor is given by

β_{R} = \frac{2 \times 0.01}{1} = 2 \times 10^{- 2} \sec .

This corresponds to a critical damping factor in the highest mode of

ξ_{\max} = \frac{ω_{\max} β_{R}}{2} = 10.

The stable time increment with damping is, thus, reduced by a factor of

(\sqrt{1 + 10^{2}} - 10) \approx 0.05,

and becomes

\begin{matrix} Δ t & \approx (2 \times 10^{- 3}) \times 0.05 \\ \approx 1 \times 10^{- 4} . \end{matrix}

Thus, introducing 1% critical damping in the lowest mode reduces the stable time increment by a factor of twenty.

However, if we use mass proportional damping to damp out the lowest mode with 1% of critical damping, the damping factor is given by

α_{R} = 2 ω_{\min} ξ = 2 \times 1 \times 10^{- 2} = 2 \times 10^{- 2} \sec^{- 1},

which corresponds to a critical damping factor in the highest mode of

ξ_{\max} = \frac{α_{R}}{2 ω_{\max}} = \frac{2 \times 10^{- 2}}{2 \times 1000} = 10^{- 5} .

The stable time increment with damping is reduced by a factor of

(\sqrt{1 + 10^{- 10}} - 10^{- 5}) \approx 0.99999,

which is almost negligible.

This example demonstrates that it is generally preferable to damp out low frequency response with mass proportional damping rather than stiffness proportional damping. However, mass proportional damping can significantly affect rigid body motion, so large $α_{R}$ is often undesirable. To avoid a dramatic drop in the stable time increment, the stiffness proportional damping factor, $β_{R}$ , should be less than or of the same order of magnitude as the initial stable time increment without damping. With $β_{R} = 2 / ω_{\max}$ , the stable time increment is reduced by about 52%.

Material options

The $β_{R}$ factor applies to all elements that use a linear elastic material definition (Linear elastic behavior) and to Abaqus/Standard beam and shell elements that use general sections. In the latter case, if a nonlinear beam section definition is provided, the $β_{R}$ factor is multiplied by the slope of the force-strain (or moment-curvature) relationship at zero strain or curvature. In addition, the $β_{R}$ factor applies to all Abaqus/Explicit elements that use a hyperelastic material definition (Hyperelastic behavior of rubberlike materials), a hyperfoam material definition (Hyperelastic behavior in elastomeric foams), or general shell sections (Using a general shell section to define the section behavior).

In the case of a no tension elastic material the $β_{R}$ factor is not used in tension, while for a no compression elastic material the $β_{R}$ factor is not used in compression (see No compression or no tension). In other words, these modified elasticity models exhibit damping only when they have stiffness.

Elements

The $α_{R}$ factor is applied to all elements that have mass including point mass elements (discrete DASHPOTA elements in each global direction, each with one node fixed, can also be used to introduce this type of damping). For point mass and rotary inertia elements mass proportional or composite modal damping are defined as part of the point mass or rotary inertia definitions (Point masses and Rotary inertia).

The $β_{R}$ factor is not available for spring elements: discrete dashpot elements should be used in parallel with spring elements instead.

The $β_{R}$ factor is also not applied to the transverse shear terms in Abaqus/Standard beams and shells.

The hybrid element stiffness matrix formulation is different than the corresponding non-hybrid formulation; therefore, the stiffness proportional damping is different for the same value of the $β_{R}$ factor in nonlinear dynamic analysis. In linear analyses Abaqus/Standard imposes equivalent stiffness proportional damping for hybrid and non-hybrid elements.

In Abaqus/Standard composite modal damping cannot be used with or within substructures. Rayleigh damping can be introduced for substructures. When Rayleigh damping is used within a substructure, $α_{R}$ and $β_{R}$ are averaged over the substructure to define single values of $α_{R}$ and $β_{R}$ for the substructure. These are weighted averages, using the mass as the weighting factor for $α_{R}$ and the volume as the weighting factor for $β_{R}$ . These averaged damping values can be superseded by providing them directly in a second damping definition. See Using substructures.