20.1.1 Material damping

**Products: **ABAQUS/Standard ABAQUS/Explicit ABAQUS/CAE

Material damping can be defined:

for direct-integration (nonlinear, implicit or explicit), direct-solution steady-state, and subspace-based steady-state dynamic analysis; or

for mode-based (linear) dynamic analysis in ABAQUS/Standard.

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 “structural” or general 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.

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 Rayleigh damping, you specify two Rayleigh damping factors: for mass proportional damping and 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. A nonstructural feature (see “Nonstructural mass definition,” Section 2.6.1) is assumed to have no damping.

For a given mode *i* the fraction of critical damping, , can be expressed in terms of the damping factors and as:

where is the natural frequency at this mode. This equation implies that, generally speaking, the mass proportional Rayleigh damping, , damps the lower frequencies and the stiffness proportional Rayleigh damping, , damps the higher frequencies.

The 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 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 is used to multiply the element's lumped mass matrix to define the damping contribution from this term. has units of (1/time).

Input File Usage: | *DAMPING, ALPHA= |

ABAQUS/CAE Usage: | Property module: material editor: Alpha |

The factor introduces damping proportional to the strain rate, which can be thought of as damping associated with the material itself. 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, is interpreted as defining viscous material damping in ABAQUS, which creates an additional “damping stress,” , proportional to the total strain rate:

where is the strain rate. For hyperelastic (“Hyperelastic behavior of rubberlike materials,” Section 17.5.1) and hyperfoam (“Hyperelastic behavior in elastomeric foams,” Section 17.5.2) materials 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, is the material's current elastic stiffness. 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 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. has units of (time).

Input File Usage: | *DAMPING, BETA= |

ABAQUS/CAE Usage: | Property module: material editor: Beta |

Artificial damping in direct-integration dynamic analysis

In ABAQUS/Standard the operator used for implicit direct time integration contains “artificial damping” in addition to Rayleigh damping. This form of damping is controlled by the value of the numerical damping control parameter, , and is not the same as Rayleigh damping. It introduces damping that grows with frequency and with the ratio of the time increment to the period of vibration of a mode. Artificial damping is never very substantial for realistic time increments. See “Implicit dynamic analysis using direct integration,” Section 6.3.2, 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,” Section 6.3.3, 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 () increases, the stable time increment for ABAQUS/Explicit decreases according to the equation

where (by substituting , the frequency of the highest mode, into the equation for given previously)

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 1 rad/sec, while for the particular mesh chosen, the highest frequency is 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

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

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

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

and becomes

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

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

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

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 is often undesirable. To avoid a dramatic drop in the stable time increment, the stiffness proportional damping factor, , should be less than or of the same order of magnitude as the initial stable time increment without damping. With , the stable time increment is reduced by about 52%.

Damping in modal superposition procedures

Damping can be specified as part of the step definition for the following modal superposition procedures:

See “Damping options for modal dynamics,” Section 2.5.4 of the ABAQUS Theory Manual. The following types of damping are provided for linear analysis by modal methods:

Fraction of critical damping

Rayleigh damping

Composite modal damping

Structural damping

You can specify the damping in each eigenmode in the model or for the specified frequency as a fraction of the critical damping. Critical damping is defined as

where

Input File Usage: | Use the following option to define damping by specifying mode numbers: |

*MODAL DAMPING, MODAL=DIRECT, DEFINITION=MODE NUMBERS Use the following option to define damping by specifying a frequency range: *MODAL DAMPING, MODAL=DIRECT, DEFINITION=FREQUENCY RANGE |

ABAQUS/CAE Usage: | Use the following input to define damping by specifying mode numbers: |

Step module: Use the following input to define damping by specifying frequency ranges: Step module: |

Rayleigh damping introduces a damping matrix, , defined as

where is the mass matrix of the model, is the stiffness matrix of the model, and and are factors that you define.

In ABAQUS/Standard you can define and independently for each mode, so that the above equation becomes

where the subscript

Input File Usage: | Use the following option to define damping by specifying mode numbers: |

*MODAL DAMPING, RAYLEIGH, DEFINITION=MODE NUMBERS Use the following option to define damping by specifying a frequency range: *MODAL DAMPING, RAYLEIGH, DEFINITION=FREQUENCY RANGE |

ABAQUS/CAE Usage: | Use the following input to define damping by specifying mode numbers: |

Step module: Use the following input to define damping by specifying frequency ranges: Step module: |

Composite modal damping allows you to define a damping factor for each material in the model as a fraction of critical damping. These factors are then combined into a damping factor for each mode as weighted averages of the mass matrix associated with each material:

where is the critical damping fraction used in mode , is the critical damping fraction defined for material

If you specify composite modal damping, ABAQUS calculates the damping coefficients in the eigenfrequency extraction step from the damping factors that you defined for each material. Composite modal damping can be defined only by specifying mode numbers; it cannot be defined by specifying a frequency range.

Input File Usage: | Use both of the following options: |

*DAMPING, COMPOSITE= *MODAL DAMPING, MODAL=COMPOSITE |

ABAQUS/CAE Usage: | Property module: material editor: Composite:
Step module: Create Step: Linear perturbation: any valid step type: Damping: Composite modal: Use composite damping data |

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. This restriction applies when the excitation is purely sinusoidal, as in steady-state and random response analysis. The damping forces are then

where are the damping forces, ,

Structural damping can be defined only for mode-based steady-state dynamic and random response procedures.

Input File Usage: | Use the following option to define damping by specifying mode numbers: |

*MODAL DAMPING, STRUCTURAL, DEFINITION=MODE NUMBERS Use the following option to define damping by specifying a frequency range: *MODAL DAMPING, STRUCTURAL, DEFINITION=FREQUENCY RANGE |

ABAQUS/CAE Usage: | Use the following input to define damping by specifying mode numbers: |

Step module: Use the following input to define damping by specifying frequency ranges: Step module: |

Material options

The factor applies to all elements that use a linear elastic material definition (“Linear elastic behavior,” Section 17.2.1) 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 factor is multiplied by the slope of the force-strain (or moment-curvature) relationship at zero strain or curvature. In addition, the factor applies to all ABAQUS/Explicit elements that use a hyperelastic material definition (“Hyperelastic behavior of rubberlike materials,” Section 17.5.1), a hyperfoam material definition (“Hyperelastic behavior in elastomeric foams,” Section 17.5.2), or general shell sections (“Using a general shell section to define the section behavior,” Section 23.6.6). It cannot, however, be used in conjunction with the viscoelastic material model (“Time domain viscoelasticity,” Section 17.7.1).

In the case of a no tension elastic material the factor is not used in tension, while for a no compression elastic material the factor is not used in compression (see “No compression or no tension,” Section 17.2.2). In other words, these modified elasticity models exhibit damping only when they have stiffness.

The factor is applied to all elements that have mass including point mass elements in ABAQUS/Standard and excluding point mass elements in ABAQUS/Explicit (where, if required, discrete DASHPOTA elements in each global direction, each with one node fixed, can be used to introduce this type of damping). For point mass and rotary inertia elements in ABAQUS/Standard mass proportional or composite modal damping are defined as part of the point mass or rotary inertia definitions (“Point masses,” Section 24.1.1, and “Rotary inertia,” Section 24.2.1). This factor is not available for rotary inertia elements in ABAQUS/Explicit.

The factor is not available for spring elements: discrete dashpot elements should be used in parallel with spring elements instead.

The factor is also not applied to the transverse shear terms in ABAQUS/Standard beams and shells.

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, and are averaged over the substructure to define single values of and for the substructure. These are weighted averages, using the mass as the weighting factor for and the volume as the weighting factor for . These averaged damping values can be superseded by providing them directly in a second damping definition. See “Using substructures,” Section 10.1.1.