**4.3 Comparison of Rayleigh damping with the damping of the kelvin-Voigt material**

The Rayleigh damping depends only on the velocities of mass points in space, and that damping also arises in the case of the movement of a rigid body due to coefficient *α*. To compare the Rayleigh damping and damping caused by the Kelvin-Voigt material model let us assume *α* ¼ 0. The equation of motion then will read as:

$$\mathbf{M}\_{\epsilon}\ddot{\mathbf{d}} + \beta \int\_{\Omega\_{\epsilon}} \mathbf{B}^{T} \mathbf{D} \mathbf{B} d\Omega\_{\epsilon} \dot{\mathbf{d}} + \int\_{\Omega\_{\epsilon}} \mathbf{B}^{T} \boldsymbol{\sigma} d\Omega\_{\epsilon} = \mathbf{f}\_{\epsilon}^{\text{ext}} \tag{46}$$

If we compare the equation with the Kelvin-Voigt material model (45) with this eq. (49), one can see that the difference between them lies only in the term with the first derivative of the deformation parameters by time, i.e., the velocities **d**\_ . If we simplify (reduce) these equations to 1D tasks, substituting Young's modulus *E* for the constitutive matrix **D** and comparing the terms with \_ **d** from the equations we obtain the relation

$$\beta E \int\_{\Omega\_{\epsilon}} \mathbf{B}^{T} \mathbf{B} d\Omega\_{\epsilon} \dot{\mathbf{d}} = \eta \int\_{\Omega\_{\epsilon}} \mathbf{B}^{T} \mathbf{B} d\Omega\_{\epsilon} \dot{\mathbf{d}} \tag{47}$$

and subsequently simple relation between the parameters *β* and *η* of both models.

$$
\boxed{\beta E = \eta} \tag{48}
$$

It is clear that the Rayleigh damping and the damping caused by Kelvin-Voight material model are identical for 1D problems.

#### **4.4 Damping caused by the plasticizing, or damaging of material**

**Figure 13** shows a loading and unloading diagram for a) elasto-plastic b) elastodamage material for the 1D stress and strain state.

$$\mathbf{a}(\mathbf{a})\,\boldsymbol{w}^{p}(\tilde{\mathbf{t}}) = \int\_{\boldsymbol{o}} \boldsymbol{\sigma}(t) : \dot{\mathbf{e}}^{p}(t)dt = \int\_{\boldsymbol{o}} \boldsymbol{\sigma}(t) : \dot{\mathbf{e}}(t)dt - \frac{1}{2}\mathbf{C}^{-1} : \overline{\boldsymbol{\sigma}}(\tilde{\mathbf{t}}) : \overline{\boldsymbol{\sigma}}(\tilde{\mathbf{t}}) \tag{49}$$

$$(\mathbf{b})\ w^p(\tilde{\mathbf{t}}) = \int\_{\boldsymbol{\theta}} \boldsymbol{\sigma}(t) : \dot{\mathbf{e}}^p(t) dt = \int\_{\boldsymbol{\theta}} \boldsymbol{\sigma}(t) : \dot{\mathbf{e}}(t) dt - \frac{1}{2} \overline{\mathbf{e}}(\tilde{\mathbf{t}}) : \overline{\mathbf{e}}(\tilde{\mathbf{t}}) \tag{50}$$

**Figure 13.**

*Loading and unloading diagram for elasto-plastic (a) and elasto-damage (b) materials, with division of energy into elastic and dissipative.*

*On the Nonlinear Transient Analysis of Structures DOI: http://dx.doi.org/10.5772/intechopen.108446*

**Figure 14.** *Friction diagram [1].*

This dissipation energy is lost from the mechanical system in the form of heat and this process manifests itself in dynamics as damping. The sum of this dissipation energy *w<sup>p</sup>* plus the elastic energy *w<sup>e</sup>* is equal to the total energy used for the deformation work.

In the case of plasticizing the damping occurs at the time when a plastic deformation, arises, and later vibration is no longer damped unless the maximum strain achieved so far is overcome once again. Plasticizing does not affect the natural frequency of the structure.

In the case of damage, the damping occurs at the time when a material damage arises, and later vibration is no longer damped unless the maximum strain achieved so far is overcome once again. Damage causes material softening, so the Eigen frequencies will be decreased.

#### **4.5 Damping caused by friction**

The Coulomb friction (see the diagram in **Figure 14**) is also a significant part of internal damping of structures. It occurs in connections of structural elements and most often comes about in screw or rivet connections in steel structures. Mechanical work and pertinent dissipation arise in the case of any relative motion in the connections. In contrast to plastic or damage behavior, mechanical work is performed with each relative forward and backward motion, and the pertinent damping is then permanent during vibration. The friction force is usually proportional to the pressure force in connections (Coulomb friction).
