Loss Factor Peak of Viscoelastic Materials: Magnitude to Width Relations

Pritz, T.

doi:10.1006/jsvi.2001.3636

Cited by 29 publications

(19 citation statements)

References 14 publications

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“…Parameter fits of experimental measurements to the fractional Zener model for biological materials include for brain [56][57][58][59], human root dentin [60], cranial bone [61], liver [57], arteries [62], breast [63], and hamstring muscle [64]. Non-biological materials are exemplified by metals [39], doped corning glass [49], rubber [51], and polymers [63,[65][66][67][68][69]. For an account of experimental fits to fractional calculus stress-strain models made up to the year of 2009, see Section 2 in [48].…”

Section: Experimental Evidencesmentioning

confidence: 99%

On a fractional Zener elastic wave equation

Näsholm

Holm

2013

fcaa

103

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This survey concerns a causal elastic wave equation which implies frequency power-law attenuation. The wave equation can be derived from a fractional Zener stress-strain relation plus linearized conservation of mass and momentum. A connection between this four-parameter fractional wave equation and a physically well established multiple relaxation acoustical wave equation is reviewed. The fractional Zener wave equation implies three distinct attenuation power-law regimes and a continuous distribution of compressibility contributions which also has power-law regimes. Furthermore it is underlined that these wave equation considerations are tightly connected to the representation of the fractional Zener stress-strain relation, which includes the spring-pot viscoelastic element, and by a MaxwellWiechert model of conventional springs and dashpots. A purpose of the paper is to make available recently published results on fractional calculus modeling in the field of acoustics and elastography, with special focus on medical applications.

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Section: Experimental Evidencesmentioning

confidence: 99%

On a fractional Zener elastic wave equation

Näsholm

Holm

2013

fcaa

103

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“…after (25) where are the arithmetic and geometric means of the relaxation parameters, and j Cl + C2 ß2 represent the arithmetic and the ponderate means of the damping rations. From the previously presented reasoning, the roots of Eq.…”

Section: Casementioning

confidence: 99%

“…Lesieutre and Mingori [17] proposed the anelastic displacement field (the ADF method) also based on internal variables. The research of Pritz [23][24][25][26][27][28] on the four-parameter model and on the behavior of complex properties of rubbers and rubberlike materials has also had a great infiuence. Such models assume constitutive relationships based on differential equations upon fractional derivatives of stress and strain.…”

Section: Introductionmentioning

confidence: 99%

Characterization of Real Eigenvalues in Linear Viscoelastic Oscillators and the Nonviscous Set

Lázaro¹,

Pérez-Aparicio

2013

Journal of Applied Mechanics

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In structural dynamics, energy dissipative mechanisms with nonviscous damping are characterized by their dependence on the time-histoiy of the response velocity, which is mathematically represented by convolution integrals involving hereditaiy functions. The widespread Biot damping model assumes that such functions are exponential kernels, which modify the eigenvalues' set so that as many real eigenvalues (named nonviscous eigenvalues) as kernels are added to the system. This paper is focused on the study of a mathematical characterization of the nonviscous eigenvalues. The theoretical results allow the bounding of a set belonging to the real negative numbers, called the nonviscous set, constructed as the union of closed intervals. Exact analytical solutions of the nonviscous set for one and two exponential kernels and approximated solutions for the general case ofN kernels are developed. In addition, the nonviscous set is used to build closedform expressions to compute the nonviscous eigenvalues. The results are validated with numerical examples covering single and multiple degree-of-freedom systems where the proposed method is compared with other existing one-step approaches available in the literature.

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“…In order to numerically differentiate these levels, the loss factor peak will be used as damping index. As known [53,54,55,56,57], the loss factor of a frequency-domain constitutive equation is a real-valued and frequency-dependent function defined as the quotient between the imaginary and the real part of the complex stiffness. Here, the complex stiffness associated with the viscoelastic links of the structure shown in Fig.…”

Section: Example 2: Multiple Degree-of-freedom Systemsmentioning

confidence: 99%

Computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed-point iteration

Lázaro

Pérez-Aparicio

Epstein

2012

Applied Mathematics and Computation

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Linear viscoelastic structures are characterized by dissipative forces that depend on the history of the velocity response via hereditary damping functions. The free motion equation leads to a nonlinear eigenvalue problem characterized by a frequency-dependent damping matrix. In the present paper, a novel and efficient numerical method for the computation of the eigenvalues of linear and proportional or lightly nonproportional viscoelastic structures is developed. The central idea is the construction of two complex-valued functions of a complex variable, whose fixed points are precisely the eigenvalues. This important property allows the use of these functions in a fixed-point iterative scheme. With help of some results in Fixed Point Theory, necessary conditions for global and local convergence are provided. It is demonstrated that the speed of convergence is linear and directly depends on the level of induced damping. In addition, under certain conditions the recursive method can also be used for the calculation of non-viscous real eigenvalues. In order to validate the mathematical results, two numerical examples are analyzed, one for single degree-of-freedom systems and another for multiple ones.

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Loss Factor Peak of Viscoelastic Materials: Magnitude to Width Relations

Cited by 29 publications

References 14 publications

On a fractional Zener elastic wave equation

On a fractional Zener elastic wave equation

Characterization of Real Eigenvalues in Linear Viscoelastic Oscillators and the Nonviscous Set

Computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed-point iteration

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