Analysis of models for viscoelastic wave propagation

Thomas S. Brown; Shukai Du; Hasan Eruslu; Francisco-Javier Sayas

Open Access

Analysis of models for viscoelastic wave propagation

Thomas S. Brown

Shukai Du

Hasan Eruslu

and

Francisco-Javier Sayas

| Oct 03, 2018

Applied Mathematics and Nonlinear Sciences

Volume 3 (2018): Issue 1 (June 2018)

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Page range: 55 - 96

Received: Dec 01, 2017

Accepted: Mar 09, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00006

Keywords
Hyperbolic PDE, viscoelasticity, Laplace transforms, semigroups of operators, stability analysis, fractional derivatives

© 2018 Thomas S. Brown et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

The window function g(t) is smoothly changing in (0:5;1:5)∪(2:5;3:5), and constant on the rest of the domain. The function h(t) has a similar shape, although the upper plateau (forced normal stress) is longer.

Effect of changing C1 using the parameters given in Table 1.

Effect of changing ν using the parameters given in Table 2.

Each plot shows the displacement of a 1D viscoelastic rod evaluated at the right endpoint, x = 1 for different values of ν. From left to right are the Zener, Maxwell, and Voigt models.

The Dirichlet boundary conditions used for the 2D spacetime simulations: single pulse and a periodic train of pulses, which will make the solution transition to time-harmonicity.

Space-time plots with parameters described in Table 4 where first and last four subplots correspond to the signals shown on the left and right of Figure 5.

Space-time plots with parameters described in Table 5 where first and last four subplots correspond to the signals shown on the left and right of Figure 5.

Snapshots for the 3D simulation showing the norm of the stress. From left to right, then from top to bottom, time-step = 9, 30, 55, 70, 100, 200, 350, 500.

The results of the same simulation as Figure 9, without the colormap. From left to right, then from top to bottom, time-step = 9, 30, 55, 70, 100, 200, 350, 500.

The parameters used to create the plots given in Figure 4.

	Zener	Maxwell	Voigt
C_o	1	0	1
C_l	1	1	1
a	0.5	0.5	0
ν	0.25,0.5,0.75,1	0.25,0.5,0.75,1	0.25,0.5,0.75,1
ρ	1	1	1

The parameters used to create the plots given in Figure 2. The first choice of C1 for the Zener and Voigt models reduce the model to linear elasticity

	Zener	Maxwell	Voigt
C_o	1.5	0	1.5
C_l	0.75,1,2.75	0.05,0.25,2	0,0.25,2
a	0.5	0.5	0
ν	1	1	1
ρ	1	1	1

The parameters used in in the fractional Zener model simulations to create the 2D spacetime plots shown in Figure 6.

	Elastic	Zener	Fractional Zener	Heterogeneous Domain
C_o	1.75	1.5	1.5	1.75 (x < 1/2), 1.5 (x ≥ 1/2)
C₁	1.75	1.75	1.75	1.75
a	1	0.5	0.5	1 (x < 1/2), 0.5 (x ≥ 1/2)
ν	1	1	0.3	1
ρ	10	10	10	10

The parameters used in the fractional Voigt model simulations to create the 2D spacetime plots shown in Figure 8.

	Elastic	Voigt	Fractional Voigt	Heterogeneous Domain
C_o	1.75	1.75	1.75	1.75
C₁	1.75	1.75	1.75	1.75
a	1	0	0	1 (x < 1/2), 0 (x ≥ 1/2)
ν	1	1	0.3	1
ρ	10	10	10	10

The parameters used in the Maxwell model simulations to create the 2D spacetime plots shown in Figure 7.

	Elastic	Maxwell	Fractional Maxwell	Heterogeneous Domain
C_o	1.75	0	0	1.75(x < 1/2), 0 (x ≥ 1/2)
C₁	1.75	1.75	1.75	1.75
a	1	1	1	1
ν	1	1	0.3	1
ρ	10	10	10	10

The parameters used to create the plots given in Figure 3.

	Zener	Maxwell	Voigt
C_o	1.5	0	1.5
C_l	1	1	1
a	0.5	0.5	0
ν	0.05,0.5,0.95	0.05,0.5,0.95	0.05,0.5,0.95
ρ	1	1	1

j.AMNS.2018.1.00006.tab.007.w2aab3b7b5b1b6b1ab1b6c15b4aAa

F(s)	X	Y	φ(x)	m	μ	λ	u
sI_H¹→L² S_f(s)	L²(Ω)	L²(Ω)	1/ψ_*(x)	0	0	f	u̇
ε ∘ S_f(s)	L²(Ω)	𝕃²(Ω)	1/ψ_*(x)	0	0	f	ε(u)
C(s)(ε ∘ S_f(s))	L²(Ω)	𝕃²(Ω)	ϕ(x)/ψ_*(x)	r	0	f	σ
s I_H¹→L² S_α(s)	H^1/2(Γ_D)	L²(Ω)	$\begin{matrix} \frac{ϕ_{⋆} (x)}{ψ_{⋆} (x) min {1, \sqrt{x}}} \end{matrix}$ $\begin{array}{} \displaystyle \frac{\phi_\star(x)}{\psi_\star(x)\min\{1,\sqrt x\}} \end{array}$	1 + r	$\begin{matrix} \frac{1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{1}{2} \end{array}$	α	u̇
ε ∘ S_α(s)	H^1/2(Γ_D)	𝕃²(Ω)	$\begin{matrix} \frac{ϕ_{⋆} (x)}{ψ_{⋆} (x) min {1, \sqrt{x}}} \end{matrix}$ $\begin{array}{} \displaystyle \frac{\phi_\star(x)}{\psi_\star(x)\min\{1,\sqrt x\}} \end{array}$	1 + r	$\begin{matrix} \frac{1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{1}{2} \end{array}$	α	ε(u)
C(s)(ε ∘ S_α(s))	H^1/2(Γ_D)	𝕃²(Ω)	$\begin{matrix} \frac{ϕ (x) ϕ_{⋆} (x)}{ψ_{⋆} (x) min {1, \sqrt{x}}} \end{matrix}$ $\begin{array}{} \displaystyle \frac{\phi(x)\phi_\star(x)}{\psi_\star(x)\min\{1,\sqrt x\}} \end{array}$	1 + 2r	$\begin{matrix} \frac{1}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{1}{2} \end{array}$	α	σ
s I_H¹→L² S_β(s)	H^−1/2(Γ_N)	L²(Ω)	$\begin{matrix} \frac{1}{min {1, x} ψ_{⋆} (x)} \end{matrix}$ $\begin{array}{} \displaystyle \frac1{\min\{1,x\}\psi_\star(x)} \end{array}$	1	0	β	u̇
ε ∘ S_β(s)	H^−1/2(Γ_N)	𝕃²(Ω)	$\begin{matrix} \frac{1}{min {1, x} ψ_{⋆} (x)} \end{matrix}$ $\begin{array}{} \displaystyle \frac1{\min\{1,x\}\psi_\star(x)} \end{array}$	1	0	β	ε(u)
C(s)(ε ∘ S_β(s))	H^−1/2(Γ_N)	𝕃²(Ω)	$\begin{matrix} \frac{ϕ (x)}{min {1, x} ψ_{⋆} (x)} \end{matrix}$ $\begin{array}{} \displaystyle \frac{\phi(x)}{\min\{1,x\}\psi_\star(x)} \end{array}$	1 + r	0	β	σ

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Analysis of models for viscoelastic wave propagation

Published Online: Oct 03, 2018

Page range: 55 - 96

Received: Dec 01, 2017

Accepted: Mar 09, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00006

Keywords
Hyperbolic PDE, viscoelasticity, Laplace transforms, semigroups of operators, stability analysis, fractional derivatives

© 2018 Thomas S. Brown et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

The parameters used to create the plots given in Figure 4.

The parameters used to create the plots given in Figure 2. The first choice of C1 for the Zener and Voigt models reduce the model to linear elasticity

The parameters used in in the fractional Zener model simulations to create the 2D spacetime plots shown in Figure 6.

The parameters used in the fractional Voigt model simulations to create the 2D spacetime plots shown in Figure 8.

The parameters used in the Maxwell model simulations to create the 2D spacetime plots shown in Figure 7.

The parameters used to create the plots given in Figure 3.

j.AMNS.2018.1.00006.tab.007.w2aab3b7b5b1b6b1ab1b6c15b4aAa

Analysis of models for viscoelastic wave propagation

Published Online: Oct 03, 2018

Page range: 55 - 96

Received: Dec 01, 2017

Accepted: Mar 09, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00006

KeywordsHyperbolic PDE, viscoelasticity, Laplace transforms, semigroups of operators, stability analysis, fractional derivatives

© 2018 Thomas S. Brown et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

The parameters used to create the plots given in Figure 4.

The parameters used to create the plots given in Figure 2. The first choice of C1 for the Zener and Voigt models reduce the model to linear elasticity

The parameters used in in the fractional Zener model simulations to create the 2D spacetime plots shown in Figure 6.

The parameters used in the fractional Voigt model simulations to create the 2D spacetime plots shown in Figure 8.

The parameters used in the Maxwell model simulations to create the 2D spacetime plots shown in Figure 7.

The parameters used to create the plots given in Figure 3.

j.AMNS.2018.1.00006.tab.007.w2aab3b7b5b1b6b1ab1b6c15b4aAa

Keywords
Hyperbolic PDE, viscoelasticity, Laplace transforms, semigroups of operators, stability analysis, fractional derivatives