Analysis of models for viscoelastic wave propagation

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_

	Zener	Maxwell	Voigt
Co	1	0	1
Cl	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
Co	1.5	0	1.5
Cl	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
Co	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
Co	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
Co	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
Co	1.5	0	1.5
Cl	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
sIH1→L2 Sf(s)	L2(Ω)	L2(Ω)	1/ψ*(x)	0	0	f	u̇
ε ∘ Sf(s)	L2(Ω)	𝕃2(Ω)	1/ψ*(x)	0	0	f	ε(u)
C(s)(ε ∘ Sf(s))	L2(Ω)	𝕃2(Ω)	ϕ(x)/ψ*(x)	r	0	f	σ
s IH1→L2 Sα(s)	H1/2(ΓD)	L2(Ω)	ϕ⋆(x)ψ⋆(x)min{1,x}$\begin{array}{} \displaystyle \frac{\phi_\star(x)}{\psi_\star(x)\min\{1,\sqrt x\}} \end{array}$	1 + r	12$\begin{array}{} \displaystyle \frac{1}{2} \end{array}$	α	u̇
ε ∘ Sα(s)	H1/2(ΓD)	𝕃2(Ω)	ϕ⋆(x)ψ⋆(x)min{1,x}$\begin{array}{} \displaystyle \frac{\phi_\star(x)}{\psi_\star(x)\min\{1,\sqrt x\}} \end{array}$	1 + r	12$\begin{array}{} \displaystyle \frac{1}{2} \end{array}$	α	ε(u)
C(s)(ε ∘ Sα(s))	H1/2(ΓD)	𝕃2(Ω)	ϕ(x)ϕ⋆(x)ψ⋆(x)min{1,x}$\begin{array}{} \displaystyle \frac{\phi(x)\phi_\star(x)}{\psi_\star(x)\min\{1,\sqrt x\}} \end{array}$	1 + 2r	12$\begin{array}{} \displaystyle \frac{1}{2} \end{array}$	α	σ
s IH1→L2 Sβ(s)	H−1/2(ΓN)	L2(Ω)	1min{1,x}ψ⋆(x)$\begin{array}{} \displaystyle \frac1{\min\{1,x\}\psi_\star(x)} \end{array}$	1	0	β	u̇
ε ∘ Sβ(s)	H−1/2(ΓN)	𝕃2(Ω)	1min{1,x}ψ⋆(x)$\begin{array}{} \displaystyle \frac1{\min\{1,x\}\psi_\star(x)} \end{array}$	1	0	β	ε(u)
C(s)(ε ∘ Sβ(s))	H−1/2(ΓN)	𝕃2(Ω)	ϕ(x)min{1,x}ψ⋆(x)$\begin{array}{} \displaystyle \frac{\phi(x)}{\min\{1,x\}\psi_\star(x)} \end{array}$	1 + r	0	β	σ