Transient propagation and wave splitting

542 M. Romeo

3. Transient propagation and wave splitting

We assume here that the subset V consists of the half-space {x, y, z ∈ R 3 | x 0} and denote by e 1 the unit normal to the plane boundary of V . Let us suppose that a uniform displacement u t and a uniform traction w t = T0, te 1 are given at the surface x = 0 for t 0, while no deformation occurs for t 6 0 throughout V . We look for solutions to the one-dimensional problem ρ∂ 2 t u x, t = ∂ x w x, t, x ∈ R + , t ∈ R + , 10 where ρ is the mass density of the medium and w = Te 1 . Equation 10 splits into a couple of scalar equations which, accounting for 2 and 7, read ρ∂ 2 t u k = ∂ x w k , 11-a ρ∂ 2 t u ⊥ = ∂ x w ⊥ , 11-b where u k = u 1 and u ⊥ represents one of the two components u 2 and u 3 , and where w k x, t = 4 3 µ + κ ∂ x u k x, t + Z t 4 3 µ + κ G ′ τ ∂ x u k x, t − τ dτ, 12-a w ⊥ x, t = µ∂ x u ⊥ x, t + Z t µG ′ τ ∂ x u ⊥ x, t − τ dτ. 12-b In the following we shall consider one of the two scalar problems given by 11-a, 12-a and by 11-b and 12-b for the quantities u and w which will be taken as the displacement and the traction of pure compressive or pure shear motions. Posing v = ∂ t u and taking into account that ∂ x ux, 0 = 0 for x ∈ R + , we can rewrite our problem in the following form ρ∂ t vx, t = ∂ x wx, t, 13-a ∂ t wx, t = σ ∂ x vx, t + Z t G ′ τ vx, t − τ dτ 13-b for x ∈ R + and t ∈ R + , together with the boundary conditions v 0, t = v t, 14-a w 0, t = w t 14-b and the initial conditions vx, 0 = 0, 15-a wx, 0 = 0. 15-b In Eq. 13 σ = 4 3 µ + κ for compressive motions and σ = µ for shear motions. According to the viscoelastic model outlined in the previous section, we assume that Gτ has a bounded first derivative in R + . Hence the integral operator K defined as Kf t = f t + Z t G ′ τ f t − τ dτ 16 Wave splitting in linear viscoelasticity 543 for any f ∈ L 1 R + , is invertible and the system 13 can be written as ∂ x v w = 0 σ −1 K −1 ρ ∂ t v w . 17 The operator K −1 can be easily derived from 16 on applying the Laplace transform. We obtain K −1 f t = L −1 1 sLGs · f t 18-a for any f ∈ L −1 R + , where the asterisk denotes time convolution, i.e. a · bt = Z t at − τ bτ dτ. 18-b In order to obtain a diagonalized form of the matrix at the right-hand side of Eq. 17, we introduce the quantities v + , v − T as follows v w = 1 1 −σ P ∗ σ P ∗ v + v − =: D v + v − 19 and equivalently v + v − = D −1 v w = 1 2 1 −σ −1 S ∗ 1 σ −1 S ∗ v w , 20 where P t and St are suitable functions in L 1 R + , to be determined according to the condition S · P · f t = f t 21 for any f ∈ L 1 R + . We observe that, in view of Eq. 15 we have v + x, 0 = v − x, 0 = 0, ∀x ∈ R + . 22 Substitution of 19 into 17 and the use of 20 allows us to obtain ∂ x v + v − = D −1 0 −σ K −1 ρ D∂ t v + v − 23-a = 1 2 − 1 c 2 S · −K −1 · P · 1 c 2 S · +K −1 · P · − 1 c 2 S · −K −1 · P ∗ 1 c 2 S · +K −1 · P · ∂ t v + v − 23-b where c 2 = σρ. In deriving Eq. 23 we have exploited the result ∂ t Dv + , v − T = D∂ t v + , v − T , which is a consequence of the convolution’s properties and of Eq. 22. We finally impose the diagonalization condition 1 c 2 S − K −1 · P · f t = 0 24 for any f ∈ L 1 R + . Equations 24 and 21, together with Eq. 18 yield 544 M. Romeo P t = ± 1 c L −1 p sLGs t, 25-a St = ±c L −1 1 √ sLGs t 25-b and Eq. 23 becomes ∂ x v + v − = − 1 c Q· 1 c Q· ∂ t v + v − 26 with Qt = L −1 1 √ sLGs t 27 and where the choice of the sign in 25 has been performed according to the meaning of v + and v − as forward and backward propagating disturbances. Explicit expressions for P t and Qt = Stc can be derived for the Maxwell model and the Maxwell–Kelvin model described in Section 2. With the help of a table of Laplace transforms Roberts and Kaufman, 1966, we have P t = δt exp−t2τ M + E t, 1 2τ M , Qt = δt exp−t2τ M + F t, 1 2τ M and P t = δt exp−t2τ 2 + E t, 1 2τ 2 + E t, 1 2 1 τ 1 + 1 τ K + E ·, 1 τ 2 · E ·, 1 2 1 τ 1 + 1 τ K t, Qt = δt exp−t2τ 2 + F t, 1 2τ 2 + F t, 1 2 1 τ 1 + 1 τ K + F ·, 1 τ 2 · F ·, 1 2 1 τ 1 + 1 τ K t respectively for the Maxwell model and the Maxwell–Kelvin model with t 0, where δt is the usual Dirac delta distribution and the functions E and F are given by Et, β = β I βt + I 1 βt exp−βt, F t, β = β I βt + I 1 βt exp−βt, where I and I 1 are modified Bessel functions.

4. Evaluation of the wave propagator