Even without any knowledge of the precise elastic properties of materials, we are able to specify the conditions for elastodynamic fields at jump disconti-nuities of materials: These conditions immediately follow from the governing Equations 3.1 and 3.2.
We refer to the sketch in Figure 3.3: The homogeneous or inhomogeneous, isotropic or anisotropic, dissipative or nondissipative material (1) contains an
“inclusion” V with material properties (2) that may equally be arbitrary than those of material (1), they should just vary discontinuously on the surface S of the inclusion; n denotes the outer normal59on S. Now we select a “very small”
piece∆S on S—it should be considered as planar—and coat it with a volume Vi with surface Si and outer normal ni; Vi simultaneously contains material (1) as well as material (2) (Figure 3.3a). In the following, we investigate the volume integrals
of the governing equations of elastodynamics (3.1) and (3.2) having in mind the limit i−→ ∞ of a series of volumes Vi similar to the transition from
ni
59We postulate that S exhibits only “rounded” edges and corners with an existing normal.
Furthermore, we assume particle motions on S so small that S can be considered as time invariant.
Figure 3.3a to b; in this limit, the volumes should approach the geometry of a flat box adapting more and more to ∆S from both sides, finally resulting in an outer surface∆S(1) and an inner surface∆S(2) whose pertinent (outer) normals n and−n originate from ni for i−→ ∞. Applying Gauss’ theorems to the first integrals on the right-hand sides of (3.80) and (3.81), we have to evaluate the limit i−→ ∞ in the equations
Let us first consider the volume integrals of elastodynamic fields on the left-hand side: If the fields are “physically reasonable,” i.e., without mathe-matical singularities, the integrals tend to be zero with vanishing integration volume.60
The surface integrals in (3.82) and (3.83) tend to integrals over∆S(1) and
∆S(2) for i−→ ∞, where the normal −n on ∆S(2) accounts for the negative
According to the mean value theorem of integral calculus (Burg et al. 1990) we always find a vector of position RS on ∆S—it equally resides on ∆S(1) and∆S(2) due to the adaptation of∆S(1) and∆S(2) to∆S—which satisfies
What remains is the investigation of the Vi-integrals over the prescribed sources f (R, t), h(R, t): In the following, we distinguish two cases.
60There is nothing to accumulate (integrate) in a zero volume.
Homogeneous transition conditions: Continuity of the traction vec-tor, the surface deformation rate tensor, and the particle displace-ment vector: The given source functions f (R, t), h(R, t) should represent volume sources without singularities; then they do not contribute to the limit of Equations (3.80) and (3.81) for i−→ ∞, and hence the governing equations of elastodynamics (3.82) and (3.83) are reduced to the homogeneous transition conditions due to (3.84), (3.85), (3.86), and (3.87):
n· T(1)(RS, t)− n · T(2)(RS, t) = 0, RS ∈ S, (3.88) n v(1)(RS, t) + v(1)(RS, t)n− n v(2)(RS, t)− v(2)(RS, t)n = 0, RS ∈ S;
(3.89) we could divide by the small but finite surface element ∆S unfolding the independence of the resulting equations from the arbitrary partial surface∆S of S ensuring that RS in (3.88) and (3.89) may finally be a vector of position of any point on S. The homogeneous transition conditions (3.88) and (3.89) therefore require the continuity of the traction vector n· T(R, t) as surface traction density and the tensor n v(R, t) + v(R, t)n as surface deformation rate if R moves from one side of S in material (1) to the other side of S in material (2), even if the material properties exhibit a jump discontinuity on S.
The governing elastodynamic equations do not tell anything regarding other field vector and tensor components.
The homogeneous transition condition (3.89) can even be simplified. We write (3.89) short-hand
n v + v n = continuous (3.90) and take subsequent projections of this tensor equation into the direction of the normal on S and tangential to S. Hence:
n· (n v + v n) = v + v · n n
= vt+ 2v· n n = continuous; (3.91) we have replaced v by the sum v = vt+ vn of the tangential vector
vt= (I− n n) · v and the normal vector
vn= v· n n.
Then, we calculate the projection
(I− n n) · (n v + v n) = (v − v · n n)
= vt
n = continuous; (3.92)
requiring the continuity of vt; therefore, the continuity of vn is required in combination with (3.91). Both facts result in the homogeneous transition condition
v(1)(RS, t)− v(2)(RS, t) = 0, RS ∈ S, (3.93) namely, the continuity of the particle velocity vector. To deduce the continuity of the particle displacement vector, we need an additional argument (de Hoop 1995):
Due to the relation
v(R, t) = ∂u(R, t)
∂t (3.94)
between particle velocity and particle displacement, the transition condition (3.93) is equivalent to
∂u(1)(RS,τ)
∂τ = ∂u(2)(RS,τ)
∂τ , (3.95)
therefore, time integration yields
t 0
∂u(1)(RS,τ)
∂τ dτ = u(1)(RS, t) + u(1)(RS, 0)
= u(2)(RS, t) + u(2)(RS, 0). (3.96) It makes sense to postulate that elastodynamic fields are “switched on” at a certain time instant being identically zero for smaller times; consequently, we choose the time origin as far in the past that u(1)(RS, 0) = u(2)(RS, 0)≡ 0 holds, i.e., we deal with causal fields. According to (3.96), we conclude the continuity of the particle displacement vector for those fields:
u(1)(RS, t)− u(2)(RS, t) = 0, RS∈ S. (3.97) Of course, the homogeneous transition conditions (3.88) and (3.97) also hold for the Fourier spectra:61
n· T(1)(RS,ω) − n · T(2)(RS,ω) = 0, RS ∈ S, (3.98) u(1)(RS,ω) − u(2)(RS,ω) = 0, RS ∈ S. (3.99) Inhomogeneous transition conditions: Definition of surface source densities: As announced, for the second case, we allow for the existence of
61At first sight, it looks as if (3.99) follows from the Fourier transformed equation (3.93) without any further assumptions; yet (3.93) leads to the Fourier transformed equation
ω
u(1)(RS, ω) − u(2)(RS, ω)
= 0, and the conclusion can only read
u(1)(RS, ω) − u(2)(RS, ω) = u0(RS)δ(ω)
with an arbitrary vector u0(RS) because ωδ(ω) = 0. An inverse Fourier transform and the comparison with (3.96) reveals that u0(RS)/2π = u(1)(RS, t = 0) − u(2)(RS, t = 0) so that only causal fields in the time domain yield u0(RS)≡ 0.
prescribed surface source densities on S besides singularity-free volume source densities. In terms of mathematics surface source densities can be considered as “amplitudes” ofδ-singular volume source densities on S62according to—we use the singular functionγS(R) of the surface S:
fS(R, t) = t(R, t)γS(R), (3.100) hS(R, t) = g(R, t)γS(R), (3.101) because, only in that case, the Vi-volume integration of fS and hS yields a finite value:
Vi
fS(R, t) dV =
Vi
t(R, t)γS(R) dV
=
∆S
t(R, t) dS
= t(RS, t)∆S; (3.102)
the last sign of equality implies the application of the mean value theorem of integral calculus. Similarly, we obtain
Vi
hS(R, t) dV = g(RS, t)∆S. (3.103)
With (3.102), (3.103), and (3.104) through (3.107), the inhomogeneous tran-sition conditions
n· T(1)(RS, t)− n · T(2)(RS, t) =−t(RS, t), RS ∈ S, (3.104) 1
2
n v(1)(RS, t) + v(1)(RS, t)n− n v(2)(RS, t)− v(2)(RS, t)n
=−g(RS, t), RS ∈ S, (3.105)
for the traction vector and the tensor of the surface deformation rate are obtained provided surface sources on S are—no matter how—prescribed. Such prescribed sources yield a discontinuity of the field quantities involved.
We might read the inhomogeneous transition conditions (3.104) and (3.105) from left to right: If the traction vector n· T(R, t) and the tensor n v(R, t) + v(R, t)n are—for any reasons—discontinuous on a surface S, such a discontinuity defines surface source densities. This interpretation will be ex-tremely helpful to understand Huygens’ principle in elastodynamics (Section 15.1.3).
The spectral versions of (3.104) and (3.105) apparently read as
n· T(1)(RS,ω) − n · T(2)(RS,ω) = −t(RS,ω), RS∈ S, (3.106)
62The dimension (of the components) of t is force/area and the dimension (of the com-ponents) of g is length/second because the dimension ofγSis length−1.
1 2
n v(1)(RS,ω) + v(1)(RS,ω)n − n v(2)(RS,ω) − v(2)(RS,ω)n
=−g(RS,ω), RS ∈ S; (3.107)
the Fourier spectrum of the surface deformation tensor (n u + u n)/2 is ob-tained from (3.106):
1 2
n u(1)(RS,ω) + u(1)(RS,ω)n − n u(2)(RS,ω) − u(2)(RS,ω)n
=−j
ωg(RS,ω), RS ∈ S. (3.108)
The “simple version” (3.99) does no longer exist in the case of inhomogeneous transition conditions.