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 Incoming links: Tensile-crack radiation pattern

Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 78 / 4 / iv / Solution Created 2026-10-03 Updated 2026-10-14

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The first two terms are the radiative P wave and S wave displacements:

u_{P}^{rad} = \frac{λ + 2 μ γ ^{2}}{4 π ρ α ^{3} R} e \dot{V} (t - R / α), u_{S}^{rad} = \frac{2 μ γ}{4 π ρ β ^{3} R} (n - γ e) \dot{V} (t - R / β) .

(1)

They decay as

R^{- 1}

and are purely transient: once the opening rate vanishes, each disappears after its respective arrival time. The P displacement is longitudinal and the S displacement transverse. An abrupt opening history makes these formulas distributions; a smooth finite-duration intrusion gives ordinary pulses.

Suppose the source starts from zero opening and ends with volume change

V_{0}

. At times later than the end of intrusion plus

R / β

, both retarded openings equal

V_{0}

and

I = \frac{V _{0} R ^{2}}{2} (\frac{1}{β ^{2}} - \frac{1}{α ^{2}}) .

(2)

The remaining terms combine to the static displacement field

u_{stat} = \frac{V _{0}}{4 π ρ R ^{2}} {[\frac{λ + μ - 3 μ γ ^{2}}{α ^{2}} + \frac{μ ( 3 γ ^{2} - 1 )}{β ^{2}}] e + \frac{2 μ γ}{α ^{2}} n} .

(3)

It decays as

R^{- 2}

, so it must not be identified with the transient far-field pulses.

For a Poisson solid,

λ = μ

α^{2} = 3 β^{2}

and

μ = ρ β^{2}

. Let

θ

be polar angle from the crack normal, with unit vector

e_{θ}

in the increasing-

θ

direction. Then

γ = cos θ

and

n - γ e = - sin θ e_{θ}

. The tensile-crack radiation pattern becomes

u_{P}^{rad} = \frac{V ˙ _{P}}{12 π 3 βR} (1 + 2 cos^{2} θ) e, u_{S}^{rad} = - \frac{V ˙ _{S}}{4 π βR} sin 2 θ e_{θ} .

(4)

At fixed

R

and positive opening rate, the P pattern is outward, has no angular node, and is three times larger along the normal than in the crack plane. The signed S amplitude has four alternating lobes: it vanishes along the normal and in the crack plane, and its magnitude is maximal at

4 5^{\circ}

to them. Neither field has an azimuthal SH component about the crack normal.

The final static field simplifies to

u_{stat} = \frac{V _{0}}{12 π R ^{2}} [(6 cos^{2} θ - 1) e + 2 cos θ n] = \frac{V _{0}}{12 π R ^{2}} [(8 cos^{2} θ - 1) e - sin 2 θ e_{θ}] .

(5)

Its radial part is outward near the poles, changes sign at

cos^{2} θ = 1/8

, and is inward in the crack plane. The tangential part has the same signed angular factor as the S radiation, but is permanent and has a different distance dependence. Its radial sign changes are not nodes of the whole vector because the tangential component generally remains nonzero there. Rotating a meridional sketch around the crack normal gives the complete three-dimensional axisymmetric pattern.

Figure 1.
Transient P and S displacement patterns and final static displacement for a tensile crack in a Poisson solid
. **Details**.

The arrows show displacement at points on a circle of constant observation radius. Each panel uses its own scale; the separate formulas retain the physical P/S prefactors and arrival times. The normal is vertical and the crack lies horizontally.

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Poisson solid 2026-10-14

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An isotropic linear elasticity material with equal positive Lamé parameters has Poisson's ratio

1/4

and the displayed compressional/shear speed ratio. This simplifying model fixes the relative amplitudes in the tensile-crack radiation pattern; it does not describe every stable isotropic solid.

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