The title problem is solved, for a crack propagating in a slab which is composed of material which can be viscoelastic, “vertically stratified” and transversely isotropic with the axis in the “vertical” direction. The result is illustrated by specialising to the case of a stationary crack in the plane x3 = 0 of an infinite isotropic elastic medium, whose leading edge is at x1 = a cos(kx2), with |k|a ≪ 1.
WillisJRMovchanAB. Dynamic weight functions for a moving crack. I: Mode I loading. J Mech Phys Solids1995; 43: 319–341.
2.
MovchanABWillisJR. Dynamic weight functions for a moving crack. II: Shear loading. J Mech Phys Solids1995; 43: 1369–1383.
3.
WillisJRMovchanAB. Three-dimensional dynamic perturbation of a propagating crack. J Mech Phys Solids1997; 45: 591–610.
4.
WillisJR. Asymptotic analysis in fracture: An update. Int J Fract1999; 100: 85–103.
5.
RamanathanSFisherDS. Dynamics and instabilities of planar tensile cracks in heterogeneous media. Phys Rev Lett1997; 79: 877–880.
6.
MorrisseyJWRiceJR. Crack front waves. J Mech Phys Solids1998; 46: 467–487.
7.
FinebergJSharonECohenG. Crack front waves in dynamic fracture. Int J Fract2003; 119: 247–261.
8.
WillisJR. Crack front perturbations revisited. Int J Fract forthcoming. DOI: 10.1007/s10704-012-9795-y.
9.
FreundLB. Dynamic fracture mechanics. Cambridge, New York: Cambridge University Press (1990).
10.
SlepyanLI. Models and phenomena in fracture mechanics. Berlin: Springer-Verlag, 2002.
11.
MovchanABWillisJR. The influence of viscoelasticity on crack front waves. J Mech Phys Solids2001; 49: 2177–2189.
12.
MovchanNVMovchanABWillisJR. Perturbation of a dynamic crack in an infinite strip. Quart J Mech Appl Math2005; 58: 333–347.
13.
WillisJRMovchanNV. Crack front waves in an anisotropic medium. Wave Motion2007; 44: 458–471.
14.
LeblondJ-BPatinetSFrelatJLazarusV. Second-order coplanar perturbation of a semi-infinite crack in an infinite body. Eng Fract Mech2012; 90: 129–142.
15.
VasoyaMLeblondJ-BPonsonL. A geometrically-nonlinear analysis of coplanar crack propagation in some heterogeneous medium. Int J Solids Struct2013; 50: 371–378.
16.
RiceJR. Weight function theory for three-dimensional elastic crack analysis. In WeiRPGangloffRP (eds) Fracture Mechanics: Perspectives and Directions (Twentieth Symposium) (ASTM STP 1020). Philadelphia, PA: American Society for Testing and Materials, 1989, pp.29–57.
17.
WillisJR. Self-similar problems in elastodynamics. Phil Trans R Soc1973; A274: 435–491.
