A. Cochard et Jr. Rice, A SPECTRAL METHOD FOR NUMERICAL ELASTODYNAMIC FRACTURE-ANALYSIS WITHOUT SPATIAL REPLICATION OF THE RUPTURE EVENT, Journal of the mechanics and physics of solids, 45(8), 1997, pp. 1393-1418
Perrin et al. (1995) and Geubelle and Rice (1995) have introduced a sp
ectral method for numerical solution of two- and three-dimensional ela
stodynamic fracture problems. The method applies for ruptures confined
to a plane separating homogeneous elastic half spaces. In this method
, the physical variables, such as the traction components of stress an
d displacement discontinuity on the rupture plane, are represented as
Fourier series in space with time-dependent coefficients. An analytica
l solution is found for each Fourier mode, in that each Fourier coeffi
cient for stress is expressed by the time convolution of the correspon
ding coefficient for displacement with a convolution kernel specific t
o the rupture mode. Once the 2D formulation of the method is known, th
e method is readily generalizable to 3D problems in that it involves o
nly linear combinations of the convolution kernels found for each rupt
ure mode in 2D. This conceptual simplicity has, however, a major drawb
ack: due to the Fourier series representations of the physical variabl
es, the problem solved is in fact an infinite and periodic replication
of rupture events on the fracture plane. So, in order to study the ev
olution of a single rupture, one has to use a spatial period large eno
ugh in order that the waves coming from the replication cracks do not
enter the zone of interest during the time duration studied, or provid
e negligible stress alteration when they do arrive. We show here how t
o rigorously offset this defect while retaining the modal independence
. Once expressed in the spatial domain, the method amounts to truncati
ng in space the space-time convolution kernels, in a manner that provi
des an exact evaluation for all positions within the rupture domain (w
here the constitutive law between stress and displacement discontinuit
y is to be imposed), but not outside. In order for the method to be id
entical in structure to the method of Perrin er al. (1995) and Geubell
e and Rice (1995), the period of the Fourier series is requested to be
only twice as large as the rupture domain of interest. The only diffe
rence, then, to the original spectral method is that the convolution k
ernels in the Fourier domain require more elaborate calculations to be
established, but this has to be done only once to allow simulations o
n a given domain. (C) 1997 Elsevier Science Ltd.