Bloch waves
There are two types of scalar waves that travel through the elastodynamic problem, the impulsive wave and the continuous wave (also known as the far-field wave).
The impulsive wave is characterized by a wavepacket that travels at a constant speed $v$ to the tip of the crack. At the tip, the local stress fields are zero. In this case, the vertical displacement is the only component of the displacement field. In the limit of a fast tip (i.e., very small amplitude), the field tends to a single point. If the tip amplitude is comparable to the width of the wavepacket, then the field is more complicated, but the displacement tends to a linear function of the x-coordinate (as seen in the normal mode solution). Since the full 3D displacement can be expressed in terms of a series of plane waves, the displacement field is simply a superposition of Bloch waves
where $u_{ij}$ is the velocity of the displacement wave. The displacement field has a plane wave representation
where are plane waves and are the transverse wave numbers of plane waves in the direction normal to the plane of motion of the wave. The $k_j$ are given by
From the above expression, it can be seen that the maximum displacement in the plane is
where $u_0$ is the maximum displacement in the plane.
The scalar displacement and strain can be written as
The in the above equation are given by
The total energy per unit volume of the crack is given by
where
and
The above expressions for the energy can be related to the velocity fields by using the following expression
Here, is the wave speed and is the shear wave speed.
The dynamic fracture criterion
The fracture of an intact solid in the presence of a continuously growing crack will occur when the energy is positive and the shear wave speed is smaller than the wave speed. To find the location of the fracture surface, we consider the energy density,
where the and are the and components of the displacement and strain respectively, is the area of the crack surface, and is the unit normal vector along the surface. In the near-tip region, there is no displacement, so we can write the displacement as a linear function of the distance 01e38acffe
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