should be zero, for all i. Here is the strain tensor.
Linearization of (x) defines affine operators
For x=xv, let be the operators corresponding to . The operator , linearization of the relative rate of movement of two blocks at xv, does not depend on the rate of movement of the coordinate system. Expression in brackets in A1 can be rewritten as the ith component of the vector ΣvVv(x0).
When the blocks are rigid, the integral in A1 vanishes, and the Saint Venant compatibility condition is satisfied if and only if the affine operator
is zero. This is an element of the Lie algebra
Geometric incompatibility for a system of deformable blocks can be defined as
It is easy to check that the value of G does not depend on the rate of movement of the coordinate system. Thus G is an invariant of the rate of deformation.
For rigid blocks, the integral in A4 vanishes, and the geometric incompatibility becomes
similar to Eq. 3. This is an element of , the external square of the Lie algebra of the orthogonal affine group. If K=0, this can be rewritten as
similar to (Eq. 4). Otherwise, Vv should be modified as in (Eq. 5) to make K=0.
For a system of rigid blocks, an analog of the Stokes formula is valid for G: its value for a contour is equal to the sum of its values over the junctions within the contour, when the Saint Venant incompatibility is zero for all these junctions.
For a system of two-dimensional rigid blocks, let x=(x, y), and Vv(x, y)=(Xv−yωv, Yv+xωv). Here (Xv, Yv) is the relative displacement rate at the origin, and ωv is the relative rotation rate of the two blocks adjacent at xv. In this case, is three-dimensional, and the Saint Venant incompatibility has three components:
The space is also three-dimensional. Accordingly, the geometric incompatibility G has three components:
Here , define the operators in Eq. A5. Expression A6 can be rewritten similarly in terms of Xv, Yv, and ωv.
This work was supported by National Science Foundation Grant no. 931650, and P.O. no. 56659 from University of Southern California, on behalf of the Southern California Earthquake Center. The first author (A.G.) was partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University, Contract DAAL03–91-C-0027, and Department of Geological Sciences at Cornell University, under National Science Foundation Grant no. EAR-94–23818.
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