should be zero, for all *i.* Here is the strain tensor.

Linearization of **(x)** defines affine operators

**[A2]**

For **x**=**x**_{v}, let be the operators corresponding to . The operator , linearization of the relative rate of movement of two blocks at **x**_{v}, does not depend on the rate of movement of the coordinate system. Expression in brackets in **A1** can be rewritten as the *i*th component of the vector Σ_{v}*V*_{v}(**x**_{0}).

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

**[A3]**

is zero. This is an element of the Lie algebra

of the orthogonal affine group. This operator is a natural generalization of the Saint Venant incompatibility . Note that

Geometric incompatibility for a system of deformable blocks can be defined as

**[A4]**

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

**[A5]**

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

**G**=*V*_{1}∧*V*_{2}+*(V*_{1}+*V*_{2}∧*V*_{3}*)*

+…+(*V*_{1}+…+*V*_{n−2}∧*V*_{n−1}), **[A6]**

similar to (Eq. **4**). Otherwise, *V*_{v} 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 *V*_{v}*(x, y)*=(*X*_{v}−*yω*_{v}*,* **Y**_{v}+*xω*_{v}). Here (**X**_{v}, **Y**_{v}) is the relative displacement rate at the origin, and *ω*_{v} is the relative rotation rate of the two blocks adjacent at **x**_{v}. In this case, is three-dimensional, and the Saint Venant incompatibility has three components:

**[A7]**

The space is also three-dimensional. Accordingly, the geometric incompatibility **G** has three components:

**[A8]**

Here , define the operators in Eq. **A5**. Expression **A6** can be rewritten similarly in terms of *X*_{v}*, Y*_{v}*,* and *ω*_{v}*.*

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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