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The use of the crack surface function as the state variable can provide a simple and efficient method to model crack growth. One can start with the crack surface in an initial configuration and use a single signed distance function to advance the crack front. The method is effective for small cracks, and the crack surface is related to the stress field in the material. The distance function can be defined as a function of the displacement field,, and the geometry of the crack surface in the present configuration. The crack is specified by a single signed distance function,, in general. It takes the form of a piecewise linear function, i.e.,. represents the crack surface and is the signed distance from the crack surface to the node, and is the direction of the crack propagation. The crack is simply a curve that is traced by the incremental crack front as the crack propagates. While the distance function is a scalar, the space that it represents is a vector surface in three dimensions and a surface in two dimensions. The crack propagation can be defined by a single degree of freedom that describes the crack front.

The crack surface function was first introduced by Kachan and Hutter (1997) to model the propagation of a crack in a semi-infinite material. In their method, the crack front was specified by a single piecewise linear distance function that defines a plane and a normal vector. The crack is propagated by moving the plane orthogonally to the normal vector. The crack area was prescribed using the two planes that were maintained parallel to each other. Crack propagation was continuous in the sense that the planes were not allowed to move. The crack area was defined by three nodes that were connected by a line segment.

To extend the critical field method to model the growth of cracks, Wei (2009) extended the interface element method from two dimensions to three dimensions. They used an interpolation formula of tetrahedral elements to calculate the crack front and its normal direction. Then a constraint algorithm was used to drive the movement of the crack front. He also noted that this method is not stable if the size of the elements is too large, as the crack tends to split into two, or even more, cracks. The current implementation of the method exhibits a stable crack growth even when the elements are relatively coarse, but the convergence of the displacement solution is much slower than in the conventional finite element method. We intend to address this issue in the near future.

As concrete is typically more ductile than the reinforcing materials (known as the reinforcement ratio), the reinforcing material is typically more critical to the resisting of crack propagation than the concrete.

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