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# Fracture Element in Rheological Models

Force-displacement curves of viscoelastic materials that exhibit failure can be described by mechanical analogs composed of springs, dashpots, and fracture elements. This Demonstration presents simple combinations of these elements and shows how their constants and the displacement rate can affect their array's deformation and rupture pattern.

### DETAILS

Snapshot 1: model A with high
Snapshot 2: model B at low displacement rate ()
Snapshot 3: model B at high displacement rate ()
Snapshot 4: model C with
Snapshot 5: model C with
The definition of a fracture (shear pin) element is: if , then , else , where is the fracture force. In an in-series array with a spring having a constant (model A), the array's constitutive definition is: if , then , else .
If the fracture element is in-series with a three-element viscoelastic solid model (model B), the array's definition is: if , then , else , where and are the springs' constants, is the dashpot's viscosity, and is the displacement rate. According to this model, therefore, the fracture's occurrence is determined by both the elements' constants and the displacement rate.
If two springs and ) with two fracture elements ( and ) in series are combined in a parallel array (model C), this array's definition is (if , then , else 0) + (if , then , else 0). Thus, according to this model, the details of the serrated force-displacement depend on the absolute and relative magnitudes of the four elements' constants.
References
[1] B. Drake, "A Quasi-Rheological Model Element for Fracture," Journal of Texture Studies, 2, 1971 pp. 265–372.
[2] R. I. Tanner, F. Qi, and S-C. Dai, "Bread Dough Rheology and Recoil," Journal of Non-Newtonian Fluid Mechanics, 148, 2008 pp. 33–40.

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