With the use of eq. (5), eq. (2) can be written as follows. Application of divergence to eq. (2) with the use of eq. (5) and the mathematical identity leads to the socalled equation of continuity. Eq. (9) indicates that is the flow of , allowing us to put it in the following form. With eq. (7), eq. (10) can further be rewritten as Here the temporal change in density can be interpreted as the generation of dislocation [10]. Being associated with the net flow into the unit volume , this change in density is in the same direction as the local velocity, or . Thus, the corresponding flow can be put in the following form. Being proportional to velocity, the longitudinal force is by nature energy dissipating. 2.3 Transition to fracture The above argument indicates the following scenario of transition from elastic deformation to fracture of initially linearly elastic materials [7,11]. When a material enters the plastic regime, it loses the longitudinal elastic force, and instead, gains the transverse restoring force represented by on the right-hand side of eq. (6). At the same time, the longitudinal force becomes energy dissipative as represented by eq. (12). Based on this observation, the transition from the elastic regime to the plastic can be characterized by and proportional to the local velocity. Note that this is a local effect; even if the stress-strain curve is before the yield point (i.e., in the linear regime) and therefore the specimen is considered to be globally elastic, it is possible that the deformation is locally plastic. There are a number of experimental observations that support this interpretation [12]. While the mechanism of energy dissipation via is effective, the work done by the external force causing the deformation, such as the work done by a tensile machine, is partially dissipated in this fashion and partially stored as the rotational elastic energy associated with the restoring force . As the deformation develops, the material tends to lose these mechanisms. Some theoretical consideration [11] and experimental observations [6] indicate that it is likely that materials loses the restoring mechanism first, causing the transverse wave to decay, and then enters the final phase where the 77
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