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# Elasticity, viscosity, solid- and liquid-like behavior, and plasticity

When one tries to deform a piece of material, some of the above properties appear depending on the amplitude and duration of the applied stress.

• Long time application of weak stress: Solids initially deform and then resist to deform. Fluids deform (flow) continuously.
• Short time application of weak stress: If deformation follows the stress, material is elastic!!!. If deformation rate follows the stress, material is viscous.
• Application of high stress: After a certain stress (yield stress), some solids start to deform irreversibly. These are called plastic solids. There are also yield stress fluids, whose threshold stress is much lower than plastic solids.

## Deborah number

A transition from a more resistant (elastic) to a less resistant behavior (viscous) has a relevant characteristic time scale: the relaxation time of the material. Correspondingly, the ratio of the relaxation time of a material to the timescale of a deformation is called Deborah number :

{\displaystyle \displaystyle \displaystyle De={\frac {\text{characteristic relaxation time of material}}{\text{time scale of deformation}}}}

Small Deborah numbers correspond to situations where the material has time to relax (and behaves in a viscous manner), while high Deborah numbers correspond to situations where the material behaves rather elastically. Water can show elastic behavior when the time scale of deformation becomes very short. For example, when one tries to jump to water from a height more than 100 meters, water feels like a solid ground at the instant of collision ( do not try). Corn starch and water mixture(suspension) is a good example with which low and high De number effects can be shown.

## Rheological Material

Fluids can be classified according to the relation between stress {\displaystyle \displaystyle \tau } and deformation rate {\displaystyle \displaystyle du/dy}. The Newtonian fluids show a linear relation{\displaystyle \displaystyle \tau =\mu {\frac {du}{dy}}\Rightarrow \mu =\mu (T,P)}

Fluids which do not follow the linear law between stress an the deformation rate are called non-newtonian and they are the subject of rheology. A dilatant (shear-thickening) fluid increases resistance with increasing applied stress. Alternately, a pseudoplastic (shear-thinning) fluid decreases resistance with increasing stress. If the thinning effect is very strong, the fluid is termed plastic. The limiting case of a plastic substance is one which requires a finite yield stress before it begins to flow. The linear-flow Bingham plastic idealization is shown in the figure, but the flow behavior after yield may also be nonlinear. Examples of a yielding fluid are toothpaste and ketchup, which will not flow out of the tube until a finite stress is applied by squeezing.

Some fluids show decreasing (thixotropic) or increasing resistance (rheopectic) in time for the same deformation rate. For example, pudding is a rheopectic fluid and some paints are thixotropic.

Types of different fluids regarding the change of the stress in time for constant a strain

Types of different fluids regarding their stress-strain dependence