What Is a Fluid? — Question 2

Question 2

Hold the displacement instead of the force

An apparatus first shifts the top face of a sample sideways by a small amount, then locks both bounding plates. The figure shows the horizontal force that the sample subsequently exerts on the top plate. Signs have been replaced by magnitudes. Sample A is modeled as an elastic solid; sample B as a fluid with a transient response. Ignore interfacial slip and edge effects.

1. Explain why B can be a fluid even though its boundary displacement is fixed and the sample eventually becomes motionless.

2. Which sample stores a persistent shear restoring effect in this test? Does zero late-time horizontal force imply that B cannot exert any force on its container?

Question 2 - Solution

Hold the displacement instead of the force

(a) Derivation and calculation

Locked plates impose$$x(t)=x_0,\qquad\dot x=0.$$The recorded limiting shear forces are$$\boxed{F_A(\infty)=2.0\,\mathrm N},\qquad\boxed{F_B(\infty)=0}.$$A fluid can relax to rest after displacement is imposed: no persistent shear load remains in B.

(b) Result and interpretation

If $A_p$ is the plate area,$$\tau_A(\infty)=2.0/A_p,\qquad\tau_B(\infty)=0.$$$$\mathbf F_{\mathrm{wall}}=F_n\hat{\mathbf n}+F_t\hat{\mathbf t},\quad F_t=0\ \not\Rightarrow\ F_n=0.$$A retains the shear restoring effect. B can still exert normal pressure on the container.

Physical check

This is a fixed-displacement relaxation test; it does not contradict continued deformation under a maintained nonzero shear load.