What Is a Fluid? — Question 6

Question 6

Two mechanical models with the same early displacement

A shear-testing machine idealizes sample S by $F=Kx$ and sample V by $F=C\,dx/dt$, where $x$ is the relative sideways displacement of its faces. Take $K=400\,\mathrm{N/m}$ and $C=2000\,\mathrm{N\,s/m}$. Apply $F=2.0\,\mathrm{N}$ from $t=0$ to $t=20\,\mathrm{s}$. Neglect inertia; V starts at $x=0$. The equations are supplied models, not assumed laws for every real material.

1. Find both displacements at $t=5.0\,\mathrm{s}$ and immediately before force removal. Explain why the first measurement alone is misleading.

2. After removing the force, predict the displacement according to each model. Identify the solid-like and fluid-like responses and one limitation of this idealization.

Question 6 - Solution

Two mechanical models with the same early displacement

(a) Derivation and calculation

$$x_S=F/K=2/400=\boxed{.0050\,\mathrm m}.$$$$\dot x_V=F/C=2/2000=.0010\,\mathrm{m/s},\quad x_V=.0010t.$$$$t=5:\quad x_S=x_V=5.0\,\mathrm{mm};\qquad t=20^-:\quad\boxed{x_S=5.0,\ x_V=20\,\mathrm{mm}}.$$

(b) Result and interpretation

After removing the force,$$F=0:\quad Kx_S=0\Rightarrow\boxed{x_S=0};\quad C\dot x_V=0\Rightarrow\boxed{x_V=20\,\mathrm{mm}}.$$S is solid-like; V is fluid-like. One matched displacement does not identify the long-time law. The instantaneous elastic return follows only from the assumed absence of inertia.

Physical check

The V model has no elastic term to restore its accumulated deformation.