FYKOS Year 35 Series 4 Problem 4 — Analogy · FYKOS 35 #35.4.4

Physics · mechanics

Problem

Assume we have two linear springs with elastic modulus

E = 2.01\,\mathrm{GPa}

and a piston with viscosity

\eta = 9.8\,\mathrm{GPa\cdot s}

. The dependence of stress

\sigma

on relative extension

\epsilon

is characterized by formula

\sigma_{\mathrm{s}} = E\epsilon_{\mathrm{s}}

for spring, and by formula

\sigma_{\mathrm{d}} = \eta \dot{\epsilon}_{\mathrm{d}}

for piston, where the dot represents the time derivative (Newton's notation). We connect a spring of length

l_{\mathrm{s}}

and a piston of length

l_{\mathrm{d}}

into series, and then we connect the other spring of length

l_{\mathrm{p}}

in parallel to them. Abruptly, we stretch the entire system into the state of

\epsilon_0 = 0.2

, and we hold the extension constant. Determine, in what time (from stretching) will the stress decrease to half of the original value, if

\frac{l_{\mathrm{s}}}{l_{\mathrm{p}}} = 0.5

holds.