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Linearity and reversibility answer different questions. The proportional limit ends the interval in which axial stress is proportional to axial strain. Elastic deformation means that deformation is recovered after unloading; a nonlinear response can still be elastic. Conversely, a short straight portion of a loading curve does not by itself demonstrate recovery.

To check recovery, unload and examine the local extensometer strain relative to its initial value, allowing for instrument resolution, drift and any time dependence. A residual extension can indicate plasticity in continuum mechanics, but grip slip can produce a residual in crosshead displacement without the same local residual. Compare the channels before assigning the effect to the specimen. Yield point concerns the onset of irreversible deformation; a specified offset proof stress is an operational convention rather than Young's modulus. For example, a $0.2\%$ offset proof stress is the intersection of the loading curve with $\sigma =E(\epsilon -0.002)$, using the correctly referenced strain. Report the fitted stress/strain window and the unloading evidence alongside the modulus, so that apparatus seating, departure from linearity and onset of plasticity in continuum mechanics remain distinguishable.

In the completely elastic exterior, radial elastic compatibility makes $q+2t$ constant. The far-field stress vanishes, so $q+2t=0$ and $t=-q/2$. Radial force balance then gives $q^{\prime }+3q/r=0$, whence $q=C/r^3$. The pressure boundary condition is $q(a)=-P$: positive applied pressure is compressive. ThereforeFor completeness, the corresponding radial displacement field is $u=P{a}^3/(4\mu r^2)$, which gives these components through Hooke's law and decays at infinity.

The Tresca yield criterion becomesFor $P\ge 0$ this is largest at the cavity wall and decreases strictly with radius when $P>0$. The first yield point is therefore at $r=a$, where $3P/4={\tau }_Y$. HenceThe normalization matters: ${\tau }_Y$ in this paper is a shear yield stress, rather than the uniaxial yield strength.

The following values are stipulated arithmetic inputs, not measurements of a material. Take $A_0=2.00\mathrm{m}{\mathrm{m}}^2$ and lengths $L_1=50.0\mathrm{mm}$, $L_2=100.0\mathrm{mm}$. At forces $F=(0,100,200,300,400)\mathrm{N}$, stipulate crosshead displacementsThe two different initial offsets deliberately test the use of intercepts. Linear regression of $x_i=a_i+q_{i}F$ gives $a_1=0.005\mathrm{mm}$, $a_2=0.008\mathrm{mm}$ andTheir reciprocal spring constants are $k_{\mathrm{obs},1}=2.85714kN/mm$ and $k_{\mathrm{obs},2}=1.66667kN/mm$. The compliance-versus-length slope isso the recovered Young's modulus and machine compliance areThe corresponding apparatus spring constant is $1/C_m=10.0kN/mm$.

Check the reconstruction at $400\mathrm{N}$. The machine contribution is $C_{m}F=0.040\mathrm{mm}$, and the two specimen contributions are $0.100\mathrm{mm}$ and $0.200\mathrm{mm}$. Adding the respective offsets gives the stipulated $0.145\mathrm{mm}$ and $0.248\mathrm{mm}$. An extensometer spanning $L_g=25.0\mathrm{mm}$ on either ideal uniform specimen would instead read $0.050\mathrm{mm}$ extension: engineering strain $0.00200$, at engineering stress $200N/m{\mathrm{m}}^2=200\mathrm{MPa}$. Their ratio is again $100\mathrm{GPa}$. Ignoring machine compliance gives apparent moduli $71.43\mathrm{GPa}$ and $83.33\mathrm{GPa}$ even after removing the offsets. No yield point or real material identity is implied by these stipulated linear inputs.

A yield point marks the onset of permanent deformation according to a stated experimental criterion. In a tensile testing experiment, low loads can produce recoverable extension, while higher loads leave a residual extension after unloading. A sharp transition may be visible in the stress-strain curve, but many specimens change gradually from mainly elastic to increasingly plastic response.

When no unambiguous sharp point exists, an offset proof stress gives an operational definition: draw a line parallel to the initial elastic slope and displaced by a specified engineering strain, then identify its intersection with the loading curve. The chosen offset must be reported. The yield point is distinct from both the limit of exact proportionality and the tensile strength at maximum load. This matters because a small measurable departure from linear response does not by itself establish a particular amount of permanent plastic strain.