Strain And Young Modulus

wrong-strain-rejects-the-modulus

For a stated stress and Young modulus only one strain closes E = sigma / epsilon exactly; any other strain is inconsistent and the modulus is rejected when the stress-strain ratio fails to match. That verdict is exact only for stated stress, strain, and Young modulus in the linear elastic regime. Real materials add calibration, geometry, temperature, loading history, and uncertainty.

A same-looking modulus claim fails when its cited strain source changes. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The rejected claim still cites real sources

The stress and strain sources are valid. The rejection comes from the claimed modulus failing their exact quotient.

σ=4 Paϵ=18\sigma=4\ \text{Pa}\qquad \epsilon=\frac{1}{8}
Rejected modulus sourcesThe source values are valid but the claim is not.stressSource=boundstrainSource=boundmodulus=16 PaacceptedBit=0 bit

Changing the strain source changes the required modulus

Two rows are internally consistent. The final row keeps the small strain source but claims the wrong modulus.

σϵEaccepted4 Pa1832 Pa14 Pa1416 Pa14 Pa1816 Pa0\begin{array}{c|c|c|c}\sigma&\epsilon&E&\text{accepted}\\4\ \text{Pa}&\frac{1}{8}&32\ \text{Pa}&1\\4\ \text{Pa}&\frac{1}{4}&16\ \text{Pa}&1\\4\ \text{Pa}&\frac{1}{8}&16\ \text{Pa}&0\\\end{array}

A wrong strain source rejects the claimed modulus

The displayed modulus is not trusted because the cited strain makes the exact quotient different.

4 Pa18=32 Pa≠16 Pa;accepted=0\frac{4\ \text{Pa}}{\frac{1}{8}}=32\ \text{Pa}\ne16\ \text{Pa};\quad \text{accepted}=0
Rejected Young modulus ledgerSame stress with a smaller strain would require a larger modulus.stressSource=boundstrainSource=boundmodulus=16 PaacceptedBit=0 bit