Mechanical Properties & Failure Workbench

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Engineering reference

Mechanical Properties & Failure: theory, method, and sources

This materials science workspace publishes 12 governing equations, 6 stated assumptions, 1 documented boundary, and 2 sources so the numbers it returns can be checked rather than taken on trust.

Calculations run locallyCalculation & source methodology

How this tool works

Eleven modules cover engineering and true stress–strain, isotropic elastic constants, modulus of resilience, toughness by curve integration, Vickers hardness, Charpy impact energy, critical crack stress, Paris law fatigue crack growth, Basquin high-cycle fatigue, Norton creep, and Hall–Petch strengthening.

Fracture, fatigue, and creep are treated with the same stress basis, so a design stress can be checked against fast fracture, cyclic life, and time-dependent deformation in sequence.

Calculators and topics covered

stress strain
fracture
fatigue
creep
hardness
tensile test
Vickers
Hall Petch
Basquin
fracture toughness

Core equations

\text{engineering stress and strain:}\quad \sigma = \frac{F}{A_{0}},\; \varepsilon = \frac{\Delta L}{L_{0}}

\text{true stress and strain:}\quad \sigma _{t} = \sigma \left(1 + \varepsilon \right),\; \varepsilon _{t} = \ln \left(1 + \varepsilon \right)

\text{isotropic elasticity:}\quad G = \frac{E}{2 \left(1 + \nu \right)},\; K = \frac{E}{3 \left(1 - 2 \nu \right)}

\text{modulus of resilience:}\quad U_{r} = \frac{\sigma _{y}^{2}}{2 E}

\text{toughness:}\quad U_{t} = \int \sigma \; d\varepsilon \; \left(\text{trapezoidal}\; \text{integration}\; of\; the\; \text{curve}\right)

\text{Vickers hardness:}\quad HV = \frac{1.854 F}{d^{2}}

\text{Charpy impact energy:}\quad E = \mathrm{mg} \left(h_{1} - h_{2}\right),\; \text{impact}\; \text{toughness} = \frac{E}{A}

\text{critical crack stress:}\quad \sigma _{c} = \frac{K_{IC}}{Y \sqrt{\pi a}}

\text{Paris law:}\quad \frac{da}{dN} = C \left(\Delta K\right)^{m},\; \text{integrated}\; \text{from}\; a_{0}\; to\; a_{1}

\text{Basquin relation:}\quad \sigma _{a} = \sigma ^{\prime}_{f} \left(2 N_{f}\right)^{b}

\text{Norton creep:}\quad \dot{\varepsilon } = A\sigma ^{n} \exp \left(\frac{- Q}{kT}\right)

\text{Hall–Petch:}\quad \sigma _{y} = \sigma _{0} + k \cdot d^{\frac{- 1}{2}}

Method and assumptions

Assumptions

Materials are homogeneous, isotropic, and free of residual stress unless stated.
Engineering stress uses the original cross-sectional area, so it diverges from true stress after necking begins.
Linear elastic fracture mechanics applies, which requires small-scale yielding at the crack tip.
Paris law integration assumes constant amplitude loading with no crack closure, overload retardation, or threshold effects.
Basquin and Norton coefficients are material- and temperature-specific and must come from test data for the exact condition.
Fatigue and creep predictions are order-of-magnitude estimates, not certified life.

Limitations and design boundaries

Educational screening models only. Material certification, process qualification, applicable standards, statistical variability, environmental conditioning, and professional engineering judgment are required for real selection or design.

Sources and references

Primary sources are preferred for ratings, standards, manufacturer data, and externally defined constants.

Source policy

Callister and Rethwisch, Materials Science and Engineering: An Introduction
Askeland and Wright, The Science and Engineering of Materials