Fracture toughness ($K_c$)

Fracture toughness measures resistance to crack advance under specified loading and environmental conditions. In linear elasticity, $K_c$ is the critical stress intensity factor. For opening-mode loading, a valid plane strain measurement is denoted $K_{Ic}$, with units $\mathrm{Pa}\sqrt{\mathrm{m}}$, commonly $\mathrm{MPa}\sqrt{\mathrm{m}}$. It differs from a tensile yield strength: a crack changes the near-tip stress concentration.

For a fixed geometry with $K_I=Y\sigma \sqrt{\pi a}$, the threshold $K_I=K_{Ic}$ givesFor example, $K_{Ic}=10\mathrm{MPa}\sqrt{\mathrm{m}}$, $\sigma =100\mathrm{MPa}$ and $Y=1$ give $a_c\simeq 3.18\mathrm{mm}$ in this idealized geometry. The crack-tip plastic zone must remain small for the elastic intensity description to apply; extensive plasticity in continuum mechanics requires a different characterization.^[ref]

A thin elastic plate model may instead prescribe a critical front curvature ${\kappa }_f$. Such a model parameter cannot be substituted numerically for $K_{Ic}$ without a relation specific to that fracture model.

Crack growth becomes energetically possible when the energy release rate reaches its critical value. For fixed crack geometry this gives a brittle-failure stress scaling as ${\sigma }_f\propto \sqrt{E{G}_c}$.

For linear antiplane elasticity with body forces and Dirichlet data proportional to time, uniqueness gives $u(s,t)=tu(s,1)$ at any fixed crack length $s$. The minimized elastic-plus-load potential and its energy release rate scale quadratically. The local Griffith fracture criterion laws are $\dot{\ell }\ge 0$, $G(\ell ,t)\le G_c$ and $\dot{\ell }(G_c-G)=0$, with positive toughness $G_c$.

If $G(s,1)$ is finite and continuous near the initial length, it is bounded there by $M$. Continuity of the crack path keeps it in that neighborhood for a positive interval. Choose the interval short enough that $t^{2}M<G_c$; complementarity then forces zero growth throughout it. This proves a positive initial waiting time without assuming the crack is stationary in advance.

The decrease in potential energy per increment of crack area under specified loading constraints. Its critical value $G_c$ is a crack-growth threshold, not energy stored per specimen volume.

Use antiplane energy $\frac{\mu }{2}\int |\nabla u{|}^2$ and normalize the tip singularity as $u=k\sqrt{r}\sin (\theta /2)$ plus a less singular term. The energy release rate is the small-circle flux $\int [\mathcal{W}{n}_x-\mu u_r{u}_x]rd\theta$. Here $\mathcal{W}=\mu k^2/(8r)$, $u_r=k\sin (\theta /2)/(2\sqrt{r})$ and $u_x=-k\sin (\theta /2)/(2\sqrt{r})$. Integrating over $(-\pi ,\pi )$ gives the displayed coefficient. The full three-dimensional antiplane stress has squared Frobenius norm $2{\mu }^2|\nabla u{|}^2$, so its leading norm ahead of the tip is $\mu |k|/\sqrt{2r}$. These constants depend on the normalization of the stress intensity factor.

The threshold energy release rate for propagation of a specified crack under specified environmental and loading conditions.

A requirement that a crack penetrate a pressure boundary and produce a detectable leak before unstable fracture. A simple screening comparison uses critical crack size versus wall thickness, but the result depends on crack geometry and on whether stress or pressure is held fixed.

A conservative material-screening upper bound obtained by requiring wall penetration before fast fracture even at a stated yield-limited stress. Substituting a thickness-dependent pressure stress instead yields a different fixed-pressure inequality; the two comparisons must not be conflated.

The reduction in a mechanical system's potential energy per incremental crack area under specified loading conditions. In small-scale-yielding linear elasticity, mode-I plane strain gives $G=K_I^2/E^{\prime }$ with $E^{\prime }=E/(1-{\nu }^2)$. The critical value includes the dissipative processes associated with crack advance.

A region of irreversible deformation near a crack tip where an elastic singular-field extrapolation would exceed the material's local yielding response. Its size depends on loading and constraint. Plastic work and blunting can raise resistance to crack extension.

An amplitude characterizing a near-tip linear-elastic crack field for a specified loading mode. The mode-I opening factor scales as the displayed expression for a suitable crack geometry factor $Y$. It is not itself the local stress, and large-scale plasticity can invalidate a linear-elastic characterization.

Crack propagation predominantly along grain boundaries. Boundary cavities, chemical segregation and environmental attack are possible contributors; fracture appearance alone does not establish which caused failure.

Brittle crack propagation associated with separation along particular crystallographic planes. Cleavage facets and river patterns can help identify the mechanism, while establishing its cause requires material, loading and exposure evidence.

A change from predominantly ductile deformation and fracture to predominantly brittle fracture as temperature decreases under specified loading and specimen conditions. A quoted transition temperature depends on its criterion: absorbed impact energy, fracture appearance and fracture toughness need not give the same number. In ferritic steel, irradiation hardening can raise the transition temperature.