Stress, Strain, Dislocations and Strengthening

Dislocation reactions in FCC metals

A dislocation reaction must conserve the Burgers vector, so the sum of the product Burgers vectors equals the sum of the reactant vectors. Whether the reaction proceeds depends on energy: line energy scales with b^2, so a reaction is energetically favourable only when the sum of the squared magnitudes of the products is smaller than that of the parent, \sum |\mathbf{b}_{\text{prod}}|^2 < |\mathbf{b}|^2. In FCC metals the perfect dislocation \frac{a}{2}\langle 110\rangle dissociates into two Shockley partials \frac{a}{6}\langle 112\rangle. These partials are imperfect and glissile on \{111\} planes, and they bound a stacking fault between them. For example, \frac{a}{2}[110]+\frac{a}{2}[\bar{1}01]=\frac{a}{2}[011] is favourable, because each reactant has |\mathbf{b}|^2=a^2/2, giving a total of a^2, while the product has only a^2/2. The product magnitude is a/\sqrt{2}. A reaction whose products have a larger total b^2 than the parent is unfavourable and does not occur spontaneously. Two leading partials on different \{111\} planes can react to form a sessile Lomer–Cottrell lock. Dissociation is checked in the same way: the partial vectors must add back to the parent vector, and their squared magnitudes must add up to less than the parent value.

Schmid law slip systems

Schmid's law states that slip begins on a system when the resolved shear stress reaches the critical resolved shear stress (CRSS). The resolved shear stress is \tau = \sigma\cos\phi\cos\lambda = m\sigma, where \phi is the angle between the loading axis and the slip-plane normal, \lambda is the angle between the loading axis and the slip direction, and m=\cos\phi\cos\lambda is the Schmid factor. A slip system whose plane contains the loading axis, or whose direction is perpendicular to it, has m=0 and cannot be activated. In FCC metals such as aluminium, the active systems are the \{111\}\langle 110\rangle family, and the system with the largest m is activated first when m\sigma \ge \tau_{CRSS}. The yield stress of a single crystal is \sigma_y=\tau_{CRSS}/m. For example, if \phi=60^\circ and \lambda=45^\circ, then m=0.5\times0.707=0.354, and a CRSS of 2 MPa gives \sigma_y=2/0.354\approx5.7 MPa. For a cylindrical specimen the applied stress is \sigma=F/A_0 with A_0=\pi d^2/4. If the slip plane normal and the slip direction each make 45^\circ with the axis, then m=0.5 and \tau_{CRSS}=0.5\sigma.

Solute–dislocation interaction phenomena

Solute atoms interact with dislocations and their stacking faults, and this produces several characteristic phenomena. A Cottrell atmosphere is a cloud of solute atoms segregated around a dislocation, which pins it and gives the sharp yield point and Lüders bands. Suzuki interaction is the segregation of solutes to the stacking fault, which lowers the stacking-fault energy locally and can pin partials. The Bauschinger effect is the decrease in yield stress when the loading direction is reversed after prior plastic deformation. Dynamic strain ageing, in which solute atoms repeatedly catch up with moving dislocations, produces serrations in the stress–strain curve. The Peierls–Nabarro stress is the lattice friction that a dislocation must overcome to glide, so it is directly linked to dislocation glide. Nabarro–Herring creep is diffusional creep, in which stress-assisted diffusion of vacancies through the grains produces plastic flow in a polycrystal, and it is not a solute–dislocation effect. Recrystallization lowers the dislocation density, and superplasticity arises from grain-boundary sliding in fine-grained materials at high temperature and low strain rate. Paris law describes fatigue crack growth per cycle, and Considère's criterion marks the onset of necking.

Tensile stress-strain behaviour

Engineering stress is S=F/A_0 and engineering strain is e=\Delta L/L_0, while true stress is \sigma=F/A and true strain is \varepsilon=\ln(L/L_0). Because the instantaneous area is smaller than the original area after yielding, true stress is always higher than engineering stress once the specimen deforms. Resilience is the strain energy stored per unit volume in the elastic region, equal to the area under the elastic part of the engineering curve. The maximum load on the engineering curve is the point where dS/de=0, which is the Considère condition. On the true curve, however, the necking condition is d\sigma/d\varepsilon=\sigma, which is not zero. Thus the maximum true stress does not coincide with the maximum load. For plastic deformation at nearly constant volume, A_0L_0=A_1L_1. If the uniform elongation is e, then A_0=A_1(1+e).

Orowan equation for strain rate

The Orowan equation relates the plastic shear strain rate to the mobile dislocation density \rho_m, the Burgers vector magnitude b and the mean dislocation velocity v: \dot{\gamma}=\rho_m b v. In tension the same relation holds with \dot{\varepsilon}. The strain rate is linear in both v and \rho_m, so \dot{\varepsilon}\propto v and \dot{\varepsilon}\propto\rho, whereas squared dependences do not follow from the relation. Rearranging gives v=\dot{\gamma}/(\rho_m b). For a BCC metal with \mathbf{b}=\frac{a}{2}\langle 111\rangle, the magnitude is b=\frac{\sqrt{3}}{2}a, which must be converted to metres before use. A modest strain rate in a crystal with a moderate mobile density therefore gives a dislocation velocity of the order of 10^{-4} m/s.

Principal stresses and maximum shear stress

Principal stresses are the eigenvalues of the stress tensor, and the principal directions are the eigenvectors. If a row and column of the tensor are zero, that direction is itself a principal axis with a principal stress of zero. The remaining 2\times2 block gives the other two principal stresses, and the largest shear stress is \tau_{max}=(\sigma_{max}-\sigma_{min})/2, where the maximum and minimum are taken over all three principal stresses including zero. In plane stress, the principal stresses are \sigma_{1,2}=\sigma_{avg}\pm R, where \sigma_{avg}=(\sigma_{xx}+\sigma_{yy})/2 and R=\sqrt{\left(\frac{\sigma_{xx}-\sigma_{yy}}{2}\right)^2+\tau_{xy}^2}. The sign of \tau_{xy} does not change R, so the magnitude of the maximum principal stress depends only on \sigma_{avg} and R.

Yield criteria (Tresca, von Mises)

The Tresca criterion says yielding occurs when the maximum shear stress reaches half the uniaxial yield strength, which is equivalent to \sigma_1-\sigma_3=\sigma_y, where \sigma_1 and \sigma_3 are the largest and smallest principal stresses. The von Mises criterion instead uses the second deviatoric invariant J_2, and yielding occurs when \sqrt{J_2} reaches a critical value. For a diagonal stress tensor, the principal stresses are simply the diagonal entries, sorted in order. Tresca's criterion is therefore more conservative than von Mises' because it gives the lower bound of the two for a given stress state.

True strain in hot forging

True (logarithmic) strain is defined as \varepsilon=\ln(h_0/h) in compression, where h_0 is the initial height and h is the final height. Its magnitude is positive for compression and negative for tension, depending on the sign convention used. A height reduction of 50% gives h=0.5h_0, so |\varepsilon|=\ln 2\approx0.69, which is smaller than the engineering reduction of 0.5. The same logarithmic form is used in hot forging, so the true strain is additive over successive passes.