The primary species of precipitation strengthening are second phase particles. These particles impede the movement of dislocations throughout the lattice. You can determine whether or not second phase particles will precipitate into solution from the solidus line on the phase diagram for the particles. Physically, this strengthening effect can be attributed both to
size and
modulus effects, and to
interfacial or
surface energy. The presence of second phase particles often causes lattice distortions. These lattice distortions result when the precipitate particles differ in size and crystallographic structure from the host atoms. Smaller precipitate particles in a host lattice leads to a tensile stress, whereas larger precipitate particles leads to a compressive stress. Dislocation defects also create a stress field. Above the dislocation there is a compressive stress and below there is a tensile stress. Consequently, there is a negative interaction energy between a dislocation and a precipitate that each respectively cause a compressive and a tensile stress or vice versa. In other words, the dislocation will be attracted to the precipitate. In addition, there is a positive interaction energy between a dislocation and a precipitate that have the same type of stress field. This means that the dislocation will be repulsed by the precipitate. Precipitate particles also serve by locally changing the stiffness of a material. Dislocations are repulsed by regions of higher stiffness. Conversely, if the precipitate causes the material to be locally more compliant, then the dislocation will be attracted to that region. In addition, there are three types of interphase boundaries (IPBs). The first type is a coherent or ordered IPB, the atoms match up one by one along the boundary. Due to the difference in lattice parameters of the two phases, a coherency strain energy is associated with this type of boundary. The second type is a fully disordered IPB and there are no coherency strains, but the particle tends to be non-deforming to dislocations. The last one is a partially ordered IPB, so coherency strains are partially relieved by the periodic introduction of dislocations along the boundary. In coherent precipitates in a matrix, if the precipitate has a lattice parameter less than that of the matrix, then the atomic match across the IPB leads to an internal stress field that interacts with moving dislocations. There are two deformation paths, one is the
coherency hardening, the lattice mismatch is : \varepsilon_{coh} = \frac{a_p - a_m}{a_m} : \tau _{coh} = 7G\left|\varepsilon_{coh}\right|^\frac{3}{2}\left(\frac{rf}{b}\right)^\frac{1}{2} Where G is the shear modulus, \varepsilon_{coh} is the coherent lattice mismatch, r is the particle radius, f is the particle volume fraction, b is the burgers vector, rf/b equals the concentration. The other one is
modulus hardening. The energy of the dislocation energy is U_{m}=G_{m}b^2/2, when it cuts through the precipitate, its energy is U_{p}=G_{p}b^2/2, the change in line segment energy is : \bigtriangleup{U} = \left(U_{p} - U_{m}\right)2r = \left(G_{p} - G_{m}\right)b^2r. The maximum dislocation length affected is the particle diameter, the line tension change takes place gradually over a distance equal to r. The interaction force between the dislocation and the precipitate is : F = {dU \over dr} = \left(G_{p} - G_{m}\right)b^2 = G_{m}b^2\frac{G_{p} - G_{m}}{G_{m}} = G_{m}b^2\varepsilon_{Gp} and \tau = \frac{F}{bL}. Furthermore, a dislocation may cut through a precipitate particle, and introduce more precipitate-matrix interface, which is
chemical strengthening. When the dislocation is entering the particle and is within the particle, the upper part of the particle shears b with respect to the lower part accompanies the dislocation entry. A similar process occurs when the dislocation exits the particle. The complete transit is accompanied by creation of matrix-precipitate surface area of approximate magnitude A = 2\pi rb \,\!, where r is the radius of the particle and b is the magnitude of the burgers vector. The resulting increase in surface energy is E = 2\pi rb\gamma_s \,\!, where \gamma_{s} is the surface energy. The maximum force between the dislocation and particle is F_{max} = \pi r\gamma_s \,\!, the corresponding flow stress should be \Delta\tau=F_{max}/bL=\pi r\gamma_{s}/bL. When a particle is sheared by a dislocation, a threshold
shear stress is needed to deform the particle. The expression for the required shear stress is as follows: : \tau = cG\varepsilon^\frac{3}{2}\left(\frac{rf}{b}\right)^\frac{1}{2} When the precipitate size is small, the required shear stress \tau is proportional to the precipitate size r^{1/2}, However, for a fixed particle volume fraction, this stress may decrease at larger values of r owing to an increase in particle spacing. The overall level of the curve is raised by increases in either inherent particle strength or particle volume fraction. The dislocation can also
bow around a precipitate particle through so-called Orowan mechanism. Since the particle is non-deforming, the dislocation bows around the particles (\phi_{c}=0), the stress required to effect the bypassing is inversely proportional to the interparticle spacing (L-2r), that is, \tau_{b}=Gb/(L-2r), where r is the particle radius. Dislocation loops encircle the particles after the bypass operation, a subsequent dislocation would have to be extruded between the loops. Thus, the effective particle spacing for the second dislocation is reduced to (L-2r') with r' > r, and the bypassing stress for this dislocation should be \tau_{b}' = Gb/(L-2r'), which is greater than for the first one. However, as the radius of particle increases, L will increase so as to maintain the same volume fraction of precipitates, (L-2r) will increase and \tau_{b} will decrease. As a result, the material will become weaker as the precipitate size increases. For a fixed particle volume fraction, \tau_{b} decreases with increasing r as this is accompanied by an increase in particle spacing. On the other hand, increasing f increases the level of the stress as a result of a finer particle spacing. The level of \tau_{b} is unaffected by particle strength. That is, once a particle is strong enough to resist cutting, any further increase in its resistance to dislocation penetration has no effect on \tau_{b}, which depends only on matrix properties and effective particle spacing. If particles of A of volume fraction f_{1} are dispersed in a matrix, particles are sheared for r and are bypassed for r>r_{c1}, maximum strength is obtained at r=r_{c1}, where the cutting and bowing stresses are equal. If inherently harder particles of B of the same volume fraction are present, the level of the \tau_{c} curve is increased but that of the \tau_{b} one is not. Maximum hardening, greater than that for A particles, is found at r_{c2}. Increasing the volume fraction of A raises the level of both \tau_{b} and \tau_{c} and increases the maximum strength obtained. The latter is found at r_{c3}, which may be either less than or greater than r_{c1} depending on the shape of the \tau-r curve. ==Governing equations==