At equilibrium between two thermodynamic phases of a pure substance, their molar Gibbs free energies are equal. Along a coexistence line,Subtracting the two differentials givesFor a nonzero volume change,The second equality uses $\Delta g=\Delta h-T\Delta s=0$ on coexistence. Total entropy and volume changes may be used instead if they refer to the same amount of material. The ratio must use one consistent transformation direction, so reversing the labels changes both signs and leaves the slope unchanged. This is the general Clapeyron equation; the ideal-vapour approximation often called Clausius-Clapeyron makes further assumptions not needed for two solids.

The original diagram is a Kelvin-Voigt model: a spring and a dashpot act in parallel. The spring is an ideal elastic element, ${\sigma }_s=E{\epsilon }_s$; under its own sinusoidal stress its strain is in phase with stress and energy is stored reversibly. The dashpot is viscous, ${\sigma }_d=\eta {\dot{\epsilon }}_d$; its strain under an oscillating stress lags by90degrees and mechanical energy is dissipated.

In parallel the strain is common and the branch stresses add:For imposed harmonic stress, the strain amplitude and phase lag areThe short- and long-timescale limits follow under the printed roman labels below.

A sudden constant stress ${\sigma }_0$ gives delayed, bounded creep:The strain is initially zero because an instantaneous change would require an infinite dashpot stress. Initially the dashpot carries the load, with finite strain rate ${\sigma }_0/\eta$; the spring gradually carries more of it as the rate decays. On unloading, the strain recovers exponentially to zero. The network does not produce the instantaneous strain jump or unbounded constant-stress creep of a series Maxwell element.

TODO: User-deferred biology. No protein or organism-specific solution or research is included.

TODO: User-deferred biology. No protein or organism-specific solution or research is included.

TODO: User-deferred biology. No protein or organism-specific solution or research is included.

Use the two partials on each plane as obtained above. The four pairwise sums, in units of $a/6$, areThe shortest resultant gives the largest line-energy reduction in the $G{b}^2$ approximation:The initial summed squared magnitude is $a^2/3$ and the final value is $a^2/18$, a sixfold reduction in that part of the energy estimate.

The line of intersection of the two original planes is parallel toIts dot product with the resultant Burgers vector is zero, so the junction is edge in character. A plane containing both line and Burgers vector has normal $[100]$, rather than an FCC close-packed $\{111\}$ glide normal. The resulting stair-rod dislocation is therefore sessile for ordinary glide and can form a Lomer-Cottrell lock. It obstructs further dislocation motion and promotes work hardening.

This energetic comparison assumes comparable line lengths and elastic prefactors. Geometry, line character and stacking-fault geometry affect a complete junction-energy calculation, but the small resultant identifies the favorable candidate requested by the vector model.

A Shockley partial dislocation in an FCC slip plane has Burgers vector of the family $a⟨112⟩/6$. Its vector must lie in the specified plane, and the two vectors must sum to the perfect dislocation vector.

For the first plane normal ${\mathbf{n}}_1=[\bar{1}11]$, one allowed dissociation isThe vectors add to $a[303]/6$, and ${\mathbf{n}}_1\cdot [211]=-2+1+1=0$, ${\mathbf{n}}_1\cdot [1\bar{1}2]=-1-1+2=0$.

For ${\mathbf{n}}_2=[111]$,Their sum is $a[\bar{3}30]/6$, and both have zero dot product with $[111]$. Interchanging leading and trailing labels does not change these vector identities.

Each perfect vector has squared magnitude $a^2/2$, whereas the sum of the two partial squared magnitudes is $a^2/3$. The line-energy estimate proportional to $G{b}^2$ therefore favors separation, opposed by the stacking-fault energy of the ribbon between the partials.

For $N$ distinguishable lattice sites occupied by $N_i$ mutually indistinguishable atoms of each species, the number of configurations isUsing Stirling's approximation,For one mole, $N=N_A$ and $k_B{N}_A=R$, soEquiatomic configurational entropy follows when the composition has $X_i=1/n$, givingThe reference is the unmixed configurational arrangement, and the model assumes random site occupation without short-range order. It counts configurational entropy, not all vibrational, electronic or magnetic entropy. A large number of elements does not by itself guarantee a stable single solid solution: competing thermodynamic phases and the enthalpy term also enter the Gibbs free energy.

TODO: Practical weapon-fabrication and optimization instructions are not provided. The general metallurgy and diffusion analysis are retained in the preceding parts; this manufacturing subpart remains explicitly limited.

The materials interpretation is a cooling-rate gradient in a near-eutectoid steel. Before transformation, the parent thermodynamic phase is austenite. A small section can cool rapidly enough to suppress diffusional decomposition and form martensite, a supersaturated, highly strained transformation product. A much thicker interior cools more slowly and can transform diffusively to pearlite: alternating ferrite and cementite lamellae grouped into colonies. Bainite or mixed products are also possible for intermediate histories.

The likely contrast is therefore a martensite-rich rapidly cooled region and a more pearlitic slowly cooled interior. Brittle fracture is promoted by the highly strained martensitic state, residual stresses and local stress concentration. A larger intact interior need not share the same microstructure or local loading. The observations are consistent with this contrast, not a measurement proving pure thermodynamic phases everywhere.

The sketches below are generic steel microstructures, rather than a component design or fabrication prescription. No cooling curve, dimensions or transformation diagram is provided, so exact thermodynamic phase fractions, plate sizes and a universally pearlitic centre cannot be inferred.

The normal to the section is $[010]$, so its in-plane principal vibration directions are $[100]$ and $[001]$. Their refractive-index difference isThe slow direction is $[001]$. The optical path retardation isAt an angle $\alpha$ between a principal direction and the polarizer, ideal crossed-polar transmission has the formwhere $I_P$ is the intensity after the first polarizer. The two components recombine parallel to the incident polarization when $\Gamma =m\lambda$, and the crossed analyser removes them. ConsequentlyThe next order is260nm, outside ordinary visible illumination. At exact principal-axis alignment, $\sin 2\alpha =0$ and every wavelength is extinguished, explaining the black appearance rather than a special single-wavelength cancellation.