The dynamics of solid-solid phase transitions 2. Incoherent interfaces

Incoherent phase transitions are more difficult to treat than their coherent counterparts. The interface, which appears as a single surface in the deformed configuration, is represented in its undeformed state by a separate surface in each phase. This leads to a rich but detailed kinematics, one in which defects such as vacancies and dislocations are generated by the moving interface. In this paper we develop a complete theory of incoherent phase transitions in the presence of deformation and mass transport, with phase interface structured by energy and stress. The final results are a complete set of interface conditions for an evolving incoherent interface.Frequently used symbols A_i,C_i generic subsurface of S_t - B_i undeformed phase-i region - C configurational bulk stress, Eshelby tensor - F deformation gradient - G inverse deformation gradient - H relative deformation gradient - J bulk Jacobian of the deformation - ¯K, K_i total (twice the mean) curvature of and S_i - Lin (U, V) linear transformations from U into V - Lin⁺ linear transformations of ³ with positive determinant - Orth⁺ rotations of ³ - Q^a external bulk mass supply of species a - ¯S bulk Cauchy stress tensor - S bulk Piola-Kirchhoff stress tensor - S_i undeformed phase i interface - U_i relative velocity of S_i - Unim⁺ linear transformations of ³ with unit determinant - ¯V, V_i normal velocity of and S_i - intrinsic edge velocity of S and A _i S - W_i volume flow across the phase-i interface - X material point - b external body force - e internal bulk configurational force - f_i external interfacial force (configurational) - ¯g external interfacial force (deformational) - grad, div spatial gradient and divergence - gradient and divergence on - h relative deformation - h^a, diffusive mass flux of species a and list of mass fluxes - ¯m outward unit normal to a spatial control volume - ¯n, n_i unit normal to and S_i - n subspace of ³ orthogonal to n - ¯q^a external interfacial mass supply of species a - s ......... - ¯v, v_i compatible velocity fields of and S_i - ¯w, w_i compatible edge velocity fields for and A_i - x spatial point - y_i deformation or motion of phase i - y^. material velocity - generic subsurfaces of - , _i deformed body and deformed phase-i region - () energy supplied to by mass transport - symmetry group of the lattice - _i, surface jacobians - lattice - () power expended on - spatial control volume - S deformed phase interface - lattice point density - interfacial power density - , A total surface stress - C configurational surface stress for phase 1 (material) - ¯C_i configurational surface stress (spatial) - F_i tangential deformation gradient - G_i inverse tangential deformation gradient - H incoherency tensor - ¯1(x), 1_i(X) inclusions of ¯n(x) and n _i (X) into ³ - K configurational surface stress for phase 2 (material) - ¯L, l_i curvature tensor of and S_i - ¯P(x), P_i(X) projections of ³ onto ¯n(x) and n_i (X) - ¯S, S deformational surface stress (spatial and material) - ¯a, a normal part of total surface stress - c normal part of configurational surface stress for phase 1 (material) - e_i internal interfacial configurational force - ¯v, v_i unit normal to and A _i - (x),_i(X) projections of ³ onto ¯n(x) and n _i (X) - _i normal internal force (material) - bulk free energy - slip velocity - _i=(–1)ⁱ _i ......... - ^a, chemical potential of species a and list of potentials - ^a, bulk molar density of species a and list of molar densities - _i normal internal force (spatial) - surface tension - , _i effective shear - referential-to-spatial transform of field - interfacial energy - grand canonical potential - l unit tensor in ³ - x, vector and tensor product in ³ - (...)^., _t(...) material and spatial time derivative - , Div material gradient and divergence - gradient and divergence on S_i - (...), (...) normal time derivative following and S_i - (...) limit of a bulk field asx ,x_i - ...],...> jump and average of a bulk field across the interface - (...)_ext extension of a surface tensor to ³ - tangential part of a vector (tensor) on and S_i