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Organogenesis involves large deformations and complex shape changes that require elaborate mechanical regulation. Models of tissue biomechanics have been introduced to account for the coupling between mechanical response and biochemical processes. Recent experimental evidence indicates that the mechanical response of epithelial tissue is strongly anisotropic, with the degree of anisotropy being correlated with the existence of long-range orientational order of cytoskeletal organization across the tissue. A theoretical framework is introduced that captures the dynamic feedback between tissue elastic response and cytoskeletal reorganization under stress. Within the linear regime for small and uniform applied strains, the shear modulus is effectively reduced by the nematic order in cytoskeletal alignment induced by the applied strain. This prediction agrees with experimental observations of epithelial response in lithographically patterned micro tissues. 

We use self-consistent field theory to study the isotropic–nematic interface and topological defects in a compressible nematic of semi-flexible molecules. 

We analyze the spontaneous chiral-symmetry breaking states of a nematic liquid crystal system confined to a cylindrical capillary with homeotropic boundary conditions. A stability diagram is constructed, and the geometry of the states investigated. 

The structure of isolated disclinations and disclination dipoles in anisotropically elastic nematic liquid crystals is exploredviaa singular potential computational model. 

An exact kinematic law for the motion of disclination lines in nematic liquid crystals as a function of the tensor order parameter $\text{Q}$is derived. Unlike other order parameter fields that become singular at their respective defect cores, the tensor order parameter remains regular. Following earlier experimental and theoretical work, the disclination core is defined to be the line where the uniaxial and biaxial order parameters are equal, or equivalently, where the two largest eigenvalues of $\text{Q}$cross. This allows an exact expression relating the velocity of the line to spatial and temporal derivatives of $\text{Q}$on the line, to be specified by a dynamical model for the evolution of the nematic. By introducing a linear core approximation for $\text{Q}$, analytical results are given for several prototypical configurations, including line interactions and motion, loop annihilation, and the response to external fields and shear flows. Behaviour that follows from topological constraints or defect geometry is highlighted. The analytic results are shown to be in agreement with three-dimensional numerical calculations based on a singular Maier–Saupe free energy that allows for anisotropic elasticity. 

We study two dimensional tactoids in nematic liquid crystals by using a Q -tensor representation. A bulk free energy of the Maier–Saupe form with eigenvalue constraints on Q , plus elastic terms up to cubic order in Q are used to understand the effects of anisotropic anchoring and Frank–Oseen elasticity on the morphology of nematic–isotropic domains. Further, a volume constraint is introduced to stabilize tactoids of any size at coexistence. We find that anisotropic anchoring results in differences in interface thickness depending on the relative orientation of the director at the interface, and that interfaces become biaxial for tangential alignment when anisotropy is introduced. For negative tactoids, surface defects induced by boundary topology become sharper with increasing elastic anisotropy. On the other hand, by parametrically studying their energy landscape, we find that surface defects do not represent the minimum energy configuration in positive tactoids. Instead, the interplay between Frank–Oseen elasticity in the bulk, and anisotropic anchoring yields semi-bipolar director configurations with non-circular interface morphology. Finally, we find that for growing tactoids the evolution of the director configuration is highly sensitive to the anisotropic term included in the free energy, and that minimum energy configurations may not be representative of kinetically obtained tactoids at long times. 

Abstract The inherent inconsistency in identifying the phase field in the phase field crystal theory with the material mass and, simultaneously, with material distortion is discussed. In its current implementation, elastic relaxation in the phase field crystal occurs on a diffusive time scale through a dissipative permeation mode. The very same phase field distortion that is included in solid elasticity drives diffusive motion, resulting in a non physical relaxation of the phase field crystal. We present two alternative theories to remedy this shortcoming. In the first case, it is assumed that the phase field only determines the incompatible part of the elastic distortion, and therefore one is free to specify an additional compatible distortion so as to satisfy mechanical equilibrium at all times (in the quasi static limit). A numerical solution of the new model for the case of a dislocation dipole shows that, unlike the classical phase field crystal model, it can account for the known law of relative motion of the two dislocations in the dipole. The physical origin of the compatible strain in this new theory remains to be specified. Therefore, a second theory is presented in which an explicit coupling between independent distortion and phase field accounts for the time dependence of the relaxation of fluctuations in both. Preliminary details of its implementation are also given.