A systematic coarse graining of microscopic atomic displacements generates a local elastic deformation tensor D as well as a positive definite scalar χ measuring nonaffinity, i.e., the extent to which the displacements are not representable as affine deformations of a reference crystal. We perform an exact calculation of the statistics of χ and D and their spatial correlations for solids at low temperatures, within a harmonic approximation and in one and two dimensions. We obtain the joint distribution P(χ,D) and the two-point spatial correlation functions for χ and D. We show that nonaffine and affine deformations are coupled even in a harmonic solid, with a strength that depends on the size of the coarse-graining volume Ω and dimensionality. As a corollary to our work, we identify the field h(χ) conjugate to χ and show that this field may be tuned to produce a transition to a state where the ensemble average <χ> and the correlation length of χ diverge. Our work should be useful as a template for understanding nonaffine displacements in realistic systems with or without disorder and as a means for developing computational tools for studying the effects of nonaffine displacements in melting, plastic flow, and the glass transition.

Coarse-graining atomic displacements in a solid produces both local affine strains and "non-affine" fluctuations. Here we study the equilibrium dynamics of these coarse grained quantities to obtain space-time dependent correlation functions. We show how a subset of these thermally excited, non-affine fluctuations act as precursors for the nucleation of lattice defects and suggest how defect probabilities may be altered by an experimentally realisable "external" field conjugate to the global non-affinity parameter. Our results are amenable to verification in experiments on colloidal crystals using commonly available holographic laser tweezer and video microscopy techniques, and may lead to simple ways of controlling the defect density of a colloidal solid.

Customarily, crystalline solids are defined to be rigid since they resist changes of shape determined by their boundaries. However, rigid solids cannot exist in the thermodynamic limit where boundaries become irrelevant. Particles in the solid may rearrange to adjust to shape changes eliminating stress without destroying crystalline order. Rigidity is therefore valid only in the metastable state that emerges because these particle rearrangements in response to a deformation, or strain, are associated with slow collective processes. Here, we show that a thermodynamic collective variable may be used to quantify particle rearrangements that occur as a solid is deformed at zero strain rate. Advanced Monte Carlo simulation techniques are then used to obtain the equilibrium free energy as a function of this variable. Our results lead to a unique view on rigidity: While at zero strain a rigid crystal coexists with one that responds to infinitesimal strain by rearranging particles and expelling stress, at finite strain the rigid crystal is metastable, associated with a free energy barrier that decreases with increasing strain. The rigid phase becomes thermodynamically stable when an external field, which penalizes particle rearrangements, is switched on. This produces a line of first-order phase transitions in the field-strain plane that intersects the origin. Failure of a solid once strained beyond its elastic limit is associated with kinetic decay processes of the metastable rigid crystal deformed with a finite strain rate. These processes can be understood in quantitative detail using our computed phase diagram as reference.

In complex crystals close to melting or at finite temperatures, different types of defects are ubiquitous and their role becomes relevant in the mechanical response of these solids. Conventional elasticity theory fails to provide a microscopic basis to include and account for the motion of point defects in an otherwise ordered crystalline structure. We study the elastic properties of a point-defect rich crystal within a first principles theoretical framework derived from the microscopic equations of motion. This framework allows us to make specific predictions pertaining to the mechanical properties that we can validate through deformation experiments performed in molecular dynamics simulations.

We describe a phase transition that gives rise to structurally non-trivial states in a two-dimensional ordered network of particles connected by harmonic bonds. Monte Carlo simulations reveal that the network supports, apart from the homogeneous phase, a number of heterogeneous "pleated" phases, which can be stabilised by an external field. This field is conjugate to a global collective variable quantifying "non-affineness", i.e. the deviation of local particle displacements from local affine deformation. In the pleated phase, stress is localised in ordered rows of pleats and eliminated from the rest of the lattice. The kinetics of the phase transition is unobservably slow in molecular dynamics simulation near coexistence, due to very large free energy barriers. When the external field is increased further to lower these barriers, the network exhibits rich dynamic behaviour: it transforms into a metastable phase with the stress now localised in a disordered arrangement of pleats. The pattern of pleats shows ageing dynamics and slow relaxation to equilibrium. Our predictions may be checked by experiments on tethered colloidal solids in dynamic laser traps.