Non-metric Connection and Metric Anomalies in Materially Uniform Elastic Solids

Abstract

Metric anomalies arising from a distribution of point defects (intrinsic interstitials, vacancies, point stacking faults), thermal deformation, biological growth, etc. are well known sources of material inhomogeneity and internal stress. By emphasizing the geometric nature of such anomalies we seek their representations for materially uniform crystalline elastic solids. In particular, we introduce a quasi-plastic deformation framework where the multiplicative decomposition of the total deformation gradient into an elastic and a plastic deformation is established such that the plastic deformation is further decomposed multiplicatively in terms of a deformation due to dislocations and another due to metric anomalies. We discuss our work in the context of quasi-plastic strain formulation and Weyl geometry. We also derive a general form of metric anomalies which yield a zero stress field in the absence of other inhomogeneities and any external sources of stress.

On Structured Surfaces with Defects: Geometry, Strain Incompatibility, Stress Field, and Natural Shapes

“…It is a consequence of the fundamental existence theorem of linear differential systems that in absence of disclinations (i.e., Ω ijkl = 0) over a simply connected U (hence V ), if the matrix fieldḡ ij := g ij − 2q ij is positive-definite for symmetric functions q ij =q ji , thenQ kij = −2q ij;k is the only solution to (28) over V . This result is proved in [60]. As the density of metric anomalies is assumed to be uniform with respect to the ζ coordinate, we will interpret this representation of the metric anomalies in absence of disclinations over simply connected patches over ω as…”

“…The second term S ij := −δA ζ ij p n p , on the other hand, is symmetric; it represents a stretching, with the three principal values of the tensor ζn = ζ ij p n p A i ⊗ A j as measures of the stretch along their respective (linearly independent) principal directions. The tensor ζ can be shown to be related to the metrical properties of M as it gives rise to a smeared out anomaly within the material structure which causes elongation or shortening of material vectors under parallel transport along loops (see [60] for details), as shown in Figure 7(a). We will assume ζ ≡ 0 in rest of the paper since, at present, we do not know of any defects in 2-dimensional materials which they would otherwise represent.…”

“…where λ := 1 2 q µ µ = 1 2 a µα q αµ is the trace of q and q µν is the deviatoric part of q µν (i.e., q µ µ = 0). The first term represents isotropic metric anomalies and the second represents anisotropic metric anomalies [60].…”

“…in terms of an invertible second order tensor field F p := F p ij G i ⊗ G j , the plastic distortion field, defined over simply connected subsets V ⊂ M [60]. The well-defined surface plastic distortion tensor F p := F p ζ=0 maps the reference base vectors A α to the crystallographic base vectors a α := F p A α over the material space.…”

Biological growth in bodies with incoherent interfaces

“…We consider, treating F 0 as an hyperelastic membrane, a free energy density per unit area of the stress-free configuration as Ψ( H, C). It is then straightforward to employ the dissipation inequality for the material points occupying F 0 to obtain, on one hand, P =ĵ∂ H Ψ G −T and µ =ĵ∂ C Ψ and, on the other, ( E + µ E 0 ) · ( G)˙ G −1 + Grad f µ · M ≤ 0, such thatĵ ∈ R + is the ratio of infinitesimal areas of the film in the stress-free configuration with respect to the reference configuration and E =ĵ Ψ1 − H T ∂ H Ψ is the elastic surface Eshelby tensor; compare these with (20)- (21) and (25)- (26). The kinetic laws which satisfy the inequality are…”