A note on $3$d-$1$d dimension reduction with differential constraints

Abstract

Starting from three-dimensional variational models with energies subject to a general type of PDE constraint, we use Γ-convergence methods to derive reduced limit models for thin strings by letting the diameter of the cross section tend to zero. A combination of dimension reduction with homogenization techniques allows for addressing the case of thin strings with fine heterogeneities in the form of periodically oscillating structures. Finally, applications of the results in the classical gradient case, corresponding to nonlinear elasticity with Cosserat vectors, as well as in micromagnetics are discussed.2010 Mathematics Subject Classification. Primary: 49J45; Secondary: 35E99, 74K15, 74Q05.

Asymptotic variational analysis of incompressible elastic strings

“…In the context of the analysis of thin objects, interesting effects may occur due to the interaction between restrictive material properties and the lower-dimensional structure of the objects. We mention here a few selected examples: thin (heterogenous) films and strings subject to linear first-order partial differential equations, which are general enough to cover applications in non-linear elasticity and micromagnetism at the same time, are studied in [28][29][30], cf. also [25,27]; pointwise constraints on the stress fields appear naturally in models of perfectly plastic plates [15,17]; for work on lower-dimensional material models that involve issues related to non-interpenetration of matter and (global) invertibility, we refer for instance to [31,39,41,47]; physical growth conditions, which guarantee orientation preservation of deformation maps, have been taken into account in models of thin nematic elastomers [2] and von Kármán-type rods and plates [16,37].…”

Asymptotic variational analysis of incompressible elastic strings

“…In the context of the analysis of thin objects, interesting effects may occur due to the interaction between restrictive material properties and the lower-dimensional structure of the objects. We mention here a few selected examples: thin (heterogenous) films and strings subject to linear firstorder partial differential equations, which are general enough to cover applications in nonlinear elasticity and micromagnetism at the same time, are studied in [26,27,28], cf. also [24,25]; pointwise constraints on the stress fields appear naturally in models of perfectly plastic plates [15,17]; for work on lower-dimensional material models that involve issues related to non-interpenetration of matter and (global) invertibility, we refer for instance to [29,35,37,41]; physical growth conditions, which guarantee orientation preservation of deformation maps, have been taken into account in models of thin nematic elastomers [2] and von Kármán type rods and plates [16,33].…”

Variational analysis of multiscale problems with differential constraints

“…In the context of the analysis of thin objects, interesting effects may occur due to the interaction between restrictive material properties and the lower-dimensional structure of the objects. We mention here a few selected examples: thin (heterogenous) films and strings subject to linear first-order partial differential equations, which are general enough to cover applications in nonlinear elasticity and micromagnetism at the same time, are studied in [129,130,131], cf. also [106,128]; pointwise constraints on the stress fields appear naturally in models of perfectly plastic plates [72,77]; for work on lower-dimensional material models that involve issues related to non-interpenetration of matter and (global) invertibility, we refer for instance to [135,171,186,205]; physical growth conditions, which guarantee orientation preservation of deformation maps, have been taken into account in models of thin nematic elastomers [3] and von Kármán type rods and plates [76,159].…”