A Dimension Reduction Result in the Framework of Structured Deformations

Dimension Reduction in the Context of Structured Deformations

“…To apply directly the results contained in [7], we would need to impose a stronger regularity on the field d, namely, we would have to require d ∈ C(ω; R 3 ). We avoid this by applying the technique presented in [32]: we approximate d by a sequence of piecewise constant functions d k ∈ L p (ω; R 3 ), and we exploit the property (3.7) of the bulk energy density W 3d,2d and the approximation result provided in [15,Lemma 2.9].…”

Structured Deformations of Continua: Theory and Applications

“…In this respect, partial results are available in the special case of the purely interfacial initial energy densities in Theorem 4. Moreover, in this case, another relaxation procedure is available in the literature: in [17] a relaxation that simultaneously defines a 2d energy on structured deformations is studied. We prove [6,Section 5] that the left-and right-hand paths provide the same relaxed energy densities, whereas those computed using the central path in Fig.…”

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“…The articles [Choksi et al 1999;2011;] also provide summaries of that treatment, and 2011; provide alternative settings for structured deformations. The summary in [Choksi et al 1999] is intended for those interested in immediate applications in continuum mechanics while ] sets the stage for applications of structured deformations to thin bodies [Matias and Santos 2014]. The article [Šilhavý 2015] reexamines the results of [Choksi and Fonseca 1997] in a broader setting while providing refinements of counterparts of the approximation theorem and the identification relation (1-4).…”

“…The homogenized energy introduced in this paper (56) is minimized by using the package "Weak Form PDE" and by introducing standard third-order Hermite finite elements. While the used code is surely not optimized for the introduced problem (we believe that the recently developed numerical methods would be more efficient, see, e.g., [Cazzani et al 2016a;2016b;Greco and Cuomo 2013;2014;2016;Turco and Aristodemo 1998;Beirão da Veiga et al 2008;Della Corte et al 2016]), its rate of convergence seems satisfactory for getting preliminary results concerning the behavior of the simplest structures; actually, it is based on the introduction of an auxiliary tensor field which appears in the deformation energy and is equated to the displacement gradient by means of suitable fields of Lagrange multipliers. Remark also that all presented numerical simulations are really and intrinsically mesh-independent, because of the properties of the introduced continuum model, where the second gradient of displacement is at the same time modeling the relevant physical properties and supplies a regularizing effect on equilibrium equations.…”

“…For this purpose, frame-indifferent variants of the specific fields obtained here are available through known factorizations of structured deformations in which the factor that tracks disarrangements is unchanged under changes in frame [Del . Our explicit formulas are also starting points for the study of examples in other contexts involving structured deformations: second-order structured deformations [Owen and Paroni 2000] in which second gradients and their limits enter into submacroscopic changes in geometry as well as processes for dimension reduction [Matias and Santos 2014] in the presence of disarrangements that describe thin structures undergoing submacroscopic slips, separations, and switches.…”