Sensitivity of the macroscopic response of elastic microstructures to the insertion of inclusions

Abstract

This paper proposes an exact analytical formula for the topological sensitivity of the macroscopic response of elastic microstructures to the insertion of circular inclusions. The macroscopic response is assumed to be predicted by a well-established multi-scale constitutive theory where the macroscopic strain and stress tensors are defined as volume averages of their microscopic counterpart fields over a representative volume element (RVE) of material. The proposed formula-a symmetric fourth-order tensor field over the RVE domain-is a topological derivative which measures how the macroscopic elasticity tensor changes when an infinitesimal circular elastic inclusion is introduced within the RVE. In the limits, when the inclusion/matrix phase contrast ratio tends to zero and infinity, the sensitivities to the insertion of a hole and a rigid inclusion, respectively, are rigorously obtained. The derivation relies on the topological asymptotic analysis of the predicted macroscopic elasticity and is presented in detail. The derived fundamental formula is of interest to many areas of applied and computational mechanics. To illustrate its potential applicability, a simple finite element-based example is presented where the topological derivative information is used to automatically generate a bi-material microstructure to meet pre-specified macroscopic properties.

Topological derivative-based topology optimization of structures subject to multiple load-cases

“…!o et al (2002); Masmoudi et al (2005)), image processing (Auroux et al (2007); Belaid et al (2008); Hintermüller (2005) ; Amstutz et al (2012); Bojczuk and Mróz (2009) ;Burger et al (2004); Giusti et al (2008Giusti et al ( , 2010b; Kobelev (2010); Leugering and Sokołowski (2008); Novotny et al (2003Novotny et al ( , 2005Novotny et al ( , 2007; Turevsky et al (2009)). See also applications of the topological derivative in the context of multiscale constitutive modeling ; Giusti et al (2010aGiusti et al ( , 2009a; Novotny et al (2010)), fracture mechanics sensitivity analysis (Ammari et al (2014); Van Goethem and Novotny (2010)) and damage evolution modeling (Allaire et al (2011)). Regarding the theoretical development of the topological asymptotic analysis, see for instance (Amstutz (2006(Amstutz ( , 2010 ;Feijóo et al (2003); Garreau et al (2001); Hlaváček et al (2009); Khludnev et al (2009); Lewinski and Sokołowski (2003); Nazarov and Sokołowski (2003a,b, 2005, 2006, 2011; Żochowski (2003, 2005)), as well as the book by Novotny and Sokołowski (2013).…”

A multiscale model for dislocations: From mesoscopic elasticity to macroscopic plasticity

“…(3.8). This will be achieved by taking local average values, in the spirit of a multiscale approach as proposed by Germain et al [14] and later developped in, e.g., [15,25]. The main result of this section is that incompatibility of the homogenized strain reads inc E =η− I 2 trη withη = 2∇×κ…”

“…where α is a non negative dimensional parameter 15 (scalar or tensorial) and where inc(·) on the RHS is an elliptic operator of order two in the sense of Legendre and Hadamard [8]. Hence, provided appropriate boundary conditions, the incompatibility driven Bravais flow is parabolic and shows existence on every times interval, as opposed to ordinary Ricci flows [9,16,26,29].…”

“…REGULAR BRAVAIS STRAINĒ 15) while the remaining plastic strain is made of two terms: a purely concentrated part on a surface Σ with unit normal n and a (fractal) Cantor part. By (4.13), we have inc E + E P,c (ũ) = − inc (Ξ), whereby inc (Ξ) = 0, since taking any smooth test-function Ψ with nonvanishing incompatibility and compact support U satisfying 0 < H 3 (U ) < ∞, it is observed that…”

“…Observe that tr ( inc E) = (Δ trE − ∇ · E · ∇) with tr inc E B = − 1 2 R = tr (κ × ∇) and ∇ · inc = inc · ∇ = 0, so that taking the trace of (5.8) shows the following evolution law for the scalar Bravais curvature. 15 With the physical dimensions of a surface flow [m 2 /s]. 16 Among optimization principles in Mechanics, maximization has proven justified in thermodynamic analysis of many material processes, as for instance the celebrated Hill-Mandel principle of maximal dissipation in crack evolution [38].…”