By David Broek
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The strain tensor ε tends toward a uniform value E∞ at inﬁnity, or equivalently, the displacement ﬁeld ξ tends toward E∞ · z. 34] where χI is the indicator function4 of I. We begin with the case E∞ = 0. 35] ∇ χI involves the derivation of a discontinuous function across the boundary ∂I. It therefore involves the Dirac 4 χI (z) = 0 if z ∈ / I; χI (z) = 1 if z ∈ I. 42 Micromechanics of Fracture and Damage distribution5 δ∂I associated with the boundary ∂I of the domain I. More precisely, n denoting the unit normal vector to ∂I oriented outward with respect to I, we have ∇ χI = −nδ∂I .
There exists a single-valued holomorphic function F (z) deﬁned on S such that: φ (z) = A log(z) + F (z) By integration, we obtain: φ(z) = A(z log(z) − z) + F(z) with F(z) = z F (u) du zo where zo is some ﬁxed point in S. 40] Fundamentals of Plane Elasticity 13 where φ∗ (z) is a single-valued holomorphic function deﬁned on S. 41] where ψ ∗ (z) is a single-valued holomorphic function deﬁned on S. 39], and take advantage of the fact that the displacement is single-valued. 43] where Fx and Fy denote the components of the resultant force acting on the contour.
As already stated, the tensor R will take a constant value RI inside an ellipsoidal (respectively, elliptic) inclusion. 2. 49] Fundamentals of Elasticity in View of Homogenization Theory 45 To begin with, the strain E∞ at inﬁnity is 0 and the strain ﬁeld ε solution is sought as a function of the polarization ﬁeld τ . 1 in the case of a uniform polarization stress in a bounded domain. More precisely, it consists of representing the polarization ﬁeld τ (z ) as an inﬁnity of superimposed elementary uniform polarization ﬁelds, each being restricted to the elementary domains dVz centered at z .