A Limit Model of a Soft Thin Joint

Let E³ be the Euclidian space referred to the orthonormal frame (0, e → 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1147.tif"/> , e → 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1148.tif"/> , e → 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1149.tif"/> ) and let Ω₋ and Ω₊ be open, disjoint, connected domains with boundaries ∂Ω₋ and ∂Ω₊ piecewise of class C ². Ω₋ and Ω₊ are bonded together on a surface S = ∂Ω₋ ⋂ ∂Ω₊. We assume that S is a set of positive 2—measure, is described by a global piecewise C ² chart, given by x ₃ = φ(x ₁, x ₂). We also set Ω ¯ = Ω ¯ + ∪ Ω ¯ − https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1150.tif"/> . In fact, we describe the bonding of the two solids occupying Ω₋ and Ω₊ by a thin adhesive layer, whose thermomechanical coefficients are small with respect of those of the adherents. More precisely, for a fixed ɛ > 0 we define the domain Ω 0 ε https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1151.tif"/> occupied by the thin layer: Ω ¯ 0 ε =   { x → s + ε z e → 3 ,       0 ≤ z ≤ 1 ,       x → s ∈   S } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq1152.tif"/>