Asymptotic Analysis of Elastic Elliptic Membrane Shells in Frictional Contact: Exploring Wear Phenomena

We consider a family of linearly elastic shells, all sharing the same middle surface, with thickness <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mn>2</mml:mn> <mml:mi>ε</mml:mi> </mml:math> , clamped along their entire lateral face, which upon deformation may enter in frictional contact with a moving foundation along its lower face. As a result of friction, material might be removed from the interface, thus causing wear. We focus in the case of an elliptic membrane, for which the orders of applied body force density, surface tractions density, and compliance functions with respect to the small parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>ε</mml:mi> </mml:math> , representing thickness, are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , respectively. We show that the solution pair <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mrow> <mml:mi mathvariant="bold">u</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> of displacements and wear fields of the three-dimensional scaled variational contact problem converges to a pair of limit functions, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mrow> <mml:mi mathvariant="bold">u</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , which can be identified with the solution pair of a limit two-dimensional variational problem, since <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="bold">u</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> is independent of the transverse variable, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> . Besides, not all the convergences happen in the same topologies, since <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>w</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ε</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→</mml:mo> <mml:mi>w</mml:mi> </mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>ε</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/