Effect of surface bending and stress on the transmission of line force to an elastic substrate (2018)

For a broad class of soft materials their surface stress can strongly influence mechanical behaviour. For example, a line force applied to the surface of an elastic substrate is locally supported by surface stress over an elasto-capillary length <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> (surface stress/elastic modulus). Surface stress regularizes the otherwise highly singular stress and strain fields. However, surface such as lipid bilayer interfaces can also resist deformation by bending. This has not been studied either by experiments or theories. We analyse a theoretical model of the response of a half-space to a line force when the surface carries both a stress and resistance to bending. We find that surface bending further regularizes the singular fields. The local stress field near the line load can be separated into three regions. Region 1 occupies distances from the line load smaller than an elasto-capillary bending length <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">b</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> (bending stiffness/elastic modulus to the 1/3 power) where surface bending dominates and the elastic stress and strains are continuous. Region 2 occupies intermediate distances between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">b</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">b</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:math> where surface stress dominates. At distances larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> we retrieve the classical elasticity solution. The size of region 2 depends on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>κ</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">b</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and vanishes for small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> .