Uniqueness theorem in coupled strain gradient elasticity with mixed boundary conditions

• DOI: 10.1007/s00161-021-01048-6
• Publication Date: 2021-08-27T10:03:13Z
• Authors: Lidiia Nazarenko; Rainer Glüge; Holm Altenbach
• Keywords: Elasticity

Abstract The equilibrium equations and the traction boundary conditions are evaluated on basis of condition stationarity Lagrangian for coupled strain gradient elasticity. quadratic form energy can be written as a function second displacement contains fourth-, fifth- sixth-order stiffness tensor $${\mathbb {C}}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> , {C}}_5$$ <mml:mn>5</mml:mn> {C}}_6$$ <mml:mn>6</mml:mn> respectively. Assuming invariance under rigid body motions balance linear angular momentum is obtained. uniqueness theorem (Kirchhoff) mixed value problem proved case elasticity (novel). To this end, total potential altered to presented an uncoupled modified vector. Such transformation leads decoupling equation density. solution in standard manner by considering difference between two solutions.