Abstract
The equilibrium equations and the traction boundary conditions are evaluated on the basis of the condition of the stationarity of the Lagrangian for coupled strain gradient elasticity. The quadratic form of strain energy can be written as a function of the strain and the second gradient of displacement and contains a fourth-, a fifth- and a sixth-order stiffness tensor {mathbb {C}}_4, {mathbb {C}}_5 and {mathbb {C}}_6, respectively. Assuming invariance under rigid body motions the balance of linear and angular momentum is obtained. The uniqueness theorem (Kirchhoff) for the mixed boundary value problem is proved for the case of the coupled linear strain gradient elasticity (novel). To this end, the total potential energy is altered to be presented as an uncoupled quadratic form of the strain and the modified second gradient of displacement vector. Such a transformation leads to a decoupling of the equation of the potential energy density. The uniqueness of the solution is proved in the standard manner by considering the difference between two solutions.
Highlights
In which the strain energy density is a function of the strain and the second gradient of the displacement vector, is a natural extension of the classical theory of elasticity
The equilibrium equation and the corresponding natural boundary conditions are re-derived on the basis of the Lagrange variational principle modified for the coupled linear strain gradient continua
Assuming rigid body motion the balance lows for the total forces and total torque are obtained from the principle of virtual power
Summary
In which the strain energy density is a function of the strain and the second gradient of the displacement vector, is a natural extension of the classical theory of elasticity. It has been introduced a block diagonalization of the composite stiffness in strain gradient elasticity By such a formal transformation, which contains a fourth-, a fifth- and a sixth-order stiffness tensor C4, C5 and C6, necessary conditions for positive definiteness and convexity of the isotropic strain and strain gradient energy, accounting for the coupling stiffness C5 have been obtained. The difference between the equations presented here and in Mindlin and Eshel [38] is in index associations of the scalar products This is the result of the presence of a coupled term in the strain and strain gradient energy Eq (1) and of the symmetry of the stiffness tensor of fifth-rank. The uniqueness theorem (Kirchhoff) for mixed boundary value problem is extended to the coupled linear strain gradient elasticity in Sect.
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