Eshel N.N., On first strain-gradient theories in linear elasticity

Abstract-This paper contains a study of the linear theory of elasticity in which the potential energy-density depends on the gradient of the strain in addition to the strain. In the first part of the paper, three forms of the theory are compared and the relations connecting the stresses in the three forms and the boundary conditions in the three forms are derived. In the second part of the paper, ambiguities in the form of the moment-equation of equilibrium and the definition of couple-stress are resolved by a derivation based on conservation principles rather than the variational principles employed previously