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We discuss ellipticity property within the linear couple-stress elasticity. In this theory, there exists a deformation energy density introduced as a function of strains and gradient of macrorotations, where the latter are expressed through displacements. So the couple-stress theory could be treated as a particular class of strain gradient elasticity. Within the micropolar elasticity, the model is called Cosserat pseudocontinuum or medium with constrained rotations. Applying the classic definitions of ordinary ellipticity and strong ellipticity to static equations of the couple-stress theory, we conclude that these equations are neither elliptic nor strongly elliptic. As a result, one should be aware of extending properties of full strain gradient models such as Toupin–Mindlin strain gradient elasticity to models with incomplete set of second derivatives. [ABSTRACT FROM AUTHOR]

In this paper, it is proven an existence and uniqueness theorem for weak solutions of the equilibrium problem for linear isotropic dilatational strain gradient elasticity. Considered elastic bodies have as deformation energy the classical one due to Lamé but augmented with an additive term that depends on the norm of the gradient of dilatation: only one extra second gradient elastic coefficient is introduced. The studied class of solids is therefore related to Korteweg or Cahn–Hilliard fluids. The postulated energy naturally induces the space in which the aforementioned well-posedness result can be formulated. In this energy space, the introduced norm does involve the linear combination of some specific higher-order derivatives only: it is, in fact, a particular example of anisotropic Sobolev space. It is also proven that aforementioned weak solutions belongs to the space H 1 (d i v , V) , i.e. the space of H 1 functions whose divergence belongs to H 1 . The proposed mathematical frame is essential to conceptually base, on solid grounds, the numerical integration schemes required to investigate the properties of dilatational strain gradient elastic bodies. Their energy, as studied in the present paper, has manifold interests. Mathematically speaking, its singularity causes interesting mathematical difficulties whose overcoming leads to an increased understanding of the theory of second gradient continua. On the other hand, from the mechanical point of view, it gives an example of energy for a second gradient continuum which can sustain externally applied surface forces and double forces but cannot sustain externally applied surface couples. In this way, it is proven that couple stress continua, introduced by Toupin, represent only a particular case of the more general class of second gradient continua. Moreover, it is easily checked that for dilatational strain gradient continua, balance of force and balance of torques (or couples) are not enough to characterise equilibrium: to this aim, externally applied surface double forces must also be specified. As a consequence, the postulation scheme based on variational principles seems more suitable to study second gradient continua. It has to be remarked finally that dilatational strain gradient seems suitable to model the experimentally observed behaviour of some material used in 3D printing process. [ABSTRACT FROM AUTHOR]