On the well posedness of static boundary value problem within the linear dilatational strain gradient elasticity

Abstract

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 $$H^1(div,V)$$ H 1 ( d i v , V ) , i.e. the space of $$H^1$$ H 1 functions whose divergence belongs to $$H^1$$ 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.

Uniqueness theorem in coupled strain gradient elasticity with mixed boundary conditions

“…In these theories it was assumed that the potential energy density depends not only on the strain, but also on higher derivatives of the displacement vector. More recently, the generalized non-classical theories have been also applied to modeling of materials at the micro-and nano-scale [10,23] to describing of phenomena like dislocations [21], to analyzing of composites with a high difference of the material properties at a lower scale [2,20,42,45,51] to describing some phenomena in regions with stress concentrations [5], to accounting for boundary and surface energies [14,28] or to removing singularities caused by discontinues of boundary conditions (e.g., [6,24,46,49]). It has been shown in numerous papers (see, for example, [22,34,35,41]) that some restrictions of the classical theory of elasticity can be overcome with such gradient expansion.…”

On nonlinear dilatational strain gradient elasticity

“…The class of deformation energies considered here is singular, in the sense given by the hypotheses demanded by the theorems presented in [67,81,97,98]. It is therefore clear that the study of mathematical well-posedness of the equilibrium and dynamical problems for dilatational strain gradient elasticity require the adaptation and/or the generalization of the standard arguments used in the theory of elasticity, in a similar way to what has been done, for the linear case, in [45,48]. In these last papers, the considered problem was linear, while here one deals with a nonlinear one.…”

“…The well-posedness of boundary-value problems within the linear dilatational strain gradient models was studied in [48].…”

“…In Sect. 4, the linear dilatational second gradient elasticity is introduced as in [48]. The comparison of the dilatational strain gradient elasticity with other models of continua with scalar microstructure as the Cowin-Nunziato poroelasticity [23,94] is then given.…”

Inverse Hooke's law and complementary strain energy in coupled strain gradient elasticity

“…These continua are called second gradient continua [4] or strain-gradient [3], where stored energy depends not only on strain, but also on higher derivatives of the displacement. More recently, the generalized continua was also applied to modeling materials at the micro-and nanometer scale [5,6], to describing phenomena like dislocations [7], to composites with high contrast (at a lower scale) in material properties [8][9][10][11][12], to catching some phenomena in regions with a stress concentration [13], to including boundary and surface energies [14,15] or to removing singularities, when discontinues appear in the boundary conditions, for example, [16][17][18][19]. It has been shown in numerous papers [20][21][22][23] that the limitations of classical elasticity theory can be overcome with such gradient expansion.…”