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), 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.
Highlights
Strain gradient elasticity deals with those models of continuum media where a strain energy density depends on the first and second gradients of placements
Le Roux and other scholars see, e.g. [21,51,52] and the reference therein, continued the line of thought started by Piola and, more recently, by using the by using a more modern formalism and the powerful tools given by functional analysis, Paul Germain gave a further impulse to generalised continuum mechanics, see [28,42,43]
We proved the existence and uniqueness of a weak solution of a specific class of equilibrium problems within the framework of newly proposed strain gradient elasticity model specified with the adjective “dilatational ”
Summary
Strain gradient elasticity deals with those models of continuum media where a strain energy density depends on the first and second gradients of placements. Page 3 of 16 182 elasticity is considered together with couple stress one, albeit in the title only the particular case of couple stress models is evoked (and the reader is explicitly warned about this circumstance) Another example of incomplete second gradient models, which already was mentioned above, is the Korteweg fluid with W = W (ρ, ∇ρ), where ρ is a mass density [8,29]. The aim of this paper is to prove the uniqueness and existence of weak solutions when considering the particular case of incomplete second gradient model given by the dilatational strain gradient continuum. Some conclusions are presented in the final section, together with some future research perspectives
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