The method of virtual power in the mechanics of continuous media, I: Second-gradient theory

Germain, P.

doi:10.2140/memocs.2020.8.153

Cited by 74 publications

(72 citation statements)

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“…The strain gradient tensor is adopted to model size and non-local effects observed in the considered continuum system [ 39 , 40 , 41 , 42 , 43 , 44 ]. The presence of

in the list of meaningful variables follows the suggestion by Biot [ 32 ] and Cowin [ 45 ] to model porous materials.…”

Section: The Model Employed To Describe Bone Remodelingmentioning

confidence: 99%

Bio-Inspired Design of a Porous Resorbable Scaffold for Bone Reconstruction: A Preliminary Study

2021

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The study and imitation of the biological and mechanical systems present in nature and living beings always have been sources of inspiration for improving existent technologies and establishing new ones. Pursuing this line of thought, we consider an artificial graft typical in the bone reconstruction surgery with the same microstructure of the bone living tissue and examine the interaction between these two phases, namely bone and the graft material. Specifically, a visco-poroelastic second gradient model is adopted for the bone-graft composite system to describe it at a macroscopic level of observation. The second gradient formulation is employed to consider possibly size effects and as a macroscopic source of interstitial fluid flow, which is usually regarded as a key factor in bone remodeling. With the help of the proposed formulation and via a simple example, we show that the model can be used as a graft design tool. As a matter of fact, an optimization of the characteristics of the implant can be carried out by numerical investigations. In this paper, we observe that the size of the graft considerably influences the interaction between bone tissue and artificial bio-resorbable material and the possibility that the bone tissue might substitute more or less partially the foreign graft for better bone healing.

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“…The strain gradient tensor is adopted to model size and non-local effects observed in the considered continuum system [ 39 , 40 , 41 , 42 , 43 , 44 ]. The presence of

in the list of meaningful variables follows the suggestion by Biot [ 32 ] and Cowin [ 45 ] to model porous materials.…”

Section: The Model Employed To Describe Bone Remodelingmentioning

confidence: 99%

Bio-Inspired Design of a Porous Resorbable Scaffold for Bone Reconstruction: A Preliminary Study

2021

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“…The used heuristic procedure will lead to a formal expression of the work functional for externally applied loads: it is proven that together with standard volume forces and surface contact forces, dilatational strain gradient continua can support, on edges already present in the reference configuration, externally applied edge forces. Moreover, see [63], one can apply on contact surfaces also external double forces, that it interactions that expend work on normal (to the contact surface) derivative of virtual displacement. Rather interestingly it has been presented a reasoning which indicates that one can apply, on the regular parts of the boundary (i.e on the boundary faces) of the considered dilatational strain gradient continua, external forces concentrated on curves that are not already edges, in the reference configuration.…”

Section: Conclusion and Research Perspectivesmentioning

confidence: 99%

“…After works by Korteweg [71], Cahn and Hilliard [15,16] dealing with so-called capillary fluids, by Mindlin [87][88][89] and Toupin [128,129] dealing with more general continua, including elastically deformable solids, and after the more general approaches proposed by Germain [62,63] and Sedov [113], who considered microstructured generalized continua, see also [39], the second gradient elasticity was discussed and enlarged in many works, e.g. [7,12,32,54,134,138].…”

Section: Introductionmentioning

confidence: 99%

On nonlinear dilatational strain gradient elasticity

Eremeyev

Cazzani

Dell’Isola

2021

Continuum Mech. Thermodyn.

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We call nonlinear dilatational strain gradient elasticity the theory in which the specific class of dilatational second gradient continua is considered: those whose deformation energy depends, in an objective way, on the gradient of placement and on the gradient of the determinant of the gradient of placement. It is an interesting particular case of complete Toupin–Mindlin nonlinear strain gradient elasticity: indeed, in it, the only second gradient effects are due to the inhomogeneous dilatation state of the considered deformable body. The dilatational second gradient continua are strictly related to other generalized models with scalar (one-dimensional) microstructure as those considered in poroelasticity. They could be also regarded to be the result of a kind of “solidification” of the strain gradient fluids known as Korteweg or Cahn–Hilliard fluids. Using the variational approach we derive, for dilatational second gradient continua the Euler–Lagrange equilibrium conditions in both Lagrangian and Eulerian descriptions. In particular, we show that the considered continua can support contact forces concentrated on edges but also on surface curves in the faces of piecewise orientable contact surfaces. The conditions characterizing the possible externally applicable double forces and curve forces are found and examined in detail. As a result of linearization the case of small deformations is also presented. The peculiarities of the model is illustrated through axial deformations of a thick-walled elastic tube and the propagation of dilatational waves.

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“…Later, 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, also 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].…”

Section: Introductionmentioning

confidence: 96%

“…This special case of second gradient continuum models have a peculiarity which made them more acceptable by those who want to postulate continuum mechanics on the primitive concept of "balance laws" for forces and couples (or torques). In facts, [21] in the framework of couple-stress theories, the admissible boundary conditions (naturally deduced from the adapted variational principle) assume a particular for which double-forces are vanishing, see [28,43] for a detailed discussion of the concept of double force and Alibert et al [5] for its interpretation at micro-scales. Let us also note that the concept of double forces and other hypermoments are known in structural mechanics, see comments in [31, pp.…”

Section: Introductionmentioning

confidence: 99%

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

Eremeyev

Lurie

Solyaev

et al. 2020

Z. Angew. Math. Phys.

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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.

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The method of virtual power in the mechanics of continuous media, I: Second-gradient theory

Cited by 74 publications

References 3 publications

Bio-Inspired Design of a Porous Resorbable Scaffold for Bone Reconstruction: A Preliminary Study

Bio-Inspired Design of a Porous Resorbable Scaffold for Bone Reconstruction: A Preliminary Study

On nonlinear dilatational strain gradient elasticity

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

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