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

2020 ◽  
Vol 71 (6) ◽  
Author(s):  
Victor A. Eremeyev ◽  
Sergey A. Lurie ◽  
Yury O. Solyaev ◽  
Francesco dell’Isola
Keyword(s):  
Strain Gradient ◽  
Weak Solutions ◽  
Variational Principles ◽  
Gradient Elasticity ◽  
Deformation Energy ◽  
Well Posedness ◽  
Strain Gradient Elasticity ◽  
Elastic Bodies ◽  
Integration Schemes ◽  
Second Gradient

AbstractIn 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.

scholarly journals On existence and uniqueness of weak solutions for linear pantographic beam lattices models

2019 ◽  
Vol 31 (6) ◽  
pp. 1843-1861 ◽  
Author(s):  
Victor A. Eremeyev ◽  
Faris Saeed Alzahrani ◽  
Antonio Cazzani ◽  
Francesco dell’Isola ◽  
Tasawar Hayat ◽  
...  
Keyword(s):  
Energy Density ◽  
Strain Gradient ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Gradient Elasticity ◽  
Well Posedness ◽  
Strain Gradient Elasticity ◽  
Derivatives Of

Abstract In this paper, we discuss well-posedness of the boundary-value problems arising in some “gradient-incomplete” strain-gradient elasticity models, which appear in the study of homogenized models for a large class of metamaterials whose microstructures can be regarded as beam lattices constrained with internal pivots. We use the attribute “gradient-incomplete” strain-gradient elasticity for a model in which the considered strain energy density depends on displacements and only on some specific partial derivatives among those constituting displacements first and second gradients. So, unlike to the models of strain-gradient elasticity considered up-to-now, the strain energy density which we consider here is in a sense degenerated, since it does not contain the full set of second derivatives of the displacement field. Such mathematical problem was motivated by a recently introduced new class of metamaterials (whose microstructure is constituted by the so-called pantographic beam lattices) and by woven fabrics. Indeed, as from the physical point of view such materials are strongly anisotropic, it is not surprising that the mathematical models to be introduced must reflect such property also by considering an expression for deformation energy involving only some among the higher partial derivatives of displacement fields. As a consequence, the differential operators considered here, in the framework of introduced models, are neither elliptic nor strong elliptic as, in general, they belong to the class so-called hypoelliptic operators. Following (Eremeyev et al. in J Elast 132:175–196, 2018. 10.1007/s10659-017-9660-3) we present well-posedness results in the case of the boundary-value problems for small (linearized) spatial deformations of pantographic sheets, i.e., 2D continua, when deforming in 3D space. In order to prove the existence and uniqueness of weak solutions, we introduce a class of subsets of anisotropic Sobolev’s space defined as the energy space E relative to specifically assigned boundary conditions. As introduced by Sergey M. Nikolskii, an anisotropic Sobolev space consists of functions having different differential properties in different coordinate directions.

scholarly journals On nonlinear dilatational strain gradient elasticity

2021 ◽  
Author(s):  
Victor A. Eremeyev ◽  
Antonio Cazzani ◽  
Francesco dell’Isola
Keyword(s):  
Strain Gradient ◽  
Deformable Body ◽  
Gradient Elasticity ◽  
Deformation Energy ◽  
Elastic Tube ◽  
Contact Forces ◽  
Specific Class ◽  
Strain Gradient Elasticity ◽  
Second Gradient ◽  
Eulerian Descriptions

AbstractWe 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.

scholarly journals Uniqueness theorem in coupled strain gradient elasticity with mixed boundary conditions

2021 ◽  
Author(s):  
Lidiia Nazarenko ◽  
Rainer Glüge ◽  
Holm Altenbach
Keyword(s):  
Quadratic Form ◽  
Potential Energy ◽  
Boundary Conditions ◽  
Uniqueness Theorem ◽  
Strain Gradient ◽  
Mixed Boundary ◽  
Displacement Vector ◽  
Gradient Elasticity ◽  
Strain Gradient Elasticity ◽  
Second Gradient

AbstractThe equilibrium equations and the traction boundary conditions are evaluated on the basis of the condition of the stationarity of the Lagrangian for coupled strain gradient elasticity. The quadratic form of strain energy can be written as a function of the strain and the second gradient of displacement and contains a fourth-, a fifth- and a sixth-order stiffness tensor $${\mathbb {C}}_4$$ C 4 , $${\mathbb {C}}_5$$ C 5 and $${\mathbb {C}}_6$$ C 6 , respectively. Assuming invariance under rigid body motions the balance of linear and angular momentum is obtained. The uniqueness theorem (Kirchhoff) for the mixed boundary value problem is proved for the case of the coupled linear strain gradient elasticity (novel). To this end, the total potential energy is altered to be presented as an uncoupled quadratic form of the strain and the modified second gradient of displacement vector. Such a transformation leads to a decoupling of the equation of the potential energy density. The uniqueness of the solution is proved in the standard manner by considering the difference between two solutions.

scholarly journals On weak solutions of the boundary value problem within linear dilatational strain gradient elasticity for polyhedral Lipschitz domains

2021 ◽  
pp. 108128652110255
Author(s):  
Victor A. Eremeyev ◽  
Francesco dell’Isola
Keyword(s):  
Strain Gradient ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Numerical Solutions ◽  
Gradient Elasticity ◽  
Lipschitz Domains ◽  
Elastic Model ◽  
Strain Gradient Elasticity ◽  
Existence And Uniqueness Theorem ◽  
Math Phys

We provide the proof of an existence and uniqueness theorem for weak solutions of the equilibrium problem in linear dilatational strain gradient elasticity for bodies occupying, in the reference configuration, Lipschitz domains with edges. The considered elastic model belongs to the class of so-called incomplete strain gradient continua whose potential energy density depends quadratically on linear strains and on the gradient of dilatation only. Such a model has many applications, e.g., to describe phenomena of interest in poroelasticity or in some situations where media with scalar microstructure are necessary. We present an extension of the previous results by Eremeyev et al. (2020 Z angew Math Phys 71(6): 1–16) to the case of domains with edges and when external line forces are applied. Let us note that the interest paid to Lipschitz polyhedra-type domains is at least twofold. First, it is known that geometrical singularity of the boundary may essentially influence singularity of solutions. On the other hand, the analysis of weak solutions in polyhedral domains is of great significance for design of optimal computations using a finite-element method and for the analysis of convergence of numerical solutions.

Sign in / Sign up

Export Citation Format

Share Document