Abstract
This article investigates the theoretical and numerical analysis as well as applications of the three-dimensional theory of first strain gradient elasticity. The corresponding continuous and discrete variational formulations are established with error estimates stemming from continuity and coercivity within a Sobolev space framework. An implementation of the corresponding isogeometric Ritz–Galerkin method is provided within the open-source software package GeoPDEs. A thorough numerical convergence analysis is accomplished for confirming the theoretical error estimates and for verifying the software implementation. Lastly, a set of model comparisons is presented for revealing and demonstrating some essential model peculiarities: (1) the 1D Timoshenko beam model is essentially closer to the 3D model than the corresponding Euler–Bernoulli beam model; (2) the 3D model and the 1D beam models agree on the strong size effect typical for microstructural and microarchitectural beam structures; (3) stress singularities of reentrant corners disappear in strain gradient elasticity. The computational homogenization methodologies applied in the examples for microarchitectural beams are shown to possess disadvantages that future research should focus on.
Highlights
During the development of modern science and technology, continuum mechanics has turned out to be a powerful physico-mathematical framework of theories and models describing the mechanics and thermodynamics of solids and fluids
In the presence of an applied point or line load or at a crack tip [8,9,10], are a noticeable class of nonphysical concepts appearing in the classical continuum mechanics; in this article we investigate the singularity at a reentrant corner as an example
We have addressed some occasions in which the classical continuum mechanics crucially fails to capture the results of virtual experiments
Summary
During the development of modern science and technology, continuum mechanics has turned out to be a powerful physico-mathematical framework of theories and models describing the mechanics and thermodynamics of solids and fluids. Convergence analysis has a small role in these studies and the convergence orders remain open, in particular On these grounds, the targeted novelties of this article on the three-dimensional version of strain gradient elasticity and isogeometric analysis are the following; (1)–(4) from the point of view of numerics, (5)–(6) from the point of view of mechanics: (1) variational formulations within a Sobolev space framework of order H 2(Ω ), Ω ∈ R3; (2) solvability via continuity and coercivity; (3) error estimates for conforming Ritz–Galerkin formulations; (4) an isogeometric implementation with a verification and confirmation for the theoretical error estimates; (5) model comparisons between the 3D solid model and the corresponding 1D beam models approving the size effects peculiar for strain gradient models; (6) demonstrations for model peculiarities within the three-dimensional framework, including a computational homogenization for a cellular structure and an example about the removal of a stress singularity.
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