Mixed finite element formulations for strain-gradient elasticity problems using the FEniCS environment

V. Phunpeng n, P.M. Baiz

Department of Aeronautical Engineering, Imperial College London, SW7 2AZ, United Kingdom a r t i c l e i n f o

Article history:

Received 4 June 2014

Received in revised form 2 November 2014

Accepted 24 November 2014

Keywords:

Mixed finite element formulation

Strain-gradient elasticity

FEniCS a b s t r a c t

This paper presents finite element implementations for the solution of gradient elasticity problems using the FEniCS project. The FEniCS Project provides a novel tool for the automated solution of partial differential equations by the finite element method. In particular, it offers a significant flexibility with regards to modelling and numerical discretization choices. The weak form of the gradient elasticity problem is derived from the Principal of Virtual Work. An equivalent mixed-type finite element formulation for the strain gradient elasticity problem is implemented in order to avoid the use of C1 continuous elements. The complete methodology and source codes (python scripts) are provided.

Numerical results are presented and compared with well-known benchmark examples, demonstrating the applicability of FEniCS for such applications. & 2014 Elsevier B.V. All rights reserved. 1. Introduction

Cutting edge composite material technology is becoming more dependent on nanosize particles and/or nanostructural systems. Nano scale systems are now commonly embedded into a standard matrix in order to obtain materials with more attractive properties than its original constituents. Carbon nanotubes (CNTs) are an example of nano-scale structures, for which a great deal of research has been carried out in order to investigate their properties and potential applications. Other nano scale aggregates that are being used in state of the art composite technology are Nanoclays and Graphene. Unfortunately, simulation technology in this area is not developing as fast as material science and manufacturing technology, potentially hampering the future progress of nano engineered materials.

To study nanocomposites, classical (local) continuum theories tend to be inadequate, due to the fact that local theories do not account for micro-structural length scales of the material. Traditional or local elasticity theory considers that stresses at a point are a function of the strains at the same point (bonding forces between atoms are not considered). This assumption is valid for most continuum/macro scale simulations but in general it does not hold for micro/nano scales. To study the issue of size effects, theories which consider the material behaviour at a point as a function of the deformation of the surrounding have been proposed, these are usually referred as non-local or strain gradient theories.

Non-local theories state that the stress at a point depends on the strain at all points in the continuum body [23,40]. There is a great deal of research dealing with non-local elastic theories [23–26,50] and gradient theories [34,40,61], as these could provide an efficient way to account for important scale effects. To study size effects in micro/nano composite materials, non-local/strain gradient theories currently represent one of the most attractive options as they provide a strong theoretical interpretation that converges to standard behaviours. For example, at the macro-scale, non-local/strain gradient theories seamlessly converge to the standard classical elasticity theory and are also consistent with continuum thermodynamics theory [44–47]. This type of theory can be extended to consider other configurations, e.g. non-singularities in dislocations and cracks [3–6], as well as dispersion and size effects in nano-objects [15,31].

The most widely used and most important computational method in engineering applications is the Finite Element Method (FEM). It has been implemented successfully in engineering problems in many areas such as solid and fluid mechanics [1,22,27,29,49,52], electromagnetics [30,37] and biomechanics [48,51], to mention just a few examples. Most engineering commercially available software packages for structural/material simulations are based in FEM (e.g. Abaqus [55]). Unfortunately very few provide support/capabilities to

Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/finel

Finite Elements in Analysis and Design http://dx.doi.org/10.1016/j.finel.2014.11.002 0168-874X/& 2014 Elsevier B.V. All rights reserved. n Corresponding author.

E-mail addresses: vp1110@ic.ac.uk (V. Phunpeng), p.m.baiz@imperial.ac.uk (P.M. Baiz).

Finite Elements in Analysis and Design 96 (2015) 23–40 model the nano/micro scale behaviour of materials (implementations of non-local/strain gradient FEM formulations are more complex than for classical theories).

Under certain conditions, there is a direct equivalence between non-local stress theories and strain gradient theories. From the two approaches, strain gradient theories are well known to be simpler to solve/implement. They have previously been considered in several publications, using either the mixed-type finite element formulation [7,53] or meshless methods [17,56]. The main difficulty of using FEM to solve strain gradient elasticity is the continuity requirement (C1 continuity). In this paper, C1 continuous finite element can be avoided by a mixed-type formulation.

Establishing mixed formulations was also not an easy task in the past [9]. Fortunately, thanks to the FEniCS project [36], development of complex mixed finite element formulations has become easier. The FEniCS Project is a modern collection of open source software components directed at the automated solution of Partial Differential Equations (PDEs) by FEM [36,57]. The FEniCS environment provides the following key components: (i) a high-level user interface (written in Python) that imitates the mathematical formulation of a finite element discretization, (ii) sophisticated computational techniques that allow a large class of finite element discretizations to be concisely defined while providing highly efficient code [32,33,43]. A very important characteristic of the FEniCS environment is its flexibility for changing the governing equations and their method of discretization, allowing the use of highly unusual function spaces. This flexibility makes FEniCS particularly useful for areas of research in which computational experimentation with different constitutive relations, different auxiliary equations and different regimes are of importance (as in the case of strain gradient elasticity). Its automation focus also allows for the automation of tedious and error-prone tasks such as the assembly of finite element matrices and vectors, encouraging rapid prototyping and development. Overall, the user-specified equations and methods of discretization remain explicit while the generic finite element implementation details are kept under the hood.