Comparison of gradient elasticity models for the bending of micromaterials

doi:10.1016/j.commatsci.2015.10.031

Computational Materials Science

Volume 116, 15 April 2016, Pages 52-61

https://doi.org/10.1016/j.commatsci.2015.10.031 Get rights and content

Highlights

•
A modified strain gradient- and micropolar continuum theory is presented.
•
An analytical and numerical solution is derived from the equilibrium formulations.
•
The finite element variational formulations are solved within the FEniCS project.
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AFM bending experiments with epoxy and SU-8 microbeams show a positive size effect.
•
The material length scale parameters are fitted in a least square approach.

Abstract

In the context of static elasticity theory for isotropic materials and small deformations, two continuum mechanical theories of higher order are considered. Thinking of generalized continua, a modification/simplification of the strain gradient- and the micropolar theory is shown. An analytical solution strategy is derived for each extended theory using the Euler–Bernoulli beam assumptions. A numerical solution of the resultant differential equations of fourth order, derived from the equilibrium formulation, is realized using a finite element implementation in an open source FE environment. In AFM experiments with microbeams, made of epoxy and of the polymer SU-8, force and deflection data is recorded. As a result, positive size effects depending on thickness are observed and quantified for the microbeams. The material length scale parameters and the elastic moduli are fitted with a least square approach and compared to literature values.

Graphical abstract

Introduction

Materials with an intrinsic microstructure may show size-dependent material behavior when the outer dimensions of a body are reduced. This is referred to as the size effect in mechanics. Micromaterials are important in micro- and nanosystem technology and denote basic materials whose properties are dependent on the microscopic design of its internal structure. Size effects in elasticity manifest themselves in a stiffer or softer elastic response to external forces applied to small structures made of these micromaterials. This has first been observed in several experiments on metals and polymer materials [14], [28], [39], [6], [7], [37], [4], [30]. Simulation of structural behavior grows in importance from the design phase up to the valuation of reliability and is driven by miniaturization, saving of materials, and targets of higher performances. Because conventional (Cauchy-) continuum theory is unable to predict size effects, several continua of higher order are applicable either in a manner that can be interpreted in terms of physics or as an alternative technique of homogenization with regard to real internal structure. Theories of higher order are for example non-local theories [36], [13], strain gradient theories [32], [40], micropolar theories [16], [10], [38], [35], [9], theories of material surfaces [17] or fractional continuum mechanics [3], [18]. In this work, a comparison of two models of higher elastic gradient theories (in terms of higher gradients of displacements) is performed. In Section 2 the higher gradient theory of elasticity is established and the modified strain gradient model, as well as the couple stress model is presented. Analytical and finite element solutions are used in order to analyze the problem of simple beam bending. Current experimental investigations of the size effect in SU-8 and epoxy are presented in Section 3. An Atomic Force Microscope (AFM) is used to apply a quasi point-force at the free end of a cantilever and to record force as well as deflection data of the microbeam bending. In Section 4 the elastic modulus is evaluated according to the Euler–Bernoulli beam assumptions and fitted to the afore-mentioned higher gradient models. The additional material parameters that stem from the incorporation of the higher gradient terms were determined following an inverse analysis using both the analytical and the finite element solution.

Section snippets

Selected higher gradient elasticity models

In order to predict physical phenomena on the micro-, nano-, or even atomic scale the use of generalized continua was proposed, for example, by Eringen [10], [11], [12], [13] and Maugin and Metrikine [29]. A history of origin of generalized continuum theories that are attributed to the initial works of Cosserat and Cosserat [5] is presented in [1]. With the development of a linearized Cosserat theory a model with couple stresses has been developed [40], [31], [19], [33] that incorporates

Identification of higher gradient coefficients

According to [21] the analytical investigation of a bending mode of deformation for different sizes of a beam structure results in a relation of the bending rigidity to the outer dimension and to the internal material parameter. In order to derive an analytical formula for simple beam bending within the MSG and the CS model, the displacement field according to the Euler–Bernoulli beam assumptions is used: $u_{x} = - z \frac{d w (x)}{d x}, u_{y} = 0, u_{z} = w (x) .$

After applying the principle of virtual displacements to the

Results and comparison

The conventional and the higher order elastic parameters are fitted by using the method of minimization of least squares of the error between the measured and predicted data, cf., Table 1. The best material parameters from the FE approaches are filtered out from a loop of FE calculations using an underlying grid of pairs of coefficients (E, $ℓ$ ) with a resolution of $Δ E = 0.05$ GPa and $Δ ℓ = 50$ nm. The results for the scale dependent elastic moduli are in good agreement to the predictions of the analytical

Conclusions

In the present work continuum theories of higher order were presented. Their potential to quantify the size effect in elastostatic bending deformations was illustrated. A numerical solution strategy using an open source finite element environment was used in order to solve the derived differential equations for the MSG and for the CS theory. The MSG theory was derived from the point of view of Mindlin’s formulations of displacement gradients of higher order, whereas the CS theory was derived

Acknowledgements

The present work is supported by (Deutsche Forschungsgemeinschaft) DFG under Grant MU 1752/33-1. The authors would like to thank the Fraunhofer Institute for Reliability and Microintegration Berlin for sample preparation and the Physikalisch-Technische Bundesanstalt Braunschweig for the help in AFM calibration.

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View full text

Comparison of gradient elasticity models for the bending of micromaterials

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Selected higher gradient elasticity models

Identification of higher gradient coefficients

Results and comparison

Conclusions

Acknowledgements

Mech. Mater.

Int. J. Eng. Sci.

Acta Metall. Mater.

Int. J. Eng. Sci.

J. Mech. Phys. Sol.

Comput. Mater. Sci.

Int. J. Solids Struct.

Int. J. Solids Struct.

Int. J. Eng. Sci.

Int. J. Solids Struct.

Arch. Appl. Mech.

Finite gradient elasticity and plasticity: a constitutive mechanical framework

Theorie des corps deformables

Phys. Rev. Lett.

Phys. Rev. B

Proc. R. Soc. A

Foundations of Micropolar Mechanics

Continuum physics

Microcontinuum Field Theories. I. Foundations and Solids