Strain gradient elasticity with geometric nonlinearities and its computational evaluation
- B Emek Abali^{1}Email author,
- Wolfgang H Müller^{1} and
- Victor A Eremeyev^{2}
https://doi.org/10.1186/s40759-015-0004-3
© Abali et al.; licensee Springer. 2015
Received: 10 February 2015
Accepted: 5 June 2015
Published: 8 July 2015
Abstract
Background
The theory of linear elasticity is insufficient at small length scales, e.g., when dealing with micro-devices. In particular, it cannot predict the “size effect” observed at the micro- and nanometer scales. In order to design at such small scales an improvement of the theory of elasticity is necessary, which is referred to as strain gradient elasticity.
Methods
There are various approaches in literature, especially for small deformations. In order to include geometric nonlinearities we start by discussing the necessary balance equations. Then we present a generic approach for obtaining adequate constitutive equations. By combining balance equations and constitutive relations nonlinear field equations result. We apply a variational formulation to the nonlinear field equations in order to find a weak form, which can be solved numerically by using open-source codes.
Results
By using balances of linear and angular momentum we obtain the so-called stress and couple stress as tensors of rank two and three, respectively. Since dealing with tensors an adequate representation theorem can be applied. We propose for an isotropic material a stress with two and a couple stress with three material parameters. For understanding their impact during deformation the numerical solution procedure is performed. By successfully simulating the size effect known from experiments, we verify the proposed theory and its numerical implementation.
Conclusion
Based on representation theorems a self consistent strain gradient theory is presented, discussed, and implemented into a computational reality.
Keywords
Background
Traditional constitutive models relating stresses and strains are independent of the size and shape of the continuous body. For example, we model the linear response at small deformations with HOOKE’s law, which has the same form for huge and small structures. Unfortunately, such a simple approach becomes inadequate at the micrometer scale. One of the basic approaches in statics, the so-called EULER-BERNOULLI beam theory, results in inaccurate solutions at very small dimensions. For example, sub-micrometer structures frequently show a stiffer response than predicted by traditional theory. This so-called size effect has been known experimentally for a long time, see, e.g., (Morrison et al. 1939). Formally, the size effect can be modeled by material properties that depend on specimen size. However, in order to include the size effect in a more rational manner, we will generalize the theory of elasticity by means of higher gradient terms. In fact, theories of higher gradients were proposed more than four decades before, cf., (Mindlin and Tiersten 1962; Mindlin and Eshel 1968). They are still under discussion. Moreover, various variants were developed over the last decades, see for an overview (Gurtin et al. (2010), §90). Especially in micromechanics an applicable theory of generalized theory of elasticity becomes necessary, as pointed out by (McFarland and Colton 2005).
We shall discuss deformation and its description in terms of higher gradients of displacement within the framework of continuum mechanics principles. First, we present the balance equations of linear momentum and angular momentum, and identify their flux terms as stress and couple stress, respectively. Second, when deriving constitutive equations for the stress and the couple stress we use tensorial relations. Balance equations in combination with constitutive equations will result in nonlinear field equations. Third, in order to solve these field equations we generate a weak form by using a variational formulation. For the weak form discretization in time is performed by making use of the finite difference method. For discretization in space the finite element method is used. Fourth, we implement a code in Python, see (Jones et al. 2001), by using a novel collection of open-source packages distributed under the FEniCS project, see (Logg et al. 2011). We publish the code in (Abali 2015) under GNU Public license as stated in (Gnu Public 2007) in order to encourage further studies.
Methods
Governing equations
In local continuum mechanics it is assumed that the particles interact within the local neighborhood, where the distance becomes infinitesimal such that the first gradient describes the behavior of material accurately. We can generalize the behavior by including the second gradient, which enables an interaction of particles in a greater neighborhood. This theory is nonlocal and we need different equations restricting the first and the second gradients.
where we have introduced the LEVI-CIVITA symbol, ε _{ ijk }. The flux of linear momentum, σ _{ jk }, is the CAUCHY stress tensor. Following (Müller (1973), Ch. II, § 2.d) we can multiply the balance of linear momentum in its local form by ε _{ ijk } x _{ j } and subtract the result from the balance of angular momentum for acquiring a balance of spin. The production term of the spin reads ε _{ ijk } σ _{ jk }. For non-polar media the spin and its production vanish, i.e., s _{ i }=0 and ε _{ ijk } σ _{ jk }=0. This assumption leads to a symmetric CAUCHY stress tensor, σ _{ ij }=σ _{ ji }. A non-polar medium has no intrinsic spin such that the continuum possesses three degrees of freedom given by the displacement, u _{ i }. For structures on the macroscale the balance of linear momentum is sufficient for calculating the displacement. The balance of angular momentum is automatically satisfied by a symmetric CAUCHY stress tensor, in other words, the flux of angular momentum is assumed to vanish. For structures on the microscale this assumption must be rediscussed and a model for the flux of angular momentum needs to be implemented.
The skew-symmetric form was presented in a similar way in (Mindlin and Eshel 1968; Toupin 1962), (Truesdell and Toupin (1960), Sect. 205). However, the starting point and the motivation are different here.
These equations of motion include supply terms, f _{ i }, l _{ ij }, to be given and flux terms, P _{ ij }, μ _{ ijk }, to be defined with respect to the displacement (or its gradient). Only then Eqs. (21) are closed and can be solved.
Constitutive relations
Hence the most general form of the couple stress or the flux of spin for linear elasticity has nine phenomenological constants. Quite often two more assumptions are made. First, one takes μ _{ ijk }≈μ _{ jik } for granted. Second, one assumes that μ _{ ijk } E _{ i j,k } is a part of the (deformation) energy, such that D _{ ijklmn }=D _{ lmnijk } holds. Under these assumptions nine constants reduce to five material constants, see (dell’Isola et al. (2009), Eqs. (3.1)–(3.7)) and for an overview of such theories refer to (Askes and Aifantis (2011), Sect. 2). We try to avoid introducing assumptions restraining the formulation to specific type of materials.
In a heterogeneous material the material parameters, α, β, γ, can depend on position and they may also depend on temperature. Here we will implement them as constants and investigate their roles in deformation. The constitutive Eq. (31) for the couple stress tensor and Eq. (26) for the stress tensor will be implemented in the numerical investigation.
Therefore, the constitutive Eq. (32) is a special choice of the proposed relation in Eq. (31). Of course the assumption α=β is difficult to justify. Thus we will use the more general formulation given by Eq. (31).
in a numerical computational environment that allows us to comprehend the role of the parameters α, β, γ.
Computational approach
The weak form, F, is of second-order in space regarding the displacement field. Therefore, we choose finite elements of the continuous GALERKIN type of second polynomial degree. In other words, the displacements and also their test functions are from a HILBERT space, u _{ i },δ u _{ i }∈H ^{2} as described in (Hilbert 1902). Moreover, their gradients have to exist, i.e., more specifically the solution space is a SOBOLEV space within the finite domain, referred to as finite elements. Elements are discrete subdomains, Ω ^{ i }∩Ω ^{ j }={}, ∀i≠j, which collectively constitute the region, \(\sum \Omega ^{e} = {\mathcal {B}_{0}}\), where the computation takes place.
This approach is fully automatized by using a symbolic derivation, see (Alnaes and Mardal 2010). Therefore, the only necessary input is the weak form given in Eq. (41). All 3D-visualizations are realized by using ParaView.^{1} All 2D-plots were created by MatpPlotLib packages, see (Hunter 2007), developed for NumPy, see (Oliphant 2007). The code used for solving the examples in the next section is published in (Abali 2015) under GNU public license as declared in (Gnu Public 2007).
Results
The initial shape is denoted by black lines. The classical beam bending (without couple stress) is colored in gray for comparison. The parameter α=−1 (red) has an insignificant effect relative to the parameters β=−1 (green) and γ=−1 (blue). The green and blue colored deformations present an additional bending, such that the amount of bending on yz-plane decreases. In other words, the beam responds stiffer to shear loading in case of existing β or γ parameters.
The initial geometry is again denoted by black lines, we have tilted the geometry for better visualization. The gray deformation is the classical stretching without couple stress. The effect of α (red) is significant again by causing an additional bending motion. Relative to the effect of α the effects of β and γ can be neglected.
The initial shape can be seen in black lines in the front view. In this case γ (blue) causes the most significant deviation from the classical solution (gray) without couple stress.
By observing the three loading cases we can conceive possibilities for measuring the parameters, α, β, γ. During shear loading the effects due to the α and γ parameters are smaller than the effect of β, such that it may be neglected. For tensile loading the effects of β and γ are smaller than α and may be ignored. In torsion the effect of α is significant and the effects of β and γ may be neglected. Under these simplifications the parameter α (red) can be measured by a tensile test by assuming that green and blue deformations are the same as the gray deformation in Fig. 2. The parameter β (green) can be measured by a shear test with the simplification that the red, blue, and gray deformations are the same in Fig. 1. The parameter γ (blue) could be measured by a torsion test under the assumption that the red and green deformations in Fig. 3 are the same as the gray deformation.
Variation of YOUNG’s modulus predicted by the EULER-BERNOULLI beam theory for the ratio ℓ/h=ℓ/b=30 in case of changing the length of the beam
ℓ in μm | \(u_{z}^{\text {max}}\) in μm | E in mN/ μm^{2} = GPa |
---|---|---|
100 | 495.19·10^{−3} | 73 |
60 | 176.05·10^{−3} | 74 |
20 | 17.72·10^{−3} | 81 |
10 | 4.11·10^{−3} | 88 |
5 | 0.96·10^{−3} | 94 |
As discussed previously the stiffening behavior is due to additional bending resulting from the couple stress. However, this bending does not affect the curvature. We have observed this behavior by plotting the normalized (with respect to the tip deflection) z-displacement of each beam. Since the curvature remains the same, we omit to present the results.
We emphasize that the material constant, E, does not change in reality. This example demonstrates that the beam when treated as a continuous body by using strain gradient elasticity responds stiffer than predicted by the EULER-BERNOULLI beam theory.
Conclusion
We have briefly outlined strain gradient elasticity from a continuum mechanics perspective. Starting from the balances of momenta we have obtained the so-called stress and couple stress tensors (of rank two and three, respectively). By applying general tensor relations we have obtained the necessary constitutive equations for the stress and for the couple stress. It is significant that we have proposed a couple stress with three material parameters, viz., α, β, and γ. In order to comprehend their impact during deformation we have implemented a numerical solution procedure where the discretization in time has been combined with the finite difference method. The discretization in space was realized with the finite element method. By simulating different loading cases we analyzed the couple stress parameters. We also verified the proposed theory qualitatively by establishing a simulation of the size effect.
There have been three main difficulties that we have overcome with some assumptions and left their discussions to further studies. The first difficulty arises by motivating a flux of spin in a non-polar medium. Since spin fails to exist in a non-polar medium and since we have assumed that the CAUCHY stress tensor is symmetric (so that the spin production vanishes), it is rather difficult to justify why the flux of angular momentum (couple stress) should exist. Nonetheless, our objective has been the modeling of couple stress for a non-polar medium. The second difficulty lies in determining a description for a measurement procedure for the material parameters in the proposed couple stress, namely α, β, γ. We have discussed their possible measurement after some assumptions, where α is determined by tensile, β by shear, and γ by torsion. However, the correctness of simplifications based on these assumptions is difficult to test. The third difficulty arises by varying the material parameters in order to comprehend their roles quantitatively. Their effects seem to be counter-intuitive and difficult to explain in a straightforward way. Numerical problems arise by choosing positive or greater values for the parameters. Unfortunately, we could not find general conditions in order to restrict the possible values of parameters. For using positive definiteness or thermodynamical laws we need to define the energy due to the spin. Spin is assumed to vanish and the stored energy is not uniquely defined for strain gradient theory. Therefore, the verification of the chosen parameters, and thus, the validation of presented results seem to be more difficult than expected. Any quantitative verifications by using experiments have been left to further research.
Endnote
^{1} http://www.paraview.org
Appendix
since the arguments are arbitrary. The constant c _{1} is called the material parameter in a constitutive equation relating two tensors of rank one.
Declarations
Authors’ Affiliations
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