A size-dependent yield criterion

Abstract

In this paper, a size-dependent non-classical yield criterion is introduced on the basis of the modified couple stress theory in order to capture the size-dependency of the micro-scale structure yielding loads where the attempts of the classical yield criteria such as the von-Mises have been in vain. In order to develop the new yield criterion, the deviatoric part of the micro-scale structure strain energy density, including both classical and non-classical parts, is equated to the deviatoric strain energy density of a macro-size tensile-test sample at the yielding point. For bending of microbeams and torsion of microbars, the size-dependent yielding moments have been determined based on the new criterion. The results of the present criterion are compared to the experimental data and the von-Mises based classical results. A good agreement is observed between the present and the experimental results while the error of using the classical von-Mises criterion is considerable. In addition, it is observed that unlike the von-Mises criterion, the new criterion successfully predicts the size-dependency of the experimental data. It is noted that the results of the new criterion approach the von-Mises results as the size of the structure increases.

Introduction

Nowadays, micro-scale structures are used as the key building blocks of Micro-Electro-Mechanical-Systems (MEMS) such as electro-statically-actuated micro-switches (Attia, Tremblay, Laval, & Hesto, 1998), micro-resonators (Tilmans & Legtenberg, 1994), micro-mirrors (Moeenfard & Ahmadian, 2012) and Atomic Force Microscopes (AFMs) (Kahrobaiyan et al., 2010, Kahrobaiyan et al., 2011) which are widely employed in various fields of technology especially in micro-engineering.

Many experimental researches performed on the mechanical behavior of micro-scale structures proved that the behavior of these components is different from that predicted by the classical continuum theory. According to these experiments, the normalized stiffness of the micro-scale structures, which the classical continuum theory predicts to be independent of the structure size, is significantly size-dependent. These experiments have also revealed that the classical continuum theory underestimates the stiffness of micro-scale structures. The most important experimental researches on this issue can be outlined as: Fleck et al., some torsion tests on the thin Copper wires (Fleck, Muller, Ashby, & Hutchinson, 1994); Stolken et al., bending test on Nickel microbeams (Stölken & Evans, 1998); Lam et al., bending of Epoxy microbeams (Lam, Yang, Chong, Wang, & Tong, 2003); McFarland and Colton, bending of Polypropylene microcantilevers (McFarland & Colton, 2005); Moreau, bending of Nickel thin foils (Moreau et al., 2005); Motz et al., bending of Copper microbeams (Motz, Schöberl, & Pippan, 2005) and Son et al., bending of Aluminum thin foils (Son, Jeong, & Kwon, 2003).

In the abovementioned experiments, the size dependency is detected and it is concluded that not only the mechanical response, but also the yield strength of the micro-scale structures is size-dependent. According to these experiments, there is a considerable gap between the yielding load predicted by the classical continuum theory and those observed experimentally; noted that the gap increases as the size of the structure decreases.

The classical continuum theory is neither able to justify the size dependency happening in micro-scale structures nor capable of modeling them accurately. However, the non-classical continuum theories such as the couple stress and the modified couple stress theory can successfully model this phenomenon in micro-scale structures. The couple stress theory has been introduced and developed by some researchers such as Koiter, Ejike, Mindlin and Tiersten in early 60s (Ejike, 1969, Koiter, 1964, Mindlin and Tiersten, 1962). In this theory beside the two classical material constants (i.e. the elastic modulus and Poison’s ratio) two additional material parameters are appearing which enables the theory to predict and model the size-dependency in micro-scale structures. Asghari et al. developed a Timoshenko beam model based on this theory to investigate the size-dependent static behavior of microbeams (Asghari, Kahrobaiyan, Rahaeifard, & Ahmadian, 2011). They concluded that the bending stiffness of the new model is greater than those evaluated based on the classical Timoshenko beam theory.

Yang, Chong, Lam, and Tong (2002) performed a modification on the couple stress theory and presented the modified couple stress theory. They utilized the equilibrium equation of moments of couples in addition to two classical equilibrium equations i.e. the equilibrium equation of forces and moment of forces. Soon after that, this new theory became a popular non-classical theory. Many researchers utilized the modified couple stress theory to develop beam and plate models as well as investigate the size-dependent phenomena in microsystems. Some of these works on developing beam and plate models can be listed as: linear homogenous Euler–Bernoulli beam model by Park and Gao (2006) and Kong, Zhou, Nie, and Wang (2008); linear homogenous Timoshenko model by Ma, Gao, and Reddy (2008); nonlinear homogenous Euler–Bernoulli beam model by Xia, Wang, and Yin (2010) and Kahrobaiyan, Asghari, Hoore, and Ahmadian (2012); nonlinear homogenous Timoshenko beam model by Asghari, Kahrobaiyan, and Ahmadian (2010); linear functionally graded Euler–Bernoulli and Timoshenko beam models by Asghari et al., 2010, Asghari et al., 2011 and linear homogenous Kirchhoff plate model by Tsiatas and Yiotis (2010).

In addition to developing beam and plate models, mechanical behavior of microsystems have also been investigated and analyzed based on the modified couple stress theory. Some of these works can be expressed as: investigating the vibration of fluid-conveying microtubes by Wang (Wang, 2010); analyzing the buckling of micro-tubules by Fu and Zhang (2010); studying the dynamic characteristics of Atomic Force Microscopes (AFMs) by Kahrobaiyan, Asghari, Rahaeifard, and Ahmadian (2010) and investigating the size-dependent static behavior of electro-statically actuated micro-cantilevers and micro-bridges by Rahaeifard et al., 2011, Rahaeifard et al., 2012.

To sum up, according to the inability of the classical continuum theory to capture the size dependency of micro-scale structures mechanical behavior and inaccuracy of the classical theory evaluations of the micro-scale structures stiffness, the non-classical continuum theories such as the modified couple stress theory have been emerged and different kinds of beam and plate models have been developed based on these theories.

Regarding the experimental observations on the size-dependent nature of the micro-scale structures yielding load (Fleck et al., 1994, Moreau et al., 2005, Motz et al., 2005, Son et al., 2003, Stölken and Evans, 1998) and the fact that the attempts of the classical continuum theory to justify the size-dependency and to accurately evaluate the yielding load of the micro-scale structures have been in vain, introducing a new yield criterion based on the non-classical continuum theories capable of capturing the size-dependency seems to be essential. Hence, in this paper, a new yield criterion is developed based on the modified couple stress theory. There are plenty of works in which failure criteria are developed on the basis of the classical continuum theory where the microstructure issues such as continuous distribution of defects, grain shape evolution, crystallographic texture, type of microstructure, grain size, dislocation density, intragranular microstructure evolution, geometrically necessary boundaries and incidental dislocation boundaries are accounted for (Carvalho Resende et al., 2013, M’Guil et al., 2011, Stünitz et al., 2010, Zattarin et al., 2004). Most of the aforementioned failure criteria are based on the empirical relations. Although these failure criteria consider the microstructure issues, they are unable to justify the size-dependent yielding happening in micro-scale-structures. It is observed that for mechanical structures whose characteristic sizes are in the order of micron and sub-microns; such as microbeams, microplates and microbars; the yielding phenomenon is extremely size-dependent, i.e. as the characteristic size of the micro-scale-structure such as the thickness or radius decreases, the yielding load exerted to the structure increases (Fleck et al., 1994, Moreau et al., 2005, Motz et al., 2005, Son et al., 2003, Stölken and Evans, 1998). The abovementioned size-dependent yielding phenomenon cannot be explained by the existing yield criteria even those that account for microstructure effects. However, contrary to the aforementioned classical yield criteria that model the microstructure important parameters, the new yield criterion is based on a well-known non-classical continuum theory i.e. modified couple stress theory, instead of empirical relations and is successfully able to justify the size-dependent yield phenomenon explained before. In order to develop the new yield criterion, the deviatoric part of the micro-scale structure strain energy density (distortion energy), including both classical and non-classical parts, is set to be equal to the deviatoric strain energy density of a macro-size experimental sample at its yielding point during a simple tensile test. The yielding loads of some micro-scale structures subjected to the torsional and bending loads are derived based on the new size-dependent yield criterion and the results are compared to the experimental data. It is indicated that there is a good agreement between the results of the new criterion and those obtained from the experimental tests while the error of using the classical yield criteria such as the classical von-Mises criterion will be significant.