Size-dependent energy release rate formulation of notched beams based on a modified couple stress theory
Introduction
Fracture toughness, i.e. resistance of a structure to fracture in presence of flaws such as cracks, voids, inclusions, is one of the most important design parameter in structural analysis. one of the main issues in fracture mechanics is determination of the strain energy release rate acting as the crack extension force from which the stress intensity factors can be calculated via Irwin’s relation [1].
In order to determine the energy release rate or stress intensity factor in practical problems, the researchers have employed some methods such as experimental [2], [3], numerical [4], [5] and analytical [6], [7] methods. Analytical methods for estimation of stress intensity factor in the linear elastic fracture mechanics approach are considered as [8]: the direct methods, the energy based methods, the singularity function methods, the superposition method and the boundary integral equation methods.
A remarkably simple method for close approximation of energy release rate, G in notched beams was discovered by Kienzler and Herrmann [9] and Herrmann and Sosa [10]. The method was derived from a certain unproven hypothesis (postulate) regarding the energy release when the thickness of the fracture band is increased [11].
Due to the vast applications of Micro-Electro-Mechanical-Systems (MEMS) such as microresonators [12], micropumps [13] and Atomic Force Microscopes (AFM) [14], many researchers have been studying the behavior of the micromechanical components used in MEMS. Since these components usually work at high operating frequencies, they are subjected to very high numbers of fatigue cycles during their life. Hence, investigation of fatigue mechanism in MEMS seems to be crucial. To that end, the calculation of strain energy release rate for MEMS mechanical components would be helpful.
The experimental observations indicate that the mechanical behavior of micro/sub-micro-scale structures is size-dependent that cannot be justified by the classical continuum mechanics [15], [16], [17]. Since the attempts of the classical continuum theories has been in vain to predict and justify the small scale effect leading to size-dependency in micromechanical elements, some non-classical continuum theories such as the non-local theory, the strain gradient theory and the couple stress theory have been emerged and developed during past years. By considering higher-order stresses known as the couple stress in addition to the classical stresses, the couple stress theory has been developed [18], [19], [20]. In this non-classical theory, there exists two material length scale parameter beside the two well-known Lamé constants which enables the theory to capture the size-effect. The couple stress theory has been employed by researchers in order to investigate the size-dependency in micro-scaled mechanical components (e.g. see [21]).
By utilizing the equilibrium equation of the moments of couples in addition to the two traditional equilibrium equations of forces and moments of forces, a modified couple stress theory has been proposed by Yang et al. [22]. The new higher-order equilibrium equation leads to the symmetry of the couple stress tensor which consequently makes the constitutive equation to have only one additional material length scale parameter. The aforementioned length scale parameter relates the couple stresses to the curvatures of the continuum. It is indeed a material characteristic different for each individual material that can be determined by performing some standard experimental tests. Some of the standard tests can be mentioned as micro-torsion test, micro-bending test and micro/nano indentation test [23], [24], [25]. By exposing the micro-scale samples with different thicknesses or diameters to the bending or torsion loads, and then curve-fitting the obtained graphs delineating the normalized bending stiffness or normalized torsional rigidity versus the sample thickness or diameter, the length scale parameter can be achieved for a specific material. In addition, the length scale parameter can also be related to the dislocation-based physical quantities such as barrier strength of the boundary for slip transmission, Burger vector length and shear modulus [17], [23], [24], [25].
The modified couple stress theory has been utilized by researchers to develop new beam, plate and bar theories capable of capturing the size effect that can be outlined as: Linear homogenous Euler–Bernoulli beam [26], [27], Linear homogenous Timoshenko beam [28], Nonlinear homogenous Euler–Bernoulli beam [29], Nonlinear homogenous Timoshenko beam [30], Linear functionally graded Euler–Bernoulli beam [31], Linear functionally graded Timoshenko beam [32], Nonlinear functionally graded Timoshenko beam [33], Functionally graded higher order beam theory [34] and Functionally graded plates [35], [36].
Moreover, the modified couple stress theory has been employed in order to investigate the characteristics of micro-scale structures such as: electrostatically actuated microbeams [37], microtubules [38], embedded microbeams carrying moving particles [39] and fluid-conveying microtubes [40].
In order to investigate the crack propagation and fatigue behavior of micromechanical components used in MEMS or macro scale structures made of material whose aggregate size is comparable with their dimensions, modeling the small scale effect and deriving the size-dependent strain energy release rate for these components seems to be inevitable. There have been many experimental investigations on the fracture toughness and energy release rate of size-dependent structures [4], [41], [42]. But only a few works have been done to study the effect of length scale parameter on fracture characteristics, analytically [43].
In this article, the size-dependent strain energy release rate for the problem of three-point bending in notched beams is obtained using the procedure established by Kienzler and Herrmann [9]. The formulation is derived for Euler–Bernoulli and Timoshenko beam theories utilizing the modified couple stress theory.
Section snippets
Preliminaries
In the modified couple stress theory, the strain energy density W for a linear elastic isotropic material in infinitesimal deformation is written as [27]wherein which σij, εij, mij and χij denote the components of the symmetric part of stress tensor σ, the strain tensor ε, the deviatoric part of the couple stress tensor m and the symmetric part of the curvature tensor χ,
Bending of a size-dependent beam with an edge crack
In this section, the strain energy release rate of modified couple stress Euler–Bernoulli and Timoshenko cracked beams are developed for bending case. To that end, as it is observed in Fig. 1, a size-dependent beam having an edge crack with length a located at x = 0 subjected to distributed external lateral force q and couple m is considered here. In the rest of this section, the stain energy release rate of modified couple stress Euler–Bernoulli and Timoshenko beams is formulated following the
Energy release rate formulation
Henceforward, during a case study, it will be delineated that how the abovementioned formulation can be applied to a real-case three-point bending problem. To that end, consider an initially straight size-dependent beam with length 2L having a uniform rectangular cross-section with height h and width subjected to a three-point bending load P (see Fig. 3). In addition, it is considered that the beam has an edge crack with length a located at x = 0. Solution of this problem is equivalent to
Conclusion
In this paper utilizing the modified couple stress theory, a non-classical continuum theory capable of capturing the size effect, the energy release rate of Euler–Bernoulli and Timoshenko notched beams has been formulated. Afterward, the normalized energy release rate is determined as a function of the ratio of the length scale parameter to the beam depth for three-point bend condition. The results can be outlined as:
- •
The fracture behavior of materials is size dependent, i.e. it is function of
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2022, Mechanics of MaterialsCitation Excerpt :The modified couple stress theory (MCST) possesses only one length-scale parameter which relates the couple stresses of a continuum to its curvature. This theory was used to present size-dependent mechanical models of bending (Ma et al., 2008), vibration (Kong et al., 2008), buckling (Akgöz and Civalek, 2011), and energy release rate (Sherafatnia et al., 2014) of small-scale structures. On the other hand, in recent years, some studies on manipulating the propagation of waves in PCs were performed at small scales.
The material length scale parameter used in couple stress theories is not a material constant
2018, International Journal of Engineering ScienceSize dependent energy release rate of notched FGM beams based on a modified couple stress theory
2016, Materials Today: ProceedingsA damped sandwich beam model based on symmetric-deviatoric couple stress theory
2015, International Journal of Engineering ScienceCitation Excerpt :Symmetric-deviatoric couple stress theory has been interested by many researchers to develop governing equations of microstructure as well as to investigate the influence of size effect phenomena on behavior of microsystems. Some of these works can be listed as: an Euler–Bernoulli beam model for static bending and dynamic analysis by Park and Gao (2006) and Kong, Zhou, Nie, and Wang (2008), respectively, a Timoshenko beam model for static bending and dynamic analysis by Ma, Gao, and Reddy (2008), a micromechanics model of hexagonal honeycomb structures by Park and Gao (2008a), a Kirchhoff plate model for static bending analysis by Tsiatas (2009), a nonlinear Euler–Bernoulli beam model for static bending, free oscillation and post buckling analysis by Xia, Wang, and Yin (2010), functionally graded Euler–Bernoulli and Timoshenko beam models for static and free vibration analysis by Asghari, Ahmadian, Kahrobaiyan, and Rahaeifard (2010, 2011), functionally graded Euler–Bernoulli, Timoshenko and Reddy beam models for buckling analysis by Nateghi, Salamat-Talab, Rezapour, and Daneshian (2012), single-layered graphene sheets in an elastic matrix for free vibration analysis by Akgöz and Civalek (2012), nonlinear formulation of functionally graded Kirchhoff and Mindlin microplates for static bending, buckling and free vibration by Thai and Choi (2013), axially functionally graded tapered Bernoulli–Euler microbeam model for free vibration analysis by Akgöz and Civalek (2013), a Mindlin plate finite element for static bending, buckling and dynamic analysis by Zhang, He, Liu, Gan, and Shen (2013), dynamic stability analysis of functionally graded higher-order shear deformable microshells by Sahmani et al. (2013), axisymmetric nonlinear free vibration of size-dependent functionally graded annular microplates by Ke et al. (2013), a size-dependent formulation for energy release rate of notched microbeams by Sherafatnia, Kahrobaiyan, and Farrahi (2013), nonlinear geometrically imperfect beam for dynamic analysis by Farokhi, Ghayesh, and Amabili (2013), dependency of material length scale parameter on higher-order elasticity theory unlike Lame’s constants by Dehrouyeh-Semnani (2015), microinertia effects on the dynamic characteristics of microbeams by Fathalilou, Sadeghi, and Rezazadeh (2014), nonlinear static bending and free vibration analysis of microbeams based on the nonlinear elastic foundation by Şimşek (2014), a comparison study between constitutive and Euler–Bernoulli beam models by Dehrouyeh-Semnani (2014), nonlinear thermal stability and vibration of pre/post-buckled temperature- and microstructure-dependent functionally graded beams resting on elastic foundation by Komijani, Esfahani, Reddy, Liu, and Eslami (2014), simulation of fluid–solid interaction in a microchannel using a combination of Lattice-Boltzmann method and finite element method by Esfahanian, Dehdashti, and Dehrouyeh-Semnani (2014a, 2014b), nonlinear modeling of a curved microtubes conveying fluid for free oscillation analysis by Tang, Ni, Wang, Luo, and Wang (2014), a shear deformable functionally graded cylindrical shell model for free vibration analysis by Beni, Mehralian, and Razavi (2014), orthotropic Kirchhoff-type skew microplate for static bending, dynamic and buckling analysis by Tsiatas and Yiotis (2014), a yield criterion model by Kahrobaiyan, Rahaeifard, and Ahmadian (2014), a functionally graded micro-ring segment model for the analysis of coupled extensional–flexural waves by Mustapha (2014), nonlinear vibrations analysis of functionally graded Mindlin microplates by Ansari, Faghih Shojaei, Mohammadi, Gholami, and Darabi (2014), axisymmetric postbuckling analysis of functionally graded annular Mindlin microplates using the physical neutral plane by Ke, Yang, Kitipornchai, and Wang (2014), a discussion on evaluation of material length scale parameter based on micro-cantilever test by Dehrouyeh-Semnani and Nikkhah-Bahrami (2015a), isotropic and functionally graded sandwich microbeam with elastic core for static bending, free vibration and buckling analysis by Thai, Vo, Nguyen, and Lee (2015), compositelaminated microbeam model based on different beam theories for buckling and dynamic analysis by Mohammad Abadi and Daneshmehr (2014, 2015), nonlinear vibration analysis of microplates by Ghayesh and Farokhi (2015b), dynamics and instability of current-carrying microbeams in a longitudinal magnetic field by Wang, Liu, and Dai (2015), the influence of size-dependent shear deformation on mechanical behavior of micro-structures-dependent beam using finite element method by Dehrouyeh-Semnani and Nikkhah-Bahrami (2015c), and internal energy transfer in dynamical behaviour of Timoshenko microarches by Ghayesh and Farokhi (2015a). The aim of this investigation is to develop the mathematical model of the damped sandwich microbeam as well as to study the influence of size effect on vibration characteristics of the microstructure.
The influence of size-dependent shear deformation on mechanical behavior of microstructures-dependent beam based on modified couple stress theory
2015, Composite StructuresCitation Excerpt :Modified couple stress theory has been widely used by many researchers to develop the governing equations of microstructures-dependent beam and study the size-dependent mechanical behavior of them. Some of works associated with microstructures-dependent beam based on modified couple stress theory can be listed as: an Euler–Bernoulli beam model for static bending analysis and comparison with experimental data by Park and Gao [21], an Euler–Bernoulli beam model for free vibration and buckling analysis by Kong et al. [22] and Akgöz and Civalek [23], respectively, a Timoshenko microbeam model for static bending and free vibration analysis by Ma et al. [24], buckling analysis of microbeams based on different beam theories with general boundary conditions by Mohammad-Abadi and Daneshmehr [25], thermal effect on natural frequencies and critical buckling load of Euler–Bernoulli and Timoshenko microbeams by Ke et al. [26], thermal effect on static deflection, natural frequencies and critical buckling load of third shear deformation microbeam by Daneshmehr et al. [27], a composite microbeam model based on a new modified couple stress theory obtained by defining an asymmetric curvature tensor by Chen et al. [28], buckling and dynamic analysis of micro composite laminated Euler–Bernoulli and Timoshenko beams by Mohammad-Abadi and Daneshmehr [29,30] based on the new modified couple stress theory proposed by Chen et al. [28], Wave propagation characteristics of a twisted micro scale beam by Mustapha and Zhong [31], energy release rate formulation of notched Euler–Bernoulli and Timoshenko microbeams and comparison with experimental data by Sherafatnia et al. [32], a generalized thermoelasticity model for Timoshenko microbeams by Taati et al. [33], in-plane and out-of-plane motion characteristics of microbeams with modal interactions by Ghayesh et al. [34], an investigation of incorporating the Poisson effect in microbeam models using available experimental data by Dehrouyeh-Semnani and Nikkhah-Bahrami [35], a comparison study between Euler–Bernoulli and constitutive plane-strain microbeam models by Dehrouyeh-Semnani [36], dynamic characteristics of Euler–Bernoulli microbeams considering micro-inertia effect by Fathalilou et al. [37], two-dimensional simulation of fluid–structure interaction in a microchannel by a coupled lattice Boltzmann-finite element approach considering Knudsen number by Esfahanian et al. [38,39], static and dynamics analysis of microbeams based on a new shear deformation beam theory by Darijani and Mohammadabadi [40], dynamics and instability of current-carrying microbeams in a longitudinal magnetic field by Wang et al. [41], three-dimensional vibration analysis of curved microbeams under fluid force induced by external flow by Tang et al. by [42], a discussion on evaluation of material length scale parameter based on micro-cantilever test by Dehrouyeh-Semnani and Nikkhah-Bahrami [43], nonlinear static bending, free oscillation and post buckling of Euler–Bernoulli microbeams by Xia et al. [44], nonlinear static bending and free oscillation of Timoshenko microbeams by Asghari et al. [45], nonlinear forced vibration of Timoshenko microbeams by Ghayesh et al. [46], nonlinear dynamics of geometrically imperfect microbeams by Farokhi et al. [47], nonlinear static bending and thermal post-buckling analysis of Euler–Bernoulli microbeams by Wang et al. [48], nonlinear free oscillation and divergence instability of micropipe conveying fluid by Yang et al. [49], nonlinear thermal stability and vibration of pre/post-buckled temperature- and microstructure-dependent functionally graded beams resting on nonlinear elastic foundation by Komijani et al. [50], nonlinear three-dimensional modeling of curved microtubes conveying fluid for vibration analysis by Tang et al. [51], nonlinear static and free vibration of microbeams on nonlinear elastic foundation by Şimşek [52], vibration of post buckled Euler–Bernoulli micro scale beams by Ansari et al. [53], functionally graded Euler–Bernoulli and Timoshenko microbeam models for static and vibration analysis by Asghari et al. [54,55], static and dynamic analysis of third-order shear deformation functionally graded microbeams by Salamat-talab et al. [56], buckling analysis of functionally graded microbeams based on different beam theories by Nateghi et al. [57], dynamics stability analysis of functionally graded microbeams by Ke et al. [58], thermal effect on buckling and free vibration of functionally graded Euler–Bernoulli and Timoshenko microbeams by Nateghi and Salamat-talab [59], buckling analysis of a functionally graded microbeam embedded in Wrinkle–Pasternak elastic medium using a unified higher order beam theories by Şimşek and Reddy [60], free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams by Akgöz and Civalek [61], static and free vibration analyses of small-scale functionally graded beams possessing a variable length scale parameter using different beam theories by Aghazadeh et al. [62], thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium by Akgöz and Civalek [63], mechanical behavior analysis of functionally graded sandwich microbeam with functionally graded and homogeneous core[64], nonlinear analysis of functionally graded piezoelectric actuator based on Timoshenko beam theory and von Kármán nonlinearity by Komijani et al. [65,66]. Moreover, Dehrouyeh-Semnani [67] showed that material length scale parameters is dependent on the higher-order continuum theory unlike Lame constants and Ashoori and Mahmoodi [68] presented a modified version of couple stress theory in general curvilinear coordinates.