On a second-order rotation gradient theory for linear elastic continua

International Journal of Engineering Science 100 (2016) 74–98

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On a second-order rotation gradient theory for linear elastic continua Mohamed Shaat, Abdessattar Abdelkefi∗ Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA

a r t i c l e

i n f o

Article history: Received 22 October 2015 Revised 23 November 2015 Accepted 25 November 2015

Keywords: Higher-order deformation Rotation gradient Couple stress Strain gradient Linear elasticity

a b s t r a c t A second-order rotation gradient theory for non-classical elastic continua is developed. This theory accounts for the higher-order deformation of the material structure where the material particle inside the elastic domain is idealized as a microvolume having three degrees of freedom, namely, a translation, a micro-rotation, and a higher-order micro-deformation. The associated strain energy density is a function of the infinitesimal strain tensor and the first and second gradients of the rotation tensor. It is demonstrated that for materials in nanoscale applications and because of some defects at the material structure level, a higher-order deformation measure may be needed. The second-strain gradient theory has the merit to account for the higher-order deformation of the material particle. However, this theory has limited applications because it depends on 16 additional material constants for isotropic elastic continua. By discussing some physical concepts relevant to the natures of material structures, crystallinity, and amorphousness, the second-strain gradient theory is reduced to the secondrotation gradient theory for certain types of materials. For isotropic materials, the developed second-rotation gradient theory only depends on three additional material constants instead of 16. A continuum model equipped with an atomic lattice model is then proposed to examine the applicability of the available non-classical theories and the applicability of the proposed theory for different types of materials. Published by Elsevier Ltd.

1. Introduction In the classical theory of elasticity, a material body is modeled as a continuum. Each material particle is treated as a mass point that has only three translational degrees of freedom. In the classical theory of linear elasticity, the strain energy density is a function of the infinitesimal strain. On the other hand, for micro-/nano-solids, the material particle has to be represented as a small volume element considering its inner structure to describe the microscopic motion and to account for the microstructure size-dependency. Some higher-order micro-continuum theories have been developed to account for the microstructure effects by introducing additional degrees of freedom with additional deformation measures and material constants to the conventional ones (Chen, Lee, & Eskandarian, 2004; Cosserat & Cosserat, 1909; Edelen, 1969; Eringen, 1966, 1999; Eringen & Suhubi, 1964; Hadjesfandiari & Dargush, 2011; Lam, Yang, Chong, Wang, & Tong, 2003; Mindlin, 1964, 1965; Mindlin & Eshel, 1968; Mindlin & Tiersten, 1962; Polyzos & Fotiadis, 2012; Shaat, 2015; Toupin, 1962; Yang, Chong, Lam, & Tong, 2002). These theories are discussed in details in Section 2.

∗

Corresponding author. Tel.: +1 575 646 6546; fax: +1 575 646 6111. E-mail addresses: [email protected] (M. Shaat), [email protected], [email protected] (A. Abdelkefi).

http://dx.doi.org/10.1016/j.ijengsci.2015.11.009 0020-7225/Published by Elsevier Ltd.

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The main motivation that makes scientists in 1960s–1970s develop the non-classical continuum mechanics theories is the inclusion of the higher-orders of the measures of the underlying classical theories from a mathematical point of view (Mindlin, 1964, 1965; Mindlin & Eshel, 1968; Mindlin & Tiersten, 1962; Toupin, 1962). Recently, few researchers have discussed these existing theories from an atomistic point of view (Chen et al., 2004; Polyzos & Fotiadis, 2012). Although these mathematical and atomistic representations show the merit of these non-classical theories to account for the continuum size effects, such as the residuals of the micro-fields and the nonlocal fields, the physical understanding and the applicability of these non-classical continuum theories for different types of materials are absolutely necessary. Due to the lack of the physical understanding of the applicability of the non-classical theories for different materials, these theories have limited contributions to reflect the mechanics of materials in micro-/nano-scale applications. Another reason, the constitutive equations in the context of these nonclassical classical theories depend on new material constants which are still experimentally unpredictable. The first contribution of the present study is to bring the suitable physical explanations for the available non-classical continuum theories and their applicability for different types of material structures. With the aid of these explanations, these theories properly can be applied for small-size materials. Furthermore, these explanations can help in investigating the suitable experimental setups to measure the additional constants presented in the context of each theory. Most of the existing non-classical continuum theories account only for the first-order deformation of the material particle/material structure. However, in some specific cases (see Section 3.1), the higher-order deformation of the material structure has to be accounted for. The necessity to account for the higher-order deformation of the material structure and the way of modeling it are presented as the second contribution of the present paper. A second-order strain gradient theory has been developed such that the strain energy density is a function of the conventional strain in addition to the first and the second gradients of the strain (Mindlin, 1965). This theory is developed based on mathematical concerns to capture some physical phenomena including surface tension and cohesive forces (Mindlin, 1965). In the present effort, the applicability of the second-strain gradient theory for materials without microstructures is discussed. We showed that the second-strain gradient theory has the merit to account for the higher-order deformation of the material particle. For isotropic materials, the second-strain gradient theory has 18 material constants which make it impractical for elastic field problems. To this end, as a third contribution of the present work, a novel linear elasticity theory in which the strain energy density is a function of the infinitesimal strain tensor and the first and second gradients of the rotation tensor is developed. In this theory, the constitutive equations only depend on three additional material constants that make this theory practical for well-defined elastic field problems with the potential to account for the higher-order deformation of the material structure. This study is organized such that the applicability of the existing micro-continuum theories for different materials is first discussed. Then, the necessity and the modeling of the higher-order deformations of the material structure are investigated. After that, the essential relations of the second-rotation gradient theory are derived and discussed from the continuum and the atomistic points of view representing the limit of application of the theory for materials. The size-dependent linear elasticity for 2D problems, anti-plane problems, and micro/nano beams are then studied. The applicability and the effectiveness of the proposed second-rotation gradient are discussed by comparing the theory to the existing non-classical theories. Finally, a deep discussion on the experimental determination of the additional material constants of the proposed second-rotation gradient theory for different materials is presented. 2. On the applicability of the existing non-classical continuum theories to specific materials The available non-classical continuum theories and their applicability for materials are presented. A continuum model equipped with an atomic lattice model is proposed to illustrate the microstructure effects on the deformation energy of linear elastic isotropic materials. Moreover, the model is used to map each non-classical continuum theory to the corresponding suitable material type. In the context of continuum mechanics, the material particle inside the continuum represents the main unit of the continuum’s material structure. Usually, the material particle represents a single crystal for polycrystalline materials, a single grain for granular materials, or a chain of molecules for amorphous materials. For continua in macro-scale sizes, the material particle (crystal, grain, or chain of molecules) is small enough, compared to the continuum size. Hence, this material particle can be modeled as a mass point and thus the classical theory of elasticity can be applied. On the other hand, when the size of the continuum reduces to micro-/nano-scale sizes, the ratio of the size of the crystal, the grain, or the chain of molecules (material particle) inside the material structure to the size of the continuum increases. Therefore, the material particle should be represented in the context of the continuum theory as a volume element which exceeds the limit of applicability of the classical theories. The proposed continuum model reveals that the elastic domain of material is considered consisting of an infinite number of material particles and each material particle is a microvolume which has a certain inner structure. To model the inner structure of the material particle, the continuum model is equipped with an atomic lattice model to account for the lattice dynamics of this inner structure. This model is used to show the applicability of the available non-classical micro-continuum theories from the material structure concerns. The material particle is considered as a rigid microvolume, or it is allowed to deform according to the applied micro-continuum theory. It should be mentioned that the proposed continuum model can be applied to single crystalline materials by considering the whole continuum consisting of a single crystal with the material particle is the unit-cell inside the crystal structure. Thus, by modeling the unit cell as a volume element and considering the internal phonons of the unit cell, the model can reflect the same results as those of molecular dynamics. Moreover, the proposed model can be applied for polycrystalline materials by modeling

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Fig. 1. The kinematical variables in the context of the proposed continuum model.

the material particle (crystals or grains) as a deformable volume body. In this way, the model can capture the internal phonons of the crystal. 2.1. Continuum model to capture first-order deformation of material structures From the basic concepts of structures of materials, the material structure usually consists of repeated units that affect the different properties of the material. In the case of materials with microstructure (such as polycrystalline materials), the single unit that forms the structure of these materials is the crystal/grain. The crystal inside the material structure consists of smaller repeated units called unit cells and each unit cell contains atoms with a certain arrangement. To illustrate the effects of the microstructure on the deformation energy of elastic continua made of materials with microstructures, a continuum model equipped with an atomic lattice model is proposed. In the context of this model, materials with microstructures are modeled consisting of an infinite number of material particles and each material particle is idealized as a single crystal, as shown in Fig. 1. Each single crystal is modeled as a volume element that contains a large number of unit cells. The unit cell is represented as a mass point because, in some cases, it has very small size in comparison with the crystal size, and the internal phonons of the unit cells have no contribution to the continuum deformation. This is acceptable for solids with one atom per primitive unit cell. For materials with microstructures, such as polycrystalline and granular materials, the micro-medium (material particle/crystal) undergoes a deformation that is completely different from the deformation induced in the surrounding macro-medium (continuum body) (Chen et al., 2004; Mindlin, 1964). Therefore, the unit cells are allowed to translate with a micro-displacement ui which is independent from the macro-displacement ui of the center of the material particle (the whole crystal). The micro-displacement field ui is introduced to capture the acoustic phonons of the unit cells inside the crystal structure. This micro-displacement field is the basic field from which the total deformation energy of the continuum can be calculated. Thus, the proposed continuum model has the merit to account for the acoustic dispersive behavior of the crystal structure. To determine the deformation energy of a continuum with a microstructure, the effects of the unit cells’ displacement (microdisplacement) ui on the deformation of the crystal (material particle) are first captured by introducing a micro-deformation field ψ ij as the micro-gradient of the micro-displacement as follows:

ψi j = ui, j

(1)

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This micro-deformation tensor ψ ij is introduced to capture the effects of the acoustic phonons of the unit cells on the deformation energy of the material particle (crystal). By decomposing ψ ij into its symmetric ψ (ij) and skew-symmetric ψ [ij] parts, a micro-strain tensor εi j and a micro-rotation tensor θij are introduced as follows:

εi j = ψ(i j ) =

 1 ψi j + ψ ji 2

(2)

θij = ψ[i j] =

 1 ψi j − ψ ji 2

(3)

The micro-strain tensor captures the deformability of the material particle/crystal and the micro-rotation tensor captures the rigid-rotations of the material particle/crystal. Thus, in the context of the present model, each single crystal undergoes a deformation (captured by the micro strain tensor εi j ), a rigid rotation (captured by the micro-rotation tensor θij ), and a translational displacement (macro-displacement ui ) as a result of the unit cells translational motions. Second, a relative deformation measure γ ij is introduced to capture the difference between the macro-displacement gradient ui, j and the micro-displacement gradient (micro-deformation ψ ij ) as:

γi j = ui, j − ψi j

(4)

This deformation measure is needed for materials with microstructure to capture the difference between fields inside and outside the crystals. Third, to capture the effects of the micro-deformation (micro-strain εi j and micro-rotation θij ) on the continuum total strain energy, a micro-deformation gradient χ ijk (a macro-gradient of the micro-deformation) is introduced such that:

χi jk = ψi j,k

(5)

Consequently, in addition to the usual macro-strain ɛij , the micro-deformation gradient χ ijk and the relative-deformation tensor γ ij contribute to the continuum total strain energy as follows:

W =F



εi j , γi j , χi jk



(6)

In the context of the present continuum model, the single crystal (material particle) is modeled having three types of degrees of freedom, namely, micro-strain (first-order deformation), micro-rotation, and macro-displacement. Furthermore, the proposed model reflects the influences of the crystal acoustic modes of vibration on the deformation energy of the continuum body. 2.1.1. Materials with microstructure In this section, the available non-classical theories for materials with microstructure are discussed based on the proposed continuum model. Starting with the Micromorphic theory (Eringen, 1999; Eringen & Suhubi, 1964) and the Microstructure theory (Mindlin, 1964), the material particle in the context of these theories undergoes three different types of degrees of freedom, namely, a rigid macro-displacement, a micro-strain, and a rigid micro-rotation. This exactly matches with the proposed continuum model, as shown in Fig. 1. Thus, the Microstructure and Micromorphic theories have the merit to account for the deformability of the crystals, inside the material structure, due to the acoustic modes of vibrations of the unit cell. It should be mentioned that the Micromorphic theory accounts for the kinetics and interactions of atoms, while for infinitesimal deformation and slow motion assumptions, the Micromorphic theory is reduced to the Microstructure theory where the kinetics of atoms is ignored. Another theory, named Cosserat theory (Cosserat & Cosserat, 1909), has been used to measure the effects of the microrotation on the continuum deformation energy with neglecting the possible deformations that could be induced in the material particle. In Cosserat theory, the material particle is considered rigid. Hence, the kinematical quantities are only the particle macro-displacements and micro-rotations, neglecting the micro-strain field εi j . In addition, the micro-rotations of the particle are modeled to be independent of the continuum macro-rotations, θ ij , i.e.

θi j =

 1 ui, j − u j,i 2

(7)

Therefore, Cosserat theory is suitable for materials with stiff, nearly rigid, microstructures accounting for the microstructure rigid rotation effects. To this aim, the micro-rotation tensor θij contributes to the Cosserat continuum deformation energy

through a gradient of the micro-rotation, Ri j = θi, j , where θi = density function for a Cosserat continuum is given by:

W =F



εi j , γ[i j] = θi j − θij , Ri j



1  2 ei jk θk j

is the micro-rotation vector. In this case, the strain energy (8)

Based on the previous discussion, the Micromorphic theory, the Microstructure theory, and Cosserat theory reflect some molecular dynamics where these theories account for the acoustic dispersive phonons for materials with microstructure (crystalline and granular materials). The possible induced deformations in the linear elastic materials with microstructures can be captured through the Microstructure theory or Cosserat theory. The Microstructure theory accounts for the micro-strain and the micro-rotation effects while Cosserat theory models the material particle as a rigid microvolume accounting only for the micro-rotation effects.

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Fig. 2. The kinematical variables in the context of the proposed model for materials without microstructure.

2.1.2. Materials without microstructures Crystalline materials are materials whose atoms, molecules, or ions are ordered and arranged inside the distinct boundary of the crystal. Due to their distinct boundary, the micro-medium inside the crystals is completely different from the surrounding macro-medium. On the other hand, non-crystalline materials have amorphous structure which has a disordered arrangement of atoms or molecules with no boundaries. In other words, for non-crystalline materials, the micro-medium is merged with the macro-medium. Thus, the proposed continuum model and the Microstructure theory could be applied for amorphous materials by putting the relative deformation measure γi j = 0. This introduces the first-strain gradient theory (Mindlin & Eshel, 1968) in which the material particle is modeled as a microvolume consisting of a finite number of molecules and each molecule undergoes a micro-displacement ui , as shown in Fig. 2. Due to the amorphous structure, the distinction between the micro-motion ui and the macro-motion ui can be neglected. Therefore, following the proposed continuum model, the strain energy density is a function of the macro-strain and the second-gradient of the macro-displacement ui ,

W =F



εi j , uk,i j



(9)

where uk, ij is introduced in the context of the first-strain gradient theory as the suitable measure for the deformations induced in the continuum due to the micro-rotations and the micro-strains of the material particle. Micro-strains and micro-rotations of the material particle (chain of molecules) can be obtained from the first gradient of the micro-displacement field. Usually, materials with amorphous structures involve long-range interactions. Eringen’s nonlocal theory (Edelen, 1969; Eringen, 1966) is the suited for. In the context of Eringen’s nonlocal theory, the material particle is idealized as a point (molecule) without microstructure; thus, each specific particle (molecule) receives energies diffused due to its long-range interactions with all other particles in the continuum volume. The summation of these diffused energies to a specific particle induces an additional residual-like stress at this particle which represents the nonlocality effects. The first-strain gradient theory and the second-strain gradient theory have some nonlocal features but with shorter nonlocal ranges. The first-strain gradient theory accounts for oneneighbor interaction at a specific particle while the second-strain gradient theory accounts for the two-neighbor interactions (Polyzos & Fotiadis, 2012). Another set of theories which account for the micro-rotation effects without the distinctions between the micro-rotations and the macro-rotations are the couple stress theories (Hadjesfandiari & Dargush, 2011; Mindlin & Tiersten, 1962; Toupin, 1962; Yang et al., 2002). By setting γ[i j] = 0, the Cosserat theory is reduced to the classical couple stress theory (Mindlin & Tiersten, 1962; Toupin, 1962). In addition to the infinitesimal macro-strain ɛij , a rotation gradient tensor Ri j = θi, j is introduced to capture the micro-rotation effects. Thus, in the context of the couple stress theory, the material particle undergoes a macro-displacement ui and a rigid micro-rotation equals to the macro-rotation θ ij . For materials without microstructures, the first-strain gradient theory and the couple stress theory can be applied where the distinction between the micro-medium and the macro-medium is neglected. For more illustration, the first-strain gradient theory accounts for the possible deformations induced in the material particle (micro-strain and micro-rotation). However, the couple stress theory is a Cosserat type theory accounts for the micro-rotation effects representing the material particle as a rigid microvolume. It should be mentioned that the strain gradient theories and the couple stress theories have the merit to account for the acoustic dispersion relations of amorphous materials.

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2.1.3. Single crystal materials Usually, crystals exhibit three types of phonons, namely, acoustic, external optics, and internal optics depending on the crystal type (Chen & Lee, 2003; Chen et al., 2004). Metallic crystals contain one atom per primitive unit cell and hence their dynamic vibration is represented by the dispersive acoustic vibrations (Chen & Lee, 2003; Chen et al., 2004). The dispersion relations of covalent and ionic crystals have both acoustic and external optical branches (Chen & Lee, 2003; Chen et al., 2004). In molecular crystals, atoms are bonded through one of the intermolecular forces, such as dispersion–forces and dipole–dipole interactions. The most important feature of molecular crystals is that fields inside the molecule are completely different from outside the molecule (Chen & Lee, 2003; Chen et al., 2004). In other words, modes of vibrations inside the molecule are different from modes of vibration of the whole molecule. Molecular crystals exhibit acoustic, internal optical, and external optical dispersion phonons (Chen & Lee, 2003). Atomistic simulations have shown that the acoustic and the external optical phonons have nonlocal effects (Chen & Lee, 2003; Demiray, 1977; Gao, 1999a, 1999b). Thus, due to their abilities to account for the effects of the long-range interatomic interactions, the nonlocal theories are the suitable candidates that can be used to model single metallic, covalent, and ionic crystal materials. Experimental investigations for nano-sized materials have shown that the effective Young’s modulus of single crystal materials is likely to decrease with the decrease in the materials’ size (Lia, Ono, Wang, & Esashi, 2003). Eringen’s nonlocal elasticity has shown great merits to match experimental results for single crystal materials with only one atom per primitive unit cell (Chen & Lee, 2003; Chen et al., 2004; Maranganti & Sharma, 2007). The Eringen’s attenuation function is detected by fitting it with the acoustic dispersion branches of materials (Chen & Lee, 2003; Chen et al., 2004; Eringen, 1972). For diatomic single crystal materials, the asymmetric nonlocal elasticity is proposed to capture the acoustic and the external optical phonons of diatomic solids (Demiray, 1977; Gao, 1999a, 1999b). In the asymmetric nonlocal elasticity, an additional attenuation function is presented to match the external optical dispersion branch of the crystal (Demiray, 1977). Because of their nonlocal characters, the second-strain gradient theory, the first-strain gradient theory, and the couple stress theories can be applied for single crystal materials that exhibit only acoustic dispersion relations. The proposed continuum model can be applied for molecular single crystal materials by modeling the continuum as a single crystal consisting of an infinite number of molecules (material particles). For large crystals, the molecules can be modeled as mass points and hence the classical theory of elasticity can be applied. However, for tiny crystals, the inner atomic structure of the molecule should be modeled accounting for the external and the internal phonons of the molecules. The Micromorphic theory and the Microstructure theory have the merit to account for the internal dispersion phonons of single molecular crystal materials (Chen & Lee, 2003). In the Micromorphic theory and the Microstructure theory, a relative deformation measure γ ij is introduced to capture the difference between the internal modes of vibration and the external modes of vibration which makes the Micromorphic theory and the Microstructure theory the most suitable candidates to model single molecular crystal materials. Furthermore, Cosserat theory can be applied for single molecular crystal materials whose molecules are rigid enough to neglect their internal deformation fields accounting only for their rotational modes of vibrations (Chen & Lee, 2003; Chen et al., 2004). 2.1.4. The applicability of the non-classical theories for materials We present in Table 1 a summary for the kinematical variables, the number of material constants, the non-classical degrees of freedom that each one of the available non-classical continuum theories have, and the material type for which each one of these theories can be applied. It is demonstrated that the second-strain gradient theory, the first-strain gradient theory, and the couple stress theories can be applied for single crystal materials and amorphous materials accounting for one-neighbor or two-neighbor interactions. Because these theories do not account for the difference between the internal modes and the external modes, these theories are irrelevant to most crystalline materials (Maranganti & Sharma, 2007). However, for the case of materials in which the difference between the internal modes of vibration and the external modes of vibration is small and for materials that exhibit small nonlocal characteristics, these theories can be applied (Maranganti & Sharma, 2007). It should be mentioned that the Microstructure theory, the first-strain gradient theory, and the couple stress theories account for only the first-order deformation of the material structure. However, in some cases the higher-order deformation of the material structure is needed to be modeled. Next, the necessity for the higher-order deformations of the material structure is discussed. In addition, the challenges to model these deformations for elastic field problems are presented. 3. Material structures with higher-order micro-deformations 3.1. Necessity for a higher-order micro-deformation measure When the size of a structure becomes comparable to the size of its material microstructure, size and microstructural effects are observed. Due to the lack of internal length scale parameters, the classical theory of linear elasticity fails to describe the sizedependent behavior of the material. However, this is possible with the use of other enhanced elastic theories where intrinsic parameters correlating the microstructure with the macrostructure are involved in the constitutive equations as well as in the equation of motion of the considered elastic continuum. Recently, many researchers (Dai, Wang, Abdelkefi, & Ni, 2015; Gao & Mahmoud, 2014; Ke, Wang, Yang, & Kitipornchai, 2012; Ma, Gao, & Reddy, 2011; Shaat & Abdelkefi, 2015; Shaat, Mahmoud, Gao, & Faheem, 2014) have used one of the micro-continuum theories to investigate the microstructure effects for linear elastic materials. Most of the micro-continuum theories only account for the first-order deformation of the material particle limiting their applications for micro-scale applications. However, in some cases, the higher-order deformation of the material particle

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Table 1 A summary of the available non-classical continuum theories and their applicability for materials. Theory

Material particle representation

Kinematical variables

Eringen’s nonlocal elasticity (Eringen, 1966)

Mass point with only translational motions.

Macro-strain and the nonlocal residual at each point.

1

- Single crystal materials that exhibit long-range acoustic phonons (Chen et al., 2004; Chen & Lee, 2003). - Amorphous materials (Maranganti & Sharma, 2007).

Asymmetric Volume element with nonlocal elasticity translational and rigid (Demiray, 1977) rotational motions.

Macro-strain, the nonlocal residual at each point and the relative rotationb .

2

- Diatomic single crystal materials that exhibit long-range acoustic and external optical phonons (Demiray, 1977). - Amorphous materials.

Microstructure theory (Mindlin, 1964)

Volume element has micro-deformations (micro-strains and micro-rotations) and macro-displacements.

Macro-strain, the gradient of the micro-deformation and the difference between the micro-deformation and the macro-deformation.

16

- Single molecular crystal materials (Chen et al., 2004; Chen & Lee, 2003). - Polycrystalline materials. - Amorphous materials with short nonlocal-range effects.

Cosserat theory (Cosserat & Cosserat, 1909).

Volume element has micro-rotations and macro-displacements.

Macro-strain, the gradient of the micro-rotation and the difference between the micro-rotation and the macro-rotation.

4

- Single molecular crystal materials with nearly rigid molecules (Chen et al., 2004; Chen & Lee, 2003). - Polycrystalline materials with rigid crystals. - Amorphous materials with short nonlocal-range effects.

Second-strain gradient theory (Mindlin, 1965)

Volume element with higher-order deformations, micro-strains, micro-rotations, and macro-displacements.

Macro-strain, the first-strain gradient and the second-strain gradient.

16

- Single crystal materials that exhibit short-range acoustic phonons (Polyzos & Fotiadis, 2012). - Amorphous materials with short nonlocal-range effects (Polyzos & Fotiadis, 2012). - It may be applied for polycrystalline materials (Maranganti & Sharma, 2007).

First-strain gradient theory (Mindlin & Eshel, 1968)

Volume element has micro-strains, micro-rotations, and macro-displacements.

Macro-strain and the first-strain gradient.

5

- Single crystal materials that exhibit short-range acoustic phonons (Polyzos & Fotiadis, 2012). - Amorphous materials with short nonlocal-range effects (Polyzos & Fotiadis, 2012). - It may be applied for polycrystalline materials.

Classical couple stress theory (Mindlin & Tiersten, 1962; Toupin, 1962)

Volume element has micro-rotations and macro-displacements.

Macro-strain and the rotation gradient.

2

- Single crystal materials that exhibit short-range acoustic phonons. - Amorphous materials with short nonlocal-range effects. - It may be applied for polycrystalline materials.

Modified couple stress theory (Yang et al., 2002)

Volume element has micro-rotations and macro-displacements.

Macro-strain and the symmetric part of the rotation gradient.

1

- Single crystal materials that exhibit short-range acoustic phonons. - Amorphous materials with short nonlocal-range effects. - It may be applied for polycrystalline materials.

Consistent couple stress theory (Hadjesfandiari & Dargush, 2011)

Volume element has micro-rotations and macro-displacements.

Macro-strain and the skew-symmetric part of the rotation gradient.

1

- Single crystal materials that exhibit short-range acoustic phonons. - Amorphous materials with short nonlocal-range effects. - It may be applied for polycrystalline materials.

a b

No. of material constantsa

Applicability for materials

The number of the additional material coefficients for isotropic materials. The relative rotation is the difference between the local rotations at two neighboring points in the elastic domain (Gao, 1999b).

has to be considered. For instance, in the near future the available micro-continuum theories will approach a point of physical limitation. The attitude of nanotechnology and nanoscience is to develop intensively small nano-devices. Due to their limits of applicability, the available micro-continuum theories will not be capable of modeling structures in these nano-scale applications. Therefore, upgraded theories should be developed to represent the higher-order deformation of the material structure in order to be compatible with the real behavior of materials at nano-scale applications. It has been detected that, for most elastic materials, once the size of the continuum body reduces to the micro-scale, the non-classical degrees of freedom of the material structure begin to affect the deformation energy of the continuum with an

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Fig. 3. Twinned crystal (http://en.wikipedia.org/wiki/Crystal_twinning).

accumulative and an increased manner as its size decreases (Dai et al., 2015; Gao & Mahmoud, 2014; Ke et al., 2012; Ma et al., 2011; Shaat & Abdelkefi, 2015; Shaat et al., 2014). In the framework of the Microstructure theory and its reduced versions for polycrystalline materials, the unit cell or the molecule inside the material particle is assumed to be small enough, and it is represented as a mass point with only translational degrees of freedom, as shown in Figs. 1 and 2. However, for very small sizes of the continuum body (for example, nano-sizes) the ratio of the particle size to the continuum size significantly increases such that the discrete structure of the unit cell or the molecule inside the particle has to be considered. In this case, the unit cell or the molecule can be represented as a nanovolume such that its possible degrees of freedom cannot be disregarded and have to be included. Hence, a higher-order micro-deformation measure is deserved to reflect the possible deformations induced in the material particle due to the new degrees of freedom of the unit cell or the molecule. The Microstructure theory and its reduced versions consider an ideal crystal structure. However, the atomic arrangements of real materials depart in many ways from that ideal case. The real material structure is featured by different types of defects which have been discussed by the theories of crystal defects and disorders (Kelly & Groves, 1970; Nelson, 2002). As an example, crystal twinning occurs when two separate crystals share portions of the same crystal lattice points in a symmetrical manner forming an irregular crystal structure called twinned crystal, as shown in Fig. 3. This twinning action has many modes of formation. In fact, forming of twins can be during formation or growth of the crystal (Growth twins), during cooling (annealing or transformation twins), or after the crystal has formed (deformation or gliding twins). Due to the real irregular shape of the crystal, the material particle has to be modeled considering these defects and irregularities. In this case, the particle can be idealized as a single twinned crystal to account for the twinning effects on the material deformation. Usually, a twin (a twinned crystal) contains two or more crystals in contact or in penetration through each other. Each individual crystal undergoes deformation and motion as degrees of freedom which cause higher-order micro-deformation in the material particle. Therefore, a higher-order micro-deformation measure is required to capture these new deformations of the material particle which comes out due to the irregularity in the crystal shape. 3.2. Modeling the higher-order deformation of the material particle To model the effects of the higher-order deformations of the material particle, the developed continuum model can be modified such that the elastic body is modeled as a continuum containing an infinite number of material particles which undergo higher-order deformations and each material particle consists of a single crystal containing a set of deformable unit cells. Each unit cell undergoes a micro-displacement ui (the displacement of the center of the unit cell) and a cell-deformation ψij = ui, j

which comes out due to an atomic-displacement ui , as shown in Fig. 4. Consequently, the material particle will have a macrodisplacement ui (the displacement of the center of the particle), a micro-deformation ψij = ui, j (the first-order deformation of

the material particle), and a micro-gradient of the cell-deformation χijk = ψij,k (the higher-order deformation of the material particle). In this case, the total strain energy of the continuum body contains five main deformation measures instead of the three measures used in the previous continuum model and the Microstructure theory such that:

W =F



εi j , γijk = ψij,k − ψij,k , γi j = ui, j − ψij , χijk , ηi jkl



(10)

where γijk is the gradient of a relative deformation tensor measures the difference between the gradient of the microdeformation ψ  of the material particle and the gradient of the cell-deformation ψ  . γ is the usual relative deformation tensor ij

ij

ij

that appears in the Microstructure theory which measures the difference between the micro-deformation ψij and the macrogradient of the macro-displacement ui, j of the material particle. ηijkl denotes the macro-gradient of χijk tensor while χijk is the

macro-gradient of ψij tensor.

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Fig. 4. The kinematical variables for a material particle consisting of deformable unit cells in the context of the developed continuum model.

The strain energy in Eq. (10) depends on 150 independent kinematical variables which is a huge number of variables. In comparing to the strain energy of the Microstructure theory which only depends on 42 independent variables, the strain energy in Eq. (10) depends on higher-order deformation measures that account for the second-order deformation of the material particle. Although the higher-order deformation measures are needed, Eq. (10) is impractical in its current form. Therefore, it is convenient to neglect the deformation measures that are relevant to the less affecting internal interactions inside the material structure. For instance, the relative deformation measures, γijk and γ ij , in Eq. (10) can be eliminated for materials without microstructure or for materials with low differences in fields between the internal micro-mediums and their surrounding macro-mediums. For more illustration, when the difference between the atomic displacement ui  , micro-displacement ui  , and macro-displacement ui fields is very small, γijk and γ ij can be neglected from Eq. (10). In this case, the strain energy becomes a function of the first, the second, and the third gradients of the macro-displacement field ui , or equivalently it becomes a function of the infinitesimal strain in addition to the first and the second gradients of strain. This typically reduces the strain energy in Eq. (10) to the strain energy of the second-order strain gradient theory (Mindlin, 1965). Based on this explanation, it can be concluded that the second-strain gradient theory is suitable to capture the deformability of the unit cell or the molecule inside the material particle for materials without microstructure. In addition, this theory reflects a simpler model than the presented atomic lattice model given in Fig. 4 or the continuum model given in Eq. (10) by reducing the number of independent kinematical variables from 150 to 54. Compared to the Microstructure theory and the first-strain gradient theory, the second-strain gradient theory captures the higher-order deformations of the material particle. However, the second-strain gradient theory is still accounting for a large number of degrees of freedom which makes it difficult to be applied for continuum mechanics. Indeed, the strain energy density depends on 54 independent kinematical variables and 18 associated material coefficients for isotropic materials. To this end, in the next section, a novel simplified theory for materials without microstructure is developed to account for the higher-order deformations of the material particle. In the context of this simplified theory, the strain energy only depends on 18 independent kinematical variables and 5 material coefficients for isotropic materials.

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Fig. 5. The induced degrees of freedom in the material particle as a result of the molecule rigid translations and rotations.

4. The second-order rotation gradient theory As previously mentioned, the strain energy in Eq. (10) and its reduced version depend on a large number of kinematical variables which makes them very difficult to be applied in elastic field problems. However, for certain types of materials and according to their crystal structure, some deformation variables can be disregarded. Molecules are made of two or more atoms either of the same element or of different elements. Intramolecular forces including ionic, covalent, and metallic bonds are the forces of attraction which hold an individual molecule as a single unit. Compared to intermolecular forces, which are the forces of attraction or repulsion acting between neighboring molecules, intramolecular forces are very strong. Molecules are likely to be rigid and need large energies to break their atomic bonding. For example, molecules and crystals with ionic bonds are very stiff, brittle, and tend to cleave rather than deform because bonds are strong (Israelachvili, 2011). Moreover, framework crystals are soft crystals composed of very rigid molecules which are linked flexibly through London dispersion forces (Chen et al., 2004). These rigid molecules tend to move rigidly inside the crystal or material structure. Consequently, representing the molecules inside the material structure as rigid volumes is convenient. In another scenario, because ionic crystals are very stiff crystals which tend to translate or rotate rigidly inside the material structure, a twinned crystal with small ionic crystals is likely to deform due to the rigid rotations of the constituent crystals. In this section, the bases of a novel second-order rotation gradient theory for materials without microstructure are derived and discussed. In this theory, the material particle is modeled as a deformable microvolume containing a set of rigid molecules which are allowed to translate and to locally rotate, as shown in Fig. 5. A rotation gradient measure is introduced to capture the effects of the local rotations of the molecules on the material particle. In addition, a micro-rotation tensor is introduced to reflect the effects of the translational motions of the molecules on the material particle. Starting from Eq. (10), the strain energy of the second-rotation gradient theory is formed by neglecting the less affecting deformation measures. First, for materials without microstructure, γijk and γ ij , are eliminated from Eq. (10). Second, the molecules are represented as rigid volumes; hence,

their strain field, εij , is also disregarded from Eq. (10). Third, with neglecting the small strain field of the material particle εi j from Eq. (10) and considering its higher-order deformation, the material particle is subjected to a displacement field ui (the translational motions of the center of the particle), a micro-rotation θ i (rotational motions of the particle about its center), and a micro-rotation gradient θ i, j (the micro-gradient of the local rotation of the molecule) as degrees of freedom. If the strain fields of the material particle and the molecule are taken into account, this theory yields the second-strain gradient theory. This proposed second-rotation gradient theory provides a simplification for the second-strain gradient theory for materials with nearly rigid

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molecules. In this case, the strain energy density is a function of the conventional infinitesimal strain and the first and the second gradients of the rotation tensor. The developed second-rotation gradient theory is compatible with the properties of the molecular structure of materials with rigid molecules. Thus, by neglecting the less affecting fields, strain fields, and accounting for the highly affecting ones, the number of the independent kinematical variables and the number of the material coefficients are highly reduced which makes the second-rotation gradient theory applicable for the elastic field problems. 4.1. The kinematic variables The kinematic variables to be employed in the context of the second-rotation gradient theory are given by:

εi j =

 1 ui, j + u j,i 2

(11.a)

θi j =

 1 ui, j − u j,i 2

(11.b)

Ri j = θi, j

(11.c)

∅i jk = θk,i j

(11.d)

where ɛij is the symmetric infinitesimal strain tensor (small deformation tensor) which is the suitable measure for the continuum deformation that is relevant to the rigid translations of the material particle. θ ij is the skew-symmetric infinitesimal rotation tensor which measures the rigid-like rotations of the continuum. It should be noted that θ ij does not contribute to the deformation energy and consequently cannot appear in the strain energy density function. As a result, in classical continuum theories, such as Cauchy elasticity, the rotation tensor is disregarded and ɛij is used only as a measure of the induced deformation in the continuum. The first gradient of the rotation tensor Rij is the suitable measure of deformation induced in the deformable body which comes as a result of the rigid rotations of the material particle (where θi = 12 ei jk θk j ). Moreover, φ ijk is the second gradient of the rotation tensor which is introduced to capture the molecules rigid rotation effects, as shown in Fig. 5. Neglecting the rigid rotations of the molecules inside the material particle and putting φijk = 0, the second-rotation gradient theory is reduced to the couple stress theory (which can be considered as the first-rotation gradient theory). In the framework of the couple stress theory (Mindlin & Tiersten, 1962; Toupin, 1962), the material particle is represented as a rigid microvolume such that only the first gradient of the rotation tensor Rij and the infinitesimal strain ɛij contribute to the continuum deformation energy. The proposed second-rotation gradient theory outweighs the couple stress theory in accounting for the deformability of the material particle by introducing the second gradient of the rotation tensor φ ijk in order to account for the local rotations of the molecules inside the material particle. Compared to the first-strain gradient theory (Mindlin & Eshel, 1968) which measures the first-order deformation (the microstrain) of the material particle, the proposed second-rotation gradient theory takes into account the second-order deformation of the material particle through a rotation gradient measure. Moreover, the proposed theory represents a simplification for the second-strain gradient theory to facilitate its applicability for the elastic field problems. In the classical couple stress theory (Mindlin & Tiersten, 1962; Toupin, 1962), there are two additional higher-order material constants which are introduced to capture the effects of the rigid rotation of the material particle on the deformation energy of elastic continua (Mindlin & Tiersten, 1962; Shaat, 2015; Toupin, 1962). Consequently, at least two experimental tests are required to find these two independent material constants (Hadjesfandiari & Dargush, 2011; Shaat, 2015; Yang et al., 2002). This motivated Yang et al. (2002) and Hadjesfandiari and Dargush (2011) to propose, respectively, the modified couple stress and the consistent couple stress theories, as two alternatives, in which only one higher-order material constant is introduced. In the context of the modified couple stress theory, Yang et al. (2002) claimed that the symmetric part of the rotation gradient tensor, χi j = 12 (Ri j + R ji ), is the only suitable measure for the deformation energy of the microstructure rigid rotation. On the other hand, in the consistent couple stress theory, Hadjesfandiari and Dargush (2011) claimed that the skew-symmetric part, ki j = 12 (Ri j − R ji ), is the only suitable measure for this deformation energy. Recently, Shaat (2015) showed that the whole rotation gradient tensor, Rij , contributes to the deformation energy and these two alternative theories represent only two simplifications for the classical couple stress theory. Indeed, some induced continuum deformations due the material particle micro-rotation can be neglected for the sake of simplicity and the applicability of the theory (Shaat, 2015). Furthermore, it has been proved that, in some elasticity problems, like plate problems, the modified and the consistent couple stress theories give the same results as those of the classical couple stress theory (Shaat, 2015). It has also been concluded that the modified and the consistent couple stress theories can provide an acceptable simplification for specific material structures and applications (Shaat, 2015). By decomposing the first gradient of rotation tensor Rij into its symmetric and skew-symmetric parts, three possible versions of the second-rotation gradient theory can be obtained. In the first one, considering the whole Rij , nine additional material constants should be considered. In the second version, only the symmetric part of Rij is utilized. In this case, six additional material constants are required in the constitutive equations. In the third version of the second-rotation gradient theory, three additional higher-order material constants are needed when only considering the skew-symmetric part of Rij . Depending on the type of the material structure and the considered application, the second and third versions of the second-rotation gradient theory can be effectively applied (Shaat, 2015). Table 2 summarizes these three possible versions of the second-rotation gradient theory.

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Table 2 The kinematical variables of the three versions of the second-rotation gradient theory. Versions of the second-rotation gradient theory

Material particle degrees of freedom

Kinematical variables

Version 1

Version 2

Version 3

Displacement Rotation Rotation gradient

ui

ui

ui

Strain 1st gradient of rotation 2nd gradient of rotation

ɛij Rij Rij, k

ɛij

χ ij χ ij, k

ɛij kij kij, k

42 11

30 8

18 5

Number of independent kinematical variables Number of material constants

θ ij

Rij

θ ij χ ij

θ ij kij

In the previous paragraphs, the bases of the second-rotation gradient theory with its three possible versions are developed and discussed for the different types of materials. Because it is impractical to determine the huge number of additional material constants in the first and second versions of the second-rotation gradient theory and because our objective is to develop a useful/applicable theory for nano-scale applications, in the rest of this research study, the essential constitutive and governing equations of the third version of the second-rotation gradient theory will be derived and applied for various elastic field problems. The same strategy can be used to form these essential equations for the other two versions of the theory. In the third version of the second-rotation gradient theory, the skew-symmetric part of the first gradient rotation tensor, kij , is considered as the measure for the continuum deformations induced due to the particle micro-rotations instead of the whole Rij tensor. Thus, the second gradient of the rotation tensor φ ijk is reduced to the gradient of the skew-symmetric rotation gradient tensor ψ ij , such that:

ψi j = ki, j

(12)

where ki = 12 ei jk kk j is the skew-symmetric rotation gradient vector. The most interesting merit of the proposed second-gradient of rotation theory is that the number of the independent kinematical variables is reduced to 18 variables instead of 54 variables when using the second-strain gradient theory for isotropic materials without microstructure. Thus, by figuring out the features of the material structure, the less affecting variables can be neglected from the deformation energy to propose an applicable theory for elastic field problems. 4.2. Mechanics of material particles The inner structure of the material particle is considered such that it contains a large number of molecules and each molecule is represented as a rigid nanovolume which has rigid rotational and translational motions as degrees of freedom. These internal rigid motions cause the deformation and the rigid translations and rotations of the material particle, as presented in Fig. 5. The effects of the internal motions on the mechanics of particles are discussed to derive the equilibrium equations for a system of discrete particles considering the inner structure of the individual particle. In the conventional mechanics of particles, the material particle is represented as a mass point, and it can only undergo three translational motions due to the applied forces Fi . However, in the present second gradient of rotation theory, as a consequence of the internal motions inside the material particle, an additional resultant couple field Li and resultant higher-order couple field Mi affect the material particle at its centroid which force the particle to rotate and to deform. Consequently, for static equilibrium, the Newton’s second law for a system of discrete particles is adjusted for the resultant couple field Li and the resultant higherorder couple field Mi to be in the following form:



Fi = 0



(13.a)



Xi × Fi + Li = 0

Mi = 0

(13.b) (13.c)

Eq. (13.b) shows the effects of the internal couples i along with the conventional couples of forces (Xi × Fi ) on the mechanics  i . Eq. (13.c) represents Mi as the resultant higher-order couples that cause the possible deformations of particles, where Li = of the material particle as a consequence of the internal couples i . 4.3. Equilibrium equations of deformable bodies in linear elasticity The equilibrium equations of linear elasticity based on the second-order rotation gradient theory can be directly derived from (Eqs. 13.a)–(13.c). To this end, a material continuum occupying a volume V bounded by a surface S as the current configuration is considered. The responsible tractions for the changes in the originality of the deformable body are the force traction vector ti(n ) , the moment traction vector μi(n ) , and the higher-order moment traction vector mi(n ) . Surface forces ti(n ) dS, surface moments

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Fig. 6. Surface tractions affecting the surface of the volume element and their corresponding degrees of freedom.

μi(n) dS, and higher-order surface moments mi(n) dS are, respectively, represented by a force-stress tensor σ ji , a couple stress tensor μji , and a higher-order couple stress tensor mij , as follows: n ti( ) = σ ji n j

(n )

(14)

μi = μ ji n j

(15)

n mi( ) = m ji n j

(16)

where nj is the unit normal vector of the continuum surface S.

The surface forces ti(n ) dS are responsible for the rigid motions of the continuum surface S. tx(n ) dS is the force aiming to rigidly

move the surfaces of the volume element parallel to x-axis, as presented in Fig. 6(a). The surface moment vector μi(n ) dS is

responsible for the rigid rotations of the surface S. For example, μx(n ) dS is the moment aiming to rotate the surface of the volume element about x-axis, as shown in Fig. 6(b). The higher-order surface moment vector mi(n ) dS is responsible for the possible deformations that can be induced in the surface of the volume element, as shown in Fig. 6(c). By integrating the equilibrium equations of a system of discrete particles over the continuum volume, the equilibrium equations for the continuum mechanics can be derived. Thus, integrating Eqs. (13.a)–(13.c), one obtains:





t (n) dS = 0     X × t (n) + μ(n) dS = 0 (X × F + C )dV + V S    m(n) dS = 0 V

S

F dv+

(17.a)

S

(17.b) (17.c)

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The higher-order body couple is neglected and the only considered body tractions are F and C which, respectively, denote the body force and the body couple vectors. To transform the surface integrals to volume integrals, the divergence theorem is used and hence Eqs. (17.a)–(17.c) can be expressed as:



V



V



V

(F + ∇ · t )dV = 0

(18.a)

(X × (F + ∇ · t ) + C− ∈: t + ∇ · μ )dV = 0

(18.b)

(∇ · m )dV = 0

(18.c)

Substituting Eq. (18.b) into Eq. (18.a), the equilibrium equations for linear elasticity are given by:

F +∇ ·t = 0

(19.a)

∇ · μ− ∈: t + C = 0

(19.b)

∇ ·m=0

(19.c)

Using index notations, these equilibrium equations are rewritten as:

σ ji, j + Fi = 0

(20.a)

μ ji, j + ei jk σ jk + Ci = 0

(20.b)

m ji, j = 0

(20.c)

Eqs. (20.a)–(20.c) represent the equilibrium equations for elastic continua according to the second-order rotation gradient theory. By eliminating the couple stress tensor μji , the higher-order couple stress tensor mji , and the body couple vector Ci , Eqs. (20.a)–(20.c) reduce to the well-known equilibrium equations of the classical linear elasticity which are:

σ ji, j + Fi = 0

(21.a)

ei jk σ jk = 0

(21.b)

On the other hand, eliminating the higher-order couple stress tensor mji from (20.a) to (20.c) yields the equilibrium equations of the couple stress theory (the first-order rotation gradient theory) which have the following form:

σ ji, j + Fi = 0

(22.a)

μ ji, j + ei jk σ jk + Ci = 0

(22.b)

4.4. Principle of virtual work The kinematical variables which are employed to form the principle of virtual work are the infinitesimal strain ɛij , the volume rigid rotation tensor θ ij , the rotation gradient tensor kij which is the skew-symmetric of the general rotation gradient tensor θ i, j , and the second gradient of the rotation tensor ψ ij . The material particle inside the volume body undergoes a displacement ui , a rigid rotation θ i , and a rotation gradient ki as degrees of freedom. By multiplying Eq. (20.a) by the virtual displacement δ ui , Eq. (20.b) by the virtual rotation δθ i , and Eq. (20.c) by the virtual rotation gradient δ ki and then integrating over the volume, the virtual forms of Eq. (20) can be expressed as:





V



V



V

 σ ji, j + Fi δ ui dV = 0

(23.a)



 μ ji, j + ei jk σ jk + Ci δθi dV = 0



m ji, j



δ ki dV = 0

(23.b) (23.c)

It should be noted that

  σ ji, j δ ui = σ ji δ ui , j − σ ji δ ui, j

(24)

and using the divergence theorem, Eq. (23.a) can be rewritten as follows:



S

n ti( ) δ ui dS−



V

σ ji δ ui, j dV +



V

Fi δ ui dV = 0

(25)

Similarly, by using the following relation:

  μ ji, j δθi + ei jk σ jk δθi = μ ji δθi , j − μ ji δθi, j − σ jk δθ jk

(26)

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Eq. (23.b) can be expressed as:



S

μi(n) δθi dS−



V

μ ji δθi, j dV +

 V

σ ji δθi j dV +

 V

Ci δθi dV = 0

(27)

Adding Eqs. (25) and (27), one obtains:



V

μ ji δθi, j dV +



V

      σ ji δ ui, j − δθi j dV = ti(n) δ ui dS+ μi(n) δθi dS+ Fi δ ui dV + Ci δθi dV S

S

V

(28)

V

Also, from Eq. (23.c) and by using



m ji, j δ ki = m ji δ ki



,j

− m ji δ ki, j

(29)

yields to the following equation:



S

n mi( ) δ ki dS−



V

m ji δ ki, j dV = 0

(30)

where kij is used as the rotation gradient tensor instead of the general θ i, j . By adding Eqs. (28) and (30), and by using δεi j =

δ ui, j − δθi j , we obtain: 

V



      μ[ ji] δ ki j + σ( ji) δεi j + m ji δψi j dV = ti(n) δ ui dS+ μi(n) δθi dS+ mi(n) δ ki dS + Fi δ ui dV + Ci δθi dV S

S

S

V

(31)

V

In Eq. (31), the essential boundary conditions are vectors ui , θ i , and ki where the natural boundary conditions are ti(n ) , μi(n ) ,

and mi(n ) . Moreover, the symmetric force stress tensor σ (ji) is conjugate to ɛij , the skew-symmetric couple stress tensor μ[ji] is conjugate to kij , and the higher-order couple stress tensor mji is conjugate to ψ ij and all of them contribute to the continuum strain energy. 4.5. Strain energy density and constitutive equations

Starting from the general form of the strain energy density given in Eq. (10), the strain energy density according to the second gradient of rotation theory is a function of the infinitesimal strain ɛij in addition to the first and the second gradients of the rotation tensor. Considering the skew-symmetric part of the first rotation gradient, the strain energy density can be written in this form:



W =F

 1 εi j , ki j , ψi j = λεii ε j j + μεi j εi j + 4a1 ki j ki j + b1 ψi j ψi j + b2 ψi j ψ ji 2

(32)

where λ and μ are the conventional Lame’s constants and a1 , b1 , and b2 are three additional material constants. a1 is the ratio of the couple stress to curvature-due to-bending of the deformable body, i.e., a couple stress modulus which has the dimension of force (Mindlin & Tiersten, 1962). Moreover, b1 and b2 are another two material moduli have the dimension of moment multiplied by meter which are introduced to capture the effects of the higher-order couple stresses (couples of couple stress) on the defor√ mation energy of the continuum. A length scale parameter l = a1 /μ has been introduced to scale the couple stress modulus a1 with the conventional shear modulus μ (Hadjesfandiari & Dargush, 2011; Mindlin & Tiersten, 1962; Yang et al., 2002). Following the same used strategy, a new higher-order length scale parameter lh = b1 /(μl 2 ) is proposed to scale the higher-order material modulus b1 with the couple stress modulus μl2 . Another nondimensional parameter η = b2 /b1 is introduced to relate the two material moduli b1 and b2 . Considering these new relations, the strain energy density can be rewritten as:

W=

  1 λεii ε j j + μεi j εi j + 4μl 2 ki j ki j + μl 2 lh2 ψi j ψi j + ηψi j ψ ji 2

(33)

For the positive definiteness of the strain energy density, the used parameters should verify the following relations:

3λ + 2μ > 0, μ > 0, l 2 > 0, lh2 > 0, −1 ≤ η ≤ 1

(34)

When the material parameter η = 1, the contribution of the skew-symmetric part of ψ ij does not have any role in the strain energy density equation. However, when considering η = −1, the contribution of the symmetric part of ψ ij is disregarded from Eq. (33). When,−1 < η < 1, a balance between the symmetric and the skew-symmetric parts is obtained. From Eqs. (31) and (33), the constitutive equations according to the second-rotation gradient theory can be expressed as:

σ( ji) =

∂W = λεkk δi j + 2μεi j ∂εi j

μ ji = μ[ ji] = m ji =

∂W = 8μl 2 ki j ∂ ki j

  ∂W = 2μl 2 lh2 ψi j + ηψ ji ∂ψi j

(35)

(36)

(37)

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The additional parameters for different materials can be experimentally determined via an experimental setup or numerically via atomistic simulations. The additional constitutive Eqs. (36) and (37) depend on three scaling parameters. These parameters are used to scale the intrinsic material constants, correlating the first and the second-rotation gradients, to the conventional material constants. Experimental investigations can be used to detect the values of these scaling parameters for different materials. However, the justifications of the physical meaning of these parameters with respect to the crystal structure parameters can only be detected via atomistic simulations. To experimentally measure the additional parameters for different materials, an experimental setup for micro/nano beams can be developed. By considering different sizes of the beam, starting from macro to nano scales, the three length scale material parameters, l, lh and η can be determined. The first-order length scale parameter, l, can be experimentally measured by making the size of the beam big enough so that the higher-order parameter, lh , has no contribution to the deformation of the beam. After that, the higher-order parameter, lh , can be measured with the aid of nano-sized beams. 4.6. Uniqueness theorem for boundary value problems The uniqueness of the linear elasticity boundary value problem is investigated based on the second gradient of rotation theory. Assuming that there exist two different solutions {u1i , εi1j , k1i j , ψi1j , σi1j , μ1i j , m1i j } and {u2i , εi2j , k2i j , ψi2j , σi2j , μ2i j , m2i j } to the same boundary value problem with identical body forces and body couples and subjected to the same boundary conditions. The difference between the two solutions is defined as:

udi = u1i − u2i

(38.a)

εidj = εi1j − εi2j

(38.b)

kdij = k1i j − k2i j

(38.c)

ψidj = ψi1j − ψi2j

(38.d)

σidj = σi1j − σi2j

(38.e)

μdij = μ1i j − μ2i j

(38.f)

mdij = m1i j − m2i j

(38.g)

This difference solution is the one of the problem with zero applied body forces, body couples, and boundary conditions and satisfies the equilibrium equations of the boundary value problem such that: d σ ji, j =0

(39.a)

μdji, j + ei jk σ jkd = 0

(39.b)

mdji, j = 0

(39.c)

Multiplying Eq. (39.a) by εidj , Eq. (39.b) by kdij , and Eq. (39.c) by ψidj , then integrating over the volume and using the divergence theorem yield:



2 V



W d dV =



V

 μdji kdij + σ jid εidj + mdji ψidj dV = 0

(40)

Substituting Eqs. (35)–(37) into Eq. (40), one obtains:





V

2W d dV =

V



  λεiid ε djj + 2μεidj εidj + 8μl 2 kdij kdij + 2μl 2 lh2 ψidj ψidj + ηψidj ψ jid dV = 0

(41)

The strain energy density of the difference solution should be positive and λ, μ, l 2 , lh2 are also positive definite. Inspecting Eq. (41), four possible scenarios can be studied separately: • For η = 0, it is clear from Eq. (41) that the boundary value problem always gives a unique solution. • For η = 1, the uniqueness of the boundary value problem should be performed by changing the difference solution of the second-rotation gradient theory from ψidj = ψi1j − ψi2j to ψ(di j ) = ψ(1i j ) − ψ(2i j ) . In this case, the boundary value problem always gives a unique solution. • For η = −1, by changing the difference solution from ψidj = ψi1j − ψi2j to ψ[id j] = ψ[i1 j] − ψ[i2 j] and by examining the uniqueness, it is observed that the boundary value problem always gives a unique solution. • For −1 < η < 1, there is no evident proof of uniqueness. Therefore, each considered boundary value problem has to be checked for uniqueness. In this case, the uniqueness of the boundary value problem highly depends on the applied boundary conditions and body forces. A similar analysis has been performed by Mindlin and Tiersten (1962) when determining the uniqueness of the classical couple stress theory.

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4.7. The nonlocal character and the applicability of the proposed second-rotation gradient theory The material particle in the context of Eringen’s nonlocal theory (Edelen, 1969; Eringen, 1966) is idealized as a mass point which has a translational degree of freedom accounting for the long-range interactions between the particles inside the continuum body. In this theory, the displacement field of a certain material particle is affected by the displacements of the other surrounding particles in the domain of interaction. The total displacement field u of a particle located at a point x has the following form:

u (x ) =



V

α (|x − x| )u(x )dV (x )

(42)

where α (|x − x| ) is a nonlocal kernel function (attenuation function) according to it the displacements of the neighboring points x affect the displacement field of the reference point x. Expanding the displacement u(x ) with respect to the point x and substituting the result into Eq. (42) yields:

u(x ) = u(x ) + C1 ∇ u(x ) +

C2 ∇∇ u(x ) + . . . 2

(43)

where C1 = V (x − x )α (|x − x| )dV (x ), C2 = V (x − x ) α (|x − x| )dV (x ). By decomposing the displacement gradient ∇ u(x) into its symmetric and skew-symmetric parts, i.e., ∇ u(x ) = ε(x ) + θ (x ) and by inspecting Eq. (43) it may be observed that the material particle undergoes different types of degrees of freedom. The first term in Eq. (43) represents the translational motion of the material particle, the second term represents the micro-strain ɛ(x)and the rigid micro-rotations θ (x)of the material particle, and the third term represents the second-order deformation of the material particle. The first-strain gradient theory is a nonlocal-type theory which accounts for one-neighbor interaction. Thus, only the first two terms in Eq. (43) are the considered ones in the context of the theory, i.e.,

u(x ) = u(x ) + C1 ∇ u(x )

2

(44)

On the other hand, the second-strain gradient theory accounts for two-neighbor interactions, thus the first three terms of Eq. (43) are considered in the theory such that:

u(x ) = u(x ) + C1 ∇ u(x ) +

C2 ∇∇ u(x ) 2

(45)

In the couple stress theory, the material particle is assumed having only translations u(x) and micro-rotations θ (x) as degrees of freedom. It can be concluded that the couple stress theory has some nonlocal character by taking into consideration a oneneighbor interaction providing a simplification for the first-strain gradient theory:

u(x ) = u(x ) + C1 θ (x )

(46)

However, in the context of the proposed second-rotation gradient theory, the material particle undergoes an additional higher-order deformation captured through a rotation gradient measure ∇θ (x). Consequently, the second-rotation gradient theory has a nonlocal character and represents a simplification for the second-strain gradient theory as follows:

u(x ) = u(x ) + C1 θ (x ) +

C2 ∇θ (x ) 2

(47)

Due to its nonlocal character, the proposed second-rotation theory can be applied for single crystal materials and amorphous materials accounting for the two-neighbor interaction of the acoustic phonons. Furthermore, the proposed theory can be applied for polycrystalline materials in which the difference between fields inside and outside their crystals is negligible. In addition to that, this theory has the merit to account for the higher-order deformation of the material structure which is needed for some structural defects, such as twinning. 5. Size-dependent linear elasticity models In this section, using the proposed second-rotation gradient theory, the size-dependent linear elasticity for 2D problems, anti-plane problems, and bending beams are investigated and discussed. 5.1. Two-dimensional linear elasticity theory The second-rotation gradient theory for 2D linear elastic materials is studied. The displacement fields for 2D problems are given by:

ux = u0 (x, y ) ; uy = v0 (x, y ) ; uz = 0

(48)

For plane strain assumptions, the infinitesimal strain tensor ɛij , the rotation vector θ i , the skew-symmetric first-rotation gradient tensor kij , the skew-symmetric first-rotation gradient vector ki , and the second gradient of rotation tensor ψ ij can be written in the following form:

1 2

εxx = u0,x ; εyy = v0,y ; εxy = (u0,y + v0,x )

(49.a)

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1 2

θx = θy = 0; θz = (v0,x − u0,y )

91

(49.b)

1 1 (u0,yy − v0,xy ) ; kxz = (u0,xy − v0,xx ) 4 4 1 1 kx = kzy = − (u0,yy − v0,xy ) ; ky = kxz = (u0,xy − v0,xx ) 4 4 1 1 1 ψxx = (v0,xxy − u0,yyx ) ; ψyy = (u0,yxy − v0,xxy ) ; ψxy = (v0,xyy − u0,yyy ) ; 4 4 4 kyz =

(49.c) (49.d)

1 4

ψyx = (u0,yxx − v0,xxx )

(49.e)

The non-zero components of the force stress, the couple stress, and the higher-order couple stress tensors are:

σxx =

 2μ  (1 − ν )u0,x + νv0,y ; 1 − 2ν

σyy =

 2μ  ν u0,x + (1 − ν )v0,y ; 1 − 2ν

  σxy = μ u0,y + v0,x

    μxz = 2μl 2 v0,xx − u0,yx ; μyz = 2μl 2 v0,xy − u0,yy ; μzx = −μxz ; μzy = −μyz

(50.a)

(50.b)

1 22 μl lh (1 + η )(v0,xxy − u0,yyx ); myy 2 1 = μl 2 lh2 (1 + η )(u0,yxy − v0,xxy ); myx 2 1 = μl 2 lh2 ((v0,xyy − u0,yyy ) + η (u0,yxx − v0,xxx )); mxy 2 1 = μl 2 lh2 ((u0,yxx − v0,xxx ) + η (v0,xyy − u0,yyy )) 2

mxx =

(50.c)

Then, from Eq. (33), the virtual strain energy for the second-rotation gradient theory can be expressed as:



 σ ji δεi j + μ ji δ ki j + m ji δψi j dV    1 1 1 1 1 1 = −σxx,x − σxy,y − μyz,yy − μxz,yx + mxx,yyx − myy,yxy + myx,yyy − mxy,xxy δ u0 2 2 4 4 4 4 V   1 1 1 1 1 1 + −σyy,y − σxy,x + μyz,xy + μxz,xx − mxx,xxy + myy,xxy − myx,xyy + mxy,xxx δv0 dv

δW =



V

2

2

4

4

4

4

(51)

and hence the governing equations in terms of stress resultants can be written as:

1 2

1 4

(52.a)

1 2

1 4

(52.b)

σxx,x + σxy,y + (μyz,yy + μxz,yx ) − (mxx,yyx − myy,yxy + myx,yyy − mxy,xxy ) = px σyy,y + σxy,x − (μyz,xy + μxz,xx ) + (mxx,xxy − myy,xxy + myx,xyy − mxy,xxx ) = py

where pi are the applied forces per unit area of the body. The governing equations in terms of displacements can be obtained by direct substitution for the stress components from Eq. (50). 5.2. Anti-plane deflection linear elasticity For an anti-plane deflection linear elasticity problem, the displacement field is assumed to have the following form:

ux = uy = 0, uz = w(x, y )

(53)

where w is the out − of − plane deflection. In this case, the non − zero components of εi j , θi , ki j , ki , and ψi j are :

1 1 2 2 1 1 θx = w,y ; θy = − w,x 2 2 1 2 kxy = ∇ w ; kyx = −kxy 4 1 kz = kyx = − ∇ 2 w 4 1 1 ψzx = − ∇ 2 w,x ; ψzy = − ∇ 2 w,y 4 4

εxz = w,x ; εyz = w,y

(54.a) (54.b) (54.c) (54.d) (54.e)

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where ∇ 2 = ∂ 2 /∂ x2 + ∂ 2 /∂ y2 . The non-zero stress components are given by:

σxz = μw,x ; σyz = μw,y

(55.a)

μxy = −2μl 2 ∇ 2 w ; μyx = −μxy

(55.b)

mxz = −

      1 2 2 2  1 1 1 μl lh ∇ w,x ; mzx = − μl 2 lh2 η ∇ 2 w,x ; mzy = − μl 2 l12 η ∇ 2 w,y ; myz = − μl 2 lh2 ∇ 2 w,y 2 2 2 2

(55.c)

Then, the stress resultants are given by:



Ni j =

σi j dz; Pi j =

Z



Z

μi j dz; Yi j =

 Z

mi j dz

(56)

From Eq. (31), the differential form of the governing equation, in terms of stress resultants, can be expressed as:

Nxx,x + Nzy,y +

1 1 (Pxy,xx + Pxy,yy ) + (Yxz,xxx + Yxz,yyx + Yyz,xxy + Yyz,yyy ) = p(x, y ) 2 4

(57)

where p(x, y) is the applied transverse mechanical load. Hence, the governing equation in terms of the deflection becomes:

1 8

μh∇ 2 w − μl 2 h∇ 4 w + μhl 2 lh2 ∇ 6 w = p(x, y )

(58)

where h is the plate thickness and ∇ 6 = ∂∂x6 + ∂∂y6 + 3 ∂ x∂4 ∂ y2 + 3 ∂ x∂2 ∂ y4 ; ∇ 4 = ∂∂x4 + ∂∂y4 + 2 ∂ x∂2 ∂ y2 . It should be mentioned that η does not appear in the displacement governing equation although it exists in the constitutive equations which shows the independency of the deflection on η for anti-plane problems. 6

6

6

6

4

4

4

5.3. Euler–Bernoulli beam Consider a beam with a cross section of area A and its centerline is x-axis. Let w is the deflection of the axis of the beam in a plane parallel to xz-plane. The displacement field from the classical elasticity is given by:

ux = −zw,x (x ) , uy = 0 , uz = w(x )

(59)

The non-zero components of the kinematical variables are:

εxx = −zw,xx

(60.a)

θy = − w,x

(60.b)

1 1 w,xx ; kyx = − w,xx 2 2 1 kz = kyx = − w,xx 2 1 ψzx = − w,xxx 2 kxy =

(60.c) (60.d) (60.e)

Consequently, the non-zero stress components can be written as follows:

 2μ 1 − ν  ( ) σxx = − zw,xx ; σyy = σzz = νσxx 1 − 2ν μxy = −4μl 2 w,xx ; μyx = 4μl 2 w,xx

(61.b)

mxz = −μl 2 lh2 w,xxx ; mzx = −μl 2 lh2 ηw,xxx

(61.c)

(61.a)

The differential equilibrium equation in terms of the stress resultants can be expressed as follows:

−Mxx,xx +

1 1 1 Pyx,xx − Pxy,xx + Yxz,xxx = p(x ) 2 2 2

(62)

where the stress resultants can be defined in the following form:



Mxx =

A

 Pyx =

A

 Pxy =

A

zσxx dz = −

2μ(1 − ν )I w,xx 1 − 2ν

(63.a)

μyx dz = 4μl 2 Aw,xx

(63.b)

μxy dz = −4μl 2 Aw,xx

(63.c)

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 Yxz =

A

mxz dz = −μl 2 lh2 Aw,xxx

93

(63.d)

where I = A z2 dA is the classical inertia of the beam. The equilibrium equation (Eq. (62)) can be rewritten in terms of the deflection as:

D∇ 4 w − Dh ∇ 6 w = p(x ) where ∇ n

(64)

n = ∂∂xn and D = D0 + Dc .D0 , Dc , and Dh are, respectively, the conventional bending rigidity of the beam that is conjugate

to the force stress tensor, the bending rigidity which is due to the couple stress tensor, and a higher-order bending rigidity which is due to the higher-order couple stress tensor. These rigidities are defined as follows:

D = D0 + Dc ; D0 =

2 μ ( 1 − ν )I 1 ; Dc = 4μl 2 A; Dh = μl 2 lh2 A 1 − 2ν 2

(65)

When lh = 0, the equilibrium equation is reduced to the one of beams based on the consistent couple stress theory (Hadjesfandiari & Dargush, 2011; Shaat, 2015) or beams based on the modified couple stress theory (Shaat, 2015; Yang et al., 2002). Thus, the proposed second-rotation gradient-based beam outweighs the ones based on the consistent or the modified couple stress theories in accounting for the higher-order couple stresses. Compared to the first-strain gradient theory (Mindlin & Eshel, 1968) and the modified strain gradient theory (Lam et al., 2003) which only depend on the first-order deformation, the proposed theory accounts for the second-order deformation of the material particle. The bending rigidities of the beam have been defined according to the modified strain gradient theory as follows (Lam et al., 2003):





D = D0 1 +

bh h

2 

; Dh = D0 δ 2

(66)

where in the context of the modified strain gradient theory, three material constants are introduced in addition to the conventional ones to capture the first-order micro-deformation of the material particle. Lam et al. (2003) defined bh and δ for the case in which the three material parameters have the same value as:

b2h = (10.6 − 15.4ν ) 2 ; δ 2 = (0.57 − 1.05ν ) 2

(67)

where ν is the Poisson’s ratio. According to the modified strain gradient theory and Eqs. (66) and (67), the higher-order deformations of the beam only depend on the bending modulus μ 2 . On the other hand, in the framework of the proposed secondrotation gradient theory, the higher-order deformations of the beam depend on a higher-order bending modulus μ 4 in addition to the first-order bending modulus μ 2 , for the case when l = lh = . 6. Effects of the higher-order rotation gradient on the bending behavior of micro/nanobeams 6.1. Analytical solution The proposed Euler–Bernoulli beam model based on the second-rotation gradient theory is studied to show the effects of the higher-order couple stress on the bending behavior of nano-beams. Consider a simple supported beam of length L, width b, and thickness h such that the boundary conditions to be applied are given by:

w(x ) = 0 at x = 0, L 1 Mxx + Pxy − Yxz,x = 0 at x = 0, L 2

(68)

The solution of the equilibrium Eq. (64) which satisfies the considered boundary conditions can be expressed as:

w(x ) = X sin

π L

x

(69)

The beam is assumed to be subjected to a sinusoidal load,

p(x ) = q0 sin

π L

x

(70)

where X is the amplitude of the beam deflection and q0 is the load intensity. Substituting Eqs. (69) and (70) into Eq. (64), the amplitude of the beam is given by:

X=

 π 4

D

L

q0

 6

+ Dh πL

(71a)

The beam bending rigidities D and Dh can be computed according to the proposed second-rotation gradient theory from Eq. (65). On the other hand, the rigidities according to the modified strain gradient theory can be computed from Eq. (66).

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Fig. 7. Effects of the beam size on the higher-order bending rigidity Dh according to the SRGT and MSGT (ν = 0.38).

6.2. Numerical results The effects of the higher-order material parameter lh on the bending rigidity and the deflection of the beam are investigated when considering the proposed Second-Rotation Gradient Theory (SRGT) and the Modified Strain Gradient Theory (MSGT). To this end, a simple supported beam of length L = 20h, width b = 2h, and thickness h is considered. For the sake of study, the beam microstructure material parameters based on the SRGT and MSGT are assumed having the same value of . The plotted curves in Fig. 7(a) and (b) show the variations of the ratio of the beam rigidities( Dh /D )/h as a function of the beam thickness for different values of the microstructure material parameter . In these plots, both the SRGT and the MSGT are considered where D and Dh are defined in Eq. (65) and Eq. (66) for the SRGT and the MSGT, respectively. Fig. 7(a) shows the results for = 10, 20, and 60 nm while the results for = 1, 5, and10 μm are shown in Fig. 7(b). For larger thicknesses, the higher-order bending rigidity Dh is very small when compared with the bending rigidity D. On the other hand, a decrease in the size of the beam is accompanied with an increase of the higher-order rigidity, Dh . The SRGT shows a continuous increase in the higher-order rigidity Dh for the continuous decrease in the beam size. However, the MSGT shows a little increase in the rigidity Dh until it reaches a peak value and then it is hypothetically fixed at this value for the continuous decrease in the beam size, as shown in Fig. 7(a). Moreover, when = 1, 5, and 10 μm in micro-scale, the higher-order bending rigidity Dh according to the SRGT significantly increases when decreasing the beam size. Furthermore, the SRGT shows an increase in the higher-order

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95

Fig 8. The distribution of the nondimensional beam deflection w(x) × (D/q0 L4 ) along the beam length ( = 1 μm) when considering different thicknesses of the beam.

bending rigidity when increasing the material parameter value, while the MSGT provides no sense for the change in the material parameter value, as shown in Fig. 7(b). Fig. 8 shows the distribution of the nondimensional beam deflection, w(x) × (D/q0 L4 ), along the beam length for a beam whose microstructure material parameter = 1 μm when considering the modified strain gradient theory (MSGT), couple stress theory (Dh = 0), and the proposed theory SRGT. Our developed theory shows a decrease in the beam defection when the beam size is decreased. This is due to the increase in the effect of the higher-order rigidity Dh . However, the MSGT provides almost the same deflection with no attention to the decrease in the beam size. Furthermore, this figure reflects the merit of including the higherorder rigidity when measuring the beam deflection. As an example, for micro/nano-resonators, it is necessary to accurately estimate the deflection of the elastic beam nearby and at its pull-in behavior (Shaat & Abdelkefi, 2015; Shaat & Mohamed, 2014). It is clear from Fig. 8 that both the couple stress and the MSGT overestimate the beam deflection which can result in disastrous situations. On the other hand, the proposed SRGT succeeds to capture the higher-order deformation of the material structure of the beam with easy and effective way in comparing to the available couple stress and strain gradient theories.

7. Experimental identification of the additional material constants of the proposed theory Some studies have been performed to determine the additional material constants presented in the non-classical continuum theories. The material constants of the local and nonlocal Micromorphic theory for single crystal silicon and single crystal diamond are determined by matching the theoretical-based phonons dispersion relations with the dispersion relations from the atomistic calculations or experimental measurements (Chen & Lee, 2003; Zeng, Chen, & Lee, 2006). The additional material length scales of the Cosserat theory and the classical couple stress theory are obtained analytically via homogenization of heterogeneous elastic materials (Bigoni & Drugan, 2007). An atomistic approach is proposed by Maranganti and Sharma (2007) to determine the material constants of the second-strain gradient theory by matching the dispersion relations extracted from the strain-gradient elasticity and those extracted from the atomistic approach. Moreover, the additional material constant of the modified couple stress theory has been determined experimentally for some materials in different studies. Based on a bending test of a micro-cantilever beam, Park and Gao (2006) estimated the material constant of a couple-stress based epoxy using the experimental setup presented by Lam et al. (2003). With the aid of the experimental results for an electrostatically-actuated micro-cantilever presented by Osterberg (1995), Baghani (2012) and Rokni, Seethaler, Milani, Hosseini-Hashemi, and Li (2013) estimated the material length scale parameter of the modified couple stress theory for silicon. With the aid of a bending test for a micro-cantilever, McFarland and Colton (2005) determined the material length scale parameter of the modified couple stress theory for a Polypropylene. It should be mentioned that atomistic-based calculations of the material constants of the non-classical theories are limited for single crystal materials assuming perfect crystal structure of the material. However, the real feature of the material structure includes defects which exceed the limits of applicability of the atomistic simulations. Furthermore, atomistic simulations are inapplicable for polycrystalline materials because of the need to model the material structure containing many crystals with each crystal has millions of atoms which needs huge computational efforts. Therefore, experimental measurements of the non-classical material constants are preferred. In most of the performed experimental studies, only one material parameter is determined where it is difficult to extract more than one material parameter from one experimental setup. However, by

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Fig. 9. Steps to determine the additional material constants of the proposed second-rotation gradient theory: (a) bending test of micro/nano-beams using AFM, (b) fitting of the discrete experimental data, and (c) determining l and lh that gives a theoretical curve that fits with the experimental curve.

considering different sizes of the test specimen starting from macro-to-nano scales more than one material parameter can be determined via the same experimental setup. In the present section, illustrations for two experimental setups that can be used for determining the material constants, l, lh , and η introduced in the constitutive equations of the proposed second-rotation gradient theory are presented. In the first experimental setup, micro/nano beams tested under bending using atomic force microscopy (AFM) can be used for determining l and lh . Another experimental setup for micro/nano plates under tension can be used to identify η. It is observed that the equilibrium equation of the beam (Eq. (64)) in terms of deflection does not depend on η. Thus, two material length scales l and lh can be determined through the bending test of micro-/nano-beams. Then, the obtained values of l and lh are used with the tension plate-based experimental setup to determine the value of η. Four main steps are needed to determine the additional material constants, l and lh , by the bending test of micro/nano beams. In the first step, lithography process can be used to fabricate beam-like-specimens (Calahorra, Shtempluck, Kotchetkov, & Yaish, 2015). These specimens are needed at different sizes and made from the material for which the additional material constants will be measured. In the second step, each specimen is tested under bending using AFM, as shown in Fig. 9(a). In this test, the specimen is subjected to a point load by the AFM tip. After that, the total descent of the AFM tip is measured and used to determine the specimen’s deflection. The total descent of the AFM tip depends on the elasticity of the specimen and the elasticity of the AFM tip. Therefore, the specimen’s deflection, ws , can be determined from the total descent, Ztip , as follows (Calahorra et al., 2015):

ws = Ztip − wtip

(71b)

where wtip = F/Ktip is the induced deformation of the AFM tip (F is the applied AFM tip force and Ktip is the AFM tip stiffness) (Calahorra et al., 2015). In the third step, the obtained experimental data of deflections of the specimens are collected and plotted versus the beam’s size. As depicted in Fig. 9(b), after measuring the deflection of each specimen, wexp , using AFM, the nondimensional difference between the measured deflection, wexp , and the corresponding theoretical deflection using the classical theory of elasticity, wcl , is calculated as follows:

w∗ =

wexp − wcl wcl

(71c)

Then, the resulted nondimensional difference in deflection is plotted against the beam’s size (red circles in Fig. 9(b)). After that, a curve fitting for the resulted discrete experimental data can be obtained (the black line in Fig. 9(b)).

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97

In the fourth step, the obtained experimental-based curve that fits the discrete experimental data is used to determine the required material length-scale parameters. More precisely, the objective is to detect the values of l and lh that gives a theoretical nondimensional difference in deflection that fits with the experimental curve, as shown in Fig. 9(c). First, the obtained experimental curve is divided into three regions, as shown in Fig. 9(c). In region 1, the beam can be modeled using the classical theory of elasticity. In region 2, because the size of the beam is not small enough to reflect the effects of the material length scale parameter lh , the beam, in region 2, can be modeled by neglecting the contributions of lh and only considering the contributions of l. In region 3, both l and lh will contribute to the beam’s deflection. Second, a theoretical nondimensional difference in deflection is calculated based on the proposed second-rotation gradient theory. In the beginning, lh is set to zero in the mathematical model and the nondimensional difference in deflection is calculated. By using try and error technique, the value of l is selected such that it gives a theoretical curve that fits most of region 2 of the experimental curve, as shown in Fig. 9(c). After that, the beam is modeled considering both l and lh and the value of lh is selected such that it gives a theoretical curve that fits with the whole experimental curve, as shown in Fig. 9(c). With these four steps, the material length scale parameters l and lh of the proposed second-rotation gradient theory can be determined for different materials using bending of micro/nano-beams. These steps will help in developing experimental setups to detect these material constants for different materials in the future. To detect the value of η, plate specimens under tensions are needed to be tested. According to Eqs. (50) and (52), the induced displacements of plates under tensions depend on l, lh , and η. The values of l and lh can be obtained via the previous beam-based experimental setup. By following the same procedure as the beam-based experimental setup, the value of η can be obtained via the plate-under-tension test. 8. Conclusions In this research study, a continuum model equipped with an atomic lattice model is proposed to investigate the applicability of the available non-classical continuum theories for different types of materials. Then, this continuum model is adopted to model the higher-order deformation of the material microstructure. After that, a linear theory of elasticity, named second-rotation gradient theory, has been developed in which the material particle is idealized as a microvolume having a rigid translation, a rigid micro-rotation, and a higher-order micro-deformation. The associated strain energy density to this theory is a function of the infinitesimal strain tensor and the first and the second gradients of the rotation tensor. The second-rotation gradient theory has introduced a new higher-order deformation measure which is needed for very small sizes of the material continuum and/or to account for the irregularities and defects in the material structure. It was shown that this proposed theory has a nonlocal feature as it accounts for the two-neighbor interactions similar to the second-strain gradient theory. However, the proposed theory is easier to apply for the elasticity field problems. It was reported that only five material constants have been introduced in the context of the second-rotation gradient theory to define the constitutive equations instead of the eighteen material constants introduced by the second-strain gradient theory. The second-rotation gradient theory outweighs the first-strain gradient theory and the couple stress theory in accounting for the higher-order deformation of the material particle instead of representing it as a rigid microvolume or considering only its first-order deformation. The effects of the higher-order deformation of the material structure on the bending behavior of the second-rotation gradient-based Euler–Bernoulli beam have been investigated. A higher-order bending rigidity has been introduced to capture the higher-order deformation of the material structure and compared to the one introduced by the modified strain gradient theory. The higher-order bending rigidity of the proposed theory showed a high dependency on the beam size and the microstructure material parameters unlike the one of the formal theory. This proposed theory is very beneficial and will have a strong usefulness in order to effectively study the performance and response of many mechanical/dynamical systems in nano-scale applications including bio-mass sensors, nano-electromechanical systems, and drug delivery systems. Acknowledgments The authors would like to thank Prof. Igor Sevostianov, New Mexico State University, USA and Prof. Fatin F. Mahmoud, Zagazig University, Egypt for the useful discussions. References Baghani, M. (2012). Analytical study on size-dependent static pull-in voltage of microcantilevers using the modified couple stress theory. International Journal of Engineering Science, 54, 99–105. Bigoni, D., & Drugan, W. J. (2007). Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. Journal of Applied Mechanics, 74, 741–753. Calahorra, Y., Shtempluck, O., Kotchetkov, V., & Yaish, Y. E. (2015). 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