Couple stress based strain gradient theory for elasticity

Transcription

1 International Journal of Solids and Structures 39 (00) Couple stress based strain gradient theory for elasticity F. Yang a,b, A.C.M. Chong a, D.C.C. Lam a, *, P. Tong a a Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, PR China b Institute of Computational Engineering and Science, Southwest Jiaotong University, Chengdu , Sichuan, PR China Received 13 January 001; received in revised form 9 January 00 Abstract The deformation behavior of materials in the micron scale has been experimentally shown to be size dependent. In the absence of stretch and dilatation gradients, the size dependence can be explained using classical couple stress theory in which the full curvature tensor is used as deformation measures in addition to the conventional strain measures. In the couple stress theory formulation, only conventional equilibrium relations of forces and moments of forces are used. The couple s association with position is arbitrary. In this paper, an additional equilibrium relation is developed to govern the behavior of the couples. The relation constrained the couple stress tensor to be symmetric, and the symmetric curvature tensor became the only properly conjugated high order strain measures in the theory to have a real contribution to the total strain energy of the system. On the basis of this modification, a linear elastic model for isotropic materials is developed. The torsion of a cylindrical bar and the pure bending of a flat plate of infinite width are analyzed to illustrate the effect of the modification. Ó 00 Elsevier Science Ltd. All rights reserved. Keywords: Couple stress; Strain gradient; Elasticity 1. Introduction The size dependence of deformation behavior in micron scale had been experimentally observed in metals (Fleck et al., 1994; St olken and Evans, 1998; Nix, 1989; Stelmashenko et al., 1993; Ma and Clarke, 1995; Poole et al., 1996) and polymers (Lam and Chong, 1999; Chong and Lam, 1999). The behavior cannot be explained by the conventional theories of mechanics and the couple stress theory (Fleck and Hutchinson, 1997) has been used to explain the size dependence of the deformation behavior. The difference between the conventional theories of mechanics and the couple stress theories of mechanics are discussed below. In the mechanics of particles (Martin and Thornton, 1995), forces applied onto the material particles are unobservable. The forces can only be determined through observations on the motions of the material particles. They can also be determined by investigating the change of kinetic energy of the material particles. In conventional Newtonian mechanics, a material particle in a deformable continuum can only undergo translations. The applied force on the material particle is the only power to alter its motion and the force is * Corresponding author. Tel.: ; fax: address: medcclam@ust.hk (D.C.C. Lam) /0/$ - see front matter Ó 00 Elsevier Science Ltd. All rights reserved. PII: S (0)0015-X

2 73 F. Yang et al. / International Journal of Solids and Structures 39 (00) related to the acceleration of the material particle through Newton s second law. (Alternately, the force can be defined from the perspective of the kinetic energy of the material particle using d Alembert s principle.) In conventional mechanics, a force drives a material particle to translate. In the classical couple stress theories (Toupin, 196; Mindlin and Tiersten, 196; Koiter, 1964; Mindlin, 1964) for linear elastic materials, the applied loads on the material particle include not only a force to drive the material particle to translate but also a couple to drive it to rotate. In this classical conception, only the conventional equilibrium relationships of forces and moments (of forces) are enforced and the couple is unconstrained in the absence of higher order equilibrium requirements. More recently, Eringen and coauthors have studied deformation behavior in the microscale continua. Their works are summarized in Eringen s published works in 1971 and in 1998 (Eringen, 1971, 1998). In their micropolar theory, they suggested that the location vector and a rigid vector representing its inner rotation define the motion of a particle. The particle s motion is characterized by the changes of the vectors. The equilibrium relations in the micropolar theory are established on the basis of conservational laws of momentum and moment of momentum. Since there is no higher order equilibrium relation in the micropolar, microstretch and micromorphic continua works, the attachment of the rigid vector for rotation to a particle is arbitrary. In this paper, the concept of representative volume element is introduced, and the force and couple applied to a single material particle are defined. A new set of equilibrium relations for a system of material particles to account for the rotations of these material particles is developed. The results are then generalized to the couple stress theory of continuum. By the introduction of a higher order equilibrium condition, the arbitrary nature of couples in the classical couple stress theory is resolved without the use of rigid vector attachment condition, as was used in the micropolar theory. The paper will also show that the new equilibrium relations dictated that the couple stress tensor must be a symmetric tensor. Since the symmetric part of curvature tensor is the additional measure of deformation that conjugates to the couple stress, this means that the antisymmetric part of curvature tensor does not conjugate to the couple stress and it does not appear explicitly in the deformation energy density function. On the basis of this result, a theoretical framework for a modified couple stress theory is developed, and a linear elastic constitutive model is developed within this framework. We will show that the number of required material length scale parameters is only one in the new modified couple stress framework, instead of two in the classical couple stress theory. Two simple applications of the modified couple stress theory, torsion of a thin cylinder and pure bending of a flat plate of infinite width, are presented.. Equilibrium relationship in conventional mechanics of particles In conventional theories of continuum mechanics, a material body is modeled as a continuum consisting of an infinite number of material particles. We regard each material particle as a geometric point with specific geometric and physical characteristics, including location, mass and motion. When the material particle is modeled as a geometric point, it can only undergo translation motion, and is changed only by the force applied onto the particle. Denoting the displacement vector of the particle as u and the force vector applied to it as F, we can write Newton s second law of motion as F ¼ m u; ð1þ where m is the mass and u is the acceleration vector of the material particle. According to d Alembert s principle (Martin and Thornton, 1995), the force can be defined from the perspective of the kinetic energy of the material particle as F ¼ D Dt ok o_u ; ðþ

3 F. Yang et al. / International Journal of Solids and Structures 39 (00) where D=Dt is the total time derivative operator. The kinetic energy of the particle k is k ¼ 1 m_u _u: ð3þ If a set of forces is applied to the material particle, the resultant force F, the vector sum of the forces, is related to the acceleration or kinetic energy of the material particle through F ¼ X F i ¼ m u or F ¼ X F i ¼ D ok Dt o_u : ð4þ For static equilibrium, the forces on the material particle are balanced. The equilibrium relation for a single material particle in conventional mechanics becomes F ¼ X F i ¼ 0: ð5þ In order to obtain the equilibrium relations for a system of material particles, an equivalent system of forces is investigated. As shown in Fig. 1, a force F A applied to a material particle at point A in a system in Fig. 1(a) is equivalent to an equal force F 0 A at another point B of the system and a couple of forces L0 A applied to the system as shown in Fig. 1(c). The equivalence is established through an intermediate step shown in Fig. 1(b), in which two equal and opposite forces F 0 A and F00 A with F0 A ¼ F A are applied to point B and the force F A and the force F 00 A form the couple L0 A. The equivalent force and couple are, respectively F 0 A ¼ F A; L 0 A ¼ðx B x A ÞF A ; ð6þ where x A and x B are the position vectors of points A and B. In conventional mechanics, a couple of forces is a free vector in the space of the material particle system. The couple can be translated and applied to any point in the system, which means that the motive effect of a couple on the system of material particles is independent of the location where the couple is applied. Thus, the forces F i and the couples of forces L i applied to a set of material particles within the system is equivalent to a resultant force and a resultant couple of forces, and the couple can be applied to an arbitrary point within the system. The resultant force F 0 and the couple of forces L 0 are, respectively, F 0 ¼ X F i ; L 0 ¼ X ðx i F i þ L i Þ; ð7þ where the origin is specified as the point onto which the equivalent resultant force is applied. If the system of the material particles is in equilibrium, the resultant force and couple vanish, giving the conventional equilibrium relations X X Fi ¼ 0; ðxi F i þ L i Þ¼0; ð8þ for a system of material particles. Fig. 1. Equivalence of force: a force at point A is equivalent to a force at point B and a couple of the force.

4 734 F. Yang et al. / International Journal of Solids and Structures 39 (00) Generalized motions and forces of a particle In general higher order theories, the material particle should possess the conventional characteristics of translation, but should also possess the characteristics of rotations and deformations. The conventional conception of material particle as a geometric point is inadequate for use in higher order theories of mechanics. Rotations of material particles in higher order theory can be realized by encasing the material particle with a representative volume element. The representative volume element is hypothesized as an infinitesimal neighborhood of the material particle consisting of an infinite number of smaller material particles. Each of the smaller material particles is also encased with an even smaller representative volume element. Thus, the behavior of a material particle at a specific scale can be characterized by the behavior of the smaller material particles in its representative volume element. For example, consider an arbitrary material particle located in the representative volume element of the material particle characterized by a location vector x and an infinitesimal volume dv. The displacement u 0 of the conventional material particle at x 0 can be represented by u 0 ¼ uðx 0 ; tþ ¼uðx; tþþhðx; tþdx; ð9þ where Dx ¼ x 0 x and h is the rotation vector. h is defined as h ¼ 1 : u r; ð10þ in which is the alternating tensor and r is the Hamiltonian differential operator. The velocity of the conventional material particle at x 0 is _u 0 ¼ _uðx 0 ; tþ ¼_uðx; tþþ _ hðx; tþdx; where h _ is the angular velocity. The kinetic energy of the representative volume element at x, is then 1 kðx; tþ ¼ dv q0 _u 0 _u 0 dv 0 ¼ 1 q 0 dv 0 _u _u þ q 0 Dxdv 0 _u h _ þ 1 h dv dv _ q 0 ½ðDx DxÞg Dx DxŠdv 0 h; _ ð1þ dv where q 0 is the mass density of the particles and g is the unit tensor. If x is the mass center of the representative volume element, the above equation becomes kðx; tþ ¼ 1 m_u _u þ 1 _ h I _ h; where m and I, m ¼ q 0 dv 0 ; dv I ¼ q 0 ½ðDx DxÞg Dx DxŠdv 0 dv are the mass and the moment of inertia tensor of the particle, respectively. The generalized forces applied to the particle encased by the representative volume element are defined as F ¼ D Dt ok o_u ¼ m u; L ¼ D Dt ok o _ h ¼ I h; where F is a force and L is a couple. F and L drive the translation and the rotation of the particle, respectively. If a set of forces F i and a set of couples L i are applied to the particle, the resultant force F and couple L are related to the kinetic energy of material particle via ð11þ ð13þ ð14þ ð15þ

5 F. Yang et al. / International Journal of Solids and Structures 39 (00) F ¼ X F i ¼ D ok Dt o_u ¼ m u; L ¼ X L i ¼ D ok Dt o h _ ¼ I h: ð16þ In static equilibrium, the forces and couples on the material particle encased by the representative volume element are F ¼ X F i ¼ 0; L ¼ X L i ¼ 0: ð17þ Thus, the equilibrium relations for a single material particle consist of the conventional equilibrium relation of forces as in Eq. (5) and the additional equilibrium relation of couples. This concept of representative volume element containing couples is now suited for use as a foundation for couple stress theories. 4. Equilibrium of a material particle system Using the equilibrium relations for a single material particle encased with a representative volume element as a basis, we can now establish the equilibrium relations for a system of material particles to account for the equilibrium of moments of couples. The equilibrium relations of a conventional system of forces applied to a system of multiple material particles are derived from a resultant force and a resultant couple of forces applied to an arbitrary point. The couple of forces is a free vector in the conventional mechanics, which means that the effect of the couple applied on an arbitrary point in the space of the system of material particles is independent of the position of the point. In other words, the couple can translate to any point in space freely and the resulting motive effects are unchanged. As a result, only the conventional force equilibrium and moment equilibrium are involved in the equilibrium relations as given in Eq. (8) (Toupin, 196; Mindlin and Tiersten, 196; Koiter, 1964; Mindlin, 1964). The equivalence of a couple that is not a free vector but a driving force that rotates the material particles is shown in Fig.. The couple vector L A at A in a system of material particles in Fig. (a) is equivalent to a couple L 0 A and a couple of couples M0 A applied to the point B in Fig. (c). This equivalence is derived through an intermediate step as shown in Fig. (b). In the intermediate step, two opposite couple vectors L 0 A and L00 A of equal magnitude, L 0 A ¼ L A ð18þ are placed at point B. The couple of couples M 0 A defined as M 0 A ¼ðx B x A ÞL A ð19þ is equivalent to the couple L A at point A and the couple L 00 A at point B. For a system of material particles, the forces F i and couples L i applied to the system are equivalent to a resultant force F 0, a resultant couple L 0 and a resultant couple of couples M 0 applied to the origin where F 0, L 0 and M 0 are given as Fig.. Equivalence of couple: a couple at point A is equivalent to a couple at point B and a moment of the couple.

6 736 F. Yang et al. / International Journal of Solids and Structures 39 (00) F 0 ¼ X F i ; L 0 ¼ X ðx i F i þ L i Þ; M 0 ¼ X x i L i : ð0þ In static equilibrium, these equations are equal to zero giving the relations, X Fi ¼ 0; X ðxi F i þ L i Þ¼0; X xi L i ¼ 0; ð1þ which are the equilibrium relations for a system of discrete particles encased by representative volume element. 5. Governingequations in couple stress theory of deformable body 5.1. Equilibrium equations The equilibrium relations in Eq. (1) are for a system of discrete particles. In this section, we shall develop the equilibrium relations for a continuum. Consider an arbitrary volume v 0 of a deformable body bounded by piecewise smooth surfaces denoted by ov 0. We denote t n and l n respectively as the force and couple vectors per unit area transmitted through the surface ov 0. The subscript n represents the direction of the external normal n to the surface. Additionally, we use f and l to denote the body force and the body couple per unit volume of the material particles respectively. For the volume of the continuum, the first two equilibrium equations of Eq. (1) can be rewritten as f dv þ t n ds ¼ 0; v 0 ov 0 ðþ ðx f þ lþdv þ ðx t n þ l n Þds ¼ 0; v 0 ov 0 where x is the position vector of a material particle in the continuum. Generalizing the conventional Cauchy s principle, Koiter (1964) proposed t n ¼ t n; ð3þ and l n ¼ l n; where t is the stress tensor and l is the couple stress tensor. Using the divergence theorem to transform the surface integrals in Eq. () to volume integrals, we obtain ðt rþfþdv ¼ 0; v0 ð5þ ½x ðtrþfþ : t þ l rþlšdv ¼ 0: v 0 Since the volume v 0 is arbitrary, the volume dependence can be eliminated which then leads to t rþf ¼ 0; ð6þ l rþl : t ¼ 0; which are the equilibrium relations of deformable body and are equivalent to the first two equations of Eq. (1). The second equilibrium relation in Eq. (6) indicates that the stress tensor t generates an equivalent body couple : t acting together with l to maintain the equilibrium of the continuum. Since the couple moment must vanishes, we have ð4þ

7 F. Yang et al. / International Journal of Solids and Structures 39 (00) x ðl : tþdv þ x l n ds ¼ 0; ð7þ v 0 ov 0 where l : t is the residual body couple and l n is the surface couple traction. This is equivalent to the third equation in Eq. (1). Using the divergence theorem, we can rewrite Eq. (7) as ½x ðl : t þ l rþ : lšdv ¼ 0; ð8þ v 0 which leads to the conclusion that : l ¼ 0; ð9þ i.e., the couple stress tensor is symmetric. The expressions in Eq. (6) can be combined through decomposition of the stress tensor and the couple stress tensor. Let us denote the symmetric and antisymmetric parts of the stress tensor, respectively, as r ¼ 1ðt þ tt Þ; s ¼ 1ðt tt Þ; ð30þ where t T is the transpose of t. Decomposition of the couple stress tensor l into the spherical part lg and the deviatoric part m gives, l ¼ lg þ m; l ¼ 1 trðlþ; trðmþ ¼0: ð31þ 3 The equilibrium relations in Eq. (6) can be rewritten as ðr þ sþrþf ¼ 0; ð3þ lrþmrþl : s ¼ 0: Eliminating the antisymmetric stress tensor s in Eq. (3) gives the final equilibrium equation r rþ 1 : ðm rrþlrþþf ¼ 0: ð33þ This is in the same form as that of the classical couple stress theory (Koiter, 1964). However, unlike the classical couple stress theory, Eq. (33) requires that the deviatoric couple stress tensor m to be symmetric. 5.. Generalized strains and principle of virtual work In this section, the deformation measures, e and v, conjugate to r and m, are derived via the application of the principle of virtual work. Consider a convex volume v 0 of the deformable body bounded by a surface ov 0 that consists of a finite number of sub-surfaces whose outer normals form a continuous vector field. The deformation energy density is assumed to be a function of the gradients of generalized motions, which are the gradients of translation and rotation in the present revised couple stress model. The principle of virtual work states that the change in the deformation energy equals the work done by the external generalized loads through the virtual displacement du and the virtual rotation dh, i.e., d wðu r; h rþdv ¼ ðdu f þ dh lþdv þ ðdu t n þ dh l n Þds; ð34þ v 0 v 0 ov 0 where w is the deformation energy density per unit volume. Substituting Eqs. (3) and (4) into Eq. (34) and applying the divergence theorem, we can rewrite the principle of virtual work as dwdv ¼ ½du ðtrþfþþdur: t þ dh ðlrþlþþdhr: lšdv: ð35þ v 0 v 0

8 738 F. Yang et al. / International Journal of Solids and Structures 39 (00) Let us denote the symmetric parts of the displacement gradient and the rotation gradient, respectively, as e ¼ 1 ðu rþruþ; ð36þ v ¼ 1 ðh rþrhþ; where e is the strain tensor and v is the symmetric curvature tensor. We also define c ¼ 1 ðh r rhþ ð37þ as the antisymmetric curvature tensor, and define as h ¼ : 1u r: Using Eqs. (35) (38) and the equilibrium equations of Eq. (6), we derive the following relation, dwdv ¼ ðde : r þ dv : mþdv; ð39þ v 0 v 0 which shows that e and v are conjugated to r and m, respectively. Note that m and v are both deviatoric and symmetric. Since the volume v 0 is arbitrary, Eq. (39) implies dwðu r; h rþ¼de : r þ dv : m: ð40þ The variation of w is dwðu r; h rþ¼dwðe; h; v; cþ ¼de : ow oe þ dh ow oh ow ow þ dv : þ dc : ov oc : Thus, we have r ¼ ow oe ; m ¼ ow ov ; ow oh ¼ 0; ow oc ¼ 0: ð4þ The above equations show that the deformation energy density w does not depend explicitly on the rotation h (the antisymmetric part of displacement gradient) and the antisymmetric curvature tensor c (the antisymmetric part of rotation gradient). That is to say, only the symmetric part of displacement gradient (conventional strain tensor) and the symmetric part of rotation gradient (symmetric curvature tensor) contribute to the deformation energy. The rotation vector and the antisymmetric curvature tensor do not contribute to the deformation energy. The strain tensor and symmetric curvature tensor are the deformation measures conjugate to the symmetric stress tensor r and the deviatoric couple stress tensor m, respectively. This conclusion will have important implications for the number of independent higher order material parameters. ð38þ ð41þ 6. Linear isotropic elasticity A linear elastic constitutive law within the present modified couple stress framework is developed in this section. A linear constitutive law for isotropic elastic couple stress materials can be derived from the deformation energy density as given in Eq. (4). For the linear isotropic materials, w is a quadratic function of the invariants of generalized strains. It can be written as w ¼ 1 kðtreþ þ lðe : e þ l v : vþ; where k and l are the Lame s constants, and l is a length scale parameter. Substituting Eq. (43) into Eq. (4), we obtain the linear constitutive relations, ð43þ

9 F. Yang et al. / International Journal of Solids and Structures 39 (00) r ¼ kg trðeþþle; ð44þ l 1 m ¼ llv: In this model, only one length scale parameter is needed in addition to the conventional Lame constants. The single material length scale parameter dependence is the direct consequence of the conclusion of Section 5. that w is a function of the strain and the symmetric curvature tensors only, and does not depend explicitly on the rotation (the antisymmetric part of the deformation gradient) and the antisymmetric part of the curvature tensor. This is a particularly useful contribution of the study as this conclusion reduced the experimental problem of finding two independent higher order material length scale parameters to just one. 7. Examples In this section, two simple examples, torsion of a thin cylindrical bar and pure bending of a plate of infinite width, are derived to illustrate the effects of strain gradients. The governing equation is Eq. (33) and the general boundary conditions are detailed in Appendix A Torsion of a thin cylindrical bar The x-axis of the Cartesian coordinate system is taken along the axis of the cylindrical bar. The displacement field in the cylindrical bar is assumed to be in the same form as in the conventional mechanics, u x ¼ 0; u y ¼ jxz; u z ¼ jxy; ð45þ where j is the twist per unit length of the cylindrical bar. From the displacement field, the components of the strain tensor, the rotation vector and the symmetric curvature tensor are e xx ¼ e yy ¼ e zz ¼ e yz ¼ 0; e xy ¼ 1jz; e xz ¼ 1jy; h x ¼ jx; h y ¼ 1jy; h z ¼ 1jz; v xx ¼ j; v yy ¼ v zz ¼ 1j; v xy ¼ v yz ¼ v zx ¼ 0: From the constitutive relation in Eq. (44), the components of the stress tensor and the couple stress tensor are r xx ¼ r yy ¼ r zz ¼ r yz ¼ 0; r xy ¼ ljz; r xz ¼ ljy; ð47þ m xx ¼ ll j; m yy ¼ m zz ¼ ll j: One can easily verify that the stresses and couple stresses satisfy the equilibrium equation (33) and the traction free boundary conditions in Eq. (A.3) on the outer cylindrical surface. The resultant torque at the ends of the cylindrical bar can be determined through the virtual work principle, " Q ¼ Q 0 1 þ 6 l # ; ð48þ a where a is the radius of the cylindrical bar and Q 0 is the torque per unit length of the cylindrical bar calculated from conventional theory, Q 0 ¼ 1 plja4 : The last term in the bracket of Eq. (48) shows the effect of rotation gradient and cylinder radius on the torsional rigidity of the cylindrical bar. The significant size effect on the torsional rigidity is shown in Fig. 3, where the horizontal axis is the ratio of the cylindrical radius to the material length scale parameter, a=l, ð46þ ð49þ

10 740 F. Yang et al. / International Journal of Solids and Structures 39 (00) Fig. 3. Effect of cylindrical radius on the torsional rigidity of the cylindrical bar. Q=Q 0 is the normalized torsional rigidity, a is radius of the cylindrical bar and l is the length scale parameter. and the vertical axis is the normalized torque, Q=Q 0. When the cylindrical radius is close to the material length scale parameter, the rigidity is about seven times of the magnitude of conventional result. For torsion of the copper wires, l was determined to be 3 lm (Chong et al., 001). Elastic deformation in copper structures in the micron-scale is strongly affected by strain gradients. 7.. Bending of a plate of infinite width We take the x y plane of the Cartesian coordinate system as the midplane of the plate. The y-axis is parallel to the direction of the width. The displacement field is assumed in the same form as in conventional mechanics, i.e., u x ¼ jxz; u y ¼ 0; u z ¼ 1 m 1 m jz 1 jx ; ð50þ where j is the curvature of the bended midplane of the plate, and m is Poisson s ratio. This displacement field satisfies the boundary conditions u y ¼ 0; h x ¼ h z ¼ 0 at y ¼1: ð51þ The corresponding strain tensor, rotation vector and symmetric curvature tensor are e xx ¼ jz; e yy ¼ 0; e zz ¼ m 1 m jz; e xy ¼ e yz ¼ e zx ¼ 0; h x ¼ h z ¼ 0; h y ¼ jx; ð5þ v xx ¼ v yy ¼ v zz ¼ v yz ¼ v zx ¼ 0; v xy ¼ 1j: From Eq. (44), the components of the stress tensor and couple stress tensor are obtained as follows: r xx ¼ l 1 m jz; r yy ¼ ml 1 m jz; r zz ¼ r xy ¼ r yz ¼ r zx ¼ 0; ð53þ m xx ¼ m yy ¼ m zz ¼ m yz ¼ m zx ¼ 0; m xy ¼ ll j: It is easy to verify that the stresses and couple stresses satisfy the equilibrium equation in Eq. (33) and the traction-free boundary conditions in Eq. (A.3) at z ¼h= surfaces with h being the plate thickness. The tractions and the couple tractions on the y ¼1surfaces are respectively,

11 F. Yang et al. / International Journal of Solids and Structures 39 (00) Fig. 4. Effect of plate thickness on the bending rigidity of the plate for m ¼ 0:3. M=M 0 is the normalized bending rigidity, h is the thickness of the plate and l is the length scale parameter. p x ¼ p z ¼ 0; p y ¼ ml 1 m jz; q x ¼ll j; q z ¼ 0; and the tractions and the couple tractions on the x ¼a= surfaces are p x ¼ l 1 m jz; p y ¼ p z ¼ 0; q y ¼ll j; q z ¼ 0: The equivalent resultant moment per unit width is thus " # l M ¼ M 0 1 þ 6ð1 mþ ; ð56þ h where M 0 is the corresponding resultant moment per unit width calculated from the classical theory M 0 ¼ lh3 6ð1 mþ j: ð57þ The last term in the bracket of Eq. (56) gives the contribution of rotation gradients to the bending rigidity and shows the effect of plate thickness on the bending rigidity. The thickness effect on the bending rigidity is shown in Fig. 4, where the horizontal axis is the ratio of plate thickness to material length scale parameter, h=l, and the vertical axis is the normalized moment, M=M 0. When the plate thickness is close to the material length scale parameter for m ¼ 0:3, this bending rigidity is about five times the conventional rigidity. The plate bending behavior is similarly affected by strain gradients as that for torsion of copper wires above-mentioned. ð54þ ð55þ 8. Discussion In classical couple stress theories (Toupin, 196; Mindlin and Tiersten, 196; Koiter, 1964; Mindlin, 1964), the equilibrium of forces and moments are the same as those in conventional mechanics. The deformation measures conjugate to the symmetric stress tensor r and the couple stress tensor l are the strain

12 74 F. Yang et al. / International Journal of Solids and Structures 39 (00) tensor e and the curvature tensor h r, which includes the symmetric part as well as the antisymmetric part of the curvature tensor. For linear elastic materials, there are two independent length scale parameters associated with the symmetric and antisymmetric parts of the curvature tensor, respectively. The two parameters cannot be determined from a single test such as the twisting of a thin cylindrical bar or pure bending of a thin film. A combination of twisting and bending tests is necessary to determine the two parameters. In the present theory, an equilibrium relation for moments of couples is introduced to remedy the freefloating nature of the couple vector. The additional equilibrium relation restricts the couple stress tensor to be symmetric. As a result, the deformation energy becomes independent of the antisymmetric part of the curvature tensor. Consequently for linear isotropic elastic materials, there is only one independent length scale parameter in the couple stress theory, and it is associated with the symmetric curvature tensor. The single length scale parameter can be determined by twisting of slim cylinders of different diameter. The implications of the additional equilibrium relation for couples in the higher order strain gradient elasticity have been investigated and are reported in another paper (Yang et al., submitted for publication). 9. Conclusion In this paper, equilibrium of the moment of couples is introduced as an additional equation for the couple stresses. The additional equilibrium relation requires the couple stress tensor to be symmetric. As a result, the only deformation measures, which contribute to the deformation energy, are the symmetric parts of the displacement gradient (strain tensor) and the rotation gradient (symmetric curvature tensor). A linear elastic constitutive law for isotropic couple stress materials is developed on this basis. The number of material length scale parameters is reduced from the two in the classical couple stress theories to only one in the present theory. The torsion of a thin cylindrical bar and the bending of a thin plate of infinite width are presented to illustrate the effect of the strain gradients in the modified couple stress theory. Acknowledgements This work has been supported by the Research Grants Council of the Hong Kong Special Administrative Region, the People s Republic of China. Fan Yang also acknowledges the support of the fund of Southwest Jiaotong University of PR-China. Appendix A. Boundary conditions The boundary conditions associated with the above modified couple stress theory are examined in this section. Koiter (1964) derived the boundary conditions of the classical couple stress theory using the principle of virtual work for the whole body. He concluded that the normal component of the rotation vector on the boundary surface could not be prescribed independently. In other words, only the three components of the displacement vector and the two tangential components of the rotation vector can be specified as the generalized displacement conditions on a boundary surface. Thus, the total number of generalized displacement conditions is five. The resulting boundary conditions are summarized below: u ¼ u; h ðhnþn ¼ h t ða:1þ

13 on the smooth part of ov i ; and F. Yang et al. / International Journal of Solids and Structures 39 (00) u k ¼ u k ; ða:þ along the intersection C i of two smooth parts of the boundary surface, where the overbar denotes a prescribed quantity, h t is the prescribed rotation vector tangent to the boundary surface, and u k is the displacement component along the direction k of the intersection C i of two smooth parts of the boundary surface. The corresponding generalized traction conditions are r n þ 1n ½mr ðnmnþr þ lš ¼r n 1n ½ðl n nþrš; m n ðnmnþn ¼ l n ðl n nþn ða:3þ on the smooth part of the boundary surface ov i and 1 ½ðn m nþ þ ðnmnþ Š¼1½ðl n nþ þ ðl n nþ Š; ða:4þ along the intersection C i of two smooth parts of the boundary surface, where r n and l n are the forcetraction and couple-traction on the boundary surface, respectively, and the subscripts þ and indicate the two opposite sides of the intersection. References Chong, A.C.M., Lam, D.C.C., Strain gradient plasticity effect in indentation hardness of polymers. J. Mater. Res. 14, Chong, A.C.M., Yang, F., Lam, D.C.C., Tong, P., 001. Torsion and bending of micron-scaled structures. J. Mater. Res. 14, Eringen, A.C., Continuum Physics. Academic Press, New York. Eringen, A.C., Microcontinuum Field Theories. Springer-Verlag, New York. Fleck, N.A., Hutchinson, J.W., Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.), Advances in Applied Mechanics, vol. 33. Academic Press, New York, pp Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W., Strain gradient plasticity: theory and experiments. Acta Metall. Mater. 4, Koiter, W.T., Couple stresses in the theory of elasticity, I and II. Proc. K. Ned. Akad. Wet. (B) 67, Lam, D.C.C., Chong, A.C.M., Indentation model and strain gradient plasticity law for glassy polymers. J. Mater. Res. 14, Ma, Q., Clarke, D.R., Size dependent hardness of silver single crystals. J. Mater. Res. 10, Martin, J.B., Thornton, S.T., Classic dynamics of particles and systems, fourth edition. Saunders College Publishing, Philadelphia. Mindlin, R.D., Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, Mindlin, R.D., Tiersten, H.F., 196. Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, Nix, W.D., Mechanical properties of thin films. Metall. Trans. A 0, Poole, W.J., Ashby, M.F., Fleck, N.A., Micro-hardness of annealed and work-hardened copper polycrystals. Scripta Metall. Mater. 34 (4), Stelmashenko, N.A., Walls, M.G., Brown, L.M., Milman, Y.V., Microindentations on W and Mo oriented single crystals: an STM study. Acta Metall. Mater. 41 (10), St olken, J.S., Evans, A.G., A microbend test method for measuring the plasticity length scale. Acta Metall. Mater. 46, Toupin, R.A., 196. Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 11, Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P., submitted for publication. Strain gradient theory for linear elasticity.