Coupling of surface effect and hyperelasticity in combined tension and torsion deformations of a circular cylinder

International Journal of Solids and Structures 85–86 (2016) 172–179

Contents lists available at ScienceDirect

International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr

Coupling of surface effect and hyperelasticity in combined tension and torsion deformations of a circular cylinder T. Sigaeva, A. Czekanski∗ Lassonde School of Engineering, York University, Toronto, Canada

a r t i c l e

i n f o

Article history: Received 31 October 2015 Revised 2 February 2016 Available online 19 February 2016 Keywords: Surface effect Residual stress Hyperelasticity Tension and torsion Poynting effect Stiffness

a b s t r a c t During the past decade there has been increasing research into the role of surface mechanics which can be relevant for many of today’s elastomeric applications. This paper incorporates surface effects into large deformation elasticity for a well-known continuum model of a circular cylinder under combined torsion and tension loadings and thus, generalizes it for application at smaller scales, such as nano- and microscales. At these scales, the effect of the surface and residual surface stresses on the overall deformation of the bulk material is of critical importance. The proposed model employs the Gurtin–Murdoch theory to represent the surface effect as a residually prestressed thin hyperelastic film of separate elasticity, perfectly bonded to the bulk. Using a constitutive model for which analytical solution can be derived, we demonstrate the pronounced effect of the surface and residual surface stresses not only on the values of resultant axial force and twisting moment, but also on the axial stiffness, the torsional rigidity and the Poynting effect. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Recently, considerable attention has been paid to surface mechanics, a theory, that accounts for differences in molecular behavior of materials near the surface and inside the bulk. The focus of ongoing research in this area has been on extending the principles of continuum mechanics for application at smaller scales, such as micro- and nano- scales. It is well known that at these scales, the relatively large surface area to volume ratio of a representative element has a critical effect on the overall deformation of the bulk material. Gurtin and Murdoch (1975) have developed a continuum-based theory to account for surface/interface energies, stresses and tension . Various versions of this theory were comprehensively discussed in a recent publication by Ru (2010). The Gurtin–Murdoch model is mathematically equivalent to the assumption of a surface coated by or reinforced with a thin, stiff solid film with different elastic properties. As a result, even in the absence of external loading, residual stresses occur in the area of bonding (Huang and Wang, 2006). Both surface and residual surface stresses have a pronounced effect on the material properties at smaller scales and cannot be ignored.

∗

Corresponding author. Tel.: +1 41 67362100. E-mail address: [email protected] (A. Czekanski).

http://dx.doi.org/10.1016/j.ijsolstr.2016.02.019 0020-7683/© 2016 Elsevier Ltd. All rights reserved.

Significance of the surface effect was clearly demonstrated in the framework of linear elasticity (small deformations) in a number of papers – see, for example, Sigaeva and Schiavone (2014a; 2014b) and the references contained therein. Fewer works address relevant examples of the contribution of the surface effect in finite deformations in the framework of hyperelasticity, although the theory is well developed (Steigmann and Ogden, 1997; Huang and Wang, 2006; Huang and Sun, 2007). Due to emerging applications and rapid advancement of nanotechnologies in rubber industry (Thomas and Stephen, 2010), more research needs to be conducted in this field. In addition, since hyperelasticity models are widely employed to predict the response of tissues, artery wall and muscels (Humphrey et al., 1992), the consideration of the surface and residual surface effect on micro-level can significantly improve the accuracy of the existing models in biomechanics. A relevant example of the contribution of the surface effect for finite deformations is given by Altenbach et al. (2013), where an axially extended rod is considered. The results obtained demonstrate how the addition of a prestressed reinforcing film will impact the extensional stiffness of the cylinder. In this paper we further explore the influence of the surface and residual surface stresses on the mechanical response of a solid circular cylinder under combined torsion and axial extension. This problem is of great theoretical and practical importance and, in addition to stiffness, has many interesting nonlinear effects such as torsional rigidity and Poynting effect (Barenblatt and Joseph, 1997; Horgan and Murphy, 2012a), which can be considered for our comparative study.

T. Sigaeva, A. Czekanski / International Journal of Solids and Structures 85–86 (2016) 172–179

F

F s∗ = F s · F s0

F s0

M

F s, F

γ∗

γ0 , κ0

γ, κ M

(a)

(b)

(c) F

Fig. 1. (a) Fictitious stress-free configuration for a thin film; common (b) reference configuration and (c) current configuration for a hyperelastic cylinder with a thin film.

For this particular example, we demonstrate how the contribution of the surface effect can change for different materials at different scales. This paper is organized as follows: in Section 2 we present the basics of the Gurtin–Murdoch theory in the framework of hyperelasticity. The implementation of the proposed model for an example of torsional deformations of a cylinder superimposed on axial extension is presented in Section 3. Finally, Section 4 provides numerical results demonstrating the pronounced effect of the surface and residual surface stresses. 2. Theoretical background To account for difference in the material behavior on the surface and inside the bulk, which can be especially significant on nano- and micro- scales, we employ a Gurtin–Murdoch theory (Gurtin and Murdoch, 1975). This theory is mathematically equivalent to the assumption of a surface coated by or reinforced with a thin solid film of different elasticity. We consider the equilibrium of a deformable solid occupying a cylindrical region whose generators are parallel to the z-axis of a cylindrical coordinate system. We assume that this cylinder is made of a homogeneous, initially isotropic and incompressible hyperelastic material. The lateral surface of the cylindrical body is coated with a thin reinforcing film representing the surface effect. This film is made of a homogeneous, isotropic, hyperelastic material with elastic constants different from those of the cylinder. Let us denote the configurations for the bulk and thin film with no external loadings by κ 0 and γ 0 , respectively, and call them ‘reference configurations’ (Fig. 1b). The so-called ‘current configurations’, denoted by κ and γ , represent the configurations for the bulk and thin film when subjected to external loadings (Fig. 1c). Deformation gradients from κ 0 to κ and from γ 0 to γ are given by F and Fs . Furthermore, the physical creation of the cylindrical body is generally generating stresses in the undeformed configuration (Huang and Wang, 2006), which are, again, different on the surface and inside the bulk. By interpreting this for Gurtin–Murdoch theory, we can state that residual stresses exist for the cylinder in κ 0 and for the thin hyperelastic film in γ 0 . Since the residual stresses in the bulk materials have already been thoroughly studied in the framework of hyperelastic theory due to their significance for biomechanical applications (Merodio et al., 2013; Merodio and Ogden, 2015), in this work we assume that residual stresses exist in the surface film only and contribute to boundary conditions of

173

the bulk. Following this interpretation, we can also say that mathematical composition of the thin elastic film on a cylindrical body resulting in a perfect bonding requires its initial prestretching or precompression and does not cause significant residual deformations in the cylinder bulk (cylinder’s residual stresses are omitted). Therefore, let us assume that there exists a ‘fictitious stress-free’ configuration γ ∗ for the thin reinforcing film and it is decomposed to the reference configuration γ 0 with the help of the deformation gradient F s0 (Fig. 1a). Due to residual surface stresses, the constitutive behavior of the surface film can not be described with a deformation gradient Fs only. Using the multiplicative decomposition approach to account for the existence of the fictitious stressfree configuration (Lubarda, 2004), we deduce that the surface hyperelastic strain energy density is a function of F s∗ = F s · F s0 , i.e., U = U (F s∗ ). Further application of the well-known principle of the material frame-indifference allows as to write the surface hyperelastic strain energy density as a function of the right Cauchy–Green strain tensor C s = F sT · F s with F s0 acting as a parametric tensor, s s i.e. U = U (F sT 0 · C · F 0 ) (Altenbach et al., 2013). Thus, the surface Cauchy stresses acting on the lateral surface of the cylinder are given by

2 s ∂U F · · F sT Js ∂Cs   sT 2 s sT s sT = s F s · U1 F s0 · F sT ·F , 0 + 2U2 F 0 · F 0 · C s · F 0 · F 0 J

σs =

(1)

where J s = det(F s ) is the surface area ratio and Ui = ∂∂W , i = 1, 2. Iis Invariants are expressed as in Altenbach et al. (2013) s s I1s = tr(F sT 0 · C · F 0 ),

s s 2 I2s = tr(F sT 0 · C · F0) .

The presence of the film in both the reference and current configurations gives rise to a non-classical Young–Laplace boundary condition, while equilibrium equations and constitutive equations of the bulk solid remain unchanged (Huang and Wang, 2006). For the current configuration in the absence of body forces they look as follows

σ = −pI + 2W1 B − 2W2 B−1 ,

(2)

∇ · σ = 0 in the bulk,

(3)

σ · n = t 1 on the edges, s s ∇ · σ    = t2 on the lateral surface.

σ ·n−

(4)

the surface effect

Here p is the Lagrange multiplier to ensure incompressibility, I is an identity tensor, B = F · F T is the left Cauchy–Green strain tensors, Wi = ∂∂W (i = 1, 2) with invariants I1 = trB and I2 = I i

− trB2 ]; σ represents the Cauchy stresses; ∇ and ∇ s are the usual (3D) and tangential differential (2D) nabla operators for the bulk cylinder and thin film, respectively. Normal to the surface is given by n; and t1 , t2 are the prescribed tractions. In the following section we specify the loading type in order to study the influence of the surface effect on the mechanical behavior of the material. 1 2 2 [ (trB )

3. Tension and torsion of a circular cylindrical bar Assume that a cylinder of an initial length L and radius A is subjected to an axial load F and twisting moment M on its ends so that it changes its length and radius to l and a, respectively. Using cylindrical coordinate system, a point in the reference configuration with coordinates (R, , Z) with respect to the basis (eR , e , eZ )

174

T. Sigaeva, A. Czekanski / International Journal of Solids and Structures 85–86 (2016) 172–179

will move to a point (r, θ , z) with respect to the basis of current configuration (er , eθ , ez ) when subjected to deformations. The semi-inverse representation for this particular type of deformation can be written as

1 r = √ R,

λ

boundary condition in the form of (4), i.e.:

σrr

θ =  + λτ Z, z = λZ,

dσrr σrr − σθ θ + = 0, dr r 1 |r=a + σθsθ |r=a = 0. a   the surface effect

where λ = l/L is an extension ratio and τ = φ /L is an angle of twist per unit length. Thus, the deformation gradient for the bulk of the cylinder is given by

The first equation yields that p is a function of r only. After integration and some simple manipulations, the Lagrange multiplier, as a function of a radial reference coordinate R, is

√ 1 F = √ (er ⊗ eR + eθ ⊗ e ) + λτ Reθ ⊗ eZ + λez ⊗ eZ .

p( R ) = 2 W 1

1

λ

λ

B−1

1



A R

√

W1 r1 dr1 +

λ

A

σθsθ |R=A .  

the surface effect

The left Cauchy–Green strain tensor and its inverse are, respectively,

B=

− 2W2 λ − 2λτ 2

er ⊗ er +

1



+ λτ 2 R2 eθ ⊗ eθ

λ λ +λ3/2 τ R(eθ ⊗ ez + ez ⊗ eθ ) + λ2 ez ⊗ ez , √ = λ(er ⊗ er + eθ ⊗ eθ ) − λτ R(eθ ⊗ ez + ez ⊗ eθ ) 1 + 2 + τ 2 R2 ez ⊗ ez . λ

Therefore, we know all components of σ and we are able to derive expressions for the resultant axial force and twisting moment. The following equation for the resultant axial force is obtained

F =

2π 0

a 0

σzz rdrdθ +

= 4π (λ − λ−2 )



A 0

2π a 0

σzzs ds

(W1 + λ−1 W2 )RdR

A πA −2π τ 2 (W1 + 2λ−1 W2 )R3 dR + √ (−σθsθ

λ

0

Finally, the nonzero components of Cauchy stresses from (2) are

σrr = −p + 2W1 σθ θ = −p + 2W1

1

the surface effect

(5)

− 2W2 λ,

λ 1

where the last term corresponds to the surface and residual surface effects contribution. Similarly, we derive the expression for the resultant torque



+ λτ R − 2W2 λ, √ σθ z = 2τ R(W1 λ3/2 + W2 λ ), 1 σzz = −p + 2W1 λ2 − 2W2 2 + τ 2 R2 .

λ

2 2

M=

λ

2π 0



A 0

σzθ r2 drdθ +

2π a 0

σzsθ ads =

(W1 + λ−1 W2 )R3 dR + 2π A2 λ−1 σzsθ |R=A .   

(6)

the surface effect

4. Discussion

√ 1 F s = √ eθ ⊗ e + λτ Reθ ⊗ eZ + λez ⊗ eZ

λ

√

and, therefore, J s = λ. The surface deformation gradient corresponding to the residual deformations in the reference configuration is given by

F s0 = x1 e ⊗ e˜ + x2 eZ ⊗ eZ˜ . Here basis (eR˜ , e˜ , eZ˜ ) belongs to the fictitious stress-free configuration. Physically, x1 and x2 represent the circumferential and axial stretches of the thin cylindrical film caused by fitting it to the lateral surface of the bulk cylinder. Thus, the nonzero components of the Cauchy surface stresses from (1) can be written as

√

σθsθ = 2U1 (x21 λ−3/2 + x22 λτ 2 R2 ) + 4U2 x41 λ−5/2

a 0

= 4π τ

Next, let us consider deformations in the thin surface film. Since deformations along the thickness are uniform, the surface deformation gradient can be expressed as



+τ 2 R2 [x42 λ5/2 + 2x21 x22 λ−1/2 ] + x42 λ3/2 τ 4 R4 ,



|R=A +2σzzs |R=A ),  



σθsz = 2x22 τ R U1 λ + 2U2 (x21 + x22 λ3 + x22 λ2 τ 2 R2 ) ,

 σzzs = 2x22 λ3/2 U1 + 2U2 λx22 (λ + τ 2 R2 ) . Now we are in the position to determine the Lagrange multiplier from equilibrium Eq. (3) and the stress-free lateral surface

An incompressible Neo–Hookean material model is the simplest hyperelastic model and it is widely employed in prominent works for demonstration of different effects in the problem of combined tension and torsion (Kanner and Horgan, 2008; Hamdaoui et al., 2014). Here we use it to establish constitutive law for the bulk and derive analytical expressions for the resultant axial force (5) and torque (6). Thus, let us assume that the energy density for the bulk cylinder is given by

W=

μ 2

(I1 − 3 ),

(7)

where μ is the shear modulus. In fact, the Neo–Hookean material model (7) can adequately describe the mechanical behavior in the intermediate extension ratio range, i.e. λ ∈ [0.5, 2]. As in Altenbach et al. (2013), we assume that the surface energy density is given by

U=

1 1 λs (I1s − 2 )2 + μs (I2s − 2I1s + 2 ) 8 4

(8)

with material parameters λs and μs of the Lame’s type. This surface energy density can be reduced to the classical relations of linear surface elasticity for infinitesimal deformations (Gurtin and Murdoch, 1975). To simplify our calculations even further, let us put λs = μs .

T. Sigaeva, A. Czekanski / International Journal of Solids and Structures 85–86 (2016) 172–179

175

Therefore, the non-dimensional resultant axial force and torque are given

F

μπ A2

= F˜ = F˜b + F˜s = λ −

1

λ2

−

1 2 ψ +α 4



 1 2

x21 x22 − 4x22 λ

3 1 1 1 − x41 3 + 2x21 2 + 3x42 λ3 + ψ 2 (2x22 − 3x21 x22 2 λ λ λ  1 3 + x42 λ2 − x22 λψ 2 − x42 λψ 2 ) , 2 2

M

μπ A3

(9)

˜ =M ˜b + M ˜s =M =

1 ψ + α x22 ψ (−4 + 3x21 λ−1 + 3x22 λ2 + 3x22 λψ 2 ) 2

(10)

μs with α = μ serving as a surface effect contribution parameter and A ψ = τ A denoting, for simplicity, the total angle of twist. Here F˜b ˜ b correspond to the resultants of the cylinder without conand M ˜ s , linear functions of tribution of the surface effect, while F˜s and M parameter α , denote the resultants necessary to maintain the surface film deformations identical to those occuring on the cylinder’s lateral surface. Furthermore, we note that for the cylinder without external ˜ = 0 (ψ = 0), an appropriate choice of the forces, i.e., F˜ = 0 and M surface residual circumferential and axial stretches x1 , x2 should lead to insignificant changes of the total value of stretch λ. For instance, x1 , x2 ∈ [0.8, 1.2] (Altenbach et al. (2013)) will result in λ ∈ [0.94, 1.06], which, in our opinion, seems a reasonable physical response. Next, let us discuss the surface effect contribution parameter α = μμAs . In Shenoy (2002), it is stated that the ratio of the surface elastic constant to the bulk elastic constant is proportional to the material length scale, i.e., μμs ∝ lμ . Thus, it will be interesting to compare the contribution of the surface effect for different materials at different scales, where a decrease of the scale of the problem is performed via the radius A → 0. Atomistic simulations have revealed that the material length scale for metals are typically of the order 0.01–0.1 nm (e.g., lμ = 10−10 m) (Miller and Shenoy, 20 0 0; Wang et al., 2006). Even though analogous data for materials exhibiting hyperelastic behavior are not available, to the authors knowledge, we can make the following assumptions about the order of their material length scale. For example, the ratio of the surface tension γ s to the bulk modulus μ for soft material such as hydrogel can vary over several orders of magnitude from nanometers to micrometers (Kang and Huang, 2010). Here we will assume that lμ for hydrogel is of γ the same order as μs and choose lμ = 10−6 m. Finally, we pick the intermediate order between the above values for soft hydrogel and hard metals to estimate the material length scale for rubbers, i.e. γ lμ = 10−8 m. In fact, this ratio for rubber μs also fluctuates over

several orders of magnitude (i.e., 10−7 − 10−9 m) depending on the rubber composition (Zhang et al., 2003). Fig. 2a and b show the fractions of the resultant force and moment due to the surface effect for three values of lμ corresponding to suggested material length scales for steel, rubber and hydrogel. It can be clearly seen that the contribution of the surface effect increases at smaller scales. The surface effect for metals does not play a significant role at the microscale level, while for rubber it becomes notable (we pick several values of α shown on Fig. 2 to distinguish between stronger and weaker surface effect contribution). Very soft materials such as hydrogels are expected to have significant surface effects even at the macroscale level. Further research, atomistic simulations and experiments should be done to determine relevant values of μμs for hyperelastic materials to support our assumptions.

Fig. 2. Fractions of the resultant (a) axial force and (b) twisting moment due to surface effects at different values of radius A for: (i) metals, (ii) rubbers and (iii) hydrogels assuming λ = 1.5, ψ = 1, x1 = x2 = 1.

One possible way to constrain values of total angle of twist ψ for our model that assumes λ ∈ [0.5, 2] is to consider the Poynting effect. For a better visual representation of the nonlinearity of this effect, we will temporarily include λ > 2 into the analysis (on Fig. 3). The Poynting effect is an essential characteristic of materials with nonlinear mechanical behavior (Poynting, 1909), where in the absence of an axial load, rods elongate under application of torsional deformation; conversely, if the rod is in pure torsion, compressive force should be applied to restrain it axially. This refers to a positive Poynting effect and is especially evident in elastomers. Recently it was discovered that biopolymer gels can have a negative or reverse Poynting effect (Storm et al., 2005), i.e., they shorten when twisted or, differently, to sustain them in pure torsional deformations, tensile force should be applied. The analysis and simulations from Horgan and Murphy (2012a), Levin et al. (2014) indicate that the negative Poynting effect can be caused by material anisotropy, residual deformations and presence of inhomogeneity. In Horgan and Murphy (2012b), for instance, it was experimentally shown that stretched rabbit papillary muscles contracted in length when twisted. Since the surface effect in our model is represented by a thin elastic film of different elasticity perfectly bonded to the body, which can be also residually deformed, it would be interesting to study how the Poynting effect in this type of ‘composition’ is reflected. The Poynting effect increases (or decreases) due to specific factors if under the same twist it changes its length more (or less) than a cylinder without these factors. Mathematically, the Poynting effect represents the relation between λ and ψ from (9), when F˜ = 0. First, we will neglect the residual deformations assuming x1 = x2 = 1. As can be seen from Fig. 3a, increasing α , the surface effect parameter, leads to a small decrease of the Poynting effect for small twist ψ < 1, followed by a considerable increase for bigger values of ψ > 1. This results from the nontrivial response of the surface and its material model on the applied deformations (8) and (9). Residual surface effects, in turn, have more apparent influence on the Poynting effect for small values of the total angle of twist ψ . Fig. 3b, c and d illustrate the influence of the uniform (x1 = x2 = x0 ), circumferential (x1 = x0 , x2 = 1) and axial (x1 = 1, x2 = x0 ) residual surface deformations for different values of x0 and fixed α = 0.1. A comparison of these results reveals that the axial residual deformations (Fig. 3d) have the strongest influence on the magnitude of the Poynting effect—precompression (x0 < 1) of the surface increases it and prestretching (x0 > 1) decreases it. For the circumferential residual deformations (Fig. 3c), in turn, these

176

T. Sigaeva, A. Czekanski / International Journal of Solids and Structures 85–86 (2016) 172–179

Fig. 3. Dependance of the axial stretch on the total angle of twist (Poynting effect curves) for a cylinder with surface effect and: (a) no residual surface deformations (x1 = x2 = 1), (b) uniform residual surface deformations x1 = x2 = x0 (α = 0.1), (c) circumferential residual surface deformations x1 = x0 , x2 = 1 (α = 0.1), (d) axial residual stretches x1 = 1, x2 = x0 (α = 0.1). Gray dash-dotted lines are used to indicate allowed values of λ ≤ 2.

Fig. 4. Stresses due to surface effect  s = (2σzzs − σθsθ ) |R=A plotted versus: (a) axial stretch λ for ψ = 0, 1, 1.5; (b) total angle of twist ψ for λ = 0.7, 1, 1.5. Influence of the surface effect parameter α on the magnitude of a non-dimensional resultant axial force plotted versus: (c) axial stretch λ in the case of tension only (ψ = 0) (Altenbach et al., 2013); (d) axial stretch λ in the case of tension and torsion for ψ = 1; (e) total angle of twist ψ in the case of tension and torsion for λ = 0.7, 1, 1.5.

T. Sigaeva, A. Czekanski / International Journal of Solids and Structures 85–86 (2016) 172–179

177

Fig. 5. Stresses due to surface prestretching (x1 = x2 = x0 = 1.2)  s = (2σzzs − σθsθ ) |R=A with (. ) = (. ) |x0 =1.2 −(. ) |x0 =1 plotted versus: (a) axial stretch λ for ψ = 0, 1, 1.5; (b) total angle of twist ψ for λ = 0.7, 1, 1.5. Influence of the residual surface effect parameter x1 = x2 = x0 (when α = 0.1) on the magnitude of a non-dimensional resultant axial force plotted versus: (c) axial stretch λ in the case of tension only (ψ = 0) (Altenbach et al., 2013); (d) axial stretch λ in the case of tension and torsion for ψ = 1; (e) total angle of twist ψ in the case of tension and torsion for λ = 0.7, 1, 1.5.

changes are opposite and more moderate. Further on, for simplicity, we will work with uniform residual stresses only (x1 = x2 = x0 ), influence of which on the Poynting effect can be found on Fig. 3b. What is more, we discover that the presence of residual deformations (for example, subfigure of Fig. 3b) leads to a negative Poynting effect (λ < 1 for x0 = 1.2 and ψ < 0.5), i.e., the cylinder tends to shorten for sufficiently small values of total angle of twist ψ , but then tends to elongate on further twisting with increasing value of ψ . In Levin et al. (2014), it was shown that initial deformations of the inner part of a composite cylinder lead to the negative Poynting effect. Analogously, in our investigations we state that at smaller scales, such as nano- and micro-, where the surface is represented by a thin film attached to the body, the negative Poynting effect can take place due to specific residual surface deformations. As mentioned earlier, choice of the Neo–Hookean material energy (7) for the bulk have constrained λ ∈ [0.5, 2]. For the total angle of twist to be consistent with this constraint, from Fig. 3 it is evident that its value should be in the interval ψ ∈ [0, 2] (gray dash-dotted lines are used to indicate allowed values of λ ≤ 2). Now, let us consider how the surface will influence the re˜ = M˜ b + M˜ s . sultant axial force F˜ = F˜b + F˜s and twisting moment M From (9) we can decide whether the total resultant force in the axial direction is increasing or decreasing due to the surface effect by looking at the sign of its surface counterpart F˜s . Providing F˜s > 0, the magnitude of F˜ is increased if the resultant axial force

associated with bulk cylinder F˜b is tensile or is decreased if F˜b is compressive. Expression (5) implies that sgn(F˜s ) = sgn( s ), where  s = (2σzzs − σθsθ ) |R=A . For the specifically chosen surface energy density (8), Fig. 4a and b demonstrate combinations of stretch λ and twist ψ , for which doubled axial surface stresses are greater s > σ s and  s (or smaller) than the circumferential ones, i.e., 2σzz θθ s s s > 0 (or 2σzz < σθ θ and  < 0). Using these data, we can observe changes in F˜ when the scale of the problem decreases and surface to volume ratio becomes considerable. Consider Fig. 4c–e, which plot the resultant axial force for increasing α , the surface effect parameter. In the case of tension only (ψ = 0), the surface effect contribution always increases the magnitude of the axial resultant force (tensile or compressive) since  s = 0 when λ = 1 (Fig. 4c) (Altenbach et al., 2013). In the case of a combined tension and torsion, the magnitude of F˜ will rise almost everywhere except the region with λ > 1 where F˜s changes its sign and the circumferential surface stresses becomes larger than axial ones ( s > 0). This is illustrated in Fig. 4d (its subfigure) and e, where F˜ is plotted, respectively, versus stretch λ for ψ = 1 and versus total angle of twist ψ for the cases of compression λ = 0.7, pure torsion λ = 1 and tension λ = 1.5. Values of stretch λ and twist ψ , for which F˜ = 0, can be also found from the Poynting effect curves shown on Fig. 3a. Additionally, one can notice on Fig. 4e that F˜ < 0 for all values of total angle of twist ψ when λ = 1. Physically, it is related to a classical Poynting effect and means that the state of pure torsion is possible only if F˜ is compressive. In the absence of this force, twisting would lead to elongation of the cylinder (Fig. 3a).

178

T. Sigaeva, A. Czekanski / International Journal of Solids and Structures 85–86 (2016) 172–179

Fig. 6. Influence of the surface effect parameter α and residual surface effect parameter x0 (when α = 0.1 and x1 = x2 = x0 ) on the magnitude of a non-dimensional resultant twisting moment plotted versus total angle of twist ψ or axial stretch λ for the cases of: (a),(c) tension and torsion with ψ fixed (ψ = 1); (b),(d) pure torsion (λ = 1).

Fig. 7. Influence of the surface effect parameter α and residual surface effect parameter x0 (when α = 0.1 and x1 = x2 = x0 ) on the values of effective axial stiffness and torsional rigidity plotted versus total angle of twist ψ or axial stretch λ for the cases of: (a),(c) tension and torsion with ψ fixed (ψ = 1); (b),(d) pure torsion (λ = 1).

Next, we will investigate how initial prestretching (x1 = x2 = x0 > 1) or precompression (x1 = x2 = x0 < 1) of the surface film will affect both total axial resultant force F˜ and its surface counterpart F˜s , for which residual surface stresses are omitted and α = 0.1. For this purpose, we need to determine sgn(F˜s ) = sgn( s ), where (. ) = (. ) |x0 =1 −(. ) |x0 =1 . For example of the surface film initially undergoing tensile deformations with x0 = 1.2, Fig. 5a and b demonstrate combinations of stretch λ and twist ψ , for which the increment of doubled axial surface stresses due to this initial prestretching is greater (or smaller) than the increment of

s > σ s s the circumferential surface stresses 2σzz θ θ and  > s s s ˜ 0 (or 2σzz < σ and  < 0). Correspondingly, if Fs > 0

θθ

and force Fb > 0 is tensile, the magnitude of the total resultant axial force F˜ increases due to surface prestretching; otherwise, if force F˜b < 0 is compressive then F˜ decreases due to surface prestretching. Resultant axial forces for α = 0.1 and different residual strains x0 are depicted in Fig. 5c–e. In general, initially precompressed surface film decreases the magnitude of F˜ required to maintain imposed deformations, while for the prestretched surface it is just the

T. Sigaeva, A. Czekanski / International Journal of Solids and Structures 85–86 (2016) 172–179

opposite. This can be clearly seen when F˜ is plotted, respectively, versus stretch λ for ψ = 0 on Fig. 5c (Altenbach et al., 2013) or ψ = 1 on Fig. 5d and versus total angle of twist ψ for the cases of compression λ = 0.7, pure torsion λ = 1 and tension λ = 1.5 on Fig. 5e. Similarly to the previous case, in zones with changing sgn(F˜s ) we can observe different behavior, where precompression of the surface increases the magnitude of F˜ and prestreching decreases it. As before, values of stretch λ and twist ψ , for which F˜ = 0, can also be determined from the Poynting effect curves shown on Fig. 3b. Apart from this, one can notice the presence of the negative Poynting effect on Fig. 5e for the cylinder in pure torsion (λ = 1) with prestretched surface x0 > 1. To ensure that such a cylinder does not change its length during pure torsion deformations, it is necessary to apply tensile axial force F˜ > 0 for small twist and compressive axial force F˜ < 0 for larger twist. In the absence of this force, the cylinder would contract when slightly twisted (subfigure of Fig. 3b). Let us now turn our discussion to the influence of the surface effect on the magnitude of the resultant twisting moment ˜ = M˜ b + M˜ s . As before, from (10) we can see that M ˜ will increase M or decrease depending on the sign of M˜ s . The first thing to take ˜ > 0 for all allowed combinations of λ, ψ , into account is that M x0 , α . Secondly, M˜ s can be a very small negative value only for significantly precompressed surface x0 = 0.8 and insignificant twist ψ < 0.5 (from (6) sgn M˜ s = sgn σθsz |R=A and σθsz > 0 almost everywhere). Thus, we can immediately conclude the following: larger ˜ is required to maintain prescribed deformatorsional moment M tions in the cylinder if the surface effect behavior is taken into ˜ with no residual account. Furthermore, value of this moment M surface stresses decreases if the surface is initially precompressed; and rises if the surface is initially prestretched. Supporting graphs are shown on Fig. 6. Finally, another well-established way to highlight the effect of the surface and residual surface stresses on the mechanical response of the cylinder is to consider its effective axial stiffness ˜ dF˜ M | and torsional rigidity ddψ |λ . In the absence of residual surdλ ψ face stresses, these characteristics will increase with increasing values of α , the surface effect contribution parameter (Fig. 7a and b). Prestretching of the surface will make the cylinder bulk stiffer, while, on the contrary, precompression will cause the body to re˜ in a softer manner (Fig. 7c sist applied axial force F˜ or torque M ˜ and d). It should be noted that the effective axial stiffness ddλF |ψ without surface effect monotonically decreases with λ, while addition of the surface effect causes its monotonical decrease followed by an increase after some value of stretch λ is achieved.

5. Conclusion This research has examined the pronounced effect of surface and residual surface stresses on the mechanical response of a cylinder under torsional and tensile deformations. Particularly, when the scale of the problem decreases, surface and residual surface stresses cause significant changes in the behavior of the resultant axial force, the twisting moment, the axial stiffness, the torsional rigidity and the Poynting effect. An important finding in this study is the possibility of a negative Poynting effect at smaller scales, where the relatively large surface area to volume ratio of a representative element as well as residual stresses have a critical effect on the overall deformation of the bulk material. Further research should be done to capture this non-trivial effect experimentally.

179

Acknowledgements The authors gratefully acknowledge the generous financial support of York University. References Altenbach, H., Eremeyev, V.A., Morozov, N.F., 2013. On the influence of residual surface stresses on the properties of structures at the nanoscale. In: Altenbach, H., Morozov, N.F. (Eds.), Surface Effects in Solid Mechanics. In: Advanced Structured Materials, 30. Springer, Berlin, Heidelberg, pp. 21–32. doi:10. 1007/978- 3- 642- 35783- 1_2. Barenblatt, G.I., Joseph, D.D. (Eds.), 1997. Collected papers of R. S. Rivlin, 1. SpringerVerlag, New York doi:10.1007/978- 1- 4612- 2416- 7. Gurtin, M.E., Murdoch, A.I., 1975. A continuum theory of elastic material surfaces. Arch. Rational Mech. Anal. 57 (4), 291–323. doi:10.10 07/BF0 0261375. Hamdaoui, M.E., Merodio, J., Ogden, R.W., Rodriguez, J., 2014. Finite elastic deformations of transversely isotropic circular cylindrical tubes. Int. J. Solids Struct. 51 (5), 1188–1196. doi:10.1016/j.ijsolstr.2013.12.019. Horgan, C.O., Murphy, J.G., 2012a. Finite extension and torsion of fiber-reinforced non-linearly elastic circular cylinders. Int. J. Non-Lin. Mech. 47 (2), 97–104. doi:10.1016/j.ijnonlinmec.2011.03.003. Horgan, C.O., Murphy, J.G., 2012b. On the modeling of extension-torsion experimental data for transversely isotropic biological soft tissues. J. Elast. 108 (2), 179– 191. doi:10.1007/s10659- 011- 9363- 0. Huang, Z.P., Sun, L., 2007. Size-dependent effective properties of a heterogeneous material with interface energy effect: from finite deformation theory to infinitesimal strain analysis. Acta Mechanica 190 (1–4), 151–163. doi:10.1007/ s00707- 006- 0381- 0. Huang, Z.P., Wang, J., 2006. A theory of hyperelasticity of multi-phase media with surface/interface energy effect. Acta Mechanica 182 (3–4), 195–210. doi:10.1007/ s00707- 005- 0286- 3. Humphrey, J.D., Barazotto, R.L., Hunter, W.C., 1992. Finite extension and torsion of papillary muscles: a theoretical framework. J. Biomech. 25 (5), 541–547. doi:10. 1016/0 021-9290(92)90 094-H. Kang, M.K., Huang, R., 2010. Effect of surface tension on swell-induced surface instability of substrate-confined hydrogel layers. Soft Matter 6, 5736–5742. doi:10. 1039/C0SM00335B. Kanner, L.M., Horgan, C.O., 2008. On extension and torsion of strain-stiffening rubber-like elastic circular cylinders. J. Elast. 93 (1), 39–61. doi:10.1007/ s10659- 008- 9164- 2. Levin, V.A., Zubov, L.M., Zingerman, K.M., 2014. The torsion of a composite, nonlinear-elastic cylinder with an inclusion having initial large strains. Int. J. Solids Struct. 51 (6), 1403–1409. doi:10.1016/j.ijsolstr.2013.12.034. Lubarda, V.A., 2004. Constitutive theories based on the multiplicative decomposition of deformation gradient: thermoelasticity, elastoplasticity, and biomechanics. Appl. Mech. Rev. 57 (2), 95–108. doi:10.1115/1.15910 0 0. Merodio, J., Ogden, R.W., 2015. Extension, inflation and torsion of a residually stressed circular cylindrical tube. Continuum Mech. Thermodyn. 28 (1), 157– 174. doi:10.10 07/s0 0161-015-0411-z. Merodio, J., Ogden, R.W., Rodriguez, J., 2013. The influence of residual stress on finite deformation elastic response. Int. J. Non-Lin. Mech. 56, 43–49. doi:10.1016/ j.ijnonlinmec.2013.02.010. Miller, R.E., Shenoy, V.B., 20 0 0. Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11 (3), 139–147. doi:10.1088/0957-4484/11/3/ 301. Poynting, J.H., 1909. On pressure perpendicular to the shear planes in finite pure shears, and on the lengthening of loaded wires when twisted. Proc. Roy. Soc. London A: Math. Phys. Eng. Sci. 82 (557), 546–559. doi:10.1098/rspa.1909.0059. Ru, C.Q., 2010. Simple geometrical explanation of Gurtin–Murdoch model of surface elasticity with clarification of its related versions. Sci. China Phys. Mech. Astronomy 53 (3), 536–544. doi:10.1007/s11433-010-0144-8. Shenoy, V.B., 2002. Size-dependent rigidities of nanosized torsional elements. Int. J. Solids Struct. 39 (15), 4039–4052. doi:10.1016/S0 020-7683(02)0 0261-5. Sigaeva, T., Schiavone, P., 2014a. Solvability of the Laplace equation in a solid with boundary reinforcement. J. Appl. Math. Phys. 65 (4), 809–815. doi:10.1007/ s0 0 033- 013- 0359- 4. Sigaeva, T., Schiavone, P., 2014b. Surface effects in anti-plane deformations of a micropolar elastic solid: integral equation methods. Continuum Mech. Thermodyn. 1–14.. doi:10.10 07/s0 0161-014-0404-3. Steigmann, D.J., Ogden, R.W., 1997. Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. Roy. Soc. London A: Math. Phys. Eng. Sci. 453 (1959), 853–877. doi:10.1098/rspa.1997.0047. Storm, C., Pastore, J.J., MacKintosh, F.C., Lubensky, T.C., Janmey, P.A., 2005. Nonlinear elasticity in biological gels. Nature 435 (7039), 191–194. doi:10.1038/ nature03521. Thomas, S., Stephen, R. (Eds.), 2010, Rubber Nanocomposites: Preparation, Properties, and Applications. John Wiley & Sons (Asia) Pte Ltd doi:10.1002/ 9780470823477. Wang, J., Duan, H.L., Huang, Z.P., Karihaloo, B.L., 2006. A scaling law for properties of nano-structured materials. Proc. Roy. Soc. London A: Math. Phys. Eng. Sci. 462 (2069), 1355–1363. doi:10.1098/rspa.2005.1637. Zhang, Y., Ge, S., Rafailovich, M.H., Sokolov, J.C., Colby, R.H., 2003. Surface characterization of cross-linked elastomers by shear modulation force microscopy. Polymer 44 (11), 3327–3332. doi:10.1016/S0 032-3861(03)0 0185-X.