1 Introduction

In recent years,nanoporous materials have attracted extensive attention for their novel mechanicalproperties and their potential applications in microelectrode array chip^[1],molecularfluids^[2],photocatalysis^[3],gas or molecular abruption^[4],catalysis^[5],energy absorption^[6],electricconduction^[7],sensor^[8],actuation^[9],etc. Due to the large amount of surface,nanoporousmaterials exhibit unique mechanical behaviors,which have been confirmed to be size-dependentin experiments^{[10, 11, 12, 13]}. Biener et al.^[11] found that nanoporous gold could be as strong as bulkAu,despite being a highly porous material,and the ligaments in nanoporous gold approachedthe theoretical yield strength of Au. Hodge et al.^[12] believed that the foam strength was governedby the ligament size at the nanoscale,in addition to the relative density. Weissm¨uller etal.^[14] found that the relation among the strain,the surface stress,and the elastic parametersdepended strongly on the geometry of the microstructure.

Up to now,the theory of surface elasticity proposed by Gurtin and Murdoch^[15] has beenextensively used to reveal the size-dependent mechanical properties of nanoporous materials^[16].Based on the theory,Feng et al.^[17] investigated the surface effects on the elastic propertiesof nanoporous materials using the cubic unit cell model. Lu et al.^[18] arrived at a similarconclusion as that of Feng et al.^[17] using the Kelvin model. Further,Xia et al.^[19] used a modifiedTimoshenko beam model to study the elastic microstructural buckling behavior of nanoporousmaterials,which exhibited a significant dependence on the average ligament width. Goudarziet al.^[20] and Moshtaghin et al.^[21] obtained a macroscopic size-dependent yield function fornanoporous materials by considering the residual surface stress and the surface elasticity. Theaforementioned works indicate that the surface elasticity plays a significant role in the elasticproperties of nanoporous materials while the residual surface stress contributes more to thebuckling strength and the yield strength.

The theory of surface elasticity proposed by Gurtin and Murdoch^[15] has also been used toinvestigate the high elastic deformation of nanostructured materials,which has been observedin experiments^{[22, 23, 24]}. Based on the theory,Liu et al.^[25] established the governing equationsof large displacement for nanowires with different boundary conditions. It showed that thefixed-fixed nanobeam would get “stiffer” under the positive residual surface stress,and viceversa,“softer” under the negative one. Wang and Yang^[26] studied the influence of surfacestresses on the postbuckling behavior of nanowires. Wang et al.^[27] presented a nonlinear rodmodel to investigate the surface effects of the superelasticity of nanohelices. Again,the aboveworks verify that the theory of surface elasticity is well adequate to describe the large elasticdeformation behaviors of nanostructured materials. To the best of our knowledge,the largeelastic deformation behaviors of nanowires or nanohelices have been extensively studied,whilethe similar work on nanoporous materials is scarce.

Moreover,the microstructural anisotropy of nanoporous materials was observed in theexperiments^[28],which would have influence on the high deformation on the macro-level. Toreveal the influence of structural anisotropy,Lu et al.^{[29, 30]} developed an anisotropic Kelvinmodel to analyze the mechanical properties of anisotropic open-cell elastic foams without surfaceeffects. However,few papers have been published to study the mechanical properties ofanisotropic nanoporous materials under high elastic deformation.

The main object of the present work is to analyze the surface effects of anisotropic nanoporousmaterials under high elastic deformation. Based on our previous work^{[18, 29, 30]},this paper studiesthe high strain behaviors of elastic nanoporous materials using the Kelvin model,whilst theinfluence of structural anisotropy is also considered. First,by an improved anisotropic Kelvinmodel,the analytical strain-stress relation is deduced for the nanoporous materials under highstrains of both compression and tension. Then,the surface effects on the nonlinear response areinvestigated. Finally,the influence of structural anisotropy on the overall response is discussed. 2 Theoretical description and derivation

2.1 Surface stress-strain relationship

As we know,surface atoms experience different local environments from bulk atoms^[16]. Based on the previous work^[31],the surface stress tensor τ_ij can be expressed as follows:

where γ is the free energy per unit area,"ε_ij is the strain tensor,and δ_ij is the Kroneckerdelta. Further,Lu et al.^[18] and Feng et al.^[17] developed a nanobeam model,which consisted ofbulk and surface with zero thickness. The isotropic surface layer and the isotropic bulk of thematerial are supposed. Further,the uniform thickness t₀ and Young’s modulus E₁ of the surface layer are also assumed. Consequently,a surface constitutive relation for the one-dimensionalform of Eq. (1) can be expressed as^[32] where τ is the surface stress,τ₀ is the residual surface stress,E_s = E₁t₀ is the surface elastic modulus,and εs is the surface strain. A mathematical description of the transverse stresses induced from longitudinal stresses of the tortuous surfaces is given by the generalized Young- Laplace equation as follows^{[33, 34]}: where △σ_ij is the stress jump across a surface,n_i is the unit normal vector,and κ_αβ is the curvature tensor. 2.2 Anisotropic Kelvin model

Nanoporous materials have sponge-like structures with open-cell network structures,whichconsist of interconnected nano-ligaments. As shown in Fig. 1(a),an improved Kelvin modelwith a surface layer and a uniform circular cross-section,which has a periodical structure in thethree-dimensional space,is developed. The anisotropic Kelvin model consists of eight hexagonsand six quadrilaterals,in which the inclination angle β denotes the orientation of the obliqueligament relative to the rise direction,and λ is the anisotropy ratio defined as λ = tan β,asshown in Fig. 1(b). When β = π/4,the model is reduced to the classic Kelvin model. Similarto the analyzed method of Lu et al.^[18],a simplified periodical structural cell under compression(under tension only changing in the loading direction) is extracted,as shown in Fig. 2.

2.3 Loading in rise direction

As shown in Fig. 3,a curvilinear coordinate system (s,θ) with the origin O located in themidpoint of strut AE is introduced to analyze the deformation of the half-strut. The coordinates is the measurement of strut length,and the coordinate θ is the angle between the tangent tothe strut and the rise direction.

Just as shown in Fig. 3,a transverse distributed force is applied on the strut bending. Basedon the results of Wang and Feng^[35],the distributed transverse force can be expressed as

where H is determined by the cross-sectional shape of the strut and the surface stress. This study assumes that the cross-section of the strut is a circle. Thus,the expression of H is^[18] where D is the diameter of the strut. Although the vertical displacement of the strut is big enough,the strain is still in the infinitesimal range,as mentioned in Ref. ^[25]. Substituting Eqs. (2) and (5) into Eq. (4) and considering the small deformation approximation,i.e.,ε_s ≈ − ,we have where the component of ( )² is neglected. The effective bending rigidity of the compositebeam is defined as follows: where E₀ denotes Young’s modulus of the bulk material.

The moment at a common position is

where x represents a variable in the x-coordinate in Fig. 3,and t is an integral variable. According to the knowledge of materials mechanics,the differential equation of the deflection curve is as follows: Based on the geometric relationship shown in Fig. 3, Differentiating Eq. (9) with respect to s,we have When H₀ =0 in Eq. (11) and E_s =0 in Eq. (7),it will be reduced to the form without the surface effects in Ref. ^[29],similar hereinafter. Multiplying Eq. (11) with dθ and integrating from the point O yield Equation (12) can also be expressed as The length of the half-strut OA in Fig. 3 is l′,which can be expressed as The integrating equation (13) gives where β denotes the inclination angle at the point A.

Clearly,P is a function of variable α. Then,the stress in the z-direction σ_z can be expressed as

Based on Eq. (10), Integrating Eq. (17) gives the length of the projection length of OA in the z-direction,which is The initial projection length of OA in the z-direction is . Hence,the strain is given as

In conclusion,the compressive stress-strain relationship in the z-direction can be evaluated from Eqs. (16) and (19) by a given value of λ. Similarly,under tension,Eqs. (18) and (19) can be expressed as

Under tension,the stress in the z-direction σ_z can also be expressed as Eq. (16). The tensile stress-strain curve in the z-direction can be obtained from Eqs. (16) and (21) under tension. 2.4 Loading in transverse direction

In Fig. 4,the deformation of OC is displayed with O located in the midpoint of the strut EC. Similarly,the differential equation of the deflection curve yields

In Fig. 4,we have

Similarly,combining Eqs. (22) and (23) yields The length l′ is similarly written as The imposed stress σ_x to the model can be expressed as Similarly,the length of the deformed strut in the x-direction is

In Fig. 2(b),the initial length of the strut in the x-direction is . Then,the strain is

Hence,the stress-strain relationship in the transverse direction can be evaluated by combination of Eqs. (26) and (28). Similarly,under tension,Eqs. (27) and (28) can be expressed as follows:

Under tension,the stress in the x-direction σ_x can also be expressed as Eq. (26). Thestress-strain curve in the x-direction can be obtained from Eqs. (26) and (30) under tension.

It is noteworthy that the present derivation can be reduced to those of Lu et al.^{[29, 30]}which have been proved by the finite element method when the surface effect is not considered.Furthermore,the present model is consistent with the form of Zhu et al.^[36] when λ is specifiedas 1. Moreover,Young’s modulus of Lu et al.^[18] can be obtained by the present results bytaking limit. 3 Examples and discussion

In this paper,the parameters of nanoporous materials are selected as follows^[27]: the bulkYoung’s modulus is E₀ =70.29 GPa,the residual surface stress is τ₀ =0.91 N/m,and the surfaceelastic modulus is E_s =5.19 N/m. In addition,we take the relative density of nanoporous materialsas 0.07 in the following discussion. To facilitate the analysis,two parameters are definedfor convenience of comparison of the results. The dimensionless stress is adopted according toLu et al.^[37],i.e.,

where E₀ denotes Young’s modulus of the bulk material,and ρ and ρ₀ are the densities of the nanoporous material and the bulk material,respectively. To highlight the surface effects,the normalized stress is defined as the ratio of stress of nanoporous materials to that of macroporous materials with the same relative density,i.e., where σ_n is the stress of nanoporous materials,and σ_m is the stress of macro-porous materials.The following results are calculated by the MATLAB software.

Figure 5 illustrates the dimensionless stress-strain curves obtained from the isotropicnanoporous materials (λ=1) with different strut diameters,in which τ₀ =0.91 N/m and E_s =5.19 N/m. To conduct a direct comparison,the results of macro-porous materials derived byZhu et al.^[36] and Lu et al.^[30] are also presented. Figure 5 shows that the dimensionless stressof nanoporous materials increases nonlinearly with the strain,and the increasing rate becomessmaller for larger compressive strain,while the increasing rate becomes larger for larger tensilestrain. It is observed that the tension/compression asymmetry is very significant on account ofgeometric nonlinearity under high strains. The curves dramatically rise up as the strut diameterdecreases,indicating that the mechanical behavior of nanoporous materials is size-dependentunder high strains of both compression and tension. Moreover,the initial slope of stress-straincurve also increases as the strut diameter decreases,which is in accord with those in the workof Feng et al.^[17] and Lu et al.^[29]. However,such surface effects would not disappear until thestrut diameter D is larger than 50 nm,and then the stress-strain curve of nanoporous materialsapproaches that of macro-porous materials.

The surface effects of nanoporous materials are similar under compression and tension. Forthe sake of brevity,we only present the compressive results for convenience. In Fig. 6(a),thevariation of dimensionless compressive stress-strain curves with surface elasticity is presentedfor isotropic nanoporous materials,τ0=0.91 N/m. As the surface elasticity increases,the initialstiffness of nanoporous materials evidently increases,and the compressive curves rise up as well.It indicates that the surface elasticity has a significant effect on the high strain behaviors ofnanoporous materials. Especially,the influence of surface elasticity on compressive behaviorbecomes more evident as the strain increases. For nanoporous materials with Es = 0,itscompressive behavior is close to that of macro-porous materials.

The influence of the residual surface stress τ0 on the dimensionless compressive stress-straincurves of nanoporous materials is shown in Fig. 6(b),where λ = 1 and Es =5.19 N/m. Thestrain-stress curve with a positive residual surface stress is higher than that with τ0=0,whilethe curve with a negative residual surface stress is lower than that with τ0=0. Especially,thestrain-stress curve is even lower than that of macro-porous materials when the negative residualsurface stress is low enough. Similarly,the influence of residual surface stress on compressivebehavior becomes more distinct with the increase in the strain. This phenomenon is consistentwith the work of Liu et al.^[25].

From the above discussion,the surface effects of nanoporous materials are determined bythe combined effects of the surface elasticity and the residual surface stress. Moreover,under high strains,both of them have significant influence on the mechanical behaviors of nanoporous materials.

To reveal the influence of structural anisotropy,the compressive strain-stress curves ofanisotropic nanoporous materials are shown in Fig. 7,and our previous works are presented^[29].The compressive stress-strain curves rise up and go down in the rise and transverse directions,respectively. The variation tendency of mechanical properties for nanoporous materials is similarto that of macro-porous materials,which was provided by Lu et al.^[29]. Furthermore,thecurves dramatically rise up as the strut diameter decreases in different directions,which have asimilar tendency with isotropic nanoporous materials. Moreover,it is obviously shown that theisotropic curve with D=2 nm of nanoporous materials is even higher than the anisotropic curveof macro-porous materials in the rise direction. Hence,both the strut size and the anisotropyhave a great effect on the mechanical behaviors of nanoporous materials. The carrying capacityof the nanoporous materials can be enhanced by decreasing the strut diameter or increasingthe anisotropy ratio.

To facilitate comparison,the compressive stress is also expressed as the form of normalized stress (see Eq. (32)) as shown in Fig. 8. It shows that the normalized stress in the transverse direction is higher than that of isotropic nanoporous materials,while the normalized stress in the rise direction is lower. It indicates that the structural anisotropy would affect the surface effectsof nanoporous materials,which are enhanced in the transverse direction but is weakened in therise direction. The reason is that the angle between the strut axis (e.g.,the line CE in Fig. 2(a))and the transverse direction is larger than that between the strut axis and the rise direction,and thus the bending deformation of strut is more significant for transverse compression. Interestingly,the distance between the isotropic curve and the one of the transverse direction islarger than that between the isotropic curve and the one of the rise direction,indicating thatthe surface effects are reinforced in the transverse direction for anisotropic nanoporous materials.In addition,it is shown that the normalized stress of transverse direction increases morerapidly with the strain than that of the isotropic compression and that of the rise direction. Inother words,the enhancement of surface effects with increasing strains is more remarkable inthe transverse direction for anisotropic nanoporous materials.

Combining Fig. 7 with Fig. 8,we can conclude that the surface effects in the transverse direction would be reinforced while the surface effects in the rise direction would be weakened for anisotropic nanoporous materials. 4 Conclusions

Based on an improved anisotropic Kelvin model,we use the theories of Euler-Bernoulli beamand surface elasticity to derive the stress-strain relationships under high strains of both compressionand tension for elastic nanoporous materials. This work studies the effect of strut sizeon the mechanical behaviors of nanoporous materials,in which the residual surface stress andsurface elasticity are incorporated. The dimensionless stress-strain curves nonlinearly rise upas the strut diameter decreases and the surface elasticity increases. The residual surface stresswould lead to an increase in the carrying capacity of nanoporous materials. For anisotropicnanoporous materials,the surface effects in the transverse direction would be reinforced,whilethe surface effects in the rise direction would be weakened.