Bifurcation of elastic solids with sliding interfaces
BBifurcation of elastic solids with sliding interfaces
D. Bigoni a , N. Bordignon a , A. Piccolroaz a , S. Stupkiewicz b a DICAM, University of Trento, Italy b Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland
Abstract
Lubricated sliding contact between soft solids is an interesting topic in biomechanics andfor the design of small-scale engineering devices. As a model of this mechanical set-up, twoelastic nonlinear solids are considered jointed through a frictionless and bilateral surface, sothat continuity of the normal component of the Cauchy traction holds across the surface, butthe tangential component is null. Moreover, the displacement can develop only in a way thatthe bodies in contact do neither detach, nor overlap. Surprisingly, this finite strain problem hasnot been correctly formulated until now, so that this formulation is the objective of the presentpaper. The incremental equations are shown to be non-trivial and different from previously(and erroneously) employed conditions. In particular, an exclusion condition for bifurcationis derived to show that previous formulations based on frictionless contact or ‘spring-type’interfacial conditions are not able to predict bifurcations in tension, while experiments – oneof which, ad hoc designed, is reported – show that these bifurcations are a reality and becomepossible when the correct sliding interface model is used. The presented results introduce amethodology for the determination of bifurcations and instabilities occurring during lubricatedsliding between soft bodies in contact.
Keywords: frictionless contact; large strains; nonlinear elasticity
Lubricated sliding along an interface between two deformable bodies is typically characterized byvery low friction and arises, for instance, in several biotribological systems (Dowson, 2012), such asthe contact-lens/cornea (Dunn et al., 2013) and the articular cartilage (Ateshian, 2009) complexes,or in various engineering devices, such as windscreen wipers, aquaplaning tires, and elastomericseals (Stupkiewicz and Marciniszyn, 2009). These soft and slipping contacts are often character-ized by large elastic or viscoelastic deformations so that it is not obvious how to formulate theReynolds equation to adequately model the fluid flow between two contact surfaces that undergolarge time-dependent deformations (Temizer and Stupkiewicz, 2016). Moreover, a distinctive fea-ture of lubricated soft contacts is that they are capable of sustaining tensile contact tractions during sliding, particularly in transient conditions, a phenomenon clearly visible when a suctioncup is moved on a lubricated substrate. Indeed, as long as the pressure does not drop below thecavitation pressure, a soft contact can be loaded in tension, possibly imposing large deformations1 a r X i v : . [ phy s i c s . c l a ss - ph ] J a n n a highly compliant solid. As an example of this situation, the sequence of photos shown in Fig. 1refers to an experiment (performed at the Instabilities Lab of the University of Trento) on tensilebuckling involving a sliding contact between two soft solids. This system has been designed andrealized to obtain a compliant sliding element, and thus to buckle in tension, without using rigidparts such as rollers or sliding sleeves. In particular, a ‘T-shaped’ silicon rubber element is clampedFigure 1: A sequence of photos showing a tensile bifurcation involving sliding contact between two soft solids. Asilicon rubber suction cup is applied on a lubricant oil film to the upper part of a ‘T-shaped’ silicon rubber (gray inthe photo), clamped at the lower end. The suction cup is pulled vertically, so that the straight configuration of the‘T’ is a trivial equilibrium configuration (photo on the left) and a tensile bifurcation occurs when this element startsbending (second photo from the left) and the suction cup slips, as shown in the sequence of photos. Note that in thissystem rigid mechanical devices such as rollers or sliding sleeves are avoided. at the lower end and connected at the upper flat end to a silicon rubber suction cup, which has beenapplied with a lubricant oil. The system is pulled in tension and displays a tensile bifurcation inwhich the ‘T’ bends while the suction cup slides along the upper flat end of the ‘T’. This bifurcationresembles that analyzed in (Zaccaria et al. 2011), but involves here soft solids.A bilateral and frictionless sliding contact condition has been often employed to model theabove-mentioned problems (for instance, in geophysics, Leroy and Triantafyllidis, 1996, or forsliding inclusions, Tsuchida et al., 1986, or roll-bonding of metal sheets, Steif, 1990), where twobodies in a current configuration share a common surface along which shear traction and normalseparation/interpenetration must both vanish, but free sliding is permitted.Another model is based on a ‘spring-like’ interface, in which the incremental nominal traction isrelated to the jump in the incremental displacement across the interface (see Suo et al. 1992; Bigoniet al. 1997). This model, in the limit of null tangential stiffness and null normal compliance shouldreduce to the sliding interface model. While these models are elementary within an infinitesimaltheory, they become complex when the bodies in contact suffer large displacement/strain (and mayevidence bifurcations, as in the case of the soft materials involved in the experimental set-up shownin Fig. 1). As a matter of fact, the freely sliding interface model has never been even formulated so far and the ‘spring-like’ model will be shown not to reduce to the freely sliding interface in theabove-mentioned limit of vanishing tangential stiffness and normal compliance.The correct formulation for a sliding interface, together with the derivation of incremental2onditions, are the focus of the present article: the former turns out to be non-trivial and the lattercorrects previously used conditions, which are shown to lead to incorrect conclusions. Moreover,a generalization of the Hill’s exclusion condition for bifurcation (Hill, 1957; see Appendix A) tobodies containing interfaces, shows that the ‘spring-like’ interface cannot explain bifurcations whichcan in fact be obtained with the correct formulation of the sliding contact and which exist in reality,as the above-mentioned experiment shows.The availability of analytical solutions for incremental bifurcations of nonlinear elastic solidsis crucial for many applications involving soft materials (De Tommasi et al. 2010; Destrade andMerodio, 2011; deBotton et al. 2013; Ciarletta and Destrade, 2014; Steigmann and Ogden, 2014;Liang and Cai, 2015; Destrade et al. 2016; Riccobelli and Ciarletta, 2017), so that the importanceof the model derived in this paper is that it allows to obtain solutions for bifurcations occurringin soft bodies in contact with a frictionless planar interface. Several of these solutions, whichare important for applications, are here obtained, while other problems which do not admit ananalytical solution are solved by employing the finite element method and a linear perturbationtechnique. The obtained solutions show that sliding conditions strongly affect bifurcation loadsand promote tensile bifurcations (such as that visible in the experiment reported in Fig. 1), whichare shown to remain usually undetected by employing previously used, but incorrect, conditions.
Two nonlinear elastic bodies (denoted by ‘+’ and ‘ − ’) are considered in plane-strain conditions,jointed through a bilateral frictionless interface, Fig. 2. Points in the reference configurations B +0 and B − are mapped to the current configurations B + and B − via the deformations g ± : B ± → B ± ,so that x + = g + ( x +0 , t ) , x − = g − ( x − , t ) , (1)where t denotes the time, the subscript ‘0’ is used to highlight the referential description. Therefore,the displacement vector u is related to the deformation through u ± = g ± ( x ± , t ) − x ± (2)where ‘ ± ’ denotes that the equation holds for both quantities ‘+’ and ‘ − ’.The interface has the form of a regular surface Σ in the current configuration and is the im-age of another regular surface Σ in the reference configuration, where it admits the arc-lengthparameterization x +0 = x ( s +0 ) , (3)so that, since the parameter s − can be expressed as function of s +0 and time, the following expressioncan be derived x − = x ( s − ) = x ( s − ( s +0 , t )) . (4)The unit tangent vectors to the surface in the reference configuration, Σ , can be expressed as t +0 = ∂ x +0 ∂s +0 | ∂ x +0 ∂s +0 | , t − = ∂ x − ∂s − | ∂ x − ∂s − | . (5)3igure 2: Deformation of two nonlinear elastic bodies under plane strain conditions and jointed through a frictionlessand bilateral interface. The interface constitutive law enforces a bilateral constraint on the displacement (so that thetwo bodies can neither detach, nor interpenetrate, during deformation) and continuity of the Cauchy traction, butwith the tangential component of the latter being null. A finite and unprescribed sliding of the two bodies can occuracross the interface.
Note that a point x on the interface Σ in the current configuration is the image of two differentpoints x +0 and x − on Σ . This condition, representing the fact that the two bodies in contact canneither detach nor interpenetrate , can be expressed as x = x + = x − so that g + ( x +0 ( s +0 ) , t ) = g − ( x − ( s − ( s +0 , t )) , t ) . (6)The above condition defines the implicit dependence of s − on s +0 (and time) that has alreadybeen exploited in Eq. (4). Introducing the deformation gradient F ± = ∂ g ± ∂ x ± , (7)taking the derivative of Eq. (6) with respect to s +0 and applying the chain rule of differentiationyields F + ∂ x +0 ∂s +0 = F − ∂ x − ∂s − ∂s − ∂s +0 , (8)finally leading to the definition of the tangent vector t in the spatial configuration on Σ at xt = F + t +0 | F + t +0 | = F − t − | F − t − | . (9)4he unit normal at x on Σ can be obtained through the Nanson’s rule of area transformation n = A +0 A + J + ( F + ) − T n +0 = A − A − J − ( F − ) − T n − , (10)so that n = ( F + ) − T n +0 | ( F + ) − T n +0 | = ( F − ) − T n − | ( F − ) − T n − | . (11)Note that while n +0 and n − , as well as t +0 and t − , are different, there is only one n and one t . The interface is assumed to maintain a frictionless sliding contact, so that the normal componentof the Cauchy traction has to be continuous and the tangential component null. These conditionscan be written as follows n · (cid:74) T (cid:75) n = 0 , t · T + n = t · T − n = 0 , (12)where T is the Cauchy stress and (cid:74) ℵ (cid:75) = ℵ + − ℵ − is the jump operator of the quantity ℵ across Σ.On introduction of the first Piola–Kirchhoff stress S = J T F − T (where J = det F ) and using theNanson’s rule (10) yields T ± n = S ± n ± ι ± , (13)where ι ± = A ± /A ± is the ratio between the spatial and referential area elements, so that Eqs. (12)can be transformed to n · (cid:18) S + n +0 ι + − S − n − ι − (cid:19) = 0 , t · S + n +0 ι + = t · S − n − ι − = 0 . (14) Before deriving the relations pertaining to the interface, the following relations are introduced whichare standard for continua and still hold for points at the left and right limit of Σ: • The material time derivative, denoted by a superimposed dot, of the tangent and normal unitvectors to the surface Σ at x is ˙ t ± = ( I − t ⊗ t ) L ± t , (15)˙ n ± = − ( I − n ⊗ n )( L ± ) T n (16)where I is the identity tensor, L ± is the gradient of the spatial description of velocity for the‘+’ and ‘ − ’ parts of the body L ± ( x ± , t ) = grad v ± , (17)and v is the spatial description of the velocity v ± ( x ± , t ) = ˙ x ± ( x ± ( x ± , t ) , t ) , (18)where x ± = x ± ( x ± , t ) denotes the inverse of x ± = g ± ( x ± , t ).5 The ratio between the deformed and the undeformed area elements can be obtained from theNanson’s rule, Eq. (10), as ι ± = J ± (cid:12)(cid:12) ( F ± ) − T n ± (cid:12)(cid:12) , (19)from which its material time derivative can be obtained in the form˙ ι ± = J ± (tr L ± − n · L ± n ) (cid:12)(cid:12) ( F ± ) − T n ± (cid:12)(cid:12) = ι ± ( I − n ⊗ n ) · L ± , (20)as well as the following material time derivative (cid:18) ι ± (cid:19) · = − tr L ± + n · L ± n J ± | ( F ± ) − T n ± | = − ι ± ( I − n ⊗ n ) · L ± . (21)A point on the sliding interface Σ has to be understood as the ‘superposition’ of the two points,one belonging to the body B + and the other to the body B − , so that x + = x − along Σ. Takingthe time derivative of the equation x + = x − at fixed s +0 , the velocities of the two points x + and x − can be related to each other through˙ x + = ˙ x − + F − ∂ x − ∂s − ˙ s − . (22)The time derivative at fixed s +0 is in fact the material time derivative for the ‘+’ part of the body,while it involves an additional term related to the variation of s − for the ‘ − ’ part of the body.Equations (5) and (9) show that F − ∂ x − /∂s − is parallel to the tangent unit vector t , so thatthe scalar product of the unit normal n with both sides of Eq. (22) yields the continuity conditionacross the interface Σ for the normal component of the velocity (cid:74) ˙ x (cid:75) · n = 0 , (23)while the scalar product with the unit tangent t yields ˙ s − , thus˙ s − = ( ˙ x + − ˙ x − ) · t | F − ∂ x − ∂s − | . (24)The time derivative of Eqs. (9) and (11) at fixed s +0 provides˙ t + = ˙ t − + ∂ t ∂s − ˙ s − , ˙ n + = ˙ n − + ∂ n ∂s − ˙ s − , (25)which using Eqs. (15) and (16) lead to ∂ t ∂s − ˙ s − = ( I − t ⊗ t ) (cid:74) L (cid:75) t , ∂ n ∂s − ˙ s − = − ( I − n ⊗ n ) (cid:74) L T (cid:75) n . (26)The scalar product of Eqs. (26) with t and n yields t · ∂ t ∂s − ˙ s − = 0 , n · ∂ t ∂s − ˙ s − = (cid:74) L nt (cid:75) , (27)6nd n · ∂ n ∂s − ˙ s − = 0 , t · ∂ n ∂s − ˙ s − = − (cid:74) L nt (cid:75) . (28)The time derivative of Eq. (14) at fixed s +0 allows to obtain n · ˙ S + n +0 ι + − n · ˙ S − n − ι − − ˙ s − (cid:18) n · ∂ S − ∂s − n − ι − + n · S − n − ∂ (cid:0) ι − (cid:1) ∂s − + n · S − ι − ∂ n − ∂s − (cid:19) = n · T n (cid:74) L tt (cid:75) , (29)while the time derivative of Eq. (14) at fixed s +0 leads to t · ˙ S + n +0 = − ˙ t + · S + n +0 (30)and t · ˙ S − n − = − ˙ t − · S − n − − ˙ s − ∂ t − ∂s − · S − n − − ˙ s − t − · ∂ S − ∂s − n − − ˙ s − t − · S − ∂ n − ∂s − , (31)so that, using Eqs. (27), (28), and (15), the following expressions are derived t · ˙ S + n +0 = − L + nt n · S + n +0 , (32)and t · ˙ S − n − = − L + nt n · S − n − − ˙ s − t − · ∂ S − ∂s − n − − ˙ s − t − · S − ∂ n − ∂s − . (33) The general interface conditions derived above are now simplified for the special case of a planarsliding interface that is assumed to satisfy the following conditions: • the interface is planar both in the reference and in the current configurations (but can incre-mentally assume any curvature), so that: n = n +0 = n − , t = t +0 = t − , ∂ n − ∂s − = 0¯; (34) • the Cauchy traction components are uniform at the interface and satisfy: T + nn = T − nn , T + nt = T − nt = 0; (35) • a relative Lagrangian description is assumed in which the current configuration is assumedas reference (so that F + = F − = I and ι + = ι − = 1 and S ± = T ± ).7t follows from the above assumptions that ∂ (cid:0) ι − (cid:1) ∂s − = 0 , ∂ S − ∂s − = 0¯ . (36)Now, introducing a reference system x – x aligned parallel respectively to the unit tangent t andnormal n to the interface, the equations governing the rate problem across the above-introducedplanar interface are the following: • continuity of normal incremental displacements, from Eq. (23),˙ x + n ( x ,
0) = ˙ x − n ( x ,
0) ; (37) • continuity of incremental nominal shearing accross the interface, from Eqs. (32) and (33),˙ S + tn ( x ,
0) = ˙ S − tn ( x ,
0) ; (38) • dependence of the incremental nominal shearing on the Cauchy stress component orthogo-nal to the interface T nn and incremental displacement gradient mixed component L nt , fromEq. (32), ˙ S + tn ( x ,
0) = − αT nn L nt ( x , , (39)where α = 1; • dependence of the jump in the incremental nominal stress orthogonal to the interface on theCauchy normal component T nn and the jump in the tangential component of the incrementaldisplacement gradient L tt , from Eq. (29),˙ S + nn ( x , − ˙ S − nn ( x ,
0) = αT nn (cid:74) L tt ( x , (cid:75) . (40)where, again, α = 1.The parameter α has been introduced in the above equations to highlight the difference withrespect to the incorrect conditions sometimes assumed at the interface (for instance by Steif, 1990)˙ S ± tn ( x ,
0) = 0 , ˙ S + nn = ˙ S − nn , (41)which correspond to α = 0. Note that the only possibility to obtain a coincidence between thecorrect α = 1 and the incorrect α = 0 conditions is when the stress normal to the interface vanishes,namely, when T nn = 0.The ‘spring-type’ interfacial conditions used by Suo et al. (1992), Bigoni et al. (1997) andBigoni and Gei (2001) do not reduce (except when T nn = 0) to the correct frictionless slidingconditions (39) and (40), in the limit when the stiffness tangential to the interface tends to zeroand the normal stiffness to infinity. In this limit case, the ‘spring-type’ conditions reduce to theincorrect equations obtained with α = 0, so that they cannot properly describe slip without friction,unless when T nn = 0. Note that the stress orthogonal to the interface, T nn has been always assumedto be null by Bigoni et al. (1997) and Bigoni and Gei (2001); all bifurcation analyses reported inthese papers are therefore different from those considered in the present paper, where the transversestress is never null. 8 .1 Plane strain bifurcation problems involving a planar interface In the following, a series of incremental bifurcation problems are solved, involving two elastic nonlin-ear solids in contact through a sliding interface aligned parallel to the x –axis. This problem set-upis similar to various situations analyzed in the literature (Dowaikh and Ogden, 1991; Cristescu etal. 2004; Ottenio et al., 2007; Fu and Cai, 2015; Fu and Ciarletta, 2015), with the variant that nowthe interfacial conditions are different. It is important to highlight that the two solids in contactmay be characterized by different constitutive assumptions and may be subject to a different stateof prestress in the x –direction. In fact, the possibility that the two bodies may freely slide acrossthe interface allows to relax the usual compatibility restrictions.The incremental constitutive equations are characterized by the following parameters (Bigoni,2012, Chapter 6.2) ξ = µ ∗ µ , η = T tt + T nn µ , k = T tt − T nn µ , (42)so that ˙ S = µ (2 ξ − k − η ) L + ˙ p, ˙ S = µ (2 ξ + k − η ) L + ˙ p, ˙ S = µ [(1 + k ) L + (1 − η ) L ] , ˙ S = µ [(1 − η ) L + (1 − k ) L ] , (43)where ˙ p plays the role of a Lagrange multiplier, because the body is assumed incompressible, L kk = 0. For the sake of simplicity, a neo-Hookean material behaviour is assumed, ξ = 1, so thatthe material always lies in the elliptic imaginary (EI) regime and − < k < , Λ = (cid:112) ξ − ξ + k = | k | , (44)together with additional definitions to be used later, β = (cid:115) | k | − | k | β = (cid:115) − | k | − | k | , Ω = iβ , Ω = iβ , Ω = − iβ , Ω = − iβ . (45) Two elastic half-spaces are now considered in contact through a sliding interface, planar in thecurrent configuration, which is assumed as reference configuration, see the inset in Fig. 3.The upper (the lower) half-space x > x <
0) is denoted with ‘+’ (with ‘ − ’) and theincremental conditions at the interface are given by Eqs. (37)–(40), plus the condition of exponentialdecay of the solution in the limits x → ±∞ . For simplicity the two half spaces are modelled withthe same material and subject to the same prestress, so that bifurcations are possible only due tothe presence of the interface.Employing the representation v ± = (cid:101) v ± ( x ) f ( c , x ) , v ± = (cid:101) v ± ( x ) f (cid:48) ( c , x ) , (46) f ( c , x ) = exp( ic x ) , f (cid:48) ( c , x ) = if ( c , x ) , (47) (cid:101) v ± ( x ) = − b ± Ω ± e ic Ω ± x − b ± Ω ± e ic Ω ± x − b ± Ω ± e ic Ω ± x − b ± Ω ± e ic Ω ± x , (48)9 v ± ( x ) = − i (cid:104) b ± e ic Ω ± x + b ± e ic x + b ± e ic Ω ± x + b ± e ic Ω ± x (cid:105) (49)for the incremental displacements (Bigoni, 2012), where c is the wavenumber of the bifurcatedmode, the decaying condition implies b − = b − = b +3 = b +4 = 0 , (50)so that the eigenvalue problem governing incremental bifurcations can be written as (cid:2) M (cid:3) b +1 b +2 b − b − = 0 , (51)where the matrix [ M ] is given by − − − η + Λ 2 − η − Λ − η − Λ − η + Λ2 − η + Λ + T nn µ α − η − Λ + T nn µ α (cid:16) − η − Λ + T nn µ α (cid:17) (cid:113) − k (cid:16) − η + Λ + T nn µ α (cid:17) (cid:113) − Λ1 − k (cid:16) − η − Λ + T nn µ α (cid:17) (cid:113) − k (cid:16) − η + Λ + T nn µ α (cid:17) (cid:113) − Λ1 − k . (52)Non-trivial solutions of the system (51) are obtained when det M = 0, to be solved for thebifurcation stress. Note that matrix M does not contain the wavenumber of the bifurcated mode,so that the critical load for bifurcation is independent of the wavelength of the bifurcation mode(even if the sliding interface is present).The resulting bifurcation condition for a sliding interface ( α = 1) can be written as √ − Λ (cid:18) T nn µ + 2 − η + Λ (cid:19) − √ (cid:18) T nn µ + 2 − η − Λ (cid:19) = 0 . (53)If, instead of the correct interface conditions, α = 1, one assumes the incorrect condition α = 0,bifurcation corresponds to √ − Λ (2 − η + Λ) − √ − η − Λ) = 0 . (54)Using Eqs. (42) and for given values of longitudinal T tt and transverse T nn prestresses, Eqs. (53)and (54) (which hold for a generic incompressible material, subject to generic prestress conditions)can be solved. Results are reported in Fig. 3 for a neo-Hookean material, ξ = 1, assuming boththe correct condition α = 1 (on the left) and the incorrect one α = 0 (on the right). The red andblue zones identify in the figure the prestress combinations for which det M assumes positive andnegative values, respectively, so that the boundary between these zones (marked with red lines)corresponds to bifurcation. The dashed lines represent failure of ellipticity, so that points situatedbeyond this line do not represent states attainable through a smooth deformation path (becauseellipticity loss corresponds to the emergence of discontinuous solution).Note that in the case of null prestress normal to the interface, T nn = 0, an interfacial bifurcationoccurs for T tt /µ ≈ − . α = 1 or α = 0. This is the only situation in which the two conditionsprovide the same bifurcation stress. 10igure 3: Interfacial bifurcation of two elastic incompressible half-spaces (made up of the same neo-Hookeanmaterial, subject to the same prestress) in contact through a planar sliding interface in the T nn − T tt plane for asliding interface α = 1 (left). The incorrect condition α = 0 is also included for comparison (right). The pointscorresponding to bifurcation are represented by red lines (at the boundary between the red and blue zones), whilethe dashed lines correspond to failure of ellipticity. Note that with α = 1 bifurcation in pure tension occurs (i.e. with T tt = 0), which is excluded for α = 0. Therefore, the (correct) sliding interface condition explains tensile bifurcation.Note also that in this case bifurcations for both negative stresses T nn and T tt do not occur (except in the domain ofslightly negative T nn ). An interesting case occurs when only a tensile prestress orthogonal to the interface T nn isapplied (and the transverse prestress is null, T tt = 0), where a tensile bifurcation occurs for T nn /µ ≈ α = 0 is used or also if the modelling wouldinvolve a perfectly bonded interface (in which case all bifurcations are excluded within the limitsof ellipticity). This simple example reveals the importance of a correct definition of the interfacialconditions.A comparison between the correct α = 1 and incorrect α = 0 conditions reveals a completelydifferent bifurcation behaviour. In fact, for positive T nn bifurcation is possible in the correct case fornegative, null and slightly positive T tt . These bifurcations do not occur in the incorrect situation.Moreover in the latter situation there is a zone of bifurcation occurring for negative T nn whichis excluded in the correct case. As an example, in the special, but interesting, case of uniaxialcompression ( T nn < T tt = 0), there is no bifurcation in the correct case α = 1, whilebifurcation occurs in the other case.To better elucidate this situation, an exclusion condition of the Hill (1957) type is derived inAppendix A. For α = 0, this condition becomes completely insensible to the presence of the slidinginterface (and reduces to the Hill’s condition obtained without consideration of any interface), so11hat bifurcation is always excluded when both conditions T nn ≥ T tt ≥ α = 1, the exclusion condition evidences a term pertaining to the interface,which allows the bifurcation to occur for both positive T nn and T tt . An elastic layer (of current thickness H ) is considered, connected to an elastic half-space througha planar sliding interface, see the inset in Fig. 4. Both the layer and the half-space are assumed toobey the same neo-Hookean material model. The system is subject to a uniform biaxial Cauchyprestress state with principal components T tt and T nn . A reference system x – x is introducedaligned parallel respectively to the unit tangent t and normal n to the interface.In addition to the incremental boundary conditions given by Eqs. (37)–(40) at the slidinginterface ( x = 0), the decaying condition as x → −∞ , plus the condition holding at the freesurface ( x = H ), have to be enforced. The latter condition differs for dead or pressure loading asfollows: • for dead loading, ˙ S + nn ( x , H ) = ˙ S + tn ( x , H ) = 0; (55) • for pressure loading,˙ S + nn ( x , H ) = − T nn L nn ( x , H ) , ˙ S + tn ( x , H ) = − T nn L nt ( x , H ) . (56)Imposing the above conditions, a linear homogeneous system is obtained for the bifurcation stress T nn /µ , when the longitudinal prestress is assumed null ( T tt /µ = 0). The bifurcation stress isreported in Fig. 4 as a function of the wavenumber of the bifurcated field, for both situations ofdead loading and pressure loading and for both correct and incorrect conditions, respectively, α = 1and α = 0.For pressure loading, a tensile bifurcation is observed, which occurs for both the correct ( α = 1,left in the figure) and incorrect ( α = 0, right in the figure) conditions at the interface. A tensilebifurcation for dead loading is possible only when the correct condition α = 1 is employed, while inthe other case the Hill’s type condition (see Appendix A) excludes bifurcations for tensile T nn andnull T tt . In any case, results are strongly different for the correct and incorrect models of interface,showing once again the importance of a correct modelling of interfacial conditions. Two layers (one denoted by ‘+’ and the other by ‘ − ’), connected through a planar sliding interfaceare considered, subject to transverse and longitudinal prestresses T nn and T tt . The transverse stressis assumed to be generated by either a dead, Eqs. (55), or a pressure, Eqs. (56), loading (see theinsets in Fig. 5). Now only the correct condition α = 1 is considered, as for α = 0 the Hill’stype condition excludes bifurcation for positive dead loading T nn and null transversal loading, seeAppendix A.As in the case of a layer on a half-space ( H − /H + → ∞ ), see Section 3.1.2, compressive pressureloading, T nn <
0, does not lead to buckling, and tensile dead loading yields a bifurcation. The12igure 4:
Bifurcation of a layer connected to an elastic incompressible half-space through a sliding interface. Bothlayer and half-space are modelled with the same neo-Hookean material and subject to the same prestress orthogonal tothe interface. Both dead and pressure loadings are considered for the two interfacial conditions α = 1 and α = 0 (thelatter condition is incorrect and included only for comparison). The normalized bifurcation stress T nn /µ is reportedversus the normalized wavenumber of the bifurcated field c H . Note that for dead load bifurcation in tension ispossible only when the correct interfacial condition, α = 1, is considered. results for H − /H + < H − /H + > cH + . A special feature characterizing the presence of sliding interfaces is the appearance of tensile bifur-cations, often excluded for other models of interfaces (for instance in the perfectly bonded case).These bifurcations are usually hard to be obtained analytically (the simple cases reported in theprevious section are of course exceptions), so that the aim of this section is to use a finite-elementmethod combined with a linear perturbation analysis to analyze tensile bifurcations occurring un-der plane strain conditions in a system of two elastic slender blocks and a hollow cylinder with aninternal coating, in both cases jointed through a sliding interface. The former mechanical systemis related to the problem of buckling in tension of two elastic rods (Zaccaria et al., 2011), while thelatter is related to a problem of coating detachment.
A mixed formulation is adopted in order to implement incompressible hyperelasticity in plane-strainconditions. Quadrilateral 8-node elements are used with quadratic (serendipity) interpolation ofdisplacements and continuous bilinear interpolation of the pressure field that plays the role of a13igure 5:
Bifurcation of two elastic incompressible layers in contact through a sliding interface. Both layers aremodelled with the same neo-Hookean material and subject to the same prestress orthogonal to the interface. Thenormalized bifurcation stress T nn /µ is reported versus the normalized wavenumber of the bifurcated field c H + , fordifferent values of the thickness ratio H − /H + . Lagrange multiplier enforcing the incompressibility constraint using the augmented Lagrangianmethod. Standard 3 × AceGen/AceFEM system(Korelc, 2009). As a verification of the computational scheme, the problem of two elastic half-spaces (Section 3.1.1) and the problem of a layer on an elastic half-space (Section 3.1.2) have beenanalyzed, and a perfect agreement with the corresponding analytical solutions has been obtained.14 .2 Tensile bifurcation of two elastic slender blocks connected through a slidinginterface
As the first numerical example, bifurcation in tension is studied for the problem of two identicalelastic rectangular blocks jointed through a frictionless bilateral-contact interface, see the inset inFig. 6. The axial displacements are constrained at one support and uniform axial displacement isprescribed at the other support. Additionally, in each block, the lateral displacement is constrainedat one point in the middle of the support. In the base state, the rods are thus uniformly stretched,while the bifurcation mode in tension involves bending of both blocks accompanied by relativesliding at the interface, as shown in the inset of Fig. 6, where the problem scheme, together withthe undeformed mesh and the deformed mesh at buckling are reported (the mesh used in the actualcomputations was finer than that shown in Fig. 6 as an illustration).Figure 6:
Two identical neo-Hookean rectangular blocks uniformly deformed in tension, jointed through a slidinginterface. The blocks have initial length L , width H , and shear modulus µ = µ +0 = µ − . The bifurcation force F cr is made dimensionless through multiplication by the square of the current length L of the blocks and division by thebending stiffness B (per unit thickness) of the blocks calculated with reference to their current width L . Note thatthe bifurcation force tends, at increasing length of the block, to the value calculated for two elastic rods in tension ofshear stiffness µ (reported with a straight red line). The present problem is, in fact, a continuum counterpart of the problem, studied by Zaccaria etal. (2011), of tensile bifurcation of two inextensible elastic Euler–Bernoulli beams clamped at oneend and jointed through a slider. For that problem, the normalized critical tension force F cr hasbeen found equal to 4 F cr L / ( π B ) = 0 .
58, where L denotes the beam length and B the bendingstiffness.Figure 6 shows the normalized critical force as a function of the initial length-to-height ratio, L /H . For consistency, the force has been normalized using the current length L = λL and thebending stiffness B = µH / H = λ − H and current incremental shear modulus µ = µ ( λ + λ − ) /
2, even though thecritical stretch λ is close to unity (e.g., λ = 1 .
006 for L /H = 4 and λ = 1 .
002 for L /H = 8).15he result in Fig. 6 shows that for slender blocks the critical force agrees well with the model ofZaccaria et al. (2011), which critical load is reported with a red straight line. For thick blocks, thetwo models differ, for instance, by 20% at L /H = 4. A hollow cylinder is now considered with an internal coating and loaded by a uniform externalpressure. The cylinder and the coating interact through a frictionless contact interface. Thegeometry is specified by the outer radius R o , the inner radius R i , and the coating thickness h thathas been assumed equal to h = 0 . R o , see the inset in Fig. 7. The shear moduli of the tube andcoating are equal. The case where the coating is absent is also investigated for comparison.Figure 7: Bifurcation pressure p cr , made dimensionless through division by the shear modulus µ , for a cylinderwith (blue line) and without (orange line) internal coating, as a function of the ratio between the inner and outer radiiof the cylinder, R i /R o . The coating is connected to the cylinder with a sliding interface. Note the strong decrease ofthe bifurcation pressure due to the presence of the coating. Figure 7 shows the critical pressure p cr normalized through division by the shear modulus µ as a function of the inner-to-outer radius ratio, R i /R o . As a reference, the critical load of a hollowcylinder without coating is also included. The bifurcation modes are reported in Fig. 8 for theuncoated and in Fig. 9 for the coated case. In the case of coating, two buckling modes are observeddepending on the wall thickness. For R i /R o greater than approximately 0.38, a global bucklingmode occurs, as illustrated in Fig. 9. This mode is also characteristic for the uncoated hollowcylinder in the whole range of R i /R o . For the same ratio of R i /R o and the same load p/µ , thebase state is identical for the cylinder with coating and for the uncoated one. However, the criticalload is different, and, in the global-mode regime, the sliding interface reduces the critical load byapproximately 11%. 16 local bifurcation mode is observed for the coated hollow cylinder when R i /R o is less thanapproximately 0.38, as illustrated in Fig. 9. In this buckling mode, the layer and the inner part ofthe tube deform in a wave-like fashion, while the outer part of the tube remains intact. This modeis thus similar to the buckling mode characteristic for the layer resting on an elastic half-space, seeSection 3.1.2, with the difference that here the substrate is curved. In the local-mode regime, thecritical load is significantly reduced with respect to the uncoated cylinder (which buckles in theglobal mode). For instance, for R i /R o = 0 .
1, the critical load is reduced by 50%.Figure 8:
Bifurcation modes for a hollow cylinder (without coating) subjected to an external pressure (dashed linesdenote the undeformed configuration, solid lines denote the bifurcation mode in the deformed configuration). Thebifurcation modes correspond to the loads indicated in Fig. 7, to which the letters are referred.
Figure 9:
Bifurcation modes for a hollow cylinder with an internal coating jointed through a sliding interface. Thecylinder is subjected to an external pressure. Bifurcation modes correspond to the loads indicated in Fig. 7, to whichthe letters are referred. Note that an enlarged detail of the inner, coated surface is reported for each geometry (dashedlines denote the undeformed configuration, solid lines denote the bifurcation mode in the deformed configuration, thesliding interface is denoted in red).
As a conclusion, the presence of a coating connected with a sliding interface is detrimental to17he stability of the system, so that the coating tends to slide and the bifurcation load is stronglylower than that calculated in the case when the coating is absent.
As mentioned in the introduction, experiments have been designed and realized (in the ‘InstabilitiesLab’ of the University of Trento), showing a tensile bifurcation which involves two soft solidsconnected through a sliding interface, Fig. 10.Figure 10:
The set-up of an experiment showing a tensile bifurcation involving two soft solids connected trough asliding interface. A vertical displacement (rotations are left free) is imposed to the head of a suction cup connectedto a ‘T-shaped’ silicon rubber element. A lubricant oil is applied, so that the suction cup can slide along the upperedge of the ‘T’ element.
In particular, a ‘T-shaped’ silicon rubber element has been manufactured with a ‘stem’ havingrectangular cross section 10 mm ×
30 mm (RBSM from Misumi, with 7.4 MPa ultimate tensilestrength) and an upper end of dimensions 160 mm ×
10 mm ×
40 mm. Three different lengths ofthe stem have been tested, namely, L = 210 mm, L = 180 mm, and L = 150 mm. The upper flatpart of the ‘T’ has been attached (through a lubricant oil, Omala S4WS 460) to a silicon rubbersuction cup. The suction cup has been pulled in tension (by imposing a vertical displacementat a velocity of 0.7 mm/s, with a uniaxial testing machine, Messphysik midi 10). The load anddisplacement have been measured respectively with a load cell (a MT1041, RC 20kg, from MetlerToledo) and the potentiometric transducer inside the testing machine. Data have been acquiredwith a system NI CompactDAQ, interfaced with Labview (National Instruments).The oil used at the suction cup contact allows the suction cup to slide along the upper part ofthe ‘T’ element. Therefore, when the suction cup is pulled, the system initially remains straightand the stem deforms axially. However, at a sufficiently high load, a critical condition is reachedand the system buckles. Consequently, the stem of the ‘T’ element bends and the suction cup slides18long its upper flat end, see Fig. 1.This is a simple experiment showing a tensile bifurcation of two soft elastic materials (the ‘T’element and the suction cup), when they are connected through a sliding interface, a phenomenonwhich is predicted by the model developed in the present paper, in particular by the use of thecorrect interface conditions (37)–(40).Note, however, that the oil does not allow a completely free sliding of the suction cup, so thatan initial relative movement at the suction cup–rubber element interface requires the attainment ofan initial force, which suddenly decreases when the relative displacement increases and eventuallybecomes negligible, thus realizing the sliding interfacial conditions analyzed in the present paper.This is evident in the load-displacement curves, shown in Fig. 11, two for each tested length. Thecurves are marked blue for L = 210 mm, green for L = 180 mm and red for L = 150 mm. Thecurves show a peak in the force, followed by steep softening and the final attainment of a steadysliding state, where the junction behaves as a sliding interface. The peak forces exhibit a significantscatter which is related to the transition from sticking friction, through mixed lubrication at theonset of sliding, to hydrodynamic lubrication during developed sliding, the latter exhibiting muchsmaller scatter.The interest in the developed soft system is that it allows the realization of an element bucklingin tension, which is essentially similar to the structural system designed by Zaccaria et al. (2011),but now obtained without the use of rollers or other mechanical devices. Two-dimensional plane-stress finite element simulations have been performed with Abaqus to val-idate the model of a sliding interface between two soft materials against the experimental resultspresented in the previous section.The geometry is shown in the inset of Fig. 11 and consists in a rectangular block of edges B = 10 mm and L = { , , } mm. The lower edge of the elastic block is clamped, whereasthe upper edge is in contact with a rigid plane which can freely rotate and is connected to an elasticspring which models the stiffness of the suction cup. Contact conditions at the interface betweenthe elastic block and the rigid plane (shown as a red line in the inset of Fig. 11) are prescribed suchthat a bilateral and frictionless interaction is realized. An initial imperfection has been introduced,that consists in a rotation of the rigid plane by an angle of 0 . ◦ . The rigid plane is modelledusing a two-dimensional 2-node rigid element (R2D2), while the rectangular block is modelledusing 4-node bilinear elements with reduced integration and hourglass control (CPS4R element inAbaqus). The material of the elastic block is a neo-Hookean hyperelastic material characterized bya shear modulus µ = 7 MPa. The spring describing the suction cup is a linear elastic spring withstiffness k s = 4 .
25 MPa. Displacement boundary conditions (vertical displacement δ = 15 mm) areprescribed at the upper end of the elastic spring.The results of the finite element simulations are shown in Fig. 11 as solid lines with markers. Itis shown that the finite element model is able to predict correctly the post-critical behaviour. Thepeak load is not predicted by the model because the effects of the lubricant at the interface (whichproduces an increase of the load before buckling) are not taken into account.19igure 11: Experimental and simulated load–displacement curves of the structure sketched in the inset for threedifferent lengths of the vertical stem, L = 210 mm (red lines), L = 180 mm (green lines) and L = 150 mm (bluelines). The model of sliding interface correctly captures the post-critical behaviour, where the lubricated contactrealized a low friction sliding condition. A model of sliding interface has been developed for soft solids in sliding contact, a problem of interestin various technologies, exemplified through the design and experimentation on a soft device, whichrealizes a compliant slider. The derived incremental equations are not trivial and differ frompreviously (and erroneously) employed interface conditions. A fundamental simplifying assumptionin the model is the bilaterality of the contact, which on the other hand is the key to obtain analyticalsolutions for several bifurcation problems. Some of these solutions have been obtained, which showthat: (i.) the interface plays a strong role in the definition of critical conditions, (ii.) the interfacepromotes tensile bifurcations, one of which has been experimentally verified, which cannot bedetected if previously used (and erroneous) interfacial conditions are used.
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Competing interests.
We declare we have no competing interests.
Authors’ contributions.
All authors contributed equally to this work and gave final approvalfor publication.
Acknowledgements.
We thank D. Misseroni for the help with experiments.20 unding.
A.P., S.S., N.B. gratefully acknowledge financial support from the ERC AdvancedGrant ‘Instabilities and nonlocal multiscale modelling of materials’ ERC-2013-ADG-340561-INSTABILITIES.D.B. thanks financial support from the PRIN 2015 “Multi-scale mechanical models for the designand optimization of micro-structured smart materials and metamaterials” 2015LYYXA8-006.
A An exclusion condition for bifurcation of two solids in contactwith a sliding interface
Following the Hill (1957) generalization of the Kirchhoff proof of uniqueness of the linear theoryof elasticity, two incremental solutions are postulated, for the problem sketched in Fig. 12, ˙ x ± α ,˙ S ± α (with α = 1 , x ± , ∆ ˙ S ± are in equilibrium with homogeneousboundary conditions and null body forces.Figure 12: Deformation of a solid containing a sliding interface. B and B denote the reference and currentconfiguration, respectively. Integration of the equilibrium equations for both bodies yields (cid:90) B ± (cid:16) Div ∆ ˙ S ± (cid:17) · ∆ ˙ x ± = (cid:90) B ± Div (cid:16) ∆ ˙ S ± T ∆ ˙ x ± (cid:17) − (cid:90) B ± ∆ ˙ S ± · ∆ ˙ F ± = 0 , (A.1)so that the divergence theorem provides (cid:90) B ± ∆ ˙ S ± · ∆ ˙ F ± = ∓ (cid:90) Σ ± ∆ ˙ x ± · ∆ ˙ S ± n . (A.2)A sum of the two Eqs. (A.2) yields the following Hill’s type exclusion condition for bifurcation (cid:90) B ∆ ˙ S · ∆ ˙ F > − (cid:90) Σ (cid:16) ∆ ˙ x + · ∆ ˙ S + n − ∆ ˙ x − · ∆ ˙ S − n (cid:17) ∀ ∆ ˙ S ± , ∆ ˙ x ± . (A.3)21efore proceeding with the assumptions employed in the present article, the exclusion condition(A.3) is specialized to the case of the ‘spring-type’ interface introduced by Suo et al. (1992) andemployed also by Bigoni et al. (1997). This interface is charaterized by: (i.) full continuity of thenominal incremental tractions across the interface and (ii.) a linear interfacial constitutive law ofthe type ˙ S − n = H (cid:74) ˙ x (cid:75) , (A.4)where H is a constitutive tensor [note that in the notation of the present paper there is a signdiffering in equation (A.4) from Suo et al. (1992)]. Using the two above conditions (i.) and (ii.) inequation (A.3), the exclusion condition becomes (cid:90) B ∆ ˙ S · ∆ ˙ F + (cid:90) Σ (cid:74) ∆ ˙ x (cid:75) · H (cid:74) ∆ ˙ x (cid:75) > ∀ ∆ ˙ S , ∆ ˙ x . (A.5)Equation (A.5) shows that for a positive-definite interfacial tensor H (in other words excludingsoftening interfaces) the term pertaining to the interface is always positive. It may be easilyconcluded that:When the incremental constitutive response of a solid is governed by a positive definitetensor (as for instance for a Mooney–Rivlin material subject to non-negative principalstresses) bifurcation is always excluded for mixed boundary conditions of dead loadingand imposed displacements even in the presence of positive-definite interfaces of thetype introduced by Suo et al. (1992) .For instance, in a case in which all principal stresses are positive or null (as it happens in atensile problem of the type experimentally investigated in this paper) bifurcations are excluded.In order to substantiate the above statement with an example, consider two elastic blocks madeup of Mooney–Rivlin material connected through a planar interface of the type proposed by Suoet al. (1992) without softening. If these blocks will be pulled in tension with a dead loading, thecondition (A.5) excludes all possible bifurcations. But the bifurcation will occur in reality, as theT-problem shows. This bifurcation is found if the interface is replaced with a sliding interface ofthe type described by equations (37)–(40).The following assumptions are now introduced: • a Lagrangean formulation is assumed with the current state taken as reference, so that B ≡ B and Σ ≡ Σ; • plane strain deformation in the plane x – x prevails; • a planar interface is assumed, so that n = n and t = t ; • the material is prestressed by a uniform Cauchy stress with principal components T tt and T nn ; • the constitutive equation of the material is incrementally linear˙ S = E [ ˙ F ] for compressible material , (A.6)˙ S = E [ ˙ F ] + ˙ p I for incompressible material . (A.7)22hen Eq. (A.3) becomes (cid:90) B ∆ ˙ S · ∆ L > − (cid:90) Σ (cid:16) ∆ v + t ∆ ˙ S + tn + ∆ v + n ∆ ˙ S + nn − ∆ v − t ∆ ˙ S − tn − ∆ v − n ∆ ˙ S − nn (cid:17) , (A.8)where v is the incremental displacement and L its gradient and repeated indices are not summed.Introducing the fourth-order elastic tensor E and using Eqs. (37) and (38), Eq. (A.8) can berewritten as (cid:90) B ∆ L · E [∆ L ] > − (cid:90) Σ (cid:16)(cid:0) ∆ v + t − ∆ v − t (cid:1) ∆ ˙ S tn + ∆ v n (cid:16) ∆ ˙ S + nn − ∆ ˙ S − nn (cid:17)(cid:17) . (A.9)Finally, using Eqs. (39) and (40), the condition for excluding bifurcation in an elastic solid con-taining a sliding interface becomes (cid:90) B grad v · E [grad v ] − αT nn (cid:90) Σ ( v n (cid:74) v t,t (cid:75) − (cid:74) v t (cid:75) v n,t ) > , (A.10)holding for all (not identically zero) continuous and piecewise continuously twice differentiablevelocity fields v satisfying homogeneous conditions on the part of the boundary where incrementaldisplacements are prescribed and assuming arbitrary values on Σ, but with the normal componentsatisfying v + n = v − n .The parameter α in Eq. (A.10) highlights the difference between the correct interface conditions( α = 1) derived in the present work and the incorrect interface conditions ( α = 0) assumed by Steif(1990).In the special case in which T nn = 0, Eq. (A.10) reduces to the Hill exclusion condition (cid:90) B grad v · E [grad v ] > , (A.11)showing that for a positive definite incremental elastic tensor E the incremental solution is unique,whenever the sliding interface is free of normal prestress, otherwise bifurcation is not a-priori ex-cluded. When the incorrect assumption α = 0 is made, condition (A.11) is obtained independentlyof the value of T nn , thus excluding bifurcation for positive definite E . Positive definiteness of E is equivalent to the requirement that the principal prestresses T , T , and T (which enter in thedefinition of E ) satisfy all the inequalities T + T > T + T > T + T > T > T = T = 0 (Hill, 1967; Bigoni, 2012). References [1] Ateshian, G.A., 2009. The role of interstitial fluid pressurization in articular cartilage lubrica-tion. J. Biomech. 42, 11631176.[2] Bigoni, D., 2012. Nonlinear Solid Mechanics. Cambridge University Press, New York.[3] Bigoni, D. and Gei, M., 2001. Bifurcations of a coated, elastic cylinder. Int. J. Solids Struct.,38, 5117-5148. 234] Bigoni, D., Ortiz, M. and Needleman, A., 1997. Effect of interfacial compliance on bifurcationof a layer bonded to a substrate. J. Mech. Phys. Solids 34, 4305-4326.[5] Ciarletta, P. and Destrade, M. 2014. Torsion instability of soft solid cylinders. IMA J. Appl.Math. 79, 804-819.[6] Cristescu, N.D., Craciun, E.M. and Soos, E., 2004. Mechanics of elastic composites. Boca Raton.Chapman & Hall/CRC.[7] deBotton, G., Bustamante, R. and Dorfmann, A. 2013. Axisymmetric bifurcations of thickspherical shells under inflation and compression.