status solidi Stability of elastic material with negative stiffness and negative Poisson s ratio

physica pss www.pss-.com status solidi asic solid state physics Editor s Choice Staility of elastic material with negative stiffness and negative Poisson s ratio Shang Xinchun 1, and Roderic S. Lakes, 3 1 Department of Mathematics and Mechanics, University of Science and Technology, Beijing, 3 College Road, Haidian, Beijing 183, P.R. China Department of Engineering Physics, Engineering Mechanics and Materials Science Program, 15 Engineering Drive, University of Wisconsin, Madison, WI 5376, USA 3 Rheology Research Center, 15 Engineering Drive, University of Wisconsin, Madison, WI 5376, USA Received 14 Septemer 5, accepted 8 June 6 Pulished online 18 January 7 PACS 46.5.+, 6..Dc Staility of cuoids and cylinders of an isotropic elastic material with negative stiffness under partial constraint is analyzed using an integral method and Rayleigh quotient. It is not necessary that the material exhiit a positive definite strain energy to e stale. The elastic oject under partial constraint may have a negative ulk modulus K and yet e stale. A cylinder of aritrary cross section with the lateral surface constrained and top and ottom planar surfaces is stale provided the shear modulus G > and G /3 < K < or K >. This corresponds to an extended range of negative Poisson s ratio, < ν < 1. A cuoid is stale provided each of its surfaces is an aggregate of regions oeying fully or partially constrained oundary conditions. phys. stat. sol. () 44, No. 3, 18 16 (7) / DOI 1.1/pss.57719 REPRINT

phys. stat. sol. () 44, No. 3, 18 16 (7) / DOI 1.1/pss.57719 Editor s Choice Staility of elastic material with negative stiffness and negative Poisson s ratio Shang Xinchun 1, *,, 3 and Roderic S. Lakes 1 3 Department of Mathematics and Mechanics, University of Science and Technology, Beijing, 3 College Road, Haidian, Beijing 183, P.R. China Department of Engineering Physics, Engineering Mechanics and Materials Science Program, 15 Engineering Drive, University of Wisconsin, Madison, WI 5376, USA Rheology Research Center, 15 Engineering Drive, University of Wisconsin, Madison, WI 5376, USA Received 14 Septemer 5, accepted 8 June 6 Pulished online 18 January 7 PACS 46.5.+, 6..Dc Staility of cuoids and cylinders of an isotropic elastic material with negative stiffness under partial constraint is analyzed using an integral method and Rayleigh quotient. It is not necessary that the material exhiit a positive definite strain energy to e stale. The elastic oject under partial constraint may have a negative ulk modulus K and yet e stale. A cylinder of aritrary cross section with the lateral surface constrained and top and ottom planar surfaces is stale provided the shear modulus G > and G/3 < K < or K >. This corresponds to an extended range of negative Poisson s ratio, < ν < 1. A cuoid is stale provided each of its surfaces is an aggregate of regions oeying fully or partially constrained oundary conditions. 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim 1 Introduction Negative structural stiffness entails a reversal of the usual relationship etween force and displacement in deformed ojects: the applied force is in the opposite direction to the displacement rather than in the same direction. Negative structural stiffness can occur in ojects with pre-strain including ojects in the post-uckling regime [1]. Such ojects contain stored energy and are unstale unless they are constrained. Negative stiffness is of interest in part ecause one can otain extreme material damping in a system with elements of positive and negative stiffness. Such effects were oserved experimentally [] in a lumped system containing a post-uckled tue. As for material stiffness, i.e. modulus, the condition of positive definite strain energy entails, for isotropic solids, G > and 1 < ν <.5 with ν as Poisson s ratio (see e.g. [3]). One can express this as G > and K > with K as ulk modulus. Positive definite strain energy implies staility for an elastic ody under stress oundary conditions, including zero stress which means no constraint. A fully constrained oject under displacement oundary conditions has a unique solution [4] and is incrementally stale [5] if the elastic moduli are strongly elliptic: for an isotropic solid, the criteria are G > and < ν <.5 or 1 < ν <. Strong ellipticity requires the tensorial modulus C 1111, which governs the speed of plane compressional waves or the stiffness under compression with lateral constraint, to e positive. The ulk modulus can, however, e negative: 4G/3 < K <. * Corresponding author: e-mail: lakes@engr.wisc.edu, Fax: +1-68-63-7451 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

Editor s Choice phys. stat. sol. () 44, No. 3 (7) 19 Unstale G < Not strongly elliptic. Domains form. Unstale ν = 3/5 K/ Ruery Constant volume ν = 1/ Spongy Stale C < 1111 Not strongly elliptic. Unstale I II III Stiff ν =.3 ν = Dilational Anti-ruery Auxetic ν = -1 Constant shape E < ν = - E > ν = -1 ν = K = -G/3-4G/3 < K < Stale if constrained G ν = 1, K = - 4G/3 Fig. 1 (online colour at: www.pss-.com) Map of ulk modulus K vs. shear modulus G, showing regions of negative Poisson s ratio ν and negative moduli for an isotropic solid. Negative stiffness is distinct from negative Poisson s ratio. Poisson s ratio ν, is defined as the negative lateral strain of a stretched or compressed ody divided y its longitudinal strain; it is dimensionless. Most materials stretched with longitudinal force elongate longitudinally ut also contract laterally, hence have a positive Poisson s ratio. For most solids Poisson s ratio ranges etween.5 and.33; the range for staility of isotropic solids is from 1 to.5; within that range all moduli are positive. Recently Lakes and co-workers have conceptualized, faricated and studied negative Poisson s ratio foams [6, 7] with ν as small as.8. These materials ecome fatter in cross section when they are stretched and they are stale. Wojciechowski [8 1] analyzed several micro-structures predicted to exhiit negative Poisson s ratio. Milton [11] showed that negative Poisson s ratio can e achieved in hierarchical laminates, and that one can approach the isotropic lower limit 1 y proper choice of constituent moduli. Negative Poisson s ratio materials have een called anti-ruer y Glieck [1], dilational y Milton [11], and auxetic y Evans and co-workers [13, 14]. Negative stiffness, y contrast, refers to a situation in which a reaction force occurs in the same direction as imposed deformation. The relationship etween the moduli and the Poisson s ratio, allowing negative values, is shown in Fig. 1. The upper right quadrant of this map was discussed y Milton to elucidate the role of negative Poisson s ratio in relation to the moduli. Negative modulus is of interest in part ecause one can otain extreme material damping [, 15] in a composite with constituents of positive and negative modulus. In such a composite, the negative stiffness inclusions are under partial constraint from the surrounding matrix. The elastic and viscoelastic ehavior of composites having a negative stiffness phase has een illustrated for composites with particulate ferroelastic inclusions [16]. Specifically, a composite with a tin matrix and a small concentration (1%) of vanadium dioxide particle inclusions was prepared. Vanadium dioxide was chosen since it exhiits a ferroelastic phase transformation at a convenient temperature 67 C. This composite exhiited a large www.pss-.com 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

11 Shang Xinchun and R. S. Lakes: Staility of elastic material with negative stiffness peak in mechanical damping and an anomaly in modulus in the vicinity of the transformation temperature. Composites with inclusions of negative stiffness may e called exterliral or liral since they are on the oundary of alance [16]. Lakes and Drugan [17] studied spherically symmetric unit cells and showed selected solutions to e well ehaved for inclusions of negative ulk modulus. Partial constraint is also of interest in the context of experimental study of materials in the vicinity of phase transformations. Negative ulk modulus [18] is possile in a pre-strained lattice and in several crystalline materials. Analysis of an Ising model [19] of a lattice predicts K < near the critical temperature. Softening [] of the ulk modulus (analogous to softening of the shear modulus in ferroelastics) has een oserved in YInCu 4 crystals at a temperature of 67 K. Composites with spherical inclusions of negative ulk moduli exhiit anomalies in the composite ulk modulus and Young s modulus (and in the corresponding mechanical damping). A partially constrained ar with negative ulk modulus is stale with respect to elementary deformation modes [18]. However, there are an infinite numer of modes in a continuum. The staility of negative stiffness elastic media under partial constraint is not well understood. It is the purpose of the present work to explore the staility of isotropic elastic cuoids and cylinders under partial constraint of some surfaces. Formulation.1 Governing equations for elastic solid Consider an isotropic homogeneous elastic solid. The elastic ody occupies a three dimensional region V in the Cartesian coordinate system ( x, y, z ), and it has a regular (ounding and piecewise smooth) surface V. The motion of elastic ody can e descried y the displacements which dependent on the space coordinates ( x, y, z ) and the time t: u = u ( x, y, z, t), u = u ( x, y, z, t), u = u ( x, y, z, t). (.1) x x y y z z On the asis of classical theory of elasticity, the infinitesimal strains are given y u u x y uz Ê uy uxˆ εx =, εy =, εz =, γ xy = +, x y z x y γ yz Ê u u z y ˆ ux uz =, γ Ê ˆ + zx = +. y z z x (.) The rotation is given y 1 Ê u u z yˆ 1 ux uz 1 Ê uy uxˆ ωx = -, ωy = Ê - ˆ, ωz = -. y z z x x y (.3) The stresses satisfy the linear isotropic elastic Hooke s law: σ = λe+ Gε, σ = λe+ Gε, σ = λe+ Gε, x x y y z z τ = Gγ, τ = Gγ, τ = Gγ, xy xy yz yz xz xz (.4) where the ulk strain e = εx + εy + εz. The Lame and shear moduli λ and G are related to Young s modulus E and Poisson s ratio ν y λ = Eν/[(1 + ν) (1 - ν)], G = E/[(1 + ν )], respectively. The displacement equation of motion ([1], p. 78) is Ê u v w G ˆ Ê ( uvw,, ) ( λ G),, ˆ Ê ˆ + + + + + + = ρ ( uvw,, ). x y z x y z x y z t (.5) 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim www.pss-.com

Editor s Choice phys. stat. sol. () 44, No. 3 (7) 111 Suppose that the ody forces are asent and the works of surface forces to displacements vanish at every point on the entire oundary of the elastic ody, i.e. we have the oundary condition: σ = (, i j = x, y, z) on V, (.6) ijnu j i where the direction cosines ( nx, ny, nz) = (cos( n, x),cos( n, y),cos( n, z)) and n is the normal unit vector to the surface V.. Dynamic staility and real frequency In view of the intuitive notion of staility, an equilirium configuration, or an equilirium state, is stale, which means that the configuration will have only a small departure from its undistured configuration after any small disturance. In the usual case, such a small disturance may either induce small oscillation aout the undistured equilirium configuration or the perturation ecomes damped out. The wellknown Liapounov s theory of dynamic staility provides fundamental methods and criteria to examine staility of an equilirium state for discrete systems with a finite numer of degrees of freedom. Liapounov s direct method in staility theory has een developed y Movchan to e more appropriate for application to continuous systems. Susequently, Leipholz introduced Movchan s staility theorem for elastic ody under conservative loads, and reached a conclusion that the staility of the equilirium position is ensured y real and positive eigenvalue ω of the prolem []. In the case that the volume force and the surface load acting on the elastic ody vanish, the eigenvalue ω is namely the square of the natural frequency of the elastic ody ecause small oscillatory motion superimposed on an initial natural state is free viration. Leipholz s work is of considerale importance in theoretical analysis of staility as he estalished a connection etween dynamic staility of natural state and natural frequency of elastic ody. According to the aforementioned conclusion, if an elastic ody has real natural frequency for all modes of free viration then its natural state is stale. Therefore, we investigate staility conditions of the natural state for elastic materials, especially for elastic materials with negative stiffness. In this way we enale the determination of conditions under which the natural state is as a guide for experiments..3 Rayleigh s Quotient To investigate staility of an elastic ody in the natural state, we consider first free viration which is regarded as a small motion superimposed on an initial natural state of the elastic ody. The displacements are assumed as iωt iωt iωt u = u( x, y, z)e, u = v( x, y, z)e, u = w( x, y, z)e, (.7) x y z where i = - 1 and ω is the resonant (natural) frequency. The displacement mode (.7) is a normal mode in which all particles of the elastic ody are moving synchronously, that is, passing through their rest positions simultaneously. In the initial natural state, the displacements, stresses and the strains of an elastic ody all vanish. The conservation of total energy requires that the sum of the kinetic energy and the potential energy ecome zero: Ú Ô Ï u u x y uz ρ È Ê ˆ Ê ˆ Ê ˆ Ô Ì Í + + + W dv = t t t V ÔÓ Ô, (.8) where the volume mass density of material ρ > and the strain energy function, i.e. the strain energy per unit volume, is given y ([1], p. 14) W = + G + + + G + + - - - (.9) 1 [( λ ) ( εx εy εz) ( γ xy γ yz γ zx 4εxεy 4εyεz 4 εzεx)]. www.pss-.com 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

11 Shang Xinchun and R. S. Lakes: Staility of elastic material with negative stiffness It can rewritten as W = + G + + + G + + 1 6 {(3λ ) ( εx εy εz) 3 ( γ xy γ yz γ zx) + - + - + - G[( εx εy) ( εy εz) ( εz εx) ]}. (.1) Sustituting (.7) and (.1) into the equation of energy conservation (.8), we otain the Rayleigh s quotient: ω dv V (,, ) = R u v w = Ú V Ú U ρ( u + v + w )dv, (.11) where the function Ï È 1 Ô Ê u v wˆ Ê v uˆ u w Ê v wˆ U = (3 λ G ) 3 G Ê ˆ Ì + + + + + + + + + 6 Í x y z x y z x z y ÔÓ ÈÊ u vˆ Ê v wˆ w u Ô + G Ê ˆ Í - + - + -. x y y z z x Ô (.1) In variational methods of elastic viration, Rayleigh s quotient plays an important role, for instance, Rayleigh s quotient provides an upper ound for the lowest eigenvalue: ω min R [3]. In view of methods for energy, Rayleigh s quotient of an elastic ody may contain all information of free viration. It is more difficult to show staility than instaility since there are an infinite numer of modes. However, Rayleigh s quotient is an expression for the square of natural frequency ω corresponding to all possile modes of free viration; therefore it is given the same symol. The meaning is to e understood y context. Moreover, the staility for elastic ody is equivalent to that the Rayleigh s quotient having a positive lower ound [4]. Thus, Rayleigh s quotient could e a preferred approach to investigate staility of an elastic ody. In general, the denominator of Rayleigh s quotient is always positive ecause it corresponds to the positive kinetic energy. In the case of conservative loads, the numerator of Rayleigh s quotient corresponds to the total potential energy, that is, the strain energy minus the work of external loads. Whether the numerator is positive or not is dependent from the material moduli, the shape of oundary and the oundary condition of the elastic ody. 3 Staility under conditions of positive definiteness and strong ellipticity: review and analysis 3.1 Staility under the positive definiteness condition From (.1), the positive definiteness condition of the strain energy function W is G >, 3λ + G >. (3.1) Oviously, this condition guarantees that the numerator of the Rayleigh s quotient (.11) is positive since the function U is positive definite. This implies that the square of natural frequencies for all modes of free viration is positive, that is, the natural frequencies are real. It follows the well-known result: an elastic ody with aritrary regular shape made of positive definite material is always stale, whether the surface is constrained or not. Also the positive definiteness condition (3.1) requires all moduli to e positive. It excludes any possiility of negative modulus. In the following analysis, surface constraint is imposed and the staility of a solid of negative modulus under constraint is studied. 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim www.pss-.com

Editor s Choice phys. stat. sol. () 44, No. 3 (7) 113 3. Staility of an elastic ody fully constrained on the surface Now we rewrite the strain energy function (.9) as following Kelvin s form ([1], p. 168): 1 { ( λ )( εx εy εz ) 4 ( ωx ωy ωz ) W = + G + + + G + + ÈÊ u uy uy u ˆ u u u u Ê uy u u uyˆ + 4 G z z Ê z x x z ˆ Í - + - + x - x. y z y z x z x z x y x y (3.) Sustituting (.7) and (3.) into the Eq. (.8), we can otain the another expression of the Rayleigh s quotient: ω U dv + I 1 1 V 1(,, ) = R u v w = Ú V Ú ρ( u + v + w )dv, (3.3) where the function Ï È 1 Ô Ê u v wˆ Ê w vˆ u w Ê v uˆ U1 = ( λ G ) G Ê ˆ Ì + + + + - + - + - Í x y z y z z x x y ÔÓ Ô Ô (3.4) and the volume integral ÈÊ w v v wˆ u w w u Ê v u u vˆ I1 = 4G Ú Ê ˆ Í - + - + - d V. y z y z z x z x x y x y (3.5) V Using integration y parts, we can transform the volume integral (3.5) in V into the closed surface integral on V as follows (to see A1. in the Appendix): Ï ÈÊ u u ˆ Ê v wˆ v v w u I = G n w + v - u + + n ÈÊ u + w ˆ - v Ê + ˆ 1 Ú Ì x Í y z y y z Í x z z x V Ó ÈÊ w w ˆ Ê u vˆ + nz Ív + u - w + d S. y x x y (3.6) Oviously, for an elastic ody with aritrary regular shape if the material oeys the condition of strong ellipticity: G >, λ + G >, (3.7) then we know from (3.4) that the function U 1 is non-negative. Indeed, a positive semi-definite form in the displacement gradients ([5], p. 1) is: U1. (3.8) Therefore, if the surface integral I 1 =, then the Rayleigh s quotient from (3.3) is non-negative. In Eq. (3.8), the equality holds only if the vector of displacement u = ( uvw,, ) has no dilatation and no rotation: u = and u =, which implies that the vector of displacement is harmonic [6]: u =. (3.9) www.pss-.com 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

114 Shang Xinchun and R. S. Lakes: Staility of elastic material with negative stiffness This equation has two situations of solution for the assumed homogenous oundary condition: either trivial solution u in V or non-trivial solution u π in V. The first situation guarantees that the Rayleigh s quotient is positive definite, so elastic ody is stale. However, the latter leads to the possiility that Rayleigh s quotient (all frequency) ecomes zero for some non-trivial solutions of displacement. In the latter situation, elastic ody has neutral staility. Therefore, we otain a result: Conclusion I An elastic ody with aritrary regular shape which material moduli satisfy the condition of strong ellipticity (3.7) is either stale or neutrally stale if its oundary condition makes the surface integral (3.6) I 1 =. Moreover, under the assumed homogenous oundary condition, if Eq. (3.9) has a unique solution (the trivial solution u ), then the elastic ody is stale; if the Eq. (3.9) has a nontrivial solution, then the elastic ody is neutrally stale. It should e pointed that the surface integral (3.6) I 1 = in the Conclusion I is different from the Chen s integral oundary condition ([7], formula (13); [8], formula (A.1)). Chen s result indicates Τ that for all fourth order skew tensor W =-W if the displacement vector u satisfies the integral condition: W: nƒ ƒuƒ u ds = (where the symol ƒ and : denote tensor product and doule Ú V scalar product, respectively), then an elastic ody with aritrary regular shape which material moduli satisfy the condition of strong ellipticity is stale. As compared with Chen s integral condition, the integral condition I 1 = in Conclusion I is more direct. However, in Conclusion I it requires the additional condition that the harmonic Eq. (3.9) has only a trivial solution u under the assumed homogeneous oundary condition. In some oundary conditions, this additional condition may hold. As a specific case of Conclusion I, when the oundary of the surface V of an elastic ody is fully constrained, i.e. the displacement oundary condition u = v= w= on V. In this case Eq. (3.9) has a unique trivial solution ecause of the uniqueness of the Dirichlet prolem, and from (3.6) the surface integral I 1 =. Thus we reach another well-known result: an elastic ody fully constrained (fixed) on the surface with aritrary regular shape is stale if its material is strongly elliptic. It is worthwhile to note that the strong ellipticity condition (3.7) is weaker than the positive definiteness condition (3.1). The positive definiteness condition limits all moduli to e only positive, however the strong ellipticity condition allows some moduli to ecome negative. 3.3 Condition of strong ellipticity and negative elastic moduli Strong ellipticity allows some moduli to e negative. Negative stiffness is not excluded y any physical law [17]. Examples of negative stiffness and the use of negative stiffness constituents are discussed in the Introduction. In the view of physics, the strong ellipticity condition (3.7) is consistent with some restrictions upon materials. First, for isotropic elastic materials, the stress and strain share the same principal axes and the greater principal stress should occur in the direction of the greater principal strain. The two restrictions require that the empirical inequality and the Baker Ericksen inequality in finite elasticity hold, respectively. In case of linear elasticity the two inequalities reduce to one of the conditions of strong ellipticity: G > [9 31]. Also, another restriction on materials is that tension produces extension when the lateral faces are fixed; this requires that the tension-extension inequality in finite elasticity must e satisfied. In linear elasticity this inequality corresponds to another condition of strong ellipticity: λ + G > [9, 31]. Moreover, the strong ellipticity condition (3.7) is a necessary and sufficient condition to ensure that the speeds of all plane waves in elastic media filling three-dimensional space are positive. Moreover, for the displacement oundary value prolem of linear elastostatics in ounded regions, the strong ellipticity condition (3.7) guarantees the uniqueness of solution [6]. For isotropic elastic materials, the strong ellipticity condition (3.7) shows that the shear modulus G and tensorial compression modulus C 1111 = λ + G oth are positive. If either of the moduli is negative, the material will undergo a deformation from its assumed initial state even though no forces are applied to it [3]. If strong ellipticity is violated, the material may exhiit an instaility associated with the for- 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim www.pss-.com

Editor s Choice phys. stat. sol. () 44, No. 3 (7) 115 mation of ands of heterogeneous deformation [33]. Bands associated with negative C 1111 were oserved [34] in open cell foams under heavy compression. Bands which may e interpreted in terms of negative G are known in ferroelastic materials [35] elow a critical phase transformation temperature. Based on these fundamental considerations, we investigate staility in the range G > and λ + G >, i.e. strong ellipticity condition. There exist three regions (Fig. 1) in the plane of elastic modulus G and K in which the strong ellipticity condition (3.7) holds: (i) Region I: G >, K > ( E >, - 1 < ν < 1/ ). In this region, the material is positive definite and all elastic moduli are positive. For all oundary conditions, the material in this region is always stale. Region I corresponds to positive definite energy and is unquestionaly stale. Negative Poisson s ratio, though counter-intuitive and not oserved in common isotropic materials, is consistent with staility within the aove range. Staility of elastic ojects corresponding to the other regions is examined in the following. (ii) Region II: G >, - G/3< K < ( E <, - < ν < - 1). In this region oth the Young s modulus E and the ulk modulus K are negative. Moreover, we have the condition: G >, λ + G >. (3.1) (iii) Region III: G >, - 4 G/3 < K < - G/3 ( E >,1 < ν < + ). In this region negative ulk modulus K occurs, ut Young s modulus E is positive. Hence the condition of strong ellipticity allows the elastic material to have the negative ulk modulus K and the negative Young s modulus E. The staility of materials condition is not yet well understood in regions II and III. 3.4 Staility in partial constraint: Ryzhak s result As discussed aove, an elastic ody with negative modulus for which all oundaries are fully constrained (fixed) is stale in the condition of strong ellipticity. By contrast, if its oundaries are fully stress-free, it is usually unstale even if the condition of strong ellipticity holds. The fully fixed and fully stress-free are the two extreme cases of oundary conditions. Staility conditions for other oundary condition cases etween the two extreme cases are of interest and are studied in the present work. By means of Fourier expansion procedure, Ryzhak [8] found a modified result: An elastic cuoid (rectangular parallelepiped) satisfying the condition of strong ellipticity (3.7) is stale if each of its surfaces has one of three types of oundary condition: (i) zero displacement (fixed); (ii) tangent directions to oundary are fixed ut the normal direction is stress-free; (iii) normal is fixed ut tangent stressfree. The Ryzhak result implies that the full constraint of surface displacement is not necessary for a strongly elliptic material to e stale. In other words, it reveals a possiility of staility under partial constraint of the surface. In the following analysis, we will deal with the questions: Does there exist a stale or neutrally stale elastic ody with negative ulk modulus (in the region II or III) if some portion of its oundary has not any constraint (stress-free)? A general answer of such question seems rather difficult. In the next two sections, we will give some positive answers to this question for elastic odies with specific geometric shapes. www.pss-.com 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

116 Shang Xinchun and R. S. Lakes: Staility of elastic material with negative stiffness 4 Staility of elastic cuoids with negative modulus 4.1 Simplification of the integral I 1 Consider an elastic cuoid with negative modulus and which oeys the strong ellipticity condition (3.7). The lengths of the cuoid are a,, c in three dimensions, respectively. The six surfaces of the cuoid are denoted as: Ω Ω 1, 3,4 5,6 = { x =, a, y, z c}, Ω = { x a, y =,, z c}, = { x a, y, z =, c}. (4.1) On each surface of the cuoid, only the normal cosine is not zero and the two tangent cosines ecome zero: nx Ω 1, = 1, ny Ω 3,4 = 1, nz Ω 5,6 = 1 (the others are zero). (4.) In this case, the closed surface integral (3.6) can e rewritten y c a ÏÔ ÈÊ u uˆ Ê v wˆ I1 = GÌÚÚ Í v+ w - u + dydz y z y z ÔÓ c a v v w u + ÈÊ w u ˆ v Ê ˆ ÚÚ Í + - + dzdx z x z x y= y= a z= a ÈÊ w w ˆ Ê u vˆ Ô + ÚÚ Íu + v - w + dxd y. x y x y z= Ô Then, using integration y parts, we can reduce the integral I 1 into the following form: c a ca c a z= c Ï y= Ô Ê v wˆ w u Ê u vˆ Ô I1 = G u dydz v Ê ˆ Ì- ÚÚ + - + dzdx- w + dxdy+ J y z ÚÚ z z ÚÚ y= x y Ô Ó z= Ô (4.3) where the line integration is Ú Ú Ú (4.4) x = a y= z c vw ( dx+ ud z) = y= z= Ryz Rzx Rxy J = u ( v d z+ w d y ) + + w ( u d y+ v d x ). In the expression (4.4) the closed curves Rxy, Ryz and Rzx are respectively the edges of the three rectangular regions which are denoted y R = { x a, y }, R = { y, z c} and R = { z c, x a} xy yz and located on the three different coordinate planes. Also all line integrals in (4.4) are along the counterclockwise direction. Calculating these line integrals, we otain the identity (see Appendix ): J. (4.5) zx 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim www.pss-.com

Editor s Choice phys. stat. sol. () 44, No. 3 (7) 117 4. Types of oundary conditions considered On the premise of the oundary condition (.6), the types of oundary su-regions are classified into four groups. In a general case, each surface of the cuoid is an aggregation of the four types of suregions, namely () i () i () i () i Ωi = Bu + Buσ + Bσu + Bσ ( i = 1,,,6). (4.6) The four independent types of su-oundary regions on the i-th surface Ωi ( i = 1,,,6) are defined as follows: () Type I: The fixed oundary su-region B i (). All components of displacement vanish in B i : u = v = w= in u () i. u B (4.7) () i () Type II: The mixed oundary su-region Bu σ. The normal direction to oundary su-region Bu i σ is fixed and the two tangent directions are stress-free. This type of oundary condition, for example, on the surface Ω 1 can e written as u u Ê v wˆ w u =, + = Ê + ˆ = z y x z in B σ. (4.8) () u 1 Type III: The mixed oundary su-region B σ u. The two tangent directions to oundary B σ u are fixed and the normal direction is stress-free. This type of oundary condition, for example, on the surface Ω 1 can e written y È u Ê v wˆ Í( λ+ G) + λ + =, v = w = x y z in (1) B σ u. (4.9) Type IV: The stress-free oundary su-region B σ. All stresses are free in B σ. This type of oundary condition, for example, on the surface Ω 1 can e written as È u Ê v wˆ Ê v wˆ w u ( λ G) λ,, Ê ˆ Í + + x + y z = + = + = z y x z in () i B σ. (4.1) 4.3 An extension of Ryzhak s result Suppose that each surface of the cuoid is an aggregation of the type I, II and III oundary su-regions, namely () i () i () i Ωi = Bu + Buσ + Bσu ( i = 1,,,6), (4.11) () where oundary su-regions i () B u, i () Bu σ and B i σ u have at least one non-empty segment. Taking into account that the partial derivative with respective to variale y or z can e exchanged in order with evaluation at x =, that is u u v w = ( u ), = ( u ), = ( v ), = ( w ), y y z z y y z z on the surface Ω 1 ( x = ) we have oundary condition: u B (1) (1) u or B uσ =, ( v (1) ) ( (1) B + w ) σu B =. σu y z www.pss-.com 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

118 Shang Xinchun and R. S. Lakes: Staility of elastic material with negative stiffness Fig. (online colour at: www.pss-.com) A cuoid constrained on its periphery so that each of its surfaces has at least one region of partial or full constraint. Constraint Thus, the integral term on the surface Ω 1 ( x = ) in the expression (4.3) ecomes c Ê v wˆ Ê ˆ Ê È ˆ u + dyd z = + u ( v ) + ( w ) dydz = Ú Ú ÚÚ ÚÚ. (4.1) y z Í (1) (1) y z Bu+ Buσ B σu Similarly, the all other integral terms in the expression (4.3) also vanish. In this case the integral I 1 =. Hence the Rayleigh s quotient (3.3) is non-negative in the condition of strong ellipticity (3.7), since from the function U 1 is non-negative. Moreover, the positive definiteness of the function U 1 for displacements ( uvw),, is dependent on the uniqueness of solution of the Eq. (3.9). It is easy to see that except for the three cases of oundary condition, the Eq. (3.9) has unique zero solution if its each surface is an aggregation of the type I, II and III oundary su-regions, i.e. (4.11) holds (proof to see A.3 in the Appendix). Thus, on the asis of Conclusion I, we otain an extension of Ryzhak s result: Conclusion II An elastic cuoid (Fig. ) satisfying the condition of strong ellipticity (3.7) is stale if each of its surfaces is an aggregation of the type I, II and III oundary su-regions, namely the condition (4.11) holds, except for three cases of oundary conditions which give neutral staility. We remark that Ryzhak s result is for anisotropic elasticity; in that sense it is more general. However, it has a limitation that each surface of the cuoid is only one of the type I, II and III oundary suregions, ut can not fit together in the three types of oundary su-regions in the same surface, namely, it requires the limitation: () i () i () i Ωi = Bu or Buσ or Bσu ( i = 1,,, 6). (4.13) Comparing with (4.11) it is clear that the condition (4.11) is weaker than the limitation (4.13). Distinctly, in the isotropic elasticity case, the aforementioned result extends Ryzhak s one. Actually, it has () no limitation to the shape, numer and location of the su-regions i () B u, i () Bu σ and B i σ u in each of the surfaces Ω i ( i = 1,,,6). 4.4 Examples of exact solutions In order to show that the square of natural frequency ω is positive, we intend to give an exact solution () i to a specific example. Consider an elastic cuoid in which all six surfaces are type II ( Ω i = Bu σ ( i = 1,,,6) ) and the condition of strong ellipticity holds. Although Ryzhak s qualitative result has covered the case of this example, however from application point of view, it is highly significant to find an exact solution of frequencies. To the end, for this example we assume displacement modes as u = A1sin αlxcos βmy cos γ nz, v = Acos αlxsin βmycos γ nz, (4.14) w= A3 cos αlxcos βmysin γ nz, where αl = lp / a, βm = mp /, γ n = np / c ( l, m, n = 1,, ). 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim www.pss-.com

Editor s Choice phys. stat. sol. () 44, No. 3 (7) 119 It is easy to validate that the displacement mode (4.14) satisfies all oundary conditions on the six surfaces. Sustituting the displacement mode (4.14) into the equation (.5) gives [ - ρω ] = B I A, where the vector A = [,, ] A A A Τ 1 3 and the coefficient matrix È( λ + G) αl + G( βm + γ n) ( λ + G) αlβm ( λ + G) αγ l n Í B = Í ( λ + G) βm + G( αl + γ m) ( λ + G) βmγ n. Í sym. ( λ + G) γ n + G( αl + βm) The frequency equation is l m n l m n f( ω) = det[ B- ρω I ] = -[ ρω - G( α + β + γ )] [ ρω -( λ + G)( α + β + γ )] =. Hence we otain λ + G G ω1 = ( αl + βm + γ n) >, ω = ω3 = ( αl + βm + γn) > ( l, m, n = 1,, ). (4.15) ρ ρ () i As an another example, all surfaces are type III ( Ω i = B σ u ( i = 1,,,6)), the displacement modes can e assumed as u = A cos α xsin β ysin γ z, v = A sin α xcos β ysin γ z, 1 l m n l m n w= A sin α xsin β y cos γ z. 3 l m n (4.16) Similarly, the same solutions of frequencies are otained as the expression (4.15). 4.5 Cuoid with stress-free surfaces: neutral staility Now we consider an elastic cuoid in which the material satisfies the condition: 4 G >, λ + G = (which is equivalent to K =- G < or E > and ν = 1) (4.17) 3 and the condition (4.6) holds, that is, each surface of the cuoid is an aggregation of the type I, II, II and IV oundary su-regions, which includes stress-free oundary su-region B σ (Type 4). Oserve that the condition on moduli and Poisson s ratio differs from the prior condition; a new result is under discussion. Under the condition (4.17), the oundary condition on su-region B σ, for example, on the surface Ω 1 ( x = ) (4.1) can e reduced into Ê v wˆ Ê v uˆ w u,, Ê ˆ + = + = + =. y z x y x z (4.18) This leads to the integral term on the surface Ω 1 ( x = ) in the expression (4.3) to vanish: c Ê v wˆ Ê ˆ Ê È ˆ u + dyd z = + + u ( v ) + ( w ) dydz ÚÚ ÚÚ ÚÚ ÚÚ y z Í (1) (1) (1) y z Bu Buσ B σu È Ê v wˆ + ÚÚ Íu + dydz =. y z (1) Bσ www.pss-.com 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

1 Shang Xinchun and R. S. Lakes: Staility of elastic material with negative stiffness Similarly, all other integral terms in the expression (4.4) also vanish, so the integral I 1 =. It should e noticed that when the condition (4.17) holds the function U 1 is non-negative: U 1. The equality holds only if all rotations ωx = ωy = ωz =, that is u =, it means that the displacements can e expressed y a potential function ψ ( x, y, z) : ψ ψ ψ u =, v =, w=. x y z (4.19) Taking a special potential function 3 3 3 3 3 3 ψ = Cx y z ( x -a) ( y -) ( z - c) (4.) it is evident that oth displacements and the components of strain vanish in the six surfaces Ω i ( i = 1,,,6), and it implies all components of stress vanish also in the six surfaces. Thus, when each surface of the cuoid is aggregation of the type I, II, III and IV oundary suregions, the solution (4.19) and (4.) is non-trivial solution of displacement in V. In this case the Rayleigh s quotient (3.3) degenerates into positive semi-definiteness ut not positive definiteness. This leads to a result: Conclusion III An elastic cuoid has neutral staility if the material satisfies the condition (4.17) and each of its surfaces is an aggregation of the type I, II, III and IV oundary su-regions. 5 Staility of an elastic cylinder with negative modulus Consider an elastic solid cylinder with aritrary cross-section, not necessarily of circular or rectangular section, and aritrary length; this includes cuoids under lateral constraint. Suppose that the lateral surface of the cylinder is fully fixed and the top and ottom surfaces are fully free, as shown in Fig. 3. Take the axis of shape-center of the cross section as the coordinate z-axis and the ottom surface in the xycoordinate plane. Denote the lateral surface of the cylinder y D, the ottom and top surfaces y Ω 1 and Ω, respectively. The whole oundary of the cylinder is V. In order to examine the staility of the cylinder, we write the strain energy function (.9) as new form: u u W = Ì + G + + + G Í + + + + - + - x y x y Ó 1 Ï È ( )( ) ( 4 ) ( ) Ê y u y 4 x ux ˆ λ εx εy εz γ zx γ yz ωz εx εy εz. (5.1) Fig. 3 A cylinder of aritrary cross section constrained on its periphery (the lateral surface). 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim www.pss-.com

Editor s Choice phys. stat. sol. () 44, No. 3 (7) 11 Sustituting (5.1) into the strain-displacement Eq. (.), incorporating energy conservation, Eq. (.8) we otain the new form of Rayleigh s quotient: ω U dv + I V (,, ) = R u v w = Ú V Ú ρ( u + v + w )dv, (5.) where the function Ï È 1 Ô Ê u v wˆ Ê w vˆ u w Ê v uˆ U = ( λ G ) G Ê ˆ Ì + + + + + + + + - Í x y z y z z x x y ÔÓ Ê u v wˆ Ô + G + - x y z Ô (5.3) and the integral Ê v u u vˆ I = 4GÚ - d V. x y x y V The direction cosines on the top and ottom surfaces ecome n = n = (5.4) x Ω1, y Ω1, and the displacement oundary condition on the lateral surface is u = v =, w =. (5.5) D D D Using integration y parts, we can transform the integral into the closed surface integral: È Ê u vˆ v u I = G nx v u n Ê y u v ˆ Ú Í - + - ds =. y y x x (5.6) D+ Ω1+ Ω Oviously, under the condition (3.1) we have the function U. The equality holds only if the displacement u, v and w satisfy the following all equations: w u w v w u v v u =, + =, + =, + =, - =. (5.7) z z x z y x y x y Integrating the front three equations in (5.7) we get the solution ϕ ϕ w= ϕ( x, y), u = - z + ψ1( x, y), v = - z + ψ( x, y), (5.8) x y where ϕ, ψ 1 and ψ are functions to e determined. Introducing the solution (5.8) into the last two equations in (5.7) and using the aritrariness of the variale z we have Ê ˆ ϕ = = + x y (5.9) www.pss-.com 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

1 Shang Xinchun and R. S. Lakes: Staility of elastic material with negative stiffness and Cauchy- Riemann equation of ψ 1 and ψ : ψ1 ψ ψ1 ψ + =, - =. x y y x This results in =, ψ1 =. (5.1) ψ Sustituting the solution (5.8) into the oundary condition (5.5), we get ϕ n Γ =, (5.11) ψ =, ψ =, (5.1) 1 Γ Γ where Γ is the closed line which is the oundary of the projective region of cross-section in the xycoordinate plane. The uniqueness of the Dirichlet prolem (5.1) and (5.1), and the Neumann prolem (5.9) and (5.11) give ψ1 = ψ and ϕ = constant, respectively. Also, from the last oundary condition in (5.5) and the first expression in (5.8), we have ϕ Γ =. Thus, ϕ. Hence, the function U. The equality holds only if u = v= w=. This implies that the Rayleigh s quotient (5.) is positive definite. Here we otain a new result: Conclusion IV An elastic cylinder with aritrary cross-section and in which the lateral surface is fully fixed and the top and ottom surfaces are fully stress-free is stale if its material satisfies the condition (3.1), namely G >, λ + G >. As shown in Section 3, the (3.1) condition is allows G >, - G/3 < K <, i.e. that the elastic parameters are located in the region II in which oth the Young s modulus E and the ulk modulus K are negative (see Fig. 1). We can also have G >, K >. 6 Discussion In the elastic cuoid, there is no assumption regarding the proportions; the cuoid could e a long ar or a thin plate. It is therefore not surprising that a constraint, over all directions or some directions must e applied to a region of each surface for staility of a solid oeying strong ellipticity. Strong ellipticity allows negative ulk modulus K and negative Young s modulus E, ut the lower modes of a long ar are governed y E. A fully free long ar therefore must therefore have a positive E to e stale. Staility conditions for cuoids with fixed proportions are not yet known. The present approach to the cuoid staility is on the asis of the Rayleigh s quotient and is totally different from the Ryzhak s Fourier expansion procedure. The present approach is more direct and allows us to deal with various much more complex assemled forms of the three types of oundary su-regions in the same surface. A particular case of the cuoid corresponds to neutral staility. Specifically, as an interesting example of the cuoid analysis, the elastic cuoid in which six surfaces are all stress-free is neutrally stale if its shear modulus G > and ulk modulus K = 4G/3 <. Usually, an elastic ody with any negative modulus is considered to e unstale if its oundaries are fully stress-free, however in this example neutral staility rather than instaility occurs. The condition (4.17) leads to vanishing of the velocity of longitudinal waves of a medium. This condition was of some historical interest in the context of early models of electromagnetism ased on a concept of space as an elastic medium [36]. For the special case that the entire oundary is fixed, Ericksen and Toupin [37] pointed that an elastic ody with aritrary regular shape has neutral staility when the condition (4.17) holds, and emphasized the importance of distinguishing etween ordinary and neutral staility. Usually, the staility ensures the uniqueness of solution, 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim www.pss-.com

Editor s Choice phys. stat. sol. () 44, No. 3 (7) 13 however the neutral staility does not imply uniqueness of solution. Indeed, for the case of stress oundary and mixed oundary, the solution is non-unique under the condition [5, 6]. As for cylinders constrained on the lateral surface, the condition (3.1) satisfies the strong ellipticity condition (3.7). In the view of physics, the condition λ + G > means the equiiaxial plane strain modulus is positive ([3], p. 17). As specific examples, the thin or thick elastic plates with the fixed side and the cuoid in which the top and ottom surfaces are fully stress-free and the other four surfaces are fully fixed are special cases of the cylinder discussed here. It is should e pointed out that the case of cuoid with two stress-free and four fixed surfaces is not covered in Ryzhak result. Referring to Fig. 1, it is interesting that isotropic materials with Poisson s ratio elow the lower staility limit of 1 can e stailized y various kinds of constraint, ut if Poisson s ratio exceeds the upper staility limit of 1/, strong ellipticity is violated, leading to an instaility which occurs regardless of surface constraint. Implications regarding experiment are as follows. Negative structural stiffness is well known in the context of post-uckled elements, and has een oserved experimentally (e.g. []). Negative elastic or viscoelastic moduli have een inferred from the ehavior of composites containing ferroelastic inclusions, ut have not een directly oserved. The present results indicate the possiility of oservation of negative moduli under constraint associated with instrumental displacement control. 7 Conclusions It is not necessary that a partially constrained elastic solid exhiit a positive definite strain energy to e stale. The elastic oject under partial constraint may have a negative ulk modulus and yet e stale. A cylinder of aritrary cross section with the lateral surface constrained and top and ottom free surfaces is stale if the shear modulus G > and G/3 < K < or K >. A cuoid is stale provided each of its surfaces is an aggregate of regions oeying fully or partially constrained oundary conditions. Appendix A1 Derivation of (3.6) Using integration y parts, the first term in volume integral (3.5) ecomes or another form: w v v w Ê w w ˆ w w Ú - = - - + z y y z y z y z y z - Ú Ú y z V V V Ê ˆ Ê ˆ dv v v dv n n vds Ê w wˆ = Ú nz -ny vds y z V w v v w Ê v v ˆ v v Ú - = - - + y z y z y z y z y z - Ú Ú z y V V V Ê ˆ Ê ˆ dv w w dv n n wds Ê v vˆ = Ú ny -nz wd S. z y V For the other terms in volume integral (3.5) have similar expressions. These expressions result in (3.6). www.pss-.com 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

14 Shang Xinchun and R. S. Lakes: Staility of elastic material with negative stiffness A Derivation of the identity (4.5) The line integration (4.4) can e rewritten as ÏÈ Ô z= c a J = GÌÍ ( wu) d y+ ( wu) dy z= ÔÍ Ú Ú Ó Rxy Ryz È a y= + Í Ú ( uv) d z + Ú ( uv) y= dz Í Ryz R zx È y= z= c Ô + Í Ú ( vw) y= d x + Ú ( vw) z= d x. Í R zx Rxy Ô (A.1) Calculating the front two terms in the aove line integrals, we have Ú Rxy z = c x = a z= + Ryz ( wu) d y ( wu) dy = ( wu) d y - ( wu) d y + ( wu) d y - ( wu) dy a a z= c z= z= c z= + ( wu) d y - ( wu) d y + ( wu) d y - ( wu) dy =. Ú Ú Ú Ú Ú Ú Ú Ú Ú a a z= z= z= c z= c Similarly, the other terms in the line integrals in (A.1) all vanish. Hence the identityj is otained. A3 The proof for uniqueness of solution of the Eq. (3.9) in the condition (4.11) On the asis of Green s first identity: u u udv = - u dv + u ds n Ú Ú Ú V V V and Eq. (3.9), we get Ú V dv = u u Ú u ds n V Ê V = ˆ 6 Â Ωi. (A3.1) i= 1 In the surfaceω 1 the integral u u v w IΩ = ds Ê u v w ˆ dyd z. 1 Ú u = + + n Ú x x x Ω1 Ω1 Since equations u = and u = still hold in the oundary V, we have u Ê v wˆ v u w u =- +, =, =. x y z x y x z 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim www.pss-.com

Editor s Choice phys. stat. sol. () 44, No. 3 (7) 15 Thus the integral Ï È IΩ = 1 Ú Ì- u Í ( v ) + ( w ) y z Ω Ó 1 È + Í( v ) ( u ) + ( w ) ( u ) dyd z. y z (1) (1) (1) Since the oundary condition in the surface Ω 1 = Bu + Buσ + Bσu is either u = for v= w= for B σ, from (A3.) we have I Ω =. Similarly, in the other surfaces the integral 1 Ú Ωi (1) u u u ds = ( i =, 3,, 6). n B (A3.) (1) (1) u Bu σ + or Hence, from (A3.1) the displacement gradients all are zero. This implies the displacements are constants, namely, the rigid shift displacements. Only for the following three cases of oundary condition: (i) The surfaces Ω 1, are type II and the others are type III; (ii) The surfaces Ω 3,4 are type III and the others are type II; (iii) The surfaces Ω 5,6 are type III and the others are type II, the rigid shift displacements are possile along x, y, z direction, respectively. Except for the three cases the Eq. (3.9) has unique zero solution. Acknowledgments Many useful discussions with W. Drugan are gratefully acknowledged. X. Shang acknowledges a scholar award from the China Scholarship Council which greatly facilitated this work. R. Lakes is grateful for support via the U.S. National Science Foundation. References [1] J. M. T. Thompson, Instaility and Catastrophes in Science and Engineering (John Wiley & Sons, London, 198). [] R. S. Lakes, Philos. Mag. Lett. 81, 95 1 (1). [3] S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd edition (McGraw-Hill, New York, 197). [4] E. and F. Cosserat, C. R. Acad. Sci. Paris 16, 189 191 (1898). [5] Lord Kelvin (W. Thomson), Philos. Mag. 6, 414 45 (1888). [6] R. S. Lakes, Science 35, 138 14 (1987). [7] R. S. Lakes, Adv. Mater. 5, 93 96 (1993). [8] K. W. Wojciechowski, Mol. Phys. 61, 147 158 (1987). [9] K. W. Wojciechowski, Phys. Lett. A 137, 6 64 (1989). [1] K. W. Wojciechowski and A. C. Branka, Phys. Rev. A 4, 7 75 (1989). [11] G. W. Milton, Modelling the properties of composites y laminates, in: Homogenization and effective moduli of materials and media, edited y J. L. Erickson, D. Kinderlehrer, R. Kohn, and J. L. Lions (Springer Verlag, Berlin, 1986), pp. 15 175; J. Mech. Phys. Solids 4, 115 1137 (199). [1] J. Glieck, Anti-ruer, The New York Times (Science Times), 14 April 1987, p. 1. [13] B. D. Caddock and K. E. Evans, J. Phys. D, Appl. Phys., 1877 188 (1989). [14] K. L. Alderson and K. E. Evans, Polymer 33, 4435 4438 (199). [15] R. S. Lakes, Phys. Rev. Lett. 86, 897 9 (1). [16] R. S. Lakes, T. Lee, A. Bersie, and Y. C. Wang, Nature 41, 565 567 (1). [17] R. S. Lakes and W. J. Drugan, J. Mech. Phys. Solids 5, 979 19 (). [18] Y. C. Wang and R. S. Lakes, J. Compos. Mater. 39, 1645 1657 (5). [19] D. J. Bergman and B. I. Halperin, Phys. Rev. B 13, 145 175 (1976). [] B. Kindler, D. Finsterusch, R. Graf, F. Ritter, W. Assmus, and B. Lüthi, Phys. Rev. B 5, 74 77 (1994). [1] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edition (Dover, New York, 1944). [] H. H. E. Leipholz, Dynamic staility of elastic systems, in: Staility, Study No. 6, Solid Mechanics Division, edited y H. H. E. Leipholz (University of Waterloo, Waterloo, Canada, 197), p. 56. www.pss-.com 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

16 Shang Xinchun and R. S. Lakes: Staility of elastic material with negative stiffness [3] L. Meirovitch, Methods of Analytical Dynamics (McGraw-Hill, New York, 197), p. 493. [4] R. J. Knops and E. W. Wilkes, Theory of elastic staility, in: Handuch der Physik, Vol. III, edited y S. Flügge (Springer-Verlag, Berlin, 1973), p. 8. [5] R. Hill, J. Mech. Phys. Solids 9, 114 13 (1961). [6] R. J. Knops and L. E. Payne, Uniqueness Theorems in Linear Elasticity (Springer-Verlag, New York, 1971), pp. 7 8. [7] Y.-C. Chen, Arch. Ration. Mech. Anal. 113, 167 175 (1991). [8] E. I. Ryzhak, J. Mech. Appl. Math. 47, 663 76 (1994). [9] C. Truesdell and W. Noll, The non-linear field theories of mechanics, in: Handuch der Physik, Vol. III, edited y S. Flugge (Springer-Verlag, Berlin, 1965), p. 159. [3] M. F. Beatty, Appl. Mech. Rev. 4, 1699 1734 (1987). [31] W. Jaunzemis, Continuum Mechanics (The Macmillan Company, New York, 1967), pp. 434 435. [3] R. S. Rivlin, Some thoughts on material staility, in: Finite Elasticity, edited y D. E. Carlson and R. T. Shield (Martinus Nijhoff Pulishers, London, 198), pp. 15 199. [33] J. K. Knowles and E. Sternerg, J. Elast. 8, 39 379 (1978). [34] P. Rosakis, A. Ruina, and R. S. Lakes, J. Mater. Sci. 8, 4667 467 (1993). [35] E. Salje, Phase transitions in ferroelastic and co-elastic crystals (Camridge University Press, 199). [36] E. Whittaker, A History of the Theory of Aether and Electricity (Dover Pulications Inc., New York, 1951), p. 144. [37] J. L. Ericksen and T. Toupin, Canad. J. Math. 8, 43 436 (1956); in: Foundations of Elasticity Theory, edited y C. Truesdell (Science Pulishers, New York, 1965), pp. 43 436. 7 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim www.pss-.com