post buckling analysis of curved beams

Zhichao Fan AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China; Center for Mechanics and Materials, Tsinghua University, Beijing 100084, China Jian Wu1 AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China; Center for Mechanics and Materials, Tsinghua University, Beijing 100084, China e-mail: wujian@tsinghua.edu.cn Qiang Ma AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China; Center for Mechanics and Materials, Tsinghua University, Beijing 100084, China Yuan Liu AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China; Center for Mechanics and Materials, Tsinghua University, Beijing 100084, China Post-Buckling Analysis of Curved Beams Stretchability of the stretchable and flexible electronics involves the post-buckling behaviors of internal connectors that are designed into various shapes of curved beams The linear displacement–curvature relation is often used in the existing post-buckling analyses Koiter pointed out that the post-buckling analysis needs to account for curvature up to the fourth power of displacements A systematic method is established for the accurate post-buckling analysis of curved beams in this paper It is shown that the nonlinear terms in curvature should be retained, which is consistent with Koiter’s post-buckling theory The stretchability and strain of the curved beams under different loads can be accurately obtained with this method [DOI: 10.1115/1.4035534] Keywords: post-buckling, stretchability, curved beam, curvature, finite deformation Yewang Su State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China Keh-Chih Hwang1 AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China; Center for Mechanics and Materials, Tsinghua University, Beijing 100084, China e-mail: huangkz@tsinghua.edu.cn Introduction Curved beams have simple geometry and facilitative fabrication and are widely used in the fields of electronic, aerospace, and architecture Recently, curved beams are treated as internal connectors of stretchable electronics due to their high stretchability with small strain [1–13] There are many functional stretchable and flexible electronics based on different shapes of curved beams, such as epidermal health/wellness monitors [14–18], sensitive electronic skins [19–23], and spherical-shaped digital cameras Corresponding authors Manuscript received November 16, 2016; final manuscript received December 15, 2016; published online January 24, 2017 Assoc Editor: Daining Fang Journal of Applied Mechanics [24–26] The post-buckling behaviors, especially lateral and outof-plane buckling, provide the flexibility and stretchability of the electronics [27], which are the essential properties of a robust technology of assembling various microstructures [28–30] There are many researchers studying up on the stability of planar curved beams Timoshenko and Gere [31] presented initial buckling behaviors of circular beams The analytical models for post-buckling behaviors of the inextensible ring under uniform radial pressure were developed by Carrier [32] and Budiansky [33] A systematic variational approach of space curved beams was developed by Liu and Lu and employed on buckling behavior and critical load of the serpentine structure [34] Many results of the post-buckling behaviors of curved beams are obtained by the semi-analytical energy approach and finite-element method C 2017 by ASME Copyright V MARCH 2017, Vol 84 / 031007-1 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org (FEM) [35,36] Koiter’s approach of energy minimization for post-buckling expanded the potential energy to the fourth power of displacement because the third and fourth power terms actually govern post-buckling [37–39], which require the elongation and curvature that at least express the third power of displacement This paper presents a systematic study on post-buckling of curved beam, the nonlinear relations between the deformation components (elongation and curvatures) and displacements are derived, and the perturbation method is used to obtain the analytical solution of the post-buckling behaviors of curved beams Deformation of Curved Beam ~ dP dS (2.1) The curvature of the curved beam was defined by Love [40] as dEi ¼ K  Ei i ẳ 1; 2; 3ị dS d~ p ds ds dS where p ¼ pi ei and q ¼ qi ei are the distributed force and moment per unit length in the deformed beam, respectively The derivative symbols have the relation, 1=kịdị=dSị ẳ dị=dsị The internal force, t3 , is conjugate with the elongation, k, and has the constitutive relation (2.8) where E and A are the elastic modulus and the cross section area of the beam, respectively The moment, m, is not conjugate with the curvature, j, but it is conjugate with the Lagrangian curvature, j^ , which is defined as dei ¼ j^  ei ði ¼ 1; 2; 3Þ dS (2.9) ^ , has the relation with curvawhere the Lagrangian curvature, j ture, j, as j^ ¼ kj The constitutive relation between Lagrangian curvatures (abbreviated hereafter simply to curvatures) and moments is j À K1 Þ m1 ¼ EI1 ð^ m2 ¼ EI2 ð^ j À K2 Þ (2.3) where s is the arc-length coordinate of the deformed beam, which is the function of the arc-length of initial curved beam, S The elongation of the deformed beam, k, can be obtained from k¼ (2.7) (2.2) where the curvature K ẳ K1 E1 ỵ K2 E2 ỵ K3 E3 , K1 and K2 are the curvatures along the axes in cross section, and K3 denotes the twist along the tangential direction of the centroid line ~ to The centroid line of curved beam is deformed from P ~ ỵ U, where U is the displacement of the beam The local p~ ¼ P triad vectors of the deformed curved beam are ei ði ¼ 1; 2; 3Þ, where e1 and e2 are the orthogonal unit vectors in the cross section of deformed curved beam, and e3 is the unit vector along the tangential direction of the centroid line of deformed beam, which is defined as e3 ẳ dm ỵ e3 t ỵ q ẳ ds t3 ẳ EAk 1ị 2.1 The Initial and Deformed Curved Beams The material points on the centroid line of the initial curved beam are denoted ~ by PðSÞ (Fig 1), where S is the arc-length of centroid line The local triad vectors at the centroid line of the curved beam are orthogonal unit vectors Ei ði ¼ 1; 2; 3Þ E1 and E2 are along two lines of symmetry of the cross section of beam (Fig 1) E3 is the unit vector along the tangential direction of the centroid line and can be given as E3 ¼ dt ỵpẳ0 ds (2.10) m3 ẳ GJ^ j K3 Þ where G is the shear modulus of the beam, I1 and I2 are the section area moment of inertia about the local coordinates, and J is the polar moment of inertia of the cross section of beam The components of the equilibrium equation expressed with the components of internal forces, moments, and the curvatures can be given as dti ^ j tk ỵ kpi ẳ i ẳ 1; 2; 3ị ỵ ijk j dS dmi ^ j mk ij3 ktj ỵ kqi ẳ i ẳ 1; 2; 3ị ỵ ijk j dS (2.11) (2.4) The curvature of the deformed curved beam also can be given as [40] dei ¼ j  ei ði ¼ 1; 2; 3ị ds (2.5) where j ẳ j1 e1 ỵ j2 e2 ỵ j3 e3 , j1 and j2 are the components of curvature along sectional vectors e1 and e2 , respectively, and j3 denotes the twist along the tangent direction, e3 Here, the twist angle, /, is defined with j3 as ð / ¼ j3 ds (2.6) where the axis of the twist angle changes with the location 2.2 Equilibrium Equations and Constitutive Relation of Curved Beam The internal force, t ¼ ti ei , and moment, m ¼ mi ei , of the curved beam satisfy the following equilibrium equations [41]: 031007-2 / Vol 84, MARCH 2017 Fig Schematic illustration of initial and deformed curved beam Transactions of the ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org where ijk are the components of Eddington tensor The Deformation Variables in Terms of Displacement for Planar Curved Beam The planar curved beam, which has only one nonzero component of initial curvature, K1 , is widely used We will focus on the planar curved beam in Secs 3–7 The displacement, U, can decompose in the local coordinates, Ei i ẳ 1; 2; 3ị, as U ẳ U1 E1 ỵ U2 E2 ỵ U3 E3 The elongation defined by Eq (2.4) can be expressed through the components of the displacement as &   dU3 ỵ U2 K kẳ 1ỵ2 dS " 2  2 #)12 2  dU1 dU2 dU3 ỵ U3 K ỵ ỵ U2 K ỵ dS dS dS (3.1) The relations between the local triads on the initial and deformed curved beams are given by the coefficients aij as ei ẳ aij Ej i ẳ 1; 2; 3ị (3.2) E3 and e3 are along the tangent directions of the initial and deformed curved beams, respectively By substituting Eqs (2.1) and (2.3) into Eq (3.2), the coefficients, a3j ðj ¼ 1; 2; 3Þ, can be derived as   dU1 dU2 a31 ¼ ; a32 ¼ À U3 K ; k dS k dS   (3.3) dU3 1ỵ ỵ U2 K a33 ẳ k dS The local coordinates, ei ði ¼ 1; 2; 3ị and Ei i ẳ 1; 2; 3ị, are orthonormal (i.e., ei Á ej ¼ dij and Ei Á Ej ¼ dij ), which lead the coefficients, aij , to satisfy the following equations: a2i1 ỵ a2i2 ỵ a2i3 ẳ i ẳ 1; 2; 3ị aik ajk ẳ i ẳ jị (3.4) The relations between the coefficients, aij i ẳ 1; 2; j ẳ 1; 2; 3ị, the displacements, Ui , and the twist angle, /, are complicated and cannot be explicitly expressed as Eq (3.3) Substitution of Eqs (3.1) and (3.3) into Eq (3.4) and expansion of the coefficients, aij , to the powers of generalized displacements lead to > = faij g ¼ > > : ; 0 dU1 > > ½1Š > > À w > > > > > > dS > > < = dU2 ẵ1 ỵ w ỵ U3 K > > dS > > > > > > > > dU dU > > : ; À U3 K dS dS n o n o h2i h3i ỵ aij ỵ aij ỵ f g i; j ẳ 1; 2; 3Þ ^ h2i ^ ih3i are the second and third power of displacewhere j and j i ments, Ui , and twist angle, /, which are given in Appendix A ^ ih2i and j ^ ih3i ) are important for the analThe nonlinear terms (i.e., j ysis of the post-buckling behaviors of the curved beams based on the Koiter’s theory As far as the authors are aware, the power expansion of deformation for Euler–Bernoulli curved beams (without assuming inextensibility of the beams) is a new result The results obtained by Su et al [41] for straight beams follow as a special case The Perturbation Solution for Post-Buckling of the Planar Curved Beam The planar curved beam will be buckling in-plane or out-ofplane due to the planar loads (e.g., p1 ¼ 0) The method of perturbation is used to solve differential equations (2.11) substituted with variables of the deformation and displacement, where a small ratio, a, of the maximum deflection to the characteristic length (e.g., the beam length, LS , or the initial curvature radius of beam, R) is introduced The generalized displacements from buckling, expanded to the powers of a, can be written as U1 ẳ aU11ị ỵ a2 U12ị ỵ a3 U13ị ỵ Oa4 ịLS U2 ẳ aU21ị þ a2 U2ð2Þ þ a3 U2ð3Þ þ Oða4 ÞLS U3 ẳ aU31ị ỵ a2 U32ị ỵ a3 U33ị ỵ Oa4 ịLS (4.1) / ẳ a/1ị ỵ a2 /2ị ỵ a3 /3ị ỵ Oa4 ị By substituting Eq (4.1) into Eqs (3.1) and (3.6), the elongation and curvatures can be also expanded to the powers of the small ratio, a, as k ẳ ỵ ak1ị ỵ a2 k2ị ỵ a3 k3ị ỵ Oa4 ị (4.2) ^ i0ị ỵ a^ ^ i2ị ỵ a3 j ^ i3ị ỵ Oa4 ịL1 ^i ẳ j j i1ị ỵ a2 j j S i ¼ 1; 2; 3Þ (4.3) ^ 1ð0Þ ¼ K1 , j ^ 20ị ẳ j ^ 30ị ẳ 0, kkị and j ^ iðkÞ are the funcwhere j tions of generalized displacements, and dU31ị ỵ U21ị K1 dS (4.4) d2 U21ị d U31ị K1 ỵ dS dS2 d U11ị ẵ1 ẳ w1ị K1 dS2 d/1ị ẳ dS (4.5) k1ị ẳ ^ 11ị ẳ j (3.5) é where wẵ1 ẳ / K1 dU1 =dSịdS, and the superscripts h2i and h2i h3i h3i in aij and aij refer to the second and third power of generalized displacements (i.e., Ui and /) They are given in Appendix A, and fÁ Á Ág are the terms of the fourth and higher power of displacements and twist angle Journal of Applied Mechanics After substituting Eqs (3.2) and (3.5) into Eq (2.9), the curvatures can be expressed in terms of generalized displacements as > d2 U2 dðU3 K1 ị > > > > > ỵ 9 > > > > > dS > dS ^ j K > > > > > > 1 o < = < = < = n h2i d U ẵ1 ^2 ẳ ^ j ỵ ỵ j i > ; > ; > : > > > dS2 À w K1 > > : > > ^3 j > > > > d/ > > > > : ; dS n o ^ h3i ỵ f g (3.6) ỵ j i ^ 21ị j ^ 31ị j é ẵ1 where w1ị ¼ /ð1Þ À K1 ðdU1ð1Þ =dSÞdS The coefficient, aij , can be expanded to the powers of the small ratio, a, as MARCH 2017, Vol 84 / 031007-3 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org aij ¼ þ a2 aijð2Þ þ a3 aijð3Þ þ Oða4 Þ ði ẳ jị aij ẳ aaij1ị ỵ a2 aij2ị ỵ a3 aij3ị ỵ Oa4 ị i 6ẳ jị (4.6) where aijkị are the functions of generalized displacements > > dU11ị ẵ > > > > w1ị À > > > > dS > > > > < = dU ị ẵ1 21 aij1ị ẳ w ị K ỵ U ị 1 > > dS > > > > > > > dU1ð1Þ dU2ð1Þ > > > > > K À U : ; 3ð1Þ dS dS (4.7) ^ ikị , and aijkị for k ẳ are given in The expressions of kðkÞ , j Appendix B Substituting Eqs (4.2) and (4.3) into constitutive relations (2.8) and (2.10) and considering the equilibrium equations, the internal forces and moments can be expanded to powers of the small ratio a as ti ẳ ti0ị ỵ ati1ị ỵ a2 ti2ị ỵ a3 ti3ị ỵ Oa4 ịEA mi ẳ mi0ị ỵ ami1ị ỵ a2 mi2ị ỵ a3 mi3ị ỵ Oa4 ịEI1 LÀ1 S (4.8) (4.9) where tið0Þ and mið0Þ are the forces and moments at the onset ^ iðkÞ of buckling, and the relation between t3ðkÞ , miðkÞ , and kðkÞ , j ðk ! 1Þ can be written as t3ðkÞ ¼ EAkðkÞ ^ 1ðkÞ m1ðkÞ ¼ EI1 j ^ 2ðkÞ m2kị ẳ EI2 j (4.10) m3kị ẳ GJ^ j 3kị After substituting the forces (4.8) and moments (4.9), equilibrium equations (2.11) can be decomposed as dti0ị ^ j0ị tk0ị ỵ pi0ị ẳ i ẳ 1; 2; 3ị ỵ ijk j dS (4.11) dmið0Þ ^ jð0Þ mkð0Þ À ij3 tjð0Þ ỵ qi0ị ẳ i ẳ 1; 2; 3ị ỵ ijk j dS Step 1: Solve Eq (4.12) for n ¼ with the corresponding boundary conditions to determine the buckling mode for the lead^ ið1Þ , and Uið1Þ , of elongation, curvatures, and dising order, kð1Þ , j placements, and the critical loads at the onset of buckling, pið0Þ and qið0Þ Step 2: Solve Eq (4.12) for n ¼ with the corresponding ^ ið2Þ , and boundary conditions to determine the second order kð2Þ , j Uið2Þ of elongation, curvatures, and displacements, and the increment of loads, pið1Þ and qið1Þ Step 3: Solve Eq (4.12) for n ¼ with the corresponding boundary conditions to determine the buckling mode for the third ^ ið3Þ of elongation and curvatures, and the increorder kð3Þ and j ment of loads, pið2Þ and qið2Þ In-Plane Post-Buckling Behavior of Elastic Ring As illustrated in Fig 2, the uniform distributed radial load, p2 , which remains normal to the centroid line during the deformation, is applied on an elastic thin ring The arc-length, S, is clockwise counted from A The ring at A is simply supported, and the displacement along tangent direction at B is constrained, these boundary conditions can be written as Uinị jSẳ0 ẳ i ẳ 1; 2; 3; n ẳ 1; 2; 3; ị (5.1) U3nị jSẳpR ẳ n ẳ 1; 2; 3; ị (5.2) where R ¼ 1=K1 is the initial curvature radius of ring The displacements, deformations, and forces/moments are periodical due to the periodical deformation of the ring, i.e., ịjSẳS0 ẳ ịjSẳS0 þ2pR ; ð0 S0 2pRÞ (5.3) ^ i, j ^ iðnÞ , ti , tiðnÞ , mi , miðnÞ , Ui , UiðnÞ , /, where ðÞ denotes k, kðnÞ , j /ðnÞ , and aij , aijðnÞ The width of the ring section, w, is much larger than its thickness, t, and the ring will be buckling in the initial plane of the p The out-of-plane compocurved beam under the load, p2 ¼ À nents of force, moment, and displacement are zero, i.e., t1 ¼ 0, m2 ¼ 0, m3 ¼ 0, U1 ¼ 0, and / ¼ The internal force and moment at the onset of buckling, t2ð0Þ , t3ð0Þ , and m1ð0Þ , satisfy Eq (4.11) and have the relation with critical load, p20ị , as and lẳn dtinị X ^ jlị tknlị ỵ klị pinlị ẳ ijk j ỵ dS l¼0 ði ¼ 1; 2; 3; n ¼ 1; 2; 3; :::ị lẳn dminị X ^ jlị mknlị ij3 klị tjnlị ỵ klị qinlị ẳ ijk j ỵ dS lẳ0 i ẳ 1; 2; 3; n ẳ 1; 2; 3; :::ị (4.12) where k0ị ẳ 1, and the loads are expanded to the power series of a as pi ẳ pi0ị ỵ api1ị ỵ a2 pi2ị ỵ a3 pi3ị ỵ Oa4 ịEAL1 S qi ẳ qi0ị ỵ aqi1ị ỵ a2 qi2ị ỵ a3 qi3ị ỵ Oða4 ÞEI1 LÀ2 S (4.13) where pið0Þ and qið0Þ are the critical loads at bifurcation point The differential equations in this section, in which internal forces and moments are substituted with deformation variables and displacements, are solved in the following steps 031007-4 / Vol 84, MARCH 2017 Fig Schematic illustration of elastic ring under uniform compression Transactions of the ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org t30ị ẳ  p 20ị R; t20ị ẳ 0; m10ị ¼ (5.4) After substituting constitutive relation (4.10) into equilibrium ^ 11ị , the difequations (4.12) for n ẳ and eliminating t2ð1Þ and j ferential equation for elongation, kð1Þ , is derived as U3ð1Þ (5.10) d3 kð1Þ dkð1Þ þ k12 ¼0 dS dS k12   S 2S ỵ c1 ịc1 p21ị R4 2S ỵ R cos cos R R R ỵ 4c1 EI ! p2ð1Þ R S R þ 4c1 2S ¼ R sin À þ c1 sin R ỵ c1 R EI1 U21ị ẳ R cos (5.5) and the parameters, C10 , C11 , and C12 , are also determined as ẳ ẵ1 ỵ p20ị R=EAị ỵ p20ị R =EI1 ị=R Solution of where Eq (5.5) is k1ị ẳ C12 cos k1 Sị þ C11 sin ðk1 SÞ þ C10 (5.6) where C10 , C11 , and C12 are the parameters to be determined The critical load, p2ð0Þ , can be determined by the periodical condition of elongation, k1ị , as p20ị ẳ 3EI1 ỵ c1 ị1 R3 (5.7) ^ 11ị is also obtained by substituting where c1 ẳ EI1 =EAR2 ị j Eqs (4.10) and (5.6) into Eq (4.12) as ( " p21ị R3 1 ỵ c1 ị ỵ 4C10 C10 ỵ c1 3Rc1 EI1  ' 2S 2S ỵ C12 cos À3 C11 sin R R p2ð1Þ R3 c1 ỵ c1 ị ; C11 ẳ 0; C10 ẳ ỵ 4c1 EI " # 3c1 3c21 p21ị R3 C12 ẳ ỵ ỵ c1 ị ỵ 4c1 ị EI1 where U21ị is assumed to be symmetrical about AB, and maxU21ị ị ẳ 2R Substitution of constitutive relation (4.10) into equilibrium ^ 1ð2Þ give equations (4.12) for n ẳ and elimination of t22ị and j the differential equation for elongation, kð2Þ , as d3 k2ị dk2ị ỵ k12 ẳ F21 Sị dS dS # (5.12) where ^ 11ị ẳ j (5.8) " F21 Sị ẳ # c1 p21ị R3 ỵ c1 ị ỵ 4c1 ỵ EI1 R3 ỵ c1 ị2 ỵ 4c1 ị2 2S 2 p 2ð1Þ R3 À > > > > > > c ỵ 3c ỵ 4c sin > > 1 < = R EI1 " # > > > > 45c1 ỵ 4c1 þ p2ð1Þ R c1 ð1 þ c1 Þ sin 4S > > > > : R; EI1 The differential equations for U2ð1Þ and U3ð1Þ can be derived from Eqs (4.4) and (4.5) as dU31ị U21ị ỵ ẳ k1ị dS R d2 U21ị dU31ị ẳ ^ j 11ị R dS dS2 (5.11) (5.9) The solution of Eq (5.12) is The solutions of U2ð1Þ and U3ð1Þ can be obtained by the boundary conditions (5.1) and (5.2) and the periodical condition (5.3) as k2ị ẳ C22 cosk1 Sị ỵ C21 sink1 Sị ỵ C20 S f sinẵk1 S nịF21 nị þ dndf k1 0 (5.13) where kð2Þ is also periodical, which determined the increment of load, p21ị ẳ 0, and the parameters, C20 , C21 , and C22 will be determined with the boundary conditions and orthogonality condiÐ 2pR tion [37,38], k1ị nịk2ị nịdn ẳ ^ 12ị , is also obtained by substitution of Eqs The curvature, j (4.10) and (5.13) into Eq (4.12) as ỵ c1 p22ị R3 ỵ 4c1 C21 2S ỵ C20 sin R 3R EI1 3Rc1 Rc1 16R1 ỵ c1 Þ " # 45 4C22 2S 9ð1 À 4c1 ị 4S ỵ cos ỵ cos 4R ỵ c1 ị2 R 16R1 ỵ c1 ị2 R c1 ^ 12ị ẳ j 323 ỵ 68c1 ị ỵ (5.14) The differential equations for U2ð2Þ and U3ð2Þ are given by substitution of Eqs (5.13) and (5.14) into Eqs (B1) and (B2) as  =p  2ð0Þ , versus the Fig The ratio of load to critical load, p normalized displacement, U2max =ð2RÞ, during post-buckling, which is consistent with the results of Carrier’s model Journal of Applied Mechanics d2 U22ị dU32ị ẳ F22 Sị R dS dS2 dU32ị U22ị ỵ ẳ F23 Sị dS R (5.15) MARCH 2017, Vol 84 / 031007-5 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org where   À Á   2 d U À U d U U ð Þ ð Þ dU dU 1 4d 2ð1Þ 3ð1Þ 31ị 21ị F22 Sị ẳ ^ j 12ị ỵ ỵ dS2 dS 2R R dS dS dS F23 Sị ẳ k2ị  2 dU21ị U3ð1Þ R dS and the underlined terms in F22 ðSÞ come from the nonlinear part of curvature (B2) The solutions of Eq (5.15) with the boundary conditions (5.1) and (5.2) are U2ð2Þ U3ð2Þ > > S > > > > 180c1 ỵ 2c1 ị 3c1 61 ỵ 124c1 ịcos > > < = R R " #   ẳ 22ị R S 4S > 801 ỵ c1 ị ỵ 4c1 ị > 3p > > > ỵ 3c1 ỵ 4c1 Þcos > À cos > > : À5 þ 16c1 ð1 þ c1 Þ EI1 R R;   3Rc1 S 4S cos cos ỵ R R ị ỵ c1 " # > > > 2ð2Þ R S> > > 3p   > > < = ð Þ ð ị 45 3c 61 ỵ 124c ỵ c ỵ 80c sin 1 1 R 3Rc1 S 4S R EI ẳ sin ỵ sin þ > R R 20ð1 þ c1 Þ2 320ð1 þ c1 ị2 ỵ 4c1 ị > > > 4S > > > > ỵ31 ỵ 4c1 ị15 ỵ 76c1 Þsin : ; R (5.16) where the underlined terms are derived from the nonlinear terms of the curvature, and U2ð2Þ is assumed to be symmetrical about AB The parameters, C20 ; C21 ; and C22 , are also determined as C20 c1 ỵ c1 ị p22ị R3 9c1 23 ỵ 68c1 ị ẳ ỵ ; ỵ 4c1 ị EI1 161 ỵ c1 ị2 ỵ 4c1 ị C21 ẳ 0; C22 ẳ 45c1 (5.17) 41 ỵ c1 ị2 d3 k3ị dk3ị ỵ k12 ẳ F3 Sị dS3 dS " 6c1 ỵ c1 ị1 ỵ 4c1 ị 27 24c1 16c21 321 ỵ c1 ị2 þ 1053c1 ðÀ3 þ 4c1 Þ 16R3 ð1 þ c1 Þ (5.19) where C30 , C31 , and C32 are the parameters, and the increment of load, p2ð2Þ , can be determined by the periodical condition of elongation, kð3Þ # À Á p2ð2Þ R3 2S À À c1 þ 4c21 sin R EI1 sin 6S R 031007-6 / Vol 84, MARCH 2017 p22ị ẳ 24c1 16c21 27EI1 ỵ c1 ị2 c1 ỵ 4c21 32R3 (5.20) The thickness, t, of the cross section of beam is much smaller than the radius, R, which indicates c1 ( The load, p2 , normalized by critical load, p2ð0Þ , can be simplified as  2 22ị p2 27a2 27 U2max 2p ẳ1ỵa %1ỵ ẳ1ỵ 32 p20ị p20ị 32 2R (5.18) where R3 k3ị ẳ C32 cosk1 Sị ỵ C31 sink1 Sị ỵ C30 S f sinẵk1 f nịF3 nị ỵ dndf k1 0 À Substitution of constitutive relation (4.10) into equilibrium ^ 13ị give equation (4.12) for n ẳ and elimination of t2ð3Þ and j the differential equation for elongation, k3ị , as F3 Sị ẳ The solution of Eq (5.18) is (5.21) where a ẳ U2max =2Rị has been used, and it is the same with the result of Budiansky [33] p 2ð0Þ , increases with Figure shows that the normalized load, p2 = the normalized maximum displacement, U2max =ð2RÞ, which is consistent with Carrier’s model [32] It indicates that the elongation can be neglected due to the inextensibility of elastic ring in Carrier’s model Lateral Buckling of Circular Beam The curved beam is widely used as interconnector, which is often freestanding and connects the sensors in the stretchable and Transactions of the ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org flexible electronics [2] The lateral buckling of the freestanding interconnector will happen because the thickness of the beam cross section, t, is much larger than its width, w [1,2,16] The thickness direction is along the radial direction of the circular beam, which is consistent with it in Sec The displacement, U1 , and rotation, /, which are the odd powers of the small ratio, a, of the maximum deflect of U1 to beam length, are the primary displacements, while the secondary displacements, U2 and U3 , are ^ and the even powers of the small ratio, a [41] The curvatures, j ^ , are the odd powers of the small ratio, a, and the elongation, k, j ^ , are the even powers of the small ratio, a and curvature, j 6.1 The Lateral Buckling of Circular Beam Under Bending Moment As shown in Fig 4, the bending moment, M, is applied on the circular beam at the ends This beam with length, LS , which subtends the angle, a, is simply supported in the plane and the out-of-plane, and beam ends cannot rotate around the centroid of beam, but the right beam end can freely slide in the plane of the beam, i.e., U2 jS¼0 ¼ U3 jS¼0 ¼    a a ðU2 E2 ỵ U3 E3 ịjSẳLS cos E2 sin E3  ¼0 2 S¼LS    a a tjSẳLS sin E2 ỵ cos E3  ẳ0 2 S¼LS (6.1) m1 jS¼0 ¼ m1 jS¼LS ¼ M U1 jS¼0 ¼ U1 jS¼LS ¼ ðe2 Á E1 ÞjS¼0 ¼ ðe2 Á E1 ÞjS¼LS ¼ (6.2) m2 jS¼0 ¼ m2 jS¼LS ¼ By substituting Eq (3.2) into the above equations, the boundary conditions can be expanded with respect to the perturbation parameter, a, as U2nị jSẳ0 ¼ U3ðnÞ jS¼0 ¼   a a  U2nị cos U3nị sin ẳ0 2 SẳLS  lẳn  X a a  tilị ai2nlị sin ỵ tilị ai3nlị cos ẳ0 2 SẳLS lẳ0 (6.3) U1nị jSẳ0 ẳ U1nị jSẳLS ẳ (6.4) m2nị jSẳ0 ẳ m2nị jSẳLS ẳ The internal forces and moments at the onset of buckling, ti0ị , mi0ị i ẳ 1; 2; 3Þ, satisfy Eq (4.11) and have the relation with critical load, M0ị , as t10ị ẳ t20ị ẳ t30ị ¼ 0; m2ð0Þ ¼ m3ð0Þ ¼ 0; m1ð0Þ ¼ Mð0Þ (6.5) where the normalized angle, a ¼ a=p, and k2 can be simplified to 1=Ra ị When a ẳ 1, one of the two values of the critical load in Eq (6.8) is zero, which corresponds to the freedom of a semicircular beam to rotate about the diameter connecting the two ends, the other value, M0ị ẳ pEI2 ỵ c3 =c2 Þ=LS , is the critical load for the semicircular beam When the curved beam is shallow, i.e., R ) LS , the critical load, Mð0Þ , will approach to the critical load pffiffiffiffiffiffiffiffiffiffiffi for the straight beam, ðpEI2 =LS Þ c3 =c2 [31] ^ 3ð1Þ can be obtained by the substitution of constitutive The j relation (4.10) and Eq (6.7) into equilibrium equations (4.12) for n ¼ as   !   a c2 Mð0Þ R S S ^ 31ị ẳ ỵ C41 sin ỵ C42 À C40 À cos j a R a R EI2 c3 (6.9) where the parameter, C42 , will be determined by the boundary condition The differential equations for U1ð1Þ and /ð1Þ can be derived from Eq (4.5) as d/1ị ^ 31ị ẳj dS where the maximum of U11ị is LS , and the parameters, C40 , C41 , and C42 , are where k22 ẳ ẵM0ị R=EI2 ị 1ẵM0ị Rc2 =EI2 ị c3 =R2 c3 ị, c2 ẳ EI2 =EAR2 ị, and c3 ẳ GJ=EAR2 ị The solution of Eq (6.6) is Journal of Applied Mechanics c3 a 1ị   ; C41 ẳ 0; M0ị R c3 ỵ a c2 EI2   M ð0Þ R c2 ða À 1ị p EI2   ẳ M R 0ị R c3 ỵ a c2 EI2 C40 ¼ C42 (6.6) (6.10) The solutions of Eq (6.10), which satisfy the boundary conditions (6.4), are   Mð0Þ R pc2 a ða À 1Þ À1 S EI2   sin /1ị ẳ M R  a R ð Þ (6.11) a c2 À À c3 EI2 S U11ị ẳ pa R sin a R Substitution of constitutive relation (4.10) into equilibrium ^ 3ð1Þ and t11ị give equations (4.12) for n ẳ and elimination of j ^ 2ð1Þ as the differential equation for j ^ 1ị d2 j ^ 21ị ẳ ỵ k22 j dS2 (6.7) where the parameters, C40 and C41 , will be determined by the boundary conditions The boundary conditions, ^ 21ị jSẳLS ẳ 0, which are derived by substitution of ^ 21ị jSẳ0 ẳ j j constitutive relation (4.10) into the boundary conditions (6.4), determine the critical load as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   EI2 c3 c3 c3 ỵ 1ỵ ỵ or M0ị ẳ a c2 2R c2 c2 (6.8) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   EI2 c3 c3 c3 ỵ 1ỵ M0ị ẳ a c2 2R c2 c2 d2 U11ị 1 ^ 21ị ỵ /1ị ỵ U11ị ẳ j R R dS2 m1nị jSẳ0 ẳ m1nị jSẳLS ẳ Mnị a21nị jSẳ0 ẳ a21nị jSẳLS ẳ ^ 21ị ẳ C41 cos k2 Sị þ C40 sin ðk2 SÞ j p a R (6.12) Substitution of constitutive relation (4.10) into equilibrium equations (4.12) for n ẳ and elimination of k2ị and t22ị give ^ 1ð2Þ , as ^ and j the differential equation for leading terms of j MARCH 2017, Vol 84 / 031007-7 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org ^ 12ị d3 j d^ j 12ị ỵ k32 ẳ F51 Sị dS3 dS (6.13) ^ 31ị ị ỵ1=R2 Þ^ ^ 3ð1Þ Š The solution of Eq (6.13) is j 21ị j j 21ị j where k32 ẳ 1=R2 , and F51 Sị ẳ c2 c3 ị=c1 ẵd2 =dS2 ị^ ^ 12ị Sị ẳ C52 cosk3 Sị ỵ C51 sink3 Sị ỵ C50 ỵ j S f 0 sinẵk3 f nịF51 nị dndf k3 (6.14) The elongation, kð2Þ , can be obtained by substitution of constitutive relation (4.10) and the above equation into equilibrium equations (4.12) for n ẳ as   M0ị R   p2 c2 c3 ðc2 À c3 Þða À 1ị2 S S 2S EI2 ỵ cos k2ị ẳ c1 R C52 cos ỵ C51 sin  ! a R R R M R ð Þ (6.15) À1 c3 a À c2 a EI2 ðS SÀn þc1 R3 cos F51 ðnÞdn R The differential equations for the leading terms, U2ð2Þ and U3ð2Þ , of displacements, U2 and U3 , are derived by substitution of Eq (6.14) into Eqs (B1) and (B2) as d2 U2ð2Þ U2ð2Þ ỵ ẳ F52 Sị dS2 R  2 dU32ị U22ị dU11ị ẳ k2ị dS R dS (6.16) where ẵ1 j 12ị ỵ k2ị =R dU11ị =dSị2 =2Rịỵw1ị d2 U11ị =dS2 ị F52 Sị ẳ ^ h 2 i ẵ1 w1ị ỵ dU11ị =dSị2 =ð2RÞ and the underlined terms are derived from the nonlinear terms of curvature The solutions of Eq (6.16), which satisfy the boundary conditions (6.3), are ðS ðS À nÞ S S F52 nịdn; U22ị Sị ẳ C53 cos ỵ C54 sin ỵ R sin R R R "  2 # ðS 1 dU1ð1Þ ðnÞ kð2Þ ðnÞ À U22ị nị U32ị Sị ẳ dn ỵ C55 R dn (6.17) and the parameters, C50 ; C51 ; C52 ; C53 ; C54 ; and C55 , are determined as C55 ¼ C53 ¼ C51 ¼     c2 c2 Mð0Þ R p2 À ða À 1Þ À1 EI2 c1 c3 C52 ẳ  ! M R c2 0ị a R À a À1 c3 EI2 a À > pa R2 pa R c2 Mð2Þ R p3 ða À 1ÞR > < = 1ỵ C54 ẳ ỵ C52 ỵ  !2 c1 EI2 8a Mð0Þ R c 2 > > À a À1 : ; c3 EI2 C50 ẳ C52 ỵ (6.18) c2 M2ị R Rc1 EI2 ^ 3ð3Þ and t1ð3Þ give the difSubstitution of constitutive relation (4.10) into equilibrium equations (4.12) for n ¼ and elimination of j ^ 2ð3Þ as ferential equation for j ^ 23ị d2 j ^ 23ị ẳ F6 Sị þ k22 j dS2 (6.19) where ^ 2ð1Þ À Rðc1 =c2 c3 =c2 ịẵd^ ^ 31ị ị=dSg=R F6 Sị ẳ f1 c1 =c2 ịẵM0ị Rc2 =EI2 c3 ị À 1Š^ j 1ð2Þ j j 1ð2Þ j 031007-8 / Vol 84, MARCH 2017 Transactions of the ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org The solution of Eq (6.19) is ^ 23ị ẳ C61 cosk2 Sị ỵ C60 sink2 Sị ỵ j S sinẵk2 S nފF6 ðnÞ dn k2 (6.20) ^ 2ð3Þ satisfies the boundary conditions j ^ 23ị jSẳ0 ẳ j ^ 23ị jSẳLS ¼ 0, which determine the load where C60 and C61 are the parameters j increment #     " Mð0Þ R c3 c3 c3 c2 Mð0Þ R c2  ị p a 1 1ỵ3 À4 À 4À À3 1À EI2 c3 c2 c1 c1 c1 EI2 EI2 (6.21) M2ị ẳ   !2   ! R Mð0Þ R c2 c3 c3 c2 þ c3 Mð0Þ R a 1À þ1 À 1ỵ EI2 EI2 c3 c1 c2 c1 For the narrow rectangular section of beam, which thickness is much larger than its width, i.e., t ) w, the stiffness ratios, c2 and c3 are much smaller than c1 , i.e., c1 ) c2 $ c3 The load increment in Eq (6.21) can be simplified to     Mð0Þ R Mð0Þ R c3 c2 1À p2 ða 1ị 1ỵ3 EI2 EI2 c3 c2 EI2 (6.22) Mð2Þ %    !2  R Mð0Þ R c2 Mð0Þ R c3  1À a ỵ1 EI2 EI2 c3 c2 The ratio of c3 to c2 is 2=1 ỵ ị, which only depends on the Poisson’s ratio, Eq (6.22) can be written as Mð2Þ % ðEI2 =RÞf1 ða; vÞ The ratio of bending moment load, M, to the critical load, Mð0Þ , is    4M0ị R M0ị R 7ỵ  2 p2 ð À 1Þða À 1Þ À M2ị M U1max 1ỵ EI2 EI2 (6.23) ẳ þ a2 %1þ    !2  Mð0Þ Mð0Þ LS M0ị R M0ị R 2M0ị R ỵ  a ỵ  ị ỵ2 EI2 EI2 EI2 1ỵ where the small ratio was defined as a ¼ U1max =LS The maximum principal strain in the beam can be obtained as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3       ð Þ2  j ỵ  1 j ^ ^ 231ị ^ 21ị ỵ j emax ẳ awịmax ð Þ  2  (6.24) The shortening ratio of the distance, cd , between the beam ends due to the bending moment is defined as     jUjS¼LS j a2  pa  cd ẳ pa ẳ 2R  U22ị ỵ U32ị cot S¼L  S 2R sin The ratio of the maximum principal strain in beam to emax R ce ¼ pffiffiffiffiffi ¼ f2 ða ; vÞ cd w pffiffiffiffiffi cd ðw=RÞ is (6.26) where the function, f2 ða ; vÞ, depends only on the shape and the Poisson’s ratio of beam As shown in Fig 5(a), the effect of Poisson’s ratio on the ratio, ce , can be neglected, especially for the curvature with the nonlinear terms But the effect of the normalized angle, a , on the ratio, ce , is significant as shown in Fig 5(b), where the Poisson’s ratio is 0.42 for gold, which is the primary material of the interconnector in the stretchable and flexible electronics The value gap between the ratio, ce , with and without the nonlinear terms in curvature would be larger than 100% for a > 5=6 Figure shows the normalized maximum principal strain, emax R=w, versus the shortening ratio of the ends distance, cd , for  ¼ 0:42, which indicates that the nonlinear terms in curvatures should be considered for the strain of beam There is 73% increase in the normalized maximum strain, emax R=w, without the nonlinear terms of curvature for a ¼ 2=3 and cd ¼ 0:3 from the Journal of Applied Mechanics (6.25) Fig Schematic illustration of boundary conditions of circular beam under bending moment load MARCH 2017, Vol 84 / 031007-9 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org Fig The ratio, ce , with and without the nonlinear terms in curvature: (a) the ratio, ce , versus the Poisson’s ratio m for the normalized angle  a 1=4, 1=2, and 2=3 (b) The ratio, a for gold (i.e., m 0:42) ce , versus the normalized angle  normalized maximum principal strain with the nonlinear terms, which indicates that the stretchability of circular beam is underestimated when the nonlinear terms are neglected Figure also shows that the relation between the normalized strain, emax R=w, and the shortening ratio, cd , depends on the normalized angle, a , and the longer circular beams have the higher stretchability with the same critical normalized strain This model can be used to analyze the stretchability of the serpentine bridge, which can be simplified as two semicircular beams [2] The maximum strain in the bridge fabricated with gold can be obtained as max ẳ 0:9365 p pe w=Rị epre =1 ỵ epre ị and emax ẳ 2:264w=Rị epre =1 ỵ epre ị for the curvatures with and without the nonlinear terms, where the prestrain, epre , which is applied on the soft substrate, is related to the shortening ratio, cd , by epre ẳ cd =1 cd ị For the design of the stretchability of the serpentine bridge, the nonlinear terms of the curvature should be considered in the analysis process of the post-buckling behavior of the curved beam The finite-element method (FEM) of the commercial software ABAQUS, where the shell element S4R is used since the width of the cross section, w, is much smaller than its thickness, t, is adopted to simulate the post-buckling behaviors of the curved beams under the bending moments The distributions of the twist angle / of the circular beams for the shortening ratios, cd ¼ 0.1, 0.2, and 0.3, are shown in Fig 7, where the elastic modulus and Poisson’s ratio are 79.5 GPa and 0.42 for gold, and the length, LS , thickness, t, width, w, and the normal angle, a , of the circular beam are 900 mm, 30 mm, mm, and 2/3, respectively Figure shows that the theoretical results are consistent with the FEM results 6.2 The Lateral Buckling of Circular Beams Under Uniform Pressure As shown in Fig 8, the uniform pressure, p2 ¼ Àp2 , which is along the thickness direction of cross section (i.e., Àe2 ) during the deformation, is applied on the circular beam, the lateral buckling will happen because the thickness of the beam section, t, is much larger than its width, w The power orders of displacements are similar to those in Sec 6.1 The beam with length, LS , which subtends the angle, a, is simply supported in the plane and out-of-plane, and the beam ends cannot rotate around the centroid of beam, but can freely slide toward the arch center in the plane, i.e., U3 jS¼0 ¼ U3 jS¼LS ¼ m1 jS¼0 ¼ m1 jS¼LS ¼ (6.27) ðt Á E2 ịjSẳ0 ẳ t E2 ịjSẳLS ẳ U1 jSẳ0 ẳ U1 jSẳLS ẳ e2 E1 ịjSẳ0 ¼ ðe2 Á E1 ÞjS¼LS ¼ (6.28) m2 jS¼0 ¼ m2 jS¼LS ¼ The deformation of the model is symmetrical about the midline (i.e., the dotted–dashed line in Fig 8) to avoid the rigid movement By substituting Eq (3.2) into the above boundary conditions, the expanded formulas with respect to the perturbation parameter, a, of the boundary conditions (6.27) and (6.28) are U3nị jSẳ0 ẳ U3nị jSẳLS ẳ m1nị jSẳ0 ẳ m1nị jSẳLS ẳ lẳn X tilị ai2nlị ịjSẳ0 lẳ0 lẳn X ẳ tilị ai2nlị ịjSẳLS ¼ (6.29) l¼0 U1ðnÞ jS¼0 ¼ U1ðnÞ jS¼LS ¼ a21nị jSẳ0 ẳ a21nị jSẳLS ẳ (6.30) m2nị jSẳ0 ẳ m2nị jSẳLS ẳ Fig The normalized maximum principal strain emax R=w with and without the nonlinear terms in curvature versus the shortening a 1=4, 1=2, and 2=3 ratio cd for m 0:42 and  031007-10 / Vol 84, MARCH 2017 The formulas and solving of the governing equations are similar to those in Sec 6.1 Replacing k2 , k3 , F51 ðSÞ; F52 ðSÞ; and F6 ðSÞ in Sec 6.1 with the following k2 , k3 , F51 ðSÞ; F52 ðSÞ; and F6 ðSÞ, which are Transactions of the ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org k2 ¼ R p20ị R3 1ỵ EI2 ! k3 " # p20ị R3 c2 ẳ ỵ ỵ c1 Þ R c1 EI2 (6.31) ^ 2ð1Þ d^ j 3ð1Þ d^ j 2ð1Þ d^ j 3ð1Þ d2 j 2 ^ 31ị ^ 31ị j ỵ R 3c2 2c3 ị ỵ R 2c2 c3 ị À2Rc3 j dS dS dS dS2  ! F 51 Sị ẳ   R c1  ^ p R d^ j d j ð Þ ð Þ ð Þ 2 ỵc j ^ 21ị ^ 31ị R c2 ỵ R c2 c3 ịj j ^ 21ị ỵ EI2 dS dS2 F52 Sị ẳ ^ j 12ị ỵ "  2  2 # k2ị dU11ị dU11ị  ẵ1 2 ẵ1 d U11ị w1 ị ỵ ỵw1ị 2R 2R R dS dS2 dS (6.32) !9 h i > d^ j 3ð1Þ > > > 2 > > ^ ^ ^ j 1ð2Þ ðc1 À c2 Þj 2ð1Þ þ Rðc3 À c1 Þ > j 2ð1Þ R ðc2 c3 ịj 31ị ỵ k2ị ỵ R^ > < dS =  F Sị ẳ   R c2 > > > d^ j 2ð1Þ d^ j 1ð2Þ > Á d À d > > > > ^ 31ị ỵ R2 c2 k2ị j k2ị j 31ị ỵ R2 2c1 ỵ c3 ị^ : c3 R ; dS dS dS dS The critical load, p20ị ẳ EI2 =R3 ða À2 À 1Þ, is obtained by the boundary conditions (6.30), which is consistent with the result of previous study [42] When ) c1 ) c2 $ c3 , the load increment can be simplified to p2ð2Þ ẳ i EI2 h f31 ; a ị ỵ f32 ð; a Þ R (6.33) where à 2 p2 ð1 À a Þ a ð17 À  ị1 ỵ  ị ỵ 163 ỵ 2 ị 2a 22 ỵ 33 ỵ  ị 8a a 4ị ỵ a ỵ  ị ! h i pa 2 p2 a ị ỵ 4 þ a ð1 þ  Þ 8ð1 À a ị1 ỵ  ị ỵ pa a 4ị3 ỵ  ịcot f32 ; a ị ẳ 16a a 4ị ỵ a ỵ  ị f31 ; a ị ¼ (6.34) The function f32 ð; a Þ is derived from the nonlinear terms of curvature The ratio of the uniform pressure to the critical load is h i  2 f31 ; a ị ỵ f32 ; a ị  p U p2 ị 1max 2 ẳ ỵ a2 %1ỵ a LS p20ị p2ð0Þ Fig The distributions of the twist angle of the circular beam for the different shortening ratios, cd 0.1, 0.2, and 0.3 Journal of Applied Mechanics (6.35) Fig Schematic illustration of boundary conditions of circular beam under uniform pressure MARCH 2017, Vol 84 / 031007-11 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org lateral post-buckling When the bending moment is applied on circular beam, the strain in beam is overvalued ($73%) due to the neglect of the nonlinear terms in curvature expression, which will lead to underestimate of the stretchability of circular beam The difference of post-buckling behavior also stems from the neglect of the nonlinear terms for the lateral buckling under the uniform pressure In summary, the nonlinear terms in curvature can not be neglected in the lateral buckling behavior of the curved beam Acknowledgment J.W and K.C.H acknowledge support from the National Basic Research Program of China (973 Program) Grant No 2015CB351900 and NSFC Grant (Nos 11172146, 11320101001, and 11672149) Appendix A  =p  2ð0Þ , with and withFig The ratio of load to critical load, p out the nonlinear terms in curvature versus the normalized outof-plane displacement, U1max =LS , for m 0:42 and a 1=4, 1=2, and 2=3 Figure shows the normalized load, p2 = p 2ð0Þ , versus the normalized maximum displacement, U1max =LS , which indicates that the nonlinear terms of curvature have significant impact on the analysis of the post-buckling behaviors of circular beam The uniform pressure increases in the buckling process, but it decreases since the nonlinear terms of curvature is neglected Conclusions and Discussion A systematic method is established for post-buckling analysis of curved beams in this paper The deformation variables of the curved beam are up to the third power of generalized displacements due to the necessity of the fourth power of generalized displacements in the potential energy for the post-buckling analysis [37–39] The currently prevailing post-buckling analyses, however, are accurate only to the second power of generalized displacements in the potential energy since some authors assume a linear displacement–curvature relation Although the effect of the nonlinear terms of curvature on in-plane post-buckling behavior of planar beam is not so critical, it has significant impact on the The second and third powers of generalized displacements in coefficients and curvatures are !  2 > > h2i ẵ >a ẳ w ỵU1 > 11 > > > < 1 h2i (A1) a ẳ U10 U20 ỵ U10 U3 K1 ỵ w½2Š > > 12 2 > >   > > > : ah2i ¼ U U wẵ1 U0 ỵ U U2 ỵ wẵ1 U3 K1 13 1 h2i > > a21 ẳ U10 U20 ỵ U10 U3 K1 À w½2Š > > 2 > > < !  2 À Á2 (A2) h2i ½1Š w a ẳ ỵ U U K > 22 > > > > À > : h2i a23 ẳ wẵ1 U10 ỵ U30 ỵ U2 K1 U20 U3 K h2i > > a31 ẳ U30 ỵ U2 K1 U10 > > > < h2i À ÁÀ a32 ẳ U30 ỵ U2 K1 U20 U3 K1 > h > > À Á2 i > h2i > : a33 ¼ À U ỵ U20 U3 K1   > ½1Š 0 ½1Š h3i ½1Š ½2Š 02 02 > > w w K1 ¼ U U w w ỵ U U ỵ U U ỵ U U a > 1 11 > 2 > > > >   > > ½1Š h3i 02 > > ỵ U10 U20 U30 ỵ wẵ3 w a ẳ U U > 12 > > > > !   > > À Á ½1Š > ½1Š > 0 > w K w þ U U U À U U þ U U þ À U À U U U < 2 3 3 K1    ! > > ½1Š > h3i ½1Š 0 ½2Š 0 02 02 02 > w U a ¼ U ỵ w U U w U U U ỵ U > 3 > > 13 2 > > > h i > Á À Á > ½2Š ẵ1 > 0 > > ỵ U3 w À w U3 U3 À U2 U2 À U1 U2 U3 ỵ 2U2 U3 K1 > > >   > > > > ½1Š > : ỵ w U2 U3 ỵ U1 U3 À U1 U2 K12 031007-12 / Vol 84, MARCH 2017 (A3) (A4) Transactions of the ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org !     > À Á ½1Š ½1Š ½1Š > h3i ½3Š 02 0 0 0 > > K w w w ỵ a ẳ U U U U w ỵ U U U U U U U ỵ U U U U þ U > 2 2 3 3 K1 21 > 4 >   > > > ½1Š 0 ½1Š > h3i ½1Š ½2Š 02 02 0 > K1 w w a ẳ U U w w ỵ U U ỵ U U U U 2U U U > 3 2 2 > 22 > 2À > > > > ỵ U30 U32 2U20 U2 U3 K12 ỵ U2 U32 K13 <    ! ½1Š h3i ½1Š 0 ½2Š 0 02 02 02 > w U a ẳ U w U U ỵ w U U U ỵ U > 3 23 > > > & ! 2 ' >  2  > > > 2 2 ½ Š ½ Š 1 0 0 0 > ỵ w U1 U2 U3 w ỵ U ỵ U ỵ U3 U U 2U2 U2 U3 K1 > > > >     > > 3 > 2 > > ỵ 2U2 U3 U3 ỵ U2 U3 U2 U2 K1 þ U2 U3 À U3 K1 : 2    ! À > Á 1 2 > h3i 02 02 02 0 0 > ỵ U U U K ẳ U U ỵ U U U ỵ 2U U U K ỵ U U U a > 1 3 31 > > 2 > > ! ! >       > > 1 h3i > 02 02 02 02 02 02 0 02 < a32 ẳ U2 U U ỵ U þ U3 U þ U þ U3 U U ỵ 2U2 U2 U3 K1 2     > > 3 > 2 2 > U U K K13 ỵ U U 2U U U U ỵ U U > 3 2 > > 2 3 > >   h   i > À Á > h3i > : a33 ¼ U30 U 21 ỵ U 22 ỵ U2 U0 21 ỵ U 22 2U20 U30 U3 K1 þ U30 U32 À 2U20 U2 U3 K12 þ U2 U32 K13 À Á0 À Á > w½1Š U100 þ U30 U20 þ U2 U20 À U3 U30 K10 > > * + > 8À Á9 Á À 02 > > 00 00 02 > > > h2i ỵ U U À U U À U U > < = 3 > ^1 ¼ j > ! >  2 > ỵ K1 U2 U3 ị0 K12 > > ẵ > 02 > > : ; w ỵ U À 2U U K À > 1 < 2   À Á ½1Š 00 ½1Š > 0 0 > K U U U w U ỵ U U w > 1 > >   > ^ 2h2i ¼ > > j > ½ Š ½ Š 0 00 > w U3 ỵ w ỵ U1 U2 ỵ U1 U2 K1 ỵ U1 U3 K1 > > > 2 > > > : ^ 3h2i ¼ j     > Á ½1Š 00 00 0 > h3i ½2Š 00 ½1Š À 00 0 00 00 02 02 00 0 > > ^ w U þ U ¼ w U þ U À w U U ỵ U U ỵ U U j > 3 2 þ U1 U1 U2 À 2U3 U2 U3 > 2 > > &  ! >  2  ' >  > > ½1Š ½1Š 2 02 0 0 > > þ U3 U À U À U3 w þ U À U2 w U1 þ 2U2 U3 K10 > > > > > Á Á À > À 0 Á0 À 0 > ½1Š ½2Š 00 0 2 > w U ỵ 4U K w 2U U U ỵ U U U U U U ỵ 2U U 2U U > 2 3 3 2 17 > > 1 > ỵ6 >  2 > 5K1 > 0 ½1Š ½1Š ½1Š > 00 00 02 03 00 > U U ỵU ỵ 2U U U À U w À U U w À 2U U U À 3U U À w > 3 3 2 3 > 2 > > >   À > À Á Á > > 00 2 02 02 > > U2 U3 U2 ỵ 2U2 U U ỵ 3U2 U3 U3 K1 > > 7K > ỵ4 > > ½1Š > 0 00 02 > U ỵ4U U U ỵ 2U U U ỵ U U ỵ U w 3 > 3 > > <  ỵ U30 U22 U32 ỵ 2U2 U20 U3 K13 >   >   > Á > h3i ½1Š 00 00 0 ½2Š 00 ½1Š À 00 > 00 00 02 02 00 0 > ^ w U j ¼ w U À U À w U U ỵ U U ỵ U U À U > 2 3 1 U2 U1 U2 ỵ 2U3 U1 U3 > 2 2 > > > ! > > Á 0 > ½1Š À ½2Š 0 0 > > U ỵ w U U U U w U ỵ 2U U U ỵ U U 2 3 3 K1 > > 2 > >  3   > > > 02 > ½1Š ½1Š 00 00 02 > U À U ỵ U2 U2 U3 U3 U > 2U1 U2 U1 U3 ỵ 2w U2 U3 K1 À w > > 4 > >   ỵ 7K1 > > À Á > 00 0 ½2Š ½ Š > 00 0 00 > U ỵ U ỵ U U w ỵ U U U ỵ 2U U 2U U ỵ U U U w > 2 3 3 2 > 2 > > >   !   >   > ½1Š > ½ Š ½ Š 1 0 0 0 > 00 2 > ỵ U1 U2 U3 U3 U1 U3 w U2 ỵ w U2 U3 ỵ 3U1 U2 U2 K1 ỵ w U3 U1 U2 U3 K13 > > 2 > > > > h3i : ^ ẳ0 j where ị0 ẳ dðÞ=dS, and Journal of Applied Mechanics (A5) (A6) (A7) (A8) MARCH 2017, Vol 84 / 031007-13 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org w wẵ3 ẵ2 ẳ ð ! Á Á0 À 00 1À 00 0 0 U U À U1 U2 U1 U3 K1 ỵ U1 U3 K1 ỵ U1 U2 K1 dS 2 (A9) & '   h  i0 À Á > > ½1Š ½1Š ½1Š > > 02 02 00 0 00 > > w ỵ w U ỵ U ỵ U U2 À U1 U U À w > > > > > >   > > > > > > ½1Š > > 0 > > w K ỵ U U ỵ U U U > > 3 > > > > > ð> < =  0 Á À Á ½1Š 0À 02 02 00 0 00 ẳ dS w U U U ỵ U ỵ U À U U U À U U U U À U 3 2 27 1 1 >   > > 62 7K1 > > > ỵ > > > > > > > > ỵU3 U10 U2 À w½1Š U3 K10 > > > > > >   !   > > > > > > À Á0 1 > > ½ Š ½ Š 1 0 2 > > w w K U K U U U ỵ U U ỵ U À U U À : ; 3 1 3 (A10) Appendix B ^ ið2Þ , and aijð2Þ are The expressions of k2ị , j k2ị ẳ U 32ị ỵ U22ị K1 ỵ i h 02 U 11ị ỵ U 21ị U31ị K1 (B1) 8 À Á0 À Á 09 À Á0 ½1Š 00 > 00 0 > > > > U ỵ U K ỵ w U þ U U þ U U À U U > ð Þ ð Þ ð Þ ð Þ ð Þ ð Þ ð Þ ð Þ 22 31 21 21 K1 > 32 21 31 > > ð1Þ 1ð1Þ > > > > > > >   > > > À Á < = > 2 00 00 0 > > > > > > ỵ U U U U U U ð Þ ð Þ ð Þ ð Þ ð Þ ð Þ < = 3 > ^ ¼ j > ð Þ À Á > > > ! > > > ỵ K1 U21ị U31ị K12 > > > >  ẵ1 2 > > > > > 02 > > > > > > > w1ị ỵ U 11ị À 2U2ð1Þ U3ð1Þ K1 < : ; : ; 2     > > ½ ẵ ẵ > U1002ị w12ị K1 þ wð11Þ U 00 2ð1Þ À U 3ð1Þ U10 1ị ỵ w11ị U31ị U10 1ị U21ị K10 > > > > >   ^ 22ị ẳ > > j > ½ Š ½ Š 0 00 > > ỵ w1ị U 31ị w2ị U11ị U21ị U11ị U 21ị K1 ỵ U1ð1Þ U3ð1Þ K1 > > 2 > > > : ^ 32ị ẳ /0 2ị j ! >  ẵ1 2 > 02 > w ẳ þ U a > 11ð2Þ 1ð1Þ ð1Þ > > > > < 1 ẵ ẵ a122ị ẳ w12ị U10 1ị U 21ị ỵ U10 1ị U31ị K1 ỵ w22ị > > 2 > > >   > > > : a132ị ẳ U 12ị ỵ U 31ị U ị wẵ1ị U 21ị ỵ U ị U21ị ỵ wẵ1ị U31ị K1 11 11 1 (B2) (B3) 1 ½ Š ½ Š > > a212ị ẳ w12ị U10 1ị U 21ị þ U10 ð1Þ U3ð1Þ K1 À wð22Þ > > > 2 > ! < Á2  ½1Š 2 w a ẳ ỵ U U K 22ð2Þ 3ð1Þ 2ð1Þ > ð1Þ > > > > À ÁÀ Á > ½ Š 0 :a 232ị ẳ U 22ị ỵ U32ị K1 ỵ w1ị U11ị ỵ U 31ị ỵ U21ị K1 U 21ị U31ị K1 (B4) ẳ U0 12ị U0 31ị ỵ U21ị K1 U10 1ị a > > > 31ð2Þ > > À ÁÀ Á < a322ị ẳ U0 22ị U32ị K1 U 31ị ỵ U21ị K1 U 21ị U31ị K1 > > > À Á i > 1h > : a332ị ẳ U 211ị ỵ U 21ị U31ị K1 2 (B5) é ẵ1 where w2ị ẳ /2ị K1 U 12ị dS !   0 1 ẵ w22ị ẳ U10 ð1Þ U 00 2ð1Þ À U100ð1Þ U0 2ð1Þ À U10 1ị U31ị K10 ỵ U10 1ị U31ị K1 ỵ U10 ð1Þ U2ð1Þ K12 dS 2 031007-14 / Vol 84, MARCH 2017 Transactions of the ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org References [1] Kim, D H., Lu, N S., Huang, Y G., and Rogers, J A., 2012, “Materials for Stretchable Electronics in Bioinspired and Biointegrated Devices,” MRS Bull., 37(3), pp 226–235 [2] Kim, D H., Song, J Z., Choi, W M., Kim, H S., Kim, R H., Liu, Z J., Huang, Y Y., Hwang, K.-C., Zhang, Y W., and Rogers, J A., 2008, “Materials and Noncoplanar Mesh Designs for Integrated Circuits With Linear Elastic Responses to Extreme Mechanical Deformations,” Proc Natl Acad Sci U.S.A., 105(48), pp 18675–18680 [3] Lacour, S P., Wagner, S., Narayan, R J., Li, T., and Suo, Z G., 2006, “Stiff Subcircuit Islands of Diamondlike Carbon for Stretchable Electronics,” J Appl Phys., 100(1), p 014913 [4] Lee, J., Wu, J., Ryu, J H., Liu, Z., Meitl, M., Zhang, Y W., Huang, Y., and Rogers, J A., 2012, “Stretchable 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Lateral–Torsional Instabilities of Circular Arches Under Follower Loads,” J Sound Vib., 320(3), pp 617–631 MARCH 2017, Vol 84 / 031007-15 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jamcav/935918/ on 01/25/2017 Terms of Use: http://www.asme.org ... is used to obtain the analytical solution of the post- buckling behaviors of curved beams Deformation of Curved Beam ~ dP dS (2.1) The curvature of the curved beam was defined by Love [40] as dEi... the analysis process of the post- buckling behavior of the curved beam The finite-element method (FEM) of the commercial software ABAQUS, where the shell element S4R is used since the width of. .. terms of curvature is neglected Conclusions and Discussion A systematic method is established for post- buckling analysis of curved beams in this paper The deformation variables of the curved