《纺织复合材料》课程参考文献（Textile Composites and Inflatable Structures）07 Wrinkles in Square Membranes

Wrinkles in Square Membranes Y.W.Wong!and S.Pellegrino2 1 SKM Consultants (M)Sdn Bhd,Suite E-15-01,Plaza Mont'Kiara,No.2,Jalan Kiara,Mont'Kiara,50480 Kuala Lumpur,MALAYSIA WYWong@skmconsulting.com.my 2 Department of Engineering,University of Cambridge,Trumpington Street, Cambridge,CB2 1PZ,U.K.pellegrino@eng.cam.ac.uk This paper investigates the wrinkling of square membranes of isotropic mate- rial,subject to coplanar pairs of equal and opposite corner forces.These mem- branes are initially stress free and perfectly flat.Two wrinkling regimes are observed experimentally and are also reproduced by means of finite-element simulations.A general methodology for making preliminary analytical esti- mates of wrinkle patterns and average wrinkle amplitudes and wavelengths, while also gaining physical insight into the wrinkling of membranes,is pre- sented. 1 Introduction Thin,prestressed membranes will be required for the next generation of space- craft,to provide deployable mirror surfaces,solar collectors,sunshields,solar sails,etc.Some applications require membranes that are perfectly smooth in their operational configuration,but many other applications can tolerate membranes that are wrinkled;in such cases the deviation from the nominal shape has to be known.The design of membrane structures with biaxial pre- tension,which would have a smooth surface,significantly increases the overall complexity of the structure and hence,for those applications in which small wrinkles are acceptable,engineers need to be able to estimate the extent, wavelength and amplitude of the wrinkles. The wrinkling of membranes has attracted much interest in the past,start- ing from the development of the tension field theory [1.Simpler formulations and extensions of this theory were later proposed [2-7].All of these formula- tions,with accompanying numerical solutions [8,9,model the membrane as a no-compression,two-dimensional continuum with negligible bending stiffness. Many studies of membrane wrinkling have been carried out during the past three years,and have been presented at the 42nd,43rd,and 44th AIAA SDM Conferences [10-12]. 109 E.Onate and B.Kroplin (eds.).Textile Composites and Inflatable Structures,109-122. 2005 Springer.Printed in the Netherlands

Wrinkles in Square Membranes Y.W. Wong1 and S. Pellegrino2 1 SKM Consultants (M) Sdn Bhd, Suite E-15-01, Plaza Mont’ Kiara, No. 2, Jalan Kiara, Mont’ Kiara, 50480 Kuala Lumpur, MALAYSIA WYWong@skmconsulting.com.my 2 Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, U.K. pellegrino@eng.cam.ac.uk This paper investigates the wrinkling of square membranes of isotropic mate￾rial, subject to coplanar pairs of equal and opposite corner forces. These mem￾branes are initially stress free and perfectly flat. Two wrinkling regimes are observed experimentally and are also reproduced by means of finite-element simulations. A general methodology for making preliminary analytical esti￾mates of wrinkle patterns and average wrinkle amplitudes and wavelengths, while also gaining physical insight into the wrinkling of membranes, is pre￾sented. 1 Introduction Thin, prestressed membranes will be required for the next generation of space￾craft, to provide deployable mirror surfaces, solar collectors, sunshields, solar sails, etc. Some applications require membranes that are perfectly smooth in their operational configuration, but many other applications can tolerate membranes that are wrinkled; in such cases the deviation from the nominal shape has to be known. The design of membrane structures with biaxial pre￾tension, which would have a smooth surface, significantly increases the overall complexity of the structure and hence, for those applications in which small wrinkles are acceptable, engineers need to be able to estimate the extent, wavelength and amplitude of the wrinkles. The wrinkling of membranes has attracted much interest in the past, start￾ing from the development of the tension field theory [1]. Simpler formulations and extensions of this theory were later proposed [2-7]. All of these formula￾tions, with accompanying numerical solutions [8,9], model the membrane as a no-compression, two-dimensional continuum with negligible bending stiffness. Many studies of membrane wrinkling have been carried out during the past three years, and have been presented at the 42nd, 43rd, and 44th AIAA SDM Conferences [10-12]. 109 E. Oñate and B. Kröplin (eds.), Textile Composites and Inflatable Structures, 109–122. © 2005 Springer. Printed in the Netherlands

110 Y.W.Wong and S.Pellegrino This paper considers a uniform elastic square membrane(which is a simple model of a square solar sail)of side length L+2r and thickness t that is prestressed by two pairs of equal and opposite concentrated forces,T1 and T2, uniformly distributed over a small length d at the corners,as shown in Fig.1. This membrane is isotropic with Young's Modulus E and Poisson's ratio v; it is also initially stress free and perfectly flat (before the application of the corner forces). T2.2 T1,δ1 Fig.1.Membrane subjected to corner forces. We use this problem to present a general and yet simple analytical method for making preliminary estimates of wrinkle patterns and average wrinkle am- plitudes and wavelengths in membrane structures.We also present a finite element simulation method for making more accurate estimates.The results from both our analytical approach and finite element simulations are com- pared with experimental measurements. The layout of the paper is as follows.Section 2 describes two regimes of wrinkling that were observed experimentally.Section 3 presents our method- ology for tackling wrinkling problems analytically,and hence derives solutions for the square membrane problem.Section 4 presents a finite-element simula- tion technique,whose results are compared with measurements and analytical results in Section 5.Section 6 concludes the paper. 2 Experimental Observations Fig.2 shows photographs of the wrinkle patterns in a Kapton membrane with L =500 mm,t =0.025 mm,and d =25 mm.For symmetric loading (T1 T2) the wrinkle pattern is fairly symmetric,as shown in Fig.2(a),with wrinkles

110 Y.W. Wong and S. Pellegrino This paper considers a uniform elastic square membrane (which is a simple model of a square solar sail) of side length L + 2r1 and thickness t that is prestressed by two pairs of equal and opposite concentrated forces, T1 and T2, uniformly distributed over a small length d at the corners, as shown in Fig. 1. This membrane is isotropic with Young’s Modulus E and Poisson’s ratio ν; it is also initially stress free and perfectly flat (before the application of the corner forces). Fig. 1. Membrane subjected to corner forces. We use this problem to present a general and yet simple analytical method for making preliminary estimates of wrinkle patterns and average wrinkle am￾plitudes and wavelengths in membrane structures. We also present a finite element simulation method for making more accurate estimates. The results from both our analytical approach and finite element simulations are com￾pared with experimental measurements. The layout of the paper is as follows. Section 2 describes two regimes of wrinkling that were observed experimentally. Section 3 presents our method￾ology for tackling wrinkling problems analytically, and hence derives solutions for the square membrane problem. Section 4 presents a finite-element simula￾tion technique, whose results are compared with measurements and analytical results in Section 5. Section 6 concludes the paper. 2 Experimental Observations Fig. 2 shows photographs of the wrinkle patterns in a Kapton membrane with L = 500 mm, t = 0.025 mm, and d = 25 mm. For symmetric loading (T1 = T2) the wrinkle pattern is fairly symmetric, as shown in Fig. 2(a), with wrinkles T1, δ1 L d T1, δ T2 1 , δ2 T2, δ2 L r1 r1 r1 r1

Wrinkles in Square Membranes 111 radiating from each corner;the central region is free of wrinkles.For a load ratio of T1/T2=2 the wrinkles grow in amplitude but remain concentrated at the corners.Then,for T1/T2=3 a large diagonal wrinkle becomes visible, whose amplitude grows further for T1/T2=4. (a) (b) (c) (d) Fig.2.Wrinkled shapes for Ti equal to (a)5 N,(b)10 N,(c)15 N,and (d)20 N; T2 =5 N in all cases. 3 Analytical Approach Our analytical approach is in four parts,as follows. First,we identify a two-dimensional stress field that involves no compres- sion anywhere in the membrane;the regions where the minor principal stress is zero are then assumed to be wrinkled and the wrinkles are assumed to be along the major principal stress directions.Ideally,both equilibrium and compatibility should be satisfied everywhere by the selected stress field,but

Wrinkles in Square Membranes 111 radiating from each corner; the central region is free of wrinkles. For a load ratio of T1/T2 = 2 the wrinkles grow in amplitude but remain concentrated at the corners. Then, for T1/T2 = 3 a large diagonal wrinkle becomes visible, whose amplitude grows further for T1/T2 = 4. (a) (b) (c) (d) Fig. 2. Wrinkled shapes for T1 equal to (a) 5 N, (b) 10 N, (c) 15 N, and (d) 20 N; T2 = 5 N in all cases. 3 Analytical Approach Our analytical approach is in four parts, as follows. First, we identify a two-dimensional stress field that involves no compres￾sion anywhere in the membrane; the regions where the minor principal stress is zero are then assumed to be wrinkled and the wrinkles are assumed to be along the major principal stress directions. Ideally, both equilibrium and compatibility should be satisfied everywhere by the selected stress field, but

112 Y.W.Wong and S.Pellegrino analytical solutions in closed-form -obtained by tension field theory-ex- ist only for simple boundary conditions.We have recently shown [13 that a carefully chosen,simple stress field that satisfies only equilibrium can provide quick solutions that are useful for preliminary design.More accurate stress fields can be obtained from a two-dimensional stress analysis with membrane finite elements,as briefly discussed in Section 4. Second,we note that the bending stiffness of the membrane is finite,al- though small,and hence a compressive stress will exist in the direction per- pendicular to the wrinkles.Because of its small magnitude,this stress was neglected in our previous analysis of the stress field.We assume that this compressive stress varies only with the wavelength of the wrinkles and set it equal to the critical buckling stress of a thin plate in uniaxial compres- sion.Thus,the stress across the wrinkles is a known function of the wrinkle wavelength. Third,we enforce equilibrium in the out-of-plane direction.Since the stress distribution is known,except for the wrinkle wavelength,a single equation of equilibrium will determine the wrinkle wavelength. Fourth,the wrinkle amplitudes are estimated by considering the total strain in the membrane as the sum of two components,a material strain and a wrinkling strain. 3.1 Stress Field Fig.3 shows three equilibrium stress fields that can be used to analyse membranes under (a)a symmetric loading,(b)an asymmetric loading with Ti/T2≤1/（√2-1），and(c)an asymmetric loading with T/T2≥1/(v2-1): In each case the membrane is divided into regions which are subject to simple stress states. The stress field in Fig.3(a)is purely radial in the corner regions,with T 0r= (1) V2rt where r<r+L/2 is the radial distance measured from the apex.Hence,or is uniform on any circular arc and all other stress components are zero.The central region,defined by circular arcs of radius R=r1+L/2,is subject to uniform biaxial stress of magnitude T/v2Rt. Note that near the point of application of each corner load a small,biaxially stressed region bounded by the radius ri=d/v2 has been defined.In these regions both normal stress components are T/dt. For moderately asymmetric loading,see Fig.3(b),we consider corner stress fields similar to those given by Eq.(1),hence T 0r= (2) V2rt

112 Y.W. Wong and S. Pellegrino analytical solutions in closed-form —obtained by tension field theory— ex￾ist only for simple boundary conditions. We have recently shown [13] that a carefully chosen, simple stress field that satisfies only equilibrium can provide quick solutions that are useful for preliminary design. More accurate stress fields can be obtained from a two-dimensional stress analysis with membrane finite elements, as briefly discussed in Section 4. Second, we note that the bending stiffness of the membrane is finite, al￾though small, and hence a compressive stress will exist in the direction per￾pendicular to the wrinkles. Because of its small magnitude, this stress was neglected in our previous analysis of the stress field. We assume that this compressive stress varies only with the wavelength of the wrinkles and set it equal to the critical buckling stress of a thin plate in uniaxial compres￾sion. Thus, the stress across the wrinkles is a known function of the wrinkle wavelength. Third, we enforce equilibrium in the out-of-plane direction. Since the stress distribution is known, except for the wrinkle wavelength, a single equation of equilibrium will determine the wrinkle wavelength. Fourth, the wrinkle amplitudes are estimated by considering the total strain in the membrane as the sum of two components, a material strain and a wrinkling strain. 3.1 Stress Field Fig. 3 shows three equilibrium stress fields that can be used to analyse membranes under (a) a symmetric loading, (b) an asymmetric loading with T1/T2 ≤ 1/( √2−1), and (c) an asymmetric loading with T1/T2 ≥ 1/( √2−1). In each case the membrane is divided into regions which are subject to simple stress states. The stress field in Fig. 3(a) is purely radial in the corner regions, with σr = T √2rt (1) where r < r1 + L/2 is the radial distance measured from the apex. Hence, σr is uniform on any circular arc and all other stress components are zero. The central region, defined by circular arcs of radius R = r1 + L/2, is subject to uniform biaxial stress of magnitude T /√2Rt. Note that near the point of application of each corner load a small, biaxially stressed region bounded by the radius r1 = d/√ 2 has been defined. In these regions both normal stress components are T/dt. For moderately asymmetric loading, see Fig. 3(b), we consider corner stress fields similar to those given by Eq. (1), hence σr = Ti √2rt (2)

Wrinkles in Square Membranes 113 T2=丁 A R T2-T1 L T=T2 L (a) (b) T 0 201 R 0 L R2 2 0 L (c) Fig.3.Stress fields. but vary the outer radii of these stress fields,in such a way that the radial stress is still uniform on the four arcs bounding the central region.Hence, we need to choose Ri and R2 such that R1/R2 =T1/T2 and R1+R2= L+2r1.This approach is valid until the two larger arcs reach the centre of the membrane,which happens for R1T11 R=五=V2-1 (3) For larger values of T1/T2 we consider the stress field shown in Fig.3(c); note that the diagonal region between the two most heavily loaded corners of the membrane is subject to zero transverse stress,and hence a single di- agonal wrinkle can form.Also note that the edges of the membrane are now unstressed.The stress in each corner region is now given by T 0r= 2rtsini (4)

Wrinkles in Square Membranes 113 Fig. 3. Stress fields. but vary the outer radii of these stress fields, in such a way that the radial stress is still uniform on the four arcs bounding the central region. Hence, we need to choose R1 and R2 such that R1/R2 = T1/T2 and R1 + R2 = L + 2r1. This approach is valid until the two larger arcs reach the centre of the membrane, which happens for R1 R2 = T1 T2 = 1 √2 − 1 (3) For larger values of T1/T2 we consider the stress field shown in Fig. 3(c); note that the diagonal region between the two most heavily loaded corners of the membrane is subject to zero transverse stress, and hence a single di￾agonal wrinkle can form. Also note that the edges of the membrane are now unstressed. The stress in each corner region is now given by σr = Ti 2rtsin θi (4) T2=T1 (a) (b) L L (c) R 0 0 0 0 L T2 T2 T1 T1 T2 T2 T1 T1 T2=T1 T1=T2 R1 R2 R2 R1 T1=T2 2θ1 2θ2 r1 L r1