University of Thessaly Volos, Greece

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1 th HSTAM International Conress on Mechanics Chania, Crete, Greece, 5 7 May, PRETWISTED BEAMS IN AXIA TENSION AND TORSION: AN ANAOGY WITH DIPOAR GRADIENT EASTICITY AND APPICATIONS TO TEXTIE MATERIAS Kordolemis A.M., Aravas N., Giannakopoulos A.E. Depatrment of Civil Enineerin University of Thessaly 84 Volos, Greece alkordol@uth.r, aiannak@uth.r Depatrment of Mechanical Enineerin University of Thessaly 84 Volos, Greece aravas@mie.uth,r Keywords: coupled tension torsion, dipolar radient elasticity, textile yarns Abstract. It is well known from St. Venant s torsion theory that when a torque is applied at the ends of a prismatic beam then the cross sections will, firstly, rotate about the centroid axis of the beam and, secondly, each cross section will warp in the lonitudinal direction. Rotation is depicted throuh the anle of twist per unit lenth, while the warpin is depicted throuh an appropriate warpin function of the unrotated cross sections. In the present study we considered a prismatic beam with constant initial twist alon its lenth and at the beam ends axial forces and torsional moments were applied. The overnin equations of equilibrium and the boundary conditions are obtained usin an enery variational statement. Focusin on the axial deformation, the results of the present study exhibit similarities with the results obtained from the analysis of prismatic beams subjected to axial tension usin a dipolar radient elasticity theory. The advantaeous aspect of the present study is that the microstructural lenths emere in a natural way from the eometrical characteristics of the beam cross section and the elastic material properties. The present results are extremely useful in modelin textile yarns with initial pretwist, as well in smart textiles where initial twist can be introduced deliberately. INTRODUCTION Textiles are used in numerous advanced technoloical applications such as airbas, seat belts and body armor vests. The widespread usae of textile composites is mainly twofold. Primarily, their favorable mechanical properties classify them as very sufficient load carryin components. Secondarily, their low cost production and their easy handlin make them very competitive structural materials. The mechanics of textile composite materials can be addressed at three different scales: a) the macroscopic scale where textile is treated as an anisotropic, non-linear continuum medium, b) the mesoscopic scale where the overall mechanical behavior of the composite is characteried by the interactions between the yarns, c) the microscopic scale where interactions between the fibers inside yarns are taken into account. The present study focuses on the microscopic scale in the sense that the micromechanical parameters of the yarns are considered. It is well known that most yarns are formed by the assemblae of a lare number of fibers, usually some hundreds, which are pretwisted toether about the lonitudinal axis, Fi.. In addition, the textile composite is subjected to tensile forces which tend to stretch the fiber in the lonitudinal direction. So, the fiber s mechanical behavior is equivalent to a prismatic bar with initial twist subjected to an axial force at the same time. Biot [] was the first who mentioned that the torsional riidity of a prismatic bar is increased when the bar is subjected to a tensile load. Chen [] ascertains that, when a prismatic or a cylindrical thin walled bar possess an initial twist, the torsional riidity of the bar is reater than the bar without initial twist. He concluded that the increase is due to the manitude of the initial twist, the shape and the thickness of the thin wall cross section, as well as the material of the bar. The main oal of Chen s study was to estimate the torsional riidity of steel thin wall prismatic bars and his findins were in ood areement with experimental results. The only handicap of his approach was that for a cyclic cross section he predicted an increase in torsional riidity which is not true. The study of pretwisted prismatic bars intensified in the 7s, when the problems in helicopter blades came in the foreround (see for example [, 4, 5] ). Rosen [6] conducted a thorouh report in which he verified the increase of torsional riidity and he attributed it to the interference of the initial twist with the axial loadin throuh the warpin function of the cross section of the bar. This conclusion comes to recover the ap in Chen s theory, because it is in accordance with the well-known result that a cyclic cross section does not warp and enerally it is not influenced by the twist of the beam.

2 Kordolemis A.M., Aravas N., Giannakopoulos A.E. KINEMATICS AND INEAR STRAIN ANAYSIS Consider a uniform bar of any cross section twisted by couples at the ends, Fi.. An orthoonal coordinate system Oxy is adjusted at the center of an end cross section in such a way that the axis of the fiber coincides with the axis. The displacement field of the cross section is defined as u y v x w w,,,, where are the displacement components in respectively, and is an additional displacement due to the application of the tension load. For a pretwisted bar with initial twist and lenth, the Cartesian coordinates ( ) are local to the cross section and are related to the lobal coordinates as sin cos xcos a y a xsin a y a x y since small deformation theory requires. From the assumed displacement field it is easy to evaluate the linear normal and shear strains in the cross sections: () () u v w w xx, yy, x y u v u w v w xy, x y, y x y x x x y y () The function can be found from solvin the classic Saint Venant problem in the ( ) system [6] : with boundary conditions (4) n n n (5) and THE EASTIC STRAIN ENERGY OF A PRETWISTED FIBER The elastic strain enery stored in the fiber will be dxdy (6) E U dxdy d G x y dxdy d (7) Substitutin the strain components in the above equation we take U A A A A EA w ak J a R a S w GJ w dxdy d EA A (8) where in the last equation the followin notations were introduced A dxdy (9)

3 Kordolemis A.M., Aravas N., Giannakopoulos A.E. ak dxdy a R dxdy dxdy as dxdy () () () J dxdy J () J y x dxdy x y where is the initial twist in the fiber, are cross sectional constants, is the areal torsional constant of the cross section, is an internal lenth related to the eometry of the fiber cross section and is the torsional constant of the cross section. Note that Also, for cross sections with at least one axis of symmetry (4) A, K, J, S, J J (5) p, (6) 4 CHARACTERISTIC EQUATIONS AND CORRESPONDING BOUNDARY CONDITIONS OF THE PROBEM The variation of the strain enery reads 4 w w U EA aes w d aek GJ EJ a 4 ES d w w EA aes w aek EJ aes GJ EJ aer (7) We suppose that there exists a distributed axial force as mass force and a distributed torsional moment as mass torsion such as N p, T m (8) where is the total axial force and is the total torque at the cross sections of the fiber. The work done by the external loads is W p w m d N w T B (9) where is the enery-conjuated quantity of ( ). The virtual work principal reads w UW EA aes p wd 4 w w a EK GJ EJ aes m 4 d EA aes N w w aek EJ aes GJ T EJ aer B ()

4 The last equation holds for all values of Kordolemis A.M., Aravas N., Giannakopoulos A.E. so from the interals one ets the equilibrium equations w EA aes p 4 w aek GJ EJ aes m 4 () () and the enery-conjuated boundary conditions at the ends of the fiber ( ) w EA aes N w w aek EJ aes GJ T EJ aer B () (4) (5) 5 CONSOIDATION OF THE EQUATIONS AND THE D AXIA MODE Differentiatin equation ( ) twice with respect to we et 4 4 p A w 4 4 Ea S a S (6) Substitutin equations ( ) and ( ) to equation ( ) yields 4 w w p aek GJ as E A p 4 m AE JGc Jc G (7) where in the last equation the followin notation is used J eff E S c a K, c J GJ A E G c (8) (9) where is the effective torsional constant of the fiber with initial twist. The restriction stems from the fact that the condition is always true. Equation ( ) shows that the torsional riidity of the fiber is increased by the presence of an initial twist as has been also experimentally verified [][6]. Also, equation ( ) shows that the internal lenth is defined by a natural way throuh which are material properties, and which are eometrical constants of the cross section of the fiber. Equation ( ) consist the overnin equation of the problem reardin. Combinin boundary condition equations ( ) and ( ), in view of ( ) stems where w AE w ase p N E K T a c GJ c G J () () Combinin boundary condition equations ( ) and ( ) in view of ( ) one ets where w AE h w ()

5 and Kordolemis A.M., Aravas N., Giannakopoulos A.E. are h c J G as ar as p N B c GA c GJA c GJA () (4) The quantities can be thouht as eneralied force like quantities which can be illustrated by a concentrated/distributed torque, a concentrated/distributed axial force, a couple stress like force or a combination of all of them. The new problem is defined by equation ( ) and the dynamic boundary condition ( ) with the conjuate kinematical condition, and the dynamic condition ( ) with the conjuate kinematical condition ( ). 6 -D STATIC DIPOAR GRADIENT EASTIC MODE Tsepoura et.al. [7] studied the problem of the response of a bar subjected to uniaxial loadin usin the linear dipolar radient elasticity theory. In this context, the strain enery of the one-dimensional bar is defined as U A d (5) where is the cross sectional area of the bar, is the axial strain, is the strain radient and are the Cauchy stress and couple stress, respectively. The constitutive equations are assumed to be Eu Eu (6) Eu Eu (7) where is the axial displacement and primes denote differentiation with respect to lonitudinal axis of the bar. The positive definiteness of the strain enery implies that The variation of the strain enery of the bar is expressed as (8) IV U AE u u ud AE u u u AE u u u (9) The variation of the work done by external forces is iven by W q ud u u (4) where are body forces, are traction forces in the classical sense, and are couple forces. The equilibrium of the dipolar radient elastic bar implies that ( ) for every virtual kinematical quantity. From this condition stems the overnin equation of the bar which reads IV u u q AE and the correspondin boundary conditions at the ends of bar ( ) (4) AE u u u AE u u u (4) AE u u u AE u u u (4) Comparison of equations( ) and ( ),( ) and ( ),( ) and ( ), results to transparent similarities indicatin the straihtforward analoy between the dipolar radient elastic theory with the present study. The internal material lenth of the present study corresponds to the internal lenth in [7].

6 Kordolemis A.M., Aravas N., Giannakopoulos A.E. 7 PROBEM SOUTION Assumin that (to simplify the problem) the overnin equation ( ) becomes 4 w w 4 (44) the latter equation has a solution of the form 4 w c e c e c c (45) where are interation constants to be determined from the boundary conditions. Assumin that the fiber at one end is built in, then is reasonable to state the displacement at that point to be ero, i.e. w c c c 4 (46) The applied axial force on the other end of the fiber causes a specific strain, locally at that end and not alon the whole lenth of the fiber, which implies dw c c e e c ce ce c d (47) Supposin that on the free end of the fiber( ), the condition ( ) holds, then it would be w w c c c c AE AE e e c e e c AE AE (48) Also, assumin that in the built in end of the fiber ( ), then w w w w h AE h h c c AE (49) where is an auxiliary parameter. Equations ( ) ( ) form a system of four equations with four unknowns, the solution of which ives / / / e e e AE AE AE AE c, c / / e e c AE, c 4 cosh AE AE cosh sinh (5) In the case where ( ) the displacement field becomes / / w d e S e S w e e / / e e (5) where AE N S (5)

7 For the strain distribution alon the fiber would be Kordolemis A.M., Aravas N., Giannakopoulos A.E. e e dw AE AE e e d AE e e (5) where is iven by equation( ). Plots of the displacement and strain field versus the ratio, for, and various values of are iven in Fi. ( ) ( ) and Fi. ( ) ( ). 8 FIGURES Fiure. The anle of pretwist of the fiber is the anle between the fiber and the yarn axis. Fiure. The rotation anle per unit lenth by couples applied at the ends. 9 ACKNOWEDGEMENTS This research has been co-financed by the European Union (European Social Fund ESF) and Greek national funds throuh the Operational Proram "Education and ifelon earnin" of the National Strateic Reference Framework (NSRF) - Research Fundin Proram: Heracleitus II. Investin in knowlede society throuh the European Social Fund.

8 Kordolemis A.M., Aravas N., Giannakopoulos A.E. ( ) ( ) Fiure. Displacement field variation alon the fiber axis for various values of, ( ). ( ) ( ) Fiure 4. Strain field variation alon the fiber versus for various of and. REFERENCES [] Biot M.A., 99, Increase of torsional stiffness of a prismatic bar due to axial tension, Journal of Applied Physics, vol., pp [] Chen C., 95, The effect of initial twist on the torsional riidity of thin prismatic bars and tubular members, Proceedins of the First U.S. National Conress of Applied Mechanics, June -6, pp [] Rosen A., 98, The effect of initial twist on the torsional riidity of beams Another point of view, Journal of Applied Mechanics, vol. 47, pp [4] Hodes D.H., 98, Torsion of pretwisted beams due to axial loadin, Journal of Applied Mechanics, vol. 47, pp [5] Krenk S. and Gunneskov O., 98, Statics of thin walled pretwisted beams, International Journal for Numerical Methods in Enineerin, vol. 7, pp [6] Rosen A., 98, Theoretical and experimental investiation of the nonlinear torsion and extension of initially twisted bars, Journal of Applied Mechanics, vol. 5, pp [7] Tsepoura K.G., Paparyri Beskou S., Polyos D., Beskos D.E.,, Static and dynamic analysis of a radient elastic bar in tension, Archive of Applied Mechanics, vol. 7, pp [8] Georiadis H.G., Vardoulakis I., ykotrafitis G.,, Torsional surface waves in a radient elastic half space, Wave Motion, vol., pp. 48.