On the Ballooning Motion of Hyperelastic Strings

Transcription

1 On the Ballooning Motion of Hyperelastic Strings S. Sarangi 1, R. Bhattacharyya, A. K. Samantaray Indian Institute of Technology, 7130 Kharagpur, India. Abstract The e ect of material nonlinearity on the ballooning motion of a hyperelastic string is investigated. The material nonlinearity is characterized by neo-hookean material model and the perfectly exible hyperelastic string is xed at one boundary, whereas the other boundary is rotating at constant angular speed. The governing equations of motion are derived by neglecting air drag. Numerical solution of nonlinear steady state equation reveals the dependence of xed eyelet tension on the length of the hyperelastic string. Linear stability of the steady sate motion is obtained by Galerkin s method for a small perturbation about the steady state con guration. This study shows that the neo-hookean material may generalize the linearly elastic string and inextensible string for a speci c range of values of speed and material constant. Keywor: Neo-Hookean model, Ballooning motion, Hyperelastic string, Material nonlinearity, Continuum mechanics. 1 Introduction A rope, rubber band, wire and polymeric ber are common examples of thin-structured, essentially one dimensional bodies which are described by a straight or curved spatial line. These one dimensional material objects, which have negligible bending sti ness, constitute a class of material known as lineal body. Often the manufacturing process of lineal bodies like polymeric bers, yarn and natural bers require material rotation. The rotation of the string induces a variety of complex motions and the imaginary surface generated by the lineal body, i.e., string, is known as a balloon. The term balloon is quite common in ring spinning, looming and lament winding processes. Present study is focused on the e ect of the material nonlinearity, which is characterized by neo-hookean material model, on the ballooning motion. Ballooning motion of a highly exible Euler s elastica in a textile application problem was studied by Zhu et al. [1]. Their inextensible model was extended to the linear elastic model by F. Zhu et al. [] for a planar balloon, in absence of air drag. Later, the various nonlinear dynamical phenomena of the complex motions considering air drag e ect was studied by Clark et al. [3] and Zhu et al. [4]. In recent past and at present, the use of synthetic polymers and polyesters are rapidly growing and several research papers are focused in the issue of its material behavior and its spinning applications. These types of material are quite di erent from the conventional material class and show relatively large elastic deformation compared to the former. However, a large number of research papers consider the ballooning motion of inextensible and isotropic and linear elastic material model; see for example [1,, 3, 4, 5, 6, 7, 8] and numerous other works referenced therein. In this work we consider the ballooning motion of a perfectly exible, homogeneous, incompressible, isotropic hyperelastic neo-hookean rubber string. The geometrical boundary conditions are the same as those of [] and we compare the numerical results with those for the linear elastic string given in []. The present work investigates the material nonlinearity on the ballooning motion of a hyperelastic string, which is a one dimensional continuum. The formulation speci es the liner extensibility model as a special case. To illustrate the material nonlinearity, we have assumed the material as homogeneous, incompressible, perfectly exible neo-hookean string. The dynamical problem of ballooning motion for a neo-hookean string is formulated 1 somsara@gmail.com

2 in Section and normalized equations of motion are obtained along with the boundary conditions. Section 3 discusses the steady state results. The stability of the steady state solutions are presented in Section 4. Finally, Section 5 presents the summary of the present work. We begin with the presentation of the mathematical formulation. Problem formulation Consider that a perfectly exible, homogeneous, incompressible and isotropic neo-hookean string of undeformed length L, circular cross-sectional area A 0 and modulus of Elasticity E is subjected to an axial end load P. After the equilibrium is reached the current length is l. Thus, for uniaxial deformation, the stretch is de ned as the ratio of the current length to the undeformed length, i.e., = l=l. The tension in the string is then given by (See Sarangi et al. [9] or Beatty et al. [10].) P = EA 0 3 1= : (1) The neo-hookean material model provides a simple functional form and shows fair agreement with experimental data for elastomeric material up to moderate stretch range in static tests. The molecular basis of the classical neo-hookean material model is addressed by Doi [13]. Since the string is a perfectly exible lineal body, the stretch is locally uniaxial and the tension acts in the tangential direction. The string is uniform in density and cross-section and we neglect the e ect of gravity force and air drag. Moreover, the e ect of the material damping in the string is not considered since the velocity in the tangential direction is small (Zhu et al. [1]). In the process of forming di erent balloon con gurations, the exible string will deform and its spatial representation is formulated through the Lagrangian formulation. Figure 1 shows the schematic representation of a ballooning hyperelastic string. In the absence of gravity force, the steady-state con guration is represented by 1 and the nal con guration is represented by with respect to undeformed homogeneous con guration 0. e 1, e and e 3 are axes in a rotating coordinate frame such that the steady state string con guration would appear stationary when observed from this rotating frame. The rotating axes are aligned with the xed axes system de ned by X; Y and Z at the instant of interest. In the steady state con guration, the spatial representation remains unchanged with respect to time T and the con guration space is de ned by the position vector as R 1 (S 0 ; T ) = X(S 0 )e 1 + Z(S 0 )e 3 ; where S 0 is the arc length coordinate in undeformed 0 con guration. The nal con guration in con guration is de ned by the position vector as R (S 0 ; T ) and the relative displacement between the two con gurations 1 and is de ned by the relative displacement U as U(S 0 ; T ) = R R 1 = U 1 (S 0 ; T )e 1 + U (S 0 ; T )e + U 3 (S 0 ; T )e 3 : The tension vector P i in the string is de ned as where P i P i (S 0 ; T ) = P i (S 0 ; T i = P i i ; 0 is the magnitude of the string tension and i = 1; for steady state and nal con gurations, respectively. The local string stretch i is a function of both space and time, de ned as i 0. The stretch is related to engineering strain " through the relation i = 1 + " i. The linear stress-strain relation is retained if one imposes the condition i 1 (i.e., i < 1:05), which physically implies in nitesimal deformation in relation (1). Other complex molecular based models like Arruda-Boyce [11] and Gent [1] show good agreement in static experiments for all range of stretch. However, for ballooning motion, the extension is limited to a low or moderate extension. Thus we loose no generality by choosing hyperelastic string as the neo-hookean material model. We will observe in the following sections that while it is simple, the results are fairly accurate and can be easily interpreted.

3 Figure 1: Schematic diagram of a ballooning string. as The tension vector in nal con guration is derived as a function of the steady state tension and recasts P = P 0 ; (3) where i = i 1= i, i = 1 and. The equation of motion through the balance of linear momentum in the absence of body force and surface traction in the exible string for both linear and non-linear elastic string material is 0 = A ; (4) where, is the mass density of the string and the kinematical relationship is given = e 3 (e 3 R 1 ) + e 3 (e 3 U) + e 3.1 Non-dimensionalization of the + : (5) The equations derived so far may be recast in the normalized forms with the use of the following dimensionless parameters in view of which equation (4) takes the form r = xe 1 + ye + ze 3 = R=a; s = S 0 =a; u = U=a; P 1 p = A 0 a ; = E A 0 a and t = p = e 3 (e 3 r) + e 3 (e 3 u) + e The steady-state stretch of a neo-hookean : (7) In the steady state, the balloon has zero relative displacement with respect to the rotating co-ordinate frame fo; e i g. This in turn means that the dependent variables are only the functions of space and they remain

4 unchanged with time. Hence substituting u = 0 in the equations of motion for non-linear elastic string, i.e., in equation (7), we obtain d p dr = e 3 (e 3 r) : (8) 1 Scalar product of equation (8) with dr= and assuming e = 1 (s = 0) yiel the cubic equation 3x e 1 4 = 0; (9) e having three roots; out of which the real positive root of value greater than unity is the desired solution..3 The boundary conditions The equation of motion of the ballooning string is to be solved subjected to boundary condition at each end of the string in the balloon, namely xed eyelet and rotating eyelet. The corresponding normalized forms of the boundary conditions are r(0; t) = 0; r(l 0 ; t) = e 1 + he 3 and u(0; t) = u(l 0 ; t) = 0; (10) where L 0 is the undeformed reference length of the string in 0 con guration, l 0 = L 0 =a and h = H=a are the normalized undeformed string length and balloon height, respectively. 3 Steady state ballooning An imaginary surface generated by a loop of string rotating rapidly about a xed axis is called a balloon. The steady state balloon is generated when the imaginary surface is not changing with time or the position vector of all points on the string with respect to the rotating coordinate frame are xed. For the equation of motion in the steady state con guration, the partial di erential of space and time transforms to ordinary di erential of space only. The corresponding scalar form of governing equations of motion for steady state are obtained form (8) as d x = 3 (1 1= 3 1) d z = 3 (1 1= 3 1) x 4 1 dz 4 The above set of equations is to be solved along with the tension relation (9). non-linear and must satisfy the constraint relation dx + dz = 1; dx d 1 (11) d 1 i:e: dr = 1 The equations (11) are : (1) The steady state equations of motion (11) shooting method. (1) are solved with the boundary conditions (10) through 3.1 Numerical results The numerical results without air drag for xed eyelet tension p e (= P e =Aa ) versus the undeformed length parameter 0 are shown in Fig. for = E=a = 100. The undeformed length parameter is de ned as 0 = (l 0 h)=l 0. In Fig. some of the balloon shapes are drawn at points marked by dots. The single loop shape may be visualized from Fig. 1 by the 1 con guration, where the ordinate represents

5 Figure : Normalized eyelet tension versus string length of planar balloon: h = 10 and = 100. The dots correspond to the balloon shapes for any parent virgin neo-hookean hyperelastic material. The circles correspond to eigenvalue plot in Fig. 3. the ballooning radius. Similarly one-and-a-half loop, two loop and higher loops are also shown. The string shapes drawn at the points are representative gures of ballooning shapes and detailed comparative shapes may be obtained easily. In the case when the values of 0 are small and negative, only single loop balloons with very high tension values are realizable. Since initial balloon length L 0 is much smaller than p a + H, the non-rotating (static) tension is considerably high in the string and thus for moderate values of speed the balloon shape is expected to form a single loop. From the gure it is not clear where various branch intersect in a smooth manner. However, eventually at higher values of tension these in fact do. It is clear from the gure that for xed value of h a multi-loop balloon nee higher initial length compared to the single loop balloon. It appears, at least numerically, that there exists a local minimum of eyelet tension p e = p e min = 14:716 at 0 = 0:08 and for p e p e min, single loop balloons are formed. As the tension reduces, there is a formation of multi-loop balloons. Many solutions are possible for a given value of eyelet tension at di erent undeformed length, i.e., at di erent values of 0. For the same eyelet tension, larger undeformed length suggests that the number of loops will be more except for the single loop balloons. For example, at p e = 5 the steady state shape is one-and-a-half loop at 0 = 0:003, double loop at 0 = 0:156 and two-and-a-half loop at 0 = 0:89: For a given value of 0 ; there may be a set of eyelet tension values, correspondingly, for di erent balloons shapes. In practice one may not obtain all the steady sate points drawn in Fig., as some of the solution branches are unstable as shown in the stability analysis given in the following section. 4 Stability analysis In order to study the stability of all possible steady state solutions of equations (11) and (1), the steady state shapes are given small perturbations; in other wor, steady state solutions are perturbed to obtain the nature of the resulting solutions. However, for planar motion, the perturbation about the steady-state is considered as non-planar and subsequently u(s; t) has three non-zero Cartesian components. In this part we shall consider the relative displacement u(s; t) to be small so that one may neglect the second and higher order terms of u(s; t) and its derivatives in (7), resulting a linear partial di erential equation for the vector u(s; t) as

6 @ dr + = e 3 (e 3 u) + @ ; (13) where, p = p and = 1 5. The vector form of equation in (13) is decomposed in scalar component form to 1 obtain three simultaneous partial di erential equations + + Ku = 0; (14) where the displacement vector u(s; t) = [u 1 ; u ; u 3 ] T. The gyroscopic and sti ness operators are obtained for planar motion in the coordinates x; z follows G = ; K = respectively, with K 11 0 K 13 0 K 0 K 13 0 K 33 K 11 = 1 p + x p ;s + x ;s x ;ss + ;s K 13 = z ;s x + [z ;s x ;s ] ;s x ;s z K = 1 p@ p K 33 = p + z p ;s + z ;s z ;ss + ;s z 3 5 ; (15) In (15) () ;s () =@s indicates the partial di erentiation with respect to s. The components of the sti ness matrices are di erent for linear and non-linear elastic strings. Galerkin s method is used to solve the eigenvalue problem associated with (14). This method essentially truncates the partial di erential equation to N number of nite dimensional solution space. superimposed displacement eld is represented by N-term separable series of the form u (s; t) = NX j=1 k=1 where i 1 = [1; 0; 0] T, i = [0; 1; 0] T, i 3 = [0; 0; 1] T, and the comparison functions The small 3X jk (t) j (s) i k ; (16) j (s) = p js sin satisfy the pinned boundary conditions u(0; t) = u(l 0 ; t) = 0. Substitution of the series expansion of u(s; t), de ned in (16), in the governing equation (14) generates an error "(s; t) which is weighted by the comparison function and integrated and equated to zero. This yiel the following equation: l 0 (17) +[C G ] _+[K G ]=0; (18) where = [ 11 ; 1 ; 31 ; :::::; N1 ; 1 ; ; 3 :::::::; N ; 13 ; 3 ; 33 :::::::; N3 ] T. The system matrix follows from (18) as A = 0 I [K G ] [C G ] ; (19) where 0 and I are, respectively, the null and the identity matrices of 3N 3N dimension. The eigenvalues n = n + j! n, with j = p 1 of the system matrix associated with (19) are obtained using MATLAB programs. The imaginary parts of the eigenvalues correspond to the linearized normalized natural frequencies

7 of the ballooning string. The lowest ten natural frequencies converge to within 0:5 percent of their nal values for N = 10. Hence this value of N is used for all calculations. The stable balloons have eigenvalues with non-positive real parts. Eigenvalues with positive real parts and non-zero imaginary parts correspond to utter instability. Only real positive eigenvalue signi es divergence instability. Figure 3: eigenvalues for the one-and-a-half loop (A to B) and double loop (B to E) balloons corresponding to steady state solutions in Fig. ( = 100): rst eigenvalues ; second eigenvalues. Complex conjugates are not shown. The numerical results show that all single loop balloons are stable for planar motion. However, for the planar motion, single loop balloons are stable with purely imaginary eigenvalues n = j! n. Figure 3 shows the eigenvalues for the planar balloons for = 100, corresponding to the steady state solutions shown in Fig.. We recall the hollow circles marked therein. It is seen that single loop balloons are always stable (results are not shown here). We have drawn only the one-and-a-half loop (A to B) and double loop (B to E). It appears from the numerical studies that for a planar balloon, the real parts of rst eigenvalues in one-and-a-half loop balloon are 1 = 1,! 1 = 0. One-and-a-half loop balloons are divergent unstable and the second eigenvalues (shown by dash dot line) are purely imaginary. At the turning point B, the two real parts of rst eigenvalue merge near the origin and the second eigenvalues are purely imaginary. Between B and C, the double loop balloons are stable as all the eigenvalues are purely imaginary. From C to D, the system is utter unstable with two pairs of complex eigenvalues with real parts having opposite sign and from D to E double loop balloons are again stable. The eigenvalues of the two-and-a-half loop and triple loop balloons behave similarly to those of the one-and-a-half loop and two loop balloons, respectively. Figure 4(a) shows the e ect of balloon length on the natural frequencies of the single-loop planar balloons with = 100. Here we recall the steady state solutions in Fig.. The frequencies usually decrease with increasing 0. Note that the range of tension associated for single loop is p e [14:716; 0] (See Fig..). The e ect of on the single loop planar balloon is illustrated in Fig. 4(b) for a xed value of tension p e = 17:5 and for the left solution among the two. The range covers both nite and small stretches. This e ectively implies that the eyelet stretch is varied in the range 1:000 e 5:86 for the neo-hookean string. The frequency values for linear elastic string are drawn for in the range for the same value of p e = 17:5(Only the left solution.). The ratio p= directly gives the strain within the elastic limit at steady state. Therefore, the strain " 1 = 0:19, turns out to be beyond the linear elastic limit of most materials. Nevertheless, the curves for linear elastic case for the low values of are shown under the assumption that the material is linear elastic with in a signi cant values of strain. The

8 (a)frequencies versus string length for the single loop planar balloon corresponding to Fig. ( = 100 ). (b)normalized frequency versus of a single loop balloon at p e = 17:5 (left solution) linear elastic string. neo-hoookean string. Figure 4: Single loop characteristics eyelet strain for the linear elastic string is varied 0:000 " e 0:19. It is shown by Zhu et al. [] that for 10 3 ; the linear elastic string frequencies are identical with the inextensible string. The same nature hol for the neo-hookean string also. Thus one may conclude that the neo-hookean string generalizes the inextensible string for However, approximately for 310, the frequency for linear elastic string matches with that for the neo-hookean string. We may, thus, readily conclude that the neo-hookean string generalizes both the linear elastic model and inextensible model in a speci c range of values. Thus, neo- Hookean string generalizes the ballooning motion, as it generalizes the linear elastic model for a very small strain " = p e = < 0:05 and even may generalize the inextensible model for " = p e = < 0:018(= 17:5=1000). 5 Concluding remarks This work concerns the ballooning motion of a neo-hookean rubber string and in particular, we investigate the stability of the steady state balloons. In order to obtain the steady state results, we used the shooting method. Further, the stability of various balloon shapes are investigated theoretically for di erent sets of parameters. The solution of the steady-state balloon as derived in (8) depen on a four dimensional parameter space and subsequently the extension at eyelet (i.e. e ). Thus the tension (i.e., p e ) in the in the balloon may be well presented through the other three parameters (i.e., l 0 ; h and ). The others parameters l 0 ; h and are preset for the particular set of data. The theoretical results and experimental evidence reveal that the single loop balloons have higher tension and they are always stable, whereas, the one-and-a-half loop and two-and-a-half loop balloons are usually found to be unstable. In the primary work by Hall et al. [5], the closed-form, analytical solution of inextensible planar steadystate balloons is presented. In contrast, the present study includes the e ect of material nonlinearity of the virgin rubber string, characterized by the neo-hookean model on the ballooning motion. Material nonlinearity arises due to large stretch or due to high rotational speed of such materials, else the linear assumption [] is good enough for the dynamical modeling of ballooning motion.

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