The Non-Linear Temperature Dependent Stiffness of Precompressed Rubber Cylinders

Transcription

1 ROHSTOFFE UND ANWENDUNGEN RAW MATERIALS AND APPLICATIONS Stiffness Shape Factor Non-linear Precompression Temperature Vibration Isolator A shape factor based non-linear model of a rubber cylinder's temperature and preload dependent static stiffness is presented. The influence of temperature, precompression, material parameters, cylinder length and diameter, are investigated; with the motion split into a homogeneous thermal expansion including a globally equivalent preload deformation. Stiffness depends strongly on preload, particularly in larger shape factors, and on temperature. The model proves superior to traditional work in typical shape factors, with results close to those of finite element models. Zur nichtlinearen temperaturabhaèngigen Steifigkeit von Gummizylindern unter Vorlast ± Ein effektives Modell mit Formfaktoren Steifigkeit Formfaktor Nichtlinear Vorlast Temperatur Schwingungsisolator Ein auf Formfaktoren basierendes, nichtlineares Modell der temperatur- und vorlastabhaè ngigen statischen Steifigkeit eines Gummizylinders wird praè sentiert. Der Einfluû von Vorlast, Temperatur, Materialparametern, ZylinderlaÈ nge und -durchmesser wird untersucht. Die Bewegung wird aufgeteilt in eine homogene Temperaturausdehnung und eine globale aè quivalente Deformation aufgrund von Vorlast. Die Steifigkeit ist stark von der Vorlast abhaè ngig ± im besonderen fuè r groèûere Formfaktoren ± aber auch von der Temperatur. Das vorgestellte Modell ist traditionellen Modellen uè berlegen. Die Resultate kommen denen von FE-Modellen sehr nahe. The Non-Linear Temperature Dependent Stiffness of Precompressed Rubber Cylinders ± An Effective Shape Factor Model L. Kari, Stockholm (Sweden) Rubber component modeling difficulties are due to complex stress strain relations ± involving not yet fully understood mechanisms ± including its capacity to sustain large strains, resulting in material and geometrical non-linearities. In addition, the great variety of commercially available rubber components and materials increases the modeling complexity. This paper investigates temperature and prestrain dependent static stiffness of the most common resilient element; that is, a cylindrical rubber isolator with bonded end plates. Although powerful computers and numerical algorithms enable more comprehensive modeling [1, 2], the methods deployed are still arduous and complex; thus not being ideal at an early design stage, where there may be quite cumbersome results that are usually difficult to analyse, as no closed form solution is obtained. The great challenge is to develop a simple but accurate analytical model at the initial stage ± a seemingly contradictory assignment. The analytic models include shape factor based approaches [3 ± 5]; two-part displacement approaches [6]; statistical elasticity based models [7] and semi-inverse approaches [8 ± 11], for which the simplest formulas are provided in standard textbooks [12 ±16]. The semi-inverse method [8] is the first materially and geometrically non-linear analytical solution to the problem, while [9] extends it to include torsion. However, the applicable shape factors in [8, 9] are rather large, not typical for vibration isolators, while also diverging at vanishing precompressions from a well known linear formula [12]. However, the semi-inverse method in [1] coincides with vanishing precompressions in the linear formula being suitable for typical shape factors while also modeling the temperature dependence. The applicable shape factors of the semi-inverse method in [11] are very small, assuming a large length-to-radius ratio; clearly excluding typical isolators. A recently comprehensive review of semi-inverse models is presented in [17]. Despite their analytical origin, the semi-inverse methods may appear too complicated; thus calling for even simpler models, where the shape factor based method [3] and the two-part displacement approach [6] are purely linear while the appealing and simple nonlinear model [4] accounts for the increasing shape factor that results from the decreasing length of the cylinder during compression, whereas [7] is based on a simple statistical elasticity model. The increased cylinder radius at precompression is not taken into consideration. This paper enlarges the non-linear shape factor model [4] to account for the altered shape factor resulting from both the decreasing length and increasing radius of the cylinder during compression, while also modeling the temperature dependence over a wide range from 6 to 6 8C. The model is superior to traditional work ± including semi-inverse approaches ± for typical shape factors with results close to those of finite element models. 76 KGK Kautschuk Gummi Kunststoffe 55. Jahrgang, Nr. 3/22

2 Method Rubber isolator The isolator in Fig. 1 at the reference temperature T consists of an L long vulcanized rubber cylinder with a shear modulus l, density R and diameter D, firmly bonded to metal plates. The applied rubber is isotropic, homogeneous, incompressible, unfilled/slightly filled natural rubber (NR) with a thermal expansion coefficient of a and a shear modulus of l T ˆ Tq T l T q ; 1 at an arbitrary temperature T where R T is the rubber density at T. Kinematics Consider Fig. 1 where a convenient representation of the geometry is in a cylindrical coordinate system; {r, f, z}, with the z-axis directed along the main axis and origin at the cylinder center. The isolator occupies the reference configurations B and B T at temperatures T and T, respectively, and a current configuration S T after precompression at temperature T, which is transformed into a globally equivalent configuration S equt suitable for a shape factor based model. Thus, the isolator undergoes a total motion u : B! S T which is split into u ˆ u T u : B! B T! S T, where a temperature shift T! T results in a thermal motion u : B! B T and mechanical loading in motion u: T B T!S T $ SequT. The thermal expansion/contraction u is assumed homogeneous with the coefficient a ˆ ; 2 being a motion between two stress-free and mechanically undeformed natural states B and B T. The motion u shifts the geometry into L T ˆ 1 adt 3 O adt 2 Fig. 1. A cylindrical vibration isolator undergoes a motion u, from the reference configuration B to B T, due to a temperature shift T to T, and a subsequent isothermal, mechanical motion u T, from B T to current on S T ; being globally equivalent to S equt. The total motion u is from B to S T L 3 and D T ˆ 1 adt 3 O adt 2 D ; 4 where DT ˆ T T and 1 < lim x! O x n =x n < 1, while neglecting the coefficient difference between rubber and steel. The mechanical, isothermal motion u T is an axi-symmetric, torsion-free motion from a cylinder length L T to l T ; L T l T, while the bonded plate diameter remains the same; D T ˆ d T. The modeling complexity arises mainly from geometrical and material non-linearities associated with finite prestrains, in addition to the induced stress singularities at the corners. The formation after the preload being quite different from reference configurations. Mechanical motion is generally large, the straight cylinder surface on which to impose the traction free boundary condition, is transformed into a ± previously unknown ± curved shape etc. These non-linearities are geometrical. In addition, the material responses at different points within the rubber cylinder generally disagree, which is a material nonlinearity. The response to a subsequent infinitesimal deformation is moreover, (seemingly) anisotropic and non-homogeneous, originating from the structure's prestrain dependence [2] ± as distinct from the classical infinitesimal theory. As the theory of prestrained isolators is inherently non-linear, the mathematical difficulties encountered in its applications are considerable. To achieve the appropriate simplified model is therefore the central issue, with shape factor based theories being particularly suitable. To this end, the configuration S T is transformed into a globally equivalent configuration S equt embodying a straight cylinder of length l T and diameter d equt, bonded to metal plates of diameter d T ˆ D T at its ends. The incompressibility assumption yields s L d equt ˆ D T T : 5 l T Consequently, the precompressed isolator is transformed into a locally isotropic and homogeneous body with straight cylinder surfaces; the geometry being most suitable for a geometrically non-linear, shape factor based stiffness formula. Non-linear stiffness model The shape factor is a geometrical stiffness influence function defined as the area ratio of one loaded rubber surface to the total free rubber surface of the component. The area of one loaded surface on S equt in Fig. 1 (marked in red) is A loadt ˆ pd2 T 6 4 and the area of the free surface (green) is A freet ˆ pd equt l T ; 7 KGK Kautschuk Gummi Kunststoffe 55. Jahrgang, Nr. 3/22 77

3 disregarding the slippy surface (pink), being neither a fully loaded nor a fully free rubber surface. Thus, the shape factor reads d2 T S T ˆ 4d equt l T and, by using (5), dt 8 S T ˆ p 4 : 9 L T l T The shape factor based stiffness formula reads k T ˆ ET A equt 1 2S 2 T ; 1 l T where the elasticity modulus E T ˆ 3l T and the cylinder cross section area A equt ˆ pd 2 equt= 4. As the study focus is on soft, unfilled/slightly filled natural rubber (NR), no carbon black correlation factor is introduced while using the simple temperature scaling law for the elasticity modulus (1). Explicitly, the stiffness (1) reads k T ˆ 3p l T D 2 T L T D 2 T 4 L T u T 2 1 ; 8L T L T u T 11 where the compression displacement u T ˆ L T l T and in terms of the reference temperature variables, k T ˆ 3p 1 DT 4 T 1 2aDT 3 l D 2 O adt 2 L L u 2 D 2 1 ; 12 8L L u o where the compression displacement u at the reference temperature is given by u T ˆ 1 adt 3 O adt 2 u : 13 Clearly, the non-linear stiffness (11) and (12) ± accounting for both the decreasing length and increasing radius of the cylinder during compression ± are dependent on the displacement (u T and u ), contrary to those of linear theories. The stiffness temperature dependence is additionally fairly complicated. The compression force is which by using (11) reads F T ˆ 3p l T D 2 T u T 4 L T u T 1 D2 T 2L T u T 16L 2 T L T u T ; 15 and, in terms of the reference temperature variables, F T ˆ 3p 4 1 DT 1 adt 3 O adt 2 l D 2 u L u T 1 D2 2L u 16L 2 L u : 16 Obviously, the compression force displays ± in line with the stiffness ± a non-linear displacement and temperature dependence. Results and discussion Test objects Three examples are studied: the reference lengths L ˆ 25, 5 and 1 mm and diameter D ˆ 1 mm at the reference temperature T ˆ 298 K (25 8C). Thus, the reference shape factor range covers the majority of commercially available isolators; S ˆ 1:,.5 and.25. The reference shear modulus l ˆ 4:5 1 5 N=m 2 corresponds to a nominal hardness of 48IRH, with a thermal expansion coefficient of a ˆ 6:6 1 4 K 1 from [18]. Model validation and comparison The results of the present model at the reference temperature are here compared to the results of those presented in the Introduction, including the results of a more comprehensive example; that is, those of a finite element model. The applied updated Lagrangian nonlinear finite element example [2] solves the weak formulations corresponding to the stiffness problem using the same geometry and shear modulus as here. The hyperelastic example applied is the neo-hookean material representation, since the study focus is on unfilled/slightly filled NR. The elastic response is calculated using 3 (5 6) axi-symmetric hybrid elements, within a mesh progressively refined towards the stress singula- u T F T ˆ k T du T ; 14 Fig. 2. Stiffness k versus compression displacement u at reference temperature T ˆ 298 K (25 8C) for shape factors S ˆ 1:,.5 and.25. Kari [1], Klingbeil & Shield [8], FEM [2], present, Lindley [4], statistical elasticity [7] and linear model [12] 78 KGK Kautschuk Gummi Kunststoffe 55. Jahrgang, Nr. 3/22

4 rities at the cylinder corners. Spatial integration of extrapolated Gauss point stresses results in a superior force estimate; where Gauss points closest to the central cylinder cross section are sufficient for the force calculation. As a result, the aá priori superior Lagrangian 9-node (biquadratic displacement-bilinear pressure) and the 4-node (bilinear displacement-constant pressure) hybrid elements, display negligible global differences. Furthermore, no stress corrupting checkerboard mode is observed for the latter elements. The stiffness of the present representation, for S ˆ 1: in Fig. 2, follows the finite element solution closely within a wide displacement range; to 2 % compression ± well covering the long-term service conditions ± while the simple non-linear example [4] neglects the cylinder radius increase during compression, and the simple statistical elasticity model [7], both underestimate the stiffness; displaying deviations under compression, the divergence being slightly more pronounced for the latter. However, the semi-inverse approach [1] overestimates the stiffness ± showing increasing deviation under compression, while [8] underestimates it ± though displaying a decreasing deviation under compression. The stiffness increases with compression compared to the linear stiffness solutions [12] ± the latter being the straight lines in Fig. 2 ± being the results from non-linear preload effects. However, all the examples, including the finite element solution but not the semi-inverse approach [8], coincide at vanishing preloads. Not surprising, since every well posed large precompression theory should reveal legitimate small motion behavior; that is, the linear model [12]. The stiffness of the present and the simple statistical elasticity model, for S ˆ :5 and.25 in Fig. 2, follow the finite element solution closely, throughout the whole displacement range, while the simple non-linear [4], the linear [12] and the semi-inverse model [8] underestimate the stiffness; the two latter models displaying the largest deviations. The stiffness results for the semi-inverse model [1] are generally superior to those of the simple non-linear [4], the linear [12] and the semi-inverse models [8] but slightly inferior to those of the present and the simple statistical elasticity models. The close agreement between the stiffness of the present and finite element models verifies ± for typical vibration isolator shape factors ± the non-linear model's ability to account for both the decreasing length and increasing radius of the cylinder during compression; thus being superior to traditional models ± including semi-inverse approaches. Temperature and preload dependence The compression force and stiffness of the present model versus the compression, from to approximately 2 %, for S ˆ 1:,.5 and.25 are shown in Fig. 3, 4 and 5, respectively, at temperatures T ˆ 213, 248, 273, 298 and 333 K ( 6, 25,, 25 and 6 8C). Clearly, they increase steeply with the temperature and compression. As a closed form solution of the analytical stiffness model is obtained, a deeper interpretation of the findings is possible as follows: The ratio of the stiffness at compression u to the stiffness at vanishing preload is k T u k T ˆ u 2 1 2S 2 1 u 2S 2 ; 17 by using (12), while introducing a non-dimensional compression u ˆ u =L and a reference shape factor at vanishing preload S ˆ D =4L. Since 1= 1 u > 1 for physically feasible compressions u : < u < 1, the stiffness ratio (17) is a monotonically increasing function of S for a constant u : < u < 1. Therefore, the relative stiffness increase with u, ascends with the shape factor S. This is clearly observed in Figs. 3± 5 where the isolator embodying the largest shape factor displays the most rapid stiffness increase, under compression u, while the smallest shape factor the least. What about the temperature dependence? To this end, the ratio of the stiffness, at an arbitrary temperature T ˆ T DT, to the stiffness at the reference temperature T is k T k ˆ 1 DT T 1 2aDT 3 O adt 2 ; 18 Fig. 3. Compression force F T and stiffness k T versus compression u T, for shape factor S ˆ 1:, at temperatures T ˆ 213, 248, 273, 298 and 333 K ( 6, 25,, 25 and 6 8C) KGK Kautschuk Gummi Kunststoffe 55. Jahrgang, Nr. 3/22 79

5 Fig. 4. Compression force F T and stiffness k T versus compression u T, for shape factor S ˆ :5, at temperatures T ˆ 213, 248, 273, 298 and 333 K ( 6, 25,, 25 and 6 8C) by using (12), which reads k T 1 DT 1 2aT k T 3 1 :13 1 DT ˆ T ; T T ˆ 1 DT T 19 using the actual parameter values, where jdtj << T. Thus, the main stiffness temperature dependence is due to the entropic nature of rubber, displayed by the temperature ratio T=T in (1), while the geometrical temperature dependencies are negligible. Not surprising, as the ratios D T =D ˆ L T =L ˆ 1 adt=3 O adt 2 1 and R T =R ˆ 1 adt O adt 2 1. The entropic nature of rubber is clearly observed in Figs. 3 ±5 where the stiffness curves are vertically shifted with a factor of approximately T=T with respect to the stiffness at reference temperature, in line with the prediction (19). To summarize, the thermal motions u : B! B 213K, B 248K, B 273K, (B 298K ), B 333K almost vanish, whereas the motions due to mechanical loadings u T : B T! S T $ S equt are generally not small; that is, the total motion u ˆ u T u u T. The stiffness is, however, not only strongly dependent on the compression but also on the temperature, mainly due to the factor of T=T in the shear modulus (1). Conclusions Fig. 5. Compression force F T and stiffness k T versus compression u T, for shape factor S ˆ :25, at temperatures T ˆ 213, 248, 273, 298 and 333 K ( 6, 25,, 25 and 6 8C) In presenting a novel, shape factor based, analytical, non-linear and temperature dependent model for the static stiffness of the most common, commercially available resilient element ± namely, the cylindrical rubber isolator with bonded end plates ± the influences of temperature, precompression, material parameters, cylinder diameter and length are readily investigated. The total isolator motion is split into a homogeneous, temperature expansion/contraction and a mechanical, isothermal motion. The latter is transformed into a ± from an engineering point of view ± globally equivalent configuration, accounting for the altered shape factor resulting from both the decreasing length and increasing radius of the cylinder during compression; the configuration being suitable for a shape factor based example. The motion due to the temperature shift, covering 6 to 6 8C, is found to be small, whereas 8 KGK Kautschuk Gummi Kunststoffe 55. Jahrgang, Nr. 3/22

6 the motion arising from mechanical loading is generally large; corresponding to to 2 % compression, covering long-term service conditions. The stiffness is, however, found to depend strongly on compression, particularly for larger shape factors, and temperature, where the shear modulus, being directly proportional to the temperature, plays the greatest role, while the temperature dependent rubber density, cylinder length and diameter, the smallest. The model is not only clear and simple, while modeling the temperature dependence and accounting for both the decreasing length and increasing radius of the cylinder during compression, it is also proved superior to traditional work, including the semi-inverse approaches, for typical shape factors, with results close to those of neo-hookean finite element models. References [1] K.N. Morman and T.Y. Pan, Rubber Chem. Tech. 61 (1988) 53. [2] L. Kari, Proceedings of Constitutive Models for Rubber II, Hannover, Germany (21) 285. [3] A.N. Gent and P.B. Lindley, Proc. Instn. Mech. Engrs. 173 (1959) 111. [4] P.B. Lindley, J. Strain Anal. 1 (1966) 19. [5] E. Haberstroh, H. Ehbing and M. Stommel, Kautsch. Gummi Kunstst. 5 (1997) 12. [6] A.N. Gent and E.A. Meinecke, Polym. Eng. Sci. 1 (197) 48. [7] P.K. Freakley and A.R. Payne, Theory and Practice of Engineering with Rubber, Applied Science Publishers, London (1978) 117. [8] W. Klingbeil and R. Shield, Z. angew. Math. Phys. 17 (1966) 281. [9] J.M. Hill and A.I. Lee, Mech. Phys. Solids. 37 (1989) 175. [1] L. Kari, J. Strain Anal., submitted. [11] H.-H. Dai and Q. Bi, Q. J. Mech. Appl. Math. 54 (21) 39. [12] A.N. Gent, Engineering with Rubber, Carl Hansen Verlag, Munich (1992). [13] P.B. Lindley, Engineering Design with Natural Rubber, The Malaysian Rubber Producers' Research Association, Brickendonbury (1992). [14] E.F. GoÈ bel, Rubber Springs Design, Newnes- Butterworths, London (1974). [15] A.R. Payne and J.R. Scott, Engineering Design with Rubber, Interscience Publishers, New York (196). [16] P.K. Freakley and A.R. Payne, Theory and Practice of Engineering with Rubber, Applied Science Publishers, London (1978). [17] J.M. Hill, Int. J. Non-linear Mech. 36 (21) 447. [18] Encyclopedia of Polymer Science and Technology, John Wiley & Sons, New York 7, 12 (197). The author Leif Kari is a member of the Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL), Department of Vehicle Engineering, Royal Institute of Technology, Stockholm, Sweden. Corresponding author Dr. L. Kari, MWL/Department of Vehicle Engineering, Royal Institute of Technology, SE-1 44 Stockholm KGK Kautschuk Gummi Kunststoffe 55. Jahrgang, Nr. 3/22 81