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Ind. Eng. Chem. Prod. Res. Dev. 1984, 23, 596-600
Mechanical Analysis of Elastomeric Seals by Numerical Methods Glanluca Medrl‘ and Antonio Strorzl Istituto di Meccanica applicata alle Macchine, Bologna University, Bologna, Italy
A procedure which permits the evaluation of the stress-strain field in an elastomeric seal is presented. Such a procedure is based upon a mechanical modeling of the elastomer and on the employ of finite elements which can cope with high strains. Finally, the above-mentioned procedure is checked against experimental results. The problems connected with the material modeling are examined. In particular, the monotonicity of the constitutive relation is discussed. The difficulties encountered in the numerical simulation of the elastomeric seal are investigated. In particular, the convergence problems are discussed. The procedure is applied to a rectangular seal with rounded edges, to an O-ring seal, and to a U-ring seal. The numerical isochromatic pattern is drawn automatically and is compared to the photoelastic one. The validity and limits of the proposed procedure are discussed with regard to the three seal configurations examined.
Introduction The aim of this paper is to present a procedure which permits the evaluation of the complete stress-strain field in an elastomeric seal. The relevance of the knowledge of the stress-strain field stems from the following considerations. (1)The contact pressure distribution in a seal is a good indicator of the seal performances as well of its sealing characteristics. In the case of static seals, the sealing is guaranted only if the maximum contact pressure is higher than the sealed pressure (Favretti and Molari, 1971,1972). In the case of dynamic seals, the static contact pressure profile is fundamental in the elastohydrodynamic lubrication (E.H.L.) studies, since both the inverse hydrodynamic theory (Dowson and Higginson, 1977), and Rohde’s formulation (Szeri, 1980), require its knowledge. (2) The presence of tensile stresses in an elastomeric seal can result in cracking, which in turn can cause the seal failure. In the case of O-ring seals, it is known that high tensile stresses due to the seal compression act at the center of the cross section and a t the shoulder of the contact zone. Consequently, a central crack or a surface crack can occur (Ebisu et al., 1983). Moreover, in the case that a gas dissolution occurs in the elastomer, a fast decompression can cause elastomer explosion, as it appeared from the discussion “Explosive Decompression of Elastomers” held a t Innsbruck during the 10th Fluid Sealing Conference. These two examples show that fracture mechanics applied to elastomers plays an important role in practical problems. (3) The evaluation of the seal lip load is an important parameter for the seal designer. In the case of multi-lip seals, it is fundamental to know the distribution of the total load on the various lips. Such a distribution depends on the functional character of the lips (Medri et al. 1984). The knowledge of the stress-strain field in an elastomeric seal can be obtained by an accurate mechanical characterization of the elastomer employed and by the development of numerical methods capable of simulating the behavior of an elastomeric seal. Last but not least, experiments should be performed to check the validity of the above-mentioned procedure. The main difficulty in the mechanical characterizaton of an elastomer is its inherent, highly nonlinear behavior, while the complexity of the numerical method stems from the need to take the variability of the material behavior into account, to simulate its small cubic compressibility, and to cope with high strains. 0196-432118411223-0596$01.50/0
The photoelasticity is a well-established experimental method for the analysis of the stress field in elastomeric units. Since the separation of the principal stresses from the isochromatic and isocline field requires a considerable amount of work, the numerical isochromatic field is derived and automatically drawn from the numerical stress field, and it is compared to the photoelastic isochromatic field. Therefore an assessment of the proposed method can be easily performed and its validity and limits can be readily estimated. 1. Material Modeling 1.1. General Aspects. A considerable number of constitutive equations has been proposed for the mechanical description of elastomeric materials (Oden, 1972), but their adaptability in simulating the experimental behavior has rarely been checked. Moreover, it has been shown recently that a linearization (simplification) of the constitutive equation can lead to appreciable modifications in the stress-strain field within the seal (Prati and Strozzi, 1984). The constitutive equation is (Truesdell, 1965)
where 1i are the three Rivlin strain -invariants, T is the Cauchy stress tensor, B is the left Cauchy-Green strain tensor, C is the Cauchy strain tensor, and W is the strain energy density per unit volume of the undeformed body. The authors’ experience suggests that W can be approximated by the following expression W = C1(I1- 3) C2(12- 3) + C3 In 12/3 + c4(12 - 3)2 + cs(I1 - 3) (12 - 3) + c6(11- 3)2 + C7(13ll2 - 1)’ - (13 - 1) (c1 2c2 2c3/3) (2)
+
where the coefficients c& describe the essentially deviatoric part, while the coefficient C, describes an approximately linear cubic compressibility. Since the deviatoric behavior of the elastomeric materials is noticeably different from the hydrostatic behavior, it is difficult to devise an experimental procedure where both the deviatoric and the hydrostatic characteristics can be assessed. Therefore two kinds of experiments are typically performed. The first kind is a biaxial tensile test (Medri, 1982),which is performed on a square thin specimen. This test assumes that the elastomer behaves as virtually incompressible, and therefore the stress-strain link can be 0 1984 American Chemical Society
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23,No. 4, 1984 597
assessed in uniaxial, biaxial, and equibiaxial tension. A least-square interpolation permits the definition of the material constants C1-C6. The second type test regards the measure of the cubic compressiblity of the elastomer, and therefore the evaluation of C,, by resorting to a piston device (Holownia, 1974), in which the change of shape of the cylindrical rubber specimen and ita effects on the stress field can be usually ignored. In other types of compressiblity testslarge distortions are coupled with finite volume changes; see Penn (1970) and Medri (1982b). These observations explain why the strain energy function W is usually formed by two distinct parts; the first part describes the deviatoric behavior; the second part characterizes the hydrostatic behavior. The last term in eq 2 is added in order to obtain that the stress field vanishes in the undeformed state. 1.2. Monotonicity of the Constitutive Relation. A necessary condition for obtaining the convergence of the nonlinear numerical models is the monotonicity of the constitutive relation. In particular, the so-called GCNO condition is referred to the fact that the transformation from the prinicipal stretch ratios to the principal forces is monotonous (Truesdell and Noll, 1965). From a practical point of view, the monotonicity is achievable by resorting to strain energy functions which exhibit strict convexity with respect to the principal stretch ratios. This can be practically accomplished by imposing that the Hessian form of W is positive-definite (with respect to the principal stretch ratios) in each experimental point. This imposition would result in a set of ancillary equations which influence the determination of the material constants by the least-squares method. Such a procedure is successful only if the experimental analysis is sufficiently exhaustive and capable of describing the foreseeable stress-strain field within the elastomeric element to be studied. A much stronger imposition on the strain energy function is that the Hessian form of W with respect to the principal stretch ratios is positive-definite for every point within an assigned domain. Nevertheless, this condition tends to originate numerical troubles in the determination of the material constants. The so-called Mooney law is approximate, but it is usually monotonous. This is seemingly one of the reasons why such a constitutive relation is still so popular and widely used. 2. The Finite Element Method 2.1. General Aspects. It is known that the finite element method based upon the displacement field, when applied to slightly compressible materials, can produce catastrophic results as far as the stress field is concerned (Cescotto and Fonder, 1979). This is due to the fact that the stiffness matrix is illconditioned (Fried, 1973); that is, the minimum of the potential energy is not well definite. A more physical explanation of this problem is the following: in an elastomer, a hydrostatic pressure of the order of lo2 MPa produces a volume change of some percentage. Therefore, if the function which describes the displacement field within an element approximates the volume change with an error of some percent, the error on the hydrostatic pressure can be of the order of lo2 MPa, while the error on the deviatoric stress is lower. The first contribution to the study of this difficulty is due to Herrmann (1964,1965). He based his study on an important formulation due to Reissner (1950), and observed that the finite element based on the displacement field pays more attention to the displacement than to the
stresses, since the stress filed is computed by differentiating the functions which describe the displacement field. It is a common experience that, even if a function carefully describes a certain behavior, its first derivative rarely approximates with the same accuracy the actual derivative. Herrmann proposed to describe with the same care both displacements and stresses, by fixing within every element both the function which approximates the displacement field and that which approximates the stress field. For example, if a triangular finite element in a plane state of stress is examined, both the function which describes the displacement field and that for the stress field can be chosen to be linear (Hughes and Allik, 1969). Since the displacements are linear, the stress-strain field, which is linked to the derivative of the displacement field, should be constant. On the other hand, if the stresses are chosen to be linear, they should derive from a quadratic displacement field. Such contradictions can be overcome by relaxing the stress-strain field, i.e., by accepting that the constitutive relation is satisfied in the mean (not locally) within an element. Mixed finite elements are based upon the above-mentioned considerations. More exactly, Herrmann noticed that the oscillations of the stress field were confined only to the hydrostatic part of the stress field, while the deviatoric part was generally much smoother. Therefore he proposed to fix an interpolating function for the displacement field and a second interpolating function for the hydrostatic pressure, which would result in a relaxation of the incompressibility condition. The results obtained by Herrmann were incouraging, but some disillusions derived from the subsequent research. Oden and Reddy (1975) clarified that some choices of the functions interpolating the displacement and the stress field would give rise to finite elements which are not actually mixed, since their behavior would coincide with that of more traditional finite elements. Skala (1977) obtained an irregular stress field with mixed finite elements, since some oscillations on the stress field were noticed when the compressibility of the material was decreased. The current tendency seems to invoke again the traditional finite elements based upon the displacement field, and to increase the number of nodes describing the mesh, as the compressibility of the material is decreased (Fried, 1983). 2.2. Numerical Aspects. A single field, constant strain, triangular finite element program in finite hyperelastic deformations has been developed (Medri et al., 1984) and employed in this study. The main numerical difficulty derives from the fact that, because of the above-examined aspects (section 2.1) the system of nonlinear equations is characterized by an ill-conditioned stiffness matrix (Fried, 1973). The solution of such a system is obtained through the following procedure: a linearized program supplies an approximate displacement vector (starting solution) which is improved through a bialgorithm, based upon the combination of a technique which exhibits good global convergence but poor local convergence, and an opposite technique (see Luenberger 1976, p 281). In particular, the relaxation method and Newton’s method were combined by Strozzi (1983). The relaxation method is a slow but efficient technique (Lindley, 1971), while Newton’s method shows quadratic convergence if the starting point is within the sphere of attraction; otherwise it may diverge (Luenberger, 1976). 2.3. The Numerical Isochromatic Pattern. A comparison between the experimental isochromatic pattern and the numerical one can give some idea of the validity of the numerical solution. A routine for drawing the
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Ind. Eng. Chem. prod. Res. Dev., VoI. 23,No. 4, 1984
isochromatic pattern was recently implemented in the fmite element program. The procedure by which the numerical isochromatic pattern is drawn automatically is exposed hereinafter. First, the elements of the finite element program employed in this paper exhibit discontinuities in stresses from one element to the next. To facilitate the interpretation of the stress field, the stress jumps are smoothed out by a stress averaging. The stresses of all the elements connected to a node are summed and divided by the number of elements, and the resulting value is attributed to that node. Secondly, the stress field is interpolated within each element through the isoparametric shape functions 1 *Q,d = (*d1 + €)(I - n) + * z (1- E ) x
Figure 1. The dimensions of the rectangular seal with rounded edges.
4
(1 + 9) + *dl - €)(I- 9) + *r(l
+ €)(I- 7 ) ) (3)
where and n are local coordinates which assume the values +1 and -1 a t the four nodes, and are the four nodal values of the function to he interpolated over the element. Thirdly, for a pair of adjacent values of [, the corresponding values for n are determined for a given In the case that both the values of q exhibit an absolute value lower than or equal to the unity, a segment is drawn which connects the two pairs of values. The same procedure should be repeated using 1) as primary variable, in order to fill the gaps in correspondence with n-oriented segments.
*
*;
+.
3. The Analysis of Elastomeric Seals The above-mentioned procedure is hereinafter applied to three types of elastomeric seals. In particular, a rectangular seal with rounded edges, an O-ring seal, and a U-ring seal are examined. In opposition to more conventional geometries, rectangular seals are an alternative seal design used for demanding applications such as aircraft actuators (Ruskell, 1980). Many studies have been performed on O-ring seals (Favretti and Molari, 1971, 1972), nevertheless an exhaustive evaluation of the tensile stress level as a function of the seal fractional compression, as well as an accurate analysis of the O-ring seal when introduced in particular groove geometries such as dovetail shaped cross sections, has not yet performed. U-ring seals are often employed with regard to dynamic sealing problems. Variations in the shape of the lip can produce appreciable changes in the contact pressure profiles, and therefore in the elastohydrodynamic performances of the seal. The photoelastic study was performed by placing a flat polyurethane model reproducing the seal cross section betwqen two parallel perspex plates (Prati and Strozzi, 19&1), so that a plane state of strain was obtained, which simulates an authentic situation in which the mean plan seal diameter is not modified by the initial compression. The aim of this section is the examination of the overall behavior of the three kinds of seal and the comparison between the numerical and the experimental isochromatic patterns, more than the analysis of particular aspects of the stress field. The photoelastic investigation and the numerical simulation are referred to a polyurethane elastomer (Hysol CP 4485). In Figure 1the dimensions of the rectangular seal are presented. In Figure 2 the automatic mesh used in the finite element analysis is displayed. Four triangles are dynamically condensed to form a quadrilateral element. In Figure 3 the numerical isochromatic pattem is presented
Figure 2. The mesh used in the study of half the rectangulsi' seal.
Figure 3. The numerical isochromatic pattern (left) and the photoelastic isochromatic pattern (right) for a 18% compressed rectangular seal.
together with the photoelastic one for a 18%compressed seal. The agreement between the two patterns is good especially as far as the position of the eyelets is concerned. In correspondence with such points, the contact pressure distribution exhibits two peaks, which derive from the hertzian effects a t the rounded edges. The lack of similarity between the numerical and the photoelastic pattern in the upper part of the seal is seemingly due to the fact that the frictional effects of the experimental study are absent in the numerical study, in which the contact is supposed frictionleas. Such effecta will produce perceivable variations on the photoelastic pattern especially where the stress gradient is lower, that is, in the upper zone of the seal. A more detailed analysis would show that tensile, horizontal, principal stresaes exist in the lower zone ot the seal, between the two contact pressure peaks.
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23. No. 4, 1984 599
-k"4 229
Figure 6. The dimensions of the U-ring seal.
L
Figure 4. The mesh used in the study of a quarter of m O-ring seal.
Figure I . The mesh used in the study of half the U-ring seal.
/ Figure 5. The numerical isochromatic pattern (right and the photoelastic isochromatic pattern (left) for a 20% compressed O-ring 8d.
In Figure 4 the automatic mesh used in the finite element study of O-ring seals is presented. In Figure 5 the numerical isochromatic pattem is displayed together with the photoelastic one for a laterally unrestrained 20% compressed seal. The agreement between the two pattems is good. The similarity between the central fringes is particularly interesting. A more detailed study would show that the contact pressure profile is of hertzian type and would permit the analysis of the level of tensile stresses at the center of the seal and at the shoulder of the contact zone. In particular, the two curves describing the peak tensile principal stresses in the two above-mentioned zones vs. the seal fractional compression do not appear to be proprotional. This fact suggests that the geometric and material nonlinearities play an important role in the description of such a seal. Finally, in Figure 6 the dimensions of the U-ring seal are presented. In Figure 7 the automatic mesh used in the finite element analysis is displayed. The mesh was obtained by defining only the nodes along the seal boundary and by computing the intemal nodes through the concept of high-degree isoparametric element. In Figure 8 the numerical isochromatic pattern is displayed together with the photoelastic one for a 13% compressed seal. The salient features of the two patterns look similar: the stress concentration a t the U cavity, the accumulaton of fringes at the contact between lip and sealing surface, and the low level of stress a t the seal heel. Nevertheless, the similarity is poorer than in the previous examples. This is seemingly due to the fact that the simulation of the seal heel requires a certain number of elements, so that the description of the seal portion which
Figure 8. The numerical isochromatic pattern (bottom) and the photoelastic isochromatic pattern (top) for a 13% compressed U-ring seal.
is more active from the viewpoint of the stress field is inevitably less accurate. The contact pressure peak is located a t the seal lip edge, while tensile stresses appear a t the transition zone between lip and heel. 4. Conclusions The above-examined examples show that a careful mechanical characterization of the elastomeric materials together with the employ of suitable finite element programs permits the analysis of the stress-strain field in an elastomeric seal. The photoelastic technique permits an assessment of the numerical results and an estimate of their accuracy. Therefore the procedure presented in this paper can prove to be of valuable help for the seal designer, since it can actually reduce that trial-and-error character inherent in the traditional seal design. Literature Cited cBscono. 3.:Fonder. G. I~I. J . sow strucf. 1979. 15. 589. Dowsan. D.: Higginron. 0. R. "Elastohydrodynamic Lubrication": Pergamon Press: Oxford. 1977. Ebiw. T.: Yamamoto. M.: MBekawa. H.; Onodera. A. "PrccBBdlngs of Packaging and Transpwt of Radb Active Materials Conterence": New Weans. 1983: p 258.
Favreni. G.: Molari. P. 0. O!ednamh-Pnem"acs
1971 No. 10-12: 1972. No. 1-5. 6. 7. 8 (in Italian). Fried. I. I~I. J . solids slruct. i w s , 9 . 323. Fried. 1. Comp. Sfrucf. 1983. t7, 161. Herrmann. L. R.: Toms. R. M. J . Appl. Mech. 1964. 140. Herrmann. L. R. AIAA J . 1965. 3, 1896. Halownla, B. P. J . Inst. Rubber I&. 1974 157. Hughes, T. J. R.; Allik. H. "Proceedings of Symposium of Ckll Engineers": Vanderbltl Unlversw, Nashville. TN. 1 9 6 9 p 27. Lindley. P. B. J . Strain Anal. 1971. 6 . 45.
Ind. Eng. Chem. Prod. Res. Dev. 1984, 23, 600-605
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Luenberger, D. G. "Optimization by Vector Space Methods"; Wiley: New York, 1976. Medri, G. "Proceedings of the 10th National Conference AIAS"; University of Calabria, Cosenza, 1982a; p 299 (in Italian). Medri, G. Plast. Rubber Proc. Appl. I982b, 2 , 293. Medri, G.; Strozzi, A.; Bras, J. C. M.; Gabelli, A. "Proceedings of the 10th Fluid Sealing Conference"; BHRA, Innsbruck, April 1984; p 421. m e n , J. T. "Finite Elements of Nonlinear Continua"; McGraw-Hill: New York, 1972. Penn, R. W. Trans. Soc. Rheol. 1970, 1 4 , 509. Prati, E.;Strozzi, A. J. Tribology 1984, in press.
Reissner, E. J. Math. Phys. 1950, 29, 90. Ruskell, L. E. J. Mech. Eng. Scl. 1980, 22, 9 Skala, D. P. "Proceedings of Symposium on Applied Computer Method for Engineering"; Los Angeles, 1977; p 1095. Strozzi, A. Plast. Rubber Proc. Appl. 1983, 3 , 4. Truesdell, C.; Noil, W. "The Nonllnear Field Theories of Mechanics", in "Encyclopedia of Physics", Springer-Verlag: Berlin, 1965.
Received f o r review June 18, 1984 Accepted July 23, 1984
Elastomeric Polyimides from a,w-Bis(aminomethyl)polyoxyperfluoroalkylenesand Tetracarboxylic Acids Ezlo Strepparola, * Gerardo Caporlcclo, and Enrlco Monza Montefluos S.p.A.. Montedison, Research and Development Center, Milano. Ita&
Diamine intermediates resulting from TFE photooxidation of perfluoropolyethers have been condensed with tetracarboxylic acid dianhydrldes. The molecular structures of both monomer intermediates have been selected in order to obtain elastomeric products. The chemical and physical propetties of polymers and the final characteristics of cured rubbers are discussed. The resutting mechanical behavior at low and high temperatures and the resistance to chemical attack are of interest.
Introduction The aromatic polyimides are thermally resistant resins characterized by high first and second-order transition temperatures. Interposition of short or medium perfluoroalkylene units into the main chain of aromatic polyimides results in a lowering of the glass transition and melting temperature and increases internal plasticization, without changing thermal and chemical stabilities, as Critchley et al. (1969, 1972) showed.
perfluoropolyperoxidic polyether precursors (111)where the
AO(CZF~~),(CF~O),(C~F~OO)~(CF~~~),B I11 A,B = -CF,COF, -CF,, -C2F, oxy- (and dioxy-) tetrafluoroethylene and oxy(and dioxy-) difluoromethylene units are randomly distributed along the chain. Reductive cleavage of 0-0 bonds (Sianesi et al., 1968) proceeds through chain fragmentation and degradation involving the oxyaifluoromethylene units adjacent to peroxidic groups; this gives largely difunctional derivatives (IV).
NO(C,F4O),(CFzO),CF~COOR -b
L I
X = 0,SO;, (CF,),; n = 3-8
Webster(1974) interposed low molecular weight perfluoropolyether and dioxyperfluoroalkylene units between aromatic structures (11).
IV N = -CF,COOR(>CF,); R = H, CH3 The dicarboxyl terminated perfluoropolyethers (IV) can be used directly as intermediates for several classes of polycondensation polymers to obtain polyamides and polyesters or, through further metathesis of functional end groups, can lead to other interesting classes of polymers: polyethers, from diols; polytriazines, from dinitriles; polyoxadiazoles,from dinitriles or dihydrazides; polyimides, from diamines. Mitsch et al. (1971) obtained a copolyimide (V) starting
V
I1 Rf = (CF,-CFO),-(CF,),-(0CF-CF2),,-O;~ +y=3 I
CF,
I
CF,
Studies of tetrafluoroethylene photooxidation by Sianesi et al. (1967) have made available high molecular weight 0196-4321/84/1223-0600$01.50/00 1984 American Chemical Society
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