Theoretical investigation on the fatigue life of elastomers incorporating material inhomogeneities A. Andriyana Laboratoire Matériaux, Endommagement, Fiabilité et Ingénierie des Procédés LAMEFIP) École Nationale Supérieure d Arts et Métiers ENSAM), 33405 Talence - France E.Verron Institut de echerche en Génie Civil et Mécanique - UM CNS 683 École Centrale de Nantes, BP90, 443 Nantes - France ABSTACT: The most common definition of inhomogeneity relates to a whole composed of dissimilar or non-identical elements or parts. It can simply be verbally described as a more or less rapid change in material properties. As it has been highlighted by experimental observations, fatigue crack nucleation in rubber is due to the propagation of microscopic flaws initially present in the virgin material. Such flaws are thought to be distributed throughout the material. However, in most studies on the prediction of rubber fatigue life, it is quite common to consider rubbers as materially homogeneous in the sense of continuum mechanics. In this study, our interest lies in determining theoretically the role of inhomogeneities, manifested by variations of material properties such as rigidity, in the fatigue life of natural rubber. In order to estimate fatigue life, only crack nucleation is considered and a new fatigue life predictor, introduced recently by the same authors and developed in the framework of Configurational Mechanics, is used. esults show that a small variation of material properties could lead to a significant drop in fatigue life. INTODUCTION In this study, our interest lies in determining the role of inhomogeneities, manifested by variations of material properties, i.e. material parameters in the fatigue life of natural rubber. More precisely, our question is to investigate whether a small variation of material properties could lead to a significant drop in fatigue life. For this purpose, the experimental database of Mars 00) is considered. The configurational stressbased predictor, recently introduced by Verron & Andriyana 007), is adopted in order to predict fatigue life. Here, only continuously linear and sinusoidal variations about a mean value of material properties are considered. FATIGUE PEDICTO There exist three fatigue predictors widely used in the literature for rubber fatigue: the maximum stretch ratio λ max Cadwell et al. 940; oberts & Benzies 977), the maximum principal Cauchy stress σ max André et al. 999; Abraham et al. 005; Saintier et al. 006) and the strain energy density W oberts & Benzies 977; Abraham et al. 005). In this study, a new fatigue life predictor, introduced recently by the same authors and developed in the framework of Configurational Mechanics, i.e. mechanics of continua expressed intrinsically on the material manifold Maugin 993; Maugin 995; Maugin 00), is adopted. The corresponding proposition is motivated by experimental observation in rubber fatigue: macroscopic fatigue crack nucleation is mainly due to the propagation of microscopic defects initially present in rubber body. By supposing that opening and closing of microscopic defects cavities) in rubber are due to only material normal traction and not due to material shear, the authors proposed that microscopic defects growth can be predicted by considering the smallest eigenvalue of this tensor see Verron & Andriyana 007) for elastic case and Andriyana & Verron 007) for its extension to inelastic case). Thus the predictor is given by: Σ = min Σ i ) i=,,3, 0] ) where Σ i ) i=,,3 are the principal configurational
stresses. When one or more) principal stress is negative, the predictor is strictly positive, and the defect tends to grow and to turn into a plane crack orthogonal to V, the eigenvector associated with Σ. When the three principal stresses are positive, the material tractions tend to shrink the flaw. In this case, the predictor is set to 0: it can be seen as the microscopic counterpart of the classical crack-closure effect. In order to incorporate non-proportional multiaxial loading conditions, the authors proposed to accumulate the increment of the configurational stress that contributes to flaw opening. In this case, the previous predictor becomes: Σ Σ ) ] d = min i, 0 i=,,3 ) where ) Σ d i are the eigenvalues of the damage i=,,3 part of the configurational stress tensor Σ d. This tensor is obtained by the integration over the cycle of: dσ d = 3 dσ d i V i V i 3) i= with: { dσ d dσi if dσ i = i < 0 and V i ΣV i < 0, 0 otherwise, 4) dσ i ) i=,,3 and V i ) being the eigenvalues and eigenvectors of the configurational stress tensor increment: dσ = dσ dc : dc. 5) It is to note that for fully-relaxing proportional loading conditions, the integration over one cycle reduces to the determination of the instantaneous value of the configurational stress tensor for the maximum strain state. In this way, Eq. -5) reduces to Eq. ). Finally, the number of cycles to crack initiation is correlated to Σ via a classical power law function: ) Σ β = 6) where Σ o and β are material parameters obtained from uniaxial tension fatigue data. 3 CONSTITUTIVE EQUATIONS In this Section, the constitutive equations for computing the predictor Σ are given and some data of the experimental program of Mars 00) are briefly presented. To simplify the discussion, only continuously linear and sinusoidal variations about a mean value of the material properties are considered. Σ o 3. Experimental program of Mars 00) In his remarkable dissertation, Mars 00) investigated fatigue life of natural rubber for a wide range of strain history: simple tension, simple torsion, proportional in phase) tension/torsion and non-proportional out of phase) tension/torsion, as shown in the Figure. In the following, λ and τ represent the stretch ratio and the twist angle per unit length respectively. Paths A and B were conducted under λ = τ = 0, where: λ = λ min λ max and τ = τ min τ max 7) Paths D, G, H and I correspond to tension/torsion tests with phase angles of 0, 45, 90 and 80 respectively. Paths F and L provide combination of compression/torsion in and out of phases. Only paths A, B, D and H are considered in this paper. The deformation is prescribed as: λt) = λ m + λ a sin πft) τt) = τ m + τ a sin πft + φ) 8) where λ m and λ a are the mean stretch ratio and its amplitude, τ m and τ a are the mean twist angle per unit length and its amplitude, f is the signal frequency which varies between and 4.0 Hz and finally φ is the phase angle between uniaxial tension and torsion. Loading conditions are said proportional when φ = 0. The specimen is a short, hollow cylinder of rubber, bonded between two steel mounting rings and will be modeled by a simple hollow cylinder in order to simplify the computation. Figure : Different type of loading conditions performed by Mars 00). Consider the rubber specimen of Mars 00) of which its cross section is illustrated in Figure a). In fact, wee assume that this specimen can be represented by a short thin cylinder with internal radius i =38.0 mm and external radius o =43.8 mm. Its
ubber C.9.8.7.6 ) a).4.3... 38 39 40 4 4 43 44 = ) C.7.65...6 5 ) Figure : a) Cross section of a simplified rubber specimens of Mars 00). b) An arbitrary variation of material parameter along cylinder radius. height is equal to 6.35 mm. An arbitrary variation of a given material property C is depicted in Figure b) where C i and C o represent the values of the material parameter at internal and external radii respectively. In the following, the neo-hookean constitutive equation will be adopted and C will be the corresponding unique material constant. Two forms of material parameter variations are considered: linear and sinusoidal. For linear variation, it is given by: C) = C ξ { i o i )}] while for sinusoidal variation, it is given by: { )}] C) = C i + ζ sin πn o i 9) 0) where C = MPa is a mean value of C, C max is the maximum value of material parameter, n is periodicity, and ξ and ζ are defined by: Co ξ = C ) ) C and Cmax ζ = C C ). ) Figures 3a) and 3b) illustrate linear and sinusoidal variations of the material parameter C for different values of ξ and ζ. b).45.4.35 38 39 40 4 4 43 44 Figure 3: Variation of the material parameter C along the specimen radius: a) linear variation with ξ=0, 0.05, 0.,, 0.], b) sinusoidal variation with n = 4 and, 0.05, 0.]. 3. Governing equations The computation of the proposed predictor is based on the analytical solution of the problem of simultaneous uniaxial extension and torsion of a hyperelastic cylinder as proposed for example by Green & Adkins 960). Consider a hyperelastic cylinder with uniform cross section subjected to tension/torsion loading. Let, Θ, Z) and e, e Θ, e Z ) be the coordinates and the cylindrical unit vectors in the initial configuration. Its counterpart in the deformed configuration are r, θ, z) and e r, e θ, e z ). The deformation of the cylinder is given by: r = λ θ = Θ + λτz z = λz 3) where λ and τ are the stretch and the twist per unit length respectively. The deformation gradient is: F = λ e r e + e θ e Θ ) + λτ e θ e Z + λ e z e Z 4) and the configurational stress tensor of an isotropic 3
and incompressible hyperelastic solid is given by Andriyana 006): ) W W Σ = W + p) I + I C + W C I I I 5) where p is an arbitrary pressure which can be determined by means of equilibrium equations. Considering a non-homogenous form of the Neo-Hookean strain energy density: W = C) I 3) 6) and supposing that the external lateral surface is traction free, the components of the configurational stress tensor become: Σ ) = Σ ΘΘ ) = C) λ + ) λ + λτ 3 + λτ Ω) Σ ΘZ ) = Σ ZΘ ) = C) τ Σ ZZ ) = C) λ + 4 ) λ λτ 3 where: + λτ Ω) Ω) = C + ξ ) i o ] o i 7) ξ C 8) 3 3 o i ) o 3] for linear variation of C and: Ω) = C o i ) with + ζκ ] n κ = sin n o i )) n o cos n o i ))] 9) sin n i )) n o cos n i ))] 0) for sinusoidal variation of C. Finally, the integration of the configurational stress tensor over a cycle requires the determination of its differential: dσ dσ = dλ λ + dσ ) dτ τ dt. ) The value of the predictor can now be easily computed using Eqs. ), 3) and 4). 4 ESULTS AND DISCUSSION In order to relate our predictor Σ to fatigue life, the material constants in Eq. 6), i.e. Σ 0 and β, are identified by fitting the simple tension data. As observed by Mars 00), cracks initiate on the specimen surface, thus Σ needs only to be computed at = o. Following Mars, the homogeneous Neo- Hookean strain energy density with C = C = MPa is adopted. The corresponding material constants are: Σ 0 = MPa and β = 3.3 ) Using these constants, the theoretical fatigue life for non-homogeneous Neo-Hookean material, denoted Nf inh, is computed for both linear and sinusoidal variations of the material parameter C, and different loading conditions. The results are presented in Figures 4 and 5 for linear variation, and in Figures 6 and 7 for sinusoidal variation. In Figures 4 and 6, two graphs which correspond to simple tension and simple torsion loading conditions are plotted. For the case of proportional and non-proportional tension/torsion loading conditions, they are depicted in Figures 5 and 7. In each graph, drop in fatigue life for different variation of material parameter is computed at each material point of the specimen, i.e. its radial position. Here, drop in fatigue life is defined by the ratio between Nf inh to. Note that the latter is computed using the homogeneous neo-hookean strain energy density with C = C = MPa. In general, it is observed that a small variation of material parameter could lead to a significant drop in the fatigue life. Using the neo-hookean hyperelastic strain energy density, it is found that drop in the fatigue life could reach almost 50% whenever the material parameter increases by the amount of 0% regardless of loading types Figures 4 and 5). Note that for simple tension, this drop does not depend on the prescribed stretch ratio. This independency is also observed by Saravanan & ajagopal 003) for stress. To conclude, it can be said that results of this theoretical study highlight the great influence of the variation of material properties on fatigue life. So, such variations should be measured or estimated to improve the design of rubber parts. 4
.5 Simple tension...4.3 Simple Tension.. a) 38 39 40 4 4 43 44.5 Simple Torsion...05. 38 39 40 4 a) 4 43 44.4.3 Simple Torsion.. 38 39 40 4 4 43 44 b) Figure 4: Drop in fatigue life due to linear variation a) simple tension with λ = and b) simple torsion with θ = 0. b).05. 38 39 40 4 4 43 44 Figure 6: Drop in fatigue life due to sinusoidal variation a) simple tension with λ = and b) simple torsion with θ = 0..5 Proportional tension/torsion...4.3 Proportional tension/torsion.. a) 38 39 40 4 4 43 44.5 Non proportional tension/torsion...05. 38 39 40 4 a) 4 43 44.4.3 Non proportional tension/torsion.. 38 39 40 4 4 43 44 b) Figure 5: Drop in fatigue life due to linear variation a) proportional tension/torsion with λ =, θ = 0 and b) non-proportional tension/torsion with λ =, θ = 0, φ = 90..05. 38 39 40 4 b) 4 43 44 Figure 7: Drop in fatigue life due to sinusoidal variation a) proportional tension/torsion with λ =, θ = 0 and b) non-proportional tension/torsion with λ =, θ = 0, φ = 90. 5
EFEENCES Abraham, F., T. Alshuth, & S. Jerrams 005. The effect of minimum stress and stress amplitude on the fatigue life of non strain crystallising elastomers. Mat. Design 6, 39 45. André, N., G. Cailletaud, &. Piques 999. Haigh diagram for fatigue crack initiation prediction of natural rubber components. Kautsch. Gummi Kunstst. 5, 0 3. Andriyana, A. 006. Définition d une nouvelle grandeur prédictive pour la durée de vie en fatigue des matériaux élastomères. Ph. D. thesis, Ecole Centrale de Nantes, France. Andriyana, A. & E. Verron 007. Prediction of fatigue life improvement in natural rubber using configurational stress. Int. J. Solids. Struct. 44, 079 09. Cadwell, M.,. Merril, C. Sloman, & F. Yost 940. Dynamic fatigue life of rubber. Indus. and Engng Chem. ), 9 3. Green, A. E. & J. E. Adkins 960. Large Elastic Deformations. Oxford: The Clarendon Press. Mars, W. V. 00. Multiaxial fatigue of rubber. Ph. D. thesis, University of Toledo. Maugin, G. A. 993. Material Inhomogeneities in Elasticity. London: Chapman and Hall. Maugin, G. A. 995. Material forces: Concepts and applications. Appl. Mech. ev. 485). Maugin, G. A. 00. Material mechanics of materials. Theor. Appl. Mech. 7,. oberts, B. J. & J. B. Benzies 977. The relationship between uniaxial and equibiaxial fatigue in gum and carbon black filled vulcanizates. Proceedings of ubbercon 77, vol.,..3. Saintier, N., G. Cailletaud, &. Piques 006. Crack nucleation and propagation under multiaxial fatigue in a natural rubber. Int. J. Fatigue 8, 6 7. Saravanan, U. & K.. ajagopal 003. On the role of inhomogeneities in the deformation of elastic bodies. Math. Mech. Solids 8, 349 376. Verron, E. & A. Andriyana 007. Definition of a new predictor for multiaxial fatigue crack nucleation in rubber. J. Mech. Phys. Solids To appear). 6