Characterizing elastomers for fatigue life prediction.
Fortunately, much is known about the physics of durability in elastomers, and modern solutions for the characterization and numerical simulation of fatigue performance provide unprecedented ability to forsee the consequences of design changes. Where traditional experiment-only approaches limit engineers to doing fatigue evaluation of only one or two design iterations at the end of a product development cycle (after most of the design decisions have already been made), modern approaches enable engineers to consider fatigue effects before ever building the first prototype (ref. 1).
A prerequisite to analyzing fatigue performance is to know the fundamental material parameters governing the development of damage. At very early stages of design conceptualization, estimates of these parameters can be obtained by consulting previously reported measurements. A great number of studies have been made for many different polymer types (refs. 2 and 3), filler and cure systems (ref. 4), and so on. On the other hand, since rubber compounds are typically very highly customized to each application, it is often justified to invest in measurements of a specific compound or set of candidate compounds.
With modern simulation software, it is possible not only to compute the operational history of stress and strain in a part (using a finite element model), but it is also possible to compute the damaging effects of a given load history and the resulting fatigue life (using fatigue life prediction software such as Endurica CL or fe-Safe/Rubber). The purpose of this article is to review the basic parameters that must be measured in order to use fatigue life prediction software.
Fatigue failure is the culmination of several simultaneous processes occurring in an elastomer, each of which has corresponding measurement and modeling options. Endurica's Fatigue Property Mapping service organizes the various characterization experiments according to purpose, so that, depending on the requirements of the application, the most relevant tests can be selected. The available tests are organized into the following six categories: 1) hyperelastic behavior, 2) core fatigue behavior, 3) intrinsic strength, 4) extended life, 5) nonrelaxing behavior, and 6) thermal behavior. Any of the first five modules can be executed for any single temperature in the range -40[degrees]C < T < 150[degrees]C. The last module is useful when more than one temperature must be considered.
A prerequisite to fatigue analysis is to run a Finite Element Analysis to determine the strain history at each point of the rubber part. The analyst must specify the stress-strain behavior of the elastomer, typically using a hyperelastic model to describe rubber's nonlinear stress-strain behavior (ref. 5). The analyst may also specify parameters to describe cyclic softening (i.e., the Mullins effect) (refs. 6 and 7).
The hyperelastic and Mullins parameters are obtained through experiments in which specimens are deformed cyclically in each of three distinct modes of deformation, including simple tension, planar tension and biaxial tension. The results of these experiments are then used to derive a curve fit, such as the one shown in figure 1. In this case, a third order Ogden model has been used to describe the monotonic stress strain curve (shown as solid lines). The Mullins model has been used to describe the steady state cyclic curves (dashed lines). Stress-strain curves from simple tension (blue), planar tension (green) and biaxial tension (red) are shown. Note that, for purposes of deriving the hyperelastic fit, an adjustment is made so that the strain at zero stress always reflects the most recent undeformed configuration. This practice ensures that all stress-strain curves begin at zero, zero, as required by the hyperelastic model. The Ogden model has the strain energy density functional form given below.
For the Abaqus finite element code, the Ogden strain energy potential W, containing N terms, is defined as a function of the principal stretches [[lambda].sub.j] with the material parameters [[mu].sub.i], and [[alpha].sub.1] according to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
The Mullins law is defined via the softening function [eta]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
[??] (J/[m.sup.3]) is the strain energy density on the primary (monotonic) stress-strain curve.
[I.sub.1] (unitless) is the instantaneous value of the first invariant (measures state of deformation).
[I.sub.1,max] (unitless) is the maximum prior value of the first invariant (memory of prior worst state of deformation).
r (unitless), m (J/[m.sup.3]) and [beta] (unitless) are the material constants determined via curve fitting.
If the stress on the monotonic stress-strain curve is [??](Pa), then the stress on the unloading curve after softening due to prior deformation is given by [sigma] = [eta][??].
Core fatigue behavior
Endurica's core fatigue experiments are aimed at obtaining the most basic parameters that are required for any fatigue analysis. There are four experiments in this module. These include two strength experiments: 1) an edge-cracked planar tension experiment (top left in figure 2), and 2) a strain-to-break experiment on a simple tension strip having no intentional precrack. The planar tension experiment measures the elastomer's critical fracture energy [T.sub.c] (J/[m.sup.2]). Results from these experiments are used to set up corresponding fatigue experiments, employing the same specimen types. The fatigue experiments are operated under fully relaxing cycles (i.e., the minimum load of each cycle is zero). The fatigue experiments measure: 1) the fatigue crack growth rate curve dc/dN = r(T) (mm/cycle) (bottom left of figure 2), and 2) the number of cycles to failure of a simple tension strip operating at fixed strain (bottom right). The fatigue crack growth rate curve is derived from digital images of the crack as it grows through the specimen. The digital images can later be overlaid to view the progression of the crack, as shown in figure 2, top right (axes show dimensions in mm). A power-law relationship (shown in bottom left) is then fit to the results, with slope F, and an intercept at the point ([T.sub.c], [r.sub.c]).
By performing these two fatigue experiments together, the initial size of crack precursors in the elastomer's microstructure (ref. 8) can be derived, and the material's strain-life curve can be computed and plotted. The precursor size is derived by finding the value of [c.sub.0] in the following calculation for the nucleation life N (cycles), which results in the best agreement between the observed points on the strain-life curve, and the strain-life curve computed (solid curve in bottom right of figure 2) from the fatigue crack growth rate curve r(T) measured in the crack growth experiment:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
The advantages in operational efficiency (many different operating conditions can be probed with a single test specimen) and test repeatability of the fracture mechanical test methods (since crack size is controlled, results are less scattered than nucleation experiments in which crack precursor sizes vary naturally) are strong when combined with a few crack nucleation measurements and with computation of the crack nucleation behavior over the range of operating conditions. The combination allows crack precursor size to be accurately calibrated, and it enables users to approach engineering design problems using both the crack nucleation (i.e., S-N) and crack propagation methods.
When an application targets a fatigue life exceeding one million cycles, then the intrinsic strength T0 of the rubber should be measured. The intrinsic strength is the minimum energy release rate required for causing any amount of crack growth (ref. 9). As shown in figure 3 (left), the intrinsic strength is a lower bound on the fatigue crack growth rate curve, so that when peak loads stay below the intrinsic strength, indefinite life may be expected of the elastomer. At least, the life will not be limited by crack development (although it may still be limited by aging considerations). Physically, the intrinsic strength represents the energy required to break all polymer chains crossing the crack plane, and excluding all viscoelastic energy dissipation in the material.
There are several methods that can be used to measure intrinsic strength, but the most convenient seems to be the cutting method proposed originally by Lake and Yeoh (ref. 10). In this method, depicted in figure 3 (right), the usual crack tip dissipations are avoided because the highly sharpened blade applies load directly to polymer chains at the crack tip. Strain in the experiment is measured via laser extensometer (i.e., the red 'beam' in the photo). The cutting force is measured via a load cell attached to the blade, and can be related to the intrinsic strength through the relation:
[F.sub.c] = [KT.sub.0] = Wh + f/t (4)
[F.sub.c] (J/[m.sup.2]) is the total cutting energy in the unstrained state with zero crack face friction.
W (J/[m.sup.3]) is the strain energy density stored in statically strained material ahead of the crack tip.
h (mm) is the specimen gauge height.
f (N) is the cutting force on the blade.
t (mm) is the thickness of the specimen.
K (unitless) is a constant of proportionality reflecting blade sharpness that is calibrated by means of a standard material.
Extended life and aging behavior
If aging is likely to play a major role in determining part durability, it becomes necessary to consider how the 1) stiffness, 2) critical fracture energy [T.sub.c] and 3) intrinsic strength [T.sub.0] evolve with heat history. By aging specimens at three different temperatures, for three different aging periods (3, 10 and 30 days), the Arrhenius relationship (shown in the bottom plot of figure 4, with parameters activation energy [E.sub.aTc], gas constant R and temperatures T and Tref) can be calibrated and used to describe and estimate how heat history will influence part life (ref. 11).
Under conditions where the minimum load of the cycle never unloads to zero (i.e., a non-relaxing load), some elastomers (natural rubber is the most famous example) exhibit a phenomenon known as strain crystallization. The fatigue performance of a strain-crystallizing elastomer under a non-relaxing load is vastly improved relative to an otherwise similar non-strain-crystallizing elastomer. For cases where designers rely on the strain crystallization effect to obtain durability under non-relaxing loading, the effect of strain crystallization on crack development must be measured.
The most direct and efficient approach to this measurement is to use a "crack arrest" procedure. In this procedure, a crack begins growing under fully relaxing cycles, and then the minimum strain is gradually increased while holding constant the maximum strain. This solicitation history results in a gradual increase of strain crystallization during the test, and a corresponding gradual decrease of the fatigue crack growth rate. The results of this experiment are presented as a plot (figure 5, top), showing for a given non-relaxation ratio R = [T.sub.min]/[T.sub.max], the value of the strain crystallization function x(R). x(R) is related to the powerlaw law slope F of the fatigue crack growth rate law at R = constant through the following relationship. Here, [F.sub.0] is the powerlaw slope of the crack growth rate curve under fully relaxing (R = 0) conditions:
x(R) = 1 - [F.sub.0]/F(R) (5)
Once the crystallization function has been measured, this information can be combined with results from the core module to compute how the fatigue life will vary with any combination of mean strain and strain amplitude (ref. 12). A color contour "map," or Haigh diagram, can then be computed showing fatigue performance over a wide range of conditions (figure 5, bottom).
Rubber is a thermal insulator, so that some rubber parts operate with significant thermal gradients caused either by dissipative self-heating, or by large differences in boundary temperature. Rubber's fatigue behavior is strongly dependent on temperature, so if the analysis involves large thermal gradients, it is important to measure the parameters needed to make a thermal Finite Element Analysis (figure 6, top). The recommended measurements include heat dissipation as a function of strain amplitude, thermal conductivity, specific heat and mass density. Also, they include measurement of the dependence of strength (figure 6, bottom) and fatigue parameters on temperature.
Comments on testing strategy
Fatigue experiments are often among the most difficult and costly experiments that developers make, so effective testing strategies are essential. One common risk in fatigue testing programs is that poorly chosen operating loads too easily result either in: 1) a rate of damage accumulation too small to produce observable failure during the time allocated for the experiment, or 2) a too high rate of damage that consumes allocated test specimens without having probed to the limiting observable rates of damage accumulation. Although it is beyond the scope of this article to present strategies in detail, suffice it to say that it is possible to successfully navigate the tension between these opposing risks to reliably obtain high quality measurements, over the broadest possible range of conditions, with a fixed total budget of test machine time.
Another common challenge is the variability intrinsic to fatigue measurements. The above described characterization approach mitigates against variability in several ways. First, the use of fracture mechanical methods is advantageous because crack size is directly controlled and measured. In contrast, crack nucleation (i.e., S-N style) measurements tend to show more scatter because crack precursors (which are not directly controlled for or measured) tend to vary in size somewhat from one specimen to the other. Second, variability can also be reduced by applying proper analysis methods to the measurements derived from an experiment. For example, raw numerical differentiation of a plot of crack length vs. cycles to obtain crack growth rate tends to amplify noise in the measurement. A much more reliable crack growth rate can be derived from the same set of experimental measurements by first fitting an analytic function to the history of crack length, and then differentiating the analytical function to obtain the crack growth rate.
The parameters obtained through the testing modules described in this article provide an accurate and full specification of an elastomer's fatigue behavior. The specification is based in sound physics, and it is fully compatible with modern fatigue life prediction software like fe-Safe/ Rubber (available from Dassault Systemes) and Endurica CL (available from Endurica LLC). Endurica's Fatigue Property Mapping service offers a convenient and cost-effective way to obtain a wealth of information about the material's fatigue performance. Using these methods, material and product developers now can anticipate the impact on durability of their design decisions much earlier than ever before possible, greatly reducing the risk of durability issues that are discovered only at the end of the development process.
(1.) W.V. Mars, "Fatigue life prediction for elastomeric structures, " Rubber Chemistry and Technology, 80 (3), pp. 481-503 (2007).
(2.) G.J. Lake and P.B. Lindley, Rubber Journal, Vol. 146, No. 11, pp. 30-39 (1964).
(3.) P.B. Lindley, International Journal of Fracture, Vol. 9, pp. 449-461 (1973).
(4.) S. G. Kim and S.H. Lee, Rubber Chemistry and Technology, Vol. 67, 649 (1994).
(5.) A.H. Muhr, "Modeling the stress-strain behavior of rubber," Rubber Chemistry and Technology, 78 (3), pp. 391-425 (2005).
(6.) R.W. Ogden and D.G. Roxburgh, "A pseudo-elastic model for the Mullins effect in filled rubber," Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455 (1988), pp. 2,861-2,877 (1999).
(7.) W.V. Mars, ".Evaluation of a pseudo-elastic model for the Mullins effect," Tire Science and Technology, 32 (3), pp. 120145 (2004).
(8.) A.N. Gent, P.B. Lindley and A. G. Thomas, "Cut growth and fatigue of rubbers. I. The relationship between outgrowth and fatigue," Journal of Applied Polymer Science, 8, 455 (1964).
(9.) G.J. Lake and A.G. Thomas, "The strength of highly elastic materials, "Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 300 (1,460), pp. 108119 (1967).
(10.) G.J. Lake and O.H. Yeoh, "Measurement of rubber cutting resistance in the absence of friction," International Journal of Fracture, 14 (5), pp. 509-526 (1978).
(11.) R.A. Pett and R.J. Tabar, "The oxidative aging of a compounded natural rubber vulcanizate," Rubber Chemistry and Technology, March 1978, Vol. 51, No. 1, pp. 1-6.
(12.) W.V. Mars, "Computed dependence of rubber's fatigue behavior on strain crystallization," Rubber Chemistry and Technology, 82 (1), pp. 51-61 (2009).
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|Date:||Mar 1, 2015|
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