Crack Growth or Continuum Damage?

The topic of whether to use a crack growth method or a continuum damage method for product fatigue and durability assessment has long been debated. Oftentimes, experts will recommend using a continuum damage approach in the initial phase, when no noticeable cracks are present, and then transition to a crack growth analysis when damage has reached a certain level where cracks are likely to appear.  In other applications, most of the product’s life is consumed in the crack or crack growth initiation phase, so a continuum damage method is deemed most appropriate.  There are also cases where products are in service with known detectable cracks; in this case fracture mechanics and crack growth analysis is employed to predict how fast the crack will propagate and when it will reach a critical size.

The simplest continuum damage analysis uses Wöhler curves, or S-N diagrams and Palmgren-Miner’s rule.  The S-N diagrams are built by running fatigue tests on un-cracked dumbbell specimens at various stress amplitudes, S, and measuring the number of cycles to failure, Nf. Typical S-N diagrams are shown in Figure 1 [1].  The quantity Sf is the Endurance Limit (or Fatigue Limit), below which no failure is predicted to occur.

 

Figure 1. Typicaly S-N Diagrams [1]

A linear damage rule like the Palmgren-Miner rule states that the amount of damage due to a certain number of cycles, ni, at a certain stress amplitude, Si, is a simple linear ratio compared to the number of cycles to cause failure at that stress amplitude, or

(1)

The incremental amount of damage can then be summed over different blocks of cycles at different stress amplitudes to predict failure when

(2)

One of the limitations of this approach is that sequence effects, for example going from a high-to-low stress amplitude vs. going from a low-to-high stress amplitude is not accounted for. Stated another way, the rate of damage accumulation does not depend on the current state of damage. There also tends to be a large amount of scatter in the results.  In finite element implementations, the amount of damage is tracked towards failure, and damage can be included as a state variable in the constitutive law to allow the stiffness to evolve as a function of damage.

The Endurica methods of fatigue analysis combine fracture mechanics, crack growth, and continuum damage methods. In most materials, there are crack precursors on the micron, or sub-micron level that serve as crack growth initiators. Filled elastomers are known to have many discontinuities at the micron level due to, for example, voids filled with air, agglomeration of fillers or clumps of additives.  These are treated as an initial “pre-cursor” crack with the size c0 with typical values between 10 and 100 microns. Crack growth analysis is used to predict the number of cycles, or number of repeats of a block of cycles until the crack reaches a length indicative of the end of life of the product or component.

Rather than using stress as the driver for damage as in the SN diagram, a fracture parameter called Energy Release Rate, or Tearing Energy is used as the driver for crack growth rate.  An example plot is shown in Figure 2.

The analogy to the Endurance Limit in the S-N diagram is the Intrinsic Strength, T0, below which no crack growth is predicted.  The power-law portion of the plot with slope “F” can be expressed as

(3)

 

where rc is the crack growth rate when T = Tc, the Critical Tearing Energy.   In metals, this is termed a Paris Law, in elastomers, it is the Thomas Law [2].

The damage rate in this case is the crack growth rate, dc/dN. Also, the “damage” is tracked as the predicted length of a growing crack.  The summation of the damage over a given set of cycles can be written as

 

(4)

 

The Tearing Energy in a single edge cracked tension specimen is given by

 (5)

 

where W is the strain energy density far from the crack and k is a constant depending on strain level. In a general three-dimensional state of deformation, Endurica uses the Cracking Energy Density, Wc such that,

(6)

In each of these cases, the Tearing Energy, and thus the crack growth rate is predicted to depend on the crack length, c.

Combining equations 6 and 3, we see that the damage rate, dc/dN, in this analysis, will depend on the current state of damage, c, and thus be able to represent sequence effects as part of the analysis.

In the finite element implementation with the Endurica software, there is typically no explicit crack in the FEA model. Thus the calculation of damage in the form of a growing crack is like a continuum damage approach on the macro-scale.  A co-simulation workflow is also available where the stiffness of each element in the FEA model evolves with the calculation of crack length in each element.

The Endurica analysis methods can be viewed as a continuum damage method on the macro-scale, while using fracture mechanics and crack-growth analysis on the micro-scale.  The use of fracture mechanics provides many advantages including a well-developed and validated theory for elastomers, less scatter in fatigue experiments, nonlinear damage evolution and sequence effects, and the easy ability to include many other aspects such as temperature, aging, and strain crystallization.

References

[1]        Stephens, R. I., Fatemi, A., Stephens, R.R., and Fuchs, H.O., Metal Fatigue in Engineering, 2nd edition, John Wiley & Sons, 2001.

[2]        Thomas, A.G., “Rupture of Rubber  IV. Cut Growth in Natural Rubber Vulcanizates,” Journal of Polymer Science, Vol 31, pp 467-480, 1958.

 

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Introducing Tom Ebbott, the new Vice President at Endurica!

Thomas G. Ebbott, Ph.D. Vice President Endurica LLC

Hello everyone. I am really excited to be writing this as the newest member to Team Endurica! I am really enjoying my on-boarding with Will and the team thus far. I continue to learn all the capabilities that the Endurica software has to offer, along with all the services that Endurica the company offers. I’m looking forward to using my knowledge and experience with modeling and simulation combined with expertise in fatigue and fracture in polymers to bring value to Endurica’s customers.

Endurica’s software and services enable customers to monitor, predict and improve the endurance of products. This has a positive impact on many of today’s contemporary questions. For example, for sustainability, customers need to evaluate the durability impact of using a material with a more sustainable source, or one with better recyclability or re-useability in place of an existing material. Even re-designing a component to use less material, or to last longer is more sustainable. For electric vehicles, many of the elastomeric components are called on to carry higher loads and higher torques in the case of tires. And, for fleet operations, Endurica can be used to monitor the health of elastomeric systems and predict when maintenance will be needed.

I feel I have a good background to help both Will and the team at Endurica as well as Endurca’s many and wide-ranging customers. As many of you know, I recently retired from Goodyear after nearly 36 years with that great company. While I was at Goodyear, I worked with many wonderful and capable people, and I was fortunate to have many fulfilling experiences and roles. Some include developing fundamental technology, developing products–specifically Aviation Tires and Retreads, various people leadership roles, and finally a high-level technical leader role responsible for technical strategy. While at Goodyear, I was able to publish several papers on topics such as fracture mechanics of rubber in tires, temperature distribution and rolling resistance prediction for tires, crack growth in twisted rubber disks, and continuum damage analysis of cord-rubber structures. I served on The Tire Society Executive Committee for 8 years as Treasurer. One of my long-term contributions at Goodyear was to the 30-year partnership with Sandia National Laboratories.

For my formal training, I spent 10 years at the University of Wisconsin-Madison that culminated in a PhD in Engineering Mechanics. My masters work focused on structural dynamics while my PhD research was on crack growth in polyethylene. The application of my PhD work was for the durability evaluation of natural gas distribution pipelines. The crack growth evaluation in (high density, high molecular weight) polyethylene required development of viscoelastic material laws and characterization as well as crack growth measurement systems, means to measure strain distributions, and use of viscoelastic crack growth theories.

On a personal note, my wife Sheri and I have two adult children. Amanda is teaching 2nd grade at a school near Columbus, OH, and Zachary is a junior pursuing a Finance degree at Regis University in Denver. One of my passions is flying, and I’ve had my private pilot’s license for many years. One of my most memorable flying trips was to New Mexico and Colorado. The photo shows my plane with the sun rising over the Sandia mountains in Albuquerque, NM.

I’m looking forward meeting and talking with Endurica’s customers in the coming months and learning about their needs and challenges concerning the use of elastomers and polymers for component design.

 

 

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