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Cracking growth prediction for a structure with creep and plasticity deformation

WANGMING LU's picture

My question is how we can predict the crack growth rate and direction for a structure under cyclic loading in such a way that both plasticity and creep has been built in it.

Let's say we have a simple beam that has gone through a number of thermo mechanical cycles. The cycle, for simplicity, is composed of two points. The first point has high mechanical and thermal loading, say, 2000F. Another point has no mechanical loading, and the beam is under uniform room temperature.  Often we see cracks initiated in the beam. My goal is to predict the crack growth rate and direction as the number of above defined cycles builds up.

I guess the first thing is to figure out a reasonable way to calculate stress intensity, or K, value. For now, I have no motivation to make this problem too complicated. Can I just measure the crack length and opening displacement from a cracked beam under room temperature and then use LEFM rationales to calculate K value? Here my assumption is that the crack is growing at mode I. We know even under the room temperature without loading applied on, the beam has both plastic and creep deformation in it due to the combination of high mechanical loading and high temperature. If the approach is correct, then I can build up a cracked FE model, and then apply the displacement on the edge of the crack to mimic the crack opening, and then run linear analysis to get K value. Quite simple.

If my approach is incorrect, what is the simplest way to get K value so I can predict the crack growth rate and direction using some commercially available finite element programs such as ANSYS?


Zhigang Suo's picture

You have raised a fundamental problem, which may not have been fully solved.  The problem is also of great practical significance.  I'd love to hear from other people on how they think about your question.  Here are notes I took when I read your post.

  1. Because the stress at the tip of a crack is high, the field at the crack tip is nearly always inelastic.
  2. When a material undergoing inelastic deformation, the field around the crack tip cannot be described by linear elasticity.  Consequently, one cannot calculate the stress intensity factor K at the crack tip.
  3. The combination of 1 and 2 may sound like to say that K-field is useless.  This impression is incorrect; see 4 and 5 below.  
  4. If the size of the crack-tip inelastic zone is small compared to the size of the specimen, the field outside the inelastic zone can be described using linear elasticity.  This condition is known as the small-scale yielding condition.  Under this condition, one may characterize the external loads with the stress intensity factor K.  In such a calculation, one simply neglects inelastic zone, and treat the whole sample as elastic.  This idea was due to Irwin.
  5. To advance the crack under the small-scale yielding condition, one needs to prescribe a kinetic model that relates the velocity of the crack to K.  A commonly used model for metals and polymers under cyclic loads is the Paris law.  Another commonly used model is based on the process of stress corrosion.
  6. If the size of the inelastic zone is comparable to the size of the sample, i.e., if the entire sample undergoes inelastic deformation, then the K-field is simply irrelevant.  This condition is known as the large-scale yielding condition.  One needs some other methods to predict the growth of the crack.
  7. Under the large-scale yielding condition, one popular method in recent years is to represent the process of fracture by a traction-displacement relation.  In this approach, the boundary-value problem is well described, although highly nonlinear.  Given enough computational resources, one can advance the crack.  One well known paper using this approach is Tvergaard, V., Hutchinson, J.W., " The Relation Between Crack Growth Resistance and Fracture Process Parameters in Elastic-Plastic Solids."J. Mech. Phys. Solids,40, 1377-1397 (1992).

Again, I'd love to hear from other people on this significant problem of crack growth in the presence of large-scale inelastic deformation.

WANGMING LU's picture

Thanks a lot, Zhigang,


You are right - the problem we are discussing has huge practical significance. Image the buckets in a heavy-duty gas turbine. The loading for a bucket is similar to what I have described for the beam, high temperature like 2000 plus F, high loading caused by pressure and centrifugal force, among others. It is subject to cyclic loading. Right now, gas turbine industry regularly does a NDE such as bore scope inspection to see if there is a crack, if yes, typically people either take risk to continue using it until the next big outage that gives a chance to replace the cracked bucket with a new, or reconditioned one, or take the cracked bucket out immediately if the “experience” or “engineering judgment” tells that a catastrophic event is imminent if leaving the cracked bucket in. Unfortunately gas turbine outage implies huge economic compromise. So, the ideal case will be to minimize the gas turbine outage. This is why it has huge financial implication to predict the bucket crack growth rate and direction so people can know the residual life of this expensive bucket - the cost of one bucket, for part only, is almost two times of that of a BMW car!


Now, looking back your points above. Obviously, a typical bucket will be subject to both plastic and creep deformation, with plastic deformation developed almost immediately after service – design people are struggling to avoid this, but found it was just difficult. The creep deformation is a function of time and can be big, say, a creep strain of 2% after only 5,000 hours. Typically the first scheduled inspection or replacement time is about after running the gas turbine for 24,000 hours. But many times, by much earlier time, say, 5,000 hours, the cracks should usually have already initiated.


With all this being said, you can see our problem unfortunately falls in the one which you said has not been fully resolved, it is about crack prediction for a structure with plastic and creep deformation. The crack tip inelastic region is big; the rest of the structure is of inelastic deformation as well. It is a large-scale yield problem.


I will read carefully the paper you have given above. Before that, I have a couple of points I want to further clarify with you:


 (1). If a structure is under large-scale yielding condition, the K value is meaningful. People cannot rely on K value and Paris law to predict the crack growth rate, is this correct?


(2). Some other methods such as you referred in point 6 and 7 have to be used to predict the crack growth rate. What is your understanding about their industrial application? As you know, the K value and Paris law have been used by many industries to predict the crack growth rate.


(3). We understand some other method you pointed could be difficult, and/or, immature at this time. So, how about the following method? Let say we run a creep analysis for an uncracked structure. Therefore, at any time point, we will get the total strain composed of elastic, plastic and creep strain component in this structure. Let's assume that the crack was initiated at the time point T. We then map the plastic and creep strain from the uncracked model to a cracked model (we can insert a crack to the uncracked model, and there are a number of computer programs capable of doing this), and then we run a typical linear analysis to get K value. After that, we can use Paris law to predict the crack growth rate.


I understand the K value calculated by above method is based on the assumption, at least, that the inelastic deformation for the cracked model is the same as that for the uncracked model. But if we focus on the small crack domain, I may think that this assumption could hold. Also, I think the K value here will be still different, but not sure by how much, from what is calculated by considering the large-scale inelastic deformation. Do you have any sense how much this difference will be?


I am an engineer, and also unfortunately, my background is not in the fracture mechanics although I love this field. So, I typically try to simplify the problem based on engineering judgment, and a number of "dirty" assumptions. The goal is to make some assumptions and get a quick answer reasonably right within a reasonable timing frame.


Thanks again and regards,


Wangming LU

Zhigang Suo's picture

Dear Wangming:  Thank you for your detailed description of the problem.  Let me get to your list point by point.

  1. Under large-scale yielding, nowhere in the structure is described by the K-field, which is an elastic field.  Thus, K is not applicable.  Paris Law relied on K, and is also inapplicable.
  2. Bridging models (or cohesive-zone model) have been implemented in commercial finite element code, such as ABAQUS.  The basic ideas were around in 1960s, and are now widely used to solve practical problems.
  3. The logic of the procedure described by you is unclear to me.  Can you clarify as to why you think the procedure might work?

Thank you for the discussion.  Have a good day!

Dear Zhigang,


Thank you very much for your interesting comment. It is very useful for my research on Carck spacing prediction for Reinforced Concrete.




Zhigang Suo's picture

Dere Phong:  For a reinforced concrete, does much of the saample remain elastic?  Do you have a case of large-scale bridging of a cack?  If so, here is a review on the mechanics of bridged crack in an elastic material:

G. Bao and Z. Suo, " Remarks on crack-bridging concepts," Applied Mechanics Review. 45, 355-366 (1992).

Dear Zhigang,

Thank you very much for the helpful paper. At the first step, we focus on strain localization and crack for concrete material.

Yes, You are right. Only a very narrow strip  of material behaves nonlinearly (i.e. Strain localization, fracture process zone) while the remaining part of the material is elastic.  Actually, We are developingg a new meshless method, namely Moving IRBFN  (submmited to  International journal for numerical method in engineering) to model this problem.  



In addition to Zhigang's post, I would like to suggest the FAD for material under large-scale yielding. As I gather the FAD, it makes a correction to K (LEFM). This diagram may not be used to calculate the cracking growth prediction directly, but maybe it gives us some inspiration: can we calculate the crack growth using the corrected K?




Die ersten hundert Jahre sind erfahrungsgemaess die Schwersten!
Alles Gute!

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