### A Physics Based Model for the Ductile Failure of Metals

From the millions of miles of aging pipelines to the intricate workings of a wind turbine, metals are ubiquitous. Of paramount importance in both the design and upkeep of these materials is a predictive capability for their failure. An improved understanding of ductile failure will offer increases in efficiency, reliability, and applicability of metals and their alloys. The use of computational testing is quickly becoming a viable alternative to experimental testing procedures. These computational testing methods, besides being much cheaper, offer complete control of testing parameters and greater insight into the inner workings of the material. This work investigates one such computational model for the ductile failure of metals. The model is based on a multi-scale approach, which means that the microstructure of the material is modeled and subsequently related to larger scales of interest. The results illustrate the importance of specific microstructural phenomena and confirm that their incorporation into the model will greatly enhance its predictive capabilities.

## Qiaobing Xu

Excellent work, Geoffrey. what other model has been used to describe the failure of metal during stretching? how to use your model to describe the different mechanical property between hard metal e.g. tungsten, and soft metal e.g. gold?

## Geoffrey Bomarito

Lead PresenterGreetings Qiaobing Xu,

Thank you!

My model is aimed specifically at describing the ductile failure mechanism. This mechanism is characterized by large amounts of plasticity and is common for many structural metals (e.g., aluminum and steel). However, some metals (e.g., tungsten) are more brittle and do not fail by the same mechanism, which means that my model would not describe its failure well. For the range of metals that do undergo ductile failure, my model is simply applied by changing the matrix properties and initial void volume fraction of the micro-scale model.

There are many models used to describe the failure of metals:

-Models that specifically tackle ductile failure include: the Brown-Embury model, the Thomason (limit load) model, and modifications to the Gurson model to account for failure (most notably the work by Nahshon and Hutchinson). The advantage to using my model instead of the aforementioned models is that my model offers a robust and complex description of failure that does not use any fitting parameters (i.e., my model makes predictions based solely on the observable microstructure).

-Other models, such as the Mohr-Coulomb failure model, could be used for more brittle materials.

Additionally, the entire field of fracture mechanics is devoted to these type of problems. Specifically, linear elastic fracture mechanics are commonly used for brittle materials. Non-linear fracture mechanics are more complex and are needed for applications to ductile materials.

Let me know if you are interested in knowing more about any of the models in a more specific context,

-Geoffrey

## Aparna Baskaran

Interesting work Geoffrey. In the model that you present here, we are looking at a macroscopic elasticity theory simulation with inclusions (voids) in the bulk material. I am wondering if you can give me a feeling for the scales at which this approach is relevant and the dominant cause for failure. Presumable at some short scale, the polycrystalline nature of the material becomes relevant and I will have to take into account disclinations and quantum effects and so forth. Also, why is the simulation “multiscale”?

## Geoffrey Bomarito

Lead PresenterHello Aparna Baskaran,

Thanks!

To clarify, the macroscopic model does also include an elastic-plastic material response (based on the Gurson constitutive model).

The large scale model in general is relevant for elements on the “structural component” level. More exactly, the notched bar I show in my video and poster has dimensions in the millimeter range. Larger sizes are also relevant here. Because an underlying assumption in my macro-scale model is that many voids contribute to failure, the macro-scale model must be much larger than the void size.

The size of the micro-scale model is on the order of 10s of microns. Yes, you are indeed correct that as we approach smaller scales other atomistic effects become important. In fact, other members of my research group have investigated this more closely. They found, through atomistic simulations, that sub-micron sized voids are resistant to growth until very high stresses or temperatures are reached.

On the scale of my void models, however, other effects may be still be important. Because my micro-scale models are on the order of the grain size (at least for my test material), the isotropic plasticity model I use may be too simple. Other researchers have shown that anisotropic plasticity affects the growth and coalescence of voids. Therefore, we may consider the use of a plasticity model, which can better account for crystal orientation in the future.

Finally, I use the word ‘multiscale’ to mean that I am using simulations conducted at a micro-scale to inform my simulation at the macro-scale. In essence, my model uses information from two different simulations, each of which corresponds to a different scale. This is what I meant by ‘multiscale.’

Let me know if I can further clarify anything,

-Geoffrey

## Natalia Noginova

Geoffrey, this is interesting work and very nice presentation. Different colors in metal correspond to a different strain, right? Do you take into account of a possible self-heating? Or other temperature effects?

In principle, will the testing results depend on how fast do you break the metal?

## Geoffrey Bomarito

Lead PresenterHi Natalia Noginova,

The colors in the videos of the macro-scale model (like the cover image of my video) represent damage i.e., how close an element is to failure. You’ll notice that at the beginning of loading all elements are blue (undamaged) and that just before each element fails it is red.

We do not take into account the possibility of self heating or any other temperature effects. Our assumption is that at our slow strain rates (quasi-static) and at stresses on the order of 100MPa that temperature effects will be small.

Finally, because the plasticity models we chose for our micro- and macro-scale models are rate independent, our results assume a quasi static loading. In order for us to match experimental results that show the rate sensitivity of failure, we could change our plasticity models to a rate dependent formulation. Inertial effects of void growth would already be included in our model because of our use of an explicit dynamic FE framework. This is an option for future refinement of the model, though we hope to better match results at quasi-static rates before we venture down that path.

Thanks,

-Geoffrey

## Qi-Huo Wei

Geoffrey: nice presentation. For the FE simulations, do you assume that the elastic property of the material does not change under different loading? In addition, do defects in the materials play any role in the ductile faillure?

## Geoffrey Bomarito

Lead PresenterHello Qi-Huo Wei,

Yes, we do assume that elastic properties of the material is independent of loading (i.e., that the material response is elastically isotropic). On the macro-scale model we base this assumption on the fact that the grain size is small compared to our elements and that the grains would be randomly oriented. On the micro-scale this assumption does not hold because the size our entire micro-scale model is about the same as the grain size (specifically for our aluminum test material). But because our test material is aluminum and the elastic anisotropy factor is relatively close to 1, the assumption is still valid. The role of anisotropic plasticity is believed to be much more important; I addressed this briefly in my response to Aparna Baskaran’s question in case you are interested.

Defects do play a role in ductile failure. It is from micro-scale defects such as inclusions, precipitates, and even grain boundaries that voids nucleate. Larger scale defects, such as cracks, also effect ductile failure based on how they change loading (e.g., stress concentrations).

Thank you for the questions,

-Geoffrey

## Hyunjoon Kong

Very nice work. What is the size of voids considered in your modeling? Can you modify your modeling to estimate effects of heterogeniety in void size on material ductility?

## Geoffrey Bomarito

Lead PresenterGreetings Hyunjoon Kong,

The voids considered in our model are on the order of 10 microns.

I sure hope so! Because lack of material heterogeneity is the main source of error in our model, we hope to tackle this first.

We have done tests on unit cells with different sized voids and saw that we would need to increase the size of the voids by 400% in order to match experimental results. These tests still assumed a constant void size (for each simulation) but we think it represents a rough estimate of the effect of void size.

There are some experimental and a few computational works in the literature that lead us to believe the heterogeneous spacing of the voids is actually more important that void size effects.

We may at some point try to model cells with multiple voids (of varying sizes) at once and see how this effects our results. But first, we would like to look more closely at void spacing effects.

Thanks and let me know if you have any more questions,

-Geoffrey