# Failure theories and uniaxial testing

• chandran
In summary, the stress in the structure is compared to the stress in the uniaxial tension test. The validity of this comparison is dependent on the correspondence between the two stresses.
chandran
in all the failure theory we compare the stress in the structure to the yield
stress in the uniaxial testing. How this is valid?

for example in a structure the max normal stress is sigma x. and we compare this with the maximum normal stress in a uniaxial tension test.

in vonmises also we compare the distortion energy/unit volume and the
corresponding distortion energy/unit volume in a uniaxial testing

This is an area in which my company does a lot of work, basically doing predictive analysis to determine how a structure behaves under static and dynamic loading, or when a structure fails, usually under dynamic loading.

We use both the conventional approach to failure analysis, i.e. as defined by code for a particular structure, which may be a comparison of maximum stress to uniaxial yield stress, or as you mentioned comparing distortion energy/unit volume with the 'critical' distortion energy/unit volume. I believe your term 'distortion energy/unit volume' is what is commonly referred to as 'Strain Energy Density' (SED) and this is usually compared to the 'Critical Strain Energy Density' (CSED).

Yield Strength is a bulk material property, whereas one normally calculates (predicts) stresses on a local level. The uniaxial tensile test is common and accepted method of determining yield strength.

The validity of comparing the stress in the structure to the yield stress in the uniaxial testing is determined by comparing the results of predictive analysis (and the constitutive model) with the results of the yield stength data. What this means is a comparison of the strains. Remember one can only measure load and dimensional changes, so stress is a derived term.

Unless fatigue or some flaw is involved, failure occurs when the local stress exceeds the ultimate strength (and local strain exceeds uniform elongation). Then one must differetiate between strain-controlled and load-controlled straining. In strain-controlled, the stress in a piece of structure may exceed ultimate strength, but if the strain is constrained it will not go to failure. On the other hand, if the situation is load-controlled, i.e. the load is not removed and strain is unconstrained, the system will proceed to failure.

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chandran said:
in all the failure theory we compare the stress in the structure to the yield
stress in the uniaxial testing. How this is valid?

for example in a structure the max normal stress is sigma x. and we compare this with the maximum normal stress in a uniaxial tension test.

in vonmises also we compare the distortion energy/unit volume and the
corresponding distortion energy/unit volume in a uniaxial testing

You have, indeed, hit the nail on the head - there isn't a sound theoretical reason why the uniaxial stress-strain curve should be applicable to 3-D loading simply by changing the stress axis to Mises stress and the strain axis to effective plastic strain. But it works - at least for most common structural materials.

Interestingly, the 'strain energy density' interpretation of the Mises yield criterion (originally proposed 50 years earlier by Maxwell in a letter to Kelvin) is a later interpretation - Mises just chose the simplest mathematical expression that fitted the basic requirements of any yield function.

...yeah, I hope he can appreciate the complexity & intricacies of the problem . It's fashinating reading from time to time papers dealing with testing of the very hypotheses of classic continuum theories, although one could say that the field is more active than ever with the work going about with respect to polycrystalline plasticity, dislocation dynamics, damage mechanics etc. modeling aspects.

I have been mulling over the original question and responses.

First of all, the uniaxial test is simple and inexpensive, especially if one wants lots of data. In that sense, it becomes an empirical relationship.

The validity will certainly depend on the state of stress one wants to analyze, uniaxial vs biaxial or triaxial stress states. The test may be 'more' valid for fcc and bcc materials, but perhaps not for hcp materials, especially those with residual cold work (texture is implicit).

The validity will depend on the constancy or correspondence of the relationship between the uniaxial and maximum stresses. One then must formulate a function, usually dependent on the geometry of relevant stress field (e.g. stress concentration factors), which relates maximum stress to the uniaxial stress.

I submit these thoughts, although incomplete, for further consideration.

The topic continues to be interesting ... some thoughts :

... from one end we've the experimental procedures, and in that sense when a uniaxial tensile test is assessed using a continuum mechanically viable description (i.e. true stress-strain) for most purposes we're squeezing the most information out of it and in a consistent manner (meaning that we acknowledge the limitations of the methods and constitutive theories we've applied in its interpretation).

... from other end we've the usage of the material properties and materials laws we've 'calibrated' this way. Sure, the fundamental derivation of e.g. J2 - plasticity is based on a hypothesis as usual, one which has been verified to be a good one, but of course has its limitations. Those limitations arise from material behavior, i.e. when we're talking about smooth continuum plasticity in no matter what dimension and when the scale we're investigating can be considered macroscopic, we're on a relatively safe ground. When we decrease the size - scale the basic theories such as von Mises break down, or when something happens not covered by the constitutive models, or which a good example would be material damage e.g. arising from a localized triaxial stress state etc. ... the number of such limitations is pretty staggering.

... this is one "prime time" issue undergoing lots and lots of research ...
:!) ... and when worked upon in-depth gets as complex as it gets.

## 1. What are failure theories in material science?

Failure theories in material science are mathematical models used to predict the failure of materials under different types of loading conditions. These theories take into account factors such as material properties, stress concentrations, and loading conditions to determine the point at which a material will fail.

## 2. What is uniaxial testing?

Uniaxial testing is a type of mechanical testing used to determine the mechanical properties of a material, such as strength and stiffness, by applying a single load in a single direction. This type of testing is commonly used to test materials under tension or compression.

## 3. How are failure theories and uniaxial testing related?

Failure theories and uniaxial testing are closely related as failure theories are often used to predict the behavior of materials during uniaxial testing. These theories can help engineers determine the maximum load a material can withstand before failure and how it will behave under different loading conditions.

## 4. What are the different types of failure theories?

There are several different types of failure theories, including stress-based theories, strain-based theories, and energy-based theories. Each theory takes a different approach to predicting failure and may be more appropriate for certain types of materials or loading conditions.

## 5. How accurate are failure theories in predicting failure?

The accuracy of failure theories in predicting failure depends on several factors such as the complexity of the material and loading conditions, and the assumptions made in the theory. While these theories can provide a good estimation of failure, they may not always accurately predict the exact point of failure in real-world situations.

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