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Understanding Failure Theories (Tresca, von Mises etc...) — Transcript · YTtoTranscript

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Understanding Failure Theories (Tresca, von Mises etc...)

The Efficient Engineer · 2,253 words · 11 min read · EN-ORIG

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Below is the complete, readable transcript of Understanding Failure Theories (Tresca, von Mises etc...) by The Efficient Engineer on YouTube. Read the full text, copy any part you need, or generate a transcript for any video with our free tool.

we know that when we apply loads to an object like this bracket if we keep increasing the magnitude of the load at some point the material will fail but how can we predict when static failure will occur what level do the stresses in the object need to reach for it to fail first we need to define what failure is

for ductile materials failure is usually considered to occur at the onset of plastic deformation and for brittle materials it occurs at fracture these points are easy to define for a uniaxial stress state like a tensile test they occur when the normal stress in the object reaches the yield strength of the material for ductile materials and the

ultimate strength for brittle materials

but for a more complex case of tri-axial stress predicting failure is much less straightforward in fact it's so difficult to predict that we don't have one universally applicable method instead we have to predict failure by selecting the most suitable one of a range of different failure theories each of which we know works relatively well under certain

circumstances based on experimentation because they fail in fundamentally different ways failure theories which apply for ductile materials usually aren't applicable for brittle materials and vice versa so what does a failure Theory do it's quite simple it allows us to predict failure of a material by comparing the stress state and the object we are assessing with material

properties that are easy to determine like the yield or ultimate strengths of the material that we can obtain by performing a uniaxial test the stress state at a point can be described using the three principal stresses so most failure theories are defined as a function of the principal stresses and the material strength probably the

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simplest failure theory is to say that failure occurs when the maximum or minimum principal stresses reach the yield or ultimate strengths of the material this is called maximum principle stress theory or ranking theory although it's simple to apply it's not actually a great failure theory particularly for ductile materials for reasons I'll explain later let's look at

some better failure theories for ductile materials any good failure Theory needs to be consistent with experimental observations we can make about how materials fail there is one key observation that failure theories for ductile materials need to capture which is the fact that hydrostatic stresses do not cause yielding and ductile materials let me explain a general tri-axial

stress state like the one shown here can be decomposed into stresses which cause a change in volume and stresses which cause shape Distortion stresses that cause a change in volume are called hydrostatic stresses because that is the type of stress acting on an object submerged in liquid for a hydrostatic stress configuration the

three principal stresses are always equal and there are no shear stresses for a tri-axial stress state we can calculate the hydrostatic component as the average of the three principal stresses the mechanism that causes yielding of ductile materials is shear deformation since there are no shear stresses for a state of hydrostatic stress this component can be very large

and still not contribute to yielding yielding is only caused by the stresses which cause shape distortion these are called deviatoric stresses the deviatoric component is calculated by subtracting the hydrostatic component from each of the principal stresses the hydro static and deviatoric components can be expressed in matrix form like this here we have described the stress

state using the principle stresses but we could also describe it for an arbitrary orientation of the stress element it can be helpful to use Mohr circle to understand the hydrostatic and deviatoric components of a tri-axial stress state for a hydrostatic stress configuration there are no shear stresses and so Mohr circle reduces to a

single point equal to the average of the three principal stresses shifting Mohr circle horizontally represents an increase in the hydrostatic component and increasing the radius of Mohr circle without changing the average stress represents an increase in the deviatoric component since failure of ductile materials only depends on the deviatoric component a good failure theory for ductile

materials should produce the same result regardless of where more circle is located on the horizontal axis this explains why maximum principle stress theory is not a good failure theory for ductile materials it isn't consistent with the observation that yielding is independent of hydrostatic stress to failure theories which are consistent with this observation are the

Tresca and von Mises failure criteria these are the two most commonly used failure theories for ductile materials let's start with the Tresca failure criterion which is also called the maximum shear stress theory it is named after the french engineer henri Tresca and it states that yielding occurs when the maximum shear stress is equal to the

shear stress at yielding in a tensile test this can be defined mathematically like this and graphically using Mohr circle like this this theory is consistent with the observation that hydrostatic stresses don't affect yield it doesn't care where Mohr circle is located on the horizontal axis it's common to express this theory as a

function of the principal stresses instead of as a function of the shear stresses we can see based on Mohr circle for a tri-axial stress state that the maximum shear stress is equal to the radius of the outer circle which is the difference between the maximum and minimum principal stresses divided by two Mohr circle for a uniaxial tensile

test at yielding looks like this the intermediate and minimum principal stresses Sigma 2 and Sigma 3 are equal to zero and the maximum principal stress Sigma 1 is equal to the yield strength of the material the shear stress at yielding is equal to half of the yield strength of the material so we can rewrite our equation

to obtain the standard formulation for the Tresca theory [Music] let's look at the von mises failure criterion next which is also called the maximum distortion energy theory it was initially developed by the Austrian scientist Richard von Mises but a number of others were involved in refining it so it is sometimes called the Maxwell

Huber Henke von Mises theory it states that yielding occurs when the maximum distortion energy in a material is equal to the distortion energy at yielding in a uniaxial tensile test so what is the distortion energy it is essentially the portion of strain energy and a stress element corresponding to the effect of the deviatoric stresses

the distortion energy per unit volume can be calculated from the principal stresses using this equation we know that at yielding during a tensile test the maximum principal stress is equal to the yield strength of the material and the two other principal stresses are equal to zero so we can calculate the distortion energy at yielding in a

tensile test by plugging in the appropriate principal stress values and by equating the distortion energy at yielding with the general equation for a tri-axial stress state and rearranging a bit we get an equation that defines the von Mises failure criterion again we can see that this theory considers the difference between principal stresses

and so is independent of the hydrostatic stress the term on the left is often called the equivalent von Mises stress if it is larger than the yield strength of the material yielding is predicted to have occurred the equivalent von Mises stress is a common output from stress analysis performed using the finite element method contour plots are

typically generated to show the distribution of the von Mises equivalent stress within a component as these allow areas at risk of yielding to be identified when comparing failure theories it can be useful to plot their yield surfaces a yield surface is the representation of a failure theory in the principal stress space which sounds more complicated than

it actually is to keep things simple let's start by considering a plane stress case where one of the three principal stresses is equal to zero by convention principal stresses are ordered from largest to smallest like this but since we don't know yet how the stresses will be ordered I'll refer to them as Sigma a

sigma B and Sigma C instead we'll say that Sigma C is equal to zero and so the two axes of our yield surface graph correspond to the two nonzero principal stresses Sigma a and Sigma B the yield surface for the maximum principle stress theory is easy to plot it says that yielding begins when either of these

principal stresses is equal to the yield strength of the material so it looks like this yielding is considered to have occurred when the stress state reaches this line next let's plot the yield surface for the Tresca theory this theory states that yielding occurs when the difference between the maximum and minimum principal stresses is equal to

the yield strength of the material but it's not as simple as taking the difference between Sigma a and Sigma B plane stress is still a three dimensional stress State in the top right quadrant of the graph both Sigma a and Sigma B are positive and so Sigma C which is equal to zero is the minimum

principle stress the yield surface looks like this but in the bottom right quadrant Sigma B is negative and so Sigma B is the minimum principal stress that gives us a yield surface that looks like this repeating this process for the other two quadrants completes the Tresca yield surface von Mises yield theory for plane

stress conditions is defined using this equation when expressed in terms of the principal stresses we can square both sides of the equation and see that is of the same form as an ellipse which gives us the von Mises yield surface let's plot some experimental data for tests performed on ductile materials and see

how each of these theories compares it's clear that the maximum principle stress theory has large areas where it's use is potentially unsafe so let's get rid of it and compare Tresca and von Mises both agree well with experimental observations although von Mises agrees slightly better the Tresca yield surface lies entirely inside the von Mises

surface meaning that Tresca is more conservative if we assume that the principal stresses will increase in proportion of one another as a load is applied and so the stress paths are straight lines the difference between the Tresca and von Mises theories is largest for these three configurations the maximum difference between the two

theories can be calculated to be 15.5% von Mises is usually preferred to Tresca because it agrees better with experimental data but Tresca is sometimes used because it is easier to apply and it is more conservative so far we have looked at a plane stress case with Sigma C equal to zero but for a

general three dimensional stress state Sigma C will be nonzero we know that the Tresca and von Mises yield surfaces aren't affected by hydrostatic stresses and so to obtain the yield surfaces in three dimensions we just need to extend the plane stress yield surfaces along the hydrostatic axis

failure of brittle materials is different to failure of ductile materials for brittle materials failure is considered to occur by fracture rather than yielding and unlike ductile materials brittle materials tend to have compressive strengths that are much larger than their tensile strengths this needs to be captured in any failure theory for brittle materials it also

means that to assess failure of brittle materials we need to know two separate ultimate strengths for tension and for compression Coulomb more theory is a failure theory often used for brittle materials unlike the theories we have seen for ductile materials it considers failure to be sensitive to hydrostatic stresses and captures the difference between

tensile and compressive ultimate strengths the easiest way to define this theory is using Mohr circle we start by drawing the Mohr circles corresponding to failure in the uniaxial tensile and compressive tests by drawing lines tangent to both circles we create a failure envelope Coulomb Moore theory states that a material will fail for a

stress state with a Mohr circle that reaches this envelope the Tresca yield criterion we saw earlier is actually a special case of the Coulomb Moore failure theory where we have the same material properties in tension and in compression the plane stress failure surface for Coulomb more theory looks like this

as you can see it doesn't agree particularly well with experimental data in the bottom right quadrant modified more theory is a slight variation on the theory which better fits experimental data it is one of the preferred general failure theories for brittle materials because failure theories for brittle materials need to account for the effect

of hydrostatic stresses these failure theories converge to a point when extended along the hydrostatic axis and that's it for this introduction to failure theories we've covered the most common theories but there are many more out there which may be more appropriate for specific scenarios if you enjoyed the video and would like me to cover

more topics please consider supporting the channel on patreon I'll put a link in the description thanks for watching