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(#110) Stress Strain and Material Failure Fundamentals · Mechanical Engineering Made Simple

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  Get ready, design junkies to torque your nuts, harden your
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  springs, and forge that mind into a bulletproof study guide.
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  We are mechanical engineering made simple.
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  Today we're doing a deep dive into the absolute bedrock of
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  engineering design. We're talking about the
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  relationship between force and deformation, material constants
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  that that govern everything, and the critical methods we use to
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  predict when and maybe more importantly, how something is
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  going to break. Exactly.
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  We've pulled from some foundational source material to
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  really get to the heart of it all.
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  We basically distilled the core concepts of stress, strain and
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  material failure. The mission here is to give you
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  a comprehensive shortcut, a way to really grasp the, you know,
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  the non negotiable relationships and tools for any kind of a
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  structural analysis. And we have to start with the
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  fundamentals. I mean, if you want to
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  understand how a bridge stays up or how an engine block survives,
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  you first have to speak the internal language of mechanics.
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  Which is stress and strain. Stress and stress.
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  So let's kick off with stress. It's not just the external force
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  you put on something, right? It's how that load gets
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  distributed on the inside. That's the key, and the sources
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  use a great analogy. Think about being underwater.
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  OK. The deeper you go, the more
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  pressure you feel. That pressure is essentially A
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  distributed force. It's a perfect simple example of
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  compressive stress. So stress, which we call Sigma,
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  is basically force over a given area.
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  Precisely. And we generally break it down
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  into two types. You have normal stress, that's
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  the pulling or tensile, and the pushing or compressive force
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  that acts perpendicular to a surface and the other one.
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  And then you have sheer stress, which we call Tau.
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  That's the rubbing or sliding force.
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  It acts parallel to the surface. But the stress at any single
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  point isn't just one number, is it?
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  I mean, the magnitude changes depending on the angle you're
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  looking. How do you possibly capture all
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  of that? And that's where the reality of
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  three dimensions kicks in. To really describe the complete
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  state of stress at a single point, you you can't just use
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  one or two values. You need to know all nine
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  components of stress that are acting on three mutually
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  perpendicular surfaces. This gives us the classic
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  symmetric 3 by three stress matrix.
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  And that symmetry is super important.
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  It's everything. If those sheer stresses weren't
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  balanced, that little element of material would just spin out of
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  control. The symmetry confirms rotational
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  equilibrium. It has to be stable.
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  OK, so if stress is the internal cause, then strain must be the
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  effect the deformation. How do engineers actually
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  measure how much something stretches or warps?
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  That's the language of strain. Normal strain, which is epsilon,
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  is just the rate of change of length.
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  It's dimensionless, so you can think of it like a percentage of
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  stretch. And shear strain is about the
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  distortion, the change in shape. Exactly.
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  Engineering shear strain gamma measures how much an element's
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  right angles change. It's measured in radians, and
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  it's the direct result of those internal shear stresses we
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  mentioned. So we've got stress, the force
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  and strain, the deformation. The magic that connects them is
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  Hook's law, but it doesn't always apply, does it?
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  No, it requires a few key things from the material.
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  It has to be linear, homogeneous and isotropic.
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  OK, break those down. Linear just means strain is
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  directly proportional to stress, a straight line.
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  Homogeneous means the material is the same everywhere and
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  isotropic means is properties like stiffness are the same in
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  every direction. And if all that holds true, we
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  get our three critical material constants.
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  Yes, the big three. First, the modulus of elasticity
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  E or Young's modulus. That's stiffness, right?
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  Then Poisson's ratio nu, which is how much it thins out when
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  you stretch it. And finally the shear modulus G
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  it's resistance to shear. But here's the really cool part,
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  The thing that saves designers a ton of work?
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  These three aren't actually independent.
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  That's the real insight for these kinds of materials.
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  If you know E and nu you automatically know G They are
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  mathematically linked by the equation g = e / 2 times the
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  quantity one plus nu. So if you test for stiffness and
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  thinning, you get twisting resistance for free.
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  Got it. It simplifies testing
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  tremendously. OK, so let's move from these
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  ideal uniform stress fields into the real world where geometry
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  gets messy. Let's talk about stress
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  concentrations. Yes, the enemy stress
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  concentrations are engineering hotspots to these tiny localized
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  areas where the stress gradient just skyrockets because
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  something has disrupted the smooth flow of stress.
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  And what are the usual suspects for causing these disruptions?
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  2 main things. The most common is a
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  discontinuity, basically any sudden change in geometry.
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  Think holes, notches, fillets, steps in a shaft and the second.
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  The second is contact forces, which create these really high
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  stresses right at the point of contact that then fade away
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  quickly. And we quantify this spike in
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  stress with the static stress concentration factor case of.
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  TK sub T is just a simple ratio, the maximum stress Sigma Max
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  divided by the nominal or average stress Sigma nom.
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  And the classic textbook example is a hole in a really, really
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  wide. Plate.
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  An infinitely wide plate? Yeah.
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  Yes. In that case, theory tells us
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  the stress right at the edge of the hole is exactly 3 times the
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  nominal stress K sub T is 3.0. A 300% spike.
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  But you know in the real world plates aren't infinite.
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  The source material you gave us had a real world example with a
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  measured K sub T of only 2.35. Why is it lower?
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  Because with a finite plate, the stress doesn't have an infinite
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  amount of material to redistribute into.
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  It kind of gets squeezed around the hole, which raises the
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  nominal stress a bit, but actually limits that extreme
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  peak you see in the theoretical case.
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  This is where we get to that counterintuitive aha moment from
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  the sources, the idea of using a flow analogy.
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  Right. If you picture stress paths like
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  streamlines of water flowing, the sharp corner or a hole is
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  like a big rock in the stream. The flow has to jam up to get
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  around. It and sometimes the solution is
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  to actually remove more material.
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  It sounds crazy, but it works. The example of adding two
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  smaller relief holes next to a main hole is perfect.
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  What you're doing is making the overall opening look more like a
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  smooth ellipse. You're giving the flow of stress
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  a gentler curve to follow. So you're smoothing out the
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  traffic jam? Exactly.
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  You reduce that sharp gradient, the K sub T factor drops
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  dramatically, and the part is suddenly much more robust.
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  It's a fantastic design trick. But K sub T is the static
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  factor. It only tells part of the story.
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  The real world involves different kinds of loading over
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  time. Absolutely.
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  The type of loading changes everything.
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  Short time static loading is what we use to find basic
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  properties like yield strength, but then you have to add time to
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  the equation. And what happens when a load
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  just sits there for a really long time, especially when it's
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  hot? That brings in creep.
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  Creep is where the material slowly deforms or flows under a
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  sustained load, even if that load is way below its yield
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  strength. Think of a jet engine turbine
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  blade. And then there's the big one,
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  the number one cause of failures.
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  Fatigue, which is caused by repeated cyclical loading.
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  A tiny scratch becomes a stress concentration, a crack starts,
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  and with every cycle it grows a little bit more until bang.
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  Catastrophic failure at a stress the part could have easily
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  handled just once. Speaking of limits, hooks law is
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  great in the elastic zone, but what about materials that don't
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  even have a straight line stress strain curve to begin with?
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  Good point. Things like concrete or cast
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  iron. For those, the modulus of
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  elasticity E isn't a constant. So what engineers do is they use
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  a secant modulus. They just draw a straight line
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  from the origin up to a specific operating stress on the curve,
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  and use the slope of that line. It's an approximation for a
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  specific load range. Let's actually define failure.
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  It means different things for different materials.
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  Completely different for ductile metals like steel, which can
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  stretch more than 5%. Failure is defined as yielding
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  the start of permanent deformation.
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  For brittle materials like glass, failure is just rupture,
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  fracture. So for those ductile metals, we
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  have two main theories to predict when they'll yield.
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  The first one is von Mises. The maximum distortion energy
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  theory, Yes, von Mises, it's a bit complex, but it basically
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  says yielding happens when the energy stored from distorting
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  the material shape reaches a critical value.
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  It's generally the most accurate theory we have.
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  So you calculate this thing called the equivalent of von
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  Mises stress Sigma sub VM and when it hits the materials yield
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  strength S sub ES. That's it.
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  That's the prediction, and it matches experimental data
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  incredibly well. But it's not the one most
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  commonly used for, say, safety critical design.
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  That's usually TRESCA, the maximum shear stress theory.
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  Why the simpler 1? Well, the maximum shear stress
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  theory or TRESCA is simpler. As you said, it predicts failure
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  when the maximum shear stress, which is just half the
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  difference between the largest and smallest principle stresses
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  S 1 -, S three reaches the shear strength at yield.
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  But the key reason it's used so much is that it's more
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  conservative. OK, how much more conservative?
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  Why would a designer choose less accurate theory?
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  Because it guarantees a bigger margin of safety in the worst
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  case scenario, which is pure sheer Cresca predicts yielding
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  at a load that's about 15% lower than what von MIS predicts.
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  I see. So it's a built in safety
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  factor. It's a huge advantage, less
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  computation and you get a guaranteed safety buffer.
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  So von Mises might be for high performance weight optimized
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  research, but Tresca is for building a safe bridge.
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  Got it. And for brittle materials, it's
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  a whole different game, using things like the Cool and More
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  theory, which needs both tensile and compressive strengths.
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  Right, because Britain materials are often much, much stronger in
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  compression than intention. You need both to define that
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  failure envelope. OK, let's.
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  Pivot to how engineers actually solve these problems in complex
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  systems. There are some powerful
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  analytical principles. For sure, the first is a total
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  lifesaver. The principle of superposition
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  for elastic systems. It means the total effect of
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  many loads is just the simple sum of the effects of each load
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  acting by itself. That sounds almost too easy.
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  What's the catch? The catch is that the
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  deformations have to be small. Superposition fails if a load
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  changes the geometry of the system so much that it affects
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  how other loads are applied. Can you give an example?
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  Sure. Think of a long skinny beam.
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  You put a load in the middle, it bends down.
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  Now, if you push on the end with an axial load, that axial load
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  is now acting on a bent shape, which creates extra bending
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  moments it wouldn't have otherwise.
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  The effects don't just add up anymore.
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  I see. So it only works when things
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  stay more or less in their original shape?
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  Exactly. And what about problems where
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  you have too many supports statically indeterminate
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  structures? For those we use the method of
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  consistent deformations. We have more unknown forces than
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  we have equilibrium equations. So we add in equations based on
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  geometry the fact that the deflections at the supports have
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  to be compatible with each other.
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  A fixed support can't move, for instance.
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  That gives us the extra equations we need to solve for
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  all the forces. OK, enough theory, let's get
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  practical. How do we actually measure
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  strain on a real part? The go to tool is the electrical
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  resistance strain gauge. The concept is brilliant in its
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  simplicity. You have a tiny metallic grid
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  and as it gets stretched or compressed, its electrical
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  resistance are changes in a very predictable way.
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  So you're just measuring a tiny change in resistance to find the
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  strain. That's all it is.
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  But if you're analyzing a complex part and you don't know
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  which way the stresses are going to run, you can't just slap 1
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  gauge on there, can you? You absolutely cannot.
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  You need 3 measurements. Remember, the general state of
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  plane stress has three unknowns, the stress in the X direction,
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  the stress in the Y direction, and the sheer stress Tau Phi.
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  To solve for three unknowns, you need 3 measurements.
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  Which is why we have the strained age rosette.
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  Exactly. A rosette is just three gauges
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  arranged in a specific pattern, usually a rectangle or a delta
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  shape. That gives you 3 independent
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  strain readings at that one single point.
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  From those 3 readings you can calculate everything you need to
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  know about the stress state. So if you just put 1 gauge on a
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  twisting shaft, you could get a reading of 0 and think
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  everything's fine. You'd.
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  Completely miss it because the principal stresses are at 45° to
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  the axis. The rosette is non negotiable
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  for unknown stress fields. And the readings aren't perfect
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  right out-of-the-box, are they? You mentioned corrections.
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  No2 corrections are critical first for transverse
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  sensitivity. The gauge picks up a tiny bit of
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  strain from the direction it's not supposed to be measuring.
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  2nd and more importantly, temperature resistance changes
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  with heat, so you have to compensate for that either with
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  special self compensating gauges or by using a dummy gauge nearby
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  that isn't under load. And briefly, there are other
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  more visual methods, right? Like photoelasticity?
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  Photoelasticity is a beautiful classic method.
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  You make a model of your part out of a special transparent
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  plastic, put it under polarized light and load it up.
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  The stress patterns appear as colorful interference fringes.
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  It gives you a full field map of the stress distribution.
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  Brings us full circle. We started with the theoretical
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  static stress concentration factor K sub T.
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  But we have to end with the thing that really matters for
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  longevity fatigue. This is maybe the most crucial
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  distinction for any practicing designer.
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  We use K sub T for a one time static load, but for repeated
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  loading for designing against fatigue we have to use the
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  fatigue stress concentration factor K sub F.
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  So K sub T is the perfect clean mathematical stress peak, but K
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  sub F is what the material actually feels over millions of
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  cycles. What's the final take away on
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  how these two numbers relate? The essential thing to burn into
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  your mind is this. K sub F is almost always less
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  than K sub T. Less so.
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  The material is actually tougher than the pure math would
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  suggest. In a way, yes.
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  This difference is a measure of the materials notch sensitivity.
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  At a microscopic level, materials just aren't fully
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  sensitive to every last peak of elastic stress.
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  If you designed your part using the higher case of T value for a
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  fatigue calculation, you'd end up with an overly conservative,
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  heavy and expensive part. So in the real world, you have
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  to look past the elegant elastic solution and account for the
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  messy reality of the material itself.
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  You have to know it's going to be a little more forgiving on a
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  repeated loads than a static calculation would tell you.
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  It's the difference between a textbook problem and a real
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  engineered component. It forces you as the designer to
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  understand that failure depends on a marriage of geometry and
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  material science, not just the peak stress from an equation A.
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  Perfect place to leave it moving from pure theory into informed
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  practice. Thanks for walking us through
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  this. My pleasure, always good to get
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  back to the fundamentals.