All Roads lead back to Energy
All Roads Lead
Back to Energy
A Holistic View of Material Behavior, Constraint, and Failure
— and why we simulate at all
Combating Engineering Mind Blindness
Joseph P. McFadden Sr.
Senior Engineering Fellow, Zebra Technologies · The Holistic Analyst
McFaddenCAE.com · McFadden@snet.net
Companion reader to the audiobook · June 14, 2026
Developed in collaboration with Claude (Anthropic)
Contents
Contents............................................................................. 2
Part one. All roads lead back to energy.............................. 3
Part five. The two doors — shape, and volume.................. 4
Part seven. The accounting of a material losing itself....... 6
Part ten. The conscience of the analysis............................ 8
Part eleven. Closing — we describe it by what it does........ 9
Developed in collaboration with Claude, Anthropic.
Part one. All roads lead back to energy.
Begin with a confession that physics itself has to make. Everything we do, build, or even are, is here because of energy. We gather it, we move it, we dissipate it, we conserve it. We have written its bookkeeping so precisely that we can track it across a galaxy or a transistor and find that not one unit of it ever goes missing. And yet, if you ask what energy actually is, no one can tell you. We do not know. We only know what it does. We describe it entirely in terms of its effects — the lifting it can perform, the heat it can shed, the motion it can carry. Energy is the most fundamental quantity in our science, and it is defined, from the ground up, by behavior rather than by essence.
I want to begin here, because that humility is exactly the right posture for everything that follows. If the deepest quantity we have is known only by what it does, then we should not be surprised — or ashamed — when the materials we build with turn out to be the same way. We are about to talk about how materials deform and fail, in great technical detail. But the whole subject rests on a quiet philosophical fact: we never really see the thing itself. We see what it does. And our entire craft is the disciplined art of reading behavior back into understanding.
Part two. We never meet the material. We only meet its answer.
Think about what you actually observe when you characterize a material. You never observe the material itself. You observe a response — the material answering a poke. You push, and you watch how it pushes back. Stiffness is an answer. Yield is an answer. Fracture is an answer. The number we write on the datasheet and call a property is really a verb caught mid-act — a frozen frame of something the material did when the world leaned on it.
So we describe a material exactly the way we describe energy: not by what it is, but by what it does when it is provoked. This is not a weakness in our knowledge; it is the nature of knowledge about deformable things. And it tells us what simulation is really for. We are not trying to capture some hidden essence of a plastic, or a metal, or a pane of glass. We are trying to predict its answer — to our poke, or to the world's poke. A drop. A shock. A clamping load. A lifetime of vibration. The question is always the same: when the world pushes on this, how will it push back, and at what point will it stop pushing back and simply break?
Part three. The whole situation answers — material, form, and environment.
Here is the move that most people miss, and it is the heart of a holistic view. The answer is never a property of the material alone. It is a property of the whole situation. The same resin, given the same poke, will give two completely different answers depending on the shape it has been formed into, and the world it sits in when you ask.
Geometry is not the stage on which the material performs. It is part of the performer. A smooth bar and a sharp notch, cut from the very same material, fail at entirely different strains — not because you changed the material, but because you changed which answers it was allowed to give. Environment is in the answer too. Lower the temperature, and a polymer that was ductile this morning turns brittle by afternoon. Speed the load up to impact rates, and the same material stiffens, strengthens, and loses its patience. Add a weld line where two flow fronts met inside a mold, and the local ductility can collapse by nearly an order of magnitude. None of these changed the chemistry. All of them changed the answer.
So the holistic truth is this: you are never testing a material. You are always testing a situation — a material, in a form, in an environment, under a particular poke. To forget any one of those is to mistake a single answer for the whole character of the thing. That mistake has a name we will return to: engineering mind blindness.
Part four. Force, work, and the energy a body quietly stores.
Let me put the mechanics into the same language of force and energy, because that is the language failure is actually written in. When we load a part, we apply forces to it. Stress is simply how we characterize the distribution of those applied forces through the interior of the body — force, spread over the internal area that has to carry it. There is nothing more mystical about stress than that: it is the accountant's map of where the force went.
And every material is deformable. Only the degree differs. Steel, rubber, glass, bone — push on any of them and the atoms shift, relative to their neighbors, away from the spacing they would rather rest at. Moving them out of that equilibrium takes work. And in an elastic body, that work is not lost. It is conserved — stored inside the material as internal strain energy, like a spring wound tight. Release the load, and the stored energy spends itself pushing the atoms back home.
That single idea organizes everything that follows. Failure, in every form we will discuss, is a question about that stored energy. When does the material stop storing it reversibly, like a spring, and begin to spend it irreversibly — bleeding it off into permanent flow, into heat, and finally into the creation of brand-new crack surface? Deformation gathers energy. Failure is how it gets spent. Keep that ledger in mind, and the rest of the subject becomes a single connected story rather than a list of formulas.
Part five. The two doors — shape, and volume.
Now to the most important idea in the whole subject, and it is an idea about doors — about the ways a stressed material is, or is not, allowed to spend its stored energy.
Any state of stress at a point can be split into two parts that do entirely different jobs. The first is the hydrostatic part — equal in every direction. It tries only to change the material's volume; it dilates or compresses, stretching the atomic spacing uniformly outward or squeezing it inward. The second is the distortional part — the deviatoric, shearing part, the part the von Mises stress measures. It tries to change the material's shape, sliding planes of atoms past one another.
Here is the key that unlocks ductile failure. Yield, and the plastic flow that dissipates energy, are driven almost entirely by the shearing, shape-changing part. Pure hydrostatic tension, on its own, does almost no plastic work. In the classical, pressure-insensitive picture of yielding it cannot by itself make the material flow; it can only store energy and pull voids open. Real polymers are not quite that clean — pressure can shift where they yield, which is part of why constraint matters so much — but the thrust holds: hydrostatic stress mostly accumulates rather than dissipates. So a stressed point has, in effect, two doors. Through the shear door, it can spend its energy by flowing — sliding, drawing, dissipating the imposed work as broad, cheap, distributed plastic deformation and heat. Through the dilatation door, it cannot spend at all; it can only accumulate, loading the bonds and the voids directly toward separation.
This is why we can summarize so much of a material's tendency to fracture with one master variable — the strongest single organizer of the fracture locus we build here. It is the ratio of the hydrostatic, mean stress to the von Mises, shearing stress — written with the Greek letter eta, and called the stress triaxiality. Triaxiality measures, in one number, how far the stress state has swung toward pure pulling-apart and away from useful shear. In plain terms, it measures how many doors are shut. High triaxiality means the shear door is nearly closed: few doors, little ability to flow, low ductility. Low — or negative — triaxiality means the shear door is wide open: the material yields freely, dissipates enormous energy, and draws out to large strains before it ever fails. Triaxiality is not the whole story — the finer shape of the stress state, the strain rate, and the temperature each leave their mark too — but it is the single strongest organizer, and it is the axis our locus is built upon.
A few landmarks worth carrying. At a triaxiality near minus zero point three three, uniaxial compression — voids squeezed shut, ductility effectively unlimited. Near zero, pure shear — the shear door wide open, high ductility. Near zero point three three, the ordinary tensile bar, the reference most people picture for strain at break. Near zero point six, plane-strain tension, where lateral constraint has begun closing the shear door. And at one and above — sharp notches, screw bosses, snap-fit undercuts, thin ribs into thick walls — severe constraint, where even a tough material can fail at modest strain.
And the micro-mechanics underneath is simply the story of where the unspent energy goes. First, nucleation: tiny voids open at imperfections. Then growth: the material around each void flows plastically, and high triaxiality biases that flow outward — inflating the void like an internal pressure, faster the higher the triaxiality. Notice the two doors are coupled here, not separate: it is still plastic deformation doing the work, only now steered toward opening rather than toward changing shape. Then coalescence: the ligaments between voids thin until they can no longer carry force, the voids link, and a crack is born. Voids are nothing more than the place the energy went when the material was denied the shear it needed to spend it any other way.
Part six. Constraint — what happens when we will not let it flow.
Now we can say precisely what constraint is, in this language. Constraint is any situation that takes the shear door off the table — that forces the material toward dilatation when it would much rather flow. And the cleanest place to see it is the one we care about most: through the thickness of a part, at a notch or just ahead of a crack.
Pull on a part, and a force acts normal to the crack face, trying to open it. By the Poisson effect, a material stretched in one direction wants to contract in the other two. Now ask whether it is allowed to. At the free surface — the outer skin of the part — nothing stops that sideways contraction. The surface is in what we call plane stress: no stress through the thickness, low constraint, the shear door open. The material can yield, flow, and blunt the crack. It answers in a ductile way.
But move inward, to the center, at mid-thickness. There, the material on either side will not allow that interior slice to contract sideways — it is held fast by its neighbors. And preventing a contraction is mechanically identical to applying a tension. So a single force, normal to the crack face, now induces lateral tensions as well. This is plane strain. The center is being pulled in all three directions at once. The constraint drives the mean stress up relative to the shearing part — the shear door is squeezed nearly shut. The material there can still deform, but it can no longer flow freely enough to relieve the load, so void growth and separation take over and it fails at far lower strain. The very same crack, in the very same part, is therefore ductile and forgiving at its surface, and far less ductile, far more constrained, at its core. Same material. Same poke. Different doors.
This is also why thickness is not a detail. A thin part is mostly free surface — predominantly plane stress, low constraint, and tough, because the crack front can yield along nearly its whole length. A thick part carries a large plane-strain core — a long stretch of crack front locked in high triaxiality, unable to dissipate by shear. Its toughness is therefore lower, and as thickness grows it falls toward a conservative floor. That floor, the plane-strain fracture toughness, is exactly what the material property called K subscript I-C is defined to be — which is why standardized toughness tests insist on a minimum specimen thickness — necessary, though not on its own sufficient, since the crack length, the remaining ligament, and the loading must satisfy the validity rules too. Met together, they demand a fully constrained, plane-strain crack front, so the number reported is the safe, thickness-independent lower bound.
And now the reason we simulate inside and out, and not just measure from the outside. The poke we can see at the surface. We can measure the force, the displacement, the moment the crack first shows. But the doors — whether the buried critical point still had its shear door open, what triaxiality it actually saw, what path it traveled to get there — those are invisible from the outside. Simulation is how we get to stand at the center of the part, in the place we cannot reach, and watch which answer the material is being cornered into giving. We simulate to see the constraint the world has imposed but kept hidden.
Part seven. The accounting of a material losing itself.
Once we accept that failure is energy being spent irreversibly, the way a solver represents damage becomes almost obvious. Picture a small cross-section carrying load. As micro-cracks and voids accumulate inside it, the area still intact and load-bearing shrinks, even though the outer dimensions have not changed. The intact ligaments that remain feel a higher stress than the force-over-original-area would suggest.
Continuum damage mechanics captures this with one honest scalar — a damage variable, conventionally called D, running from zero in pristine material to one in fully failed material. The nominal stress we report equals one minus D, times the true, effective stress carried by the surviving ligaments. When D is zero, the two are identical and the material is whole. As D climbs toward one, the same true stress in the shrinking intact material shows up as an ever-smaller nominal stress — which is precisely the softening we measure. At D equal to one, the material carries nothing. The whole job of a damage model reduces to two questions: when does D leave zero — that is initiation — and how fast does it climb to one — that is evolution.
Evolution is where the energy ledger has to be kept with real care. The tempting way to describe softening would be in terms of more strain — fully fail after another few percent. But a softening material localizes its deformation into a narrow band, typically one element wide in a model. Tie the softening to strain, and a finer mesh localizes into a thinner band, spends less energy, and predicts a different fracture. The answer would float with the mesh — useless for a decision. So instead we pose evolution in terms of an energy per unit area, or a displacement, and let the solver use each element's own characteristic length to convert that into the right local softening. The point is not numerical hygiene. The point is that we are insisting each failing element spend the true energy the material demands to make that crack — no more, and no less — regardless of how we happened to mesh it. Deletion, when an element finally reaches full damage and is removed, is just the last entry in that ledger: the energy fully spent, the load path gone.
Part eight. The bridge to fracture mechanics — energy, again.
That energy per unit area is not a modeling convenience we invented. It is the central quantity of classical fracture mechanics, and seeing the connection is what unites the element-level damage model with a century of theory.
It begins with Griffith, a hundred years ago, asking when a crack grows. His answer was an energy balance. Extending a crack creates new surface, which costs energy. But extending it also relieves the stressed material around it, which releases stored strain energy — that wound-spring energy from Part four. A crack advances when the energy released per unit of new crack area is at least the energy required to create it. That released energy, per unit area, has a name: the energy release rate, written capital G, for Griffith. Its threshold value, the point at which the crack actually runs, is the critical energy release rate, capital G subscript c — the material's fracture energy, its toughness in the currency of energy. And here is the quiet unification — with one honest qualification about how far it reaches. G-c and the fracture energy you hand a damage model are the same kind of thing: an energy per unit area of crack. When the failing region is sharp and the material around it stays mostly elastic, they are effectively the same number. But in a bulk damage model of a ductile material the link is looser, because the energy you set on an element is only what that element spends as it softens — while much of the real fracture energy is spent as plastic work in the material around the crack, work your plasticity model already accounts for. So a G-c read from a datasheet is the right place to begin — an energetic anchor — not a number to paste in untested.
There is a second language for the same physics, due to Irwin, the one most engineers meet first. Near a sharp crack tip the stress rises with a characteristic inverse-square-root intensity, and the strength of that field is captured by the stress intensity factor, capital K, which bundles the applied stress with the crack size. The crack runs when K reaches the fracture toughness, capital K subscript I-C — the very plane-strain property we met in Part six. The two languages are tied together exactly: for a linear-elastic material, the energy release rate equals the stress intensity squared over an effective modulus — G equals K squared over E-prime, where E-prime is Young's modulus in thin plane stress, and Young's modulus over one minus Poisson's ratio squared in thick plane strain. So a datasheet toughness converts cleanly between K and G — one measurement, two currencies, a fixed exchange rate — and gives you a principled first estimate of the energy a damage model wants. Whether that estimate stands is exactly what the calibration in Part nine is for. Energy, in any case, sits underneath both.
Two honest edges. Cohesive-zone models make the link most directly of all: a cohesive element carries a traction that rises and then falls to zero over an opening, and the area under that curve — a work, an energy per area — is exactly G-c. And a caution on scope: the K and G of Griffith and Irwin are a linear-elastic theory, valid only when the yielded zone at the crack tip is small. Tough polymers and ductile metals violate that — their plastic zones are not small — which is exactly why we turn to continuum damage and the triaxiality-driven locus for them. The energy idea survives the generalization, through the J-integral, its large-yielding successor. But notice that even the generalization keeps the same name. It is energy, all the way down.
Part nine. Listening to the material — calibration as conversation.
If a material is known only by its answer, then calibrating a failure model is not really measurement. It is listening. And there is an unintuitive fact at its heart: you cannot measure a fracture locus directly. There is no gauge that reads out fracture strain at a given triaxiality. Both are hidden — triaxiality is an internal, three-dimensional quantity, and the fracture strain lives in the single buried element that fails first. Neither is visible from outside. Only the consequence is visible: a displacement, and the instant a crack appears.
So we proceed by inference, which is the honest form of listening. We poke the material with a real test — a drop, an impact — and we record the answer with a high-speed camera, because at impact rates the whole event is over in a few milliseconds and the strain rates reach hundreds to thousands per second. At ordinary frame rates the failure happens between two frames and is simply invisible; here, the frame is the clock. Then we build a simulation of that exact test, but with damage switched off — deliberately — so that nothing perturbs the stress and strain fields we are trying to read. Up to the observed instant of fracture, that undamaged state is a sound approximation — close to correct, because in reality the material had barely begun to fail either, though a little microscopic damage may run ahead of the visible crack. We find the element at the crack-initiation site, at the fracture instant, and we read out what the solver can give us: the plastic strain — the apparent fracture strain for this material model and this mesh — and the triaxiality. Apparent, because that strain is mesh-conditioned; the discipline that keeps it honest is to calibrate and then run on the same mesh, so the conditioning cancels.
And one refinement separates a defensible calibration from a noisy one. You must not read the triaxiality at the instant of fracture. You must average it over the whole loading path, weighted by plastic strain — the convention of Bao and Wierzbicki — because triaxiality is not constant as a specimen deforms, and damage accumulated along the entire journey, not just at the end. The result is one honest pair: one path-averaged triaxiality, matched to one fracture strain. One geometry yields one point, because each shape probes only one region of triaxiality. So you sweep geometries — shear coupons low, smooth bars in the middle, notched bars high — and you sweep rates, and you connect the points into a locus that spans exactly the range your real event visits. That is the conversation: you let the material answer many carefully chosen pokes, and from its answers you infer which doors it had open, and how wide.
Part ten. The conscience of the analysis.
A damage analysis produces vivid output — a part splits, a crack runs, fragments scatter — and that vividness is exactly why it must be checked with more rigor, not less. The screen is persuasive long before the physics is earned.
A handful of quantities keep us honest. The initiation variable shows how close each point is to beginning to fail; confirm it concentrates where stress reasoning says it should. The stiffness degradation maps the softening front; you want a plausible progression, not scattered, random element loss. The element status shows what has been removed; deletion should happen where, and when, physical reasoning and the high-speed video agree. And above all, the energy balance — the conscience of the whole analysis, and the natural close of everything we have said. Total energy should be conserved. The artificial energy that numerical controls introduce should stay a small fraction of the real internal energy. And the energy dissipated by damage should be a sensible share of the work done. If the energy ledger does not balance, the fracture picture is not to be trusted, however convincing it looks. Then the final discipline: refine the mesh, vary the parameters within your calibration uncertainty, and see whether the engineering conclusion survives. One that holds can be reported with confidence. One that swings is reporting your modeling choices, not your structure.
Underneath all of it is one rule, and it is the antidote to engineering mind blindness. Trust nothing you cannot explain. Know which of your numbers are measured, and which are merely plausible, and exactly what test would tell the difference.
Part eleven. Closing — we describe it by what it does.
So we come back to where we began. All roads lead back to energy. We gather it, dissipate it, conserve it — and we still cannot say what it is. We know it only by what it does. Materials are no different. We never meet the thing itself; we meet its answer to a poke, and that answer belongs not to the material alone, but to the whole situation — the material, its form, its environment, and the doors that situation leaves open or shuts.
To simulate well is to honor all of that. To stand at the center of the part, where the constraint is hidden and the shear door has quietly closed. To keep the energy ledger honestly, from the first stored strain to the last spent crack. And never to forget which of our answers were measured, and which we assumed. That is not merely good practice. It is what it means to see — and helping more engineers see is the whole point.
Energy we cannot define, and a material we cannot meet, and yet, by reading their behavior with care and humility, we can predict, and design, and protect. That is the quiet wonder of the work. Describe the thing by what it does, do it honestly, and you will rarely be led astray.
Thank you for reading. This has been Joe McFadden.
Combating engineering mind blindness.
Engineer. Lifelong Learner. Holistic Analyst.
www.McFaddenCAE.com McFadden@snet.net
Have a thoughtful and wonderful day.
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