Hyperelastic material models describe the stress-deformation relationship for flexible materials undergoing large deformations by using a strain energy density function W(F), where stress is derived as the derivative of W with respect to the deformation gradient F. This thermodynamic formulation ensures physical consistency through requirements including material frame indifference (objectivity), isotropy, and proper energy storage behavior. The Neo-Hookean model exemplifies this approach, decomposing into deviatoric and volumetric contributions controlled by shear modulus G and bulk modulus K, respectively.
Install our extension to search inside any video instantly.
Hyperelasticity
Added:If we want to compute the deformation of objects under external influences, we need mechanical material models that describe the relationship between stress and strain within the material.
Over the last century, engineers have developed different models depending on the type of material and the applied loading.
In this video, we take a closer look at a specific class of material models that is designed to describe the behavior of flexible materials undergoing large deformations.
I'm talking about hyperelastic material models.
If we imagine cutting out a tiny volume element inside a deforming object, the deformation of this infinitesimal element is described by the deformation gradient.
And the tractions acting on the surface of the element are described by stress tensors.
For example, the first Piola-Kirchhoff stress tensor.
It's a bit easier to visualize this in 2D.
The deformation gradient F determines how infinitesimal line elements transform during deformation.
This helps us understand how an infinitesimal element deforms.
The first Piola-Kirchhoff stress tensor P determines the traction acting on the surface of that infinitesimal element.
In previous videos, we discussed the meaning of different deformation tensors and stress tensors in detail.
But what we have not discussed so far is how much stress results from an imposed deformation.
This is exactly where hyperelastic models come into play.
Before we discuss hyperelasticity in detail, it makes sense to first discuss which material behaviors can be described by hyperelastic models and which cannot.
In hyperelastic material modeling, we assume that the stress at a particular time depends only on the deformation at the same time.
Mathematically, we can write that the stress is a function of the deformation.
And it's crucial that the stress depends only on the current deformation and not how the material was deformed at earlier times.
This means that if we deform the material and then release the deformation, the stress follows the same path back to the initial stress-free configuration.
The stress being a function of only the current deformation is a simplifying assumption that does not hold for all materials and loading conditions.
For some materials, if we apply a deformation and then hold it constant, the stress gradually decreases over time.
Such materials are called viscoelastic.
As you can see, for these materials, the same deformation can correspond to different stress values.
Therefore, the stress cannot be written as a function of the deformation, and the material is not hyperelastic.
Other materials develop permanent deformation.
If we apply a load and then remove it, these materials do not return to their original shape.
This behavior is called plasticity.
Again, the same deformation can correspond to different stresses, which means that the stress cannot be written as a function of the deformation, and the material is not hyperelastic.
The hyperelastic assumption is only valid for certain materials and loading conditions, such as rubber or biological tissue under moderate deformations.
In hyperelastic material modeling, we try to find the function that describes how stress depends on deformation.
In this video, we will focus on the relationship between the first Piola-Kirchhoff stress P and the deformation gradient F.
But note that hyperelastic models can also be formulated using other measures of stress and deformation.
We will discuss this in a bit more detail at the very end of the video.
At this point, it is important to emphasize that there's no single universal function that describes the relationship between stress and deformation.
Different materials exhibit different mechanical responses in experiments.
The goal of mechanical material characterization is to test a material experimentally, collect data, and then determine the form of the function P of F for that specific material.
Now, what we could theoretically do is just try out different functions P of F and see whether they match our experimental observations.
However, there's a catch. Not every function we could come up with is physically reasonable.
If we simply guess the form of the function P of F, it is quite likely that it violates fundamental physical principles that we know must be satisfied.
For example, material behavior should be independent of the observer.
If we describe the same deforming object in a different coordinate system, nothing should change at the physical level.
This is called material frame indifference or objectivity.
But, this is just one example of the physical requirements that a hyperelastic material model should satisfy.
Throughout this video, we'll come back to several of these and look at them in more detail.
In principle, it is possible to define P directly as a function of F in a way that satisfies the physical requirements.
But, this gets very ugly very fast.
Let me give you an example.
This is what the function P of F looks like for one of the simplest hyperelastic material models, the neo-Hookean model.
Yes, you heard right. This is one of the simplest hyperelastic material models.
Even though this is still a relatively simple model, the expression for P of F looks very complex, and it is difficult to understand the physical meaning of the individual terms.
The reason this expression is so complicated is that it is carefully constructed to satisfy all the physical requirements.
Throughout this video, we will discuss these requirements and see why the Neo-Hookean model satisfies them.
And with each requirement we discuss, the structure of the model and the physical meaning of its terms will become clearer.
So, that by the end we will arrive at a representation of the Neo-Hookean model that looks far less complicated.
The first physical requirement we will look at is that the material behavior must be consistent with our understanding of thermodynamics.
From this requirement, we will find another more elegant way to formulate hyperelastic material models.
Instead of formulating P directly as a function of F, we introduce another function W of F, and then we let P be the derivative of that function with respect to F.
In this new formulation, the relationship between P and F is implicitly defined through the function W of F.
To better understand where this new formulation comes from, we have to look at some essential ideas from thermodynamics.
First of all, to avoid any confusion, note that the term thermodynamics can be a bit misleading.
In very, very, very simple terms, you can think of thermodynamics as the study of energy.
And energy is relevant to virtually every physical process.
So, when we talk about thermodynamics, it does not necessarily mean that we want to model thermal effects such as heat conduction.
In fact, in this video, we assume that temperature is constant in space and time.
Thermodynamics also does not necessarily mean that we are considering a dynamic mechanical system. When we speak of material thermodynamics, we simply mean that the material behavior is modeled in a way that it is consistent with our general understanding of energy and how it behaves in physical systems.
One of the key equations in material thermodynamics captures how different forms of energy are exchanged and converted.
In this video, rather than diving into the full thermodynamic derivation of this equation, we'll develop an intuitive understanding of the physical meaning of each term in the equation.
First of all, there's the so-called mechanical power density.
It is defined as P times F.
The colon denotes a special type of product, which is defined as the sum over I and J of PIJ times F.IJ.
You can think of this operation as a generalization of the inner product to second order tensors.
The mechanical power density represents the rate at which mechanical energy is supplied to the material at the material point per unit volume.
In simple terms, it's the external mechanical energy being fed into our infinitesimal material element.
It has units of energy per unit volume per unit time.
Recall that power has units of energy per unit time.
The term density simply tells us that we are considering power per unit volume.
Next, we have the so-called stored energy density rate W.
It describes how the energy stored inside the material per unit volume changes over time.
And finally, there's the so-called dissipation density rate D.
It describes how much energy per unit volume and time is converted into forms that cannot be mechanically recovered.
For example, when energy is converted into heat.
Now, a fundamental concept in material thermodynamics is that when we perform mechanical work on an infinitesimal element, the supplied energy is either stored in the material or dissipated or a combination of both.
Mathematically, this means that the mechanical power density must be equal to the stored energy density rate, plus the dissipation density rate.
This is a fundamental equation in material thermodynamics.
In hyperelasticity, we make one key assumption.
We assume that there's no energy dissipation.
All mechanical power supplied to the material is converted into stored energy and can be fully recovered.
This means that for hyperelastic materials, our equation simplifies to P times F dot equals W dot.
And another key assumption of hyperelasticity is that at any time the stored energy density can be expressed as a function of the deformation gradient F only.
In other words, if we know the deformation, we know how much energy is stored in the material.
Note that the stored energy density is often called the strain energy density, since it represents the energy associated with a particular state of strain.
Under the latter assumption, we can use the chain rule to find that P times F dot equals the derivative of W with respect to F times F dot.
This equation must hold for any deformation gradient rate F dot that we apply to the material.
This is only possible if P equals the derivative of W with respect to F.
Okay, let's step back for a moment and see what we have achieved.
Earlier, I mentioned that formulating P directly as a function of F can be problematic, since it is easy to violate physical requirements.
From our thermodynamic arguments, we found that P is equal to the derivative of W with respect to F.
This means that if we know the function W of F, we can differentiate it and obtain P as a function of F.
In hyperelastic material modeling, everything comes down to finding the function W of F for a material.
Once this function is known, we also know the relationship between stress and deformation.
Earlier I showed you what P of F looks like for the neo-Hookean material model.
With our new formulation, we can write the same model in a different way.
This is what W of F looks like for the neo-Hookean model.
If we differentiate it with respect to F, we recover the same function P of F that we saw earlier.
Okay, let's visualize this using our infinitesimal element.
In the initial undeformed state, the stored energy density W is zero.
When we deform the element, W increases.
The material now stores energy that can potentially be transformed back into mechanical power.
When we release the deformation, W returns to its initial value.
No matter how we deform the infinitesimal element, once the deformation is released, the stored energy density always returns to its original value.
In this way, we ensure that in a deformation cycle that starts and ends with the same state, no energy is lost, and most importantly, no energy is artificially generated, which would be unphysical.
If this were not the case, we could use the material to generate unlimited amounts of energy.
Our thermodynamically motivated formulation of the material model prevents such unphysical behavior.
We also say that the material model is thermodynamically consistent.
Before we move on, it's worth mentioning that the thermodynamic formulation with the stored energy density is not specific to large deformations.
The same arguments apply in small strain elasticity.
In a previous video, we discussed the relationship between stress and the infinitesimal strain tensor in isotropic elasticity.
The same model can also be formulated in terms of a stored energy density.
The stress tensor is then obtained by differentiating the stored energy density, resulting in exactly the same relationship between stress and strain.
So, small strain elasticity is thermodynamically consistent.
We will return to this model later and discuss its similarities to the neo-Hookean model.
Okay, let's go back to the large deformation setting.
By formulating our hyperelastic material model through the function W of F, we ensure thermodynamic consistency.
But, this is not enough to obtain a physically meaningful material model.
So, next we look at the other physical requirements that need to be fulfilled and see why the neo-Hookean model satisfies them.
For example, we have to make sure that if we describe the same deformation in a different coordinate system, nothing should change on a physical level.
Or equivalently, if we first deform our infinitesimal element and then apply a rotation, the stored energy density should not change during this rotation.
This is called material frame indifference or objectivity.
Mathematically, this means that W of F must be equal to W of Q times F, where Q is any proper orthogonal rotation matrix.
To fulfill this property, we use a little trick.
Instead of formulating W directly as a function of F, we formulate it as a function of the right Cauchy-Green tensor C.
Note that W still depends on F because C itself depends on F.
But, by letting W depend on F through C, we automatically guarantee objectivity.
This is because W of C of F equals W of F transpose F, and because Q is a proper orthogonal rotation matrix, we can write this as W of F transpose Q transpose QF.
This is equal to W of C of QF, which means that the property W of F equals W of Q times F is fulfilled.
The stored energy density of the Neo-Hookean model can be written in terms of C.
And therefore, objectivity is fulfilled.
Some materials show the same resistance to deformation in different directions.
This is called material symmetry.
Note that material symmetry is not really a physical requirement.
It's a modeling assumption that is appropriate only for certain materials.
In the special case of isotropic behavior, the material behaves identically in every direction.
Applying a deformation directly or first rotating the material and then applying the same deformation yields the same stored energy density.
Mathematically, we define an isotropic material as one for which W of F equals W of F Q transposed, where Q is again any proper orthogonal rotation matrix.
We can fulfill this property by making the stored energy density depend only on the invariants of C.
This is because any invariant of C has the property that the invariant of C of F equals the invariant of Q C Q transposed.
With a few rearrangements, this can be written as the invariant of C of F Q transposed, which means that the condition for isotropy is satisfied.
The stored energy density of the Neo-Hookean model can be written as a function of invariants of C.
Therefore, the Neo-Hookean model is isotropic.
Before we move on, note that the term I3 to the power of minus 1/3 times I1 has a special meaning.
It is an invariant of C that changes only when the shape of the material changes, but not when only the volume changes.
To see this, recall that we can decompose the deformation gradient into volumetric and deviatoric contributions, as discussed in one of the previous videos.
Then we can define the deviatoric right Cauchy green tensor, which depends only on the deviatoric deformation gradient.
And then we compute the first invariant of this deviatoric right Cauchy green tensor, which is exactly the term that appears in the neo-Hookean model.
Because this invariant depends only on the deviatoric deformation gradient, it changes only when the shape of the material changes, but not when only the volume changes.
In the other term of the neo-Hookean model, we have I3 to the power of 1/2.
This is equal to J, which only changes when the volume changes, but not when only the shape changes.
It is therefore a purely volumetric invariant.
This means that the neo-Hookean model consists of two distinct parts, a deviatoric contribution and a volumetric contribution.
This gives us a clear physical interpretation of the terms. The first term describes the material's resistance to changes in shape, while the second describes its resistance to changes in volume.
If we apply a volume-preserving deformation, only the deviatoric contribution to W changes.
And if we apply a shape-preserving deformation, only the volumetric contribution to W changes.
The material parameter G is the so-called shear modulus. It determines the material's resistance to changes in shape.
And the bulk modulus K determines how strongly the material resists changes in volume.
You might remember that in small strain elasticity, we encountered a very similar decomposition into deviatoric and volumetric contributions.
In fact, for very small deformations, the neo-Hookean model has exactly the same material response as the model we discussed for small strains.
This can be shown by linearizing the function P of F about the undeformed configuration.
Okay, let's take a look at the rest of the physical requirements.
Next, we have the balance of angular momentum.
This is satisfied if the Cauchy stress tensor is symmetric.
We won't go into the details here, but this condition is automatically satisfied when W is formulated as a function of the invariants of C.
So, we don't have to worry about this condition.
Next, we introduce a few additional requirements on the stored energy density.
We usually choose it to be zero in the undeformed state where F is the identity and non-negative for all deformations.
For the neo-Hookean model, these conditions are fulfilled.
But note that these conditions are not strict physical requirements because shifting the stored energy density by a constant does not change the material behavior.
But these conditions provide convenient normalizations.
Other common requirements are that the stored energy density grows as the material volume tends to infinity or zero.
These are called growth conditions.
For example, under uniform expansion, the stored energy density should tend to infinity.
Mathematically, we express this requirement by stating that W should tend to infinity as the determinant of F tends to infinity.
This condition is satisfied by the neo-Hookean model.
Likewise, W should also tend to infinity as the determinant of F tends to zero.
Interestingly, the neo-Hookean model does not satisfy this growth condition.
Nevertheless, it is still widely used because in many practical applications, the determinant of F remains sufficiently far from zero.
But there are modified neo-Hookean models that satisfy both growth conditions.
Finally, we have the requirement that the stress is zero if there's no deformation. That is when F is the identity.
This is again fulfilled by the neo-Hookean model.
At the beginning of the video, we saw a very complicated expression for P of F for the neo-Hookean model.
But now we can better understand where this expression comes from.
We formulated the model in terms of the stored energy density W to ensure thermodynamic consistency.
And we made this energy depend on the invariants of C to guarantee objectivity and isotropy.
By choosing a deviatoric and a volumetric invariant, we can model different responses to changes in shape and changes in volume.
The terms minus three and minus one ensure that W and P vanish in the undeformed configuration.
You can see that hyperelastic material modeling is not easy at all.
We have to take several physical requirements into account.
But by treating the material as a thermodynamic system and formulating the model through the stored energy density, it becomes much easier to develop models that are physically reasonable.
For isotropic hyperelastic materials, the simplest approach is to formulate the stored energy density in terms of the invariants of the right Cauchy-Green tensor.
Back in the day, material modeling was largely a trial and error process.
Researchers would propose new models and check whether they matched experimental data.
Nowadays, there's a growing trend to use computers to automatically discover these models or learn them using neural networks.
I'll link some papers in the video description if you're interested.
Finally, there's one more condition that we have not discussed in this video.
In general, it makes sense to impose some form of convexity condition on the stored energy density.
More specifically, in hyperelasticity, there's is known as polyconvexity.
Together with a so-called coercivity condition, polyconvexity guarantees that the boundary value problem that we want to solve when simulating hyperelastic materials has at least one solution.
Polyconvexity is not an easy topic, and not all material models used in practice satisfy this condition.
Therefore, we will not cover it in this video.
Maybe we can discuss it in a separate video.
Throughout this video, we have focused on the relationship between the first Piola-Kirchhoff stress P and the deformation gradient F.
But, as I said earlier, we can also formulate hyperelastic models using other measures of stress and deformation.
For example, if we know P as a function of F, it is easy to express the Cauchy stress or the second Piola-Kirchhoff stress as a function of F using the formulas discussed in the previous video.
Also, recall that in our thermodynamic derivations, we have expressed the mechanical power density in terms of P and F. But, but this is not the only way to express the mechanical power density.
For example, it can be expressed in terms of the second Piola-Kirchhoff stress S and the Green-Lagrange strain E.
Following the same thermodynamic arguments as before, we find that S is equal to the derivative of the stored energy density with respect to E.
This gives us alternative formulations for the relationship between stress and deformation.
From these relations, we can also easily determine how the different stress measures transform under rigid body rotations of the deformed object.
We find that S is a Lagrangian tensor, P is a two-point tensor, and sigma is an Eulerian tensor.
In previous videos, I said that the stress does not change when we rotate the deformed object. This wasn't very precise. Only the second Piola-Kirchhoff stress dress unchanged. Sorry again for this confusion.
That's all for this video. Thanks to everyone who supports the channel through a channel membership. And a special thanks to the great supporters Francis Gerardo and Nathan Yang.
Thanks for watching and see you soon.
Bye.
Related Videos
BMW Built A Radial Engine So Good It Made The Spitfire Obsolete Overnight
MachineTitans999
123 views•2026-06-18
UÇAK MOTOLARI ÇALIŞMA PRENSİMİ
PistonTV
428 views•2026-06-17
The Bizarre Design Flaw That Ruined The Convair 990
Jet-Deck
631 views•2026-06-19
Why Are Rocket Nozzles Bell-Shaped? Propulsion | Aerospace engineering | GATE | Viru Sir IITian
conceptlibrary
189 views•2026-06-15
US Navy's Helios laser tech
Striketech0310
6K views•2026-06-18
NEW ENGINEERING DESIGN FOR IAM MARWA APPALOOSA FARM @iammarwa
findingian001
443 views•2026-06-17
The Air Force Built a Jet With Wings Swept the Wrong Way
TheAbsurdArchiveYT
639 views•2026-06-16
China Is Building a Machine the World Can’t Stop
TechAIVision-f6p
192 views•2026-06-15
