Assumed Natural Strain

The simulation of thin structures, such as car panels, aircraft wings, and electronic components, is a cornerstone of modern engineering. However, accurately capturing their behavior presents a significant computational challenge. A notorious numerical problem known as "locking" can render these simulations useless, causing virtual models to become pathologically stiff and fail to deform realistically. This issue arises from the mathematical limitations of simple finite elements when modeling the complex interplay between bending and shear in thin objects.

This article explores the Assumed Natural Strain (ANS) method, an elegant and powerful technique developed to overcome the problem of locking. It is a foundational concept in computational mechanics that enables reliable and efficient analysis of thin-walled structures. By delving into this method, readers will gain a deep appreciation for the sophisticated interplay between physics, mathematics, and engineering that underpins modern simulation software.

The following chapters will guide you through this topic. "Principles and Mechanisms" will first explore the physical origins of shear and membrane locking and then explain the ingenious philosophy behind the ANS method, detailing how it circumvents the problem by assuming a more physically realistic strain field. "Applications and Interdisciplinary Connections" will then demonstrate the profound practical impact of ANS, showing how it is a critical component in everything from the static analysis of beams and shells to the high-speed dynamics of crash simulations, and how it connects structural mechanics with materials science.

Imagine trying to build a digital twin of a car door, an airplane wing, or even a crisp potato chip. These are all thin structures, where their thickness is vastly smaller than their length or width. You might think that simulating them would be easy—after all, they're mostly empty space! But as physicists and engineers discovered, the world of the thin and slender is full of strange and beautiful paradoxes. In the realm of computer simulation, this leads to a notorious problem, a numerical trap so effective it was given the name ​​locking​​. To understand the elegant solution to this problem, we must first appreciate the problem itself.

When you deform a thin object, like a plastic ruler, you can do it in a few ways. You can bend it, which is quite easy. You can also try to stretch it or shear it—imagine trying to slide the top surface of the ruler relative to the bottom one. This is incredibly difficult. There's a deep physical reason for this difference, and it all comes down to how the energy of deformation scales with the object's thickness, which we'll call $t$.

The energy required to bend the ruler, the ​​bending energy​​ ($U_b$), is proportional to the cube of its thickness ($t^3$). The energy required to shear it, the ​​transverse shear energy​​ ($U_s$), is proportional to the thickness itself ($t$).

Let's think about that. Suppose the ruler is 1 mm thick ($t=1$). Then $t^3$ is also 1. Now imagine a much thinner sheet, say 0.1 mm thick. The bending energy now scales with $(0.1)^3 = 0.001$, while the shear energy scales with $0.1$. The shear energy is now 100 times more significant than the bending energy for a comparable amount of deformation! As the object gets thinner, the resistance to shear becomes astronomically larger than the resistance to bending.

Here is where the trouble begins in a computer simulation. We build these virtual structures out of small building blocks called ​​finite elements​​. A simple and common choice is a quadrilateral element, which looks like a slightly distorted brick. When we try to simulate the pure bending of a thin plate made of these elements, the simple geometry of the elements can't perfectly capture the smooth, curved shape of bending without also producing a tiny, artificial amount of shear strain. This parasitic shear is a ghost in the machine; it's not there in the real physics, but the simulation "sees" it.

And because the shear energy ($U_s \sim t$) is so much more potent than the bending energy ($U_b \sim t^3$), this tiny ghost of a shear strain completely overwhelms the true bending physics. The simulation, trying to minimize total energy, finds that the "cheapest" thing to do is to avoid this massive (and fake) shear penalty. How? By simply refusing to bend. The element becomes pathologically stiff, or "locks." This phenomenon is aptly named ​​shear locking​​. A similar pathology, ​​membrane locking​​, can occur in curved shells when the elements cannot bend without artificially stretching their mid-surface, another high-energy action they desperately try to avoid.

How do you fight a ghost? You don't fight it with brute force. You use a clever trick. This is the philosophy behind the ​​Assumed Natural Strain (ANS)​​ method. The reasoning is as simple as it is profound: if the strains calculated directly from the element's distorted shape are wrong and are causing locking, then let's not use them. Let's assume a better strain field.

To do this, we first need to think about the element in its own language. Instead of using the global Cartesian coordinates ($x, y, z$) of our simulation world, we use the element's own internal, or ​​natural coordinates​​ ($\xi, \eta, \zeta$). Imagine a perfect reference cube, stretching from -1 to +1 in each of three directions. Our distorted element in the real world is just a mapping of this perfect cube. The natural coordinates are the coordinates on that reference cube. They are the element's native tongue.

The strains measured along these natural directions are called ​​natural strains​​. For instance, $\gamma_{\xi\zeta}$ is the shear strain between the $\xi$-axis and the $\zeta$-axis of the element's internal grid. We can get these natural strains by projecting the physical Cartesian strain tensor onto the element's own base vectors—vectors that point along its grid lines.

Here comes the ANS magic. It turns out that while the calculated shear strains are wrong at most places inside the element, there exist a few "magic" points—or, more accurately, wisely chosen sampling points—where the calculated values are much more accurate. For a four-node quadrilateral shell element, these points are often the midpoints of the element's edges. The ANS procedure for shear locking is therefore stunningly simple:

1. Go to a few special sampling points (e.g., the edge midpoints).
2. Calculate the troublesome transverse shear strains ($\gamma_{\xi\zeta}$ and $\gamma_{\eta\zeta}$) at these points only.
3. Construct a new, much simpler "assumed" strain field by interpolating from these few reliable values. For instance, we might assume the shear strain varies linearly across the element.
4. Use this new, simplified, assumed strain field to calculate the shear energy, completely ignoring the "locking" field calculated everywhere else.

Let's make this concrete. Imagine we have a four-node element and we want to find the assumed shear strain $\gamma_{13}^*$. We sample the "raw" strain at the middle of the left edge ($\xi=-1$) and get a value $s_L = 0.004$. We do the same at the right edge ($\xi=1$) and get $s_R = 0.010$. The ANS method simply says, "Let's assume the strain field is a straight line drawn between these two points." The resulting field is $\gamma_{13}^*(\xi) = 0.003\xi + 0.007$. We do a similar procedure for the other shear direction. This new, ridiculously simple field replaces the complex one that was causing all the trouble.

The beauty of this method is that it is constructed to be exact for constant states of strain. This means it passes the fundamental ​​patch test​​, a benchmark which ensures that if you model a simple block under uniform strain, your elements will give you the exact, uniform answer. This guarantees the method's consistency and convergence.

The Assumed Natural Strain method is a beautiful example of physical intuition leading to a powerful numerical solution. It's part of a larger family of "element technologies" designed to overcome the limitations of simple formulations. It's useful to contrast it with a few others to appreciate its unique character.

One relative is the ​​Enhanced Assumed Strain (EAS)​​ method. While ANS is often motivated by physical reasoning, EAS is derived more formally from a deeper variational principle (the Hu-Washizu principle). EAS augments the strain field by adding new, independent "enhanced" strain modes, which are governed by internal parameters that are solved for and eliminated at the element level. Think of it as giving the element extra "knobs" to turn to find a better strain state. ANS, by contrast, adds no new variables; it simply re-interprets the ones it already has.

Another approach often contrasted with ANS is the strategy of ​​reduced integration with hourglass control​​. To fight locking, one could just use a single integration point at the center of the element to calculate the shear energy. At this one point, the parasitic shear strains for pure bending happen to be zero, so locking disappears! But this is a brutish solution. By ignoring what's happening elsewhere, the element becomes blind to certain "hourglass" deformation modes—wiggly, non-physical patterns that have zero energy and can wreck a simulation. One must then add an artificial "hourglass control" stiffness to penalize these modes. ANS is a more surgical approach; it eliminates locking without creating the hourglass disease in the first place.

Our journey to build the perfect shell element has one final twist. A shell element in 3D space has nodes that can translate. To describe bending, we also need to describe how the orientation of the shell's cross-section changes. This is typically done with rotation variables. We need rotations about two in-plane axes to describe bending and shear. But what about the third rotation, the one about the axis normal to the shell's surface? This is the ​​drilling rotation​​.

In the physics of a simple continuum (a "Cauchy" continuum), there is no moment or force that corresponds to this spinning motion. A pure spin doesn't stretch or shear the material, so it generates no strain energy. This means that in our finite element model, this drilling degree of freedom has no natural stiffness. The element offers no resistance to being spun about its normal. This leads to a "zero-energy mode," which results in a singular stiffness matrix—a fatal flaw for the simulation.

The solution, once again, is a clever fix. We introduce a small, artificial stabilization stiffness that penalizes this drilling rotation. However, we must do it carefully, so it doesn't add fake stiffness to real, physical deformations. A robust method doesn't penalize the drilling rotation itself, but rather penalizes the difference between the independent drilling rotation variable and the spin that is kinematically calculated from the in-plane translations. This stabilization is designed to vanish for constant strain states, thereby passing the patch test and leaving the pure bending and membrane physics untainted.

And so, our story is complete. By starting with the physical paradox of thin structures, identifying the numerical demon of locking, and banishing it with the elegant philosophy of Assumed Natural Strain, we arrive at a robust and beautiful computational tool. The final touch—the careful taming of the phantom drilling rotation—leaves us with an element that is not just a crude approximation, but a small piece of numerical art, embodying a deep understanding of the interplay between physics, mathematics, and computation.

Now that we have explored the intricate mechanics of the Assumed Natural Strain (ANS) method, we might be tempted to file it away as a clever but esoteric mathematical trick. Nothing could be further from the truth. This is not just an academic exercise in chasing away numerical gremlins; it is the very key that unlocks our ability to simulate the physical world with fidelity and efficiency. To see this, we must take a journey, starting with the simplest of structures and ending with the complex, dynamic, and interconnected systems that define modern engineering. It is a journey that reveals a beautiful unity between physics, mathematics, and computation.

Imagine a simple, thin ruler. If you hold one end and push down on the other, it bends into a graceful curve. This is a state of nearly pure bending. A fundamental consequence of this bending is that the top surface stretches a little, the bottom surface compresses, and somewhere in the middle, there is a neutral line that does neither. Crucially, the cross-sections of the ruler remain almost perfectly flat and perpendicular to this curved line. This means there is practically no transverse shear strain. The material is not trying to slide past itself vertically.

Now, let's try to build a virtual version of this ruler in a computer using the finite element method. The most straightforward approach is to describe the bending and shear behavior at every single point within our discrete elements. This is called a "fully integrated" or "consistent" approach. And it fails, spectacularly. For a thin beam, the computer model becomes unnaturally, ridiculously stiff. Instead of a flexible ruler, we get a virtual block of steel. This pathology has a name: ​​shear locking​​. The mathematical constraints of our simple element are too rigid; they cannot properly represent a state of pure bending and zero shear simultaneously. The element incorrectly generates enormous, spurious shear energy, "locking" itself into a nearly undeformed state.

This is where the magic of ANS comes in. Instead of trying to enforce the physics at every infinitesimal point, the ANS formulation has the wisdom to ask a simpler, more physically motivated question. For a simple two-node beam element, it says, "Let's not worry about the complicated (and likely wrong) shear strain everywhere. Let's just sample it at one special place: the midpoint of the element.". By doing this, the element no longer tries to enforce zero shear everywhere, but rather in an averaged, more forgiving sense. This simple act of judicious sampling is enough to exorcise the demon of shear locking. The virtual beam now bends beautifully, just like the real one. The numerical result is not only qualitatively correct, but vastly more accurate.

This principle scales up with remarkable elegance. Consider a two-dimensional plate or shell. One of the most fundamental checks for any shell element is the "patch test" for pure bending. If we impose a displacement field that corresponds to pure bending, a good element should report zero shear energy, just as physics dictates. A standard, fully integrated element fails this test miserably, filling up with parasitic shear energy. An ANS-based element, however, is specifically designed to pass this test. By carefully choosing where to "look" at the shear strains—typically at the midpoints of the element's edges—it correctly calculates zero shear energy.

Passing the patch test is not just an academic badge of honor. It is a guarantee of quality. It ensures that as we refine our computational mesh, making our elements smaller and smaller, the numerical solution will reliably converge to the true physical solution. An element that fails the patch test offers no such guarantee; it may give you the wrong answer, no matter how fine your mesh. Thus, ANS is the foundation upon which reliable simulations of plates and shells are built.

So, how is this profound physical insight translated into the millions of lines of code that make up modern engineering software? The ANS method provides a clear recipe for constructing a high-performance "solid-shell" element, which uses 3D solid element kinematics to model a thin structure. This recipe involves a fascinating division of labor within the element's mathematical formulation.

The total stiffness of the element is assembled from several distinct parts, each treated with a tailored approach:

1. ​​Membrane and Bending Stiffness:​​ These components capture the in-plane stretching and the primary bending behavior. They are often calculated using a standard approach, but sometimes with a reduced number of integration points to avoid other forms of locking.
2. ​​Transverse Shear Stiffness:​​ This is the term that causes shear locking. Here, we apply the ANS procedure, replacing the kinematically derived shear strains with a field reconstructed from the special sampling points.
3. ​​Drilling Stabilization:​​ A peculiar problem with many shell elements is that they have no inherent stiffness against spinning in-plane around their own normal vector (the "drilling" degree of freedom). This is a non-physical mode that can make the global system singular. To fix this, a small, artificial penalty stiffness is added to gently resist this spinning motion.

This "mix-and-match" strategy, where different physical behaviors are handled by different numerical techniques, is a hallmark of advanced computational mechanics. An engineer running a simulation of a complex structure, like the classic benchmark of a pinched cylinder with end diaphragms, must be aware of these ingredients. They must select element formulations that properly implement ANS (or equivalent methods) to prevent locking, and they must ensure that any necessary stabilization (like for hourglassing or drilling) is active. The Assumed Natural Strain concept is not just for element developers; it is a principle that informs the daily practice of computational engineering.

The true beauty of a deep physical principle is that its echoes are heard in many different fields. The ANS method is no exception, serving as a bridge connecting structural mechanics to materials science, high-speed dynamics, and multi-physics modeling.

Connecting Structures and Materials

Real-world structures are often assemblies of different components—beams welded to plates, for instance. A major challenge in computational modeling is ensuring that the connection, or "seam," between different element types is physically consistent. Imagine modeling a curved shell stiffened by a beam running along its edge. The shell, a 2D object, must smoothly transfer forces and moments to the beam, a 1D object. The ANS formulation, by its nature of sampling strains at element boundaries, proves to be remarkably adept at ensuring this coupling is handled correctly. It helps guarantee that the shear strain is compatible across the interface, preventing spurious stresses from arising at the connection between disparate parts of a model.

The connection deepens when we consider modern composite materials, like carbon fiber laminates used in aerospace. These materials are orthotropic—their stiffness depends on the direction. A sheet of carbon fiber is incredibly strong along the fiber direction but much less so perpendicular to it. When modeling a shell made of such a material, the ANS element must not only handle the geometry correctly but also the complex material behavior. Here we find a truly sublime connection: the drilling stabilization, that seemingly ad-hoc penalty to prevent spinning, can be intelligently scaled based on the properties of the composite itself. The optimal penalty factor is related to the material's maximum stiffness, a quantity known as the spectral radius of the constitutive matrix, $\rho(D^{\ell})$. This is a beautiful piece of physics-based numerics: the fundamental properties of the material are used to tune the numerical parameters of the simulation for maximum robustness and accuracy.

Connecting Accuracy and Speed

Perhaps one of the most surprising and powerful applications of ANS is in the field of explicit dynamics—the simulation of high-speed events like car crashes or impacts. In these simulations, the solution is advanced in time through a series of very small time steps. The maximum size of this stable time step, $\Delta t_{crit}$, is limited by the Courant-Friedrichs-Lewy (CFL) condition. In essence, $\Delta t_{crit}$ is determined by the time it takes for a sound wave to travel across the smallest element in the mesh.

Here is the twist: shear locking makes an element artificially stiff. An artificially stiff material has an artificially high speed of sound. Consequently, a model with locked elements has a spuriously high numerical speed of sound, which in turn forces the simulation to take cripplingly small time steps. The simulation becomes not only inaccurate but also incredibly slow.

By removing the non-physical stiffness of shear locking, ANS does something wonderful. It lowers the artificial wave speed in the element to a more physically realistic value. This lowers the highest frequency of the discrete system, which directly increases the maximum stable time step, $\Delta t_{crit}$. The result is a win-win of breathtaking proportions: the Assumed Natural Strain method simultaneously makes the simulation more accurate and computationally faster. This is one of the key reasons why modern crash simulation software relies on element technologies built upon the foundation laid by ANS.

From a simple beam to a crashing car, the principle of Assumed Natural Strain is a golden thread. It is a testament to the idea that a deeper, more intuitive understanding of physics can lead to more elegant, more powerful, and ultimately more truthful methods for describing our world. It is not a trick; it is a discovery.