A shape memory alloy is a material that undergoes a phase transformation when it experiences a mechanical stress or temperature change. When the conditions return to normal, the SMA “remembers” its original shape and reverts to it.

*Watch the video for a closer look at shape memory behavior!*

The two crystal structures of SMA materials are known as *austenite* and *martensite*. Austenite is the SMA’s structure at higher temperatures, while martensite is the structure at lower temperatures. The transformation from austenite to martensite or vice versa is the cause of this “memory” behavior.

*The basic phase transformation process in an SMA.*

Common materials for SMAs include copper-aluminum-nickel and nickel-titanium alloys. The latter is often referred to as *nitinol*, which refers to its elemental makeup (*ni* for nickel and *ti* for titanium) as well as where it was first discovered (*nol* for Naval Ordnance Laboratory, which we will discuss later in this blog post).

*A coil made out of a nickel-titanium alloy, an SMA.*

If you search “nitinol” on various ecommerce sites, it is not hard to find SMA wires and other types of memory materials available for purchase.

The discovery of the unique behavior of SMAs is a story in itself. Early research into SMAs took place in the 1930s, when scientists looked into certain unexpected behaviors exhibited by different metals. Swedish chemist Arne Ölander noticed and described a pseudoelastic behavior when observing a gold-cadmium alloy. However, the term “shape memory alloy” did not come into play until a laboratory accident happened around 30 years later…

In the late 1950s and early 1960s, the U.S. Naval Ordnance Laboratory was working on metallurgy research, as described in Ref. 1. A scientist named William J. Buehler was melting and casting bars of nickel titanium. While waiting for the bars to cool, he dropped one of the cooled bars on the concrete floor and noticed that it caused a dull thudding sound. He found this odd and dropped a bar that was still hot, which produced a lighter sound, like a bell. Worried that something had gone wrong in the casting process, Buehler ran to a drinking fountain and cooled the hot nickel-titanium bar under the water. When he dropped the now-cool bar, it produced the thudding sound.

*A happy accident: Dropping a nitinol bar on the floor led to the discovery of its unique memory behavior.*

The effect was later demonstrated in a Naval Ordnance Laboratory meeting. Buehler’s assistant passed around a thin strip of nickel-titanium alloy that had been stretched, bent, and folded like an accordion. When the object reached Dr. David S. Muzzey, he took out his pipe lighter and heated it. The alloy quickly unfolded and reverted back to the original thin strip shape. After realizing the unique qualities and behaviors of the nickel-titanium alloy under different temperature conditions, the material came to be known as nitinol, an SMA.

More recently, and looking to the future, the development of shape memory materials has moved beyond alloys. Shape memory polymers, and other variations of shape memory materials, have been developed and even released for different commercial uses.

The unique behavior of SMAs makes them an attractive choice of material for manufacturing products and components in a wide range of industries. (Ref. 2)

In the aerospace industry, SMAs are used to develop lightweight, quiet, and efficient designs: three factors that are always a point of focus for aircraft. Components such as variable area fan nozzles, vibration dampers, and actuators are created with SMA materials. These devices are austenitic at their normal temperature and then transform to martensitic (and the desired shape) when cooled via temperature change due to airflow around the aircraft or even the ambient temperature change that occurs during a normal flight.

The change in temperature used to induce the phase change can be brought about in different ways. There can be an electronic component that heats the SMA device, or the temperature change can be caused by excess air from the aircraft’s other parts.

*A sample of a shape memory material being used in aircraft research and development. Image by Science Museum London/Science and Society Picture Library. Licensed under CC BY-SA 2.0, via Wikimedia Commons.*

A more recent technological advancement for shape memory materials in aircraft is wing morphing. SMAs are being used to develop an adaptable aircraft wing that can bend and change shape while in flight.

Back on the ground, automotive vehicles also benefit from SMAs, although the reasons have more to do with comfort and ease of use than operation. For example, some cars include an SMA valve for the pneumatic bladders in the seats. At a certain pressure, the lumbar support in a seat contours to the driver or passenger.

SMAs have also been used to build actuators that make vehicle trunks easier to close as well as noise, vibration, and harshness (NVH) valves to control noise and vibration in engines (an important performance indicator in the automotive industry).

Building design is another application area in which SMAs are useful. Including SMA rods in concrete beams, for example, can help prestress a bridge or building. On a smaller scale, shape memory materials can be used for reliable pipe fittings in a pipe network.

The use of SMAs for biomedical applications can reduce the need for medical intervention in patients. For example, medical stents can be implanted in arteries as a minimally invasive way to improve blood flow in cardiac patients. Microactuators and artificial muscles rely on SMAs for robotic prosthetics, which helps give patients with amputated limbs more freedom of motion.

*Stents, a minimally invasive cardiac treatment method, are commonly made out of SMAs. (Note: This image, taken from the Plastic Deformation During the Expansion of a Biomedical Stent tutorial model, does not include SMAs and is purely meant to illustrate the application discussed above.)*

On a smaller scale, SMAs are also used in orthodontia, such as braces; and optometry, such as glasses. If glasses frames are made with a shape memory material, they don’t have to be replaced if they get bent out of shape. Instead, they can be heated and return to their original shape.

Recently, shape memory materials have been applied to consumer electronics. For instance, autofocus components for smartphone cameras as well as certain mobile antennas can be made out of SMAs.

Certain craft projects and toys also use SMAs. One example is “bendy bracelets”, which are made with a shape memory material that enables them to bend and twist before easily returning to their original bracelet shape. (The Slinky® toy, unfortunately, is made out of a type of steel and won’t return to a tightly wound coil after getting tangled — a common struggle of children everywhere.)

When developing a design or component made out of an SMA, there are certain factors and risks to consider. A major drawback of SMAs is the risk of fatigue failure. There are only so many times certain SMAs can be bent and deformed before they return to a shape that’s a bit different from the original one (or they break).

Another drawback is the considerable lag time of the phase change for certain SMAs. If you search for “shape memory alloy” videos online, you can see that the time for the material to return to its original shape can be slow and unpredictable.

*Drawbacks such as lag time and fatigue can cause issues during the phase transformation cycle for SMAs.*

From a manufacturing standpoint, SMAs can be expensive to produce, which limits their accessibility to manufacturers and consumers. Also, since most of these materials rely on temperature to undergo changes, it could be risky to use SMAs as part of a device that operates under uncontrolled or unstable temperature conditions. An SMA used for an automotive application, for instance, must be able to perform under all possible temperature conditions that a vehicle might experience.

The mechanics of an SMA are difficult to describe because of the complexity of the phase transformations that take place. This complexity can make modeling SMAs quite the endeavor.

As of version 5.3a of the COMSOL® software, the Nonlinear Structural Materials Module includes two of the most common material models for SMAs: Lagoudas and Souza-Auricchio. Using these material models in your simulation, you are able to define the austenite and martensite properties as well as the phase transformation properties of an SMA. You are also able to easily account for heat transfer in an SMA with a built-in coupling between the *Heat Transfer in Solids* and *Solid Mechanics* interfaces.

The Uniaxial Loading of a Shape Memory Alloy tutorial demonstrates using an SMA material model in the COMSOL Multiphysics® software.

In the tutorial model, a nitinol cylinder is subject to axial tension and three separate studies are performed:

- Parametric sweep, showing the pseudoelasticity effect at different fixed temperatures
- Prescribed displacement sweep, showing that the pseudoelasticity effect is a partially unloading and partially loading loop
- Shape memory effect, shown after increasing the temperature

The model indicates that there is a temperature-dependent stress limit for the SMA. When the axial tension reaches the stress limit, the material transforms from the austenite to martensite structure; i.e., a “forward” transformation (deformation) occurs.

*Stress and strain of the SMA at different temperatures.*

During the unloading of the axial stress, the reverse transformation occurs. This “backward” transformation occurs at a lower stress level than the stress limit for the forward transformation and shows the material returning to its original shape.

*The stress and strain curves, which illustrate the shape memory effect in the alloy.*

Learn about the specialized features and functionality for mechanical analyses in the Structural Mechanics Module, an add-on to COMSOL Multiphysics, by clicking the button below.

Note: The Lagoudas and Souza-Auricchio material models for SMA also require the Nonlinear Structural Materials Module, an add-on to the Structural Mechanics Module.

- G.B. Kauffman and I. Mayo, “Chemistry and History: The Story of Nitinol: The Serendipitous Discovery of the Memory Metal and Its Applications,”
*Chem. Educator*, 2(2), 1997. - J.M. Jani et al., “A review of shape memory alloy research, applications and opportunities,”
*Materials and Design*, 56, 2014.

*Slinky is a registered trademark of POOF-SLINKY, LLC.*

One common treatment for atherosclerosis is a procedure called *percutaneous transluminal angioplasty*, which removes or compresses unwanted plaque that has built up in a patient’s coronary artery. This procedure sometimes relies on stents, placed within a blocked artery by an angioplasty balloon.

After reaching the intended location, the balloon inflates the stent, which locks into an expanded position. The balloon is then deflated and removed, while the stent remains in the artery. The expanded stent functions like a scaffold, keeping the blood vessel open and enabling blood to flow normally.

*A stent example. Image by Lenore Edman — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons.*

Of course, for the angioplasty procedure to be a success, the tools used must perform as intended. If the ends of the stent expand more than its middle — a common defect known as *dogboning* — the artery can face serious damage. Another potential issue is *foreshortening*, which makes it challenging to position the stent and can also damage the artery.

To avoid these issues and make the angioplasty a success, it’s necessary to evaluate stent designs. One step in this process is analyzing the deformation experienced by a stent.

For this example, let’s examine a Palmaz-Schatz stent model, the geometry of which is seen below. This model looks at the stress and deformation in a stainless steel stent that is expanded via a radial outward pressure on the tube’s inner surface. (The pressure represents the balloon expansion.) The original diameter of the stent is 0.74 mm, but after the expansion period, the middle section has a diameter of 2 mm.

Thanks to the inherent symmetry of the stent’s geometry, we can minimize the computational costs of this simulation by reducing the size of the model to 1/24 of its original geometry.

*The full stent geometry. The reduced geometry used in this example is represented by the darker meshed area.*

First, let’s look at the various stresses and strains experienced by the stent during operation. Below, we see the stress distribution in the stent at maximum balloon inflation (left) and the residual stress in the stent after balloon deflation (right). As expected, stress in the stent is reduced after the balloon deflates.

*Stress in the stent during balloon expansion (left) and after balloon deflation (right).*

Moving on, we analyze how the effects of dogboning (blue) and foreshortening (green) change in relation to pressure during balloon inflation. Using this plot, we can check for potentially harmful effects in the stent design and optimize its performance.

*Dogboning and foreshortening in the stent vs. the pressure in the angioplasty balloon.*

We also examine the effective plastic strains in the tube at maximum dogboning, as seen in the following image.

*Effective plastic strains and deformation at the time of maximum dogboning. The peak value is about 25%.*

In regard to the recoil parameters, note that the longitudinal recoil is around −0.9%, the distal radial recoil is about 0.4%, and the central radial recoil is approximately 0.7%. These parameters provide more details on how the stent behaves when the inflated balloon is removed.

With the information provided by simulations like this one, engineers can improve the design of stents and optimize their use in biomedical applications. To try this example for yourself, click on the button below.

- Learn about other medical uses of simulation by reading these blog posts:

Biological cells are essential for life as we know it. They not only store and replicate hereditary information in the form of DNA but also are instrumental in biological processes. In most, if not all, of these processes, the mechanical behavior of cells is a main factor in ensuring normal physiological functions.

It goes without saying that we would not exist or function without cells. Vertebrates utilize the circulation of red blood cells, erythrocytes, to deliver oxygen to body tissue. Fibroblasts use their contractile machinery to migrate to — and start the healing process of — wounds. The endothelial cells lining our blood vessels serve as filtration barriers. These cells not only rely on biochemical/transport mechanisms but also on their mechanical behavior to ensure normal physiological functions.

The structural entity responsible for providing cellular stiffness is an interconnected network known as the cytoskeleton, visualized in the image below. This cytoskeleton primarily consists of three types of polymerized filaments, each with their own distinct structure and mechanical characteristics:

- Actin
- Intermediate filaments
- Microtubules

This complex foundation provides cells with the ability to adapt their mechanical properties to the environment, both instantaneously and over time.

*A fibroblast cell with the cytoskeleton visualized, including actin (blue), intermediate filaments (green), and microtubules (red). Used with permission from Rathje et al. from the paper “Oncogenes induce a vimentin filament collapse mediated by hdac6 that is linked to cell stiffness”.*

Both cells and cytoskeletal networks are highly viscoelastic, as you can see in the plot below of a relaxation curve from a cell indentation experiment by atomic force microscopy (AFM).

*Force relaxation curve of a fibroblast cell.*

Numerous examples exist in which a diseased cell exhibits abnormal mechanical properties, promoting the progression of pathology. The cytoskeleton found in these cells is often found to behave differently compared to healthy cells. For example, cancer cells are known to exhibit significant stiffness variations compared to control cells. In many cases, this can be linked to the cytoskeleton. The intermediate filament network could be collapsing around the nucleus or there could be increased cell spreading (closely linked to the actin cytoskeleton through focal adhesions).

As mentioned, the cytoskeleton is a dynamic entity with the capability of remodeling itself on a time scale from milliseconds to hours. A consequence of this is a pronounced viscoelastic behavior, due to the nature of the constituent networks. For example, a solution of actin filaments behaves like a solid at short time scales and a liquid at longer time scales. This is due to the link between the thermal fluctuations of semiflexible filaments and their propensity to slide between each other; i.e., they are more or less kinematically constrained at short time scales. The temperature is also an important factor, partly because it affects the thermal behavior, but also because of various linking proteins in the solution.

Taken together, the mechanical behavior of this type of underlying polymer network, together with other cell constituents (e.g., cell nucleus and membrane), it is clear that a detailed analysis accounting for all of these factors is nearly impossible. However, it is possible to circumvent this challenge and obtain results by considering the cell at a macroscopic level.

By creating a finite element model in the COMSOL Multiphysics® software, you can essentially ignore the heterogeneous intracellular structure and instead view it as a continuum; i.e., the displacement field is continuous. This is an acceptable approximation if your goal is to quantify the macroscopic cell response to external stimuli.

The computational model described in this blog post is that of a relaxation test. A rigid indenter is pressed into the soft, viscoelastic cell, and the resulting relaxation of the indentation force is measured and compared to experimental data.

A model of a cell with typical dimensions is seen below. Note that the domain is created around the centerline. The semicircular section is the cell nucleus, which will also influence the mechanical response. We also create an indenter in the geometry and neglect the cell membrane in this analysis. For simplicity, we perform a 2D analysis by assuming the cell is axisymmetric.

The model is meshed with 2D elements and refined under the indenter.

The choice of material model for the cell cytoplasm and nucleus should reflect both the instantaneous and long-term response of the material. A linear elastic model is far too simple, as cells can typically withstand large strains and exhibit significant strain hardening. For the cytoplasmic response, we can choose a simple hyperelastic material model, the neo-Hookean model, in which stresses and strains are computed from a strain energy density function Ψ on the form

\Psi={\frac{\mu}{2}} (\overline{I}_1-3)+{\frac{\kappa}{2}}〖(J_el-1)〗^2

In this form, where the material is assumed (nearly) incompressible, the shear modulus *µ*, elastic volume ratio *J*_{el}, bulk modulus *κ*, and isochoric first invariant are included. To incorporate the viscoelastic behavior, two generalized Maxwell branches are also included. The nucleus has been found to be mainly elastic and is therefore modeled without viscoelastic branches.

The chosen material parameters are shown in this table:

Domain | Shear Modulus | Bulk Modulus | Energy Factor 1 | Relaxation Time 1 | Energy Factor 2 | Relaxation Time 2 |
---|---|---|---|---|---|---|

Nucleus | 5 kPa | 5000 kPa | N/A | N/A | N/A | N/A |

Cytoplasm | 0.07 kPa | 1000 kPa | 10 | 0.5 s | 10 | 50 s |

The bottom of the cell is constrained vertically. While in reality, the cell adheres to the substrate through focal adhesions, this should be a local effect and not significantly influence the force response.

Contact between the indenter and the cell is enforced by a penalty formulation, using the indenter as the source boundary. The indenter domain is prescribed a velocity of 0.1 µm/s, until the total vertical displacement is 4.6 µm. It is subsequently held fixed for the remainder of the analysis, up to a total time of 30 s.

The local deformation of the cell after indentation is shown in the plot below.

*Deformation of the cell under the indenter.*

The equivalent von Mises stresses at times 0.5 s and 30 s are shown below. Naturally, the stresses decrease due to stress relaxation because of the inclusion of viscoelastic branches for the cytoplasmic material model.

*Stress distribution at 0.5 s (left) and 30 s (right).*

The vertical reaction force on the indenter can be extracted from COMSOL Multiphysics and compared with experimental data.

*Results for the indentation force of the cell, both experimental (blue) and computed (red).*

The relaxation, as measured by experiments, typically exhibits at least two distinct regimes. These values are reasonably well predicted by the simple neo-Hookean model, along with its two viscoelastic branches. It should be noted that the initial indentation regime exhibits severe strain hardening prior to the constant slope (apparent in the plot above).

As discussed, COMSOL Multiphysics can be easily used to replicate the viscoelastic behavior of cells by (comparatively) simple material models. Naturally, an increasing level of complexity can be obtained by using more complicated material models. In this case, using other hyperelastic models, such as the Mooney-Rivlin or Ogden models, in combination with a greater number of viscoelastic branches may yield even more accurate results. Keep in mind that as more material parameters are needed, more experimental data points must be available for the material in question.

The cell is in reality a far more complex system than modeled here. There is a constant exchange of mechanical and biochemical signals that constantly alter the intracellular structure, cell shape, and locomotion behavior. Suffice to say, modeling the cell as a continuum is a major simplification, but such an approximation can serve us well in many cases. If we were to analyze metastasizing cells, for example, it may be enough to characterize their macroscopic stiffness in order to assess their capability to squeeze through tissue or arteries. In such a case, the stiffness of the cell as a whole in comparison to the obstacle would be the determining factor, not the detailed interactions of, say, the cytoskeleton and cell nucleus.

It should also be mentioned that the cell is not only a complex system but also far from deterministic and not uniquely characterized by a set of geometrical and material parameters. The response between individual cells varies depending on their health, state of locomotion, and cell cycle state, among other factors. To properly assess the mechanical cell response of a cell type experimentally, a greater number of individual cells would need to be probed. However, we are content with evaluating the capability of modeling the response of an individual cell.

In general, not only cells but also other biological materials can often be modeled by utilizing hyperelastic material models. Depending on the particular material and time scale, viscoelastic behavior can also be included. This opens up some interesting opportunities in the field of biomechanical modeling.

For example, a common type of cardiovascular condition is atherosclerosis in which white blood cells accumulate on the arterial wall, reducing blood flow and increasing the risk of a heart attack due to a blood clot. A common procedure to alleviate this condition is angioplasty, when a balloon is inserted into the artery and inflated. A mechanical stent is then often used to stabilize the artery section. Using COMSOL Multiphysics, we could capture the hyperelastic-viscoelastic behavior of the arterial wall, as well as composite characteristics due to collagen fiber directions, and compute the instantaneous and transient development of stresses and strains.

Björn Fallqvist is a consultant at Lightness by Design working with product development based on numerical analysis. He obtained a PhD from the Royal Institute of Technology in 2016, working with developing constitutive models to capture the mechanical behavior of biological cells. His main professional interest and specialization is in the fields of material characterization and using various material models to capture physical phenomena.

- Rathje et al, “Oncogenes induce a vimentin filament collapse mediated by hdac6 that is linked to cell stiffness”,
*Proceedings of the National Academy of Sciences*, 111, pp. 1515–1520, 2014. - B. Fallqvist et al., “Experimental and computational assessment of F-actin influence in regulating cellular stiffness and relaxation behaviour of fibroblasts”,
*Journal of the Mechanical Behavior of Biomedical Materials*, Vol 59, pp. 168–184, 2016.

To keep the model simple and take advantage of the symmetry of a balloon, we can build a 2D axisymmetric geometry that consists of only a rectangle and an ellipse, together with slightly larger double versions to account for the rubber balloon. Our goal is to see what happens if we let the same amount of water into different-sized balloons. To do so, we can parameterize the geometry and use a scaling factor to change the initial size of the balloon, while the material thickness and the neck radius remain constant.

*Geometry of two deflated water balloons of different sizes. The size is controlled by the stretching factor* fact: fact = *1 (left) and* fact = *2 (right).*

The water balloon model makes use of new features in version 5.3a of COMSOL Multiphysics, including improved fluid-structure interaction (FSI) functionality and a realigned moving mesh.

As of COMSOL Multiphysics version 5.3a, FSI is modeled via a *Multiphysics* node. The node connects the physics from *Fluid Mechanics* and *Structural Mechanics* interfaces. In contrast to earlier versions of the software, where there was a separate *Fluid-Structure Interaction* interface, we can now use all of the available features from the two-way coupled physics.

*The interfaces and the moving mesh after adding the FSI physics.*

In this example, it is easy to take gravity into account. All we need to do is place a checkmark in the *Laminar Flow* interface settings. This activates the gravity of the earth, which in turn has an effect on the mechanical behavior due to the hydrostatic pressure in the water. We can expect that there will be a noticeable effect of gravity on the results, and this effect will be more significant in the larger water balloon, because there is more mass in the beginning.

On the mechanical side, the physics settings can likewise be set up quickly. We only have to define a suitable material model that describes the hyperelastic properties of the material of the balloon correctly. In the Application Library, the Inflation of a Spherical Rubber Balloon model contains a variety of hyperelastic materials. We can use the Ogden model here, because it reproduces the analytical solution most accurately.

Interested in details about fitting measured data to different hyperelastic material models? Check out this previous blog post.

By the way, copying model interfaces between different models is now very simple. Since version 5.3a of the COMSOL® software, interfaces and components can be exchanged via the copy-paste functionality — even between two running COMSOL Multiphysics® simulations! This means that we can efficiently insert material settings from another model into the water balloon model.

*Parameters of the hyperelastic Ogden material model used for the balloon.*

Another improvement in COMSOL Multiphysics version 5.3a is the new positioning of the *Moving Mesh* interface. It is now found at a more prominent position under *Definitions*. One advantage of the new structure is that it helps avoid accidental overlaps between deforming and nondeforming areas. For the water balloon model, this improvement means that we have only two tasks for the mesh: selecting the balloon’s interior water as a *Deforming Domain* and adding a *Prescribed Normal Mesh Displacement* on the symmetry axis (to avoid unwanted movement away from this axis due to numerical inaccuracies).

The final step before solving the water balloon model is to set the water flow timing. Turning the tap on and off quickly with a defined amount of time in between can be expressed by a rectangle function. This function is multiplied to the inlet velocity of 15 cm/s, creating a flow of about 1.4 l/min.

*The water inlet velocity is controlled via a rectangle function.*

We can carry out a parametric sweep study to compare the simulation results for three different initial balloon sizes. All three balloons are filled with the same amount of water because the inlet velocity and the filling time period are the same. By far, the largest stress occurs in the smallest balloon. This is as expected, because the small balloon has the smallest surface and the largest relative volume increase.

*Von Mises stress distribution in the balloon material after inflation for three different initial sizes. (Note: These plots were created with the Cividis color table, a color table that is optimized for people with color vision deficiency, new as of COMSOL Multiphysics version 5.3a.)*

These results call for some animations! If we take a look at the transitional behavior of the inflation, we clearly see the influence of gravity on the largest balloon, because it oscillates before the water injection even starts. There is no prestress in the balloon, so it starts falling down a bit until the counterforce from the material is large enough to compensate for the gravity.

*Animation of the von Mises stress in the smallest water balloon during inflation.*

*Animation of the von Mises stress in the medium-sized water balloon during inflation.*

*Animation of the von Mises stress in the largest water balloon during inflation.*

The FSI functionality in COMSOL Multiphysics version 5.3a includes useful enhancements and is more user friendly than in previous software versions. With surprisingly little effort, it’s possible to set up a complex FSI model and solve it in a short time.

I am very curious to see how you use these new features to master your modeling challenges!

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Canadian Nuclear Laboratories aims to improve nuclear fuel because it limits the efficiency of power generation in nuclear reactors. As Andrew Prudil said in his keynote talk: “If we can increase the power rating of the reactors, that’s worth millions of dollars per day.” Optimized nuclear fuel also enables more green energy on a power grid and reduces the risk of nuclear accidents. Plus, the improved fuel can be used in existing reactors to enhance their performance.

Before engineers can develop improved nuclear fuel, they have to understand its behavior. This is no simple matter, as nuclear fuel experiences multiple physical phenomena during nuclear reaction fission, including high temperatures, radiation, mechanical loading, thermal expansion, the creation of fission products such as xenon and krypton, and more.

To learn more about nuclear fuel behavior during a reaction, in which “everything depends on everything else,” CNL turned to the COMSOL Multiphysics® software.

First, Prudil discussed a multiphysics model — created for his PhD thesis — that studies the behavior of nuclear fuel (or pellets, in this case). The Fuel and Sheath Modeling Tool (FAST) simulates a long row of pellets separated by small gaps inside a metal sheath. Each part of the model involves multiple types of physics. For instance, sheaths in nuclear reactors typically use zirconium-based alloys, which consist of anisotropic crystal structures. For accurate results, the model must account for how the crystals behave when pulled in different directions.

*From the video: Results for the FAST simulations.*

The simulations show how the ends of the pellets push outward to make room for the hot material at the center. The “hourglassing” phenomenon causes the ends of the pellets to create a wavy pattern in the cladding (exaggerated in the image above). FAST can also plot the radial displacement and various stress and strain fields, such as the hydrostatic pressure, von Mises stress, and axial creep. Prudil noted that the results show “very interesting, very rich spatial fields.”

With FAST, it’s possible to look at how nuclear fuel behaves in a continuum — both in terms of a temperature gradient and mechanical loading.

Prudil then discussed a model created at Canadian Nuclear Laboratories that simulates how fission gas forms bubbles on the boundary of a single grain of uranium oxide, a process that involves fission gas products such as xenon and krypton. At the grain boundary, these insoluble gases try to diffuse pressure by forming bubbles. The bubbles grow larger and larger and let gases escape.

The CNL model simulates this process for individual bubbles. Instead of using the traditional phase field method, which can be computationally expensive, they created the included phase technique to model the phase interface.

*Simulation results for the included phase technique. Animation courtesy Andrew Prudil and can be found in the paper: “A novel model of third phase inclusions on two phase boundaries“.*

Initially, the simulations show a random distribution of bubbles on the grain boundary. As time progresses, the bubbles combine to minimize the surface energy before collecting at the edges and vertices. CNL validated their approach, determining that they could control the contact angle of a single bubble on an infinite plane.

Wrapping up, Prudil mentioned that COMSOL Multiphysics could also be used to investigate other interesting multiphysics phenomena (e.g., columnar grain growth). With these capabilities, engineers can learn more about nuclear fuel and continue to advance the field.

To learn more about how CNL uses multiphysics modeling to understand the behavior of nuclear fuel, watch the keynote video at the top of this post.

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In *powder compaction*, a metal powder enters a die and is compacted through applied pressure. This high pressure comes from a punching tool inside the die cavity (often at the bottom surface). The powder is ejected from the cavity once it has been compacted and molded into a certain shape.

*Through powder compaction, metal powders are transformed into solid components. Image by Alchemist-hp — Own work. Licensed under CC BY-SA 3.0 DE, via Wikimedia Commons.*

With production rates averaging between 15 to 30 parts per minute, the powder compaction process enables manufacturers to quickly design strong components. Another benefit of this process is that it saves costs, as the component doesn’t need much additional work.

From a simulation standpoint, we need to perform a highly nonlinear structural analysis on powder compaction that accounts for:

- Contact between moving parts
- The elastoplastic constitutive law applied to the metal powder
- Geometric nonlinearity resulting from large displacements

As we demonstrate here, COMSOL Multiphysics® version 5.3 is ideal for handling such analyses.

For our example, let’s consider the fabrication of a cup-shaped component via powder compaction. The model geometry includes the workpiece (metal powder in this case) and the die. Note that the punch tool is not part of the model setup. We instead apply a prescribed displacement in the normal direction to the upper and lower faces of the powder in order to compact it. Because of the axial symmetry of the model, we can reduce its size to a 2D model, thus reducing the computational time of the simulation.

*The model geometry for a powder compaction analysis.*

The latest version of COMSOL Multiphysics includes five new porous plasticity models that cover various porosity values.

- Shima-Oyane
- Gurson
- Gurson-Tvergaard-Needleman
- Fleck-Kuhn-McMeeking
- FKM-GTN

These models are important for simulating powder compaction, as they allow us to accurately represent the porosity of the workpiece and produce reliable results. In this case, we combine the Fleck-Kuhn-McMeeking and Gurson-Tvergaard-Needleman models to describe an aluminum metal powder. Note that the die’s material properties assume it to be rigid.

In addition to the *Prescribed Displacement* boundary condition mentioned above, we also set the inner and outer dies as fixed domains.

From our simulation results, we can assess various properties of the metal powder at the end of compaction. To start, let’s look at the volumetric plastic strain. The strain at the center of the fillet appears to be minimal, while the strain near the ends is high. At the corner points of the workpiece, the strain is around 12% — likely the result of friction against the die.

*The volumetric plastic strain of the workpiece as the compaction process ends.*

During compaction, the porosity of the aluminum powder decreases, while the density and strength of the component increases. Based on the geometry and loading used in this scenario, we can expect that the changes to the porosity will be nonuniform.

The plot below shows the current void volume fraction contours of the powder; i.e., the porosity of the powder. Compared to the middle and top portions of the workpiece, the metal powder in the thin lower portion is more compact. Near the central area of the fillet, the powder is less compact because of material sliding on the rounded corner. The following animation illustrates how the volume fraction evolves over time.

*The current void volume fraction as the compaction process ends.*

*Changes in the volume fraction over time.*

Lastly, let’s consider the von Mises stress in the workpiece. The results indicate that the stress is greater in the areas where more compaction occurs.

*The von Mises stress within the workpiece.*

When simulating powder compaction, it is important to access the appropriate plasticity model that is preferably predefined in your analysis tool and available for direct use. To meet your modeling needs, COMSOL Multiphysics® version 5.3 brings you five new models that cover a wide range of porosity values.

For a helpful introduction to using these porous plasticity models, try out the example from today’s blog post.

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Arteries are blood vessels that carry oxygenated blood from the heart throughout the body. They are layered structures consisting of intima, with media and adventitia (also called externa) on the outside layers. The media and adventitia are the layers that are primarily responsible for mechanical behavior in healthy arteries.

Both of these layers are made up of collagenous soft tissues that exhibit noticeable strain stiffening behavior. Each layer has anisotropic properties because of families of collagen fibers. This means that these fiber-reinforced structures allow the blood vessels to experience large strains.

*An artery wall’s structure. Image by BruceBlaus — Own work. Licensed by CC BY 3.0, via Wikimedia Commons.*

A reliable constitutive model for arterial wall mechanics is essential to investigate changes in the arterial system due to age and disease, and for designing prostheses, among other uses (Ref. 3). The Holzapfel-Gasser-Ogden (HGO) constitutive model (Ref. 2) captures the anisotropic mechanical response described above, which has also been observed in lab experiments on excised arteries. Typical experiments in the lab measure the response of arterial sections subject to combined axial stretch and internal blood pressure, and numerical examples try to match this data to better understand its mechanics.

As Bower points out in Applied Mechanics of Solids: “*Finite strain viscoelasticity is not as well developed as finite strain plasticity, and a number of different formulations exist.*” In COMSOL Multiphysics version 5.2a, we implemented the Holzapfel model for large-strain viscoelasticity (Ref. 1, Ref. 3), which is well suited to combine with any of the predefined hyperelastic material models available in the COMSOL® software.

The author proposes a generalized Maxwell model based on the splitting of the strain energy density into volumetric and isochoric contributions

W(C,\Gamma_m) = W_{vol}(J) + W_{iso}(\bar{C}) + \sum_{m} \Psi_m(\bar{C},\Gamma_m)

Here, stands for the right Cauchy-Green deformation tensor and stands for its isochoric counterpart. The free energy associated to the nonequilibrium state, , is a function of the isochoric right Cauchy-Green tensor and internal strain-like variables (Ref. 1, Ref. 3).

The strain energy in the pure-elastic branch is normally denoted with the superscript to denote the long-term equilibrium (as ).

Then, Holzapfel derives the expression for the second Piola-Kirchoff stress in the hyperelastic and viscoelastic branches

S = 2\frac{\partial W}{\partial C} = S_{vol}^\infty + S_{iso}^\infty + 2\sum_m \frac{\partial \Psi_m}{\partial C}

and also defines the auxiliary stress tensors from thermodynamic considerations

Q_m =2 \frac{\partial \Psi_m(\bar{C},\Gamma_m)}{\partial C} =-2 \frac{\partial \Psi_m(\bar{C},\Gamma_m)}{\partial \Gamma_m}

The total second Piola-Kirchoff stress (hyperelastic and viscoelastic branches) is then given by

S = 2\frac{\partial W}{\partial C} = S_{vol}^\infty + S_{iso}^\infty + \sum_m Q_m

*A schematic representation of the second Piola-Kirchoff stress for a generalized Maxwell model in large-strain viscoelasticity.*

The evolution of the stresses in the viscoelastic branches is given by solving the rate equations

\dot{Q}_m+\frac{1}{\tau_m}Q_m = \dot{S}_{iso,m}

where is the relaxation time of the branch and is the corresponding second Piola-Kirchoff stress tensor.

Holzapfel also assumes that there should be an isochoric strain energy density associated to the spring on each branch, , so

S_{iso, m} = 2\frac{\partial W_{iso,m}}{\partial C}

The main assumption in the Holzapfel formalism is that the isochoric strain energy per branch depends on the isochoric strain energy of the main hyperelastic branch

W_{iso, m}= \beta_\alpha W_{iso}^\infty(\bar{C})

Here, the dimensionless coefficients are called *strain energy factors*.

The second Piola-Kirchoff stress per branch then becomes

S_{iso, m} = 2\frac{\partial W_{iso,m}}{\partial C} = \beta_mS_{iso}^\infty

and the rate to solve becomes

\dot{Q}_m+\frac{1}{\tau_m}Q_m = \beta_m \dot{S}_{iso}^\infty

The generalized Maxwell viscoelastic model in version 5.2a of COMSOL Multiphysics is available for all of the hyperelastic materials, and it also contains the same options for modeling thermal effects as implemented for linear viscoelasticity.

*With the User defined option, you can neglect thermal effects, use the predefined William-Landel-Ferry or Arrhenius shift functions, or define your own shift function.*

Let’s take a look at how large-strain viscoelasticity can be applied to biomechanics modeling.

To model the behavior of arterial walls after sudden changes in axial stress, we need to use a hyperviscoelastic material model, which goes beyond the HGO material model.

Take a look at the Arterial Wall Mechanics tutorial model for more details on how to set up an anisotropic hyperelastic material.

To start, let’s add viscoelastic behavior to this particular material model. As described in Ref. 3, a generalized Maxwell model of five branches added to the HGO model is suitable to model relaxation times ranging from 1 ms to 10 seconds. This model is suitable for quantitatively representing the viscoelastic response of circumferential segments of arteries (Ref. 3). To do so, we right-click on the *Hyperelastic material* node and add a *Viscoelasticity* node (we can also combine this with thermal expansion or other effects).

By default, we get a generalized Maxwell model with one branch. However, we can also use the standard linear solid (SLS) model or the Kelvin-Voigt viscoelastic model.

Then, we add the five branches with the corresponding energy factors and relaxation times as described in (Ref. 3). We are able to retrieve (and also save) such parameters from a text file.

After refurbishing the HGO hyperelastic material with five viscoelastic branches, we can simulate what happens in an artery section when subject to a constant axial strain for four minutes.

*The stresses in the viscoelastic branches. Note the different relaxation times for the stresses in the five branches of the generalized Maxwell viscoelastic material.*

The axial stress relaxes to the steady state for times longer than the highest relaxation time. The ability to model large-strain viscoelasticity in COMSOL Multiphysics enables us to easily investigate and understand different biomedical materials and applications.

- Try it yourself: Download the Arterial Wall Viscoelasticity tutorial model and related information
- See how linear viscoelasticity can be used to reduce vibrations in structural dampers

- G. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering,
*John Wiley & Sons*, 2000. - G. Holzapfel, T. Gasser, and R. Ogden, “A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models,”
*J. Elasticity*, vol. 61, pp. 1–48, 2000. - G. Holzapfel, T. Gasser, and M. Stadler, “A Structural Model for the Viscoelastic Behavior of Arterial Walls: Continuum Formulation and Finite Element Analysis”,
*European Journal of Mechanics A/Solid*, vol.21, pp. 441–463, 2002.

Squeeze-off is a procedure used for gas pipeline maintenance and repair. The process involves compressing a pipe to completely stop the flow of gas with a squeeze-off mechanism. Squeeze-off must be performed slowly to avoid damaging the pipes.

*A diagram (background) and photo (inset) of a general pipe squeeze-off procedure.*

The American Society for Testing and Materials (ASTM) has set standards for the squeeze-off procedure for gas pipelines. The location of the squeeze-off tool from the nearest pipe fitting must be either three lengths of the pipe diameter long or about 30 cm (twelve inches) long, whichever distance is greater.

This creates problems in the pipeline maintenance industry, as most average consumer pipes have a diameter of about 6 cm (2.375 inches). Although three lengths of this common pipe diameter is about 18 cm (seven inches), the standards say that the squeeze-off distance must be whichever length is greater, meaning that the squeeze-off tool would still have to be placed about 30 cm away from the pipe fitting. This can lead to digging into main roads and detours, causing costly and inefficient maintenance procedures.

These ASTM standards prompted researchers at GTI, sponsored by OTD, to investigate a question: Is the 30-centimeter distance for squeeze-off locations really necessary for smaller pipes, or can the standards be updated to allow smaller pipes to use the three-pipe-length diameter?

The team at GTI, Oren Lever and Ernest Lever, researched their question by implementing a fully parametric, time-dependent model of a polyethylene (PE) pipeline with a squeeze-off and fitting. They used both the Structural Mechanics and Nonlinear Structural Materials modules in COMSOL Multiphysics to define the numerical and mechanical properties of two sets of structural contacts: internal pipe-to-pipe contact and external pipe-to-squeeze-off contact.

The simulation enabled the researchers to analyze large deformations in the pipe at each of the different stages of the squeeze-off process, which include:

- Pressurization of the pipe
- Squeeze-off
- Hold
- Release
- Relaxation

*A simulation of the five steps of the squeeze-off process.*

*The total displacement of the pipe when it is fully squeezed off.*

The team also meshed the pipe under the squeeze bars of the squeeze-off mechanism to analyze the very large deformations that occur when the pipe is fully squeezed off and the gas flow is completely shut down. With the meshing capabilities of the COMSOL software, this process could be easily scaled to different pipe sizes besides the common 6-centimeter-diameter pipe.

The polyethylene material that makes up the pipe in question exhibits unique properties and behaviors, which the researchers needed to capture by way of a customized viscoelastic-plastic constitutive model. For this, they turned to Veryst Engineering, a COMSOL Certified Consultant, for assistance with implementing the chosen material model into their COMSOL Multiphysics simulation.

First, the team at Veryst chose the experimental material tests that were needed to calibrate the material model typically used for polyethylene, a thermoplastic. Then, they fit the parameters of this material model to the stress-strain response of the polyethylene material. Finally, they implemented a set of ordinary differential equations (ODEs) needed to use the customized material model in the simulation.

For the material tests, GTI used a medium-density polyethylene (MDPE) pipe material. They tested it in tension and compression at different temperatures; strains; and strain rates, especially at high levels of strain (such as when the pipe is completely shut off). They also performed loading and unloading tests on the chosen material.

*Tensile response of the MDPE material from experimental data and custom material model implemented by Veryst Engineering.*

They then used their own MCalibration optimization tool to find actual values of material parameters that fit the experimental data, ensuring that the custom material model was a very good fit for GTI’s simulation.

Through their research, GTI found that for smaller pipe diameters, such as those smaller than about 9 cm (3.5 inches), the closer squeeze-off distance of three pipe diameters would not cause strains that went beyond the current strain limits accepted in the pipeline industry.

To further validate their results, the researchers used accelerated lifetime testing. They found that under their updated standard squeeze-off distance, pipes would have an 80-year lifetime at an average operation temperature of 20ºC. This is actually the current industry-accepted standard life expectancy for these pipes.

*Results from the accelerated lifetime testing for the polyethylene pipe.*

With their question answered through simulation and testing, GTI is now helping to revamp standard gas pipeline maintenance procedures to be more efficient and cost effective, without sacrificing the most important element: safety.

- Read the full story about GTI and gas pipeline maintenance on page 18 of
*COMSOL News*2016 - Browse the COMSOL Blog for more information on simulation applications involving nonlinear structural materials
- Explore the latest functionality and tools for nonlinear structural mechanics modeling in COMSOL Multiphysics® version 5.2a

To begin, let’s take a look at a common geometrical configuration for a pressure vessel.

Pressure vessels are made in different shapes and dimensions. Cylindrical shapes are a common choice due to the trade off between manufacturing cost and performance.

Typically, cylindrical pressure vessels are made out of a cylinder capped by two heads. Different shapes for the heads are available, with the torispherical head often selected. Torispherical heads are composed of a crown, a toroidal shape knuckle, and a straight flange.

The two most common torispherical heads in use are the *Klöpper head* and the *Korboggen head*. What differentiates these two heads is the geometric relationships that are used to design the crown and the knuckle dimensions. In a Klöpper head, and . In a Korboggen head, and . Here, is the radius of the cylindrical part of the vessel, is the crown radius, and is the knuckle radius.

*Geometrical dimensions of a cylindrical pressure vessel with a torispherical head.*

Now that we’ve got our geometry set up, let’s briefly discuss some important elements when it comes to material plasticity and how it affects the design of pressure vessels.

Rolled metal sheets are commonly used in the manufacturing of pressure vessels. Most designs are such that a material operates within the elastic limit, as is the case when moderately large stresses act in them. The material then exhibits a deformation that is proportional to the load. Further, the shape returns to the original configuration once the force is removed. However, if the force exceeds a limiting value known as the *yield limit*, the body does not return to its original shape, even when the loading is completely removed. It instead remains partly deformed.

In such a case, we say that the body has undergone plastic deformation. For a pressure vessel made of rolled steel, this yield stress can be of the order of 500 MPa. Due to the high internal pressure, parts of the vessel in the vicinity of the transition at the knuckle may exceed the elastic limit and operate in plastic conditions.

In this example, a rather sophisticated plasticity model is incorporated into the mix: *Hill’s orthotropic plasticity*. For rolled steel plates, the yield limit in different directions is not the same due to the rolling process. Such behavior can be captured using Hill’s orthotropic plasticity model. The Nonlinear Structural Materials Module, an add-on to the Structural Mechanics Module and COMSOL Multiphysics, is the internal backbone for the elastoplastic analyses of pressure vessels. Now, with the Application Builder, we can make such physics available in an easy-to-use app that can be accessed by a wide range of users.

When building a simulation app, your task as the designer is to identify the important parameters that influence the outcome of the simulation. Once these are determined, the flexibility and functionality of the Application Builder enables you to create a specialized user interface (UI) that includes only those selected parameters. End-users can then easily run simulation tests and interpret the results without having to understand how the underlying physics are actually implemented within COMSOL Multiphysics.

Knowing that we are going to perform plasticity analyses on a pressure vessel, the question becomes this: Which parameters are relevant for selection as user inputs? The options to vary the geometry and material properties are obvious candidates. Another desirable feature is the ability to modify the operating conditions; for instance, the operating pressure inside the vessel and allowable yield fraction values based on safety considerations. Such parameters are included as inputs in our app and can be entered through the customized UI.

*The UI of the Stress Analysis of a Pressure Vessel app.*

After computation, the quantities of interest can be displayed in predefined plots:

- Von Mises effective stress
- Plastic strains
- Yielded region
- Yielded volume fraction as a function of the internal pressure

The Stress Analysis of a Pressure Vessel app is just one example of how the Application Builder can be used to encapsulate complex simulation studies within a user-friendly layout. Suppose, for instance, that you want to meet a certain safety condition for a given operating pressure inside the vessel. With an app, it is easy to alter the geometry and material properties to identify the right combination that works within the safety limits. Users are able to modify the parameters in the UI without having to understand the working principles of COMSOL Multiphysics.

With apps, as we’ve highlighted here, performing repeated computations with varying parameter settings is faster and easier than ever before. Start building apps today and experience how they can optimize your own design workflow.

- Download the demo app presented today: Stress Analysis of a Pressure Vessel
- Watch this video to learn how to build and run simulation apps

The human body, as we know, is a complex machine comprised of individual systems that work together. Soft tissue plays an important role in helping these systems thrive, connecting and supporting other structures and organs within the body. This type of tissue is comprised of collagen, elastin, and ground substance. While the ground substance helps keep the tissue hydrated, the elastin and collagen fibers determine the tissue’s mechanical behavior.

When the soft tissue undergoes deformation, the elastin fibers cause the tissue to stiffen, storing the majority of the strain energy. The collagen fibers, meanwhile, are comparatively inextensible and loose. These fibers increase the tissue’s stiffness when pulled tight, limiting the degree of deformation and protecting the tissue from injury. Because each of these collagen fiber families tend to have preferred directions, soft tissues possess anisotropic properties.

Consider arterial wall mechanics, for instance. Arteries are blood vessels comprised of three layers: the *intima*, the *media*, and the *adventitia*. The two outer layers — the media and the adventitia — are primarily responsible for the mechanics behind healthy arteries and are both made up of collagenous soft tissues. The collagen fibers give each layer anisotropic properties and allow the blood vessels to sustain rather large elastic deformations.

*The anatomy of an artery. Image by Maksim, via Wikimedia Commons.*

An important step in understanding and describing the mechanical behavior of arteries is designing a model that accurately reflects their anisotropic nonlinear properties. Let’s take a look at such an example from our Application Gallery.

In our Arterial Wall Mechanics tutorial model, the model geometry is designed to represent part of a rabbit’s carotid artery. To model the media and adventitia, we use a layered cylindrical tube, which is reduced to a 10° sector.

*A model of a carotid artery section. Here, the length (L) is 2.5 mm, the inner radius (R _{i}) is 0.71 mm, and the outer radius (R_{o}) is 1.1 mm. The media thickness is 0.26 mm and the adventitia thickness is 0.13 mm.*

The boundary conditions used in this example are meant to replicate typical experiments measuring arteries’ responses to a combination of axial stretch and internal blood pressure. With the use of roller boundary conditions, the bottom section of the artery is able to freely expand in the radial direction. Meanwhile, prescribed displacements in the axial direction address the impact of axial stretching on the top surface. Lastly, a pressure boundary load applies internal pressure on the inner surface.

Here, we consider axial stretching within the range of 1.5 to 1.9 and internal pressures within the range of 0 and 160 mmHg. Stretch refers to the ratio between the current length and the original length. Within such ranges, the mechanical response is highly nonlinear and produces large elastic deformations — a behavior that can be described mathematically by the theory of hyperelasticity.

To accurately account for the arteries’ mechanical response, we can implement the Holzapfel-Gasser-Ogden (HGO) material model in COMSOL Multiphysics. This nearly incompressible anisotropic hyperelastic material model is based on the article “A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models”, by G. Holzapfel, T. Gasser, and R. Ogden. The HGO model describes the mechanical behavior of the elastic ground substance and the collagen fiber network within the artery, accounting for the deformation and volumetric strain energy density in each fiber family.

To begin, we can first identify the static response of the artery to the applied boundary conditions. The figure below illustrates the fiber layout for the different fiber families. Red represents the media and blue represents the adventitia. The angles between the different fiber families vary.

*Fiber layout prior to deformation.*

Let’s now evaluate radial stress distribution through the thickness of the arterial wall. Here, we apply an axial stretch of 1.9 and an internal pressure of 160 mmHg.

*Radial stress distribution.*

Lastly, we can compare the internal blood pressure against the expansion of the inner radius. The analysis covers three different axial stretches within our initial range. Our results show good agreement with the findings from the article, which are illustrated by circles in the plot below.

*Comparing internal pressure with inner radius expansion.*

- Try it yourself: Download the Arterial Wall Mechanics tutorial model

Geometric nonlinearity may not even be explicitly introduced in a fundamental course on structural mechanics. In fact, geometric linearity is often tacitly assumed. In a geometrically linear setting, the equations of equilibrium are formulated in the undeformed state and are not updated with the deformation. This may sound a bit alarming at first, since computing deformations is what structural mechanics is all about.

However, in most engineering problems, the deformations are so small that the deviation from the original geometry is not perceptible. The small error introduced by ignoring the deformations does not warrant the added mathematical complexity generated by a more sophisticated theory. This is why a vast majority of analyses are made with an assumption of geometric linearity.

There are a number of cases where the deformation cannot be ignored, and not all of these cases comprise deformations that you would intuitively think of as being large.

The most important effects on the mathematics when you include geometric nonlinearity in COMSOL Multiphysics are:

- A distinction is made between the
*Spatial*and the*Material*frame. The spatial coordinates of a certain point () differ from the material coordinates of the same point () by the the displacement vector (), so that . It will thus matter whether you use uppercase or lowercase coordinate names in expressions. - The strains are represented by the Green-Lagrange strain tensor instead of the engineering strains.
- The stresses are represented by the Second Piola-Kirchoff stress tensor.
- Pressure loads take the deformation into account. Normals to boundaries are updated and area changes caused by stretching are taken into account.

You can read more about different stress and strain tensors in this previous blog post, but a small digression into the strain measures is needed. To this end, let us look at the difference between linear and the full nonlinear strain by considering some components of the strain tensor.

The *X*-direction Green-Lagrange normal strain can be written as

\epsilon_X = \frac{\partial u}{\partial X} + \frac{1}{2}(\frac{\partial u}{\partial X})^2 + \frac{1}{2}(\frac{\partial v}{\partial X})^2 + \frac{1}{2}(\frac{\partial w}{\partial X})^2

If the quadratic terms are omitted, the familiar engineering strain is retrieved:

\epsilon_X = \frac{\partial u}{\partial X}

Similarly, for a shear strain, the Green-Lagrange strain component is

\epsilon_{XY} = \frac{1}{2}(\frac{\partial u}{\partial Y} + \frac{\partial v}{\partial X} + \frac{\partial u}{\partial X} \frac{\partial u}{\partial Y} + \frac{\partial v}{\partial X} \frac{\partial v}{\partial Y} + \frac{\partial w}{\partial X} \frac{\partial w}{\partial Y} )

Again, the engineering strain is obtained by ignoring the nonlinear terms:

\epsilon_{XY} = \frac{1}{2}(\frac{\partial u}{\partial Y} + \frac{\partial v}{\partial X})

When a structure rotates significantly, the engineering strains used in basic theory will no longer give a useful representation. Rigid body rotations will cause nonzero components of the engineering strain tensor. This will, through the constitutive law, cause stresses that for physical reasons should not appear in a rigid body. Another way of viewing this is that any useful strain tensor must be able to reflect the fact that there is no stretching or change of relative angles in a rigid body motion.

Consider a 2D body rotating rigidly in the *xy*-plane around the origin. A simple linear plane stress model in which a rectangular steel plate is rotated 10° is shown below.

*Effective stress in a rectangular steel plate at a 10° rotation with no geometric nonlinearity.*

The result is an effective stress of 572 MPa, which is above the yield limit for the most common steel qualities. To see why this happened, let’s study the analytical solution:

A point originally placed at (X,Y) will then have moved to a new location (x,y), given by

\begin{matrix}

x = X \cos(\phi)-Y \sin(\phi) \\

y = X \sin(\phi) + Y \cos(\phi)

\end{matrix}

x = X \cos(\phi)-Y \sin(\phi) \\

y = X \sin(\phi) + Y \cos(\phi)

\end{matrix}

This means that the displacements (u,v) are

\begin{matrix}

u = x-X = X (\cos(\phi)-1)-Y \sin(\phi) \\

v = y-Y = X \sin(\phi) + Y (\cos(\phi)-1)

\end{matrix}

u = x-X = X (\cos(\phi)-1)-Y \sin(\phi) \\

v = y-Y = X \sin(\phi) + Y (\cos(\phi)-1)

\end{matrix}

The engineering strains will then be

\begin{array}{l}

\epsilon_X = \frac{\partial u}{\partial X} = \cos(\phi)-1 \\

\epsilon_Y = \frac{\partial v}{\partial Y} = \cos(\phi)-1 \\

\epsilon_{XY} = \frac{1}{2}(\frac{\partial u}{\partial Y}+\frac{\partial v}{\partial X}) = -\sin(\phi)+ \sin(\phi) = 0

\end{array}

\epsilon_X = \frac{\partial u}{\partial X} = \cos(\phi)-1 \\

\epsilon_Y = \frac{\partial v}{\partial Y} = \cos(\phi)-1 \\

\epsilon_{XY} = \frac{1}{2}(\frac{\partial u}{\partial Y}+\frac{\partial v}{\partial X}) = -\sin(\phi)+ \sin(\phi) = 0

\end{array}

For a rigid body rotation, all strains should be zero, but clearly two of these strain components are not. A metal will often yield at a strain that is of the order of 0.001. A fictitious strain of this size will already occur at a rotation of 2.5°. To keep the strain lower than 0.0001, there must not be rigid rotations larger than 0.8°. This means that even at angles where you often would expect a “small angle approximation” to be sufficient, the geometrically nonlinear approach must be used.

Using the same rigid body rotation as above, but using Green-Lagrange strains instead, gives

\epsilon_X = \frac{\partial u}{\partial X} + \frac{1}{2}(\frac{\partial u}{\partial X})^2 + \frac{1}{2}(\frac{\partial v}{\partial X})^2 = \cos(\phi)-1 + \frac{1}{2}(\cos(\phi)-1)^2+ \frac{1}{2}(\sin(\phi))^2 = 0

Now this strain tensor component is zero for any value of the rotation. This property can be shown for the whole Green-Lagrange strain tensor and also for arbitrary rotations.

By using a geometrically nonlinear formulation, you can avoid having these kinds of stress artifacts. This is confirmed by solving the same problem with geometric nonlinearity enabled. The stress levels are now pure numerical noise; 12 orders of magnitude lower than the yield limit.

*Effective stress at a 10° rotation while using geometric nonlinearity.*

Consider the two beams in the sketch below:

*Beams with different end conditions.*

At the right end, the upper beam is free to translate horizontally, while the lower beam is not. In a linear theory, these two end conditions are equivalent if the beam is subjected to a vertical load. There is no coupling between axial and bending action. However, in a geometrically nonlinear analysis, the different end conditions will lead to quite different results:

- When the end is free to move axially, the vertical displacement of the beam is almost the same as in the geometrically linear case.
- When the axial displacement is constrained, the vertical displacement will be smaller than in the linear case and have a strong nonlinear dependence on the load.

As the beam deflects, its center line will be stretched if the end cannot move inwards. This will introduce a significant axial force that will make the beam act similar to a wire in tension — the higher the tensile force, the more it will resist a transverse force.

*Midpoint deflection of a beam with a square cross section of 0.05 x 0.05. The red line indicates the load where the deflection in the linear analysis is 0.025 (half the height of the beam).*

The same ideas also apply to plates and shells. If the boundary conditions are such that deflection will cause in-plane tension, then the plate will become significantly stiffer with increasing deflection.

There is a rule of thumb saying that if the deflection of the beam or plate in a linear analysis exceeds half of its thickness, then geometrically nonlinear effects should be considered. This is indicated by the red line in the figure above.

As seen in the previous example, the stiffness of a structure can sometimes change significantly due to geometrically nonlinear effects. This is sometimes referred to as *stress stiffening*. The term is somewhat misleading, since it is also possible that the stiffness could decrease. If we were to add a compressive axial load to the beam above, its transverse stiffness would actually decrease.

Stress stiffening is important in, for example, rotating systems where the centrifugal forces can introduce significant tensile stresses. This causes the eigenfrequencies of the system to increase with the RPM.

*Campbell diagram showing how the natural frequencies of a rotating blade change with speed of rotation.*

Often, the loads that cause the prestress are not the same as the one for which you actually perform the analysis. So there may be two distinct load systems that must be analyzed separately.

In COMSOL Multiphysics, there are two predefined study types specifically intended for the analysis of prestressed systems:

- Prestressed Analysis, Eigenfrequency
- Prestressed Analysis, Frequency Domain

*Study types intended for the analysis of prestressed structures.*

These study types consist of two study steps in which step one is used for computing the prestress state. That study can be linear or nonlinear. The second study step is linear in itself, but includes the nonlinear terms caused by the geometric nonlinearity when setting up the stiffness matrix.

If you are interested in examples in which stress stiffening is important, please check out:

*Buckling*, or the loss of stability when the load reaches a certain critical value, is caused by geometrically nonlinear effects. In COMSOL Multiphysics, there is a specific study type called Linear Buckling for computing the first order approximation to the critical load.

*The Linear Buckling study type.*

In the linear buckling study, an approximate buckling load is obtained by solving an eigenvalue problem.

As an alternative, you can trace the full nonlinear response up to the point of collapse, and even past it. In this case, you must increase the load in smaller steps. This approach is significantly more computationally expensive, but more accurate.

*Load-deflection history with a buckling collapse at point A.
*

You can read more about buckling in this previous blog post.

Geometric nonlinearity is a property of the Study step. For those study types for which it is relevant, a check box is available in the settings for the study.

*Settings for a stationary study.*

Sometimes this check box is preselected and you cannot change it. This happens when you include certain physics nodes in the model tree that cannot be used in a linear context, such as:

- Hyperelastic material
- Large strain plasticity
- Contact

Note that most nonlinear material models, such as nonlinear elasticity or creep, do not assume geometric nonlinearity.

Geometrically nonlinear problems are often strongly nonlinear, and you need to consider that when supplying settings for the solver.

Think of the beam with the fixed end mentioned above. When solving the nonlinear problem, the solution after the first iteration will be the same as the solution to the linear problem so that all points on the beam move only vertically under a transverse load.

After the first iteration, there will thus be a significant axial elongation of the beam. Such an elongation is related to an axial force. As there is no net axial force (there is no external load in that direction), this force will end up as a residual for the next iteration. This unbalanced force may be larger than the applied load. To the nonlinear solver, this looks like a very nasty problem and the solver will often introduce damping.

Fortunately, these problems are often more well behaved than the numerics would indicate. You can then speed up the solution significantly by using a more aggressive iteration scheme than the default.

*Settings for the Fully Coupled solver.*

Using the Constant Newton scheme instead of the automatic adaptive scheme will cause the solver to make larger updates. The damping factor can be set to 1 (no damping) or possibly 0.9.

A problem where geometrical nonlinearity is the only source of nonlinearity will, in most cases, possess a unique solution for a certain load level. In this sense, it is possible to analyze the problem using a stationary analysis with a single load only. For convergence reasons, it is sometimes better to gradually increase the load using the parametric continuation solver.

An example of how the solver can be set up for a severely nonlinear problem is shown in our Pinched Hemispherical Shell tutorial model.

As we have shown above, there are several cases in which geometric nonlinearities must be considered when solving structural mechanics problems. So why don’t we always include this effect in our models to be on the safe side?

- Even if the nonlinear effect is very small, invoking the nonlinear solver will give you a significantly longer solution time. This is not an issue for small models, but when you are working with several million degrees of freedom, a reduction of the solution time by a factor of two really matters.
- Sometimes you want to be able to compare to an analytical solution, and such solutions are often based on linear theory.
- You may need to follow a standard or analysis procedure where it is assumed that a linear approach is used.
- In a geometrically nonlinear problem, it is necessary to use the actual load. If you just want to do a conceptual study of a structural response, the solution may not converge if the estimated load was too large.