We have been interested in cloaking for years and have covered this topic in various ways in previous blog posts. Although there are many different types of cloaking, one common theme is how complex the phenomenon is to achieve mathematically (and physically…).

*An ideal cloak, which is modeled as a spherical shell with a smaller sphere inside. In this optical cloaking example, light waves bend around the smaller sphere, causing it to seem invisible.
*

The concept starts with *metamaterials*. Metamaterials are artificial materials that depend on a certain structure and arrangement to work. *Cloaking* devices use these metamaterials to bend waves (such as thermal, electromagnetic, acoustic, and mechanical waves) around an object in order to hide or protect it.

Theoretically, different cloaking devices can perform different functions. For instance, electromagnetic cloaking can render things invisible from the human eye, while mechanical cloaking can hide an object from mechanical vibrations and stress. In reality, it’s not a simple task to cloak something — and this is especially true in structural mechanics. However, researchers are taking leaps forward in the realm of cloaking design.

For instance, you might recall reading about cloaking advancements for flexural waves in Kirchhoff-Love plates here on the COMSOL Blog. The research group that led this study overcame limitations that were previously associated with the cloaking of mechanical waves in elastic plates. They created a new theoretical framework for designing and building these invisibility cloaks and used COMSOL Multiphysics software to simulate and analyze the quality of their cloak.

More recently, researchers at the Karlsruhe Institute of Technology in Germany developed a very simple mathematical approach to cloaking based on a direct lattice transformation technique.

The team of scientists began by considering a 2D discrete lattice comprised of one material. Initially, an electrical analogy was studied, in which the lattice points within this structure were connected by resistors. These resistors were designed to act as a metamaterial, bending the electromagnetic waves and creating a cloak.

In the direct lattice approach, the lattice points of the structure were subjected to a coordinate transformation and the properties of the resistors were kept the same. Because the resistors and the connections between them were the same, the hole in the middle of the lattice and the distortion surrounding it could not be detected from the outside. Thus, in just one simple step, a cloak was successfully created.

The research team’s initial findings demonstrated the success of this simple and straightforward technique for cloaking in heat conduction, particle diffusion, electrostatics, and magnetostatics. Then, by replacing the resistors in the lattice structure with linear Hooke’s springs, the researchers found that their transformation approach was successful in cloaking elastic-solids as well.

To visualize and test the performance of the lattice-transformation cloak, the researchers used COMSOL Multiphysics simulation software. In the simulations, constant pressure was exerted onto the structure and the resulting strain was analyzed. The direct lattice approach was found to result in less error and less strain under various loading conditions — an indicator of very good performance.

Although mathematically *perfect* cloaking will never exist in reality, mechanical cloaking still has a lot of potential uses in the civil engineering and automotive industries. Using this technique, engineers could create strong materials that maintain their strength and durability, even when forming complex shapes. Constructing buildings of such material would help protect them from earthquake damage, for instance.

*Civil engineers could use mechanical cloaking to design support structures for bridges. (By Alicia Nijdam. Licensed under Creative Commons Attribution 2.0 Generic, via Wikimedia Commons).*

With mechanical cloaking, we could also see complex yet lightweight architecture, carbon-enforced cars, and tunnels with better stress protection in the future. Check out the links below for more information about this fascinating topic.

- Read more about the Karlsruhe Institute of Technology team’s mechanical cloaking technique from
*Phys.org* - Learn how fractals contribute to the magic of metamaterials
- Can you print an invisibility cloak with a 3D printer?
- Cloaking in science and fiction

3D printing has emerged as a popular manufacturing technique within a number of industries. The growing demand for this method of manufacturing has prompted greater simulation research behind its processes. Engineers at the Manufacturing Technology Centre (MTC) have identified their customers’ interest in a particular additive manufacturing technique known as shaped metal deposition. By building a simulation app, the team is better able to meet the demands of their customers while delivering more efficient and effective simulation results.

Designers and manufacturers are usually interested in testing various design schemes to create the most optimized device or process. As a simulation expert, you will often find yourself running multiple tests to account for each new design. The Application Builder, however, has revolutionized this process. By turning your model into a simulation app, you can enable those without a background in simulation to run their own tests and obtain results with the click of a button.

When designing an app, you can opt to include only those parameters that are important to your end-user’s particular analysis, hiding the model’s complexity while still including all of the underlying physics. As modifications are made to the design, app users can change specific inputs to simulate the performance of the new configurations. The result: A more efficient simulation process that allows engineers to focus on the design outcome rather than the physics behind the model.

Over the past few weeks, we’ve blogged about several of our own demo apps that are designed to help you get started with making apps. Today, we will share with you how a team at the MTC built their own app to analyze and optimize *shaped metal deposition* (SMD), an additive manufacturing (3D printing) technique. Let’s begin by exploring what prompted the development of this app.

*The MTC team behind the creation of the simulation app.*

The 3D printing industry has experienced tremendous growth within the last several years. As new initiatives have further developed the technology, 3D printing has emerged as a favorable method of manufacturing components for medical devices, automobiles, and apparel, to name a few.

At the MTC — which has recently become home to the UK National Centre for Net Shape and Additive Manufacturing — simulation engineers recognized their customers’ interest in additive manufacturing, with particular regards to shaped metal deposition. In contrast to powder-based additive manufacturing techniques, SMD is valued for its capability to build new features on pre-existing components as well as use a number of materials on the same part.

Similar to welding, this manufacturing technology deposits a mass of molten metal that is applied gradually on a surface. A cause for concern within this process is that the thermal expansion of the molten metal can deform the cladding as it cools. Thus, the final product can differ from the expected result.

*A simulation of temperature heating in the manufactured part, created by the MTC team. *

*Visible deformation on the manufactured part after six deposited layers.*

Using COMSOL Multiphysics, a team at the MTC created a model to better predict the outcome of the design by minimizing deformations or changing the design to account for such deformations. Responding to the growing popularity of this manufacturing technique, the MTC turned their model into a simulation app that could be shared across various departments within their organization.

The simulation app built by the MTC is based on a thermomechanical analysis of thermal stresses and deformation resulting from SMD thermal cycles. The app was designed to predict if the deposition process would create parts that fell within a specific range of tolerances. In some cases, this could require many tests to be run before arriving at an acceptable final deformation. With the app’s intuitive and user-friendly interface, app users are able to easily modify various inputs to test out each new design and analyze its performance.

*The MTC app’s user interface.*

Within the app, the MTC team has given users the ability to easily test out different geometries, change materials, apply meshing sequences, and experiment with various heat sources and deposition paths. The app also includes two predefined parametric geometries, as well as the option to import a custom geometry.

*Running a simulation using the app. This plot represents the temperature field.*

In *COMSOL News* 2015, Borja Lazaro Toralles, an engineer at the MTC, discussed the advantages of taking this approach to analyzing and optimizing SMD. “Were it not for the app, our simulation experts would have to test out each project we wanted to explore, something that would decrease the availability of skilled resources,” Lazaro Toralles noted in the article.

Since its development, the app has been shared with other members of the MTC team who do not possess simulation expertise. Distributing this easy-to-use tool throughout the organization has offered a simple way for team members to test and validate designs, expediting the simulation process and providing customers with faster results. Additionally, the availability of the app to the MTC engineers means that they are able to respond to companies who want to explore the use of this additive technology very rapidly and at a low cost.

The team at the MTC has already begun making updates to their simulation app, further enhancing its functionality and adding new resources for the end-users. Using the Physics Builder, the engineers have started designing a customized physics interface that will enable the modeling of more complex tool paths and melt pools. Tailored to their design needs, this interface will offer engineers an easier and faster method of implementation that is less prone to error.

To further improve the usability of the app, the MTC is planning to offer more contextual guidance through the card stack tool provided by the Application Builder. For increased accuracy, they have plans to add the capability of modeling the evolution of the microstructure on a macroscopic level to predict heat-affected zones.

Recognizing the advantages of building simulation apps, the MTC is looking to create additional apps to evaluate topology optimization as well as the modeling of hot isostatic pressing (HIP). They are also interested in potentially linking COMSOL Server™ with their own cluster to provide a secure environment for managing, running, and sharing simulation apps. This would be especially beneficial for those companies that do not possess high computational power.

- Read a related article in
*COMSOL News*2015: “Optimizing 3D Printing Techniques with Simulation Apps“ - To learn more about creating your own simulation apps, watch this video: Introducing the Application Builder in COMSOL Multiphysics
- Check out our series of blog posts on 3D printing

Miniature devices have many applications and researchers are constantly finding new uses for them. One such use, which we’ve blogged about before, is a microfluidic device that could let patients conduct immune detection tests by themselves. But to work in the microscale, devices like this one, of course, rely on even smaller components such as micropumps.

Let’s turn to a tutorial model of a valveless micropump mechanism that was created by Veryst Engineering, LLC using COMSOL Multiphysics version 5.1.

The micropump in the tutorial model creates an oscillatory fluid flow by repeating an upstroke and downstroke motion. The fluid flow enters a horizontal channel containing two tilted microflaps, which are located on either side of the micropump. The microflaps passively bend in reaction to the motion of the fluid and help to generate a net flow that moves in one direction. Through this process, the micropump mechanism is able to create fluid flow without the need for valves.

*The geometry of the micropump mechanism tutorial.*

Please note that the straight lines above the microflaps are there to help the meshing algorithm. Check out the tutorial model document if you’d like to learn how this model was created.

The tutorial calculates the micropump mechanism’s net flow rate over a time period of two seconds — the amount of time it takes for two full pumping cycles. The Reynolds number is set to 16 for this simulation so that we can evaluate the valveless micropump mechanism’s performance at low Reynolds numbers. The *Fluid-Structure Interaction* interface in COMSOL Multiphysics is instrumental in taking into account the flaps’ effects on the overall flow, as well as making it an easy model to set up.

*Left: At a time of 0.26 seconds, the fluid is pushed down and most of it flows to the outlet on the right. Right: At a time of 0.76 seconds, the fluid is pulled up and most of it flows from the inlet on the left.*

The simulation starts with the micropump’s downstroke, which is when the micropump pushes fluid down into the horizontal channel. This action causes the microflap on the right to bend down and the microflap on the left to curve up. In this position, the left-side microflap is obstructing the flow to the left and the flow channel on the right is widened. This naturally causes the majority of the fluid to flow to the right, since it is the path of least resistance.

During the following pumping upstroke, fluid is pumped up into the vertical chamber. Here, the flow causes the microflaps to bend in opposite directions from the previous case. This shift doesn’t change the direction of the net flow, because now the majority of the fluid is drawn into the flow channel from the inlet on the left.

Due to the natural deformation of the microflaps caused by the moving fluid, both of these stages created a left-to-right net flow rate. But how well did the micropump mechanism do at maintaining this flow over the entire simulation time period?

*The net fluid volume that is pumped from left to right.*

During the two-second test, the net volume pumped from left to right was continually increased, with a higher net flow rate during peaks of the stroke speed. This valveless micropump mechanism can function even at a lower Reynolds number.

The valveless micropump mechanism could have many future applications, one of which is to work as a fluid delivery system. In such a scenario, a micropump mechanism could take fluid from a droplet reservoir on its left and move it through a microfluidic channel to an outlet on its right. In this post we have shown just one set of simulation results. By experimenting with the tutorial model set up by Veryst Engineering, you can visualize how a valveless micropump may work in different situations and use this information to discover new uses for micropump mechanisms.

- Download the tutorial model: Micropump Mechanism

First proposed by Adrian Bejan in 1996, the *constructal law* is a theory that summarizes the generation of design and evolution phenomena within nature. With the foundation that designs universally evolve in a particular direction in time, the law states that the shape and structure of a flow system will change over time to facilitate an easier flow. In other words, design configurations will evolve in a direction that enables greater access for those currents flowing throughout it.

Take a plant, for instance. For a plant to thrive, it is important that its structure promotes the flow of nutrients and water. The plant will thus reorient its branches over time to ensure that its design facilitates the proper flow. Another example is a river. When spreading into the sea, a river will often encounter obstacles resulting from settling sediment. As such, the river will change the direction of its flow to avoid running into these obstructions.

While common in nature, these tree-like architectures are evident in manmade designs as well. Aircraft designs, for instance, have evolved over time to transport a greater number of people and goods across further distances. As noted by research from Bejan and his colleagues, maintaining the proportionality of the engine mass to the body of the aircraft has been important to aircraft success. When comparing this to statistics from various mammals, insects, and birds, a nearly identical pattern was found between mass and speed — a reflection of the connection between the law of design in nature and the evolution of manmade technologies.

*The constructal law has shown similarities between the pattern of manmade designs and elements of nature. Left: Image by Altair78, via Wikimedia Commons. Right: Image by Dan Pancamo, via Wikimedia Commons.*

Recently, Bejan and a team of researchers from Duke University and Université de Toulouse in France used the theory behind the constructal law to enhance the performance of phase change energy storage systems by maximizing the melting rate of the phase-change materials. Before diving into their research, let’s take a closer look at phase change energy storage technology.

Energy efficiency is an important consideration in the design of modern technologies. In an effort to reduce environmental impact and save on costs, designers and manufacturers often turn to energy storage techniques as a solution.

One common method used to reduce energy consumption is thermal energy storage technology. Used since the late 19^{th} century, phase change energy storage technology has become a valued approach to energy storage in refrigeration systems as well as commercial buildings. This energy storage technique involves the heating or cooling of a storage medium. The thermal energy is then collected and set aside until it is needed in the future.

Phase-change materials are often used as a storage medium within the thermal energy storage process. When undergoing phase change, a *phase-change material* (PCM) absorbs a great deal of heat at a near average temperature. As it continues to absorb heat, the material does not experience a significant rise in temperature until it is fully melted.

Once the ambient temperature of the environment around the material decreases, the PCM becomes solid again, releasing its stored heat. The use of phase-change materials as a medium is particularly valued, as these materials are able to store high capacities of thermal energy and exhibit isothermal behavior throughout the charge and discharge process.

*A schematic demonstrating how a phase-change material works. Image by Pazrev — Own work, via Wikimedia Commons.*

Looking to improve the performance of phase change energy storage systems, Bejan and his team of researchers applied the constructal law to their analysis of energy storage through the melting of a phase-change material. In their study, heating was applied along invading lines with the ability to freely morph.

Using COMSOL Multiphysics in their simulation research, the team found that the melted material evolved into an S-shaped flow. By enabling the heat to spread freely through the cold material as in a tree-like architecture, the material was able to melt more quickly. The findings also showed that increasing the tree structure’s complexity and varying its branching angle and stem length simultaneously helped to further accelerate the melting process.

While a heating and cooling coil is traditionally embedded in a phase-change material, the researchers found that the most effective way to spread the heat within the volume was to allow a natural flow to develop and evolve over time. As the constructal law indicates, giving a design the freedom to morph naturally ultimately enhances the performance of the system. Applying the theory behind the constructal law to phase change energy storage systems paves the way for advancing the energy efficiency of this technology, enabling it to continue to evolve and improve well into the future.

Solar energy is created when photovoltaic cells made up of a semiconducting material, such as silicon, transform sunlight (photons) into electricity (voltage). In photovoltaic cells, microthin layers of silicon make up integrated circuits that conduct electricity.

Photovoltaic cells range from a single cell to a group of cells called a *module*, or a group of modules called an *array*. Different amounts of solar cells can power anything from a small electronic device, such as a cellphone, to a whole building. Solar cells are used in a variety of applications, including transportation, telecommunication, space exploration, and more.

The power derived from solar cells provides a clean and sustainable source of energy; solar energy is a renewable resource. The operation of solar cells doesn’t require any moving parts or emissions, which makes them even more energy efficient.

As much as solar energy is a preferable electricity source, there are still some issues with how it is manufactured. Although the operation of solar cells is efficient, manufacturing them costs a high amount of power in itself. The silicon needed for solar cells also has to be extremely pure. The purity of silicon directly affects the amount of electricity produced from sunlight, which is an amount known as its *photovoltaic conversion efficiency*. While common metallurgical silicon is 99.9% pure, the silicon used for photovoltaic cells must be 99.9999% pure.

To fill the need for highly pure silicon for solar cells, a research team at EMIX developed the cold crucible continuous casting (4C) process, which transforms metallurgical silicon into a substance ready for solar cell use. First, silicon is fed into a water-cooled crucible and inductively heated to a melting temperature of 1414°C. Then, the silicon is electromagnetically mixed inside the crucible. Lorentz forces prevent contact between the silicon and the crucible walls.

The mixture homogenizes at the solid-liquid interface, which enhances crystallization conditions for a high purity. The melted substance is pulled through the bottom of the crucible, cooled, and solidified into a rod through an annealing process. Finally, the high-purity silicon is sawed into ingots and sold to solar cell manufacturers, where it is sliced into 200-micrometer thick wafers for use in photovoltaic cells.

*The geometry of the crucible used in the 4C process. Copyright © EMIX.*

EMIX has been using COMSOL Multiphysics software nearly as long as they’ve been manufacturing photovoltaic-quality silicon. Using the *Heat Transfer in Fluids* interface and the *Laminar Flow* interface in the Heat Transfer Module, the research team can adjust certain variables to ensure that the 4C process optimizes production rates and is as energy efficient as the solar cells it produces.

*The research team at EMIX from left to right: Julien Givernaud, Elodie Pereira, Nicolas Pourade, Florine Boulle, Alexandre Petit. Copyright © EMIX.*

By simulating the 4C process, the EMIX team can test a variety of different variables, including:

- Cooling method
- Pull rate
- Crucible and coil shapes
- Characteristics of the furnace
- Effect of the electromagnetic field
- Shape of the solid-liquid interface
- Effect of elastic stresses on crystallization behavior

*A simulation of the 4C process using COMSOL Multiphysics simulation software. Copyright © EMIX.*

Using COMSOL Multiphysics simulation software to enhance the 4C process has proved to be extremely beneficial for EMIX. They were able to estimate both inductance and impedance in their silicon manufacturing, improve their crucible design for electrical efficiency, and test different parameters for continuous casting. This led to higher production rates, lower stresses in the silicon ingots, energy savings of 15%, and a 30% increase in pull rate.

Thanks to an innovative, optimized, and energy efficient 4C process, EMIX has been able to streamline silicon production. With simulation, they have even identified processes that they will soon test on an industrial scale. For now, we can be sure that the future of photovoltaic cells and solar energy is looking bright.

- Read more about how EMIX uses simulation on page 20 of
*COMSOL News*2015 - Run a similar simulation: Download our Continuous Casting tutorial to get started

Marangoni convection — also called *thermocapillary convection* — is important in a number of processes, including welding, crystal growth, and electron beam melting. Due to the types of metals used and the extremely high temperatures involved, performing experiments to analyze Marangoni convection often proves to be rather challenging. The impact of gravity, which mixes up this convective effect with the Marangoni effect, also adds to the difficulty of studying this phenomenon.

At NASA, researchers analyzed Marangoni convection to see how mass and heat move within a fluid under microgravity conditions. Conducting the experiment in microgravity enabled the research team to create silicone oil columns much larger than those that could be studied on Earth, offering a more detailed look at the flow and instability within them. Additionally, suppressing the influence of gravity helped eliminate the possibility of gravity-induced deformation, thus enhancing the accuracy of their results.

With numerical experiments, it is very easy to separate effects that are simply impossible to remove in an experiment on Earth. Our Marangoni Effect tutorial uses a transparent liquid at ambient temperatures to find the velocity field induced through the Marangoni effect in a fluid with known thermo-physical properties. The transparency of the silicone oil makes it easy to implement and compare our simulation results with the microgravity experimental findings.

To begin, we must solve the Navier-Stokes equations to model the velocity field and pressure distribution in the fluid. Keep in mind that variations in temperature affect the velocity and cause a buoyancy force that needs to be represented in the equations. This can be done by using the Boussinesq approximation in the Navier-Stokes equations.

With the *Laminar Flow* interface, we can solve the momentum balance equations. To solve for heat transfer, we use the *Heat Transfer in Fluids* interface. Finally, we use the *Non-Isothermal Flow* multiphysics coupling to set the convective term in the heat equation and the *Marangoni Effect* multiphysics coupling to impose that the shear stress is proportional to the temperature gradient.

*The setup of the tutorial model. The Multiphysics node contains both the nonisothermal coupling and the Marangoni effect.*

This simulation presents three multiphysics couplings that must be solved using the nonlinear solver:

- Because of the temperature dependency of the fluid density, \rho, accounted for following the Boussinesq approximation, the gravity force, -\rho \textbf{g}, is given by an expression that includes temperature.
- Convective heat transfer depends on the velocity of the momentum balance.
- The Marangoni effect relates the shear stress applied at the free surface to the surface temperature gradient.

In our simulations, we analyze a gradual increase in temperature difference between vertical walls. For an almost unnoticeable temperature increase of 1 mK, the temperature field and velocity field have only a slight relation, and the decrease appears linear from left to right.

*The results of a Marangoni effect simulation after only a small change in temperature. The background color represents the temperature field and the red arrows indicate the velocity field. The black lines are isotherms.*

With an increase of 50 mK, Marangoni convection increases the fluid flow and temperature distribution. The temperature decrease is no longer linear across the plot.

*The results of the simulation after a temperature increase of 50 mK.*

Finally, we test a temperature difference of 2 K. The temperature and velocity fields are distinctly coupled and the fluid accelerates at the surface where the temperature gradient is highest.

*The results of the simulation when the temperature difference is raised to 2 K.*

As indicated by the simulation results, the Marangoni effect becomes predominant as the difference in temperature increases.

For the same temperature difference of 2 K, we can easily remove the gravity contribution and keep the Marangoni effect. With the same objective of understanding how buoyancy forces compare with the Marangoni effect, we can simply disable the Marangoni contribution at the surface, leaving the surface free of stress. The results show that the Marangoni effect is predominant versus buoyancy forces. The shape of the curve shows a peak close to the cold right wall, which is characteristic of the fluid behavior of high Prandtl numbers.

*The results of the horizontal velocity at the surface versus the horizontal coordinate (m) for a temperature difference of 2 K. Blue represents both the Marangoni effect and the buoyancy effect; green represents only the Marangoni effect; and red represents only the buoyancy effect.*

In this blog post, we have demonstrated how to set up a model representing an experiment combining gravity and Marangoni effects. Separating these two effects is challenging in an experimental setting. In numerical simulations, this process is straightforward, facilitating an understanding of each effect.

You can reproduce the results shown here by downloading the Marangoni Effect tutorial from our Application Gallery. This example uses the Non-Isothermal Flow and Marangoni Effect multiphysics couplings available in the Heat Transfer Module.

While we have focused our attention here on single-phase flows, it is worth mentioning that the Marangoni effect is also handled in the two-phase flow interfaces, which are available in the CFD Module and the Microfluidics Module.

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In the vast majority of simulations involving linear elastic materials, we are dealing with an isotropic material that does not have any directional sensitivity. To describe such a material, only two independent material parameters are required. There are many possible ways to select these parameters, but some of them are more popular than others.

Young’s modulus, shear modulus, and Poisson’s ratio are the parameters most commonly found in tables of material data. They are not independent, since the shear modulus, G, can be computed from Young’s modulus, E, and Poisson’s ratio, \nu, as

G = \frac{E}{2(1+\nu)}

Young’s modulus can be directly measured in a uniaxial tensile test, while the shear modulus can be measured in, for example, a pure torsion test.

In the uniaxial test, Poisson’s ratio determines how much the material will shrink (or possibly expand) in the transverse direction. The allowable range is -1 <\nu< 0.5, where positive values indicate that the material shrinks in the thickness direction while being pulled. There are a few materials, called *Auxetics*, which have a negative Poisson’s ratio. A cork in a wine bottle has a Poisson’s ratio close to zero, so that its diameter is insensitive to whether it is pulled or pushed.

For many metals and alloys, \nu \approx1/3, and the shear modulus is then about 40% of Young’s modulus.

Given the possible values of \nu, the possible ratios between the shear modulus and Young’s modulus are

\frac{1}{3} < \frac{G}{E} < \infty

When \nu approaches 0.5, the material becomes incompressible. Such materials pose specific problems in an analysis, as we will discuss.

The bulk modulus, K, measures the change in volume for a given uniform pressure. Expressed in E and \nu, it can be written as:

K = \frac{E}{3(1-2\nu)}

When \nu= 1/3, the value of the bulk modulus equals the value of Young’s modulus, but for an incompressible material (\nu \to0.5), K tends to infinity.

The bulk modulus is usually specified together with the shear modulus. These two quantities are, in a sense, the most physically independent choices of parameters. The volume change is only controlled by the bulk modulus and the distortion is only controlled by the shear modulus.

The Lamé constants \mu and \lambda are mostly seen in more mathematical treatises of elasticity. The full 3D constitutive relation between the stress tensor \boldsymbol \sigma and the strain tensor \boldsymbol \varepsilon can be conveniently written in terms of the Lamé constants:

\boldsymbol \sigma=2\mu \boldsymbol \varepsilon +\lambda \; \mathrm{trace}(\boldsymbol{\varepsilon}) \mathbf I

The constant \mu is simply the shear modulus, while \lambda can be written as

\lambda = \frac{E \nu}{(1+\nu)(1-2\nu)}

A full table of conversions between the various elastic parameters can be found here.

Some materials, like rubber, are almost incompressible. Mathematically, a fully incompressible material differs fundamentally from a compressible material. Since there is no volume change, it is not possible to determine the mean stress from it. The state equation for the mean stress (pressure), *p*, as function of volume change, \Delta V, as

p = f(\Delta V)

will no longer exist, and must instead be replaced by a constraint stating that

\Delta V = 0

Another way of looking at incompressibility is to note that the term (1-2\nu) appears in the denominator of the constitutive equations, so that a division by zero would occur if \nu= 0.5. Is it then a good idea to model an incompressible material approximately by setting \nu= 0.499?

It can be done, but in this case, a standard displacement based finite element formulation may give undesirable results. This is caused by a phenomenon called *locking*. Effects include:

- Overly stiff models.
- Checkerboard stress patterns.
- Errors or warnings from the equation solver because of ill-conditioning.

The remedy is to use a *mixed formulatio*n where the pressure is introduced as an extra degree of freedom. In COMSOL Multiphysics, you enable the mixed formulation by selecting the *Nearly incompressible material* checkbox in the settings for the material model.

*Part of the settings for a linear elastic material with mixed formulation enabled.*

When Poisson’s ratio is larger than about 0.45, or equivalently, the bulk modulus is more than one order of magnitude larger than the shear modulus, it is advisable to use a mixed formulation. An example of the effect is shown in the figure below.

*Stress distribution in a simple plane strain model, \nu = 0.499. The top image shows a standard displacement based formulation, while the bottom image shows a mixed formulation.*

In the solution with only displacement degrees of freedom, the stress pattern shows distortions at the left end where there is a constraint. These distortions are almost completely removed by using a mixed formulation.

In general cases of linear elastic materials, the material properties have a directional sensitivity. The most general case is called anisotropic, which means all six stress components can depend on all six strain components. This requires 21 material parameters. Clearly, it is a demanding task to obtain all of this data. If the stress, \boldsymbol \sigma, and strain, \boldsymbol \varepsilon, are treated as vectors, they are related by the constitutive 6-by-6 symmetric matrix \mathbf D through

\boldsymbol \sigma= \mathbf D \boldsymbol \varepsilon

Fortunately, it is common that nonisotropic materials exhibit certain symmetries. In an orthotropic material, there are three orthogonal directions in which the shear action is decoupled from the axial action. That is, when the material is stretched along one of these principal directions, it will only contract in the two orthogonal directions, but not be sheared. A full description of an orthotropic material requires nine independent material parameters.

The constitutive relation of an orthotropic material is easier when written on compliance form, \boldsymbol \varepsilon= \mathbf C \boldsymbol \sigma:

\mathsf{C} =

\begin{bmatrix}

\tfrac{1}{E_{\rm X}} & -\tfrac{\nu_{\rm YX}}{E_{\rm Y}} & -\tfrac{\nu_{\rm ZX}}{E_{\rm Z}} & 0 & 0 & 0 \\

-\tfrac{\nu_{\rm XY}}{E_{\rm X}} & \tfrac{1}{E_{\rm Y}} & -\tfrac{\nu_{\rm ZY}}{E_{\rm Z}} & 0 & 0 & 0 \\

-\tfrac{\nu_{\rm XZ}}{E_{\rm X}} & -\tfrac{\nu_{\rm YZ}}{E_{\rm Y}} & \tfrac{1}{E_{\rm Z}} & 0 & 0 & 0 \\

0 & 0 & 0 & \tfrac{1}{G_{\rm YZ}} & 0 & 0 \\

0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm ZX}} & 0 \\

0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm XY}} \\

\end{bmatrix}

\begin{bmatrix}

\tfrac{1}{E_{\rm X}} & -\tfrac{\nu_{\rm YX}}{E_{\rm Y}} & -\tfrac{\nu_{\rm ZX}}{E_{\rm Z}} & 0 & 0 & 0 \\

-\tfrac{\nu_{\rm XY}}{E_{\rm X}} & \tfrac{1}{E_{\rm Y}} & -\tfrac{\nu_{\rm ZY}}{E_{\rm Z}} & 0 & 0 & 0 \\

-\tfrac{\nu_{\rm XZ}}{E_{\rm X}} & -\tfrac{\nu_{\rm YZ}}{E_{\rm Y}} & \tfrac{1}{E_{\rm Z}} & 0 & 0 & 0 \\

0 & 0 & 0 & \tfrac{1}{G_{\rm YZ}} & 0 & 0 \\

0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm ZX}} & 0 \\

0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm XY}} \\

\end{bmatrix}

Since the compliance matrix must be symmetric, the twelve constants used are reduced to nine through three symmetry relations of the type

\tfrac{\nu_{\rm YX}}{E_Y} = \tfrac{\nu_{\rm YX }}{E_X}

Note that \nu_{\rm YX} \neq \nu_{\rm XY}, so when dealing with orthotropic data, it is important to make sure that the intended Poisson’s ratio values are used. The notation may not be the same in all sources.

Anisotropy and orthotropy commonly occur in inhomogeneous materials. Often, the properties are not measured, but computed using a homogenization process upscaling from microscopic to macroscopic scale. A discussion about such homogenization — in quite another context – can be found in this blog post.

For nonisotropic materials, there are limitations to the possible values of the material parameters similar to those described for isotropic materials. It is difficult to immediately see these limitations, but there are two things to look out for:

- The constitutive matrix \mathbf D must be positive definite.
- For a general anisotropic material, the only option is to check if all of its eigenvalues are positive.
- For an orthotropic material, this is true if all six elastic moduli are positive and \nu_{\rm XY}\nu_{\rm YX}+\nu_{\rm YZ}\nu_{\rm ZY}+\nu_{\rm ZX}\nu_{\rm XZ}+\nu_{\rm YX}\nu_{\rm ZY}\nu_{\rm XZ}<1

- If the material has low compressibility, a mixed formulation must be used.
- It is possible to make an estimate of an effective bulk modulus and the values of the shear moduli.
- In cases of uncertainty, it is better to take the extra cost of the mixed formulation to avoid possible inaccuracies.

When working with geometrically nonlinear problems, the meaning of “linear elasticity” is really a matter of convention. The issue here is that there are several possible representations of stresses and strains. For a discussion about different stress and strain measures, see this previous blog post.

Since the primary stress and strain quantities in COMSOL Multiphysics are Second Piola-Kirchhoff stress and Green-Lagrange strain, the natural interpretation of linear elasticity is that these quantities are linearly related to each other. Such a material is sometimes called a St. Venant material.

Intuitively, one could expect that “linear elasticity” means that there is a linear relation between force and displacement in a simple tensile test. This will not be the case, since both stresses and strains depend on the deformation. To see this, consider a bar with a square cross section.

*The bar subjected to uniform extension.*

The original length of the bar is L_0 and the original cross-section area is A_0=a_0^2, where a_0 is the original edge of the cross section. Assume that the bar is extended at a distance \Delta so that the current length is L=L_0+\Delta=L_0(1+\xi).

Here, 1+\xi is the axial stretch and \xi can be interpreted as the engineering strain. The new length of the edge of the cross section is a=a_0+d=a_0(1+\eta), where \eta is the engineering strain in the transverse directions.

The force can be expressed as the Cauchy stress \sigma_x in the axial direction multiplied by the current cross-section area:

F = \sigma_x A = \sigma_x A_0 (1+\eta)^2

To use the linear elastic relation, the Cauchy stress \boldsymbol \sigma must be expressed as the Second Piola-Kirchoff stress \mathbf S. The transformation rule is

\mathbf \sigma = J^{-1} \mathbf F \mathbf S \mathbf F^T

where \mathbf F is the deformation gradient tensor, and the volume scale is defined as J = det(\mathbf F). Without going into details, for a uniaxial case

\sigma_x = \frac{F_{xX}}{F_{yY}F_{zZ}}S_X= \frac{(1+\xi)}{(1+\eta)^2}S_X

Since for a St. Venant material in uniaxial extension, the axial stress is related to the axial strain as S_X = E \epsilon_X, we obtain

F = S_x A_0 (1+\xi) = E A_0 (1+\xi)\varepsilon_X

Given that the axial term of the Green-Lagrange strain tensor is defined as

\varepsilon_X = \frac{\partial u}{\partial X} + \frac{1}{2}(\frac{\partial u}{\partial X})^2 = \xi+\frac{1}{2}\xi^2

the force versus displacement relation is then

F = E A_0 (1+\xi)(\xi + \frac{1}{2}\xi^2)=E A_0 (\xi+\frac{3}{2}\xi^2+\frac{1}{2}\xi^3)

The linear elastic material furbished with geometric nonlinearity actually implies a cubic relation between force and engineering strain (or force versus displacement, since \Delta =L_0\xi), as shown in the figure below.

*The uniaxial response of a linear elastic material under geometric nonlinearity.
*

As can be seen in the graph, the stiffness of the material approaches zero at the compression side, \xi = \sqrt{{1}/{3}}-1 \approx -0.42. In practice, this means that the simulation will fail at that strain level. It can be argued that there are no real materials that are linear at large strains, so this should not cause problems in practice. However, linear elastic materials are often used far outside the range of reasonable stresses for several reasons, such as:

- Often, you may want to do a quick “order of magnitude” check before introducing more sophisticated material models.
- There are singularities in the model that cause very high strains in a point.
- Read more about singularities here.

- In contact problems, the study is always geometrically nonlinear.
- Often, high compressive strains appear locally in the contact zone at some time during the analysis.

In all of these cases, the solver may fail to find a solution if the compressive strains are large. If you suspect this to be the case, it is a good idea to plot the smallest principal strain. If it is smaller than -0.3 or so, we can expect this kind of breakdown. The critical value in terms of the Green-Lagrange strain is found to be -1/3. When this becomes a problem, you should consider changing to a suitable hyperelastic material model.

Compression may not be the only problem. In the analysis above, Poisson’s ratio did not enter the equations. So what happens with the cross section?

By definition in the uniaxial case, the transverse strain is related to the axial strain by

\varepsilon_Y = -\nu \varepsilon_X

When these strains are Green-Lagrange strains, this is a nonlinear relation stating that

\frac{\partial v}{\partial Y} + \frac{1}{2}(\frac{\partial v}{\partial Y})^2 = -\nu (\frac{\partial u}{\partial X} + \frac{1}{2}(\frac{\partial u}{\partial X})^2)

Thus, there is a strong nonlinearity in the change of the cross section. Solving this quadratic equation gives the following relation between the engineering strains

\eta = \sqrt{1-\nu(\xi^2+2\xi)}-1

The result is shown in the figure below.

*Transverse displacement as a function of the axial displacement for uniaxial tension of a St. Venant material. Five different values of Poisson’s ratio are shown.*

As you can see, the cross section collapses quickly at large extensions for higher values of Poisson’s ratio.

If another choice of stress and strain representation had been made — for example, if the Cauchy stress were proportional to the logarithmic, or “true” strain — it would have resulted in quite a different response. Instead, such a material has a stiffness that decreases with elongation, where the force-displacement response does depend on the value of Poisson’s ratio. Still, both materials can correctly be called “linear elastic”, although the results computed with large strain elasticity can differ widely between two different simulation platforms.

We have illustrated some limits for the use of linear elastic materials. In particular, the possible pitfalls related to incompressibility and to the combination of linear elasticity with large strains have been highlighted.

If you are interested in reading more about material modeling in structural mechanics problems, check out these blog posts:

- Introducing Nonlinear Elastic Materials
- Obtaining Material Data for Structural Mechanics from Measurements
- Part 2: Obtaining Material Data for Structural Mechanics from Measurements
- Fitting Measured Data to Different Hyperelastic Material Models
- Yield Surfaces and Plastic Flow Rules in Geomechanics
- Computing Stiffness of Linear Elastic Structures: Part 1
- Computing Stiffness of Linear Elastic Structures: Part 2

After obtaining our measured data, the question then becomes this: How can we estimate the material parameters required for defining the hyperelastic material models based on the measured data? One of the ways to do this in COMSOL Multiphysics is to fit a parameterized analytic function to the measured data using the Optimization Module.

In the section below, we will define analytical expressions for stress-strain relationships for two common tests — the *uniaxial test* and the *equibiaxial test*. These analytical expressions will then be fitted to the measured data to obtain material parameters.

Characterizing the volumetric deformation of hyperelastic materials to estimate material parameters can be a rather intricate process. Oftentimes, perfect incompressibility is assumed in order to estimate the parameters. This means that after estimating material parameters from curve fitting, you would have to use a reasonable value for bulk modulus of the nearly incompressible hyperelastic material, as this property is not calculated.

Here, we will fit the measured data to several perfectly incompressible hyperelastic material models. We will start by reviewing some of the basic concepts of the nearly incompressible formulation and then characterize the stress measures for the case of perfect incompressibility.

For nearly incompressible hyperelasticity, the total strain energy density is presented as

W_s = W_{iso}+W_{vol}

where W_{iso} is the isochoric strain energy density and W_{vol} is the volumetric strain energy density. The second Piola-Kirchhoff stress tensor is then given by

S = -p_pJC^{-1}+2\frac{\partial W_{iso}}{\partial C}

where p_{p} is the volumetric stress, J is the volume ratio, and C is the right Cauchy-Green tensor.

You can expand the second term from the above equation so that the second Piola-Kirchhoff stress tensor can be equivalently expressed as

S = -p_pJC^{-1}+2\left(J^{-2/3}\left(\frac{\partial W_{iso}}{\partial \bar{I_{1}}}+\bar{I_{1}} \frac{\partial W_{iso}}{\partial \bar{I_{2}}} \right)I-J^{-4/3} \frac{\partial W_{iso}}{\partial \bar{I}_{2}} C -\left(\frac{\bar{I_{1}}}{3}\frac{\partial W_{iso}}{\partial \bar{I}_{1}} + \frac{2 \bar{I}_{2}}{3}\frac{\partial W_{iso}}{\partial \bar{I}_{2}}\right)C^{-1}\right)

where \bar{I}_{1} and \bar{I}_{2} are invariants of the isochoric right Cauchy-Green tensor \bar{C} = J^{-2/3}C.

The first Piola-Kirchhoff stress tensor, P, and the Cauchy stress tensor, \sigma, can be expressed as a function of the second Piola-Kirchhoff stress tensor as

\begin{align}P& = FS\\

\sigma& = J^{-1}FSF^{T}

\end{align}

\sigma& = J^{-1}FSF^{T}

\end{align}

Here, F is the deformation gradient.

Note: You can read more about the description of different stress measures in our previous blog entry “Why All These Stresses and Strains?“

The strain energy density and stresses are often expressed in terms of the stretch ratio \lambda. The *stretch ratio* is a measure of the magnitude of deformation. In a uniaxial tension experiment, the stretch ratio is defined as \lambda = L/L_0, where L is the deformed length of the specimen and L_0 is its original length. In a multiaxial stress state, you can calculate principal stretches \lambda_a\;(a = 1,2,3) in the principal referential directions \hat{\mathbf{N}_a}, which are the same as the directions of the principal stresses. The stress tensor components can be rewritten in the spectral form as

S =\sideset{}{^3_{a=1}}

\sum S_{a} \hat{\mathbf{N}_{a}} \otimes \hat{\mathbf{N}_{a}}

\sum S_{a} \hat{\mathbf{N}_{a}} \otimes \hat{\mathbf{N}_{a}}

where S_{a} represents the principal values of the second Piola-Kirchhoff stress tensor and \hat{\mathbf{N}_{a}} represents the principal referential directions. You can represent the right Cauchy-Green tensor in its spectral form as

C = \sideset{}{^3_{a=1}}

\sum\lambda_a^2 \hat{\mathbf{N}_a}\otimes\hat{\mathbf{N}_a}

\sum\lambda_a^2 \hat{\mathbf{N}_a}\otimes\hat{\mathbf{N}_a}

where \lambda_a indicates the values of the principal stretches. This allows you to express the principal values of the second Piola-Kirchhoff stress tensor as a function of the principal stretches

S_a = \frac{-p_p J}{\lambda_a^2}+2\left(J^{-2/3}\left(\frac{\partial W_{iso}}{\partial \bar{I_{1}}}+\bar{I_{1}} \frac{\partial W_{iso}}{\partial \bar{I_{2}}} \right) -J^{-4/3} \frac{\partial W_{iso}}{\partial \bar{I}_{2}} \lambda_a^2 -\frac{1}{\lambda_a^2}\left(\frac{\bar{I_{1}}}{3}\frac{\partial W_{iso}}{\partial \bar{I}_{1}} + \frac{2 \bar{I}_{2}}{3}\frac{\partial W_{iso}}{\partial \bar{I}_{2}}\right)\right)

Now, let’s consider the uniaxial and biaxial tension tests explained in the initial blog post in our Structural Materials series. For both of these tests, we can derive a general relationship between stress and stretch.

Under the assumption of incompressibility (J=1), the principal stretches for the uniaxial deformation of an isotropic hyperelastic material are given by

\lambda_1 = \lambda, \lambda_2 = \lambda_3 = \lambda^{-1/2}

The deformation gradient is given by

\begin{array}{c} F = \\ \end{array} \left(\begin{array}{ccc} \lambda &0 &0 \\ 0 &\frac{1}{\sqrt{\lambda}} &0 \\ 0 &0 &\frac{1}{\sqrt{\lambda}}\end{array}\right)

For uniaxial extension S_2 = S_3 = 0, the volumetric stress p_{p} can be eliminated to give

S_{1} = 2\left(\frac{1}{\lambda} -\frac{1}{\lambda^4}\right) \left(\lambda \frac{\partial W_{iso}}{\partial \bar{I}_{1_{uni}}}+\frac{\partial W_{iso}}{\partial \bar{I}_{2_{uni}}}\right) ,\; P_1 = \lambda S_1\; \sigma_1 = \lambda^2 S_1,\;\;\;\;

The isochoric invariants \bar{I}_{1_{uni}} and \bar{I}_{2_{uni}} can be expressed in terms of the principal stretch \lambda as

\begin{align*}

\bar{I}_{1_{uni}} = \left(\lambda^2+\frac{2}{\lambda}\right) \\

\bar{I}_{2_{uni}} = \left(2\lambda + \frac{1}{\lambda^2}\right)

\end{align*}

\bar{I}_{1_{uni}} = \left(\lambda^2+\frac{2}{\lambda}\right) \\

\bar{I}_{2_{uni}} = \left(2\lambda + \frac{1}{\lambda^2}\right)

\end{align*}

Under the assumption of incompressibility, the principal stretches for the equibiaxial deformation of an isotropic hyperelastic material are given by

\lambda_1 = \lambda_2 = \lambda, \; \lambda_3 = \lambda^{-2}

For equibiaxial extension S_3 = 0, the volumetric stress p_{p} can be eliminated to give

S_1 = S_2 = 2\left(1-\frac{1}{\lambda^6}\right)\left(\frac{\partial W_{iso}}{\partial \bar{I}_{1_{bi}}}+\lambda^2\frac{\partial W_{iso}}{\partial \bar{I}_{2_{bi}}}\right),\; P_1 = \lambda S_1,\; \sigma_1 = \lambda^2 S_1\;\;\;\;

The invariants \bar{I}_{1_{bi}} and \bar{I}_{2_{bi}} are then given by

\begin{align*}

\bar{I}_{1_{bi}} = \left( 2\lambda^2 + \frac{1}{\lambda^4}\right) \\

\bar{I}_{2_{bi}} = \left(\lambda^4 + \frac{2}{\lambda^2}\right)

\end{align*}

\bar{I}_{1_{bi}} = \left( 2\lambda^2 + \frac{1}{\lambda^4}\right) \\

\bar{I}_{2_{bi}} = \left(\lambda^4 + \frac{2}{\lambda^2}\right)

\end{align*}

Let’s now look at the stress versus stretch relationships for a few of the the most common hyperelastic material models. We will consider the first Piola-Kirchhoff stress for the purpose of curve fitting.

The total strain energy density for a Neo-Hookean material model is given by

W_s = \frac{1}{2}\mu\left(\bar{I}_1-3\right)+\frac{1}{2}\kappa\left(J_{el}-1\right)^2

where J_{el} is the elastic volume ratio and \mu is a material parameter that we need to compute via curve fitting. Under the assumption of perfect incompressibility and using equations (1) and (2), the first Piola-Kirchhoff stress expressions for the cases of uniaxial and equibiaxial deformation are given by

\begin{align*}

P_{1_{uniaxial}} &= \mu\left(\lambda-\lambda^{-2}\right)\\

P_{1_{biaxial}} &= \mu\left(\lambda-\lambda^{-5}\right)

\end{align*}

P_{1_{uniaxial}} &= \mu\left(\lambda-\lambda^{-2}\right)\\

P_{1_{biaxial}} &= \mu\left(\lambda-\lambda^{-5}\right)

\end{align*}

The stress versus stretch relationship for a few of the other hyperelastic material models are listed below. These can be easily derived through the use of equations (1) and (2), which relate stress and the strain energy density.

\begin{align*}

P_{1_{uniaxial}} &= 2\left(1-\lambda^{-3}\right)\left(\lambda C_{10}+C_{01}\right)\\

P_{1_{biaxial}} &= 2\left(\lambda-\lambda^{-5}\right)\left(C_{10}+\lambda^2 C_{01}\right)

\end{align*}

P_{1_{uniaxial}} &= 2\left(1-\lambda^{-3}\right)\left(\lambda C_{10}+C_{01}\right)\\

P_{1_{biaxial}} &= 2\left(\lambda-\lambda^{-5}\right)\left(C_{10}+\lambda^2 C_{01}\right)

\end{align*}

Here, C_{10} and C_{01} are Mooney-Rivlin material parameters.

\begin{align}\begin{split}

P_{1_{uniaxial}}& = 2\left(1-\lambda^{-3}\right)\left(\lambda C_{10} + 2C_{20}\lambda\left(I_{1_{uni}}-3\right)+C_{11}\lambda\left(I_{2_{uni}}-3\right)\\

& \quad +C_{01}+2C_{02}\left(I_{2_{uni}}-3\right)+C_{11}\left(I_{1_{uni}}-3\right)\right)\\

P_{1_{biaxial}}& = 2\left(\lambda-\lambda^{-5}\right)\left(C_{10}+2C_{20}\left(I_{1_{bi}}-3\right)+C_{11}\left(I_{2_{bi}}-3\right)\\

& \quad +\lambda^2C_{01}+2\lambda^2C_{02}\left(I_{2_{bi}}-3\right)+\lambda^2 C_{11}\left(I_{1_{bi}}-3\right)\right)

\end{split}

\end{align}

P_{1_{uniaxial}}& = 2\left(1-\lambda^{-3}\right)\left(\lambda C_{10} + 2C_{20}\lambda\left(I_{1_{uni}}-3\right)+C_{11}\lambda\left(I_{2_{uni}}-3\right)\\

& \quad +C_{01}+2C_{02}\left(I_{2_{uni}}-3\right)+C_{11}\left(I_{1_{uni}}-3\right)\right)\\

P_{1_{biaxial}}& = 2\left(\lambda-\lambda^{-5}\right)\left(C_{10}+2C_{20}\left(I_{1_{bi}}-3\right)+C_{11}\left(I_{2_{bi}}-3\right)\\

& \quad +\lambda^2C_{01}+2\lambda^2C_{02}\left(I_{2_{bi}}-3\right)+\lambda^2 C_{11}\left(I_{1_{bi}}-3\right)\right)

\end{split}

\end{align}

Here, C_{10}, C_{01}, C_{20}, C_{02}, and C_{11} are Mooney-Rivlin material parameters.

\begin{align}

P_{1_{uniaxial}} &= 2\left(\lambda-\lambda^{-2}\right)\mu_0\sum_{p=1}^{5}\frac{p c_p}{N^{p-1}}I_{1_{uni}}^{p-1}\\

P_{1_{biaxial}} &= 2\left(\lambda-\lambda^{-5}\right)\mu_0\sum_{p=1}^{5}\frac{p c_p}{N^{p-1}}I_{1_{bi}}^{p-1}

\end{align}

P_{1_{uniaxial}} &= 2\left(\lambda-\lambda^{-2}\right)\mu_0\sum_{p=1}^{5}\frac{p c_p}{N^{p-1}}I_{1_{uni}}^{p-1}\\

P_{1_{biaxial}} &= 2\left(\lambda-\lambda^{-5}\right)\mu_0\sum_{p=1}^{5}\frac{p c_p}{N^{p-1}}I_{1_{bi}}^{p-1}

\end{align}

Here, \mu_0 and N are Arruda-Boyce material parameters, and c_p are the first five terms of the Taylor expansion of the inverse Langevin function.

\begin{align}

P_{1_{uniaxial}} &= 2\left(\lambda-\lambda^{-2}\right)\sum_{p=1}^{3}p c_p \left(I_{1_{uni}}-3\right)^{p-1}\\

P_{1_{biaxial}} &= 2\left(\lambda-\lambda^{-5}\right)\sum_{p=1}^{3}p c_p \left(I_{1_{bi}}-3\right)^{p-1}

\end{align}

P_{1_{uniaxial}} &= 2\left(\lambda-\lambda^{-2}\right)\sum_{p=1}^{3}p c_p \left(I_{1_{uni}}-3\right)^{p-1}\\

P_{1_{biaxial}} &= 2\left(\lambda-\lambda^{-5}\right)\sum_{p=1}^{3}p c_p \left(I_{1_{bi}}-3\right)^{p-1}

\end{align}

Here, the values of c_p are Yeoh material parameters.

\begin{align}

P_{1_{uniaxial}} &= \sum_{p=1}^{N}\mu_p \left(\lambda^{\alpha_p-1} -\lambda^{-\frac{\alpha_p}{2}-1}\right)\\

P_{1_{biaxial}} &= \sum_{p=1}^{N}\mu_p \left(\lambda^{\alpha_p-1} -\lambda^{-2\alpha_p-1}\right)

\end{align}

P_{1_{uniaxial}} &= \sum_{p=1}^{N}\mu_p \left(\lambda^{\alpha_p-1} -\lambda^{-\frac{\alpha_p}{2}-1}\right)\\

P_{1_{biaxial}} &= \sum_{p=1}^{N}\mu_p \left(\lambda^{\alpha_p-1} -\lambda^{-2\alpha_p-1}\right)

\end{align}

Here, \mu_p and \alpha_p are Ogden material parameters.

Using the *Optimization* interface in COMSOL Multiphysics, we will fit measured stress versus stretch data against the analytical expressions detailed in our discussion above. Note that the measured data we are using here is the *nominal stress*, which can be defined as the force in the current configuration acting on the original area. It is important that the measured data is fit against the appropriate stress measure. Therefore, we will fit the measured data against the analytical expressions for the first Piola-Kirchhoff stress expressions. The plot below shows the measured nominal stress (raw data) for uniaxial and equibiaxial tests for vulcanized rubber.

*Measured stress-strain curves by Treloar.*

Let’s begin by setting up the model to fit the uniaxial Neo-Hookean stress to the uniaxial measured data. The first step is to add an *Optimization* interface to a 0D model. Here, *0D* implies that our analysis is not tied to a particular geometry.

Next, we can define the material parameters that need to be computed as well as the variable for the analytical stress versus stretch relationship. The screenshot below shows the parameters and variable defined for the case of an uniaxial Neo-Hookean material model.

Within the *Optimization* interface, a *Global Least-Squares Objective* branch is added, where we can specify the measured uniaxial stress versus stretch data as an input file. Next, a *Parameter Column* and a *Value Column* are added. Here, we define lambda (stretch) as a measured parameter and specify the uniaxial analytical stress expression to fit against the measured data. We can also specify a weighing factor in the *Column contribution weight* setting. For detailed instructions on setting up the *Global Least-Squares Objective* branch, take a look at the Mooney-Rivlin Curve Fit tutorial, available in our Application Gallery.

We can now solve the above problem and estimate material parameters by fitting our uniaxial tension test data against the uniaxial Neo-Hookean material model. This is, however, rarely a good idea. As explained in Part 1 of this blog series, the seemingly simple test can leave many loose ends. Later on in this blog post, we will explore the consequence of material calibration based on just one data set.

Depending on the operating conditions, you can obtain a better estimate of material parameters through a combination of measured uniaxial tension, compression, biaxial tension, torsion, and volumetric test data. This compiled data can then be fit against analytical stress expressions for each of the applicable cases.

Here, we will use the equibiaxial tension test data alongside the uniaxial tension test data. Just as we have set up the optimization model for the uniaxial test, we will define another global least-squares objective for the equibiaxial test as well as corresponding parameter and value columns. In the second global least-squares objective, we will specify the measured equibiaxial stress versus stretch data file as an input file. In the value column, we will specify the equibiaxial analytical stress expression to fit against the equibiaxial test data.

The settings of the Optimization study step are shown in the screenshot below. The model tree branches have been manually renamed to reflect the material model (Neo-Hookean) and the two tests (uniaxial and equibiaxial). The optimization algorithm is a Levenberg-Marquardt solver, which is used to solve problems of the least-square type. The model is now set to optimize the sum of two global least-square objectives — the uniaxial and equibiaxial test cases.

The plot below depicts the fitted data against the measured data. Equal weights are assigned to both the uniaxial and equibiaxial least-squares fitting. It is clear that the Neo-Hookean material model with only one parameter is not a good fit here, as the test data is nonlinear and has one inflection point.

*Fitted material parameters using the Neo-Hookean model. Equal weights are assigned to both of the test data.*

Fitting the curves while specifying unequal weights for the two tests will result in a slightly different fitted curve. Similar to the Neo-Hookean model, we will set up global least-squares objectives corresponding to Mooney-Rivlin, Arruda-Boyce, Yeoh, and Ogden material models. In our calculation below, we will include cases for both equal and unequal weights.

In the case of unequal weights, we will use a higher but arbitrary weight for the entire equibiaxial data set. It is possible that you may want to assign unequal weights only for a certain stretch range instead of the entire stretch range. If this is the case, we can split the particular test case into parts, using a separate *Global Least-Squares Objective* branch for each stretch range. This will allow us to assign weights in correlation with different stretch ranges.

The plots below show fitted curves for different material models for equal and unequal weights that correspond to the two tests.

*Left: Fitted material parameters using Mooney-Rivlin, Arruda-Boyce, and Yeoh models. In these cases, equal weights are assigned to both test data. Right: Fitted material parameters using Mooney-Rivlin, Arruda-Boyce, and Yeoh models. Here, higher weight is assigned to equibiaxial test data.*

The Ogden material model with three terms fits both test data quite well for the case of equal weights assigned to both tests.

*Fitted material parameters using the Ogden model with three terms.*

If we only fit uniaxial data and use the computed parameters for plotting equibiaxial stress against the actual equibiaxial test data, we obtain the results in the plots below. These plots show the mismatch in the computed equibiaxial stress when compared to the measured equibiaxial stress. In material parameter estimation, it is best to perform curve fitting for a combination of different significant deformation modes rather than considering only one deformation mode.

*Uniaxial and equibiaxial stress computed by fitting model parameters to only uniaxial measured data.*

To find material parameters for hyperelastic material models, fitting the analytic curves may seem like a solid approach. However, the stability of a given hyperelastic material model may also be a concern. The criterion for determining material stability is known as *Drucker stability*. According to the Drucker’s criterion, incremental work associated with an incremental stress should always be greater than zero. If the criterion is violated, the material model will be unstable.

In this blog post, we have demonstrated how you can use the *Optimization* interface in COMSOL Multiphysics to fit a curve to multiple data sets. An alternative method for curve fitting that does not require the *Optimization* interface was also a topic of discussion in an earlier blog post. Just as we have used uniaxial and equibiaxial tension data here for the purpose of estimating material parameters, you can also fit the measured data to shear and volumetric tests to characterize other deformation states.

For detailed step-by-step instructions on how to use the *Optimization* interface for the purpose of curve fitting, take a look at the Mooney-Rivlin Curve Fit tutorial, available in our Application Gallery.

While many different types of laser light sources exist, they are all quite similar in terms of their outputs. Laser light is very nearly single frequency (single wavelength) and coherent. Typically, the output of a laser is also focused into a narrow collimated beam. This collimated, coherent, and single frequency light source can be used as a very precise heat source in a wide range of applications, including cancer treatment, welding, annealing, material research, and semiconductor processing.

When laser light hits a solid material, part of the energy is absorbed, leading to localized heating. Liquids and gases (and plasmas), of course, can also be heated by lasers, but the heating of fluids almost always leads to significant convective effects. Within this blog post, we will neglect convection and concern ourselves only with the heating of solid materials.

Solid materials can be either partially transparent or completely opaque to light at the laser wavelength. Depending upon the degree of transparency, different approaches for modeling the laser heat source are appropriate. Additionally, we must concern ourselves with the relative scale as compared to the wavelength of light. If the laser is very tightly focused, then a different approach is needed compared to a relatively wide beam. If the material interacting with the beam has geometric features that are comparable to the wavelength, we must additionally consider exactly how the beam will interact with these small structures.

Before starting to model any laser-material interactions, you should first determine the optical properties of the material that you are modeling, both at the laser wavelength and in the infrared regime. You should also know the relative sizes of the objects you want to heat, as well as the laser wavelength and beam characteristics. This information will be useful in guiding you toward the appropriate approach for your modeling needs.

In cases where the material is opaque, or very nearly so, at the laser wavelength, it is appropriate to treat the laser as a surface heat source. This is most easily done with the *Deposited Beam Power* feature (shown below), which is available with the Heat Transfer Module as of COMSOL Multiphysics version 5.1. It is, however, also quite easy to manually set up such a surface heat load using only the COMSOL Multiphysics core package, as shown in the example here.

A surface heat source assumes that the energy in the beam is absorbed over a negligibly small distance into the material relative to the size of the object that is heated. The finite element mesh only needs to be fine enough to resolve the temperature fields as well as the laser spot size. The laser itself is not explicitly modeled, and it is assumed that the fraction of laser light that is reflected off the material is never reflected back. When using a surface heat load, you must manually account for the absorptivity of the material at the laser wavelength and scale the deposited beam power appropriately.

*The Deposited Beam Power feature in the Heat Transfer Module is used to model two crossed laser beams. The resultant surface heat source is shown.*

In cases where the material is partially transparent, the laser power will be deposited within the domain, rather than at the surface, and any of the different approaches may be appropriate based on the relative geometric sizes and the wavelength.

If the heated objects are much larger than the wavelength, but the laser light itself is converging and diverging through a series of optical elements and is possibly reflected by mirrors, then the functionality in the Ray Optics Module is the best option. In this approach, light is treated as a ray that is traced through homogeneous, inhomogeneous, and lossy materials.

As the light passes through lossy materials (e.g., optical glasses) and strikes surfaces, some power deposition will heat up the material. The absorption within domains is modeled via a complex-valued refractive index. At surfaces, you can use a reflection or an absorption coefficient. Any of these properties can be temperature dependent. For those interested in using this approach, this tutorial model from our Application Gallery provides a great starting point.

*A laser beam focused through two lenses. The lenses heat up due to the high-intensity laser light, shifting the focal point.*

If the heated objects and the spot size of the laser are much larger than the wavelength, then it is appropriate to use the Beer-Lambert law to model the absorption of the light within the material. This approach assumes that the laser light beam is perfectly parallel and unidirectional.

When using the Beer-Lambert law approach, the absorption coefficient of the material and reflection at the material surface must be known. Both of these material properties can be functions of temperature. The appropriate way to set up such a model is described in our earlier blog entry “Modeling Laser-Material Interactions with the Beer-Lambert Law“.

You can use the Beer-Lambert law approach if you know the incident laser intensity and if there are no reflections of the light within the material or at the boundaries.

*Laser heating of a semitransparent solid modeled with the Beer-Lambert law.*

If the heated domain is large, but the laser beam is tightly focused within it, neither the ray optics nor the Beer-Lambert law modeling approach can accurately solve for the fields and losses near the focus. These techniques do not directly solve Maxwell’s equations, but instead treat light as rays. The beam envelope method, available within the Wave Optics Module, is the most appropriate choice in this case.

The beam envelope method solves the full Maxwell’s equations when the field envelope is slowly varying. The approach is appropriate if the wave vector is approximately known throughout the modeling domain and whenever you know approximately the direction in which light is traveling. This is the case when modeling a focused laser light as well as waveguide structures like a Mach-Zehnder modulator or a ring resonator. Since the beam direction is known, the finite element mesh can be very coarse in the propagation direction, thereby reducing computational costs.

*A laser beam focused in a cylindrical material domain. The intensity at the incident side and within the material are plotted, along with the mesh.*

The beam envelope method can be combined with the *Heat Transfer in Solids* interface via the *Electromagnetic Heat Source* multiphysics couplings. These couplings are automatically set up when you add the *Laser Heating* interface under *Add Physics*.

*The* Laser Heating *interface adds the* Beam Envelopes *and the* Heat Transfer in Solids *interfaces and the multiphysics couplings between them.*

Finally, if the heated structure has dimensions comparable to the wavelength, it is necessary to solve the full Maxwell’s equations without assuming any propagation direction of the laser light within the modeling space. Here, we need to use the *Electromagnetic Waves, Frequency Domain* interface, which is available in both the Wave Optics Module and the RF Module. Additionally, the RF Module offers a *Microwave Heating* interface (similar to the *Laser Heating* interface described above) and couples the *Electromagnetic Waves, Frequency Domain* interface to the *Heat Transfer in Solids* interface. Despite the nomenclature, the RF Module and the *Microwave Heating* interface are appropriate over a wide frequency band.

The full-wave approach requires a finite element mesh that is fine enough to resolve the wavelength of the laser light. Since the beam may scatter in all directions, the mesh must be reasonably uniform in size. A good example of using the *Electromagnetic Waves, Frequency Domain* interface: Modeling the losses in a gold nanosphere illuminated by a plane wave, as illustrated below.

*Laser light heating a gold nanosphere. The losses in the sphere and the surrounding electric field magnitude are plotted, along with the mesh.*

You can use any of the previous five approaches to model the power deposition from a laser source in a solid material. Modeling the temperature rise and heat flux within and around the material additionally requires the *Heat Transfer in Solids* interface. Available in the core COMSOL Multiphysics package, this interface is suitable for modeling heat transfer in solids and features fixed temperature, insulating, and heat flux boundary conditions. The interface also includes various boundary conditions for modeling convective heat transfer to the surrounding atmosphere or fluid, as well as modeling radiative cooling to ambient at a known temperature.

In some cases, you may expect that there is also a fluid that provides significant heating or cooling to the problem and cannot be approximated with a boundary condition. For this, you will want to explicitly model the fluid flow using the Heat Transfer Module or the CFD Module, which can solve for both the temperature and flow fields. Both modules can solve for laminar and turbulent fluid flow. The CFD Module, however, has certain additional turbulent flow modeling capabilities, which are described in detail in this previous blog post.

For instances where you are expecting significant radiation between the heated object and any surrounding objects at varying temperatures, the Heat Transfer Module has the additional ability to compute gray body radiative view factors and radiative heat transfer. This is demonstrated in our Rapid Thermal Annealing tutorial model. When you expect the temperature variations to be significant, you may also need to consider the wavelength-dependent surface emissivity.

If the materials under consideration are transparent to laser light, it is likely that they are also partially transparent to thermal (infrared-band) radiation. This infrared light will be neither coherent nor collimated, so we cannot use any of the above approaches to describe the reradiation within semitransparent media. Instead, we can use the radiation in participating media approach. This technique is suitable for modeling heat transfer within a material, where there is significant heat flux inside the material due to radiation. An example of this approach from our Application Gallery can be found here.

In this blog post, we have looked at the various modeling techniques available in the COMSOL Multiphysics environment for modeling the laser heating of a solid material. Surface heating and volumetric heating approaches are presented, along with a brief overview of the heat transfer modeling capabilities. Thus far, we have only considered the heating of a solid material that does not change phase. The heating of liquids and gases — and the modeling of phase change — will be covered in a future blog post. Stay tuned!

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My parents love each other to death, but their habits can sometimes clash. My mom enjoys watching late-night television talk shows, while my dad prefers a good night’s sleep whenever he gets the chance. As they will eventually need to downsize, I decided to help them plan a home where they could stay friends.

From past experience, I knew there was no sense in trying to optimize the location of every plant, rug, or bookcase. My dad is constantly moving furniture around to, as he says, “feel the space.” My mom, meanwhile, tends to pull the couch closer and closer to the television rather than admitting that she needs glasses.

In short, they are bound to mess around with the input data and thereby remove digit after digit from the precision of any *a priori* sound level estimate. Luckily, accuracy was not a primary concern. I just needed to establish that my dad would get his beauty sleep.

The obvious start for a “quick and dirty” model of the acoustics in an apartment is the *Acoustic Diffusion Equation* interface. This interface is very easy to use and, in most situations, it is a lot faster than the more accurate *Pressure Acoustics* or *Ray Acoustics* interfaces.

For my simulation, I created a simple drawing of the relevant rooms and included the larger pieces of furniture. With the drawing complete, I set out to create my first acoustic diffusion model. This was a breeze — two power sources representing the stereo speakers connected to the television, absorption coefficients assigned to the walls and the furniture, and…that was it.

Approximate absorption coefficients for common materials are easy to find online. If you want to be more thorough, you can use different values in different frequency bands, or even specify them as arbitrary functions of the frequency. I selected a constant low value for the walls (including the floor and the ceiling) and a higher one for the soft parts of the furniture. To compensate for the lack of carpets and the relatively sparse decoration, I then nudged the wall coefficient slightly in the upward direction. If you decide on a similar time-saving measure, I suggest that you be open about it. My parents understand that acoustic diffusion is not an exact science, and they appreciate my honesty.

*Distribution of the sound pressure level (dB) without a door between the rooms. The red dots indicate my mom’s viewing position and my dad’s head while he is trying to sleep.*

My first solution shows a sound pressure level decreasing by a quite modest 11 dB between the living room couch and the bedroom. Luckily, two important elements that would increase the difference were still missing.

The first element is a door between the rooms. If your door manufacturer cites a transmission loss in dB, make sure to check whether it concerns only transmission through the door itself, or if it was measured with the door in its frame. This makes a difference because a significant amount of sound may sneak through the space between the door and the floor unless you install a fitting. If you have access to a drawing of the door and know the material that was used to make it, you can, of course, run an acoustic-structure interaction analysis to get a second opinion. It is trivial to include a door with a specified transmission loss in an acoustic diffusion simulation.

The second element is the direct sound. The acoustic diffusion equation only deals with the part of the sound that has already struck the walls or the furniture and has become diffuse. With my mom sitting directly in front of the television, there is also a significant direct sound reaching her. By approximating the sound sources as points and neglecting shadowing from the table, it is quite simple to add the direct sound as an analytical expression in terms of the emitted sound power and the local coordinates.

In my second — and final — simulation, I added the direct sound hitting the couch and put up a door between the rooms. The total loss between the couch and the bed was now a much more acceptable 23 dB. I provided my parents with a nice printed report and gave them the thumbs up to move into the home.

*Sound pressure level distribution (dB) with a door added and an approximation of the direct sound included in the living room.*

Diffusion is often discussed as a description of the motion of particles in a gas. The particles travel in straight lines except when, at random intervals, they bounce off the gas molecules. The diffusion coefficient is a function of the mean free path between two consecutive collisions.

*Mean free path between collisions in a gas (left) and for sound particles in a room (right).*

The acoustic diffusion equation deals with conceptual “sound particles”, with a density proportional to the local sound energy. These particles do not bounce on the air molecules, but rather on the walls of the room. The mean free path \lambda and, with that, the diffusion coefficient D relate to the proportions of the room. It holds that \lambda = 4V/S, where V is the volume of the room and S is the total surface area of the room’s walls, floor, and ceiling. In turn, D=\lambda c/3, where c is the speed of sound.

The implementation of the acoustic diffusion equation in COMSOL Multiphysics is

\frac{\partial{w}}{\partial{t}}+\nabla \cdot (-D_t \nabla w) + c m_a w = q(\textbf{x},t)

The equation is solved for the acoustic energy density w, from which you can derive the sound pressure level and other important measurables. If you drop the time derivative, you get the stationary form. The volume absorption coefficient m_a accounts for the air dissipation, which is often negligible but sometimes important in very large spaces. D_t = D is the diffusion coefficient and q is an optional volumetric sound source. With the alternative formulation

\frac{\partial{w}}{\partial{t}}+\nabla \cdot (-D_t \nabla w) + c (m_a + \frac { \alpha_f } {\lambda_f}) w = q(\textbf{x},t), ~ ~ D_t = \frac {D_f D}{D_f+D}

you can also include an averaged description of the furnishing. Here, \alpha_f is the average absorption coefficient of the furniture (the fittings). The diffusion coefficient D_f and the mean free path \lambda_f derive from the number density and the average cross section of the furniture.

Say, for example, that my parents had wanted to invest in a furniture store. In that case, I would have used this formulation rather than draw each individual item.

The boundary conditions include various ways of specifying the local absorption coefficient and applying sound sources. Point sources are also available.

Like ray acoustics, the acoustic diffusion equation does not account for low-frequency behaviors, such as standing waves or diffraction around corners. These are chiefly important below the *Schroeder frequency*, which you can learn about more in the blog post “Modeling Room Acoustics with COMSOL Multiphysics“. In my parents’ new living room and bedroom, the Schroeder frequencies are 167 Hz and 183 Hz, respectively.

For events to turn into statistics, you need to monitor your system for a while. When compared to ray acoustics, the main limitation in the acoustic diffusion equation is that it does not include early sound. This means that it will systematically underestimate the sound pressure level in the vicinity of the sound sources — the case for my mom’s viewing position right in front of the television. You can take my approach and often at least partially compensate for this limitation by calculating the direct sound analytically and adding it to the diffuse solution. It can, however, become rather difficult or impossible to do so if there are obstacles near the sources acting to reflect or absorb the sound.

While it can be argued that acoustic diffusion is the least accurate of the three acoustics analysis methods available in COMSOL Multiphysics, acoustic diffusion is easier to set up and is often orders of magnitude faster to solve than other methods. The solution time required to produce the plots presented here was approximately 2.5 seconds on a regular desktop computer. For a ray tracing model with a high number of rays, obtaining good statistics would take at the very least a few minutes — and possibly hours — to solve. Pressure acoustics is the only game in town for the low-frequency, resonance-dominated range. But, for frequencies much greater than the Schroeder frequency, this approach would be out of the question due to the quickly increasing solution time and memory usage.

At the end of the day, if your parents ask you for an opinion on the sound environment in their living quarters, I’d wholeheartedly recommend that you run an acoustic diffusion simulation for them. Alternatively, if you are in the early stages of designing a concert hall or an office space, acoustic diffusion can still be a great tool for obtaining an initial assessment of the high-frequency sound distribution. You can then add ray acoustics to predict the early sound and get a more accurate result, as well as pressure acoustics to investigate the low-frequency behavior.

The simulation referenced in this blog post is available for download here. To learn more about room acoustics, I also encourage you to download the One-Family House Acoustics tutorial from our Application Gallery.

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Advanced composites are used extensively throughout the Boeing 787 Dreamliner, as shown in the diagram below. Also known as carbon fiber reinforced plastic (CFRP), the composites are formed from a lightweight polymer binder with dispersed carbon fiber filler to produce materials with high strength-to-weight ratios. Many wing components, for example, are made of CFRP, ensuring that they can support the load imposed during flight while minimizing their overall contribution to the weight of an aircraft.

*Advanced composites are used throughout the body of the Boeing 787. Copyright © Boeing.*

Despite their remarkable strength and light weight, CFRPs are generally not conductive like their aluminum counterparts, thus making them susceptible to lightning strike damage. Therefore, electrically conductive expanded metal foil (EMF) is added to the composite structure layup, shown in the figure below, to dissipate the high current and heat generated by a lightning strike.

*The composite structure layup shown at left consists of an expanded metal foil layer shown at right. This figure is a screenshot from the COMSOL Multiphysics® software model featured in this blog post. Copyright © Boeing.*

The figure also shows the additional coatings on top of the EMF, which are in place to protect it from moisture and environmental species that cause corrosion. Corrosive damage to the EMF could result in lower conductivity, thereby reducing its ability to protect aircraft structures from lightning strike damage. Temperature variations due to the ground-to-air flight cycle can, however, lead to the formation of cracks in the surface protection scheme, reducing its effectiveness.

During takeoff and landing, aircraft structures are subjected to cooling and heating, respectively. Thermal stress manifests as the expansion and compression — or ultimately the displacement — of adjacent layers throughout the depth of the composite structure. Although a single round-trip is not likely to pose a significant risk, over time, each layer of the composite structure contributes to fatigue damage buildup. Repetitive thermal stress results in cumulative strain and higher displacements, which are, in turn, associated with an increased risk of crack formation. The stresses in a material depend on its mechanical properties quantified by measurable attributes such as yield strength, Young’s modulus, and Poisson’s ratio.

By taking the thermal and mechanical properties of materials into account, it is possible to use simulation to design and optimize a surface protection scheme for aircraft composites that minimizes stress, displacement, and the risk of crack formation.

Evaluating the thermal performance of each layer in the surface protection scheme is essential in order to reduce the risks and maintenance costs associated with damage to the protective coating and EMF. Therefore, researchers at Boeing Research & Technology (BR&T), pictured below, are using multiphysics simulation and physical measurements to investigate the effect of the EMF design parameters on stress and displacement throughout the composite structure layup.

*The research team at Boeing Research & Technology from left to right: Patrice Ackerman, Jeffrey Morgan, Robert Greegor, and Quynhgiao Le. Copyright © Boeing.*

In their work, the researchers at BR&T have developed a coefficient of thermal expansion (CTE) model in COMSOL Multiphysics® simulation software. The figure shown above that presents the composite structure layup and EMF is a screenshot acquired from the model geometry used for their simulations in COMSOL Multiphysics.

The CTE model was used to evaluate heating of the aircraft composite structure as experienced upon descent, where the final and initial temperatures used in the simulations represent the ground and altitude temperatures, respectively. The *Thermal Stress* interface, which couples heat transfer and solid mechanics, was used in the model to simulate thermal expansion and solve for the displacement throughout the structure.

The material properties of each layer in the surface protection scheme as well as of the composites are custom-defined in the CTE model. The relative values of the coefficient of thermal expansion, heat capacity, density, thermal conductivity, Young’s modulus, and Poisson’s ratio are presented in the chart below.

*This graph presents the ratio of each material parameter relative to the paint layer. Copyright © Boeing.*

From the graph, trends can be identified that provide early insight into the behavior of the materials, which aids in making design decisions. For example, the paint layer is characterized by higher values of CTE, heat capacity, and Poisson’s ratio, thus indicating that it will undergo compressive stress and tensile strain upon heating and cooling.

Multiphysics simulation takes this predictive design capability one big step forward by quantifying the resulting displacement due to thermal stress throughout the entire composite structure layup simultaneously, taking into account the properties of all materials. The following figure shows an example of BR&T’s simulation results and presents the stress distribution and displacement throughout the composite structure.

*Left: Top-down and cross-sectional views of the von Mises stress and displacement in a one-inch square sample of a composite structure layup. Right: Transparency was used to show regions of higher stress, in red. Lower stress is shown in blue. Copyright © Boeing.*

In the plots at the left above, the displacement pattern caused by the EMF is evident through the paint layer at the top of the composite structure while a magnified cross-sectional view shows the variations in displacement above the mesh and voids of the EMF. The cross section also makes it easy to see the stress distribution through the depth of the composite structure, where there is a trend toward lower stress in the topmost layers. Transparency was used in the plot shown at the right to depict the regions of high stress in the composites and EMF, which is noticeably higher at the intersection of the mesh wires. Stress was plotted through the depth of the composite structure layup along the vertical red line shown in the center of the plot. The figure below shows the relative stress in each layer of the composite structure layup for different metallic compositions of the EMF.

*Relative stress in arbitrary units was plotted through the depth of the composite structure layups containing either aluminum (left) or copper EMF (right). Copyright © Boeing.*

The samples vary by the presence of a fiberglass corrosion isolation layer when aluminum is used as the material for the EMF. The fiberglass acts as a buffer resulting in lower stress in the aluminum EMF, when compared with the copper.

From lightning strike protection to the structural integrity of the composite protection scheme, it all relies on the design of the expanded metal foil layer. The design of the EMF layer can vary by its metallic composition, height, width of the mesh wire, and the mesh aspect ratio. For any EMF design parameter, there is a trade-off between current-carrying capacity, displacement, and weight. By using the CTE model, the researchers at BR&T found that increasing the mesh width and decreasing the aspect ratio are better strategies for increasing the current-carrying capacity of the EMF that minimize its impact on displacement in the composite structure.

The metal chosen for the EMF can also have a significant effect on stress and displacement in the composite structure, which was investigated using simulation and physical testing. Two composite structures, one with aluminum and the other with copper EMF, underwent thermal cycling with prolonged exposure to moisture in an environmental test chamber. In the results, shown below, the protective layers remained intact for the composite structure with copper EMF. However, for the layup with aluminum, cracking occurred in the primer, at the edges, on surfaces, and was particularly substantial in the mesh overlap regions.

*Photo micrographs of the composite structure layup after exposure to moisture and thermal cycling. A crack in the vicinity of the aluminum EMF is contained within the red ellipse. Copyright © Boeing.*

Simulations confirm the experiment results. Shown below, displacements are noticeably higher throughout the composite structure layup when aluminum is used for the EMF layer, where higher displacements are associated with an increased risk for developing cracks. The higher displacement is easiest to observe in the bottom plots, which show displacement ratios for each EMF height.

*Effect of varying the EMF height on displacement in each layer of the surface protection scheme. Copyright © Boeing.*

The larger displacements caused by the aluminum EMF can be attributed in part to its higher CTE when compared with copper, which exemplifies how important the properties of materials are to the thermal stability of the aircraft composite structures.

In the early design stages and along with experimental testing, multiphysics simulation offers a reliable means to evaluate the relative impact of the EMF design parameters on stress and displacement throughout the composite structures. An optimized EMF design is essential to minimizing the risk of crack formation in the composite surface protection scheme, which reduces maintenance costs and allows the EMF to perform its important protective function of mitigating lightning strike damage.

Refer to page 4 of *COMSOL News* 2014 to read the original article, “Boeing Simulates Thermal Expansion in Composites with Expanded Metal Foil for Lightning Strike Protection of Aircraft Structures”.

This article was based on the following publicly available resources from Boeing:

- The Boeing Company. “787 Advanced Composite Design.” 2008-2013.
- J.D. Morgan, R.B. Greegor, P.K. Ackerman, Q.N. Le, “Thermal Simulation and Testing of Expanded Metal Foils Used for Lightning Protection of Composite Aircraft Structures,” SAE Int. J. Aerosp. 6(2):371-377, 2013, doi:10.4271/2013-01-2132.
- R.B. Greegor, J.D. Morgan, Q.N. Le, P.K. Ackerman, “Finite Element Modeling and Testing of Expanded Metal Foils Used for Lightning Protection of Composite Aircraft Structures,” Proceedings of 2013 ICOLSE Conference; Seattle, WA, September 18-20, 2013.

To learn more about adding material property data to your COMSOL Multiphysics® simulations, read the following blog post series on *Obtaining Material Data for Structural Mechanics Simulations from Measurements* by my colleague Henrik Sönnerlind:

General information about aircraft design and structures can be found in chapter 1 of this handbook on aircraft maintenance from the Federal Aviation Administration.

*BOEING, Dreamliner, and 787 Dreamliner are registered trademarks of The Boeing Company Corporation in the U.S. and other countries.*