Pole vaulting is a sport with a storied history. What began as an ancient competition for Greeks, Celts, and Cretans has evolved into a medaled event in the Olympic Games. Several tournaments, including the upcoming IAAF World Indoor Championships, are also hosted throughout the year, giving pole vaulters the opportunity to showcase their skills.

The sport itself, recognized as one of the major jumping events, involves the use of a long, elastic pole to clear a bar. In the past few decades, carbon fiber and fiberglass poles have arrived on the pole vaulting scene. These advancements are helping to bring athletes to new heights and break previous world records. While the pole has an important impact on performance, there are many other elements to consider that can affect the overall jump.

When it comes to clearing a height in pole vaulting, the general approach taken by athletes can be broken down into a series of phases. Each of the phases, listed here, places different constraints on the body:

- Run up
- Pole plant and takeoff
- Pole bend and swing
- Pull and release
- Clearance

In each phase, athletes control several of the initial conditions. Such conditions include: speed; grip height (the height at which the pole vaulter grips the pole); stiffness, which differs between different pole categories; the angle of attack (the angle between the pole and the ground at takeoff); and body position while airborne.

*Angelica Bengtsson sets the Swedish pole vaulting record in 2015, achieving a 4.68 m clearance. Later that year, Bengtsson increased the national record to 4.70 m and finished in 4 ^{th} place in the 15^{th} IAAF World Championships.*

Here, we’ll provide some more details about the individual phases.

The run up phase refers to when an athlete holds the pole in an upright position and successively tilts it forward while approaching the box, the hole in the runway where the pole is placed. By holding the pole close to the body, the torque created by the weight of the pole decreases. The muscular strength thus becomes less fatigued, with most of the muscular energy retained in the body. While approaching the box, the athlete maximizes his or her speed in order to maximize the kinetic energy, *E _{K}*, which is transferred to the next phase.

During pole plant and takeoff, the pole is initially placed in the box. The athlete then bends the pole and jumps up. What we have here is a multibody system, a combination of the pole itself and the pole vaulter. To get the pole into a vertical position, the system must rotate forward. Several variables can affect the angular position of the pole, *θ*, including the jump force, *F*; the jump velocity, *v*; and the body mass, *m*.

The jump force is transferred through the body to the pole at the hand grip. This pole force creates a forward-rotating torque at the takeoff and provides a positive contribution to the forward rotation. The athlete’s velocity affects the angular momentum, which further adds to the forward rotation. The body mass, assisted by the gravity, *g*, creates a counteracting gravitational torque throughout the entire movement that decelerates the rotation. Additionally, the pole vaulter rotates around the hand grip, *φ*, and moves his or her body parts. Such motion alters the position of the body mass and the rotational inertia, influencing the pole rotation.

*The take-off phase. The double dots denote rotational acceleration.*

Let’s now walk through a few pole vaulting scenarios.

At a high angle of attack — when the pole vaulter’s body is straight, with arms stretched and hands held high in the air — the torque leverage, the distance between the ground and the hand grip, is maximized. As a result, the pole rotates forward. If an athlete bends his or her arms, the leverage might not be sufficient enough to produce the amount of torque needed to drive the pole vaulter forward. Because of this, the pole will not reach the vertical position; instead, it will spring the athlete back to the runway. The same situation will occur if the speed of the athlete is not fast enough.

The grip height has a major influence on the take-off phase as well. On one hand, with increasing grip height, the pole vaulter will come higher up along the pole in its straight vertical position. On the other hand, an increased grip height will result in a lower angle of attack, while also increasing the horizontal distance between the pole plant and the body mass, which is the leverage of the counteracting torque from the body mass. However, as an athlete becomes stronger and faster, it is possible to increase the angular momentum, compensating for the additional counteracting torque due to higher grip height.

To maximize the energy transfer to the pole, it is also important that the athlete has a pretensed body. With a looser trunk, as well as shoulders and arms, some of the energy will be dissipated in the body. Body tension has a strong influence on the variables of the pole rotation as well. At takeoff, the athlete pushes backward with his or her leg and generates a forward-acting force. The pole counteracts, rotating the athlete backward. With a loose body, the pole vaulter will come down further on the runway, closer to the pole, and tilt backward. Such a position not only gives the athlete a smaller angle of attack, but it creates a lower jump force and velocity as well — all of which reduce the desired forward rotation of the pole.

At takeoff, the pole vaulter jumps up. This results in a vertical upward and horizontal forward velocity and force. If the angle of the jump is too low, the forces on the pole will bend it substantially. Once the tensile strength of the material has passed, the pole will snap, sending the athlete straight into the landing mat and unfortunately, below the bar. The most common reason for a pole to break is surface damage. When a pole is thrown on the ground or stepped on by track spikes, surface scratches can develop. These small surface marks can be large enough to initiate a pole fracture. Since the materials used in poles (carbon fibers and fiberglass) are brittle, they have a poor tolerance to damage.

Once an athlete has jumped, he or she can no longer utilize the runway that previously helped to increase the kinetic energy and counteract the initial pole bending. In this phase, the athlete rotates around the hand grip on the pole, *φ*, and generates a centripetal force, *F _{C}*, which further bends the pole. Since the elastic energy of the pole,

As we discussed earlier, too much bending of a pole can cause it to break. An athlete can opt to use a pole with a higher stiffness, *k*, to increase the energy and force, but a stiffer pole exerts a greater stress on the body during the pole plant and takeoff.

*Bending of the pole. The dots indicate rotation velocity.*

During the swing, a pole vaulter lifts his or her legs, followed by the torso, to place them above the head when the pole reaches an upright position. The motion reduces the radius between the center of mass and the hand grip, thus increasing the rotation around the hand grip on the pole and sending the athlete higher up into the air. Moreover, the spring force from the pole now comes into play, as it catapults the pole vaulter upward.

With the ability to position the body in a certain shape, an athlete can control the inertia and position of the center of mass. Since both variables affect the angular motion around the hand grip, the athlete can optimize the angular motion of the pole; the elastic energy stored in the pole; and the spring force in the pole (theoretically, the sequence of motion that prompts an increase in jumping height). This involves considering several variables, from the position of multiple body parts to the dynamics of the pole vault. In reality, a pole vaulter’s body must respond to the dynamic changes during the vault, and with perfect timing.

When the pole is in an upright position, muscular energy and the arms are used to pull the body higher up. The velocity of the pull affects the generated power and the work done by the athlete. By increasing the velocity, more work is added to the potential energy at the grip height. This increases the potential energy of the pole vaulter, *E _{P}*, and therefore enables the clearing of heights above the grip height,

From the point at which the athlete releases the pole, he or she is moving as a free body, with the center of gravity following a parabolic path. The initial velocity is mainly directed upward and the gravitational force is acting downward. The pole vaulter’s legs clear the bar. As they are pulled downward, the legs generate a downward force, *F _{L}*, which is assisted by Newton’s third law of motion. As this happens, the hips are influenced by a counteracting upward force,

In a simple analysis of the pole vault, all of the kinetic energy from the run is transferred to the potential energy at clearance. The kinetic energy is E_K= \frac{mv^2}{2}. Here, *m* is the mass of the athlete and *v* is the velocity. The potential energy, meanwhile, is E_P= mgh, where *g* is the acceleration of gravity and *h* is the height of the elevation. A perfect energy conversion results in a maximum achievable height difference for the center of mass: \Delta h = \frac{v^2}{2g}.

An elite male athlete can reach 9.5 m/s during the run up, while an elite female athlete can reach 8.4 m/s. This corresponds to \Delta h = 4.5\, \mathrm m and \Delta h = 3.5 \, \mathrm m, respectively. Since the center of mass is initially about 1 m above the ground, it is evident that even a perfect conversion of kinetic energy into potential energy brings the pole vaulters to 5.5 m and 4.5 m, respectively. In reality, the best male athletes clear about 6 m and the best female athletes clear about 5 m. The athlete’s muscles supply additional energy during the jump.

Pole vaulting consists of many phases. By improving the details behind the technique, centimeter by centimeter and inch by inch, an athlete can work their way up to the limitations of the laws of physics and muscular strength. For many elite athletes, however, such success comes after more than 15 years of training.

Typically, there are two approaches to developing a successful jumping technique. Some people believe that a certain jumping sequence is the perfect approach and thus try to mimic it. Others, however, do not believe that one jumping sequence is the best option for everyone. Instead, they set out to develop their own technique. Incremental improvement can help athletes find local maximum in their height clearance, but to reach higher levels, they must make a significant change. Coping with this modification, which introduces a different response on an athlete’s body, requires the pole vaulter to not only be mentally and physically strong, but also to have a feeling for the physics underlying the sport.

- Interested in modeling pole vaulting? Read more about the Multibody Dynamics Module and the Structural Mechanics Module, which can be coupled to model such mechanisms in COMSOL Multiphysics.
- You can find several other blog posts pertaining to the physics of sports on the COMSOL Blog. Have a look here.

Over millions of years, evolution has enabled fish to quickly and easily move through the aquatic environments that they call home. Such environments, as we’ve previously noted on the blog, pose challenges for manmade vehicles and robots, as murky waters and lack of light make them difficult to navigate. To address these issues, fish are often used as a source of inspiration in the development of aquatic designs.

*Exploring the swimming patterns of fish helps to optimize the design of aquatic technology. Image by Jim and Becca Wicks — Snap of Snapper!. Licensed under CC BY 2.0, via Wikimedia Commons.*

Researchers at Cornell, for instance, are developing a soft robot that is designed to swim like a lamprey fish (shown below). The fish-inspired robot uses the simple structure of a lamprey to offer the potential for self-sustaining space exploration, particularly in the oceans of Jupiter’s moon, Europa.

*A lamprey is one type of fish whose behavior is inspiring the development of aquatic robots. Image by Tiit Hunt — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.*

To build these robots, researchers need to learn how fish move in their aquatic environments. They can do so with FSI studies that account for fluid velocity, pressure, and the stresses and strains on the fish. By looking at the effects of the fluid environment on the movement of the fish, and vice versa, researchers can obtain accurate and useful results.

Today, we’ll explore how a team from the Università Roma Tre used the FSI simulation capabilities of COMSOL Multiphysics to investigate fish locomotion, which is outlined in their paper “The Virtual Aquarium: Simulations of Fish Swimming“. They presented their research at the COMSOL Conference 2015 Grenoble, earning a Best Paper award and the popular choice Best Poster award at the event.

For their analysis, the researchers used a 2D approach to evaluate the movement of a fish swimming freely in a fluid domain. They simulated the interaction between the fish body (the solid) and the surrounding salt water (the fluid). They also simulated the fish’s muscle contractions using the notion of distortions. Here, we’ll look at the FSI techniques that they applied to their research.

The simulation study focused on *carangiform swimming*, a category of fish body movement in which a fish’s muscles contract from head to tail in a wavelike pattern. As the animation below illustrates, the movement causes the tail to function as a propeller and produce a localized thrust wake and momentum jet. This results in a force that propels the fish forward.

*The carangiform swimming pattern of a fish.*

In their simulation studies, the research team sought to analyze carangiform swimming for different periods of time. However, when the fish moved inside of the virtual aquarium, it deformed the surrounding mesh, as indicated below. The movement eventually produced such a large deformation that an automatic remeshing method was required. With the *Fluid-Structure Interaction* interface, the researchers were able to solve the problem by applying a moving mesh technique for short periods of time and a remeshing technique for longer intervals of time.

*Meshes used to analyze the fish’s swimming patterns. Images by M. Curatolo and L. Teresi and taken from their COMSOL Conference 2015 Grenoble presentation.*

The meshing techniques mentioned above allowed the team to accurately study the effects of fish motion on the fluid environment, including wake creation. When a solid such as a fish moves in a fluid environment, it can create a wake, or an area of disturbed flow, behind itself. The research team used COMSOL Multiphysics in this case to observe and analyze the wake pattern and vortices generated by carangiform swimming. They found that every stroke of the fish’s tail released vortices and that the mutual distance between the cores of the vortices didn’t change.

*Vortices and a wake appear as the fish starts to swim. Image by M. Curatolo and L. Teresi and taken from their COMSOL Conference 2015 Grenoble presentation.*

*The pattern of the vortices and the wake.*

To further understand how the surrounding salt water and fish body interact with one another, the researchers also computed the lift and drag. The results showed strong similarities between the tail velocity components and the lift and drag forces.

*Left: A plot comparing tail velocity forces. Right: A plot comparing the lift and drag. Images by M. Curatolo and L. Teresi and taken from their COMSOL Conference 2015 Grenoble presentation.*

When comparing the model results to actual measurements, the team of researchers found satisfactory agreement between the two cases. They hope that their simulation results will inspire future advancements in the study of fish locomotion.

Simulating swimming fish is just one example of a complex problem that can be addressed with the FSI capabilities of COMSOL Multiphysics. Such tools enable you to simulate a variety of elements, from mixers and pipe flow to vibrating structures in fluids and poroelastic media.

Interested in learning about the other types of FSI problems you can solve in COMSOL Multiphysics? Be sure to check out this blog post, which offers a helpful overview. We also encourage you to watch our archived webinar on FSI simulation, as well as download one of the fluid-structure interaction tutorial models, including this example, from the Application Gallery.

- Read the full paper: “The Virtual Aquarium: Simulations of Fish Swimming“
- Take a look at the research poster

Since 3D printing first emerged as a technology, its impact has been felt across a wide range of industries. Take automotive manufacturers, for instance. 3D printing capabilities have extended the freedom of vehicle design, enabling the development of highly customized parts while saving production time and costs. Within the medical field, the technology has led to greater innovation in the design of customized implants and devices as well as the creation of exact replicas of organs.

*A 3D printer creates a product component.*

Additive manufacturing, as we can see, is already making its mark as an efficient approach to product development. As the technology continues to advance, new freedom and flexibility is emerging, furthering its applications. One particular area of promise is the field of material design.

Material design typically tailors fine-scale structures organized in a repeating pattern to create a product with optimal performance. Depending on the requirements, the design of a single microstructure, otherwise known as a unit cell, can range from a simple monolithic geometry to a complex multimaterial geometry. In theory, the complexity, and thus the design freedom of the unit cell, is only limited by the creativity of the designer and the manufacturing capabilities.

Traditionally, 3D printing has only been capable of printing a single material product. Now, recent developments in 3D printing are showing potential for the multimaterial printing of small-scale structures. Such capabilities would provide designers with a finer control over the microstructures, with the option to combine and customize the microstructures based on their specific needs. Further, engineers would be able to select the proportion and arrangement of the individual materials included in the structure.

Researchers at TNO are using multiscale modeling and multiphysics simulation to investigate virtual material design in 3D printing. In the next section, we’ll take a closer look at their innovative research.

To begin their simulation studies, the team first designed a single unit cell with twice the stiffness in one direction as the other and analyzed the material behavior for a given geometry. The optimization capabilities in COMSOL Multiphysics allowed them to derive the appropriate value of stress corresponding to an applied strain, so as to fit their desired stiffness matrix. The simulation results were verified with a printed sample tested for the expected material behavior.

*Left: Unit cell geometry. Center: Mechanical stress for the optimized design. Right: The 3D-printed samples.*

The researchers then performed a similar study on a highly anisotropic material. In the simulation, both the spatial distribution of the material and the orientation of the anisotropic fibers could be controlled.

Speaking to the research team’s greater goal, the simulation was extended to microstructures made up of a combination of different materials. The composition and arrangement of the various materials in the structure were adjusted until the optimal level of thermal conductivity was achieved.

*The multimaterial composition for the ideal anisotropic thermal conductivity. The color white indicates regions of high conductivity, orange represents regions of low conductivity, and red depicts nonconductive material and voids.*

After optimization at the microlevel, the team at TNO focused on optimization for objects of a larger scale. While a necessary step for developing actual products, extending the results of the microstructure simulations to real-life sizes can be quite computationally expensive. Multiscale modeling provides a solution, allowing designers to simulate at the micromaterial and product scales simultaneously. The team extracted parameters for the effective structural behavior of several multimaterial cells, which could then be applied as input in the full-scale model of a device.

While topology optimization is a powerful tool for creating designs for 3D printing, there are also limitations to address for specific additive manufacturing techniques. In selective laser melting (SLM), for example, powdered material is melted into a desired shape via a laser beam. The powder that is not used must be removed from the printed object afterward, and large overhangs are not conducive to SLM designs as they have the tendency to warp.

To address such issues, the research team designed unit cells featuring different densities and then combined them to create the desired properties at the product level. Various techniques were combined in COMSOL Multiphysics, from stiffness homogenization for individual unit cell types to topology optimization at the product scale. The procedure as a whole was later applied to the development of a polymer hammer handle, shown below. The design was comprised of different unit cell types at the microlevel and optimized for proper stiffness and minimal material use.

*Left: Topology optimization simulation results. Center: Optimized hammer handle design. Right: Pattern containing multiple cell types. The densest cells, with small holes, are located near the top, and the least dense cells are located near the bottom. A few intermediate shapes can be found in between these areas.*

As the researchers at TNO demonstrated, the current limitations of additive manufacturing can be effectively addressed with multiscale modeling and multiphysics simulation. Such advancements help to extend the power and reach of 3D printing beyond conventional techniques. This enables the development of new and more complex products, further spreading the benefits of 3D printing technology.

- Read more about TNO’s simulation research on page 22 of
*COMSOL News*2015 - Browse additional blog posts on the topic of 3D printing

In COMSOL Multiphysics, you can access a variety of predefined materials to model mechanical deformation in solids. Material models for plasticity, viscoelasticity, creep, and hyperelasticity are just some of those that are available.

By using the built-in constitutive laws as a starting point, you have the ability to create your own material models based on stress or strain invariants, flow rules, or creep laws directly in the user interface. Extra PDEs or distributed ODEs can also extend a given material law. But what if your material model includes nonlinear expressions that are impossible to express in terms of standard variables, invariants, or additional PDEs?

The latest version of COMSOL Multiphysics — version 5.2 — features a new way for you to specify user-defined material models. In structural mechanics analyses, you are now able to completely define a nonlinear stress-strain relationship, or include an inelastic strain contribution with an existing elastic material. Two new features in the *Solid Mechanics* interface complement this functionality: the *External Strain* subnode under the *Linear Elastic Material* node and the *External Stress-Strain Relation* material model.

*The External Material, External Strain, and External Stress-Strain Relation nodes in the model tree.*

With the added capabilities, implementing external material functions coded in the C programming language is possible. If you write a wrapper function in C code, material functions can also be written in other programming languages, making it easy for you to reuse your legacy code.

Along with programming your own material models, distributing your models to colleagues and customers as add-ons is now an option as well. You can even create easy-to-use apps, by using the Application Builder and incorporating your external material functions, and distribute these to your colleagues and customers as well.

To show you how the functionality works, we have added a tutorial to our Application Gallery that features a series of relevant demonstrations. The examples include a model file, a source C file, and a shared dynamic-link library (DLL) compiled and linked for a 64-bit Windows® operating system. (Running the models on Linux® operating systems and Mac OS X requires additional compilation and linking.)

In the first case, we explain how to write the C code for an isotropic linear elastic material and compare our results to the built-in *Linear Elastic Material* for a simple uniaxial test. The second, and more realistic, case shows how to implement a nonlinear material model that computes damage in concrete.

Let’s take a closer look at the latter of these two examples.

The deformation of brittle materials under mechanical loads is characterized by an initial elastic deformation. Upon unloading, the material will return back to its original state. However, if a critical stress or strain level is exceeded, a nonlinear fracture phase will follow the elastic phase.

As the critical value is reached, cracks will begin to grow and spread until the material fractures. The occurrence and growth of the cracks play an important role in the failure of concrete structures, and there are a number of theories used to describe such behavior. In the continuum damage mechanics formalism, a “damage” variable represents the amount of deterioration due to crack growth. The damage variable controls the weakening of the material’s stiffness.

Mazars’ model for concrete damage characterizes the fracture behavior within concrete using an isotropic scalar damage variable d. The variable enters the constitutive stress-strain relationship as

\sigma = (1-d)C:\epsilon

Here, \sigma is the stress tensor, C is the elasticity matrix, and \epsilon is the strain tensor.

The material, which behaves as a linear-elastic solid, is undamaged when the damage variable equals zero and fully damaged as the variable approaches one.

The inception and evolution of the damage variable is quite tricky to compute with standard variables in COMSOL Multiphysics. How so? It requires memorizing previous steps and conditional manipulation of variables based on principal stresses and principal strains. To overcome this limitation, we implemented Mazars’ damage model in an external material function, adding the C code for inspection and modification. The C code includes the definition of a number of variables, for loops, and the use of state variables to memorize damage from previous steps.

*The external material function to compute Mazars’ damage model contains just over one hundred lines of code.*

The plot below shows the computed uniaxial stress-strain response using Mazars’ damage model. The results are in excellent agreement with those findings presented in the book *Mechanical Behavior of Concrete*.

*Uniaxial stress-strain response plot.*

- Download the tutorial: External Material Examples, Structural Mechanics
- For more details on how to compile the C code for different operating systems, see the section “Working with External Materials” in the
*COMSOL Multiphysics Reference Manual* - Interested in learning more about Mazars’ damage model? Check out the following articles:
- J. Reynouard et al., “Modeling the Macroscopic Behavior of Concrete”, in
*Mechanical Behavior of Concrete*, ed. J. Reynouard, J. Torrenti, and G. Pijaudier-Cabot. 63-119, Wiley 2010 - J. Mazars et al., “Local Second Gradient Models and Damage Mechanics: 1D Post-Localization Studies in Concrete Specimens”, in
*Bifurcations, Instabilities, Degradation in Geomechanics*, ed. G. Exadaktylos and I. Vardoulakis. 127-142, Springer 2007

- J. Reynouard et al., “Modeling the Macroscopic Behavior of Concrete”, in

*Microsoft and Windows are either registered trademarks or trademarks of Microsoft Corporation in the United States and/or other countries. *

Linux is a registered trademark of Linus Torvalds.

*Mac OS is a trademark of Apple Inc., registered in the U.S. and other countries.*

Think about the first architects who designed a bridge above water. The design process likely included several trials and subsequent failures before they could safely allow people to cross the river. COMSOL Multiphysics and the Optimization Module would have helped make this process much simpler, if they had computers at the time, of course. Before we start to discuss building and optimizing bridges, let’s first identify the best design for a simple beam with the help of topology optimization.

In our structural steel beam example, both ends of the beam are on rollers, with an edge load acting on the top of the middle part. The beam’s dimensions are 6 m x 1 m x 0.5 m. In this case, we stay in the linear elastic domain and, due to the dimensions, we can use a 2D plane stress formulation. Note that there is a symmetry axis at x = 3 m.

*Geometry of the beam with loads and constraints.*

Using the beam geometry depicted above, we want to find the best compromise between the amount of material used and the stiffness of the beam. In order to do that we need to convert this into a mathematically formal language for optimization. Every optimization problem consists of finding the best design vector \alpha, such that the objective function F(\alpha) is minimal. Mathematically, this is written as \displaystyle \min_{\alpha} F(\alpha).

The design vector choice defines the type of optimization problem that is being solved:

- If \alpha is a set of parameters driving the geometry (e.g., length or height), we are talking about
*parametric optimization*. - If \alpha controls the exterior curves of the geometry, we are talking about
*shape optimization*. - If \alpha is a function determining whether a certain point of the geometry is void or solid, we are talking about
*topology optimization*.

Topology optimization is applied when you have no idea of the best design structure. On the one hand, this method is more flexible than others because any shape can be obtained as a result. On the other hand, the result is not always directly feasible. As such, topology optimization is often used in the initial phase, providing guidelines for future design schemes.

In practice, we define an artificial density function \rho_{design}(X) , which is between 0 and 1 for each point X = \lbrace x,y \rbrace of the geometry. For a structural mechanics simulation, this function is used to build a penalized Young’s modulus:

E(X)= \rho_{design} (X) E_0

where E_0 is the true Young’s modulus. Thus, \rho_{design}= 0 corresponds to a void part and \rho_{design}= 1 corresponds to a solid part.

As mentioned before, in regards to the objective function, we want to maximize the stiffness of the beam. For structural mechanics problems, maximizing the stiffness is the same as minimizing the compliance. In terms of energy, it is also equivalent to minimizing the total strain energy defined as:

\displaystyle Ws_{total}=\int_\Omega Ws \ d\Omega=\int_\Omega \frac{1}{2} \sigma: \varepsilon\ d\Omega

Our topology optimization problem is thus written as:

\min_{\rho_{design}}\int_\Omega \frac{1}{2} \sigma (\rho_{design}): \varepsilon\ d\Omega

Now that our optimization problem has been defined, we can set it up in COMSOL Multiphysics. In this blog post, we will not detail the solid mechanics portion of our simulation. There are, however, several tutorials from our Structural Mechanics Module that help showcase this element.

When adding the *Optimization* physics interface, it is possible to define a *Control Variable Field* on a domain. As a first discretization for \rho_{design}, we can select a constant element order. This means that we will have one value of \rho_{design} through all the mesh elements.

After this step is completed, a new Young’s modulus can be defined for the structural mechanics simulation, such as E(X)=\rho_{design} E_0.

As referenced above, the objective function is an integration over the domain. In the *Optimization* interface, we select *Integral Objective*. The elastic strain energy density is a predefined variable named *solid.Ws*. Thus, the objective can be easily defined as \int_\Omega Ws \ d\Omega.

Our discussion today will not focus on how optimization works in practice. Basically, the optimization solver begins with an initial guess and iterates on the design vector until the function objective has reached its minimum.

If we run our optimization problem, we get the results shown below.

*Results from the initial test.*

The solution is trivial in order to maximize the stiffness. The optimal solution shows the full amount of the original material!

After this initial test, we can conclude that a mass constraint is necessary if we want to make the optimization algorithm select a design. With a constraint of 50 percent, this could be written as:

\int_\Omega \rho \leq 0.5M_0 \Leftrightarrow \int_\Omega \rho_{design} \leq 0.5V_0

In COMSOL Multiphysics, a mass constraint can be included by adding an *Integral Inequality Constraint*. Additionally, the initial value for \rho_{design} needs to be set to 0.5 in order to respect this constraint at the initial state.

Let’s have a look at the results from this new problem, which are illustrated in the following animation.

*Results with the addition of a mass constraint.*

While this result is better, a problem remains: We have many areas with intermediate values for \rho_{design}. For the design, we only need to know if a given area is void or not. In order to get mostly 1 or 0 for the \rho_{design}, the intermediate values must be penalized. To do so, we can add an exponent p in the penalized Young’s modulus expression:

E(X)=(\rho_{design})^p E_0

In practice, p is taken between 3 and 5. For instance, if p = 5 and \rho_{design}= 0.5, the penalized Young’s modulus will be 0.03125 E_0. The contribution for the mass constraint, meanwhile, will still be 0.5. As such, the optimization algorithm will try lending to 0 or 1 for the design vector.

With our new penalized Young’s modulus, we get the following result.

*Results with the new penalized Young’s modulus.*

A beam design has started to emerge! There is, however, a problematic checkerboard design, one that seems to be highly dependent upon the chosen mesh. In order to avoid the checkerboard design, we need to control the variations of \rho_{design} in space. One way to estimate variations of a variable field is to compute its derivative norm integrated on the whole domain:

\int_\Omega |\nabla \rho_{design}|^2 \ d\Omega

A new question arises: How can we minimize both the variation of \rho_{design} and the total strain energy?

Since a scalar objective function is necessary, these objectives must be combined. We can think about adding them, but first, the two expressions need to be scaled to get values around 1. Concerning the stiffness objective, we simply divide by Ws0, which is the value of the total strain energy when \rho_{design} is constant. In regards to the regularization term, we can take the following scaling factor \frac{h_0 h_{max}}{A}, where h_{max} is the maximum mesh size, h_0 is the expected size of details in the solution, and A is the area of the design space. Our final optimization problem is now written as:

\min_{\rho_{design}} \ \ {q\int_\Omega \frac{Ws}{Ws0} \ d\Omega + (1-q)\frac{h_0 h_{max}}{A}\int_\Omega |\nabla \rho_{design}|^2 \ d\Omega}

s.t. ~ \int_\Omega ρ_{design} \leq 0.5V_0

where the factor q controls the regularization weight.

Finally, the discretization of \rho_{design} needs to be changed to Lagrange linear elements to enable the computation of its derivative.

By solving this final problem, we obtain results that offer helpful insight as to the best design structure for the beam.

*Results with regularization.*

Such a design scheme can be seen at different scales in the real world, as illustrated in the bridge below.

*A warren-type truss bridge. Image in the public domain, via Wikimedia Commons.*

Now that we have set up our topology optimization method, let’s move on to a slightly more complicated design space. We want to answer the question of how to design a bridge above water. To do so, a road zone in the geometry must be defined where the Young’s modulus is not penalized.

*Design space for a through-arch bridge.*

After a few iterations, we obtain a very good result for the through-arch bridge, one that is quite impressive. Such a result could provide architects with a solid understanding of the design that should be used for the bridge.

*Topology optimization results for a through-arch bridge.*

While the mathematical optimization algorithm had no guidelines on the particular design scheme, the result depicted above likely brings a real bridge design to mind. The Bayonne Bridge, shown below, is just one example among many others.

*The Bayonne Bridge. Image in the public domain, via Wikimedia Commons.*

It is important to note that this topology optimization method can be used in the exact same way for 3D cases. Applying the same bridge design question, the animation below shows a test in 3D for the design of a deck arch bridge.

*3D topology optimization for a deck arch bridge.*

Here, we have described the basics of using the topology optimization method for a structural mechanics analysis. To implement this method on your own, you can download the Topology Optimization of an MBB Beam tutorial from our Application Gallery.

While topology optimization may have initially been built for a mechanical design, the penalization method can also be applied to a large range of physics-based analyses in COMSOL Multiphysics. Our Minimizing the Flow Velocity in a Microchannel tutorial, for instance, provides an example of flow optimization.

- “Optimal shape design as a material distribution problem”, by M.P. Bendsøe.
- Topology Optimization: Theory, Methods, and Applications, by M.P. Bendsøe and O. Sigmund.

In mechanical vibration theory, the *vibrations nodes* are defined as the points that never move when a wave is passing through them. Because of a wave created by the impact of a ball hitting a racket, the racket will, in turn, begin to oscillate and vibrate. By looking at the mode shapes of the racket — held by a player at the end of the grip — we can identify points where the vibration motion is zero (i.e., where the magnitude is zero at any time during vibration). Here are the first three mode shapes of a tennis racket computed with COMSOL Multiphysics:

*The first three mode shapes of a tennis racket, from left to right and top to bottom. The fundamental mode is at 15 Hz, the second mode is at 140 Hz, and the third mode is at 405 Hz.*

As illustrated above, many different points feature this behavior. So why am I talking as if there is only one vibration node? In reality, there is actually an infinite number of vibration nodes. Upon impact, the ball generates an infinite number of harmonic series at different frequencies. An infinite number of frequencies are excited at one time, but which vibration node is the “sweet spot”? Is it the fundamental mode shape vibration node or is it a node that results from the crossing of different harmonics?

The fundamental mode vibration node cannot be the sweet spot for an obvious reason: It is located at the grip. Try hitting the ball with the grip to pass it over the net. If you are very lucky, you may succeed, but most likely, you won’t. The second vibration mode, meanwhile, has two nodes: one at the grip and one on the strings near the frame head. The latter is considered the sweet spot. Any player that hits the ball at this point will feel almost no vibration during impact.

There are, of course, vibration nodes on the strings for higher modes, as depicted in the third mode from the simulations above. However, as the natural frequency of the mode increases, the magnitude of the vibration drastically *decreases*. The graph below shows the frequency response of a sinusoidal load of 5 ms — approximately the duration of a ball’s impact upon hitting a racket — on a beam-like structure. For frequencies higher than 300 Hz, the magnitude is almost zero. That is, the influence of the third mode or higher is negligible. No matter where the ball strikes, even at points where the magnitude of the mode shape has reached its maximum, the higher modes will not have any influence at all because they are not excited.

*A plot showing the frequency response of a sinus load of 5 ms.*

When the ball hits the tennis racket near one end, with no other force acting on it, the racket will rotate about an axis toward the other end. As the point where the ball strikes the racket becomes closer to the center of mass, the distance from the axis of rotation will decrease. In a case where the ball hits the center of mass, the racket will translate without any rotation. The center of rotation is, from a mathematical point of view, at an infinite distance from the racket.

That said, it is possible to find an impact location that produces a center of rotation near the end of the grip where the player holds the equipment. We can find a location at a certain distance from the center of mass where the ball hits the racket and results in a center of rotation near the end of the grip. Referred to as the *center of percussion (COP)*, this location is sometimes considered a sweet spot as well. No force is applied to the hands, as the racket rotates about a center of rotation near the end of the grip, avoiding the player’s hands.

*Compared to older wooden tennis rackets from the 1970s, modern forms of this equipment feature a much larger head. This new design element has been used to move the center of percussion near the middle of the strings rather than by the racket’s frame. Image by CORE-Materials, via Wikimedia Commons.*

Let’s now take a quick look at what happens from a mechanical standpoint. For this purpose, we assume that the racket can be modeled as a rigid beam-like structure.

*Sketch of the beam-like structure. The parameters used in the following equations are defined in this figure. *

A force F applied to a free beam of mass M at a distance b from the center of mass implies that the center of mass translates at a speed V_{cm}. From Newton’s second law,

F=M \frac{d V_{cm}}{dt}

Moreover, a torque is generated by the force F about the center of mass:

Fb = I\frac{d\omega}{dt}

where I is the beam moment of inertia along the rotation axis and \omega is the angular velocity. Consider P, a point at a distance c from the center of mass. The speed v of this point is v=V_{cm}-c\omega, leading to:

\frac{dv}{dt}=(\frac{1}{M}-\frac{cb}{I})F

Since the center of rotation corresponds to the point where there is no translational acceleration, the COP is at a distance b_{cop} from the center of mass, which is given by

b_{cop}=\frac{I}{c_{cr}M}

where c_{cr} is the distance between the center of rotation and the center of mass. Given that the distance between the center of mass and the ideal center of rotation (at the grip end) is known, it is rather straightforward to determine the position of the COP for a particular racket shape.

The *power point*, sometimes called the third sweet spot, is the best bouncing point. In other words, this is where the ball achieves the most bounce upon contact. From a mathematical standpoint, the power point is defined as the point with the highest *coefficient of restitution (COR)*, the ratio of the rebound height to the incident height of the ball. The coefficient of restitution is quite useful in the sense that it is the result of *all* of the design elements that affect the speed of the ball. Design engineers do not need to know the influence of each parameter, as the COR provides a combined overview of all of these factors.

The power point is located at the throat of the racket, near the center of mass. The closer the point is to the throat, the greater the stiffness and the lower the energy loss during racket deformation. When a ball hits a racket, the impact energy is divided into kinetic energy and elastic energy (energy of deformation) throughout the ball, racket, and strings. At the power spot, deformation is very small, causing the racket to give almost all of the kinetic energy back to the ball.

The power spot is very useful when returning a fast serve. Indeed, if you must return a fast serve, you do not have much time to move your racket and prepare your stroke, so you will return the ball as it comes. However, it is important to note that the closer the ball comes to the power spot, the better your stroke will be.

One last interesting spot on the racket that I’d like to mention is the *dead spot*. When a ball strikes the dead spot, the ball will not bounce at all. All of the ball’s energy is given to the racket and no energy is given back to the ball. This is due to the fact that the effective mass of the racket at the dead spot — usually close to the tip — is equal to the mass of the ball. Mechanically speaking, the ratio between the resulting force and the acceleration at the dead spot is equal to the mass of the ball.

To better understand the physical phenomena at hand, let’s imagine the ideal collision between a rigid ball at an initial speed V_0 and another rigid ball, initially at rest, that features the same mass m. The conservation of energy and the conservation of momentum lead to:

\frac{1}{2}mV_0^2 = \frac{1}{2}mV_1^2+\frac{1}{2}mV_2^2

mV_0=mV_1+mV_2

Therefore, it turns out that:

V_1=0 \ \text{and} \ V_2=V_0

If a ball collides with another ball that is of the same mass but at rest, the ball will stop dead and give all of its energy to the other ball. Thus, when a ball hits the dead spot of a racket at rest, the ball will not bounce at all. This would be a very bad spot to use when trying to return a serve. On the other hand, when you actively hit a stationary ball, as in your own serve, the dead spot will provide a high momentum transfer of energy from the racket to the ball.

Then, when it is your turn to serve, what is the optimal point? This is not only determined by the mathematics of sweet spots. In most cases, the answer would be rather close to the tip. Because of the way you move your arm, the racket will feature a significantly higher speed at the tip than at the throat. Thus, the optimal point is determined by a combination of high impact speed and good momentum transfer properties.

We have now gained insight into the physical meaning behind the three well-known sweet spots of a tennis racket. At the vibration node, the uncomfortable vibration that tennis players feel over their hand and arm is minimal. At the center of percussion, the initial shock to the player’s hand is also minimal. Lastly, at the power point, the ball rebounds with the maximum level of speed.

*The location of the sweet spots on a tennis racket.*

Perfect your game; check out these additional resources for improving your tennis skills:

*Tennis Science for Tennis Players*, H. Brody.- “Physics of the tennis racket“, H. Brody.
- “Physics of the tennis racket II: The “sweet spot“, H. Brody.
- “Physics of the tennis racket III: The ball-racket interaction“, H. Brody.

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

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.