With the use of COMSOL Multiphysics, Tingcheng Wu, Guillaume Escamez, Clement Lorin, and Philippe J. Mason from the Department of Mechanical Engineering at the University of Houston were able to perform simulations to analyze how individual components of the structure impacted its overall performance. By applying various parameters to the structural design, they were able to conclude which factors had the greatest effect on the machine, both structurally and thermally. Thus, they could determine how to achieve a balance between the two.
The team of researchers modeled the structure as a rotor shaft separated into five individual parts connected by bolts. As a means to provide thermal insulation, different materials were used on particular areas of the shaft, as you can see below.
Bolts were used to connect stainless steel with G10, a glass fiber material characterized by low thermal conductivity and a high yield stress. Image by T. Wu, G. Escamez, C. Lorin, and P. Mason, and taken from their poster submission.
Seeking to analyze the issues of heat transfer and solid mechanics in these machines, the researchers used the Heat Transfer Module and Structural Mechanics Module to create their simulations. The figure below highlights the team’s findings regarding temperature, depicting that the G10 components take on the greatest temperature gradient.
In the next figure, the connection bolts underwent the most stress within the structure, a factor that was found to decrease as the cross-section area was decreased.
Simulations highlight the thermal and structural pressure that the shaft endured, especially along its connection bolts. Image by T. Wu, G. Escamez, C. Lorin, and P. Mason, and taken from their presentation.
With the continued funding and efforts of NASA and other research teams, progress continues to be made in the design of aircraft. As torque transfer components within fully superconducting rotating machines continue to be optimized, researchers gain momentum in their quest for developing structures with greater power densities and the potential for electric propulsion. In addition to making air travel a quieter and more energy efficient process, implementing this technology paves the way for its potential use within modes of ground transportation as well.
Modular orthopedic devices, common in replacement joints, allow surgeons to tailor the size, material, and design of an implant directly to a patient’s needs. This flexibility and customization is counterbalanced, however, by a need for the implant components to fit together correctly. With parts that are not ideally matched, micro-motions and stresses on mismatched surfaces can cause fretting fatigue and corrosion. Researchers at Continuum Blue Ltd. have assessed changes to femoral implant designs to quantify and prevent this damage.
Take a few steps and see how your hips rotate. You’ll find that your body weight is continuously shifting between the left and right sides, while your legs bend, swing, and then straighten out with each step. Thus, a good modular hip replacement system will need to be able to freely allow for the natural motions of the human body — walking, running, or going up and down stairs. In addition to this, it has to be durable enough to take the continually changing, and sometimes excessive, loads placed on it during these movements, while being comprised of lightweight materials that fit and interact well with the body.
Modular implants often include stems, heads, cups, or entire joint systems. A range of materials from steel and titanium alloys to polymers and ceramics offer the surgeon many options depending on the needs of the patient. However, material and geometric selections affect the amount of wear and tear that will occur over time, so certain combinations of components are better than others. With so many different factors at play, it is not surprising that these assemblies require tight tolerances and the right material combinations to function properly and last a lifetime.
Virtual implantation of hip replacement in resected patient femur.
Studying how a modular combination of parts will behave under dynamic loads and stresses is a crucial part of the design and decision-making process. In order to understand the available combinations better and aid medical professionals in decisions, engineers at Continuum Blue have modeled three combinations of modular femur stem and head implants to investigate the fretting fatigue; the fatigue wear caused by the repeated relative sliding motion of one surface on another.
The femur head contains an angled channel for the neck of a femur stem, which in turn must be tapered correctly to fit the channel. The engineers studied three different geometric configurations using different materials for the head and stem to determine which of the three was best for minimizing fretting fatigue.
Different stem and head configurations with an ideal fit, positive mismatch, and negative mismatch.
Using kinematic load data from Bergmann et al. and based on averages from four patient sets, Continuum Blue created a COMSOL Multiphysics simulation to analyze the cyclic loading on a femur head. They used their model to determine the loading at different points during a walking gait cycle, knowing that the load would change at different locations in the rotation, and validated their results against the kinematic data.
Simulation results showing the dynamic loads and stresses during the gait cycle.
Material fatigue can be determined by studying the mean stress and stress amplitude that occur during the cyclic loading of the joint. Like the loading in the femur head shown earlier, the stresses in the femur stem will change over the course of a gait cycle. With regular leg movements, the stresses observed will take on an oscillation that reflects the repeated motion of the person walking.
SN curves for the titanium stem and cobalt chromium head used in the study.
Continuum Blue assessed the three configurations with two different materials: a cobalt chromium alloy for the head and a titanium alloy for the stem of the modular implant. For each material domain, they calculated the stresses observed over a single gait cycle and related these to both the SN curves of the material and the micro-motions of the contact surfaces. This allowed them to predict the number of cycles the device could undergo before fretting fatigue became an issue.
Areas where fretting fatigue occurs over gait cycle for each configuration.
Their results showed a surprising fact: the “ideal” fit, where the femur head channel is exactly aligned to the sides of the femur stem, was not found to be the best configuration for minimizing fretting fatigue. Rather, the configuration with a slight positive misalignment turned out to be a better choice, exhibiting lower stresses and overall fretting fatigue.
Through their simulation, Continuum Blue was able to predict the stress, contact pressure, and areas most susceptible to fretting fatigue at different points in a gait cycle. There are many other factors that will be accounted for in future research, such as the sensitivity of the implant to varying degrees of misalignment; additional designs and geometric changes; different materials; and the effects of surface finishes, coatings, or roughness that may impact the results. However, their modeling work offers a unique promise for evaluating the lifetime of a modular implant device. It was validated as an accurate way to predict the wear and tear that will occur for these three configurations of the implant. If you ever need a joint replacement analysis — you’ll know who to call.
Ever since the first offshore wind farm was built off the Danish coast in 1991, offshore wind has been gaining in popularity. Just over two decades later, at the end of 2012, the European Union was producing enough electricity from offshore wind farms to power approximately five million households. In the coming decade, offshore wind farms are expected to generate nearly one fifth of the European Union’s power consumption, jumping from about 6.04 GW in 2013 to over 150 GW by 2030, according to a report by the European Wind Energy Association.
Windmill park in Oresund between Copenhagen, Denmark and Malmo, Sweden. Photo credit: Ziad, Wikipedia Commons.
With this huge increase in wind power expected, engineers are being called in to investigate the effect that offshore turbines could have on marine life. In a recent report conducted by Xi Engineering Consultants for the Scottish Government, Brett Marmo, Iain Roberts, and Mark-Paul Buckingham investigated how different types of wind turbine foundations affect the vibrations that propagate from the turbine into the sea, and ultimately how these vibrations could affect surrounding marine life. Also involved in the project were Ian Davies and Kate Brookes of Marine Scotland, who helped define the water depth, turbine size, and foundation types of the turbines modeled in the study based off of the types of turbines submitted to the Scottish Government for licensing permits. Additionally, Davies and Brookes helped identify the marine species most likely to be affected by offshore wind.
I recently interviewed Brett Marmo about the project. “In our research, we explored how different bases affect the noise that is produced by offshore turbines, and whether or not this noise was loud enough to be heard by marine life,” Marmo explained. “We studied three different wind turbine bases and examined the possible effect that noise could have on various types of local whales, porpoise, seals, dolphins, trout, and salmon.”
Vibrations produced by offshore turbines travel from the tower into the turbine foundation and are released as noise into the surrounding marine environment. “Because the noise is emitted at the interface between the foundation and seawater, it’s likely that the intensity and frequency of the noise will vary with the type of foundation used,” described Marmo. “Using finite element analysis, we modeled three identical wind turbines, only altering the structure of the foundation.”
Below, you can see the three most common foundation types: the gravity base, jack foundation, and monopile foundation. Generally, the jacket and gravity base are used in water 50 meters or deeper, while the monopile is generally not used at depths exceeding 30 meters. Due to the different structures, materials, and size of each of these bases, the vibrations that propagate through the base behave very differently, leading to noise produced with different frequencies and sound pressure levels (SPL).
Three different foundation types are shown: a gravity base structure sitting on the seabed (left), a jacket with pin pile connections to the seabed (middle), and monopile placed onto the seabed with a transition piece (right).
“Using simulation allowed us to model the noise produced by the foundations under identical operating conditions — something that we wouldn’t have been able to achieve by just taking measurements of in-service wind turbines,” says Marmo. “Without simulation, the different environments and wind loads that these turbines experience would have made it very difficult to determine if it was truly the foundation that was affecting the noise produced and not another unaccounted for variable.”
Before delving into the simulations, let’s first explore where it is that the noise itself comes from. Noise from wind turbines can come from two places; aerodynamic noise is produced by the blades slicing through the air, and mechanical noise is generated by machinery housed in the gearbox. Almost all of the noise produced by the blades themselves gets reflected back from the water’s surface due to the large refractive difference between the air and water, and does not enter the marine environment.
Therefore, the majority of noise is created by mechanical vibrations produced in the turbine’s gearbox and drivetrain by rotational imbalances, gear meshing, blade pass, and by electromagnetic effects between the poles and stators in the generator. Each of these noise sources produce vibrations with a different frequency, which then transmit down the turbine pole and into the foundation. Here is a table of the different frequencies produced and their origin:
Frequency | |
---|---|
Rotational imbalance of rotor | 0.05 to 0.5 Hz |
Rotational imbalance of high-speed shaft between gearbox and generator | 10 to 50 Hz |
Gear teeth meshing | 8 to 1000 Hz |
Electro-magnetic interactions in the generator | 50 to 2000 Hz |
Frequency bands likely to contain vibration tones produced in the drive train of wind turbines. Table courtesy of Xi Engineering and adapted from their report.
Once the vibrations enter the foundation, the amplitude of the noise emitted is affected by the size of the excitation force, the frequency of structural resonance, and the amount of damping in the structure. Additionally, higher wind speeds lead to increased torque acting on the rotor, likely meaning that higher noise is emitted.
“Understanding the effect of damping — the dissipation of vibration energy from a structure — was one of the key analyses conducted in our project,” described Marmo. “In general, steel structures such as the jacket foundation have less damping than those built from granular materials, such as the gravity base, which is made of concrete.” The amount of internal damping taking place within a structure will therefore affect the noise emitted by different structures. In order to determine how these factors affected the noise produced, Marmo and the team turned to simulation with COMSOL Multiphysics.
Noise is produced at the interface between the wind turbine foundation and seawater, where the vibration of the foundation oscillates water molecules to produce a pressure wave that radiates from the foundation as sound. Geometric spreading and absorption reduce the intensity of the sound as it propagates farther from the foundation, with high frequency sound being absorbed more quickly and low frequency sound absorbing slower and therefore propagating further.
Marmo analyzed each of the three foundations at three different wind speeds (5 m/s, 10 m/s, and 15 m/s) and found that typically, the higher the wind speed the louder the noise produced. A comparison of the average sound pressure level at a wind speed of 15 m/s at different frequencies for each of the three foundation types is shown below.
At frequencies lower than 180 Hz, the monopile produces the largest amount of noise. Of the three foundation types, the monopile continues to produce larger SPL values up to 500 Hz. Around 600 Hz, all three foundation types become comparable in average 30 m SPL with the trend of the jacket foundation rising to become the noisiest at frequencies greater than 700 Hz.
As the graph shows, the jacket base demonstrates the lowest sound pressure level of the three at low frequencies (around 200 Hz and lower). However, at high frequencies, the jacket produces the highest sound pressure level. The monopile and gravity base exhibit comparable sound pressure levels at lower frequencies, while at higher frequencies the gravity base produces the lowest sound pressure level of the three bases. The images below illustrate the sound pressure level around each of the three foundation types at the frequency at which the foundation produces the loudest noise.
Marmo and the team also created a far-field model that used a Gaussian beam trace model to analyze the distances at which a wind farm containing 16 turbines could be heard. As mentioned above, sound at lower frequencies tends to propagate farther than sound at higher frequencies. Additionally, ambient noise can mask the sound produced by wind turbines, making them nearly impossible to hear. This was also taken into account in Marmo’s analyses.
“We found that each of the different bases produced the loudest sound in the far-field at different frequencies,” described Marmo. “At a wind speed of 10 and 15 m/s, the monopile and gravity bases are audible at least 18 km away at most frequencies below 800 Hz, while the jacket is audible at 250 Hz 10 km away and 630 Hz at least 18 km away.” Here is a summary of these results:
The next step in the project was to determine the frequencies at which marine species could detect the sound and over what distances. Each of the different foundation types emitted different sound pressure levels at different strengths and frequencies. Since various marine animals have different hearing thresholds, this also had to be taken into account.
Cormac Booth and Stephanie King of SMRU Marine at St. Andrews University were the key marine biologists who analyzed the hearing thresholds of different marine species and determined whether or not the noise produced could affect the animal’s behavior.
Hearing thresholds for dolphins, minke whales, porpoises, and seals.
Of the species examined, the minke whale had the most sensitive hearing at low frequencies (less than 2000 Hz) and was able to hear the turbine from the farthest distances. “We predict that minke whales will be able to detect wind farms constructed of either monopile or gravity foundations up to 18 km away at most frequencies below 800 Hz and for all three wind speeds,” says Marmo. “On the other hand, bottlenose dolphins and porpoises are less sensitive to low frequencies. Dolphins can detect a wind farm on a gravity base 4 km away at wind speeds above 10 ms, but can only detect jackets and monopiles at close ranges of less than 1 km.”
You can view an example of the results found in Marmo’s report, showing the hearing threshold of a seal for different wind speeds and frequencies:
Determining behavior responses was harder to predict. Using a sensation parameter, Booth and King estimated the upper and lower ranges around the hearing threshold of each of the species. Then, they determined what percentage of animals could be expected to move away from the turbines within a certain sound pressure range.
Neither seal species nor bottlenose dolphins were predicted to exhibit a behavioral response to the sounds generated under any of the operational wind turbine scenarios. However, between around 4 kilometers and 13 kilometers, 10 percent of minke whales encountering the noise field produced by the monopile foundation were expected to move away. Overall, jacket foundations appear to generate the lowest marine mammal impact ranges when compared to gravity and monopile foundations.
What does this mean for the future of offshore wind power? Marmo and his team’s report found that there were little to no detrimental effects from wind turbine noise on marine species. Although more studies still need to be conducted, these findings demonstrate that the future of offshore wind is looking positive.
Keeping that perfectly round tire shape is important for more than just aesthetics. The lower the air pressure in a tire gets, the harder it is for the car to move forward. Every tiny leak is a source of strain for the vehicle, which can be fixed — as long as the driver is aware of the low pressure in the first place.
Tire pressure monitoring sensors, designed by Schrader Electronics, are meant to mount directly on the wheel assembly and when measurement gets below a certain pressure, a warning goes off.
I was lucky enough to speak with the researchers who work on the Schrader sensor models (their company actually builds 45 million every year). If you find yourself in a fairly new vehicle, chances are it contains a pressure sensor from Schrader. Designing these sensors for functionality and longevity is a huge part of keeping cars effective and safe.
The Schrader research team, led by Christabel Evans, tested out different shapes and components so that the sensors would last and still be able to relay measurement information back to the automobile’s dashboard without external interference. They relied on COMSOL Multiphysics together with the Structural Mechanics Module and CAD Import Module to help them choose the right parameters.
The geometry of their model consists of a circuit transmitter in a solid enclosure. The enclosure takes the brunt of the force and pressure while the wheel spins. For this reason, Schrader needed to model everything the sensor might encounter on the road, including tire fitment, vibration, and shock. They also had to factor in all of the important natural parameters, such as pressure and crush load, centrifugal force, and temperature change.
Schrader’s tire pressure sensor fits directly into the rim of the wheel assembly. Image courtesy of Adam Wright, Schrader Electronics.
A model showing the enclosure illustrates that different applied forces can cause deformation in the device over time. By using finite element analysis (FEA) and simulation throughout the process of product design and testing, Schrader was able to isolate or couple variables as needed, allowing them to work towards the best design. This gave them the flexibility to build and refine as they went.
10x amplification of stress and deformation on the transmitter housing as a result of centrifugal loading, which is produced by the wheel’s rotation.
As they narrowed down the best design, Schrader also simulated the equipment used for testing the device rotating on the tire. They mapped where the greatest stress occurred and found that it mostly took place along the bolts of the collar. This allowed them to make all of the necessary adjustments to reinforce those areas and continue to optimize the design as they worked.
The spin test works at a very high speed, simulating conditions the sensors are exposed to. The model shows an increase in stress along all of the sections of the model where bolts are located.
Perhaps the most useful feature in COMSOL Multiphysics (for the Schrader engineers) was the ability to test various parameters, shapes, and designs. It was much easier to get the most out of their computational power by running several models simultaneously. As they continue their research, they plan to keep focusing on failure analysis to fortify their design and continue to improve their product’s accuracy and life span.
The outcome of your golf stroke is basically determined by the movement of the club head just prior to impact with the ball. Considering this, we should be able to see how your golf swing could be improved based on a multibody analysis.
Here, I will show you how I went about modeling various body parts, a golf club, and the connections among them using the Multibody Dynamics Module.
A simple way to simulate a golf swing is by using a two-link model, where the arm and club are the two links connected together by a hinge joint. In this model, the arm rotates about a fixed point, located at the base of the neck, and the club rotates about the wrist joint relative to the arm. The two-link model does not allow a sufficiently long backswing and is not actually a true representation of a real-life golf swing.
A better representation is the three-link model, which also includes the shoulder as a separate link. Adding one more link eliminates the problem related to the backswing. Hence, we will use this three-link model in our analysis.
Diagram of the two-link and three-link swing models.
This analysis focuses on maximizing the club head speed just prior to impact with the ball, by understanding the mechanics of a golf swing. The torque profile, applied by different body parts (shoulder, arms, and wrist) is assumed. It is limited by the maximum torque capacity of the respective parts. Among all applied torques, the wrist torque has quite an important role to play in getting the strike right.
Modeled geometry of the three-link swing model.
While simulating the downswing of the club, the entire swing can be divided into two phases. In the first phase, arm and club rotate about the fixed point as a rigid assembly. In this phase, the arm and club are folded to minimize the inertia about the center of rotation, which allows the development of maximum angular velocity for the given arm-torque capacity. Here, the wrist is cocked to the maximum possible angle (the amount it can be cocked before you become uncomfortable or the angle is detrimental to your swing) and the applied wrist torque tries to hold back the club in this position against the other two torques.
In the second phase, the wrist torque starts helping the shoulder and the arm torque by pushing the club forward to increase the club head speed to its maximum. The instance when the wrist torque changes its role is a crucial parameter in determining the stroke quality. To see its effect on the club head speed, we vary the wrist torque parametrically.
Time history of torque applied by the shoulder, arm, and wrist for (t_w = 0.19 s).
The driving torque, applied by the shoulder, arm, and wrist, has a maximum capacity and can vary within the defined range. The applied shoulder torque is assumed to start at its maximum positive value, after a short build-up time. The applied arm torque, which acts on the arm and reacts on the shoulder, builds linearly with time to its maximum positive value with the specified rate. The applied wrist torque, which acts on the club and reacts on the arm, is fully negative to start and switches to its maximum positive value at the specified time (t_w).
On the arm and wrist joint, the rotation is not fully free. It is limited in the forward and backward directions by the ligaments, muscles, joint shape, or a combination of all these. In our golf-swing analysis, rotation limit in the backward direction is more important and this limiting value may vary from person to person.
In the beginning of the downswing, due to inertial forces on the body parts, these rotations try to go below the limiting value. Hence, additional torque is applied by the equivalent stiffness and damping of the stop. This makes the effective torque applied by the arm and wrist more than what is actually applied.
Golf club head speed during the downswing for different wrist torque switch times (t_w).
Above, I have plotted the club head speed for various wrist torque switch times (t_w) for the entire duration of approximately 0.25 seconds. It can be observed that for t_w = 0.15 s, we reach the maximum speed before impact — this leads to early hitting. On the other hand, for t_w = 0.23 s, the club head speed couldn’t even reach its maximum value.
For t_w = 0.19 s, the club head speed is higher than the other two cases and close to the optimum value for the given geometrical parameters and muscle strength.
Comparison of the golf club trajectory for different values of t_w (results are displayed in the increasing values of t_w).
Motion of links and the trajectory of arm joint, wrist joint, and the golf club head.
Maximum arm torque throughout the swing and very high arm speed in the beginning of the downswing can cause an early release, with the club head reaching its maximum speed before actually hitting the ball.
We can also deduce that for the given torque capacity, it’s potentially advantageous to have a long arm swing as well as a large wrist-cock limit angle. Furthermore, the extent to which the wrist can hold back the release is limited by its torque capacity. Therefore, your golfing skills are also strongly associated with the delayed release and the late hit.
In the downloadable model, we also consider the shaft flexibility by dividing the club into two parts: the grip and the shaft. These are connected through a hinge joint with finite stiffness and damping. You can see that the effect of the shaft flexibility to the swing is negligible compared to other parameters.
Residual stresses are self-equilibrating stresses that remain after performing the unloading of an elastic-plastic structure. During the manufacturing process of a mechanical part, residual stresses will be introduced. These will influence the part’s fatigue, failure, and even corrosion behaviors.
Indeed, uncontrolled residual stresses may cause a structure to fail prematurely. Although residual stresses may alter the performance, or even lead to the failure of manufactured products, some applications actually rely on them. For instance, brittle materials, such as glass in smartphone screens, are often manufactured so that compressive residual stresses are induced on the surface to avoid crack-tip propagation.
For these reasons, residual stresses play an important role in mechanical projects as a whole. Only through qualitative and quantitative analysis of these stresses is it possible to determine the most suitable machining processes for a given application. These types of analyses also help you discover the optimal amount of material to be used for their reliability or the most suitable shape that needs to be designed, in order to avoid malfunctions and failures.
Let’s consider the following slender beam with a rectangular cross section, depth a, and width b. The beam is fixed at the left-hand side and a bending moment is applied on the free end.
Based on the beam theory, it turns out that the bending moment is constant in this case and the stress can be written as:
(1)
where I_z is the moment of inertia about the z-axis.
As M_\mathrm{b} increases, the beam first behaves in an elastic manner, but after reaching its yield moment, M_y, it begins to take on plastic behavior. This leads to an elastic-plastic cross section. Once the plastic zone has propagated through the entire cross section, the ultimate bending moment, M_\mathrm{ult}, that the beam can carry is determined. Here, it is assumed that the beam will collapse at such a moment and that it has a perfectly plastic behavior.
The outer fibers of the beam will reach the yield point first, while the core fibers remain elastic. Thus, the previous equation applied to the outer fibers of the beam provides the first yielding moment:
(2)
where \sigma_\mathrm{yield} is the yield stress.
Under an elastic-plastic moment, M_\mathrm{ep} < M_\mathrm{ult}, the plastic zone propagates through the thickness by a distance of h_\mathrm{p} at each side of the beam, as shown below.
Plastic zone penetration in a rectangular cross-section beam.
The total moment can be divided into an elastic part, M_e, and a plastic part, M_p, such that:
(3)
where I_\mathrm{e}=\frac{a(b-2h_\mathrm{p})^3}{12} is the elastic core moment of inertia along the z-axis.
Combining the last two expressions, we get the following:
(4)
When an elastic-perfectly plastic beam is unloading from M_\mathrm{ep}, a state of residual stress, \sigma_r, remains in the beam cross section. The beam attempts to recover its initial shape following recovery of elastic bending stress, \sigma_\mathrm{e}. Here, it is assumed that purely elastic unloading occurs after being loaded at M_\mathrm{ep}, corresponding to a state of elastic-plastic stress, \sigma. The residual stresses can be computed from the difference between the elastic-plastic stress and the purely elastic stress — i.e., the stress you would have if plastic behavior was not involved.
(5)
The elastic bending theory gives the recovered elastic stress as:
(6)
Assuming a perfectly plastic behavior, the stress \sigma in the plastic zone (in other terms, \frac{b}{2}-h_\mathrm{p} \le |y| \le \frac{b}{2}) remains constant and equal to \sigma_\mathrm{yield}. Therefore, according to the Equation (5), the residual stresses can be written as:
(7)
In the elastic zone (in other terms, 0 \le |y| \le \frac{b}{2}-h_\mathrm{p}), the beam theory provides the applied stress as:
(8)
Therefore, the residual stress is then deduced as:
(9)
Note that after the external moment has been removed, the beam will still have some permanent displacement due to plastic deformation, but it will also have recovered some of the displacement that was present at the peak load. This springback effect is important when you want to achieve a controlled plastic deformation.
When modeling the beam in 2D, we could choose a plane stress assumption taking Poisson’s ratio, \nu=0, to match with the 1D beam theory, which does not account for the Poisson effect. In COMSOL Multiphysics, you can model 2D plane stress by selecting a 2D space dimension and choosing the Solid Mechanics interface.
Here, we will show how to use the Solid Mechanics interface in 2D to compute the residual stresses in the beam cross section.
A snapshot of the 2D beam model using the Solid Mechanics interface.
According to the snapshot above, we define variables to evaluate the theoretical residual stresses we worked out in the section above. Those values will be used to compare the computed results with the theoretical ones.
The applied bending moment is ramped progressively. A Plasticity node is added to account for the uniaxial plastic behavior that may occur through the beam thickness. Plastic flow begins once \sigma_x reaches the critical value \sigma_\mathrm{yield}. Any fiber that has reached this value will remain at a constant state of stress during loading.
In the graph below, you can see the stress distribution along the Y-axis of the cross section. The applied bending moment has been computed from Equation (4) for a plastic zone with depth h_\mathrm{p}=\frac{b}{4}=0.01 \ \mathrm{m}. According to the blue curve, COMSOL Multiphysics results match perfectly with this value. The red curve represents the residual stresses after one loading-unloading cycle. It is worth noting that the residual stresses obtained may also be found by subtracting the elastic curve (green) from the elastic-perfectly plastic curve (blue).
Stress value after elastic-plastic loading, elastic loading, and unloading.
Equations (7) and (9) have been defined as variables and compared to the solution computed in COMSOL Multiphysics. As shown in the previous screenshot, you can create a “switch” using the if() operator, so that the two expressions representing the analytical residual stresses are gathered together in one expression. The next graph shows both analytical and computed residual stresses after two loading-unloading cycles.
Analytical vs. computed residual stresses.
COMSOL Multiphysics enables you to model the hysteresis cycle of a given material. In the case of perfectly plastic behavior, as depicted below, the second load cycle already provides a stable stress-strain response that is representative of each consecutive load cycle. For instance, you can use these load cycles to carry out a fatigue analysis.
Hysteresis behavior after three loading-unloading cycles.
Last but not least, let’s find out how strain-hardening behavior influences residual stresses and loading-unloading cycles. So far, we have been dealing with a perfectly plastic material. The yield stress remains constant, no matter the number of cycles or whether a tensile or a compressive is applied. Equation (5) is only valid as long as reverse yielding does not occur. Since reverse plastic deformation during unloading has a negative effect on the performance, it is quite important to figure out under which condition reverse yielding is likely to occur.
A ductile material that is subjected to an increasing stress in one direction (in tension, for instance) and then unloaded, will behave differently when loaded in the reverse direction. It is found that the compressive yield stress is now lower than that measured in tension. This is called the Bauschinger effect. Similarly, an initial compression provides a lowered tensile yield stress. The figure below displays this effect over two stress cycles:
Hysteresis behavior with kinematic strain hardening.
Now, let’s move on to a more sophisticated mechanical process in which residual stresses are of great importance: the sheet metal forming process.
Die forming is a widespread sheet metal forming manufacturing process. The workpiece, usually a metal sheet, is permanently reshaped around a die through plastic deformation by forming and drawing processes. A blankholder applies pressure to the blank, leading the metal sheet to flow against the die.
In order to avoid cracks, tears, wrinkles, and too much thinning and stretching, you can turn to simulations. They can also be useful to estimate and overcome the springback phenomenon. This refers to how the workpiece will attempt to recover its initial shape once the forming process is done and the forming tools are removed. Springback can lead the formed blank to reach an unexpected state of warping. To cope with this effect, the sheet can be over-bent. Thus, the die, punch, and blankholder must be manufactured not only to match the actual shape of the object, but also to allow for springback.
In this study, the sheet is made of aluminum. A Hill’s orthotropic elastoplastic material model with isotropic hardening is used to characterize the plastic deformation. It has been observed that metal sheets in deep drawing process no longer behave isotropically. There tends to be less plastic deformation through the thickness. Therefore, in die forming and deep drawing of sheets, we need a kind of anisotropy where the sheet is isotropic in-plane and has an increased strength in the perpendicular direction, called transverse isotropy.
Below, we have illustrated the forming tools that are used in the process.
Forming tools: The die is shown in red, punch in blue, blankholder in pink, and the blank in gray.
As mentioned above, simulations can allow for handling several tasks that need to be taken into account whenever such a mechanical process is worked out. For instance, optimization of the corner radius of both the die and the punch can be carried out properly to prevent tearing of the metal sheet. It may also be useful to carry out simulations in order to get the clearance that is needed between the punch and the die, to avoid shearing or cutting of the metal blank.
One of the most challenging aspects is to figure out how much of the metal sheet should be over-bent. When the sheet has been formed, the residual stresses cause the material to spring back towards its initial position, so the sheet must be over-bent to achieve the desired bend angle. Therefore, you have to properly model residual stresses as not to over- or underestimate the springback phenomenon.
The two animations below show the sheet metal forming as well as the springback of the metal blank.
Representation in the RZ-plane of the spingback phenomena.
Simulation of sheet metal forming.
When subjecting the structure to other mechanical loads, the superposition of the residual stresses can reduce the reliability of the structure or even cause irreversible damages. Therefore, the residual stresses must be released as much as possible or be managed so that the structure can withstand the external loads that may be applied. The plot below shows the Hill effective residual stresses that remain around the bend regions after the deep-drawn cup process.
Today, we studied residual stresses in structural mechanics. We introduced a conventional definition, which was first applied to a bending beam example. We simulated this bending example using COMSOL Multiphysics and compared our results to the analytical solution from the beam theory. Then, we explored the importance of the residual stresses in a sheet metal forming example. We saw that any mechanical process induces residual stresses and particular care must be given to release them properly or, at least, be certain that they will not cause any damage.
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Power transmission systems need to deliver a required level of current without overheating the cables. Cable structure, internal electric losses, installation system geometry, and environmental conditions — such as the ambient temperature or the properties of the surrounding material — all contribute to system’s thermal response. Even external loads, such as those caused by solar radiation (in the case of an aboveground cable) or in proximity to other systems (e.g., other cables crossing the route), must be accounted for in the design.
While simple mathematical models and calculations suffice for many cases, cable systems are more commonly being installed in environments with many external influences that make it hard to predict performance and side effects.
That’s where multiphysics simulation enters the scene.
COMSOL Multiphysics simulation showing temperature distribution in the cross section of a double-armored umbilical cable.
Engineers at Prysmian use the COMSOL software for optimizing the geometry and arrangements of components in their high-tech cable solutions. Such systems may require a careful combination of geometry, materials, and positioning of individual parts. For example, composite cables may contain power conductors, hoses for fluid delivery, and cables for signal transmission.
The team has carried out simulations that analyze the physical effects in individual cables and entire power transmission systems by accounting for the relevant loading and environmental conditions. This allowed them to test, in a virtual sense, their designs before building and running prototypes. In one case, they had to take mechanical loads into account and simulated impact testing in medium-voltage cables. They used their results to optimize the thickness and material choices used in the external layers.
The simulations proved particularly useful for understanding the coupled structural, thermal, and electromagnetic problems that are challenging to solve. Massimo Bechis, a modeling and simulation specialist at Prysmian, comments that they’ve used COMSOL Multiphysics to “do extensive transient analyses to account for daily variations in solar irradiation and ambient temperature conditions […] multiphysics simulation really solves these kinds of problems that were very difficult or even impossible to do before.” They have simulated a range of events and phenomena, including impact testing, temperature distribution in double-armored umbilical cables, and computational fluid dynamics (CFD) analysis in high-voltage systems with limited ventilation.
Simulation results showing a thermal and fluid flow coupled analysis of a high-voltage cable system installed inside a tunnel with only natural ventilation.
Bechis explains that simulation has improved the way products are developed, and that using the COMSOL software has helped them take a big step forward in the level of services they are able to offer customers and designers. “COMSOL is able to solve these kinds of problems because we can build a parametric model to optimize the geometry, the laying of the cables, and we can include the physics needed to account for the convection in the air […] we can account for current load changes instead of considering constant operating conditions. This allows us to satisfy requests to consider transient conditions due to load changes.”
The superb results at Prysmian show strong promise for future developments. Simulation has opened up a whole world of possibilities for design, testing, and optimization prior to prototyping. It has reduced their costs, prototyping time, the number of lab tests, and has provided a straightforward way to optimize designs ahead of time, affirming their already-spectacular product record. The team expects that these methods will eventually lead to changes not only in the design phase of development, but also in manufacturing processes.
Simulation has also given them an avenue for communicating their design choices. As Bechis concluded, “Now we are able to optimize, among other things, the structure of our cables and still meet the specifications. We can also explain why we use a certain amount of material in a certain layer and show how we came to our decisions based on the modeling [...] we have improved procedures for designing our cables and power transmission systems. We have an additional and powerful way to respond to requests from clients.”
Before going into the physics, we should briefly recall the system of frames used in COMSOL Multiphysics. When geometric nonlinearities are considered, the Solid Mechanics interface makes the distinction between material and spatial frames. The material frame expresses the physical quantities in the coordinates of the initial state \mathbf{X} = (X, Y, Z), while the spatial frame uses the coordinates \mathbf{x} = (x, y, z) of the current state.
The two figures below present the example of a square submitted to compressive strain. The square is ten centimeters long and its bottom-left corner is initially located at (X, Y) = (1~\textrm{cm}, 1~\textrm{cm}). It is then compressed by boundary loads at its left and right sides. This deformation modifies the position of almost all points of the square. For instance, the bottom-left corner moved to a new location (x, y) = (1.54~\textrm{cm}, 0.82~\textrm{cm}).
A deformed square represented in material coordinates, initial state on the left and final state on the right.
A deformed square represented in spatial coordinates, initial state on the left and final state on the right.
The material coordinates always refer to the same particle in time, which was initially at a given point (X, Y, Z). The momentum equation of Solid Mechanics is formulated in this coordinate system. On the other hand, a point (x, y, z) in spatial coordinates refers to any particle that would be located there at the current state. The heat equation is formulated in this coordinate system.
In these two frames, volume-related physical quantities have different values. For instance, without any mass source, the density in material coordinates remains constant before and after transformation while the density in spatial coordinates changes according to the volume change. Hence, in order to couple an equation formulated on the material frame (structural mechanics) and another equation formulated on the spatial frame (heat transfer), these values need to be properly evaluated on each frame. The following table provides a list of conversions for some thermal physical quantities from spatial to material frame. These conversions involve the deformation gradient \mathbf{F} = {\partial \mathbf{x}} / {\partial \mathbf{X}} and its determinant, J. Both are evaluated using the displacement field computed by the Solid Mechanics interface.
Quantity | Material | Spatial |
---|---|---|
Temperature | T | T |
Density | \rho_0 | \rho = J^{-1} \rho_0 |
Thermal conductivity tensor | \bold{k}_0 | \bold{k} = J^{-1}\mathbf{F}^T \bold{k}_0 \mathbf{F} |
Pressure work | W_{\sigma, 0} = \boldsymbol{\alpha} T : \frac{\mathrm{d} \mathbf{S}}{\mathrm{d} t} | W_\sigma = J^{-1} W_{\sigma, 0} |
Heat source | Q_0 | Q = J^{-1} Q_0 |
Conversion of thermal physical quantities from material to spatial frame.
These conversions also reflect the fact that stress and strain affect the heat transfer by modifying the geometrical configuration (represented in the spatial frame). For example, a stretched boundary is more likely to receive a higher amount of heat by radiation (Q_\mathrm{r} > Q_{\mathrm{r}, 0}), as shown below.
Radiative heat flux received at the top surface of a solid, initial state (left) and after stretching the top surface (right).
Another example, the thermal conductivity expression in the spatial frame, usually using the initial state value \bold{k}_0, involves the quantities \mathbf{F} and J related to solid strain.
Modification of the thermal conductivity on the spatial frame after deformation of a solid.
The equations of Solid Mechanics are defined in the material frame. They relate the displacement, \mathbf{u}, the second Piola-Kirchhoff stress tensor, \mathbf{S}, and the elastic strain tensor, \mathbf{E}_\mathrm{el}, by a linear momentum balance equation and a stress-strain relation:
(1)
(2)
Here, \mathbf{C} is the elasticity tensor, which is often defined from the Young’s modulus and the Poisson coefficient. It may depend on the temperature as it is the case for Carbon Steel 1020.
Young’s modulus of Carbon Steel 1020, depending on the temperature.
Without any plastic effects, the elastic strain tensor, \mathbf{E}_\mathrm{el}, carries the temperature dependence via the thermal strain tensor, \mathbf{E}_\mathrm{th}, according to:
(3)
(4)
(5)
The coefficient of thermal expansion, \boldsymbol{\alpha}, characterizes the ability of the material to contract and expand because of temperature variations. It is often scalar but may more generally take a tensor form. The table below shows a list of typical values of isotropic \boldsymbol{\alpha}.
Material | Coefficient of Thermal Expansion (10^{-6} K^{-1}) |
---|---|
Acrylic plastic | 70 |
Aluminum | 23 |
Copper | 17 |
Nylon | 280 |
Silica glass | 0.55 |
Structural steel | 12.3 |
Coefficients of thermal expansion for some materials.
In addition, \boldsymbol{\alpha} can, itself, depend on the temperature as shown by the example below.
Coefficient of thermal expansion of Carbon Steel 1020, depending on the temperature.
As seen in these examples, the values of \boldsymbol{\alpha} are most often of the order of 10^{-5} K^{-1}. Hence, for \mathbf{E}_\mathrm{th} to become significant, a high temperature difference from the reference state is necessary. For instance, aluminum needs to reach about 500 K above the reference temperature to show a thermal elongation of only 1.2%.
Example of thermal expansion of a constrained aluminum beam heated 500 K, using a deformation scale of 1:1.
Note that in the formulation of Equations (3)-(5), the thermal strain is subtracted from the total strain. This is an appropriate approximation for small strains, which the thermal strains normally are, due to usually low values of \boldsymbol{\alpha}. The more accurate multiplicative formulation, valid for large thermal strains, is shown below but not discussed further. This formulation is used for the hyperelastic materials in COMSOL Multiphysics.
(6)
(7)
(8)
The heat equation is an energy balance equation deduced from the First Law of Thermodynamics. For solids, it takes the following form when formulated on the spatial frame:
(9)
The coupling term W_\sigma is the heat source due to compression or expansion of the solid and is defined by:
(10)
which, in the case of \boldsymbol{\alpha} being independent from temperature, reduces to:
(11)
Here, \boldsymbol{\alpha} is the same coefficient of thermal expansion as in \mathbf{E}_\mathrm{th}. The low value of \boldsymbol{\alpha}, as seen in the table above, has to be compensated for by high enough values of T {\mathrm{d} \mathbf{S}} / {\mathrm{d} t} to make W_\sigma a significant heat source, that is:
We have now described four key contributions to the multiphysics coupling between Heat Transfer and Solid Mechanics:
Next, we will illustrate the last two coupling contributions and show how to handle them in COMSOL Multiphysics with a couple of modeling examples.
My colleague Nicolas previously described in more detail how to model thermal stress in a turbine stator blade. Here, we display only the results in order to show the effects of J_\mathrm{th}. Because this is a steady-state model, the pressure work, W_\sigma, can be ignored.
Temperature field on the blade surface, representation in the material frame.
Due to a hot environment, the temperature field shows values between 870 K and 1100 K compared to the reference temperature of 300 K that the shape of the stator blade is initially. Such high temperatures make the material more prone to thermal deformations. The average coefficient of thermal expansion and temperature being around 1.2·10^{-5} K^{-1} and 1070 K, \mathbf{E}_\mathrm{th} is around 0.9%.
The volume expansion, due to thermal effects, for large deformations is \Delta V/V_0 = J_\mathrm{th}-1 (where J_\mathrm{th} was introduced in Equation (8)). It is still a good approximation for a small strain, giving an expansion of around 2.80%. In postprocessing, the actual volume expansion is found to be 2.76%.
Temperature field and deformation of the stator blade, exaggerated plot with a scale factor of 3 for more visibility.
The Bracket — Transient Analysis model is available both in the Structural Mechanics Module Model Library and the Model Gallery. In this model, the arms of the brackets move according to rapid time-dependent loads. Consequently, small variations of temperature should occur.
The existing model neglects these thermal effects, so we need to add a new Heat Transfer in Solids interface.
Then, we add the two multiphysics features below to couple the Heat Transfer in Solids and Solid Mechanics interfaces:
Finally, we add the Pressure Work subfeature to handle the thermoelastic heat source, W_\sigma.
The study can also be extended to 30 milliseconds to observe more load periods.
Starting from an isothermal profile of 20°C everywhere, the small temperature variations lead to a negligible thermal strain tensor. The main contribution to thermal effects is now the thermoelastic heat source due to rapid stress variations.
Temperature profile of the bracket over time, exaggerated plot with a scale factor of 10 for more visibility.
Differences of about 0.8 K can be observed between the extreme temperatures in the bracket. The heating and cooling process is, as expected, located at corners where the stress is more important and its variations stronger.
Volume-related quantities in both frames have different values and need a conversion from each other, in particular for specific energies and density.
The two governing equations each contain coupling terms that makes the solid motion dependent on the temperature and the heat transfer dependent on the solid deformation. As shown in the previous two examples, COMSOL Multiphysics provides appropriate functionalities to conveniently account for them.
When temperatures remain near the reference state and without too rapid stress variations, these coupling effects are negligible. Otherwise, they shall be added to the formulation on the model.
To delve deeper into this topic, you can download the files related to the models mentioned here and read a couple of related blog posts via the links in the section below.
The step-by-step video tutorial features a wrench and bolt model to provide a straightforward example of how to perform a structural analysis using COMSOL Multiphysics. The video details how you can use the Solid Mechanics interface in COMSOL Multiphysics to understand how a mechanical load would affect your design.
Think you’ve seen this video before? The one shown here is a recently updated version, which reflects the COMSOL Multiphysics version 4.4 ribbon user interface.
As usual, you can also read the video transcription below.
In this simulation, we’ll import a CAD file of a combination wrench and bolt and solve a standard structural mechanics problem. We’ll grab the wrench by its socket end and apply a force to turn it clock-wise. However, since the bolt is fixed in place, the wrench will bend.
As you can see, we’ve already opened COMSOL Multiphysics and we’ll start with a blank model. The ribbon, up top, displays the logical sequence of steps your workflow should take, beginning by specifying the spatial dimension, where we select 3D.
It’s good practice to estimate what numbers we would expect for our results, like stress and moment, through hand computations. To do so, we can create parameters for our calculations and use classical beam theory by idealizing this wrench as a cantilever beam. I’ve done the computations for this already and you can see them here, in the Parameters table.
“Ftotal” is the downward force that we will apply to the socket wrench, and will be used later in the model. The rest of the parameters are used to calculate the maximum stress “MaxS”. “ht”, the height of the beam, is unknown, and is necessary to calculate the max stress, so I will show you how to measure this directly off of an imported CAD file.
Let’s add the model geometry. In the Geometry section of the Home tab, add an Import node. Use the Browse function, and specify the file path to the CAD file. Clicking the “Import” button brings the geometry into view.
So, we’ll click the Geometry tab in the ribbon, and clicking the “Select Points” button in the Graphics window toolbar and zooming into the arm of the wrench near the socket end, we select two points and measure the distance between them with the “Measure” button. Now, you can copy and paste the computed distance into the parameters field and with this final parameter, we see that the result of the maximum stress comes out to 251 Megapascals.
The next thing to do is to specify the material properties. I will do so by choosing a material from the Material Library. Through the “Add Material” button, I go to the “Built-in materials” branch and choose Structural Steel. I can check by clicking the two sub-domain IDs to ensure that these properties are indeed assigned to the wrench and bolt.
It is now time to add the physics to this model by selecting the Home tab in the ribbon and clicking “Add Physics”. We are working with solid mechanics so that choice is clear. I want to constrain this model by placing a Fixed constraint on the bottom face of the bolt. Selecting the boundary transfers the ID of this particular boundary, 35, into the boundary selection list. Furthermore, I will add a boundary load and zoom into the socket end of the wrench. Here, I choose to apply a downward force of negative “Ftotal”. Remember that “Ftotal” comes from the parameters list shown earlier – a force equal to 100 Newtons.
With the physics completely configured, I need to discretize the model. Select the Mesh tab in the ribbon and you’ll see that there are a number of meshing techniques and options available. In this case, I will simply choose “Free Tetrahedral”. “Building All” will produce a mesh very quickly. You’ll notice we receive two warning messages regarding the mesh built, because the model geometry contains small edges and faces that are smaller than the value specified for the minimum element size parameter, so we’ll decrease the value and re-build the mesh.
To solve the model, we will add a study and choose a stationary study, since we are computing the steady-state solution after the wrench has been turned. Click the “Compute” button to solve the finite element problem. Since this is a small model, it will only take seconds to solve.
The COMSOL software generates a default plot based on the physics of your simulation, in this case, Structural Mechanics, and we are shown the von Mises stress results. However, we’d like to compare our results with our hand computation. So, I will add another plot group in which the first principal stress, sp1, is shown. In the Results tab, click the “3D Plot group” button, and add a Contour plot. Click “Replace Expression” and select first principal stress. Then, change the Contour type to Filled and plot the graph. Then, under the “More Plots” button, add a Max/Min Volume plot and again replace the expression with the principal stress. Click “Plot”, and the location and magnitude of the maximum principal stress is added to the plot. Right-click the node for each of these to add deformation to the plot.
Now, let’s take a look at some of the qualitative data of this model. We have tensile and compressive stresses in the upper and bottom portions of the beam, respectively, and I would like to point out that the kink in the socket end of the wrench causes a small moment around the x-axis as well, causing the socket to twist around the x-axis, which is not taken into account by the hand computation. This twist out toward the y-axis around the x-axis produces a stress maximum towards the left of the beam, which is what we would expect. If we examine the bolt, we can see that the shear stress within it grows radially, which is also something that you would expect from this problem.
Finally, we’d like to compare the tensile stress of our simulation to our hand-calculated value. We see, here, that we have a maximum stress of 247 Megapascals, which compares quite well with the hand-computed value of 251 Megapascals.
Learn more about this model, and others like it, in our Model Gallery.
Consider a rubber balloon, completely filled with water and resting on a surface within a hole, while being pushed from the top by an indenter. The deformation of the balloon is due to the weight of the fluid as well as the indenter pushing down from the top, see below. The rubber material is modeled with a hyperelastic material model. We will use the technique explained in the previous entry to keep the volume of the cavity constant as it deforms.
The deformation of the balloon is due, in part, to the weight of the fluid, which causes it to bulge outwards and into the depression. It also deforms due to the compression from above, which causes it to bulge outwards and upwards. As a consequence of this compression, the depth of the fluid inside the balloon will change. We want to solve for this change in depth without having to solve the Navier-Stokes equations for the fluid flow, since we are only interested in the static (time-invariant) solution.
A rubber balloon filled with water is compressed at the center. As the balloon is squeezed, the location of the highest point and the depth of fluid changes, altering the hydrostatic pressure distribution.
A container of fluid will exert a hydrostatic pressure on its walls:
where \rho is the density of the fluid, g is the force of gravity, z_0 is the location of the top of the container, and p_0 is the pressure of the fluid at the top of the container. Since the balloon is filled with an incompressible fluid, the pressure, p_0, will increase as we squeeze it with the indenter.
We can also see, from the image above, that the depth of the fluid changes as the balloon is compressed. Furthermore, it appears as if computing the depth requires knowing the location of the top and bottom of the container. So, how do we incorporate this change in depth? Let’s find out…
As shown below, there are two components to the pressure loads applied inside the balloon. The first part of the load is computed from the Global Equation. The second pressure load is due to the hydrostatic pressure. Ideally, this second pressure load would be based upon the depth of the fluid, but this depth is a variable that we don’t know. So instead, let’s enter a hydrostatic load based only upon the z-location, which could have an arbitrary zero level.
The applied pressure load on the inside boundary of the balloon is the sum of the pressure load computed by the Global Equation and the hydrostatic pressure. The hydrostatic pressure is ramped up during the solution.
The Global Equation constrains the volume to remain constant during deformation.
So, it appears here as if we are applying a pressure load to constrain the volume and a load that is directly proportional to the z-location, but we are not correctly computing the hydrostatic pressure, since we do not know z_0. As it turns out, however, the Global Equation does a little bit more than you might first expect.
To see this, let us slightly re-write the equation for the pressure inside the balloon:
We can see right away that this almost exactly matches the equation we entered as the pressure load, p(z)= P_0-\rho g z, except that the pressure we are computing via the global equation is the pressure at the top of the container plus an offset due to the unknown z-location of the top. So, although we are only solving for a single additional variable, P_0, it accounts for two physical effects: the change in pressure due to the volume constraint as well as the change in the z-location of the top of the fluid.
Since this model contains both geometric and material nonlinearities and a nonlinearity due to the contact, converging to the solution can be difficult. To address this, we will use load ramping to slowly increase the effect of gravity on the model, and to gradually squeeze the balloon. A 2D-axisymmetric model is used to exploit the symmetry of the structure.
The Maximum Coupling Operator is used to find the highest point inside the cavity for postprocessing.
After we solve the model, we can postprocess the magnitude of the hydrostatic pressure by using the Maximum Coupling Operator to compute the maximum z-location along the inside boundary of the balloon.
The solution where the arrows indicate the hydrostatic pressure load that varies with depth.
The plot above shows the hydrostatic pressure load on the inside of the balloon. The length of the arrows is given by the expression: WaterDensity*g_const*(maxop1(z)-z), where maxop1(z) gives the z-location at the top of the deformed cavity.
In this example, we have modeled the varying depth of a fluid in a deformable container (a balloon, in this case). The Global Equation that is used to solve for the fluid pressure that keeps the volume constant also accounts for the change in the depth of the fluid as the balloon deforms.
By using this approach, we solve a fluid-structure interaction problem without explicitly having to solve the Navier-Stokes equations, thus saving significant computational resources. If you are interested in this type of modeling, or would like more details about this model, please contact us.
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In Part 1, we showed you how the shear effect introduced by the Timoshenko Theory affected the stiffness. Now, let’s try to incorporate another realistic effect in this model such that when the beam is pulled along its length and elongates, its cross-sectional area will be reduced. This is known as the Poisson effect. The ratio between the transverse and axial strains is denoted by the material property known as Poisson’s ratio (\nu ).
In general, the Poisson effect is incorporated in the physical description of a linear elastic deformation through the material’s stiffness matrix, [C], which relates the stresses, {σ}, and strains, {ε}. Hooke’s law is modified to account for both axial and shear components along different directions. In its generalized form, it can be written like this:
Note that the material’s stiffness matrix, [C], is a material property, as opposed to the structural (or device) stiffness (k) that we had introduced earlier.
For isotropic linear elastic materials, the components of the material’s stiffness matrix, [C], can be evaluated using only the material’s Young’s modulus and Poisson’s ratio, as the shear modulus is a function of these two parameters. For orthotropic materials, we would need to specify unique values for the Young’s modulus, Poisson’s ratio, and shear modulus. For a general anisotropic linear elastic material, the stiffness matrix could consist of up to 21 independent material parameters that take care of both Poisson’s effect and the shear effect along different directions. Although our example model uses an isotropic material, the ideas discussed in this blog entry hold true for orthotropic and anisotropic materials as well.
Now, let us find out how these effects can be incorporated in our beam model in different space dimensions. We will use a Poisson’s ratio of 0.3 (~ shear modulus of 77 GPa) for all our computations.
When modeling the beam in 2D, we could choose between a plane strain or a plane stress assumption. In COMSOL Multiphysics, you can model the 2D plane stress and plane strain cases by selecting a 2D space dimension and choosing the Solid Mechanics interface. The interface provides a drop-down menu to switch between “plane stress” and “plane strain” conditions.
The “plane strain” option is suitable when there are only in-plane axial, shear, and bending forces acting on a structure, producing in-plane strains. The out-of-plane strain components are assumed to be zero. A typical example would involve a structure that is fully constrained in the out-of-plane direction. Hence, this is not an appropriate choice for modeling our beam example. In the “plane strain” formulation, the COMSOL software solves for the in-plane displacements, u and v.
The “plane stress” option is suitable when there are only in-plane axial, shear, and bending forces acting on a structure, producing in-plane stresses. The out-of-plane stress components are assumed to be zero. In the “plane stress” formulation, COMSOL Multiphysics solves for the in-plane displacements, u and v, as well as the out-of-plane strain, (wZ ). For an anisotropic material, it additionally solves for the out-of-plane displacement gradients, uZ and vZ . That’s why, in COMSOL Multiphysics, you can use the 2D “plane stress” modeling interface even for anisotropic materials, as long as the boundary conditions support the plane stress assumption. There are several combinations of boundary conditions that would support this assumption. One such example is a beam subjected to a roller boundary condition at one end and free at the other.
Boundary conditions showing “ideal” plane stress conditions on the beam subjected to axial load. The roller boundary on the left helps in obtaining a constant value of the axial stress σ_{xx}. A geometric point at the center of the roller boundary has been used to constrain the y-displacement (v) to prevent in-plane translation.
The same “ideal” plane stress beam subjected to transverse load. The bending stress, σ_{xx}, shows a smooth expected variation, but the shear stress, σ_{xy}, is singular around the point where v is constrained.
Boundary conditions showing fixed-free beam subjected to axial load modeled using plane stress assumptions. The axial stress, σ_{xx}, is singular at the corners as a result of constraining the transverse displacement. Due to the same constraint, we also receive a non-zero σ_{yy} because of the inhibited contraction in the vertical direction.
The same fixed-free beam subjected to a transverse load. The bending stress, σ_{xx}, shows a smooth expected variation, but its maximum value is slightly higher than what we get from the “ideal” beam. This is because of the additional stiffness arising from constraining the transverse displacement. As a result of the same constraint, we also receive singular values of σ_{xy} above and below the mid-plane of the beam.
The pictures above show that our fixed-free beam example can only be “approximately” modeled using the 2D plane stress assumption. Note that in the 2D model, the local y-axis can correspond to either the y-axis or the z-axis of a Cartesian coordinate system representing a 3D space, depending on whether we are representing the xy-plane or xz-plane in 2D.
Similarly, the transverse displacement, v, in the 2D model could represent either of the transverse displacements v or w of the 3D model, depending on whether we are representing the xy- or xz-plane. We can use this information to compute both the bending stiffness k_{yy} and k_{zz} by solving the model twice: once with a height of b (0.2 m) and then with a height of t (0.1 m).
Let’s look at the effect of these ideal and real boundary conditions on the stiffness computed by the 2D model:
k_{xx} [N/m] | k_{yy} [N/m] | k_{zz} [N/m] | |
---|---|---|---|
Roller-Free | 4×10^{9} | 3.86×10^{7} | 9.91×10^{6} |
Fixed-Free | 4.01×10^{9} | 3.89×10^{7} | 9.94×10^{6} |
These values show that a realistic fixed constraint, as opposed to an idealized roller constraint, would lead to slightly higher values of stiffness due to localized stiffening effects near the fixed end. Note that for both these cases, the bending stiffness is lower than that of the Euler-Bernoulli Beam due to additional shear flexibility (i.e. accounting for shear deformation) in 2D models. Hence, these results are closer to the 1D Timoshenko beam models. (You can find the results from the 1D simulation in our previous blog post.)
The 2D modeling approach is useful as long as there are no out-of-plane forces acting on the structure and the in-plane forces do not vary along the out-of-plane direction. For more general loading conditions and constraints on the structure, a 3D model could provide more accurate information, but be more computationally taxing. For a true 3D model, you would need to choose a 3D space dimension and the Solid Mechanics interface.
A 3D representation of the fixed-free beam subjected to axial and transverse loads. Solving the model for these three load cases allows us to evaluate the axial and bending stiffness.
Summary of axial stresses for the three load cases. Note the stress concentration at the fixed end that arises from a combination of the fixed constraint boundary condition and coupling of strains along different directions via the Poisson effect.
Next, let’s look at the axial and bending stiffness computed by the 3D model. We will compute the stiffness for two cases: first by setting Poisson’s ratio to 0.3, then by setting it to 0. This will allow us to compare the 3D results with the 1D beam theory results.
Poisson’s ratio | k_{xx} [N/m] | k_{yy} [N/m] | k_{zz} [N/m] |
---|---|---|---|
ν = 0 | 4×10^{9} | 3.91×10^{7} | 9.94×10^{6} |
ν = 0.3 | 4.02×10^{9} | 3.92×10^{7} | 1.006×10^{7} |
Note that for a Poisson’s ratio of 0, the results match perfectly with those when using the 1D Timoshenko beam theory. For a Poisson’s ratio of 0.3, the Timoshenko theory predicts a lower bending stiffness as a result of accounting for the shear flexibility. However, the 3D model predicts a slightly higher axial and bending stiffness as a result of the fixed constraint boundary condition, which produces an additional stiffening effect that offsets the shear flexibility, especially when bending in the least shear-flexible direction.
The realistic effect we have seen here could unfortunately have been neglected, had we not modeled the structure in 3D.
Now we’ll revisit the method that we have used so far to compute the stiffness.
We fixed one end of the beam and applied a force at the other. We varied one component of the force vector at a time by making it non-zero (and left the other force components as zero) and evaluated the resulting average tip displacement of the beam. Under these conditions, the force vector (F) and displacement vector (u) should strictly be correlated using a compliance matrix, [s], such that \bold{u}=[s]\bold{F}.
The same equation can be written in a matrix formulation:
Here, the compliance component, s_{ij}, denotes the linearized compliance relating the amount of incremental displacement you get along the i ^{th} direction (where i can be x, y, z) when we apply an incremental force along the j ^{th} direction (where j can be x, y, z). We will call this approach as the force-controlled method.
Based on our previous definition of stiffness, we can now define a general stiffness matrix such that \bold{F}=[k]\bold{u}.
We can represent that in a matrix formulation, too:
In the stiffness matrix, the diagonal terms correspond to axial and bending stiffness and the off-diagonal terms denote any stiffness due to extension-bending coupling. In absence of such extension-bending coupling (as in our case), we receive a diagonal compliance matrix. Therefore, we could say that k_{xx} = 1/s_{xx}, k_{yy} = 1/s_{yy}, and k_{zz} = 1/s_{zz}. This is what we have used so far.
Note that the input to the second matrix formulation is equal to the different components of displacement, while the output of interest is equal to the components of the reaction force that is “felt” at the boundary with the prescribed displacement. So, in order to find the different components of the stiffness matrix, we need to vary one component of the displacement vector at a time, by making it non-zero (and setting the other displacement components to zero), and evaluate the resulting reaction force.
The so-called “free end”, where we were applying the axial and transverse loads, is not free anymore. Based on this, we can now say that the stiffness component, k_{ij}, denotes the linearized stiffness relating the amount of incremental reaction force that you get along the i ^{th} direction (where i can be x, y, z) when we apply an incremental displacement along the j ^{th} direction (where j can be x, y, z). We will call this approach the displacement-controlled method.
So far, we have limited our discussion to axial and bending stiffness that were obtained using forces and displacements. However, in reality, at any point in space, a structure can have six degrees of freedom, three of which correspond to translations while the other three correspond to rotations. Similarly, instead of only applying a force on a boundary in one of the directions parallel to x, y, or z, it is also possible to apply moments about these three axes.
For a cantilevered beam like ours, a moment about the x-axis would produce torsion, whereas moments about y- and z-axes would produce bending. All this information can be represented using the following matrix formulation:
This shows that we can have a general six-by-six stiffness matrix that could accommodate several types of stiffness terms such as axial, shear, bending, and torsion, as well as coupled terms between these modes .
With the new information in hand, let’s revisit our beam model:
The beam model defined in 3D showing how we can prescribe the displacement at the tip of the beam.
In order to implement the displacement-controlled method in COMSOL Multiphysics, we would need to do the following:
If we use the Beam theory models (for 1D analysis), they would allow us to work explicitly with all six degrees of freedom (displacements and rotations). So, we could replace the Point Load with a Prescribed Displacement/Rotation, where we can set the displacement to some non-zero value (say, 1 mm) and at the same time not impose any constraint on the rotation at the tip of the beam.
The Prescribed Displacement/Rotation feature applied at the tip of a 1D beam. A free rotation is allowed.
As shown in the previous blog post, we can use the if()
operator and the names (such as root.group.lg1
) associated with the Load Groups, such that only one component of the displacement vector can be made non-zero at a time when you are solving the same model for several load cases.
A snapshot showing how you can set up three load cases and solve the model with only one active load group at a time.
When using the Solid Mechanics model (in 2D and 3D), we can only specify or compute the three translational degrees of freedom (displacements), which are in turn used to compute the rotations. This also means that when we set up a displacement-controlled stiffness computation, the Beam theory easily allows us to either constrain the rotation or allow free rotation at a point.
In the Solid Mechanics model, on the other hand, it is not possible to freely prescribe or constrain the rotation everywhere on that surface by decoupling the rotation from the displacement. That is because we are dealing with spatial variation in displacement on a boundary. This means that if we simply assign a set of prescribed transverse displacements (say u = 0, v = 1 mm, and w = 0), it will fully constrain the rotation (i.e. Φ_{x} = Φ_{y} = Φ_{z} = 0), thereby increasing the effective bending stiffness of the beam. This is where the COMSOL software’s Rigid Connector boundary condition comes in and helps us overcome that problem in 2D and 3D solid models.
The Rigid Connector feature applied at the boundary of a 3D beam. Free rotation is allowed at the centroid of the boundary.
In our model, we could use similar expressions as shown earlier to prescribe the displacements in the Rigid Connector boundary condition to vary each of the displacement components, one at a time, using Load Groups and Load Cases. The Rigid Connector will introduce the same kind of local stiffening and stress disturbances as the Fixed constraint at the other end of the beam.
The reaction force can be computed by setting up an Integration Coupling Operator that can be used to sum up the reaction force over all the nodes on a boundary. COMSOL Multiphysics provides predefined variables such as solid.RFx
, solid.RFy
, and solid.RFz
that allow you to access the reaction force. The stiffness can then be computed as the ratio of the total reaction force and the value of the prescribed displacement at the tip of the beam. Alternatively, using an average of the displacement at the tip of the beam can be useful if the beam has an extension-bending coupling. In that case, the average displacement in a direction perpendicular to the load application will not be zero.
Screenshot of the Integration Coupling Operator and variables defined to compute the stiffness.
Finally, here’s a summary of all axial and transverse stiffness values that we have computed in different space dimensions by either neglecting or including the Poisson and shear effects. The stiffness values obtained from both the force-controlled and the displacement-controlled methods agree closely with each other, thereby providing a consistency check:
Space dimension (Poisson’s ratio) | k_{xx} [N/m] | k_{yy} [N/m] | k_{zz} [N/m] |
---|---|---|---|
1D Euler-Bernoulli | 4×10^{9} | 4×10^{7} | 1×10^{7} |
1D Timoshenko (ν = 0) | 4×10^{9} | 3.91×10^{7} | 9.94×10^{6} |
1D Timoshenko (ν = 0.3) | 4×10^{9} | 3.88×10^{7} | 9.92×10^{6} |
2D Plane Stress (ν = 0) | 4×10^{9} | 3.91×10^{7} | 9.94×10^{6} |
2D Plane Stress (ν = 0.3) | 4.02×10^{9} | 3.89×107 | 9.94×10^{6} |
3D (ν = 0) | 4×10^{9} | 3.91×10^{7} | 9.94×10^{6} |
3D (ν = 0.3) | 4.04×10^{9} | 3.92×10^{7} | 1.01×10^{7} |
We have now explored the concept of axial and bending stiffness of structures in more detail. We started with a conventional definition of stiffness, which is suitable for 0D models, and expanded it to fit assumptions and theories available to model structures in 1D, 2D, and 3D that involve multi-axial loading. We saw that the stiffness of a structure can rarely be captured by a single-valued number.
Structural stiffness could be affected by several other factors that we have not explored here. Some of these include:
If you have any questions about computing stiffness using COMSOL Multiphysics, please contact us.
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