In powder compaction, a metal powder enters a die and is compacted through applied pressure. This high pressure comes from a punching tool inside the die cavity (often at the bottom surface). The powder is ejected from the cavity once it has been compacted and molded into a certain shape.
Through powder compaction, metal powders are transformed into solid components. Image by Alchemist-hp — Own work. Licensed under CC BY-SA 3.0 DE, via Wikimedia Commons.
With production rates averaging between 15 to 30 parts per minute, the powder compaction process enables manufacturers to quickly design strong components. Another benefit of this process is that it saves costs, as the component doesn’t need much additional work.
From a simulation standpoint, we need to perform a highly nonlinear structural analysis on powder compaction that accounts for:
As we demonstrate here, COMSOL Multiphysics® version 5.3 is ideal for handling such analyses.
For our example, let’s consider the fabrication of a cup-shaped component via powder compaction. The model geometry includes the workpiece (metal powder in this case) and the die. Note that the punch tool is not part of the model setup. We instead apply a prescribed displacement in the normal direction to the upper and lower faces of the powder in order to compact it. Because of the axial symmetry of the model, we can reduce its size to a 2D model, thus reducing the computational time of the simulation.
The model geometry for a powder compaction analysis.
The latest version of COMSOL Multiphysics includes five new porous plasticity models that cover various porosity values.
These models are important for simulating powder compaction, as they allow us to accurately represent the porosity of the workpiece and produce reliable results. In this case, we combine the Fleck-Kuhn-McMeeking and Gurson-Tvergaard-Needleman models to describe an aluminum metal powder. Note that the die’s material properties assume it to be rigid.
In addition to the Prescribed Displacement boundary condition mentioned above, we also set the inner and outer dies as fixed domains.
From our simulation results, we can assess various properties of the metal powder at the end of compaction. To start, let’s look at the volumetric plastic strain. The strain at the center of the fillet appears to be minimal, while the strain near the ends is high. At the corner points of the workpiece, the strain is around 12% — likely the result of friction against the die.
The volumetric plastic strain of the workpiece as the compaction process ends.
During compaction, the porosity of the aluminum powder decreases, while the density and strength of the component increases. Based on the geometry and loading used in this scenario, we can expect that the changes to the porosity will be nonuniform.
The plot below shows the current void volume fraction contours of the powder; i.e., the porosity of the powder. Compared to the middle and top portions of the workpiece, the metal powder in the thin lower portion is more compact. Near the central area of the fillet, the powder is less compact because of material sliding on the rounded corner. The following animation illustrates how the volume fraction evolves over time.
The current void volume fraction as the compaction process ends.
Changes in the volume fraction over time.
Lastly, let’s consider the von Mises stress in the workpiece. The results indicate that the stress is greater in the areas where more compaction occurs.
The von Mises stress within the workpiece.
When simulating powder compaction, it is important to access the appropriate plasticity model that is preferably predefined in your analysis tool and available for direct use. To meet your modeling needs, COMSOL Multiphysics® version 5.3 brings you five new models that cover a wide range of porosity values.
For a helpful introduction to using these porous plasticity models, try out the example from today’s blog post.
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Let’s say you just finished creating a CAD assembly of a fitting for the threaded steel pipe referenced above. Now, you want to analyze stress in your assembly in order to better understand how this portion of the pipe system performs. With our LiveLink™ interfacing products, you can perform such analyses by integrating the COMSOL Multiphysics® software into your design workflow.
Threaded pipes are common in fire sprinkler systems. Image in the public domain, via Wikimedia Commons.
Threaded geometries include a large number of details. The complex nature of these CAD assemblies causes additional preprocessing work and takes up more computing resources during analysis. One solution is to assume that the thread is symmetric and compute the solution in a 2D section cut from the 3D object.
In previous versions of the COMSOL® software, selections from the original geometry had to be manually redefined after synchronization — a process that can be time consuming. Thanks to improvements in version 5.3, setting up CAD assembly selections is now a more efficient process. All of the relevant selections are automatically loaded and properly assigned in the COMSOL Multiphysics environment. This makes it possible to run parametric studies as well as improve 3D designs from 2D analyses.
Want to see a firsthand example? Good news: There’s a new tutorial model in the Application Gallery that highlights this functionality.
Note: While today’s example uses LiveLink™ for SOLIDWORKS®, this functionality is also available for LiveLink™ for Inventor®. For more details, see the 5.3 Release Highlights page.
In this example, you can synchronize a full threaded pipe fitting geometry built in SOLIDWORKS® software into the COMSOL Desktop® environment via LiveLink™ for SOLIDWORKS®. To compute a reduced stress analysis, you obtain a 2D section from the 3D geometry via the Cross Section node. The analysis assumes that a torque of 5000 Nm is applied to the male thread part (shown below). This part is made up of the same steel material as the other parts in the design.
Left: Full 3D assembly synchronized in COMSOL Multiphysics. Right: 2D section cut for the stress analysis.
To compute the force transmission between each part of the assembly, the model uses structural contact. In SOLIDWORKS® software, these contact surfaces are defined as face selections. After synchronizing the assembly, all of the selections are automatically transferred over to the 2D axisymmetric model. This simplifies the process of setting up the contact pair, as it is no longer necessary to manually and individually select boundary entities in contact with one another. In particular, when it comes to the thread, you only need to create a selection for two surfaces in SOLIDWORKS® software instead of selecting fifteen edges in the 2D axisymmetric model.
Looking at the results of our stress analysis, we can see the von Mises stress when the maximum torque (5000 Nm) is applied. The plot indicates that the maximum value of stress is less than that generally reported for using a class 10.9 alloy steel, highlighting the potential of using this material in this pipe fitting design.
Simulation plot depicting the von Mises stress with the maximum applied torque.
In version 5.3 of the COMSOL® software, you can combine your complex CAD assemblies and COMSOL Multiphysics analyses for an efficient modeling workflow.
Ready to try this tutorial yourself?
SOLIDWORKS is a registered trademark of Dassault Systèmes SolidWorks Corp.
Autodesk, the Autodesk logo, and Inventor are registered trademarks or trademarks of Autodesk, Inc., and/or its subsidiaries and/or affiliates in the USA and/or other countries.
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A gearbox assembly generally consists of gears, shafts, bearings, and housing. When operated, a gearbox radiates noise in its surroundings for two main reasons:
Out of all of the components in a gearbox, the primary source of vibration or noise is the gear mesh. A typical path followed by the structural vibration, seen as the noise radiation in the surrounding area, can be illustrated like this:
The noise generated due to gear meshing can be classified into two types: gear whine and gear rattle.
Gear whine is one of the most common types of noise in a gearbox, especially when it runs under a loaded condition. Gear whine is caused by the vibration generated in a gear because of the presence of transmission error in the meshing as well as the varying mesh stiffness. This type of noise occurs at the meshing frequency and typically ranges from 50 to 90 dB SPL when measured at a distance of 1 m.
Gear rattle is observed mostly when a gearbox is running under an unloaded condition. Typical examples are diesel engine vehicles such as buses and trucks at idle speed. A gear rattle is an impact-induced noise caused by the unloaded gear pairs of the gearbox. Backlash, required for lubrication purposes, is one of the gear parameters that directly impact the gear rattle noise. If possible, simply adjusting the amount of backlash can reduce gear rattle.
We know that transmission error is the main cause of gear whine, but what exactly is it? When two rigid gears have a perfect involute profile, the rotation of the output gear is a function of the input rotation and the gear ratio. A constant rotation of the input shaft results in a constant rotation of the output shaft. There can be various unintended and intended reasons for modifying the gear tooth profile, such as gear runouts, misalignment, tooth tip, and root relief. These geometrical errors or modifications can introduce an error in the rotation of the output gear, known as the transmission error (TE). Under dynamic loading, the gear tooth deflection also adds to the transmission error. The combined error is known as the dynamic transmission error (DTE).
Reducing gear whine or rattle to an acceptable level is a big challenge, especially for modern complex gearboxes, which consist of many gears meshing simultaneously. By accurately simulating these complex behaviors, we can design a quieter gearbox. COMSOL Multiphysics gives designers the ability to accurately identify problems and propose realistic solutions within the allowable design constraints. With such a tool, we can optimize existing designs to reduce noise problems and gain insight into new designs earlier in the process, well before the production stage.
A gearbox model in the COMSOL Desktop®.
Let’s consider a five-speed synchromesh gearbox of a manual-transmission vehicle in order to study the vibration and radiation of gear whine noise to the surrounding area. The gearbox is in a car and used to transfer power from the engine to the wheels.
Geometry of a five-speed synchromesh gearbox of a manual transmission vehicle.
In order to numerically simulate the entire phenomenon of gearbox vibration and noise, we perform two analyses:
In the multibody analysis, we compute the dynamics of the gears and housing vibrations, performed at the specified engine speed and output torque in the time domain. For the acoustic analysis, we compute the sound pressure levels outside the gearbox for a range of frequencies using the normal acceleration of the housing as a source of noise.
First, we look into the gear arrangement in the synchromesh gearbox. Here, helical gears are used to transfer the power from the input end of the drive shaft to the counter shaft and further from the counter shaft to the output end of the drive shaft.
The gear arrangement in the five-speed synchromesh gearbox, excluding the synchronizing rings that connect the gears with the main shaft.
The gears used in the model have the following properties:
Property | Value |
---|---|
Pressure angle | 25 [deg] |
Helix angle | 30 [deg] |
Gear mesh stiffness | 1e8 [N/m] |
Contact ratio | 1.25 |
All of the gears on the counter shaft are fixed to the shaft, whereas the gears on the drive shaft can rotate freely. Only one gear at a time is fixed on the shaft. In real life, this is achieved with the help of synchronizing rings. In the model, hinge joints with an activation condition are used to conditionally engage or disengage gears with the drive shaft.
Looking at the shafts, they are assumed rigid and rested on the housing through hinge joints, whereas the housing is assumed flexible, further mounted on the ground, and connected to the engine at one of its ends. The driving conditions considered for the simulation in terms of engine speed, load torque, and the engaged gear are as follows:
Input | Value |
---|---|
Engine speed | 5000 [rpm] |
Load torque | 1000 [N-m] |
Engaged gear | 5 |
With these settings, it is possible to run a multibody analysis and compute the housing vibrations as shown in this animation:
The von Mises stress distribution in the housing together with the speed of different gears.
In order to have a better understanding of the variation of normal acceleration as a function of time, we can choose any point on the gearbox housing. The time history of the normal acceleration at that point is shown below. Let’s transform this result to the frequency domain using the FFT solver. In this way, we can find the frequency content of the vibration. It is clear from the frequency response plot that the normal acceleration of the housing contains more than one dominant frequency. The frequency band in which the housing vibration is dominant is 1000–3000 Hz.
Time history and frequency spectrum of the normal acceleration at one of the points on the gearbox housing.
Once we have simulated the vibrations in a gearbox, let’s see how to model the noise radiation in COMSOL Multiphysics. To begin, we create an air domain outside the gearbox to simulate the noise radiation in the surrounding.
In order to couple multibody dynamics and acoustics, we assume a one-way coupling, as the exterior fluid is air. This implies that the vibrations from the gearbox housing affect the surrounding fluid, whereas the feedback from the acoustic waves to the structure is neglected. It is a good assumption that the problem is one-way coupled.
The acoustic analysis is performed for a range of frequencies. As the multibody analysis is solved in the time domain, the FFT solver is used to convert the housing accelerations from the time domain to the frequency domain.
The air domain enclosing the gearbox for acoustic analysis. The two microphones placed to measure noise levels are shown.
As a source of noise, the normal acceleration of the gearbox housing is applied on the interior boundaries of the acoustics domain. In order to avoid any reflections from the exterior boundaries of the surrounding domain, we apply a spherical wave radiation condition. With these settings, we can solve for the acoustic analysis and look at the sound pressure level in the near field as well as on the surface of the gearbox housing at different frequencies. For a better understanding of the directivity of the noise radiation, we can create far-field plots in different planes at different frequencies.
The sound pressure level in the near field (left) and at the surface of the gearbox (right).
The far-field sound pressure level at a distance of 1 m in the xy-plane (left) and xz-plane (right).
After visualizing the sound pressure level in the outside field, it is interesting to find out the variation of sound pressure with frequency at a particular location. For this purpose, two microphones are placed in specific locations.
Microphone | Placement | Position |
---|---|---|
1 | Side of the gearbox | (0, -0.5 m, 0) |
2 | Top of the gearbox | (0, 0, 0.75 m) |
These microphone locations are defined in the Parameters node in the results and can be changed without updating the solution every time.
The frequency spectrum of the pressure magnitude at the two microphone locations.
The pressure response plot at the microphone locations gives a good idea of the frequency content present in the noise. However, wouldn’t it be nice if we could actually listen to the noise recorded at the microphone, just like in a physical experiment? This is possible by writing Java® code in a model method using the magnitude and phase information of the pressure as a function of frequency.
Let’s listen to the sound files corresponding to the noise received at the two microphones…
We have already looked at the acoustics results for various frequencies. It would also be nice to see them in the time domain. Let’s transform the results from the frequency domain to the time domain using the FFT solver so that we can visualize the transient wave propagation in the surrounding area of the gearbox.
Animation showing the transient acoustic pressure wave propagation in the surrounding area of the gearbox.
The above approach describes a technique to couple multibody analysis and acoustics simulation in order to accurately compute the noise radiation from a gearbox. This technique can be used early in the design process to improve the gearbox in such a way that the noise radiation is minimal in the range of operating speeds of the gearbox. Additionally, model methods — new functionality as of version 5.3 of the COMSOL Multiphysics® software — enable us to actually hear the noise generated by the gearbox — making the simulation one step closer to a physical experiment.
Some devices require a very high degree of frequency stability with respect to changes in the environment. The most common parameter is temperature, but the same type of phenomena could, for example, be caused by hygroscopic swelling due to changes in humidity. In very high precision applications, the frequency stability requirements might specify a precision at the ppb (parts-per-billion, 10^{-9}) level. Setting up simulations that accurately capture such small effects can be a challenging task, since several phenomena can interact.
Consider a rectangular beam with the following data:
Property | Symbol | Value |
---|---|---|
Length | L | 10 mm |
Width | a | 1 mm |
Height | b | 0.5 mm |
Young’s modulus | E | 100 GPa |
Poisson’s ratio | ν | 0 |
Mass density | ρ | 1000 kg/m^{3} |
Coefficient of thermal expansion, x direction | α_{x} | 1·10^{-5} 1/K |
Coefficient of thermal expansion, y direction | α_{y} | 2·10^{-5} 1/K |
Coefficient of thermal expansion, z direction | α_{z} | 3·10^{-5} 1/K |
Temperature shift | ΔT | 10 K |
The beam geometry and mesh used in the example.
The material parameters have values that are of the same order of magnitude as those for many other engineering materials. To better separate the various effects, Poisson’s ratio is set to zero, but this assumption does not change the results in any fundamental way. Orthotropic thermal expansion coefficients are used to highlight some properties of the solution.
To analyze the effect of thermal expansion, add a Prestressed Analysis, Eigenfrequency study.
Adding the Prestressed Analysis, Eigenfrequency study.
This study consists of two study steps:
The two study steps shown in the Model Builder tree.
To compute the reference solution, you either add a separate Eigenfrequency study or run the same study sequence, but without thermal expansion.
The eigenfrequencies of the beam have been calculated for two different types of boundary conditions:
The doubly clamped beam results are shown below.
Mode Type | Eigenfrequency, Hz |
Eigenfrequency, Hz ΔT = 10 K |
Ratio |
---|---|---|---|
First bending, z direction | 50713.9 | 50425.1 | 0.9943 |
First bending, y direction | 97659.6 | 97526.2 | 0.9986 |
First twisting | 266902 | 266917 | 1.00006 |
First axial | 500000 | 500025 | 1.00005 |
Mode shapes for the doubly clamped beam.
The following table shows the cantilever beam results.
Mode Type | Eigenfrequency, Hz |
Eigenfrequency, Hz ΔT = 10 K |
Ratio |
---|---|---|---|
First bending, z direction | 8063.79 | 8066.92 | 1.00039 |
First bending, y direction | 16049.1 | 16053.7 | 1.00028 |
First twisting | 132233 | 132265 | 1.00025 |
First axial | 250000 | 250050 | 1.0002 |
Mode shapes for the cantilever beam.
The first thing to note is that the bending eigenmodes for the doubly clamped beam stand out and have a strong temperature dependence. The change is 0.6% in the first mode. For all other modes, the relative shift in frequency is significantly smaller. If you make the beam thinner, this difference would be even more pronounced. The reason for this behavior is discussed in the following sections.
In the case of the doubly clamped beam, the thermal expansion causes a compressive axial stress. With the given data, the stress is -10 MPa (computed as Eα_{x}ΔT). This stress causes a significant reduction in the stiffness of the beam — an effect often called stress stiffening, since it typically occurs in structures with tensile stresses. However, compressive stresses soften the structure.
Another way of looking at this is by performing a linear buckling analysis. You can do so by adding a Linear Buckling study to the model and using the thermal expansion caused by ΔT = 10 K as a unit load. You will then find that the critical load factor is 80.
The first buckling mode.
With a linear assumption, the beam becomes unstable at an 800 K temperature increase. At the buckling load, the stiffness has reached 0. Assuming that the stiffness decreases linearly with the compressive stress, the stiffness at ΔT = 10 K should be reduced by a factor of
Since a natural frequency is proportional to the square root of the stiffness, you can estimate the decrease to , which matches the computed value of 0.9943 well.
Stress softening also affects the twisting and axial modes, but the effect is not as obvious as it is in the bending modes.
In the cantilever beam, no stresses develop when it is heated, as it simply expands. In this case, the frequency shift is due solely to the change in geometry — an effect that is much smaller than the stress-softening effect.
The natural frequencies for the bending, torsional, and axial vibration of a beam have the following dependencies on the physical properties:
Here, the following variables have been introduced:
It is assumed that the initial dimensions of the beam are L_{0} x a_{0} x b_{0}, where a_{0} > b_{0}. After thermal expansion, the size is L x a x b.
The expansions (strains) in the three orthogonal directions are called ε_{x}, ε_{y}, and ε_{z}; respectively. In this case, they are linearly related to the thermal expansion by ε_{x} = α_{x}ΔT, ε_{y} = α_{y}ΔT, and ε_{z} = α_{z}ΔT; but in principle, it could be any type of inelastic strain.
The geometric properties scale as:
The mass density also changes. Since the same mass is now confined in a larger volume,
By introducing these expressions into the formulas for the natural frequencies, you arrive at the following expected eigenfrequency shifts:
Since the thermal expansions are very small, the approximate first-order series expansions can be expected to be accurate.
For the torsional vibrations, the situation is slightly more complicated, since the powers of a and b are mixed in the expression for the polar moment J. But if you make use of the fact that a = 2b for this geometry, then it is possible to derive a similar expression.
Now, compare the computed frequency shifts with the analytical predictions for the cantilever beam. The results are shown in the table below and the agreement is very good.
Mode Type | Computed | Predicted |
---|---|---|
First bending, z direction | 1.00039 | 1.00040 |
First bending, y direction | 1.00028 | 1.00030 |
First twisting | 1.00025 | 1.00025 |
First axial | 1.00020 | 1.00020 |
The fixed constraints at the ends of the beam cause local stress concentrations when the temperature is increased, as the transverse displacement is constrained.
The axial stress in the doubly clamped beam caused by a 10 K temperature increase.
This can have two effects:
To determine what effects the constraints should have, you must rely on your engineering judgment. Usually, the component and its surroundings are subject to temperature changes. In this situation, the possibility to add a thermal expansion to constraints in COMSOL Multiphysics comes in handy. Let’s see how the solution is affected.
Thermal expansion added to the fixed constraints for the doubly clamped beam.
For the cantilever beam, the results now change so that they perfectly match the analytical values.
Mode Type | Fixed Constraints | Stress-Free Constraints | Predicted |
---|---|---|---|
First bending, z direction | 1.00039 | 1.00040 | 1.00040 |
First bending, y direction | 1.00028 | 1.00030 | 1.00030 |
First twisting | 1.00025 | 1.00026 | 1.00025 |
First axial | 1.00020 | 1.00020 | 1.00020 |
In the analysis above, it is assumed that the material data does not depend on temperature. When looking at constrained structures (dominated by the stress-softening effect), this might be an acceptable approximation. However, with the small frequency shifts caused by geometric changes, the temperature dependence of the material must also be taken into account.
In this guide, you can see the temperature dependence of Young’s modulus for a number of metals. The stiffness decreases with temperature. For many materials, the relative change in stiffness is of an order of 10^{-4} K^{-1}. This means that for a temperature change of 10 K, you can expect a relative change in material stiffness that is of the order of 0.1%. This effect might actually be larger than the geometric effect computed above.
A small note of warning: When measuring the temperature dependence of Young’s modulus, it is important to know whether or not the geometric change caused by thermal expansion has been taken into account. In other words, you must know whether the Young’s modulus is measured with respect to the original dimensions or the heated dimensions.
When it comes to mass density, the situation is easier. When performing structural mechanics analyses in COMSOL Multiphysics, the equations are formed in the material frame. Thus, the mass density should never be given an explicit temperature dependence, since that violates mass conservation.
The coefficient of thermal expansion (CTE) usually increases with temperature. The relative sensitivity is often of the order of 10^{-3} K^{-1}. This sounds large, but it isn’t usually important when looking at the way the CTE enters the equations.
Most materials in the Material Library in COMSOL Multiphysics come with temperature-dependent material properties. In this example, you manually add a linear temperature dependence to the Young’s modulus with the following steps:
Alternatively, you can create a function and call it, with T as the argument.
Adding a linear temperature dependence to the material.
In the settings for the Linear Elastic Material, the Model Input section is now active. You then provide a temperature to be used by the material.
Adding the temperature to the material using Model Input.
After including a reduction of Young’s modulus by 1·10^{-4} K^{-1}, the resulting frequency shift turns out to be negative, rather than the positive shift observed with a constant Young’s modulus (shown in the table below).
Mode Type |
Stress-Free Constraints Constant E |
Stress-Free Constraints Temperature-Dependent E |
Difference |
---|---|---|---|
First bending, z direction | 1.00040 | 0.99990 | -0.00050 |
First bending, y direction | 1.00030 | 0.99980 | -0.00050 |
First twisting | 1.00026 | 0.99976 | -0.00050 |
First axial | 1.00020 | 0.99970 | -0.00050 |
The shift is exactly as expected for all modes — Young’s modulus is reduced by a factor 1·10^{-3} and the natural frequencies are proportional to its square root. Actually, you can include the change in Young’s modulus in the linearized expressions for the frequency shifts as:
Here, it is assumed that . The value of the coefficient β is usually negative; In this case, β = -10^{-4} K^{-1}.
For the common case of isotropic thermal expansion, these expressions simplify to:
We are looking for frequency changes that are at the ppm (parts-per-million) level. This means that it is important to avoid spurious rounding errors. There are some actions that you can take to ensure optimal accuracy.
In the settings for the Eigenfrequency node, set Search for eigenfrequencies around to a value of the correct order of magnitude.
The updated settings in the Eigenfrequency node.
Then, decrease the Relative tolerance in the settings for the Eigenvalue Solver node.
The decreased Relative tolerance in the settings for the Eigenvalue Solver node.
Change only the parameters necessary for capturing the physics. For example, use the same mesh for all studies.
If you have reason to believe that the problem is ill-conditioned, as can be the case for a slender structure, select Iterative refinement in the settings for the Direct solver.
The settings for the Direct solver, showing the option for Iterative refinement.
In version 5.3 of COMSOL Multiphysics®, the method for how inelastic strains are handled under geometric nonlinearity has been changed. A multiplicative decomposition of deformation gradients is the current default, rather than the subtraction of strains that was used in previous versions. This is one key concept to understand why it is now possible to perform this type of analysis with a very high accuracy. Let’s look at a (somewhat artificial) case where the temperature increase is 3·10^{4} K and there are no temperature dependencies in the material properties. This means that the stretches are
resulting in the volume changing by a factor of 3.952.
You can then compare the results from the prestressed eigenfrequency analysis with a standard eigenfrequency analysis on a bigger beam with L = 13 mm; a = 1.6 mm; b = 0.95 mm; and lower density, scaled by a volume factor of 3.952, ρ = 253.036 kg/m^{3}. This of course leads to large increases in the natural frequencies, as the heated object is much larger but with a lower density. The relative changes in frequency for the two approaches are shown in the following table.
Mode Type |
Thermal Expansion and Prestressed Eigenfrequency |
Larger Geometry and Lower Density |
---|---|---|
First bending, z direction | 2.2309 | 2.2308 |
First bending, y direction | 1.8759 | 1.8759 |
First twisting | 1.6702 | 1.6695 |
First axial | 1.5292 | 1.5292 |
As can be seen above, the correspondence is in excellent agreement. There is a slight difference in the twisting mode, but that disappears with a refined mesh. Actually, refining the mesh shows that the best prediction is from the prestressed eigenfrequency analysis.
We have discussed how to accurately determine changes in eigenfrequencies caused by temperature changes with COMSOL Multiphysics, as well as the effects of stress softening, geometric changes, and the temperature dependence of material properties.
Artificial ground freezing is a construction technology that involves running an artificial refrigerant through pipes buried underground. As the refrigerant circulates through the pipe network, heat is removed from the ground and ice begins to form around the pipes. This in turn causes the soil to freeze. In other words, the process converts soil moisture into ice. Once the soil is frozen, it is both stronger (sometimes as hard as concrete) and has a greater resistance to water. This allows the soil to provide effective support to the relative infrastructures, particularly those that are larger and more complex.
Once frozen, soil becomes stronger and more resistant to water.
For the AGF method to be effective, we need to know the temperature distribution inside the system. Of the physical processes that occur in AGF, the most prominent is the phenomenon of transient heat conduction with phase change. Further, it is also important to consider the relationship between this phase change and the groundwater flow — particularly when there is a higher flow velocity. These elements can impact the development of the freezing wall and thus the strength and reliability of the AGF method.
To study the AGF method, a team of researchers from Hohai University turned to the COMSOL Multiphysics® software. Their case study involves using the method to strengthen soil at a metro tunnel entrance in Guangzhou, China.
For this specific example, the refrigerant that circulates throughout the pipe system is -30ºC brine. The subsurface temperature is reduced until the pore water is frozen and the freezing wall forms. The formation within the frozen area is made up of muddy sand, and the direction of the groundwater flow is primarily horizontal and normal in relation to the axial direction of the tunnel.
To simplify modeling heat transport in a saturated aquifer, the researchers used a 2D model based on a coupling of temperature and flow fields. The model, shown below, is 20 m in both length and height. Note that five monitoring points are included. These points are used to verify the accuracy of the model by comparing the calculated temperature results with in situ measurements.
The AGF model’s geometry, with the monitoring points highlighted (left), and the model grid’s mesh (right). Images by Rui Hu and Quan Liu and taken from their COMSOL Conference 2016 Munich paper.
In this analysis, the following assumptions are made:
According to previous temperature monitoring data from the frozen area, there is an initial ground temperature of 15°C. The figure below shows the initial temperatures in various holes of thermal observation.
The initial temperatures in different holes for thermal observation. Image by Rui Hu and Quan Liu and taken from their COMSOL Conference 2016 Munich paper.
The cooling source of the freezing system is the lateral wall of the freezing pipe. Changes in the temperature of the lateral wall have the greatest impact on the temperature distribution within the system. It is possible to use the values from the temperature monitoring of the main pipe as approximations for the estimated temperature of the lateral wall. The plot below shows the fitting function and curve for the lateral wall temperature of the main pipe after a monitoring period of 40 days.
The fitting function and curve for the lateral wall temperature. Image by Rui Hu and Quan Liu and taken from their COMSOL Conference 2016 Munich paper.
With regards to groundwater flow, a flow velocity of 0.2 m/d is obtained via field tests. Between upstream and downstream, the head difference is calculated as 0.8 m.
Now onto the results. Let’s consider the temperature distribution and permeability coefficient for a range of times. In terms of temperature, when the freezing time increases, the cold temperature from the freezing pipes is primarily led downstream — with less of an influence upstream. The permeability coefficient results, which illustrate the formation of the freezing wall, indicate that the top and bottom walls form at a faster rate than those walls at upstream and downstream. Note that the freezing wall is entirely closed after 35 days.
The temperature distribution (left) and permeability coefficient results (right) at various points in time. Images by Rui Hu and Quan Liu and taken from their COMSOL Conference 2016 Munich paper.
When comparing the closure of the freezing wall and flow velocity, the closing time increases nonlinearly as the flow velocity increases. The time of closure dramatically increases when the velocity is greater than 1.5 m/d. As for the average wall thickness in all directions and relative flow velocity, the influence of the flow velocity on the thickness of the upstream wall is most prominent.
The successful validation of this model offers guidance for the metro tunnel project in Guangzhou, China. With plans to further develop this model, the researchers hope to use it as a resource for improving applications of the AGF method.
When performing a structural analysis with plate or shell elements, there is an underlying assumption that the variation of the in-plane stresses through the thickness is linear. In a local coordinate system, where z is oriented along the normal to the shell surface, it is thus possible to write
where d is the thickness. The indices i and j can be either x or y. In this decomposition, is called the membrane stress and (or ) is called the bending stress.
The decomposition of a linear stress distribution into a membrane stress and bending stress.
For the other stress components, shell theory implies that
and
Unless the shell is thick, the transverse shear stresses, σ_{iz}, are significantly smaller than the in-plane stresses.
A membrane stress has the same value all through the section. If the material is assumed to be elastoplastic with no hardening, then all points reach the failure stress at the same time. The load that causes incipient plasticity is thus also the failure load.
The stress-strain curve for an elastoplastic material with no hardening. The variable σ_{y} is the yield stress.
Now, consider pure bending with a uniaxial stress state, as in a beam. As long as the material is elastic, the stress distribution is linear through the section, with the value being zero at the midsurface. As the load increases, the stress in the outermost fibers reaches the yield limit. However, the rest of the section is still elastic. It is thus possible to further increase the load without a complete failure.
The stress distribution at incipient yielding (left), partly through yielding (middle), and collapse (right).
The bending moment at failure is 1.5 times the bending moment at initial yield. Thus, if the allowed stress only takes the maximum stress into account, the risk of collapse is larger for a membrane state than it is for a bending state.
If we consider a state of mixed bending and tension, it is possible to compute the combinations of moment, M, and axial force, N, which cause failure.
The stress state at collapse for combined tension and bending.
The membrane and bending stresses are, for an elastic case, related to the moment and axial force through
and
By writing the moment and axial force in terms of membrane and bending stresses, we arrive at the following interaction formula:
In a full 3D case, the stress distribution differs significantly from linear in the vicinity of geometric discontinuities. This is where the concept of stress linearization becomes important. The sum of the membrane and bending stress provides a linear approximation to the true stress distribution, having the property that the resultant force and moment are preserved.
The linearization of a stress tensor component from a 3D solution.
In the graph above, the maximum computed stress is 305 MPa. If the stress state is uniaxial — and the yield stress of the material is 350 MPa — this means that 87% of the load giving initial yield has been reached. However, the linearized stress predicts only 64% of the yield stress. The membrane stress contributes 32% of the yield stress.
If we want to compute a safety factor against collapse, the actual stress distribution does not matter. At failure, the stress everywhere is equal to the yield stress, either in tension or in compression. The relation between tensile and compressive stresses is uniquely determined by force and moment equilibrium.
In the figure below, we can see an example of how the stress is distributed along a stress linearization line as the load is increased in an elastoplastic analysis. The yield stress is first reached when the load parameter rises slightly above 0.38. When the load parameter reaches 0.76, a collapse ensues.
The stress distribution over a cross section as the external load is increased. The load parameter value is the ratio between the membrane stress and yield stress.
In this example, the values have been chosen so that σ_{m} = σ_{b}. Using the interaction formula above, this means that collapse should occur when
This value matches the final parameter value of 0.76 rather well. The difference can be explained by the fact that a small plastic hardening is used in the model to stabilize the analysis.
The conclusion is that for determining safety conditions within plastic collapse, the linearized stress is the relevant parameter, since it is proportional to the axial force and bending moment. Using the true peak stress gives an overly conservative design. The safety factor, which is implicit in the bending collapse, must also be taken into account.
If the structure is subjected to cyclic loading, the peak stresses are of utmost importance, as they determine the risk of fatigue crack initiation at the surface.
The concept of stress linearization is an important part of the qualification of pressure vessels, as described in ASME Boiler & Pressure Vessel Code, Section III, Division 1, Subsection NB. Here, we are required to classify stresses as either primary or secondary.
A primary stress is a stress that is required for maintaining force and moment equilibrium. Secondary stresses are caused by other effects. Typically, secondary stresses are local effects caused by either geometric discontinuities or displacement-controlled loading. Secondary stresses do not lead to a collapse when they exceed elastic limits, since they are just redistributed.
During the analysis, the stress is studied along a number of lines through the section, referred to as stress classification lines (SCLs). The choice of SCL is not unique, so here we must use our engineering judgment to find the critical locations.
Although not fully correct (but conservative), the linearized stresses are sometimes viewed as equivalent to the primary stresses. Without going into detail, the basic requirements of the code are:
Interestingly enough, this means that if the membrane stress is at the limit allowed by the first criterion, it is still allowed to add a certain amount of bending stress. The discussion above tells us why: The bending stress reduces the stress over part of the section.
As noted above, the detailed stress state is not important when it comes to static failure, as the stress distribution in the collapse state is fully determined by the force and moment equilibrium. In the figure below, the collapse interaction curve is compared with the stress limits imposed by the code.
The fundamental ASME criteria for primary stresses. The stresses are normalized by the yield stress.
It should be noted that because pressure vessels often operate at elevated temperatures, room temperature values of allowed stresses might not be sufficient.
The requirement on the secondary stresses is set to avoid cyclic plastic deformation upon repeated loading–unloading cycles. The purpose is to avoid plastic strains accumulating in each load cycle, which can lead to a fast failure due to low-cycle fatigue.
Some rules for qualifying structural elements are based on the stresses being “hand calculated” or the result of a shell or plate analysis. When we do a full 3D analysis, the effect can be that we get results that are “too good”. The effects of local stress concentrations are already taken into account by providing low allowable nominal stresses. Because of this, we might end up in a situation where using the accurate results of a full 3D analysis leads to a highly conservative design. In this case, stress linearization can provide a useful tool for converting the 3D stress state back into a set of nominal stresses.
For instance, this situation can occur when analyzing welds. Typically, the local geometry at the weld is not even well defined (unless it is a very high-quality weld that has been ground smooth). Thus, the actual local stress is not even meaningful to compute, so we must resort to methods based on nominal stresses.
A weld in a pipe used for district heating. Image by Björn Appel, Benutername Warden. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
A stress linearization does not affect the analysis as such; it is a type of result presentation. The variables to be used are set up in the Solid Mechanics interface. We add a line for stress linearization either under Variables in the context menu for the Solid Mechanics interface or under Global on the Physics tab in the ribbon.
Adding a Stress Linearization node from the context menu.
Adding a Stress Linearization node from the ribbon.
Depending on whether the component is in 3D or not, the definition of the stress linearization line comes in two different flavors. In either case, we select an edge (or set of edges) that forms a straight line through the thickness of the component that we are evaluating. In 3D, we must also define the axis orientation of the local coordinate system in which the stresses along the SCL are represented.
The settings for stress linearization in 3D.
The stress tensor components along an SCL are represented in a local coordinate system, where 1 is the direction along the line. The 2 direction is perpendicular to the line and has the following orientations:
For the last bullet point, note that the Second Axis Orientation section of the Stress Linearization node provides several options for entering the orientation.
If we have defined the SCLs prior to the analysis, then one edge data set is generated for each SCL. At the same time, a default plot called Stress Linearization is added.
The default data sets and graph plot group.
The stress linearization plot contains three graphs along the selected SCL:
An example of a default stress linearization plot.
In the stress linearization plot, we can change to another SCL by selecting the corresponding edge data set. In the default plot, the 22 stress tensor component is displayed. Of course, we can change to other components. Usually, 33 and 23 are the most important.
If we add Stress Linearization nodes after running the analysis, we must click on the Update Solution button to make the newly created variables accessible for result presentation. No default plots or data sets are automatically generated in this case.
Graphing along the SCLs is important for understanding the stress state at different locations, but at the end of the day, it is the stress intensity that is important. The maximum stress intensity for each SCL can be presented by adding a Global Evaluation node. When computing the stress intensity for the bending stress plus the membrane stress, the bending part of the out-of-plane stress components (which are supposedly small) is ignored. This approach is customary in this type of analysis.
The result quantities for stress linearization when selecting data for a Global Evaluation node.
In addition to the stress intensities, the peak stress tensor at the two ends of the SCL is available. We can also directly access the section forces and moments corresponding to the linearized stresses.
As of version 5.3 of COMSOL Multiphysics® and the Structural Mechanics Module, the functionality for stress linearization provides us with a set of built-in tools for converting a 3D stress state to one of pure bending and tension. This makes it much easier to produce results that comply with various design codes.
On any given day, we can hear the noises from car tires or trains as they come to a stop. While the sounds themselves are familiar, the phenomenon behind them might not be.
The stick-slip phenomenon is present in many applications, such as when a train comes to a stop. Image by DozoDomo. Licensed under CC BY-SA 2.0, via Flickr Creative Commons.
Stick-slip — common in mechanical contact applications — describes the motion that occurs when two surfaces alternate between sticking to one another and sliding over each other. As this motion occurs, corresponding changes take place in the force of the friction. Collectively, these interactions can impact the stresses, strains, and deformations that occur close to where the two bodies are in contact. This in turn influences the efficiency of the system as well as its safety.
There is a new tutorial available in the Application Gallery that handles a transient contact problem involving stick-slip friction transition. Let’s have a look at the setup of this model and the results that it produces.
Note: This example is currently available via the Application Library update.
For this example, our model geometry consists of a halfpipe and a section cut from a hollow pipe. The halfpipe features a transition length of 50 cm and a radius of about 1 m. Meanwhile, the pipe has a thickness of 2 cm and a radius of 15 cm.
The model geometry.
Subjected to a gravity load, the pipe is released at the top of the halfpipe with its centroid 75 cm above the horizontal plane. These two bodies remain in contact with one another at all times. Depending on the velocity of the pipe and its location in the halfpipe, the motion of the pipe fluctuates between sliding and rolling. We define the friction coefficient as a function of the slip velocity via the exponential dynamic Coulomb friction model.
For this simulation study, the values of interest are pipe displacement and energy balance — the latter of which is used to verify the accuracy of the results. The solution is computed for a time of four seconds.
The plot below depicts the von Mises stress distribution in the pipe at the final step and the trajectory of a point located on its outer surface. It is clear that the pipe deforms due to gravity and that the trajectory path transitions between stages of stick and slip friction. In the stick stage, the trajectory is smoothly parabolic in correlation with the rotation of the pipe. In the slip stage, the trajectory is slightly more elongated. From the following animation, we can see the evolution of the pipe’s motion over time.
Left: The stress distribution in the pipe and the point trajectory. Right: The motion of the pipe over four seconds.
Let’s now check the energy balance. As expected, the potential energy (green) becomes lower as the kinetic energy (blue) rises. Because of frictional dissipation energy (red), the pipe is never able to reach its initial height on the halfpipe. The majority of the energy is lost due to the friction that occurs when the pipe arrives at the steeper slope area of the halfpipe. After two seconds, the pipe stays in the region with the lower slope and rolls rather than slides. Because of the pipe’s deformable behavior, some of the total energy (pink) is stored as elastic strain energy (light blue). In the plot below on the right, we can visualize the friction coefficient as a function of time. As the results indicate, the exponential dynamic Coulomb friction coefficient causes the friction to drop exponentially as the slip velocity increases.
Left: Energy balance versus time. Right: The friction coefficient as a function of time.
In many contact problems, it is necessary to address the phenomenon of stick-slip friction transition. As this example illustrates, COMSOL Multiphysics® version 5.3 provides us with the capabilities to handle such analyses as well as verify the accuracy of our solution with new variables for energy quantities. From these findings, it is possible to design safer and more energy-efficient systems.
Ready to give this new tutorial a try? Simply click the button below.
Baking is designed to not only heat a product, such as a cake, but also to prompt the biochemical reactions of the recipe’s ingredients. The combination of dry and wet ingredients creates a mixture that gives the cake flexibility, allowing it to expand while still holding the mixture together.
Cake batter baking in the oven (left) and the finished cake (right).
Although ensuring the right amount of each ingredient is important to this end, it is also key to account for the heat and mass transfer phenomena that take place during the baking process. These underlying mechanisms can have a large impact on the temperature and moistness of the cake as well as the degree to which it swells. This in turn affects the overall quality and taste of the baked good.
In an effort to better understand and predict heat and mass transfer phenomena during cake baking, one team of researchers created a numerical model with COMSOL Multiphysics and ran a series of simulation studies. Here’s a taste of what they found.
For their analysis, the researchers created a 2D axisymmetric model. The medium was assumed to be deformable and porous, containing three phases:
To address this problem, a system of five coupled partial differential equations was solved. The five variables included in the analysis were:
To predict the swelling of the batter (a result of the increase in total gas pressure), the researchers used a viscoelastic model from the Structural Mechanics Module, an add-on product to COMSOL Multiphysics.
The physical phenomena that occur during cake baking. Image by R. Cutté, P. Le Bideau, P. Glouannec, and J.F. Le Page and taken from their COMSOL Conference 2016 Munich presentation.
As a means of validating the model, the researchers carried out a series of experimental tests. These experiments involved baking a cake for 18 minutes in an oven with a floor wall temperature of 175ºC and top wall temperature of 195ºC. Instruments used within the setup provided information about the batter’s thermal and moisture content as well as the boundary conditions. A camera was used to track the swelling of the cake.
In the simulation analysis, the researchers plotted the temperature and moisture content within the cake at three different intervals:
The plots below depict the results for each instance, also showing the swelling of the cake. These results indicate that the evaporation-condensation phenomenon causes the water content at the core of the cake (crumb) to increase. On the other hand, the water content decreases at the surface of the cake (crust). As observed in other baking processes, this physical phenomenon contributes to the formation of large moisture content gradients. These gradients create heterogeneity between porosity and thermal, hydric, and mechanical properties. Such heterogeneous characteristics are further driven by the heating mode.
The temperature and moisture content inside the cake at various time intervals. Image by R. Cutté, P. Le Bideau, P. Glouannec, and J.F. Le Page and taken from their COMSOL Conference 2016 Munich abstract.
When comparing these simulation results to the experimental findings, there is agreement with respect to temperatures, mass losses, and global deformation. Note that because of the model that is used, the expansion-reduction effect is not accounted for in this case. However, there are plans to improve this by testing other mechanical constitutive laws in the future. To make the model even more accurate, the researchers also plan to add a gas phase with three species (water, CO_{2}, and air) to the model; implement reaction kinetics; and predict the brownness of the cake.
The simulation results and experimental findings for the temperature (left) and moisture content (right) inside the cake. Images by R. Cutté, P. Le Bideau, P. Glouannec, and J.F. Le Page and taken from their COMSOL Conference 2016 Munich poster.
Left: The simulation results and experimental findings for the swelling of the cake. Right: Comparison between the final deformed meshing and the real geometry. Images by R. Cutté, P. Le Bideau, P. Glouannec, and J.F. Le Page and taken from their COMSOL Conference 2016 Munich presentation.
Baking a cake is not just an art; it’s a science. With COMSOL Multiphysics, you can create a simple yet realistic model to describe this complex process, specifically the heat transfer and mass transfer phenomena that are involved. The results generated from this model provide a better understanding of the cake baking process as a whole.
I tend to associate tricycles with children and picture them as small, brightly painted toys that are built more for enjoyment than practicality. However, tricycles are also a sustainable method of passenger transport and can even be used to transport bulk loads.
This application inspired the United Parcel Service (UPS) to use electric tricycles to deliver packages in Portland, Oregon, as well as other parts of the world. Although tricycles can’t carry as much as a typical UPS van, they are an alternative to emission-heavy vehicles and also help delivery drivers avoid traffic.
Tricycles can be used for many purposes, such as transporting loads and passengers. Left: Image by Gary J. Wood — Own work. Licensed under CC BY-SA 2.0, via Flickr Creative Commons. Right: Image by Pedro Szekely — Own work. Licensed under CC BY-SA 2.0, via Flickr Creative Commons.
We can use simulation to study a tricycle’s structure and ensure that it meets safety requirements. As an example, let’s consider a MUR-A tricycle developed by researchers from the Costa Rica Institute of Technology. To detect and address possible weak points in the tricycle’s design, the team used the Structural Mechanics Module with COMSOL Multiphysics to evaluate the mechanical performance of its frame. This enabled the team to find faults in the early stages, thereby optimizing the tricycle’s design before creating a physical prototype.
The research team’s model consists of an aluminum 6063-T83 tricycle frame that uses steel 4130 for the handlebars and bottom bracket. As we can see in the following schematic, the frame is made of standard tricycle parts and includes a rear passenger or load zone. The team imported the 3D tricycle frame design into COMSOL Multiphysics using the CAD Import Module.
Tricycle components (left) and mesh (right). Images by A. Rodríguez, B. Chiné, and J. A. Ramírez and taken from their COMSOL Conference 2016 Munich paper.
To analyze their design, the researchers applied loads to different areas of the geometry. While the frame is the only tricycle part modeled, the team used other parts — seat tube, fork, handlebar, etc. — to define the loading conditions. These conditions include:
The locations of the various loads applied to the geometry. Image by A. Rodríguez, B. Chiné, and J. A. Ramírez and taken from their COMSOL Conference 2016 Munich poster.
Using different combinations of the loading conditions, the team studied three distinct loading cases, shown in the table below. You can easily combine several sets of loads in COMSOL Multiphysics by using load groups and load case superpositions.
Impact Force | Pushing and Pulling on the Handlebars | Pedaling Force on the Bottom Bracket | Driver’s Weight | Passenger’s Weight | |
---|---|---|---|---|---|
Acceleration | X | X | X | ||
Steady Pedaling | X | X | |||
Horizontal Impact | X | X |
In regards to the horizontal impact case, it represents a sudden impact against a wall and assumes that the driver is removed from their seat while the passenger remains on the tricycle. As such, this case only accounts for the impact force and the passenger’s weight.
For each of these loading cases, the team performed a simple evaluation of the model’s stress and deformation distributions, enabling them to identify design issues and develop a safer tricycle.
Overall, the simulation results show that in every loading case, there are regions in the tricycle frame design that are susceptible to stresses above the tensile yield strength of 214 MPa and the fatigue limit of 69 MPa. The researchers did not analyze the horizontal impact case for fatigue strength, as this is not (hopefully) a continuous condition.
In the steady-state pedaling case, there are critical areas that exceed the material’s elastic limit, located where the seat and horizontal tubes meet. This is expected, as the rider’s weight causes compression in these areas. Other areas of concern are the places where the horizontal tube and down tube intersect with the cage.
The steady pedaling case. Red indicates the areas where the von Mises stresses are greater than the material’s elastic limit (left) as well as the areas that are stressed more than the material’s fatigue limit resistance (right). Images by A. Rodríguez, B. Chiné, and J. A. Ramírez and taken from their COMSOL Conference 2016 Munich paper.
As for the fatigue evaluation in this scenario, one area (extending from where the seat and down tube meet to the front of the frame) fails when exposed to a static load behind the seat tube. This static load is a potential source of fatigue failure, since it is sometimes active and sometimes not.
There are similar weak areas where the horizontal and down tubes intersect with the cage. The results indicate that additional critical areas are located at the unions of the reinforcement tube, the cage area before the rear axle, and the intersection of the head and down tube.
The acceleration loading case has the same fatigue areas as the steady-state case, but spread over a smaller area. However, there is one difference. The intersection of the head and down tubes has a critical area that is slightly larger than the steady pedaling case, extending to the bottom of the down tube.
The acceleration case. Red indicates the areas that are stressed more than the material’s fatigue limit resistance. Image by A. Rodríguez, B. Chiné, and J. A. Ramírez and taken from their COMSOL Conference 2016 Munich paper.
The team then investigated the horizontal impact case. When comparing their numerical results with the material’s elastic limits, they saw that although the frame can withstand these loads, there are critical areas in the cage.
Examining the frame’s fork region for this loading case, the team saw that it behaves similarly to the frame as a whole. The results show that when the fork is exposed to the impact force, there are only a few deformation areas, shown in the following image. Despite this, the fork region may need to be redesigned, since it does not hold up under a fatigue analysis.
The fork region for the horizontal impact case. Red indicates the areas where the von Mises stresses are greater than the material’s elastic limit. Image by A. Rodríguez, B. Chiné, and J. A. Ramírez and taken from their COMSOL Conference 2016 Munich paper.
Through their work, the researchers gathered helpful insight into the mechanical performance of their tricycle frame design. For instance, the simple fatigue analyses show that while most of the frame withstands static loads, it is compromised when it comes to long-term durability. As such, the tricycle frame needs to be strengthened.
According to existing research, one way to improve this design is to combat the tricycle frame’s low fatigue life by changing the material from aluminum 6063, with a fatigue limit of 69 MPa, to aluminum 6061-T6, which has a higher fatigue limit of 96 MPa.
While the simple analyses discussed today are a good starting point for improving the tricycle frame design, further studies (such as more fatigue and impact simulations) are required. Through this, the researchers can fine-tune their tricycle frame design, ensuring the safety of riders and passengers.
For many years, landmarks like the Hoover Dam and the Pantheon have attracted the attention of tourists. While these structures are most recognized for their famous history, they have something else in common: Both were built using concrete.
Concrete was used to build some of the world’s most famous landmarks, including the Hoover Dam and the Pantheon. Left: Image by Mobilus In Mobili. Licensed under CC BY-SA 2.0, via Flickr Creative Commons. Right: Image by Roberta Dragan — Own work. Licensed under CC BY-SA 2.5, via Wikimedia Commons.
This manmade material — the most widely used in the world — is the foundation for many modern buildings and structures. In recent years, there has been a growing trend of embedding sensors within these structures as a means for measuring the concrete’s condition. This provides an efficient way of monitoring how important parameters like temperature, humidity, and pressure impact the strength and stability of the material.
To design reliable sensors for this purpose, engineers must have an understanding of the properties of concrete and their influence on sensor performance. But due to its unique properties, concrete can be a rather complex material to analyze. For instance, because concrete is viscoelastic, it experiences a time-dependent strain increase when a constant load is applied. This effect, known as viscoelastic creep, is particularly prominent when concrete is subjected to forces for a long period of time. Even when the temperature conditions are steady, concrete structures may undergo a volumetric change. When this occurs without environmental thermal exchange, it is called shrinkage.
The unique properties of concrete make it a complex material to study. Image by Les Chatfield. Licensed under CC BY 2.0, via Flickr Creative Commons.
With the flexibility of COMSOL Multiphysics, a research team from STMicroelectronics in Italy was able to successfully model the complex phenomena in concrete and predict their influence on the performance of an embedded sensor design. This shows an example of a technology group that is versed in one type of application — sensors and the optimization of their designs — using COMSOL Multiphysics to study another application that directly affects their application — the deformation of concrete.
The researchers began by first addressing concrete viscoelasticity. To realistically describe the concrete’s viscoelastic behavior, the team implemented their own Kelvin chain model in COMSOL Multiphysics and solved strains within each of the eight Kelvin branches.
To validate the model, the researchers used a theoretical model of a simple system featuring a concrete cylinder under uniaxial pressure. The cylinder, which is 7 cm in height and diameter, has a compressive strength of 48 MPa. The analysis includes the assumption that the environment is comprised of 50% relative humidity.
The researchers built a model in COMSOL Multiphysics of this same system, comparing their simulation results for the creep with trends observed in the theoretical model. As highlighted below, the findings show perfect agreement with one another.
Comparison of the results for the creep experiment. A constant load of 1 MPa is applied in the simulation study. Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
The team then shifted gears to modeling the time-dependent concrete shrinkage. This involves using a strategy based on thermal phenomena. After calculating the shrinkage profile via equations, a unitary thermal coefficient is applied to the concrete material. For the theoretical model, a temperature profile in agreement with the computed shrinkage is used. The resulting thermal strain is meant to imitate the actual shrinkage.
Using the same benchmark model from the creep test, the team calculated the shrinkage strain. As before, the two results show perfect agreement.
Comparison of the results for shrinkage strain. In the simulation study, the time frame is around 1500 days. Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
After validating their COMSOL Multiphysics model for both viscoelastic creep and shrinkage, the researchers sought to address a more complex scenario — when a silicon sensor is embedded within the concrete. While concrete creep and shrinkage can affect all sensor types, analyzing their effect is especially relevant for pressure sensors.
To represent a simplified sensor structure for measuring pressure, a cylindrical sensor with a height of 600 μm and diameter of 2 mm is used. The membrane (the sensing portion of the configuration) is 10 μm thick with a radius of 700 μm and an internal cavity depth of 50 μm. At the top of the cylinder, a constant load of 10 MPa is applied.
The geometry of the simulated sensor structure (a), with a zoomed-in version depicting the sensor’s axial displacement (b). Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
Initially, with only creep equations applied, the creep greatly affects membrane displacement over time. Its impact is particularly noticeable at the center of the membrane and close to one of its edges.
The difference between the vertical displacement at the center of the membrane and near one of its edges. Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
The next aspect evaluated is the potential relationship between creep-induced changes and stress inside the membrane. The results show that stress does increase over time. The example below highlights this with respect to radial stress distribution.
The radial stress distribution on the sensor when the time span begins (a) and when it ends (b). Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
The following plots provide even more detail, showing the radial and angular stress components along the membrane’s radius when the analysis first starts and when it ends. Due to the creep effect, these components vary over time. Assuming that stress-sensing piezoresistive elements have been fabricated on the membrane, it is possible to observe the impact of a time-dependent creep-induced variation on sensor performance. Note that the positioning of the piezoresistive elements impacts these results.
Radial stress (a) and angular stress (b) along the radius. Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
The behavior of the membrane is also influenced by shrinkage. When adding shrinkage to the model that only previously accounted for creep, a small change in deformation occurs near the center of the membrane. On the other hand, shrinkage has a large effect on stress distribution.
The two stress component distributions at the end of the time frame are then compared, with one analysis that includes shrinkage and one that doesn’t. When only creep is considered, there are changes in stress distribution on the membrane of both components. It is therefore assumed that there is also a shrinkage-induced effect on sensor performance. Once again, the location of the piezoresistor elements impacts the results. Note that the shrinkage effect has a greater influence on the angular stress than on the radial stress.
Comparison of the radial stress (a) and angular stress (b) along the radius — both with and without shrinkage. Image by A. A. Pomarico, G. Roselli, and D. Caltabiano and taken from their COMSOL Conference 2016 Munich paper.
As the results indicate, creep and shrinkage — two of concrete’s unique properties — change the deformation and stress within a sensor’s membrane. This in turn affects sensor output, specifically the output voltage of the piezoresistors that are implemented on the membrane. Such findings are critical in the design of reliable pressure sensors for monitoring the condition of concrete.
When designing a microwave transmitter, system engineers must ensure that there are no undesirable frequencies in its output. A common solution for this is to use a microwave filter, which is often placed between the transmitter’s antenna and nonlinear power amplifier. This amplifier generates harmonics that engineers can eliminate by applying one or more narrow passband filters on the output.
A microwave transmitter tower. Image by Tom Page — Own work. Licensed under CC BY-SA 2.0, via Flickr Creative Commons.
This solution is not without its own challenges. When a transmitter is exposed to high-power loads and harsh environmental conditions — like being in a cellular base station exposed to extreme desert heat — thermal drift can occur.
Being subjected to harsh environments, like the desert heat, can cause thermal drift in microwave transmitters. Image licensed under CC BY 4.0, via ESO/C. Malin.
Thermal expansion of the structure can distort the frequency response of filters in microwave systems. Therefore, to obtain reliable filter designs, we must not only perform an accurate electromagnetics analysis, but also study the structural deformation caused by the temperature rise. As today’s example demonstrates, we can accomplish this by using the RF Module and Structural Mechanics Module with the COMSOL Multiphysics® software.
Let’s start by taking a look at our model, which contains a cylindrical post inside a copper box that is covered in a thin silver layer to help reduce losses. The electromagnetic cavity between the post and box is closed and filled with air. Although real filters tend to contain multiple cascaded cavities, our model focuses on one cell.
To help us compare how different designs affect the filter’s performance, we created two variations of this model:
The geometry of the microwave cavity filter.
When the cavity walls experience a uniform temperature increase due to factors like external heating and power dissipation in the surrounding electronics, thermal expansion and the related eigenfrequency shift occur. Here, we model the thermal expansion using the Solid Mechanics interface in the Structural Mechanics Module. The thermal expansion causes the filter’s geometry to distort, which we account for with the Deformed Geometry interface. The resulting distorted shape is used for an electromagnetics analysis.
As for the eigenfrequency analysis of the microwave cavity, we turn to the 3D Electromagnetic Waves, Frequency Domain interface in the RF Module. Next, let’s review the results of these studies.
We use the design of the copper filter to calculate the filter’s thermal expansion and perform an electromagnetic resonance mode analysis. With this analysis, we can find the lowest eigenfrequency of the filter and the post’s normal quarter-wave resonance. Looking at our results, we spot a strong capacitive coupling between the post’s top and the box’s adjacent face.
Left: The thermal expansion when the temperature is 100°C more than the reference temperature. Right: An electromagnetic mode analysis depicting both the fundamental mode’s surface current patterns and the electric field.
Moving on, we repeat these structural and electromagnetics analyses for different operating temperatures and use the resulting data to plot the eigenfrequency vs. temperature curve. With this, we compare the designs of the filter with copper only and both copper and steel.
The eigenfrequency vs. temperature curve for the designs of the copper filter and copper-steel filter.
The results indicate that the filter using both copper and steel is the better of the two design variations. This is because its two materials have different coefficients of thermal expansion and as a result, the capacitive coupling is reduced between the post’s top and the box’s adjacent face. The capacitive coupling greatly influences the resonance frequency and can combat the effect of increasing the overall cavity size when reduced.
Further, in the copper-steel filter design, we can compensate for the majority of the thermal drift by applying a temperature-driven adjustment to the distance between the end of the cylinder and the box.