Many of today’s motor vehicles rely on reciprocating piston engines as their source of power. In an internal combustion reciprocating engine, fuel combines with an oxidizer in a combustion chamber. The combustion causes the gases to expand, applying pressure to the engine’s piston, pushing it out of the chamber. The linear movement of the piston is converted to a rotating motion by way of a connecting rod, which joins the piston to the crankshaft. This continual motion exerts a great deal of stress on the connecting rod — a force that becomes greater with increasing engine speeds.

Within reciprocating engines, it is crucial that each component is analyzed critically, since one part’s failure often means replacing the entire engine. To optimize the design of this engine and ensure a long operational lifetime, we can analyze the connecting rods from the fatigue viewpoint.

The High-Cycle Fatigue of a Reciprocating Piston Engine model is based on an example of a three-cylinder reciprocating engine from the Multibody Dynamics Module. In this engine, a flywheel is mounted on the crankshaft, with the assembly supported on both ends by journal bearings. The model also features three sets of cylinders, pistons, and connecting rods that are identical. Hinge joints are used to connect the bottom ends of the connecting rods to the common crankshaft as well as to connect the pistons to the top ends of the rods. A prismatic joint is used to connect each of the cylinders to a piston.

*Reciprocating engine geometry.*

Aside from the flexible central connecting rod, it is assumed that the engine components are rigid. The cylinders are fixed and the other parts of the engine are able to freely move in space. The engine as a whole operates at 1,000 RPM and the material data is derived from structural steel, which shows a fatigue limit at 210 MPa.

Our analysis begins with the stress history in the connecting rod fillet, as a stress concentration resulting from geometrical change is assumed in this area. After a few revolutions, the engine reaches a steady-state behavior. Following the third cycle, the stress history seemingly repeats itself for each cycle, as shown in the plot below. The third principal stress dominates the connecting rod’s stress history as the part is in compression during this time. Because the first and second principal stresses are small compared to the third one, we can consider the stress state at the fillet to be uniaxial. Since the von Mises stress would be better suited in a *multiaxial* loading, we use the principal stress as the amplitude stress in the Basquin relation.

*The stress history in a connecting rod fillet.*

The following plot addresses fatigue life prediction in the connecting rod. Here, the point of focus is the fillet near the top end of the rod. According to the Basquin model, the fatigue life is predicted to be more than twenty-five billion cycles, which is a notably long operational lifetime. Although the endurance limit is not defined in the Basquin model, the relation can be used to back-calculate the fatigue life at the endurance stress to 245 million cycles. Since the model prediction gives a greater life than the back-calculated fatigue life at the endurance limit, we can assume then that the stress within the engine’s assembly is beneath the endurance limit, which as previously noted is 210 MPa for the used material, and that the connecting rod has an infinite operational life.

*The connecting rod’s fatigue life prediction.*

The initial stress history plot also indicates that the rod is designed for infinite life. With a principal stress range around 110 MPa, the stress amplitude is near 55 MPa, which is lower than the material’s endurance limit.

- Download the model: High-cycle fatigue of a reciprocating engine

Fatigue models are based on physical assumptions and are therefore said to be phenomenological. Since different micromechanical mechanisms govern fatigue under various conditions, many analytical and numerical relations are needed to cover the full spectrum of fatigue. These models, in turn, require dedicated material parameters.

It is well known that fatigue testing is expensive. Many test specimens are necessary since the impurities responsible for fatigue initiation are randomly distributed in the material. The difference in the fatigue life is clearly visible when you visualize all the test results in an S-N curve.

*An S-N curve. The black squares represent individual fatigue tests.*

Since the *S-N curve* — also called the Wöhler curve — is one of the oldest tools for fatigue prediction, there is a good chance that the material data is already available in this form. Many times, the data is given for a 50% failure risk. If you do not have access to the material data, you are faced with a testing campaign.

When you are done, pay attention to the statistical aspect and, at each load level, select the same reliability when constructing an S-N curve. This is important since the S-N curve is expressed in a logarithmic scale where a small difference in input has a large influence on the output. Then, S-N curves for different reliability levels fall under each other and you should select an appropriate level for your application. For noncritical structures, a failure rate of 50% might be acceptable. However, for critical structures, a significantly lower failure rate should be chosen.

Always pay attention when you combine fatigue data from different sources. Make sure that the testing conditions and the operating conditions are the same.

Another aspect of fatigue testing considers the mean stress that has a substantial influence on the fatigue life. In general, fatigue tests performed at tensile mean stress will give a shorter life than tests performed at a compressive mean stress. This effect is also frequently expressed using the *R-value* (the ratio between the minimum and maximum stress in the load cycle). Thus, with decreasing mean stress (or R-value), the fatigue life increases.

In the Fatigue Module, the *Stress-Life* models do not take into account this effect. When using these models, you need to choose material data obtained under the same testing conditions as the operating one.

In the cumulative damage model, the Palmgren-Miner linear damage summation uses an S-N curve. However, in this model, the S-N curve is specified with the R-value dependence and the mean stress effect is accounted for.

*The mean stress effect.*

In case you use a material library and the fatigue data is specified using the maximum stress, you can easily convert it to the stress amplitude using

\sigma_a=\frac{\sigma_{\textrm{max}}(1-R)}{2}

where \sigma_a is the stress amplitude, \sigma_{max} is the maximum stress, and R is the R-value.

The stress-based models seem to be fairly simple. For example, the Findley and the Matake models use the expressions

\left(\frac{\Delta\tau}{2}+k\sigma_{\textrm{n}}\right)_{\textrm{max}} =f

and

\left(\frac{\Delta\tau}{2}\right)_{\textrm{max}}+k\sigma_{\textrm{n}} =f

respectively. They depend on only two material constants: f and k. These material parameters are, however, nonstandard material data that can be related to the endurance limit of the material.

Note that the actual values of f and k differ between the two models. The analytical relation is somewhat cumbersome to obtain since the stress-based models are based on the critical plane approach and you need to find a plane where the left-hand sides of the above relations are maximized. This is basically done by expressing the shear and the normal stress as a function of the orientation using the Mohr’s stress circle, maximizing by setting the derivative to zero, and simplifying the resulting relation.

The different steps of the data manipulation will not be shown here. For the Findley model, the material parameters are related to the standard fatigue data using

\frac{f}{\sigma_U(R)} = \frac{(1-R)^2+2k\beta+4k^2}{2\beta(1-R)},\ \ \ \beta=\sqrt{(2k)^2+(1-R)^2}

Here, R is the R-value and \sigma_U(R) is the endurance limit. The argument of the endurance limit indicates that the stress is R-value dependent. For the Matake model, the relation is somewhat simpler and given by

\frac{f}{\sigma_U(R)}=0.5+\frac{k}{1-R}

Since both relations have two unknown material parameters, you need endurance limits from two different types of fatigue tests. To illustrate this, consider a case where one endurance limit is obtained by alternating the load between a tensile and a compressive value, R=-1. In the second case, the load is cycled between a zero load and a maximum load, R=0. For the Findley model, this leads to

\left\{

\begin{array}{lr}

\frac{f}{\sigma_U(-1)}=\frac{1}{2}\left(k+\sqrt{1+k^2}\right)\\

\frac{f}{\sigma_U(0)}=\frac{1}{2}\left(2k+\sqrt{1+4k^2}\right)

\end{array}

\right.

\begin{array}{lr}

\frac{f}{\sigma_U(-1)}=\frac{1}{2}\left(k+\sqrt{1+k^2}\right)\\

\frac{f}{\sigma_U(0)}=\frac{1}{2}\left(2k+\sqrt{1+4k^2}\right)

\end{array}

\right.

The pair of equations must be solved numerically. Here is the strategy:

- Eliminate f between the two equations. This is trivial since it always appears as a linear term.
- Now, you have a nonlinear equation for k only. Since k has a rather small variation (usually between 0.2 and 0.3), it is easy to solve even by pure trial and error.
- Given the computed k, evaluate f using either of the original equations.

For the Matake model, the two fatigue tests lead to

\left\{

\begin{array}{lr}

\frac{f}{\sigma_U(-1)}=\frac{1}{2}+\frac{k}{2}\\

\frac{f}{\sigma_U(0)}=\frac{1}{2}+k

\end{array}

\right.

\begin{array}{lr}

\frac{f}{\sigma_U(-1)}=\frac{1}{2}+\frac{k}{2}\\

\frac{f}{\sigma_U(0)}=\frac{1}{2}+k

\end{array}

\right.

which you can solve analytically.

I would like to share a few examples where the discussed fatigue models are used:

- Findley and Matake models are used to predict fatigue in the example of High-Cycle Fatigue Analysis of a Cylindrical Test Specimen.
- The S-N curve is used in the tutorial model from the Structural Mechanics Module of a bracket.
- The S-N curve with R-value dependence is used in the fatigue prediction of a model of a frame with a cutout.

A fatigue model can be selected in different ways. Expert knowledge is a good starting point. It may so be that, within your organization, there is prior knowledge on the topic if a similar application has been analyzed already. Alternatively, you may also find expert knowledge through a literature search. Since about 90% of all structural failures are caused by fatigue, there is a great chance that another engineering team has already analyzed a similar application to yours.

When there is no prior knowledge on the fatigue case, a suitable fatigue model can be proposed based on a few questions regarding loading conditions and expected fatigue failure. In the diagram below, I have summarized the key questions you should ask when evaluating fatigue using the Fatigue Module.

*Selection of the fatigue model type.*

First, you need to determine whether the external load is random or if your application is subjected to a constant cycle. A load that is not truly random, but has sequences of non-constant load cycles, could also fall into this category.

The stress history for random loads introduces a complex load scenario in the structure that requires an advanced evaluation technique to quantify the stress response. If your application is subjected to random loading, you can evaluate fatigue using the Cumulative Damage feature, where the random load is converted into a stress range distribution, rather than the single constant stress cycle — which is assumed for the other evaluation techniques.

You can find more details about this computation method in my previous blog post “Random Load Fatigue“.

At constant load cycles, the structure is affected by a repeatable load sequence. In this case, you need to determine whether the loading is proportional or non-proportional.

In proportional loading, the orientation of the principal stresses and strains does not change during the load cycle. Another way to discriminate between these two cases is to consider the characteristics of the external load. With one source of the external load, the structural response is defined by a stress tensor where all components change *in phase*. When the external load is applied in multiple points or if you have a traveling load, the components of the stress tensor can change *out of phase*. These two types of load cycles require different techniques for fatigue evaluation.

In proportional loading, the direction of the largest stress or strain that controls fatigue is clear. This was probably the type of application you worked with when you took your first class in fatigue. Back then, the load was always sinusoidal and classical methods such as the *S-N curve*, also called the Wöhler curve, were used.

In the Fatigue Module, the *Stress-Life* and *Strain-Life* models can evaluate fatigue at proportional loading. These models are based on a fatigue-life curve, which provides a direct relation between the fatigue life and the applied stress or strain amplitude.

One model in the Stress-Life family requires extra attention: The *Approximate S-N curve* (see figure below). In the model, you specify two points on the S-N curve. The first one is the transition between the high- and low-cycle fatigue, while the second defines the endurance limit. The advantage of this model is that it does not require any substantial knowledge of the fatigue material data, since the two required points can be related to the ultimate tensile strength. Although it is a rough approximation, it is a good starting point when you lack material data.

*The approximate S-N curve model. The index t denotes the transition point, while the index e denotes the endurance limit point.*

The Stress-Life models are suitable for simulating high-cycle fatigue, while the Strain-Life models are frequently used in the low-fatigue regime. The transition between the low- and high-cycle fatigue varies, but is usually somewhere in the span of 1,000 to 10,000 cycles.

The challenge for non-proportional loading is to determine the range of the fatigue-controlling parameter. Since the direction of principal stresses and strains changes, so does the direction of the parameter that gives the highest impact on fatigue life.

In the Fatigue Module, this type of application can be assessed with the *strain-based* and *stress-based* models that I discussed in the blog entry “Fatigue Prediction Using Critical Plane Models“. These are called *critical plane models* because they evaluate many orientations in space in search for the critical plane where fatigue is expected to occur.

The strain-based models are suitable for fatigue prediction at low-cycle fatigue, while the stress-based models are frequently used to predict high-cycle fatigue. Most of the fatigue models predict the number of cycles until failure. The stress-based models predict a fatigue usage factor, which is the fraction between the applied stress and the stress limit. This indicates to the user whether the stress limit has been exceeded and failure is expected or if the component will hold for the expected fatigue life. You can view the fatigue usage factor as the inverse of a safety factor.

In some cases, the stress or strain alone is not sufficient to characterize the fatigue properties. You can then use the *energy-based* models. These combine the effect of stress and strain into energy, which is released or dissipated during a load cycle.

The energy-based models are frequently used in nonlinear materials in the low-cycle fatigue regime. Since the energy can be calculated in different ways, the energy-based models can be used in proportionally and non-proportionally loaded applications.

The blog post titled “Modeling Thermal Fatigue in Nonlinear Materials” demonstrates the use of the energy-based models.

I would like to share a few examples to demonstrate how the different model types are used:

- In the example Random Load Fatigue in a Frame with a Cutout, the fatigue model takes into account a random load consisting of 1,000 load events.
- The classical SN-curve is used in fatigue evaluation of a proportionally loaded bracket.
- In the example model High-Cycle Fatigue Analysis of a Cylindrical Test Specimen, fatigue is predicted in a non-proportionally loaded test specimen. The model also shows how to obtain material parameters from two types of fatigue tests.
- An energy-based fatigue criterion is used in the Thermal Fatigue of a Surface Mount Resistor model, where the dissipated energy in a viscoplastic solder joint eventually leads to failure.

If you have any questions about your fatigue modeling application, please contact us.

]]>Using COMSOL Multiphysics, we implemented a wear model and validated it by simulating a pin-on-disc wear test. We then used the model to predict wear in an automotive disc brake problem. The results we found showed good agreement with published wear data.

*Wear* is the process of the gradual removal of material from solid surfaces that are subjected to sliding contact. It is a complex phenomenon that is relevant to many problems involving frictional contact, such as mechanical brakes, seals, metal forming, and orthopedic implants. The rate of wear depends on the properties of the contacting materials and operating conditions.

Archard’s law is a simple but widely used wear law that relates the volume of material removed due to wear W to the normal contact force F_N, sliding distance L_T, material hardness H, and a material-related constant K

W=\frac{KF_N L_T}{H}

In our work, we considered a modified version of Archard’s law:

\.{w}=k(H,T)p_N V_T

This modified law relates the wear depth w at any point to the normal contact pressure p_N, magnitude of sliding velocity V_T, and a constant k that is a function of the material and temperature. The wear constant k may be computed from experimental wear data, which is typically in the form of weight loss for a specific contact pressure and velocity.

Wear equations are not directly available in finite element analysis (FEA) codes, although their implementation in COMSOL Multiphysics is straightforward. We incorporated the wear equations within our simulations by defining boundary ordinary differential equations (ODEs) on the destination contact surfaces with the wear depth w as the independent variable. The wear depth w is then used as an offset between contacting surfaces (e.g., brake pad and disc) within the contact formulation in COMSOL Multiphysics. In particular, contact is enforced when the penetration between the contact surfaces is equal to the wear depth w, as shown in image below.

*Modification of contact gap calculation: w is the wear depth, g is the gap, and \lambda is the contact pressure.*

This wear algorithm is very efficient since it does not involve altering the nodal locations to account for material loss due to wear. It is only suitable, however, for cases where the wear depth is significantly less than the width of the contact surface.

You can enhance this wear algorithm by including more sophisticated effects, such as anisotropic wear behavior, dependence on the mean and deviatoric stresses in the solid (not just the contact pressure), threshold pressure/stress below which no wear occurs, and more. The assumption of small wear depth must still hold for this modeling approach to be accurate.

We validated the new, contact-offset-based wear model implementation by simulating a pin-on-disc wear test. Only a small section of the disk is modeled, as shown below.

*Pin-on-disc wear test model.*

The disc in this model is much stiffer than the pin and all the wear is assumed to occur in the pin. A force is applied to the pin, resulting in a circular, Hertzian-type contact pressure distribution. A constant tangential velocity is then applied to the disc. The graph below shows how the wear depth varies radially along the pin at four time instances. The total volume loss, calculated as the integral of wear depth over the contact surface, was similar to the value calculated using Archard’s law.

*Wear depth vs. radial distance in the pin-on-disc model.*

We also used the model to predict wear in an automotive disc brake problem, which is similar to the Heat Generation in a Disc Brake model that can be downloaded from the COMSOL Model Gallery. We developed a 3D thermal-structural disc brake model involving representative brake disc/rotor and brake pads.

*Disc brake model used in the COMSOL Multiphysics wear simulation.*

The structural and thermal processes are coupled through frictional heat generation, thermal expansion, and thermal contact. Both physics fields are also coupled to the wear depth evolution boundary ODE. We used a fully-coupled direct solver that converged rapidly, keeping solution times similar for problems with and without wear.

The results for both the pin-on-disc validation example and the disc brake problem were in good agreement with published wear data. In the disc brake example, the model captured the non-uniform wear rate that is typically observed on brake pads; it was higher near the outer radius and leading edge, as shown below.

*Typical brake pad wear depth profile.*

We will present more of our results, including contact pressure and wear contours, at both the Cambridge and Boston stops of the COMSOL Conference 2014.

Nagi Elabbasi, PhD, is a Managing Engineer at Veryst Engineering LLC. Nagi’s primary area of expertise is modeling and simulation of multiphysics systems. He has extensive experience in finite element modeling of structural, CFD, heat transfer, and coupled systems, including fluid-structure interaction, conjugate heat transfer, and structural-acoustic coupling. Veryst Engineering provides services in product development, material testing and modeling, and failure analysis, and is a member of the COMSOL Certified Consultant program.

]]>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.

- COMSOL Conference 2012 presentation: “Fretting Wear and Fatigue Analysis of a Modular Implant for Total Hip Replacement“

The numerical simulation of applications containing the aforementioned challenges can be tackled using the Nonlinear Structural Materials Module, which provides a collection of predefined nonlinear material models, in combination with the Fatigue Module, which contains fatigue models for many different applications.

When the temperature changes, materials want to expand or contract. In applications consisting of several different parts, this thermal deformation will be constrained, since the thermal expansion coefficients differ between various materials. The situation is more challenging in the presence of *nonlinear* materials.

Material nonlinearity implies that the deformation is not proportional to the loading. The nonlinearity of different materials can be roughly divided into *reversible* and *irreversible* nonlinearity. Reversible nonlinearity is also called elastic nonlinearity, which means that the strain state returns back to the initial state once the external load is back at its starting point.

Materials that exhibit irreversible nonlinearity can sustain permanent damage when loaded and will not return to the initial state upon unloading. An example of this phenomenon is shown in the figure below, where a surface mount resistor with a nonlinear solder material is subjected to a thermal cycle.

*Displacement in a surface mount resistor at the end of a thermal load cycle. Blue color denotes zero displacement.*

The material nonlinearity is a creep mechanism that deforms the material once it is subjected to a stress field — even when the stress field is held constant. Since the thermal expansion of the different parts of the surface mount resistor is non-uniform (greater in the printed circuit board on the bottom and smaller in the resistor on top), the assembly is stressed during a thermal load cycle.

Once the thermal load has reached the end of a load cycle, and returned to the initial temperature, a permanent deformation (creep strain) is left in the solder joints on both ends of the resistor. The permanent deformation in the solder joints prevents the remaining parts from returning to the initial state. You can see this in the figure where the resistor is compressed and bulges, while the printed circuit board is elongated.

Another type of material nonlinearity occurs when the permanent deformation only depends on the applied load and does not deform at a constant stress. This is called *plasticity* and can be demonstrated simply by bending a paper clip back and forth. If the applied force is too high, the paper clip will remain in a deformed state that does not change with time. A combination of plasticity and creep is called *viscoplasticity* and is yet another nonlinear material behavior.

Repeated loading and unloading can cause fatigue cracks. Before the fatigue life can be evaluated, you must obtain a stable load cycle. When working with nonlinear materials, many load cycles are often required before the material’s response stabilizes. Generally speaking, the nonlinear material response to a cyclic load can be summarized by three cases: immediate stability, shakedown, and ratcheting.

- In the case of the
*immediate stability*, the second load cycle will already give a stable stress-strain response that is representative for each consecutive load cycle. This is demonstrated with the dotted black line in Case (a) in the figure below. - At
*shakedown*, the elongation stops first after a certain number of cycles. Therefore, a large number of cycles may need to be simulated. See Case (b). - In
*ratcheting*, Case (c), the material experiences a continuous elongation until failure. This case is the most challenging from a fatigue point of view since a stable load cycle is never obtained. In this case, you must generally simulate all cycles from initial state to failure.

*Material response to a repeated load cycle: (a) immediate stability, (b) shakedown, and (c) ratcheting.*

There is no universal model that predicts fatigue for all nonlinear materials, and many models have been proposed over time. In the 1950s, Coffin and Manson examined fatigue in metals and proposed an exponential relation between the fatigue life and the plastic strain for the low-cycle fatigue regime.

Following this pioneering work, many researchers proposed slightly modified models, where the plastic strain has been replaced with a different strain measure, such as creep strain, plastic shear strain, total shear strain, and others. Below, you can see a comparison between two strain measures (effective creep strain and the shear creep strain) in a surface mount resistor model, which was taken from our Model Gallery:

*Development of the creep strain in a solder joint. Effective creep strain, to the left, and shear creep strain, to the right.*

Both strain measures are highest at the interface between the solder and the resistor, which coincides with the position of a thermal fatigue crack in real applications.

For many applications, strain alone is not sufficient for fatigue predictions. Instead, energy might be more suitable since it combines the effect of stress and strain. In the 1960s, Morrow proposed an exponential relation between the fatigue life and the cyclic plastic strain energy. This model has later been modified to depend on other energy quantities, such as creep strain energy, total strain energy, stress-strain hysteresis energy, viscoplastic strain energy, and others.

Many times, the fatigue-controlling energy quantity is a nonstandard energy variable that requires a separate computation. This can be done in COMSOL Multiphysics, as demonstrated in the example of accelerated life testing, where the nonlinear material has two creep mechanisms. The first one controls strains at low stresses and the second one controls strains at high stresses. The fatigue, on the other hand, is controlled only by the energy dissipation caused by the creep development at high stresses.

The strain development as well as the energy dissipation by different mechanisms is calculated in individual distributed ODE interfaces:

*Model set-up for evaluating user-defined creep strains and energies using ODE interfaces (to the left). A comparison of the results between the user-defined constitutive relations and the predefined material model from the Nonlinear Structural Materials Module (to the right). The green line is the dissipated energy at low stresses, the red line is the dissipated energy at high stresses, the dotted turquoise line is the combined dissipation by both mechanisms, and the blue line is the dissipated energy calculated with the material model from the Nonlinear Structural Materials Module.*

Fatigue cracks are frequently encountered at interfaces of sharp geometrical changes and in corners. Those places are also well-known for causing numerical singularities. Thus, a point evaluation there can give misleading results.

Darveaux proposed a model that uses an energy volume average. This approach reduces the sensitivity to meshing in critical places and predicts life based on the surrounding state. In the figure below, we use the Darveaux model to predict fatigue life based on the dissipated viscoplastic strain energy in a ball grid array.

*Fatigue life based on the average dissipated creep energy. All joints in two ball grid arrays are analyzed in a full model on the left-hand side, and to the right, a detailed study of the critical solder joint in a submodel is shown.*

At first, all solder joints are analyzed in order to identify the critical one. Then, the critical joint is reanalyzed in a detailed study using a submodeling technique described in a previous blog post. The fatigue life in the thin layers at the interface with other materials, where cracking is expected, is finally predicted. Since the model evaluates a volume average, the results are calculated per domain.

We can evaluate the Coffin-Manson model with different strain options in the *Strain-based* fatigue feature. The Morrow and the Darveaux models with different energy options can be evaluated using the *Energy-based* fatigue feature.

To wrap this up, I’d like to share a few examples where the thermal fatigue of nonlinear materials is simulated:

- The Thermal Fatigue of a Surface Mount Resistor model demonstrates how to perform a fatigue evaluation based on creep strain and the dissipated creep energy using the Coffin-Manson and Morrow type relations.
- In the Energy-Based Thermal Fatigue Prediction in a Ball Grid Array example, a microelectronic microchip containing several viscoplastic solder joints is analyzed. The fatigue life is based on the Darveaux energy volume average. This model also demonstrates how to analyze large models using the concept of submodeling.
- Fatigue life prediction, based on a more exotic energy and strain representation, is modeled in the Accelerated Life Testing example. Here, a material behavior with two creep mechanisms is evaluated and fatigue life, based on one mechanism, is predicted. The separation of strains in the two mechanisms requires recalculation of individual strains using separate ODE interfaces.

You can find all of these example models in the Fatigue Module Model Library.

*If you are interested in learning more about fatigue modeling in nonlinear materials, join the Nonlinear Structural Materials Modeling and Fatigue Evaluation webinar on May 15 ^{th}.*

The idea behind critical plane models is that failure is caused by a crack. The crack will form and run on a plane, a *critical plane*, that has the most favorable stress/strain conditions for either crack growth, crack propagation, or both events. Planes that experience the highest normal stresses and strains are usually good candidates for a critical plane.

The stress state in a point in a structure can be described with a two-dimensional tensor with three normal and three shear components. The magnitude of those stresses changes once the examined volume element is oriented in a different direction. This means that if we make a cut through a volume element and evaluate stresses on the newly created plane, its stress state will change depending on the orientation. In case of plane stress conditions, the stress state reduces to two normal stresses and one shear stress that also differ depending on the surface normal.

*Stress conditions on different planes.*

A plane in a volume element has one normal and two shear stress/strain components. A critical plane model utilizes those stress/strain components to define the critical plane in its own specific way. For example, the *Normal stress* criterion considers a plane with the largest normal stress range, the *Findley* model searches for a plane where the combination between the normal and the shear stress ranges is maximized, while the *Matake* criterion, on the other hand, evaluates planes with the highest strain range. From the picture above, it is clear that all of the planes have different orientations.

For plane stress conditions, the critical plane can be obtained with analytical expressions. The situation becomes more challenging when the load is non-proportional and the stress state is multiaxial. We must then search for the critical plane numerically and evaluate the load history in each examined plane orientation. This is done in the Fatigue Module with the *Stress-based* and the *Strain-based* models.

*Fatigue evaluated with critical plane criteria: Normal stress, Findley, and Matake. Note from the editor, 2/24/14: This image has been updated with results from COMSOL Multiphysics version 4.4.*

In the Stress-based models we can calculate *Normal stress*, *Findley*, and *Matake* criteria. These are evaluated according to the fail-safe philosophy — calculating the *fatigue usage factor* that determines whether the experienced fatigue load is above or below a fatigue limit. The material parameters for those models can be easily calculated from the results of standard fatigue tests. *Stress-based* models are usually used in the high-cycle fatigue domain where plasticity is very limited.

The *Strain-based* models evaluate strains or combination of strains and stresses when defining a critical plane. Those models can be seen as a modified, combined Basquin and Coffin-Manson strain-life relation. They predict the number of cycles to failure. In the Fatigue Module, there are three *Strain-based* models: *Smith-Watson-Topper (SWT)*, *Wang-Brown*, and *Fatemi-Socie*. Those models are suitable for low-cycle fatigue where strains are usually large.

I’d like to share three examples of fatigue evaluation based on the critical plane evaluation. Two of them evaluate high-cycle fatigue, and the last low-cycle fatigue. You can find these in the Fatigue Module.

- The Cylindrical Test Specimen model evaluates all three stress-based criteria on a non-proportionally loaded cylindrical test specimen.
- The Structural and Fatigue Analysis of a Shaft with Fillet model demonstrates how to perform high-cycle fatigue. It also show how to calculate fatigue material data based on fatigue tests in reversed axial tension and pure torsion.
- The Low Cycle Fatigue Analysis of Cylinder with a Hole model shows how to perform fatigue study in presence of plasticity. In such a case it is important to first obtain a stable load cycle before a fatigue analysis can be performed.

*If you’re interested in learning more about fatigue prediction modeling, watch the archived webinar Fatigue Modeling with COMSOL.*

Random loads introduce a variety of stresses, with different magnitude, into a structure. It is therefore important to identify overall trends in the stress history. *Rainflow cycle counting* is a popular method to transfer the variable load history into a discrete stress distribution that is characterized by certain mean stress and stress amplitude. In COMSOL Multiphysics, the stress distribution of the Rainflow counting is visualized in a new plot type, called *Matrix Histogram*.

*Stress distribution based on the Rainflow cycle counting method.*

A classic way of obtaining fatigue life is via the S-N curve. It relates the stress amplitude to the number of loading cycles a material can withstand. In variable loading however, the stress amplitude is not constant and, instead, you must use an alternative model that calculates damage contribution of each cycle. You might use the Palmgren-Miner linear damage rule, a widely used method, to capture this. In the Fatigue Module, the Palmgren-Miner rule processes the stress distribution of the Rainflow counting and relates it to the limiting S-N curve. In order to capture the mean stress effect, so that damage increases with the increasing mean stress, the S-N curve is specified with an argument for the R-value.

The fatigue analysis consists of two steps. First, you calculate the structural response of a load cycle. Next, you perform a fatigue evaluation. When the number of load events is large in a random load analysis, the simulation of the load cycle is time-consuming, but the calculation time can be greatly reduced if the nonlinear effects are not present in the simulation. In that case, the stress cycle can be prescribed with help of superposition. This is selectable with the *Generalized loads* analysis type in the Cumulative Damage feature. There, the load cycle is not prescribed load-step by load-step, but instead the history of an external load is decomposed into few generalized loads with corresponding load histories.

*The external load simulated using three generalized loads and corresponding time histories.*

The Cumulative Damage calculation, based on the generalized loads, can be summarized in following steps:

- Define generalized loads
- Prescribe generalized loads in a structural study
- Compute structural response to generalized loads
- Define load histories for all generalized loads
- Prescribe load histories to corresponding generalized loads
- Compute fatigue analysis

The first three steps are done in a structural prestudy, while the last three are done in a fatigue study.

I’d like to share two examples of simulating Cumulative Damage, with you. Both can be found in the Fatigue Module. In one of the examples, the load cycle is prescribed step-by-step, and in the other one, superposition is used via the Generalized loads option.

The “Frame with Cutout” example uses the Generalized loads option. Here, the fatigue response to 1,000 load events is simulated. The total computation time with the Generalized loads option is 8 minutes, while the load event by load event calculation takes 1.5 minutes for each load event, thus the total load cycle would require a full day of computation time. Moreover, large amount of data needs to be saved in order to be processed by the fatigue study. With the Generalized loads option you don’t need to spend this much time on your computations.

The model “Cycle Counting in Fatigue Analysis — Benchmark” compares results of the Rainflow counting against an ASTM standard. The results based on the Palmgren-Miner are compared against hand calculations.

*If you’re interested in learning more about fatigue modeling, tune into our Fatigue Modeling with COMSOL webinar on July 30 ^{th}, 2013.*

Before we go into the infamous failures of fatigue, let’s come to terms with what the phenomenon is. Guest author and Certified Consultant, Kyle Koppenhoefer of AltaSim, explains it well in his blog post “Fatigue — When Cyclic Stress Produces Fractures“. Basically, fatigue is structural damage accumulated over many cycles of applied stress. “Many” in this case refers to thousands (strain-based; low-cycle) or millions (stress-based; high-cycle) of cycles. Furthermore, the fractures that develop due to repeated loading and unloading can be microscopic in size until the day the structure fails. Scary stuff.

Folklore has it the Boston neighborhood of North End smells like molasses in the summer. In 1919, the poorly designed tank containing molasses undergoing fermentation burst one day, resulting in a 35 mph, sticky flood of molasses rushing through the city streets. Why did the tank all of a sudden break? Apparently, several factors played a part in this structural failure. In addition to the tank being ill-constructed and not tested, there was a huge change in the outside temperature over the course of a day or so (it went from -2F to 41F, or -17C to 5C). The fermentation process produced carbon dioxide, which increased the internal pressure of the tank, putting it under additional stress. These factors may have led to the final failure, but the tank had experienced cyclical loads over several years. Reportedly, a fatigue crack had developed at a manhole cover close to the tank’s base where the hoop stress is greatest. When the tank was filled with molasses completely on this fateful day, it simply broke.

Other infamous failures include an oil platform, trains, and a multitude of airplane tragedies. The Alexander L. Kielland oil platform capsized due to a fatigue crack in a 6 mm fillet weld, increased stress concentrations due to a weakened flange plate, and cyclical stresses. The Versailles train crash of 1842 was caused by a broken axle in the locomotive, from repeated loading.

While it’s obvious that fatigue is a problem not to be ignored when designing a structure, it’s also obvious that testing millions of load cycles would take a very long time — far too much time. Even so, wouldn’t it be great if you could visualize what starts to break, where, and when, *before* it actually happens? Our answer to this is the Fatigue Module; you can visualize fatigue life based on the number of cycles until failure and the calculation of fatigue usage factor. The fatigue life plot will show you what starts breaking and where — virtually, thus saving you both time and money. An excellent blog post on why you should model fatigue will provide you with more details on the topic.

*Fatigue life plot of a cylinder with a hole.*

Next time you’re in Boston you can visit the area where the molasses disaster took place, although these days it has been turned into a recreational complex called Langone Park. The only evidence left of the disastrous event today is a small plaque in commemoration. Unless you can smell the molasses, of course.

This was one of the most interesting keynotes I had ever heard, and was presented by Dr. William Vetterling — of the *Numerical Recipes* fame. Dr. Vetterling now works for ZINK Imaging, which is a spin-off from the Polaroid Corporation, and has invented an original way for providing instant photographs directly from a camera. As you can see from the above video, he is an avid COMSOL user and has taken advantage of its many applications throughout ZINK’s manufacturing and design processes. This includes modeling heat transfer, free-surface fluid flow, structural mechanics in dried paper, electrostatic-charge removal, and the bending of a blade support. View his slides here.

Moving on to the novel part of the keynote presentation, *The Library of Babel* is a short story by Jorge Luis Borges, where all the information in the universe is to be found in 410-page books in this library. However, the information is interspersed with books containing every possible permutation of letters, punctuation marks and spaces that exists. A Shakespearian classic can be found there, just as a version of it can be found with one spelling mistake. As it holds all the information in the world, there is a book there containing the cures for all the diseases in the world.

The library is not infinite as it builds on the finite characters available, such that the librarians there feverishly leaf through the books, one after another, trying to find the ones that would transform or benefit humankind.

After describing the Library of Babel, Dr. Vetterling went on to describe how COMSOL Multiphysics is similar to it, how users of the software are like the librarians. This is a fascinating analogy, and I encourage you to find out how the two are similar by watching the video.

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*The Fatigue Module provides Findley, Matake, and Normal stress methods for high-cycle, or stress-based, fatigue and the Smith-Watson-Topper, Wang-Brown, and Fatemi-Socie methods for low-cycle, or strain-based, fatigue. In strain-based fatigue, Neuber’s rule and the Hoffmann Seeger method are available for approximate solutions of the fatigue of elastoplastic materials. In combination with the Fatigue Module with the Nonlinear Structural Materials Module, it is also possible to consider complete elastoplastic fatigue computations. Enlarge image.*

Just before the release, I was on the premises of one of COMSOL’s Certified Consultants, Veryst Engineering. There I had a great conversation with Stuart Brown who has a lot of experience with structural mechanics applications and their modeling. I asked the question “why should you simulate fatigue?”, and his simple answer was that your average doctoral student does not have the time to wait for fatigue testing. Particularly for the millions of cycles that need to be run in stressed-based fatigue.

Companies, of course, still do run these test, specifically to verify their simulations (have you seen the machines that continually open and shut drawers and doors in your local IKEA store?). Yet another reason mentioned by Stuart is that often manufacturers don’t have the time to wait for a full-scale experimental testing of fatigue to be performed. The product needs to go into service as soon as possible for a variety of reasons, such as the fact that the usefulness of the product means it should be deployed immediately.

Although regulatory constraints still require testing, the design cycle does not have the luxury of waiting around for ten years of testing to see when these devices should be replaced before failure. Design decisions have to be made with the extensive assistance of simulations. A good example of fatigue analysis is this model of a cylinder with a hole. Because the stress-levels do not tell the entire story in this scenario, the structural damage due to repeated loading and unloading must be analyzed.

We have a blog entry from our very first guest author, Kyle Koppenhoefer of AltaSim Technologies, with a great description of what fatigue actually is. Something I noticed when reading his entry was that he used an application from biomedical engineering to illustrate the concept; it’s not only large structures and furniture that are affected by this phenomenon, but biomedical devices too.

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