We all are familiar with stress analysis plots on the surfaces of structural castings, with hot colors splotches indicating high stress and cool color splotches indicating low stress. Those splotches are actually parts of planes of equal stress that are cross-sections in the casting at various angles, depending on the load case, as shown in Figure 1. However, the colors of those splotches are meaningless without a reasonable value for Allowable Stress input into the stress simulation software.

When designing a forging, for example, from a uniform block or bar (not welded together), determining Allowable Stress is very easy. The metal integrity is very high, the microstructure for sustaining stress is pretty simple, and published mechanical properties are consistent and easy to find.

Not so for castings. Typically there is a lot of digging for applicable, hard-to-find data plus assumptions, guesswork, and safety factors in trying to determine a reasonable Allowable Stress value. Most often, and too simplistic, is use of the casting alloy’s yield stress plus a safety factor for unknown contingencies. Either approach has significant risk when relied upon as the basis of the stress colors plotted on the surface of the casting’s solid model. That risk is the unfortunate and expensive failure of prototype castings in durability testing. It is obvious that the stress plot colors need to be as realistic as possible.

The idea of Fracture Toughness applies well to casting structural design, eliminating the data digging, minimizing assumptions, and enabling smaller safety factors. Fracture Toughness as the Allowable Stress basis for castings is much closer in accuracy to the relatively simple wrought metal scenario. How to use Fracture Toughness to determine Allowable Stress in structural casting design is our subject.

Definition of Terms

Stress in structural design is a word used more casually than it should be. The stress that causes structural components to fail is the same stress that causes tensile test bars to fracture, as shown in Figure 2. 

Accurately defined, it is “Transformed Stress,” and for ductile metals, specifically, it is “Transformed Shear Stress.”  Almost all casting alloys are ductile, and only the brittle gray iron and white irons have a different form of “Transformed Stress.”  In the software used to simulate stress in a ductile alloy casting design, the term for “Transformed Shear Stress” has become von Mises Stress. They are synonyms.

Defects in castings is another word used more casually than it should be. Castings have “imperfections,” which is a better word. Even better is a word coined by the Steel Founders’ Society of America, “Quality Detail,” defined in Figure 3. From a structural or even cosmetic perspective, a “flaw,” “defect,” or “imperfection” is none of those if a design specification is not violated. So, “Quality Detail” implies that a detail in the surface, near surface, or interior of a casting may or may not be a consideration in design and quality acceptance.

Quality Detail is an important idea in Allowable Transformed Shear Stress because some Quality Details would lower the value of Allowable Transformed Stress. In the definition of Fracture Toughness, the Quality Detail is a pre-existing crack, which is a very serious, detrimental Quality Detail.

So, in the context of both an introduction and definition of terms, the concepts of Transformed Shear Stress and Quality Detail are important to defining Allowable Stress in a new casting design. The consequence of a significant Quality Detail is shorter cyclic life. In structural casting design, being able to account for the possibility of a detrimental Quality Detail in stress simulation is really important. That means that the software input value for Allowable Transformed Stress that considers Quality Details is fundamental to the reasonableness of stress simulation.

Introduction—Fracture Toughness: What and Why

Fracture Toughness is a facet of Fracture Mechanics. In Fracture Mechanics, K is the Stress Intensity Factor that characterizes a crack in a material. K is fundamental to the physics of crack propagation (Figure 4), and the change in K per load cycle, ∆K, is defined as Crack Driving Force (Figure 5).   

Shown in Figure 6 is the version of the Stress Intensity Factor equation, K, that defines Fracture Toughness, in general, KC . 
Fracture Toughness, K1C, is a specific version of Stress Intensity Factor, and K1C is measured from specifically designed test coupons. K1C, a specific test value based on the test coupon thickness, it is an actual material property. Values for Fracture Toughness K1C are a published property for many materials, including many casting alloys. For castings, that specific value depends on parameters beyond the chemistry of the alloy alone, such as cooling rate of solidification, microstructure quality, and heat treatment. Table 1 shows a few published values for A356 cast aluminum with some of the parameters that qualify the Fracture Toughness Value.

Fracture Toughness, K1C, is not well known in the broader spectrum of structural design, probably because it is applied in more narrow design applications, especially pressure vessels, steel bridges and steel building structures, and aluminum aircraft skins. 
If the stress level in a structural design can withstand a small, pre-existing crack in service, then the design’s safety is more certain. That’s the idea behind using Fracture Toughness in the ASME Boiler Code. 

When using Fracture Toughness more broadly in structural casting design, it is useful to understand how the material property K1C is measured. A close look at Figures 7 and 8 will help greatly to understand how the measurement test coupon is designed and how it is fractured to obtain the K1C material property’s value.

K1C is based on destructive testing of coupons made of the chosen casting alloy (in our case) with a notch to initiate a crack.  The coupon’s notch is fatigue loaded until a crack of a defined length is formed. Then the coupon with its defined crack length is overloaded until it fractures. The transformed shear stress to fracture the coupon becomes a value in the equation to calculate K1C.
Aside from the measured K1C value of Fracture Toughness, there is a range of values for the general Fracture Toughness parameter, KC. Those values for the same exact casting alloy with its same metallurgical parameters depend on the local casting thickness at a high load case cross-section, the surrounding transformed shear stress, and the size and position of a pre-existing crack at that high load case cross-section. 

For castings, the provision for a pre-existing crack is often too severe compared to more likely Quality Details, such as non-metallic inclusion pits on the surface or near-surface porosity. Those less severe Quality Details can be approximated by the design engineer using cracks of a short length and a position away from a cross-section edge.

Estimating Allowable Transformed Stress Using Fracture Toughness: Steps

1.    The first solid model: There are considerations in the design of “Castability Geometry” that are foundational to design geometry that the chosen alloy “likes” to flow into and solidify with specified integrity all by itself. How that is done is beyond our subject, but a resource can be found here1. On that well-chosen “Castability Geometry,” structural geometry that controls the load case can be overlaid. The secret to that successful overlay is the use of Area Moment of Inertia, which is a form of cross-section stiffness that easily controls the load case. Since the most powerful Area Moment of Inertia cross-sections also have a larger perimeter, those cross-sections provide large boundaries for heat transfer to assist solidification cooling.

All of that is beyond this subject, but once the solid model’s first iteration geometry is completed, begin the process of loading the model into stress simulation software:

A.    Find the published Fracture Toughness value for the chosen alloy, heat treatment, if any, and strength grade.

B.    Choose a Quality Detail type and size that could occur at a high load case cross-section. The basis of a Fracture Toughness value is a pre-existing crack, and in the casting arena that type of crack would be a hot tear. However, casting producers are on high alert for hot tears, and the likelihood of having a hot tear in a production casting is low. Hot tears are a problem in a few casting alloys, and they are prevented in production by careful tooling design and construction and manufacturing engineering of the mold process. If they should occur in production, they are watched for and easily detected and repaired by in-process weld rework.

Even a heat treat crack is less severe than the crack formed in the Fracture Toughness test coupon, so the design engineer can use engineering judgement to size and position the Quality Detail defined as a crack in the Fracture Toughness equation. Most common among Quality Details that the design engineer can classify and convert to a less severe crack are: (1) Heat treat cracks; (2) Non-metallic inclusion pits; (3) Near surface microporosity from solidification shrinkage or gas; or (4) Near surface larger shrinkage porosity.

C.    Rearranging the Fracture Toughness material property equation (the one based on the actual notched test coupon, subjected to fatigue forming the crack and then overloaded to fracture) as shown in Figure 9, calculate the Allowable von Mises Stress for the first solid model’s geometry. This is a simple calculation with just three values: The published value, K1C, the Quality Detail equivalent Crack Length, A, chosen by the designer—expressed in meters, and the Safety Factor, chosen by the designer. 

It is important to remember that K1C is measured with a test coupon 25mm thick. Most casting thicknesses at high load case cross-sections are less than 25mm thick. Recalling the thickness relationship in Figure 8, if a casting wall thickness is only 12mm, the value of KC from the actual von Mises stress can be higher than the published Fracture Toughness value K1C.

D.    Input the Calculated Allowable von Mises Stress into the stress simulation software, using the first solid model.

E.    Run the stress simulation and note the magnitude and cross-section positions of the highest von Mises Stress values.

F.    Now, using the General Equation for Fracture Toughness, Figure 10, calculate the actual value of Fracture Toughness, KC. If KC is lower than the measured, published value of Fracture Toughness, K1C, then the stress level simulated is safe for static stress (as well as an undetermined level of fatigue life), assuming an appropriate Safety Factor value was used.

As noted in C, if the casting wall thickness at a high load case cross-section is less than 25mm, then the value of KC can be safely higher than the K1C published value.  

The values for the [a/L] Factor, which are based on the size and position of the Quality Detail judged by the design engineer to be appropriate for Crack Length, a, can be determined from Figures 11 and 122. These are two typical equations, one for a crack in the middle of larger section and one for a crack at the edge of a larger section, both common in casting wall thicknesses.

For a 25mm or larger casting wall thickness, if a KC value is higher than the measured, published value of Fracture Toughness, K1C , then the geometry at the high von Mises stress cross-sections must be modified to lower the von Mises stress. For wall thicknesses less than 25mm, Figure 13 provides an engineering sense of scale for allowing a larger, safe value of KC.

On the other hand, if the actual KC is very low compared to K1C, then the geometry at the low von Mises stress cross-sections can be lightened to reduce mass. Generally that means making the revised geometry thinner and less stiff.  

The most powerful way to adjust stiffness, up or down, in casting structural geometry is the use of Area Moment of Inertia. Area MOI cross-sections especially helpful in casting structural design are the I-beam, hollow tube, and omega. All are very stiff, and they directly reduce stress. Since the most powerful Area Moment of Inertia cross-sections also have a larger perimeter, those cross-sections provide large boundaries for heat transfer to assist solidification cooling. That solidification cooling from the Area Moment of Inertia cross-section perimeter is how increased structural stiffness to control stress can avoid upsetting the fundamental Castability Geometry already established in the casting design.

2.    The second (possibly 3rd & 4th) solid models:

A.    The little images of the K1C test coupon design and the test method in Figure 7 are really important. Fracture Toughness measurement is really a fatigue test, starting with a machined V-shaped notch in the coupon, and then it is fatigue loaded at high strain to form a crack at the tip of the notch. Fatigue is continued until the crack grows to a defined length, about half of the test coupon thickness. Then the load on the coupon is increased dramatically until the overload causes the fracture.

That means that the actual KC value calculated for the first solid model already considers fatigue life. We just don’t know how many cycles of fatigue would translate from the K1C coupon test to this casting solid model.

The next solid model tweaks will home in on an acceptable value of actual KC in the context of the casting’s intended cyclic life.

B.    Software is readily available to simulate cyclic life, either fatigue or strain, coupled with the solid model’s geometry. In some cases the cyclic life simulation package is an “Add-In” to the solid model software suite. To enable the simulation, allowable von Mises Stress or Allowable Total Strain Amplitude values have to be input for selected numbers of cycles or strain reversals.

For fatigue, the input data looks like the curve in Figure 13 for a design life of 500,000 cycles. 

As always for castings, finding a curve like that for the exact alloy, heat treatment (if applicable), allowable Quality Details, and solidification integrity is very difficult. However, the Fracture Toughness method provides an easy solution that doesn’t require finding data. 

Using Fracture Toughness as an established material property means that the Allowable von Mises Stress based on K1C already includes some fatigue in its value. That means that the curve for fatigue life data input should be shallower and higher than its original curve in Figure 13, more like the tail end of those curves. That idea is illustrated in Figure 14.

Now, the “easy solution” from using Fracture Toughness can be understood. The casting designer can use engineering judgment to choose the shape and values from the Figure 14 curve. Those estimates can be input into the fatigue analysis software because the fatigue simulation results will be judged by the actual Fracture Toughness value, KC, calculated from the high stress regions in the fatigue simulation.

C.    At this point the solid model will have been revised based on the evaluations of the actual Fracture Toughness KC in comparison with the published Fracture Toughness property value, K1C. So, the structural geometry at the high load case cross-sections will have been adjusted to make KC lower than K1C , but not too low. Unnecessarily low KC values make the casting unnecessarily heavy.

The fatigue (or strain) life simulation has been completed based on the “flatter curve” guesstimated as shown in Figure 14. Assuming that the number of cycles (or strain reversals) simulated will be too many or too few, revise the “flatter curve” and repeat the simulation until the cyclic life predicted is at or slightly above the cyclic life objective. This iteration is illustrated in a concluding example.

The value of von Mises stress at that number of cycles (or reversals) on the revised “flatter curve” now is the final Allowable von Mises Stress, including a prudent safety factor.

D.    What has been accomplished is an estimate of Allowable von Mises Stress for a casting that has a real data foundation, not far from the quality of data that can be readily found for a forging.  Further, the fatigue basis of the published material property Fracture Toughness, K1C, enables the Allowable von Mises Stress estimate to consider the intended cyclic life of the casting. Given a thoughtful choice of safety factor to allow for remaining unknown contingencies, a casting design engineer can reasonably expect the component to be successful in durability testing.   

Estimating Allowable Transformed Stress Using Fracture Toughness: Example

To demonstrate the power of the foregoing to resolve the long-standing and continuing problem of estimating Allowable Transformed Shear Stress (von Mises) stress for casting design confidence, here is an example. The casting, shown in Figure 15, is a rear suspension component for an 18-wheeler truck, and the example will demonstrate the simplicity and reasonableness of using Fracture Toughness to assure that the casting will sustain the design intent of 1 million fatigue cycles. 

Today’s software to simulate von Mises stress and predict fatigue or strain life is very impressive. The algorithms to accomplish those simulations are elegant, fast, and accurate. But the results displayed as colors on the solid model surfaces of a structural design are dependent on one number. That number, the Allowable von Mises Stress, is input by the design engineer, and the realism of the results—those colors—is dependent not on the algorithms (which have been demonstrated to be accurate). The results and their colors are dependent on that one number. 

Without repeating all the foregoing principles and steps, the example will show that Fracture Toughness principles will provide that reasonable allowable stress number and do so quite simply.

The casting design shown in Figure 15 is intended as a steel sand casting made in ASTM A-148 Grade 90/60. The specified cast low alloy steel is 8630. In the suspension assembly, the casting is fixed at its center by welding it to an axle housing, and the suspension loads the casting at each end with a force of 23,340 Newtons. The force at each end is “down” only. There is no “up” force. The solid model, loaded into von Mises stress simulation software, with the loads and constraints is shown in Figure 16.
For the Allowable von Mises Stress input into the stress simulation software, use the published value for cast 8630 Grade 90/60 steel in the equation shown in Figure 9.

Figure 18 shows a reasonable von Mises Stress result for the first stress simulation run on the solid model.

The cyclic life design intent for this suspension casting is 1 million fatigue cycles. Since the von Mises Stress analysis shows the highest bending load cross-sections to be only slightly above the first Fracture Toughness-calculated allowable stress, a simulation of the fatigue life would be worthy.

As the fatigue life simulation is being evaluated, it is important to remember one of the major advantages of using Fracture Toughness as the basis for Allowable von Mises Stress in castings. Castings have Quality Details that can affect cyclic life, and the published property of Fracture Toughness, K1C, has as its most significant attribute a pre-existing crack.

Among possible casting Quality Details, a pre-existing crack on at a high load case cross-section’s surface is the worst. The Fracture Toughness test coupon for establishing K1C has a 12mm crack formed by high strain fatigue along the length of a 25mm thick test coupon. This fatigue crack formation is illustrated in Figure 19, and when the crack has propagated to the 12mm length, the test coupon is overloaded until the coupon fractures in two. Although not a K1C test coupon, also shown in Figure 19 is a coupon of medium thickness, about 12mm, with its final fracture surface. Shown in the fracture is the initial machined notch, the formation of the fatigue crack right behind the notch, and the final overload fracture surface. As an aside, notice the “shear lips” at the edges of the final fracture. Just like the “cup/cone” of tensile test coupon fracture, the shear lips demonstrate that in the fracture of ductile materials, like almost all casting alloys, the stress transforms to shear exhibiting 45-degree shear planes in the fracture surface. 

Since fatigue is a prominent attribute of the K1C value, it is reasonable to expect that the fatigue life of the von Mises stress simulation shown in Figure 18 would be longer than an Allowable von Mises stress not based on prior fatigue.  We see that expectation in the fatigue life simulation shown in Figure 20. It is longer than the 1x106 design intent.  

Now we can see the practical significance of the fatigue curves, blue dotted fatigue curve, and commentary about Figure 14. In Figure 21, we see that the Allowable von Mises stress originally based on a K1C value of 65 MPa-Meter1/2 resulted in a fatigue life longer than required. Now, Figure 14 has been updated with the real facts from the simulations, and the Allowable von Mises Stress may be able to be raised from 205 MPa to about 250 MPa. With that increase comes the opportunity to reduce the mass of the beam not only where the von Mises stresses are low but also at the highest load case cross-section. Repeated simulations are necessary, but now we can see the significant benefits of using Fracture Toughness principles for establishing Allowable Transformed Shear Stress (von Mises Stress).

Although the fracture mechanics principles and equations seem complicated, and the physics behind them IS complicated, using them for this purpose is not only far more accurate in structural casting design but simple to apply.