The invention relates to materials science and engineering, and in particular to the prediction of the service life of a metallic structure exposed to cyclic loading.
All structures and machinery components undergoing fatigue loading are prone to crack formation and its subsequent growth that increases with time. When a crack is formed, the strength of the structure or the component is decreased and can no longer function in the intended manner for which it was designed for. Moreover, the residual strength of the structure decreases progressively with increasing crack size. Eventually, after a certain time the residual strength becomes so low that the structure fails [1]. It is, therefore, of paramount importance to be able to predict the rate of decline in the component's residual strength and the remaining life of the system.
Fracture mechanics is a branch of science that provide insights into the mechanism of failure and help predict the service life of structures and machinery components [1]. As depicted in
In order to make a successful use of fracture mechanics in an engineering application, it is essential to have some knowledge of the total field shown in
The thermoelastic effect, which governs the relationship between the temperature variation and stress (or strain) change in the elastic range, has been well documented, and has been utilized to characterize the elastic stress field. Different means—such as thermocouples, thermistors, and thermography techniques—have been employed to monitor the temperature changes during mechanical tests [3-6]. The thermoelastic stress analysis by thermography is now an advanced full-field stress measurement method. In materials undergoing cyclic loading, most of the dissipated energy due to hysteresis effects manifests itself as heat, and the heat is removed from the material by heat transfer.
Heat can be transferred by three processes: conduction, convection, and radiation. Conduction is the transfer of heat along a solid object. Convection transfers heat from the “wetted area” of a solid through the exchange of hot and cold molecules, e.g., air, water, etc. Radiation is the transfer of heat via electromagnetic (usually infrared, IR) radiation. Although these three processes can occur simultaneously, it is not unusual for one mechanism to overshadow the other two. If the fatigue experiment is rapid enough, which is generally true for low-cycle fatigue testing, the temperature rise can be surprisingly high. For fatigue tests at 1,000 Hz, for example, the temperature could increase 200° to 400° K. above the initial temperature, depending on the material tested and specimen geometry [3, 4].
The temperature evolution resulting from the heat generated during the fatigue process is utilized to monitor the fatigue-crack propagation [5-8], to measure the energy required to produce a unit area of a fatigue crack by propagation [8], to determine the endurance limit of some materials [10, 11], and to characterize the evolution of cumulative damage in the fatigue process [3, 4, 12, 13].
In the present invention, a novel approach of nondestructive thermographic technique is used to characterize the fatigue behavior of metals. Specially, laboratory tests were conducted with Aluminum alloy and Stainless Steel undergoing cyclic bending and torsion loads. The same trend is expected to persist in multiaxial loading involving the combination of bending, tension and compression as well as torsion. In the laboratory tests, detailed temperature distributions on the specimen surface, and temperature changes as a function of time (cycles) were obtained. A two-dimensional form of a thermal-mechanical coupling model for a low-cycle bending fatigue was formulated to ensure the validity of the experimental results and to provide insight into the complex fatigue behavior. The results of the experimental and analytical works were used to develop a new method for predicting the fatigue life. The predictions of temperature changes during fatigue were found to be in good agreement with the experimental results.
In materials undergoing cyclic loading, most of the dissipated energy due to hysteresis effect manifests itself as heat and causes an increase in the mean temperature. An abrupt temperature rise in the first few cycles, followed by a steady state in later cycling, is a characteristic of metals that undergo the high-stress level fatigue testing.
In particular, we have determined that slope of the temperature-versus-time curve at the beginning of the test can be effectively utilized as an index for fatigue life prediction. This invention is expected to be applicable for the axial tension/compression loading and torsion of solid specimens of variety of shapes, as well as a thin-walled tube. Therefore, a temperature sensor, either contacting (e.g., thermocouple) or non-contacting (e.g., fiber optic, IR camera), can be used to measure the surface temperature of the specimen under cyclic loading. Test results obtained using the invention used a non-contacting sensor. In this arrangement, the need for measuring the dissipation energy due to plastic deformation from the hysteresis loop is eliminated. Also, this invention can provide an early prediction of the service life of machinery components under cyclic loading. The material properties and thermal boundary conditions are the input parameters and the service life time of the specimen is the output. Furthermore, for a system already in service, this device enables us to determine the remaining life.
Laboratory experimental results conducted at the Center for Rotating Machinery at Louisiana State University have confirmed the validity of this invention for the case of cyclic bending and torsion loads. A thermographic technique that utilizes an IR-camera (i.e., non-contacting method) was used to measure the temperature increase in the specimen due to hysteresis heating during fatigue testing. Similar results can be obtained using fiber optic temperature sensors where temperature can be recorded from a machine remotely.
A miniature electronic chip may be attached to the surface of a specimen under cyclic loading to measure its temperature and process the data to predict the onset of catastrophic failure. This device will be capable of measuring the slope of the temperature curve at the very early stages of the cyclic loading and rapidly estimate the specimen's fatigue life. For a new component, this information would pertain to the fatigue life; for an existing machine in service, it would provide estimate of the remaining life. This instrument provides a very fast and reliable method for the determination of the service life of the machinery components under cyclic loading and torsion. In practical applications, wireless technology provides compact, lightweight, reliable data transfer from the device that can be remotely monitored and processed in real time to predict the number of cycles for fatigue failure. An illustration showing the use of a wireless sensor and a data acquisition unit is shown in
The surface temperature is related to the number of cycles to failure. In particular, we have determined that the slope of the temperature curve at the beginning of the test can be effectively utilized as an index for fatigue life prediction. Using this technique, the remaining life of a machine can be predicted and catastrophic failure can be avoided. This technique can be applied by installing a sensor, in-situ, and testing the component while it is undergoing the fatigue load. The life expectancy estimate is obtained while the object is in use, which provides an advantage over techniques that require stopping the operation of the machine.
It is expected that this invention is applicable to other types of loadings like axial tension/compression loading, repeated bending and torsion of thin-walled tube.
Also, it is expected that a miniature wireless temperature sensor attached to the surface of the component under the fatigue load can be used to take data and the results can be monitored remotely. In this fashion, temperature of the component is collected and at the same time transferred to a signal receiver. Other types of temperature sensors either contacting (e.g., thermocouple) or non-contacting (e.g., fiber optic, IR camera) can be used to measure the temperature of the surface of specimen under fatigue loading.
A data analyzer unit to convert the temperature data to fatigue-life span. Such a unit could calculate the slope of the temperature evolution at the beginning of the test and then convert the data to determine the service life of the component. This device may consist of an integrated electronic circuit which is programmed to process the temperature data. Processing the temperature data by this device may be carried out at the same time as the component is under the fatigue life. This type of arrangement is illustrated schematically in
In a preferred embodiment, the present invention is a method of predicting the service life of an object subject to cyclic loading, including monitoring the surface temperature of the object; determining the slope (Rθ) of the increase in surface temperature of the object during a first phase, wherein the first phase is defined as a period of relatively rapid increase in surface temperature of the object; and, predicting the service life of the object in number of cycles to failure (Nf) using the equation:
2Nf=c1Rθc
An extensive experimental program has been performed that involves testing pertinent types of materials of interest. These materials are used in systems that experience bending fatigue, torsion fatigue, tension/compression, as well as combined mode. A fatigue testing apparatus was used to apply cyclic bending and torsion load. The specimens used were fabricated from Aluminum and Stainless Steel. A full field surface temperature was monitored by means of an infrared camera.
The fatigue testing apparatus used is a compact bench mounted unit with a variable speed motor, variable throw crank connected to the reciprocating platen, a failure cut off circuit in a control box, and a cycle counter. An available option is a torsion and axial tester.
Torsion Fatigue.
The results of a series of bending fatigue tests are shown in
This empirical analysis revealed that a material undergoing a fatigue test is subjected to an increase of the surface temperature, such that the higher the applied stress amplitude the higher the temperature increases. We have established that (
The first phase of the temperature increase is limited to a very low number of cycles compared to the number subsequently required to reach failure (in general, in order of 10% of the entire lifespan of the specimen for loads not close to the yield stress). The second phase, of “stabilized temperature”, varies considerably. For applied loads close to the yield stress this phase is extremely limited, while for loads only slightly above the fatigue limit (σ0), it extends over almost the whole lifespan of the specimen. For loads greater than the fatigue limit, the rate of temperature increase with the number of cycles in phase 1 and the stabilization temperature in phase 2, are higher the greater the load with respect to the fatigue limit. This phenomenon is schematically shown in
Based on this observation, we have developed a fatigue failure criterion based on the rate of change of temperature in the Phase 1 of the fatigue life curve. For a given set of materials parameters and the initial slope of the temperature-cycle curve, we can readily predict remaining fatigue life.
Thermal Analysis. A thermal analysis was carried out to predict the temperature of the specimen under the fatigue bending load. A two-dimensional heat conduction model was developed to analyze the problem. The present model is restricted to an isotropic bar with constant thermal conductivity. It was analytically treated using the integral transform technique [14]. The solution of the temperature distribution inside the bar is found to be as:
where T(x, y, t) is the temperature at point (x, y) and time t.
A more detailed description of the thermal analysis follows. We modeled the fluctuating beam with a finite rectangle 0≦x≦a, 0≦y≦b initially at room temperature, θ0. For time 0<t heat is generated within the solid beam at a rate g(x, y, t), while the boundary conditions are shown in the model below.
The governing equation, boundary and initial conditions for this problem are as follow:
where T=θ−θ0 is the temperature difference at each point and surrounding temperature. The solution of the present problem is given by [14] as follow:
K(βm, χ) and βm are kernel and eigenvalues along x coordinate; and, K(vn, y) and vn are the kernel and eigenvalues in y direction.
If the heat generation term, g(x, y, t), is only a function of space variable, i.e. g(x, y, t)=w(x, y) one can reduce Eq. (7) to the following:
Therefore, the final solution could be found from:
θ(x,y,t)=T(x,y,t)+θ0 (13)
The heat generation term is assumed to be constant with time and varies only with spatial coordinate. The heat generation term inside the solid is associated with plastic deformation and can be calculated from the hysteresis loop. The dissipation energy during fatigue manifests itself as heat and causes an increase of the mean temperature. The dissipated energy density is equivalent to a constant heat source distributed in the specimen. Because the elastic stress does not contribute to the increase in the mean temperature, the elastic-stress field in the specimen could be neglected. The heat generation is mainly due to the inelastic (plastic) deformation.
It is to be noted that the room temperature is assumed to be constant and does not vary with space and time. Also, the convective heat transfer coefficient, h4=h3, over the fluctuating beam is considered to be constant. It should be mentioned that one expects that the free end of the beam where the vertical displacement is greater than the clamped end, naturally experiences greater heat convection. However, it is a good approximation for the fixed end of the beam where the air over the surface is stationary and heat transfer is due to natural convection. The radiation heat transfer from the surface has been neglected. The heat generation term inside the solid is associated with the plastic deformation and is assumed to be constant during fatigue life as shown experimentally by Marrow [15]. The heat generation is mainly due to the inelastic deformation. In mathematically analyzing the hysteresis loops, it is best to discard the elastic strain and deal solely with the plastic strain. In this analysis, the heat generation term was calculated using the expression derived for plastic strain energy per cycle Δw, derived by Marrow [15]:
where σa is the stress amplitude, n′ is the cyclic strain hardening exponent, ε′f and σ′f are cyclic ductility and strength of the material, respectively. The stress amplitude was found from the following linear elasticity relation:
where M is the momentum and I is the second moment of inertia. Specimens for use in cantilever-beam loading are fabricated with tapered diameters proportioned to produce nominally constant stress along the test section. However, in our analysis, we assumed that the dissipated energy is equivalent to a constant heat source uniformly distributed in the specimen. Hence, the temperature distribution inside the solid bar can be found from the following:
It should be mentioned that in mathematical analysis the elastic strain was discarded and plastic strain was solely taken into account. Therefore, the simulation is generally acceptable for low-cycle fatigue tests. On the other hand, in aforementioned equation (Eq. 15) which describes bending stress, a simple linear elasticity relationship was used. This assumption was made to simplify the analytical model. In the absence of any available formulation for plastic deflection of a cantilever beam, this assumption is considered to be adequate for mathematical modeling, since the main objective of presenting the analytical thermal analysis is to better understand and cross check the experimental trends.
The surface temperature of the specimen suddenly increases just before the specimen reaches failure. This is due to occurrence of a macrocrack at the fracture point. When the macrocrack occurs, the plastic deformation at the crack tips is large. Thus, the larger the plastic deformation, the larger energy dissipation and temperature rise become.
Technology for determination of fatigue life. Based on our experimental observations, a material undergoing a fatigue test is subjected to an increase of the surface temperature such that the higher the applied stress amplitude, the greater the slope of the temperature rise. We have determined that (
In the present invention, the slope of the temperature evolution curve during the first phase of the test (Phase 1) was effectively utilized to predict the fatigue life of the component. Both experimental tests and analytical predictions show that the number of cycles for failure, Nf can be correlated to the slope of the temperature curve, Rθ, as follow:
2Nf=c1Rθ2
where c1 and c2 are constants and dependent on the material properties and thermal boundary conditions. Experimental work of Morrow [15] showed that the value of c2 for most metals is almost constant. We confirmed that the value of c2 is almost identical for both several Aluminum and Stainless Steel samples (
Once material properties and type of load are known, fatigue-life of specimen can be found at very beginning of cyclic load. The test procedure disclosed herein may be used to determine the values of c1 for a wide variety of materials and load conditions. As expected, increasing the slope, results in a decrease in the fatigue life, since higher temperature slope of the temperature rise during phase one corresponds to a larger applied load and consequently lower life duration. The correlation presented in
Determination of remaining life of a machine. This invention provides a technique to predict the remaining service life of a structure already undergoing fatigue load. In the experiments conducted, we utilized specimen that were initially intact. If the material undergoing fatigue test is initially damaged or has experienced loading, its behavior under the fatigue test is different from the intact material.
Tzou [16] studied the thermal response of the solids which have existing cracks in their structures. He concluded that in companion with the degradation of elastic moduli due to microcracks, degradation of thermal conductivity may significantly increase the temperature established in the solid. Due to large volumetric strain developed in the neighborhood of a macrocrack tip, for example, the number density of microcracks dramatically increases in the strain history. The resulting degradation of thermal conductivity in the local area prevents heat from dissipating into the surrounding media and a localized temperature spike is thus found. He proposed a relation between thermal conductivity (K) of the solid and the damage parameter (D) as follows:
where Cd is the microcrack density parameter, v the Poisson's ratio, and (Ko) is the intact value of thermal conductivity.
Considering the governing equation of heat conduction, Eq. (3), the temperature rise (or slope of the temperature) can change by changing the value thermal conductivity due to microcrack initiation and propagation. Therefore, with the methodology reported in our invention, i.e. by measuring the slope of the temperature curve, during phase one, we are able to determine the integrity of the material and consequently estimate the remaining fatigue life of an existing structure or component.
Accelerated testing technique. This invention provides a rapid and effective technique for conducting accelerated testing for evaluation of degradation in materials. Development of an accelerated testing methodology has long been a major challenge in applied science and technology. It refers to the development of an enabling technology for predicting the long-time range behavior of a structure or a component based on tests that are conducted over a short length of time. Traditional life data analysis involves analyzing times-to-failure data (of a product, system or component) obtained under normal operating conditions in order to quantify the life characteristics of a specimen, a system or a component. In many situations, for many reasons, such life data analysis (or times-to-failure data) is very difficult—if not impossible—to obtain. Two methods are available: usage rate acceleration and overstress acceleration. These methods have been used to obtain time-to-failure data at an accelerated pace. For products that do not operate continuously, one can accelerate the time it takes to induce failures by continuously testing these products. This is called usage rate acceleration. For products for which usage rate acceleration is impractical, one can apply stress(es) at levels which exceed the levels that a product will encounter under normal use conditions and use the times-to-failure data obtained in this manner to extrapolate to use conditions. This is called overstress acceleration.
The method reported in our invention would be applicable to any mechanical machine application that has cyclic fatigue crack initiation and crack propagation potential to determine useful life. Since over 90% of mechanical failures are due to fatigue damage, the present invention will be useful in many industries which employ dynamically loaded mechanical systems. Of particular significance are applications in the aircraft industries, military, marine, automotive, sensitive components such as pressure vessels, and bridges/structural applications where fatigue failure is a major concern.
Proposed Method for Determining the Fatigue Life
Based on the experimental observation, surface temperature of a metallic object undergoing a fatigue test experiences three distinct phases: an abrupt rise in temperature (phase one), followed by a steady state trend (phase two), and finally a sharp increase in temperature immediately prior to failure (phase three).
Using the thermal model discussed above, equation (19) gives us transient temperature distribution inside the bar as follow:
By setting y=b or y=0 we can find the temperature variation over the beam surface. Making the time derivative of Eq. (19), gives the expression for the rate of temperature rise over the top surface of the beam. On the other hand, we can find a relationship between the heat generation term in the above equation, w(x, y), and the number of cycles for fatigue failure, as proposed by Marrow [15]:
In this equation all the parameters except
Eventually, the number of cycles for fatigue failure can be expressed as a function of slope of the curve:
In order to check the accuracy of our thermal analysis, we compared the experimental results and the predicted number of cycles for bending fatigue load.
Constants c1 and c2 in Eq. (24) are dependent on the material properties and thermal boundary conditions. Experimental work of Morrow [15] showed that the value of (b+c) for most metals is almost constant. Subsequently, c2 is identical for both Aluminum and Steel, as seen by the results shown in
The value of c1 for Aluminum in bending and torsion is 204 and 8120 respectively. The c1 values determined for Stainless Steel for bending and torsion are 14,102 and 183,830, respectively. Values of c1 can be determined for other materials and other loading conditions using the test procedure and analytical relationships set out above.
The value of c1 may be estimated using the analytical methods described above. In particular, equation 24 shows that
where all parameters are as defined above.
If the material properties and thermal boundary conditions are known, equation 25 may be solved for c1. When this was done for an Aluminum sample under bending load, the results were consistent with the empirically derived values. Similar calculations could be performed for other materials and loading conditions.
The c1 value is derived empirically by plotting the data points as shown in
The value for c2 used in this method can be varied, but it is preferred to begin with a value of approximately −1.22, which is the empirically derived value found for Aluminum and Stainless Steel subject to either bending or torsion loading. Because Aluminum and Stainless Steel have significantly different properties in many respects, the fact that both materials resulted in the same value for the constant c2 strongly suggests that other metals would have the same, or at least a very similar, value for c2.
After an initial curve fit analysis is done with the value of c2 set at −1.22, additional curve fits may be performed with different values for c2. The range of −1.10 to −1.35 was found to encompass almost all test results for the constant c2. This range, therefore, should be sufficient to perform additional curve fit analyses, in order to obtain the overall best curve fit using the empirical data plotted as described above. Through use of this method, the values of the constants c1 and c2 may be accurately determined for any material and any cyclic loading conditions. Once these constants are determined, equation 17 may be used to predict the service life. The present invention, therefore, is applicable to a broad range of materials and real-world conditions.
The value of c2 remains nearly constant for most metals and loading conditions. Aluminum and Stainless Steel are quite different metals in many respects, yet the c2 value for these metals is nearly constant. As
This application claims the benefit of U.S. Provisional Application No. 60/956,292, filed on Aug. 16, 2007.
Number | Date | Country | |
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20090048788 A1 | Feb 2009 | US |
Number | Date | Country | |
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60956292 | Aug 2007 | US |