251 Chapter 10 Engineering Methods for Product Duration Design and Evaluation One of the primary tasks of product design for the environment consists of harmonizing the requisites of environmental performance with those of conventional design (functionality, safety, duration). To do this, methods and tools must be available to the designer that allow the evaluation and optimi- zation of design parameters determining a product’s performance (conven- tional and environmental) over its entire life cycle. The defi nition of strategies for extension of useful life and recovery at end- of-life is conditioned by several factors that limit their effectiveness. The evaluation of these factors is essential for a correct implementation of these strategies in product development. Being able to predict, in the design phase, the extension of a product’s useful life and the reuse or remanufacture of parts of it depends on the expected duration of components and on their residual life. As a consequence, any study of the environmental aspects of a product must include the consideration of parameters such as the predicted duration of a component, its resistance to the operating load, and the esti- mated damage suffered by it. Accordingly, this chapter briefl y treats certain signifi cant aspects of conven- tional design. In particular, after a short review of material fatigue and damage phenomena, attention is focused on the rapid methods currently used for the fatigue characterization of materials. 10.1 Durability of Products and Components Deterioration in the functional performance of products and their compo- nents, which greatly affects the possibility of applying the environmental strategies for extension of useful life and recovery at end-of-life, is princi- pally due to phenomena conditioning the properties of duration over time: • Structural obsolescence determined by the physical–mechanical deterioration of materials 2722_C010_r02.indd 2512722_C010_r02.indd 251 11/30/2005 1:47:56 PM11/30/2005 1:47:56 PM © 2006 by Taylor & Francis Group, LLC 252 Product Design for the Environment • Damage due to improper use or accidental events • Deterioration due to external factors and operating environment Contrary to what might be supposed, the durability of components and the constructional system (understood as their capacity to maintain the required operating performance) must not be maximized indiscriminately, but opti- mized in relation to the feasibility of using the product or reusing its compo- allow the judicious calibration of product durability (which determines the span of its physical life) in relation to: • The limits imposed on the effective useful life by the external factors previously defi ned (expressed by the span of replacement life) • The range and typology of intervention to be operated through useful life extension and end-of-life strategies A complete structural durability analysis directed toward the prediction of physical life of components requires the integration of several engineering tools and techniques, and large amounts of data collection and computation (Youn et al., 2005). Nevertheless, the durability of components and systems can be defi ned and quantifi ed with good approximation in the design stage, using established methods and mathematical tools for design for durability, the result of exhaustive studies on phenomena such as fatigue and damage. In this context, there are clear and simple rules of design for appropriate durability: • Design equal duration for components similar in terms of function- ality and intensity of use. • Design duration as a function of the product’s effective useful life. • Design heightened duration for components diffi cult to repair and maintain, and for those intended for reuse. • Design limited duration (as close as possible to the effective life required) for components needing substitution during use, and for those intended for recycling or disposal. With these premises, it could be appropriate to consider some aspects of conventional engineering design, paying particular attention to phenomena of performance deterioration (fatigue and damage), design for component durability, and methods for the evaluation of residual life. These are the basis of the modern computer-aided engineering design processes, developed to carry out design optimization for structural durability and aimed at realizing durable, manufacturable, and cost-effective products. 2722_C010_r02.indd 2522722_C010_r02.indd 252 11/30/2005 1:48:00 PM11/30/2005 1:48:00 PM © 2006 by Taylor & Francis Group, LLC nents. Graphs such as those shown in Chapter 9, Figures 9.2 and 9.3 can Engineering Methods for Product Duration Design and Evaluation 253 10.2 Fatigue of Materials Studies on material fatigue began in the nineteenth century, when, with the daily use of machines, tools, and vehicles, it was observed that working parts subjected to loads that varied over time were damaged and eventually broke, despite the fact that at no time during their use did the stresses reach the safety values determined using normal techniques for studying the resis- tance of materials. In particular, the earliest scientifi c investigations on fatigue behavior concerned railway structures. In his fi rst paper, the German engi- neer August Wöhler reported on the fatigue resistance of railway tracks, the fi rst attempt at a quantitative description of fatigue with the introduction of the concept of fatigue limit. The research undertaken by Wöhler between 1852 and 1870 produced an enormous quantity of data that he presented in graphical form, known as the Wöhler curve and still frequently used today (Wöhler, 1870). Researchers agree in describing fatigue as a localized phenomenon evolv- ing in four distinct phases: • Nucleation • Subcritical propagation of the defect • Critical propagation of the crack, which can be characterized using the theories of elastic, elastic–plastic, or completely plastic fracture mechanics • Unstable propagation The nucleation of the crack occurs in a critical zone of the component or specimen, characterized by an elevated value of local stress different from the stress value measured macroscopically on the same component. This is due to the presence of discontinuities in the material at the structural level (nonhomogeneities, microcracks) or geometric level (notches, irregularities). At the apex of the crack, the material is subjected to a localized plastic defor- mation. As the dimensions of the crack increase, there is a resulting decrease in the resisting cross-section with a consequent increase in the stress on the material. Large zones of plasticization lead to a decrease in ductility and a reduction of resistance. Thus, fatigue failure always has its origins in plastic deformations occurring at the microscopic level. According to the American Society for Testing and Materials, the phenom- enon of fatigue can be defi ned as that process that “triggers a progressive and localized permanent structural transformation in the material, when- ever it is subjected to loading conditions that produce, in some points of the material, cyclical variations in the stresses or strains” (ASTM E606–92, 2004). These cyclical variations, after a certain number of applications, can 2722_C010_r02.indd 2532722_C010_r02.indd 253 11/30/2005 1:48:00 PM11/30/2005 1:48:00 PM © 2006 by Taylor & Francis Group, LLC 254 Product Design for the Environment culminate in the presence of cracks or in the failure of the component. To study the fatigue behavior of a component it is, therefore, necessary to know the loading history, the characteristics of the material comprising the component, and the geometry of the component itself. 10.2.1 Loading History Design for component fatigue requires information on the time history of the loading the element will undergo. These loading histories are obtained using experimental techniques on preexisting components or on scale specimens. The stresses ␴ measured in this way must be representative of those to which the element under examination will actually be subjected. The time histories can be classifi ed as periodic or aleatory, following the scheme proposed in Figure 10.1. In general, actual loading histories are treated by arranging them in constant amplitude sinusoidal cycles using the Fourier series. Sinusoidal • ␴ max , maximum stress • ␴ min , minimum stress • s ss m min ϭ ϩ max 2 , mean stress FIGURE 10.1 Classifi cation of signals. 2722_C010_r02.indd 2542722_C010_r02.indd 254 11/30/2005 1:48:00 PM11/30/2005 1:48:00 PM © 2006 by Taylor & Francis Group, LLC loading is described using the variables in Figure 10.2: Engineering Methods for Product Duration Design and Evaluation 255 • ⌬ϭ Ϫ s ss minmax 2 , load amplitude • N, number of cycles 10.2.2 Design for Fatigue The theories of fatigue can be applied using three distinct approaches: • Design for infi nite life • Design for fi nite life • Design for critical dimensions of defects Of the three approaches, the fi rst is based on Wöhler’s theories. The under- lying hypothesis is that of the perfect integrity of the material (i.e., the absence of defects or cracks before loading) and that nucleation occurs after the application of the load. It is commonly used for metals, particularly steel, but it is not always applicable to other types of material. Using appro- priate damage hypotheses, it is also possible to determine the residual life. In the 1940s and 1950s, there was considerable development in the design of machines for fatigue testing. By allowing the application of greater loads, such devices made it possible to investigate the behavior of materials under more extensive regimes of plasticization. Since the phenomenon of fatigue is essentially expressed at a local level, it seemed more appropriate to describe this phenomenon through the use of strains rather than stresses. Experimental data were, therefore, represented in terms of stain ␧ versus number of cycles N. With this approach (design for fi nite life), it is possible to consider the effects of plasticity, and it is also more adaptable to variations in the test parameters. It is also more suited for application on different materials and different component geometries. However, it is more complicated to apply than the previous approach and requires greater processing power for the FIGURE 10.2 Representation of a dynamic load. 2722_C010_r02.indd 2552722_C010_r02.indd 255 11/30/2005 1:48:01 PM11/30/2005 1:48:01 PM © 2006 by Taylor & Francis Group, LLC 256 Product Design for the Environment elaboration of the data. Furthermore, given its more recent introduction, there is less data available in the literature. Also, here it is assumed that the material subjected to loading is perfectly integral with no initial defects and that the end of its useful life coincides with the formation of a crack. Conversely, the third approach (design for critical dimensions of defects) assumes that there are always internal defects present in every material and that their characteristic dimensions increase following the application of load. Therefore, a component’s useful life does not end when a defect arises but, rather, when this defect reaches critical dimensions. This approach, developed in the 1960s, led to the introduction of complex variables referring to fracture mechanics, such as the stress intensity factor (⌬K I ). This factor is a function of the orientation of the defects and of the dimensions and geometry of the part containing the defect. The growth of the crack under a variable load is usually described using diagrams of the type daրdN (velocity of crack growth) versus ⌬K I . Clearly, it is a considerable advantage to be able to assess components already damaged; however, this approach has the disadvan- tages of increased calculation times in that it requires nondestructive testing (NDT) in order to evaluate the effective dimensions of the defects present in the component. 10.2.3 Infi nite Life Approach Design for infi nite life developed between the end of the nineteenth and beginning of the twentieth century as a result of the Industrial Revolution giving rise to greater complexity of machinery subjected to dynamic loading and, therefore, susceptible to fracture. Often called Design for High Cycle Fatigue (DHCF), design for infi nite life is directed at ensuring that the speci- men, component, or subassembly under examination remains inside the elastic region throughout its useful life. More explicitly, in a component designed for infi nite life the applied loading always remains below the fatigue limit, defi ned by Wöhler as: “That stress value which does not result in the failure of the component in question whatever the number of applica- tion cycles.” In the Wöhler diagram, this value corresponds to the slope of the curve ␴ each value of dynamic load it is possible to determine the number of cycles that will lead to failure. The number of cycles to failure N r increases when the applied load decreases, to arrive at a given value ␴ 0 corresponding to a number of cycles of infi nite life. In testing, since it would be impossible to conduct a test for an infi nite number of cycles, it is possible to defi ne a number of cycles (corresponding to the elbow of the Wöhler curve) after which the material can be considered to have an infi nite residual life. This number is a characteristic of the type of material. In the case of steel, the elbow is well-defi ned by the 2722_C010_r02.indd 2562722_C010_r02.indd 256 11/30/2005 1:48:01 PM11/30/2005 1:48:01 PM © 2006 by Taylor & Francis Group, LLC versus N, also known as the “elbow” of the curve (Figure 10.3). In fact, for Engineering Methods for Product Duration Design and Evaluation 257 asymptotic trend of the curve ␴ versus N, beginning from the fatigue limit at around 10 6 cycles. Because of this characteristic, steels are particularly suited to this approach. Conversely, many other materials do not present such a clear trend and even at high numbers of cycles (from 10 6 to 10 9 ), the ␴ versus N curves continue to exhibit steep slopes. Wöhler curves are obtained from controlled loading tests, typically plot- ting the number of cycles along the x-axis and the load (maximum load ␴ max , or load amplitude ⌬␴) along the y-axis. In order to interpret the diagrams correctly, other load characteristics are specifi ed (e.g., the cycle ratio R ϭ ␴ min ր␴ max ). The data obtained from experimental tests are highly dispersed, so the construction of the curve requires a large number of specimens for each load- ing level. Furthermore, this dispersion gradually increases as the load nears the fatigue limit. Interpolating the points with the same probability of failure at different load levels gives the “different probability of failure” curve. The highest curve of the diagram represents 95% probability of failure within the corresponding number of cycles, while the lowest curve represents 5% prob- ability of failure. Wöhler diagrams allow an infi nite life component to be dimensioned in terms of resistance to fatigue, referring to the values of the fatigue limit or, in temporal terms, referring to the number of cycles to failure relative to the stress considered. With regard to the frequency of the applied loads, experience has shown that this has a negligible effect on the relation between the stresses and the number of cycles. In experimental trials on specimens under rotating bend- ing load, with frequencies up to 170 Hz, the value of frequency had no effect. Higher frequencies, up to 500 Hz, produced an increase in fatigue resistance varying between 3% and 13%. It should be noted that the frequency has no effect only when the material under examination does not reach tempera- tures high enough to alter its structure. Given that experimental trials are generally performed on simple speci- mens, to determine the actual fatigue resistance of the component to be FIGURE 10.3 Wöhler diagram for a steel. 2722_C010_r02.indd 2572722_C010_r02.indd 257 11/30/2005 1:48:01 PM11/30/2005 1:48:01 PM © 2006 by Taylor & Francis Group, LLC 258 Product Design for the Environment designed it is necessary to take into account its shape, surface fi nishing, heat treatment, (Shigley and Misehke, 1989 and so on). To do so, coeffi cients are used that evaluate the reduction in resistance due to: • The type of loading applied • The stress concentration • The surface fi nishing • The dimensions (scale effect) These factors are usually evaluated experimentally, as summarized in Table 10.1. The effective fatigue limit ␴ 0 is lower than that obtained on a specimen ␴ 0 I : ss 0 CCC o I LGS ϭ (10.1) The effect of the dimensions, or scale effect, is associated with the probability of fi nding a critical defect in the material; the greater the volume of material subjected to fatigue forces, the higher this probability will be. Also, the type of loading must be seen in terms of the probability of creating conditions of microplasticization in the material. In the case of traction, where all the points of the specimen are subjected to the same stress, a point of discontinuity would reach plasticization and trigger a crack. In the case of torsion, the points with greatest stress are on the external surface of the specimen, and there is, therefore, a lower probability that conditions of microplasticization are generated. The phenomenon is less probable under bending loads, where points of greatest stress are those along the opposite generatrices of a cylin- drical specimen. The surface fi nishing of parts is extremely important in elements subjected to fatigue. It is possible to show the coeffi cient of decreased fatigue resistance in relation to the failure load R, for various degrees of surface fi nishing. From TABLE 10.1 Fatigue limit reduction factors BENDING TRACTION TORSION C L Load Factor 1 1 0.58 C G Size Factor Diameter Ͻ 10 mm 10 mm Ͻ Diameter Ͻ 50 mm 1 0.9 from 0.7 to 0.9 from 0.7 to 0.9 1 0.9 C S Surface Finishing Factor See (Shigley and Mischke, 1989, pp. 282–286) 2722_C010_r02.indd 2582722_C010_r02.indd 258 11/30/2005 1:48:02 PM11/30/2005 1:48:02 PM © 2006 by Taylor & Francis Group, LLC Engineering Methods for Product Duration Design and Evaluation 259 diagrams like these, it can be seen that the various curves show a decrease on the y-axis for an increase on the x-axis, and thus steels with the highest failure loads are more susceptible to the effect of surface irregularities. This effect can be explained by considering the phenomenon used to determine fatigue failure. Given that the existence of microscopic cracks is inevitable in a mechanical element, all processes that can lead to an increase in their extension will lower the fatigue limit, while those limiting their extension will raise this limit. In general, those processes that generate residual compression stresses in the element are those that increase the fatigue limit, while those that generate resid- ual traction stresses result in a decrease in the fatigue limit. Heat treatments improve, to a greater or lesser extent, the fatigue resistance of the element. Finally, it is necessary to take into account the effects produced by a varia- tion in the cross-section of the component in question (e.g., coves, notches, or holes near which there is a very steep stress gradient and a maximum stress peak, as shown in Figure 10.4). This phenomenon is defi ned as Stress Concentration and is more marked as the size of the radius of curvature of the cove, notch, or hole decreases. The application of St. Venant’s torsion theory can only give approximate values of the maximum stresses. To determine the actual stress in each point of the material requires, therefore, the direct application of the general elasticity equations. In the case of moderately simple geometric shapes, Neuber provided some solutions of the stress state along the entire contour, evaluating the maxi- mum stress value (Neuber, 1958). FIGURE 10.4 Stress gradients corresponding to (a) coves and (b) notches. 2722_C010_r02.indd 2592722_C010_r02.indd 259 11/30/2005 1:48:02 PM11/30/2005 1:48:02 PM © 2006 by Taylor & Francis Group, LLC 260 Product Design for the Environment The value of the nominal stress acting on the component is thus increased by a force concentration factor K t : K t theor nom ϭ ss/ (10.2) K t is calculated using the theory of elasticity and the results are presented in Peterson diagrams (Peterson, 1959). The coeffi cient K t is theoretical because the effect of a notch also depends on the type of material and on the type of static or fatigue loading applied on the notched element. If this element is composed of a ductile material and subjected to fatigue loading, there is a redistribution of the stresses due to the plasticity of the material and to metallurgical instability caused by the fatigue process itself. In the fatigue characterization of a material, this effect is taken into account by introducing, at the experimental level, a dynamic or fatigue stress concentration factor. The fatigue notch factor K f is defi ned as: K feffnom ϭրss (10.3) where ␴ eff takes account of the distribution of the stresses within the material at the microplasticizations forming in the proximity of zones with concen- trated stresses. The two factors are interrelated: 1ՅK f ՅK t . When the material is perfectly fragile, the stresses are not redistributed and the preceding inequality becomes K f K t . The factor K f can be calculated using empirical relations that take into account the radius of curvature and the properties of the material (e.g., Heywood’s equation): K 1 K a r f t ϭϩ Ϫ ϩ 1 1 (10.4) where r is the radius of curvature and a is a constant, function of the properties of the material, with the magnitude of one length. In practice, the value of K f can be obtained as a ratio between the high cycle fatigue resistance of the mate- rial determined on an unnotched specimen and that on a notched specimen. In conclusion, it is possible to defi ne the notch sensitivity factor q, by the ratio between the increase ineffective stress due to notch and that in theoreti- cal stress due to notch: q K K f t eff nom theor nom ϭ Ϫ Ϫ ϭ Ϫ Ϫ 1 1 ss ss (10.5) 2722_C010_r02.indd 2602722_C010_r02.indd 260 11/30/2005 1:48:02 PM11/30/2005 1:48:02 PM © 2006 by Taylor & Francis Group, LLC [...]... phase of crack propagation, Miller and Zachariah introduced an exponential relation between the length of the crack and the life consumed (Miller and Zachariah, 1977) In the calculation model, damage is normalized as: a af D ϭ (10. 33) where a and af are, respectively, the instantaneous and final length of the crack The model of Later and Ibrahim, based on the propagation mechanism of very small cracks, is... constant amplitude Subsequently, numerous authors sought to develop theories of damage In particular, a distinction can be made between theories formulated before and after the 1970s The former are based on a more phenomenological approach, the latter on an analytical treatment 10. 3.2.1 Phenomenological Approach The phenomenological approach is based on three main concepts: • Damages produced at different... described mathematically by: ( da ϭ ⌽ ⌬␥p dN ) ϰ a (10. 34) where ⌽ and ␣ are constants of the material and ⌬␥p is the amplitude of the tangential plastic strain In subsequent studies, Ibrahim and Miller correlated the parameters Ni and ␣i to values of the amplitude of the tangential plastic strain ⌬␥p, using an exponential function (Ibrahim and Miller, 1980) Thus, the exponential of damage for the first stage... theories In particular, those of the last 20 years have shown that damage parameters based on energy can correlate different damages (thermal, creep, and fatigue) caused by different types of applied load Craig, Kujawski, and Ellyin developed a model of preliminary damage using a parameter that takes into account the energy density of plastic formation ⌬wp, calculated by integrating the area of the hysteresis... Damage Parameter The parameter used for the analytical measurement of damage is the percentage ratio of the area (or volume) of the cavities within the elementary cell to its nominal area (volume) The value of this parameter grows in each part of the material during its strain-history due to the effect of the two contributions noted above: the growth of preexisting cavities and the nucleation of new cavities... material, because the threshold value of this energy is constant, by measuring the value experimentally for the one-dimensional case it is possible to determine the damage triggering strain for any other value of triaxiality by imposing that the plastic energy not dissipated as heat has a single common value 10. 3.2 Cumulative Damage Fatigue and Theories of Lifespan Prediction In general, fatigue damage... parts that “announces” the failure of a ductile material at the macroscopic level, is caused by the usually extremely rapid propagation of a crack that, in turn, derives from the growth and coalescence of cavities or porosities These may already be present in the virgin material as it leaves the foundry, or be formed (nucleated) later as a result of strain 10. 3.1 Definition of Ductile Damage and Damage... based on measuring the strain The specimen is subjected to a small, symmetrical, alternating axial load, with an increasing static load superimposed, and the mean strain is measured This increases with the load, and the corresponding graph is a straight line until, at a certain load, it presents an “elbow” after which the strain increases more rapidly than before The load at this elbow is defined as... traveled through the atmosphere These collateral effects are satisfactorily eliminated by the analysis software and internal compensation systems The emissivity factor ␧ is a parameter that must be taken into consideration in the measurement of the infrared temperature, since the actual objects analyzed are rarely black bodies The values of emissivity can be measured or obtained from appropriate tables... (Palmgren, 1924) was subsequently translated into mathematical form by Miner, according to the law: D ϭ ∑r i ϭ ni ∑N (10. 26) fi This is a Linear Damage Rule (LDR), based on the principle that for every loading cycle there is a constant absorption of energy and that every material has a characteristic value of absorbed energy to reach failure According to Miner’s hypothesis, each cycle consumes a part . the virgin material as it leaves the foundry, or be formed (nucleated) later as a result of strain. 10. 3.1 Defi nition of Ductile Damage and Damage Parameter The parameter used for the analytical. develop theories of damage. In particular, a distinction can be made between theories formulated before and after the 1970s. The former are based on a more phenomenological approach, the latter. design, paying particular attention to phenomena of performance deterioration (fatigue and damage), design for component durability, and methods for the evaluation of residual life. These are