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Residual Stresses: Macro- and Microstresses
Residual Stresses: Macro- and Microstresses$ RA Winholtz, University of Missouri, Columbia, MO, USA M Rogante, Rogante Engineering Office, Civitanova Marche, Italy r 2016 Elsevier Inc. All rights reserved.
1 2 3 4 5 References Further Reading
Definitions Origins Equilibrium Relations Separation of Macro- and Microstresses Effects
1 2 3 3 4 4 5
Macro and micro residual stresses (RS) differ in character and in their effects, but always arise from material misfits of various kinds that arise in processing. RS can be categorized by the length scale of the misfit, which determines the length scale of their variations. It is important to distinguish between these different types of RS and to be able to separate them, because different measurement techniques sample and record them in different ways. Furthermore, it is necessary to avoid systematic instrumental effects. This article will discuss the definitions of RS types, their different origins and their effects on material properties, as well as methods to separate them.
1
Definitions
Stress is defined as the force supported by a material divided by the area of material supporting the force. In a continuum mechanics approach to materials, we define the stress at a point by taking the limit as the area tends to zero. Real materials, however, are composed of atoms and the concept must break down as the area approaches atomic dimensions. While it is useful to define the stress at a point, we must recognize that in reality we are defining the stress over some finite area. Real materials, in addition to being inhomogeneous on the atomic scale, can be inhomogeneous on larger scales as well. Dislocations, precipitates (e.g., carbides), separate grains, reinforcing particles or fibers, gas bubbles, inclusions, and pores are all inhomogeneities, which disrupt the stress field. Measured stress, thus, may be recorded as uniform for larger areas, but it becomes highly nonuniform as the sampling area approaches the size of the inhomogeneities. Since RS exist in the absence of external loads, they must balance to zero within a component, stresses of one sign being balanced by stresses of opposite sign elsewhere. RS must obey the equilibrium relation: Z A
sij dA ¼ 0
½1
Here, A is the area over which the stresses will balance to zero. The dimensions of the smallest area, A, over which the stresses balance defines a characteristic length which can be used to define different types of RS. Macrostresses are those that balance to zero when integrated over a cross section of a component. The characteristic length for macrostresses is thus on the order of the component’s dimensions. They can be measured by dissection techniques (Lu, 1996), where cutting part of a component relaxes the stresses elsewhere. When characterizing macrostresses, the material can usually be considered homogeneous. Residual microstresses, in contrast, arise from the fact that most real materials are inhomogeneous on a small scale. Microstresses are stresses between phases in multiphase materials (e.g., between matrix and reinforcement in composites), between grains in polycrystalline materials, or around other microstructural features such as dislocations or precipitates. Microstresses arise due to the inhomogeneity of properties in the microstructure as the material is processed. Macrostresses, by definition, are uniform across the microstructural inhomogeneities. Microstresses are then defined as the local variations of stress from the macrostress value within the inhomogeneities of the material. Microstresses will thus balance to zero over a characteristic length of the order of the microstructural features giving rise to them. Thus, they will not be revealed by dissection methods. Examples can, of course, be found where the microstructural scale is on the order of the component dimensions, which blurs these definitions; however, they remain useful for most situations. ☆ Change History: July 2015. M. Rogante edited the Definitions section by removing an unnecessary link, extended the end of the Effects section and the References.
Reference Module in Materials Science and Materials Engineering
doi:10.1016/B978-0-12-803581-8.03038-1
1
2
Residual Stresses: Macro- and Microstresses
σ
〈ασII〉 σI 〈βσII〉
βσII
σIII
Figure 1 Illustration of residual stresses of types I, II, and III. Type I stresses, sI, are constant over the microstructural scale, while type II stresses, sII, vary from grain to grain. The type III stresses, sIII, vary on an even smaller scale and balance to zero within a single grain.
Residual macrostresses are occasionally referred to as RS of type I. Microstresses that balance over a scale comparable to the grain size are similarly referred to as RS of type II. Microstresses that originate from dislocations and point defects within grains, and will balance to zero over a characteristic length smaller than the grain dimensions, are then referred to as RS of type III. RS of type I will cause shifts in diffraction peaks. Those of type III will only cause diffraction peak broadening (Schwartz and Cohen, 1987), since their characteristic length is smaller than typical diffracting volumes. In single-phase materials, type II RS will cause peak broadening for the same reason. In two-phase materials, type II RS can also cause peak shifts, because the counterbalancing stresses for one phase can occur in the other phase whose effect does not appear in a diffraction peak for the first phase. Figure 1 illustrates the various types of RS across a two-phase microstructure.
2
Origins
Residual macrostresses originate from nonuniformities in plastic deformation, temperature, or composition. In all cases, they arise from shape misfits between different regions of the material. Many manufacturing operations will produce residual macrostresses in materials. Operations such as rolling, drawing, forging, bending, straightening, shot peening, and grinding will plastically deform selected regions of the material. After removal of the loads causing the plastic deformation, the regions subject only to elastic deformation will spring back, producing RS in the component. In shot peening, for example, the surface layers are stretched by the impinging shot but are constrained by the undeformed layers below, resulting in a compressive surface layer that can improve the fatigue life. Temperature gradients can produce RS by generating thermal stresses large enough to produce plastic deformation, inducing phase transformations at different times in different locations, or by developing stresses in solidification. When a material is quenched, the outside cools first, which can generate thermal stresses large enough to deform the softer interior. When the interior subsequently cools and contracts, it puts the surface layers into compression. This is the mechanism for tempering glass and for residual macrostresses developed during heat treatment. If, during quenching, solid-state phase transformations occur with an associated volume change, the situation is more complicated. On cooling, volume changes occur due to both thermal contraction and phase transformation. Welding is another of the most significant sources of RS, usually creating large tensile stresses whose maximum value is roughly equal to the yield strength of the materials being joined, balanced by lower
Residual Stresses: Macro- and Microstresses
3
compressive RS elsewhere in the component (Rogante et al., 2006). RS on the surface can be either tensile or compressive, depending on the size and sign of the volume change with transformation. When materials solidify in a temperature gradient, residual macrostresses can be generated, most notably during welding. As material solidifies, it cools and begins to contract. The fluid in unsolidified regions cannot support stresses and accommodates the contracting material around it. When this material later solidifies, it will try to contract more than the cooler areas around it, leading to residual stresses. In a similar way, castings can develop residual stresses as different portions solidify at different rates. Again, solid-state phase transformations can produce additional complications. Residual macrostresses can also be generated by composition variations in a material. Carburizing or nitriding steel will swell the surface layers as atoms diffuse in. These layers will develop compressive RS as they are constrained by the bulk of the material. Operations such as brazing and ion implantation can also lead to residual macrostresses due to composition variations. Residual microstresses arise from the inhomogeneity of properties in the microstructure. A common source of microstresses is the difference in the coefficients of thermal expansion in multiphase materials. A composite consolidated at elevated temperatures will develop stresses between the matrix and reinforcement as they contract at different rates. In the same way, second-phase particles will generate microstresses as they cool from a precipitation temperature. Plastic deformation will also generate residual microstresses. The individual phases will have different deformation behaviors leading to microstresses. Hard particles in a softer matrix, for example, will not plastically deform and the load will preferentially transfer to them. Plastic deformation can thus produce both residual macrostresses and microstresses at the same time. Another source of residual microstresses is elastic inhomogeneities. Different phases usually have different elastic moduli, and thus a load will distribute unevenly among the phases. The same is true of a single-phase polycrytalline material, as the anisotropic grains are oriented differently. Microstresses can also be generated in maintaining coherency in precipitates such as in the precipitation of G.P. zones.
3
Equilibrium Relations
Stress gradients must obey the equilibrium relation of elementary elasticity: sij;j ¼ 0
½2
This must hold for both macro and microstresses at any point. Macrostresses vary slowly enough that gradients within a component can be measured and seen to obey this relation. The steep gradients in stress are, by definition, microstresses. The gradients in microstresses vary too rapidly to be measured by most measurement techniques and thus, only the average microstress can be measured at a location in a component. To maintain equilibrium, the stresses must balance between phases. For any area somewhat larger than the microstructural dimensions, the average phase stresses must sum to zero when weighted by their volume fraction (Noyan, 1983). For a two-phase material we have: f 〈a sIIij 〉 þ ð1 f Þ〈b sIIij 〉 ¼ 0
½3
Here the superscript II is used to denote type II stresses, the stresses in the individual phases are represented by the superscripts a and b, and f represents the volume fraction of the a phase. The microstress terms have been enclosed in brackets to emphasize that they are an average. Figure 1 illustrates these average phase stresses in a two-phase material. Equation [3] holds for each component of stress.
4
Separation of Macro- and Microstresses
Diffraction methods offer a unique way to measure microstresses in crystalline materials, because each phase will have its own diffraction pattern giving information on the stresses in that phase . Using diffraction to measure interplanar spacings in different directions, the complete strain tensor may be determined (Noyan and Cohen, 1987; Hauk, 1997). With an appropriate elastic model, the stresses can be computed from the strains. The stress tensor measured by diffraction in a particular phase is the sum of the type I and II stresses. For a two-phase material, we may write (Noyan, 1983): a total sij
¼ sIij þ 〈a sIIij i
½4
b total sij
¼ sIij þ 〈b sIIij i
½5
Here the superscripts ‘total’, I, and II denote the total stress measured by diffraction for a phase, the macrostress, and the microstress in that phase, respectively. RS of type III will be recorded as peak broadening and do not contribute to peak shifts. Note
4
Residual Stresses: Macro- and Microstresses
that the macrostress is, by definition, the same in both phases and hence is not given a phase designation. The microstresses are enclosed in brackets to indicate that a diffraction average is taken, i.e., the average over the grains in the irradiated volume oriented to diffract. Equations [4] and [5] are also valid for each component of stress. By measuring the stresses in each phase astotal and ij b total sij , and using eqns [3], [4], and [5] we may determine the macrostress, sIij, and the average microstress in each phase, 〈asIIij 〉 and 〈bsIIij 〉. The macrostress is the weighted average of the individual phase stresses: sIij ¼ f a stotal þ ð1 f Þb stotal ij ij
½6
The microstress components are then the total phase stresses less the macrostresses from eqns [4] and [5]. These equations can be generalized to include more phases.
5
Effects
RS affect many aspects of materials’ properties and performance (Rogante and Rosta, 2005; Rogante, 2008). Residual macrostresses can improve or worsen a component’s fatigue performance (ASM, 1996; Suresh, 1991; Hauk, 1997). Compressive macrostresses may delay crack initiation and slow crack growth. Tensile stresses, on the other hand, can accelerate crack initiation and growth, leading to more rapid fatigue failure. Similarly, residual macrostresses can affect the fracture behavior of components, enhancing or diminishing the fracture resistance, or the propensity for stress corrosion cracking (ASM, 1987). Shot peening and autofrettage have long been used to improve components’ performance by imparting beneficial compressive stresses where cracks would form. The strength of tempered glass is greatly improved by the presence of residual compressive macrostresses at the surface, where tensile stresses from bending lead to failure (Almen and Black, 1963). Residual macrostresses also lead to problems of distortion and dimensional control in components, particularly in welding and heat-treating (Radaj, 1992; ASM, 1991; Rogante et al., 2006). Components can also distort when material containing residual stresses is removed by machining operations. Microstresses can also affect the properties of materials in beneficial or deleterious ways. Microstresses are responsible for the strength differential effect in metal matrix composites, where the yield strength is lower in tension than in compression because the matrix contains tensile thermal microstresses (Clyne and Withers, 1993). Bauchinger effects in metals can also arise due to microstresses arising from nondeforming phases (Clyne and Withers, 1993; Suresh, 1991). Microstresses can help improve the toughness of ceramics through transformation toughening or by promoting microcracking (Lee and Rainforth, 1994). The strengthening in some metal matrix composites is achieved, in part, by the generation of dislocations from the thermal microstresses developed on cooling from consolidation (Taya and Arsenault, 1989; Clyne and Withers, 1993). Dedicated procedures have been developed to evaluate macro- and microstresses by using neutron diffraction, especially in the case of industrial applications (Rogante, 2008). For these applications, as the assesment of RS is always related to the stress free material state, the availability of carefully measured zero-strain standards is fundamental to confirm the absence of systematic instrumental effects determining the diffraction profile at a chosen scattering angle. Several efforts are currently under way, thus, to introduce new methods allowing more and more precise and practical evaluations of the unstressed lattice parameters, hence of the residual strains and stresses (Rogante, 2000; Rogante et al., 2014).
References ASM International, 1987. Corrosion. In: ASM Handbook, tenth ed., vol. 13. Materials Park, OH: ASM International. ASM International, 1991. Heat Treating. In: ASM Handbook, tenth ed., vol. 4. Materials Park, OH: ASM International. ASM International, 1996. Fatigue and Fracture. In: ASM Handbook, tenth ed. vol. 19. Materials Park, OH: ASM International. Almen, J.O., Black, P.H., 1963. Residual Stresses and Fatigue in Metals. New York: McGraw-Hill. Clyne, T.W., Withers, P.J., 1993. An Introduction to Metal Matrix Composites. Cambridge, UK: Cambridge University Press. Hauk, V., 1997. Structural and Residual Stress Analysis by Nondestructive Methods. Amsterdam: Elsevier Science. Lee, W.E., Rainforth, W.M., 1994. Ceramic Microstructures: Property Control by Processing. London: Chapman & Hall. Lu, J., 1996. Handbook of Measurement of Residual Stresses. Society for Experimental Mechanics. Lilburn, GA: Fairmont Press. Noyan, I.C., 1983. Equilibrium conditions for the average stresses measured by X-rays. Metallurgical and Materials Transactions A 14A, 1907–1914. Noyan, I.C., Cohen, J.B., 1987. Residual Stress: Measurement by Diffraction and Interpretation. New York: Springer-Verlag. Radaj, D., 1992. Heat Effects of Welding: Temperature Field, Residual Stress, Distortion. Berlin: Springer-Verlag. Schwartz, L.H., Cohen, J.B., 1987. Diffraction from Materials. New York: Springer-Verlag. Rogante, M., 2000. Physica B: Condensed Matter 276−278, 202–203. Rogante, M., Rosta, L., 2005. Proceedings of SPIE 5824, 294–305. Rogante, M., Lebedev, V.T., Kralj, S., Rosta, L., To+ ro+ k, Gy., 2006. Multidiscipline Modeling in Materials and Structures 2 (4), 419–433. Rogante, M., 2008. Applicazioni Industriali delle Tecniche Neutroniche. Proc. 1st Italian Workshop for Industry "Industrial Applications of Neutron Techniques", Civitanova Marche, Italy, 12−14 June 2008. Rogante Engineering, pp. 40−120 Rogante, M., Mikula, P., Vrána, M., 2014. Key Engineering Materials 592−593, 465–468. Suresh, S., 1991. Fatigue of Materials. Cambridge, UK: Cambridge University Press. Taya, M., Arsenault, M.J., 1989. Metal Matrix Composites. Oxford: Pergamon.
Residual Stresses: Macro- and Microstresses
Further Reading Arsenault, R.J., Taya, M., 1987. Thermal residual stress in metal matrix composite. Acta Metallurgica 35, 651–659. ASM International ASM Handbook, 10th edn, Vol. 11: Failure Analysis and Prevention 1986. ASM International Metals Park, OH ASTM Residual Stress Effects in Fatigue. ASTM STP 776 1982. American Society for Testing and Materials Philadelphia, PA Cohen, J.B., 1986. The measurement of stresses in composites. Powder Diffraction 1 (2), 15–21. Hertzberg, R.W., 1996. Deformation and Fracture Mechanics of Engineering Materials, 4th edn New York: Wiley. Thelning, K.-E., 1984. Steel and Its Heat Treatment, 2nd edn London: Butterworths.
5
1 2 3 4 5 References Further Reading
Definitions Origins Equilibrium Relations Separation of Macro- and Microstresses Effects
1 2 3 3 4 4 5
Macro and micro residual stresses (RS) differ in character and in their effects, but always arise from material misfits of various kinds that arise in processing. RS can be categorized by the length scale of the misfit, which determines the length scale of their variations. It is important to distinguish between these different types of RS and to be able to separate them, because different measurement techniques sample and record them in different ways. Furthermore, it is necessary to avoid systematic instrumental effects. This article will discuss the definitions of RS types, their different origins and their effects on material properties, as well as methods to separate them.
1
Definitions
Stress is defined as the force supported by a material divided by the area of material supporting the force. In a continuum mechanics approach to materials, we define the stress at a point by taking the limit as the area tends to zero. Real materials, however, are composed of atoms and the concept must break down as the area approaches atomic dimensions. While it is useful to define the stress at a point, we must recognize that in reality we are defining the stress over some finite area. Real materials, in addition to being inhomogeneous on the atomic scale, can be inhomogeneous on larger scales as well. Dislocations, precipitates (e.g., carbides), separate grains, reinforcing particles or fibers, gas bubbles, inclusions, and pores are all inhomogeneities, which disrupt the stress field. Measured stress, thus, may be recorded as uniform for larger areas, but it becomes highly nonuniform as the sampling area approaches the size of the inhomogeneities. Since RS exist in the absence of external loads, they must balance to zero within a component, stresses of one sign being balanced by stresses of opposite sign elsewhere. RS must obey the equilibrium relation: Z A
sij dA ¼ 0
½1
Here, A is the area over which the stresses will balance to zero. The dimensions of the smallest area, A, over which the stresses balance defines a characteristic length which can be used to define different types of RS. Macrostresses are those that balance to zero when integrated over a cross section of a component. The characteristic length for macrostresses is thus on the order of the component’s dimensions. They can be measured by dissection techniques (Lu, 1996), where cutting part of a component relaxes the stresses elsewhere. When characterizing macrostresses, the material can usually be considered homogeneous. Residual microstresses, in contrast, arise from the fact that most real materials are inhomogeneous on a small scale. Microstresses are stresses between phases in multiphase materials (e.g., between matrix and reinforcement in composites), between grains in polycrystalline materials, or around other microstructural features such as dislocations or precipitates. Microstresses arise due to the inhomogeneity of properties in the microstructure as the material is processed. Macrostresses, by definition, are uniform across the microstructural inhomogeneities. Microstresses are then defined as the local variations of stress from the macrostress value within the inhomogeneities of the material. Microstresses will thus balance to zero over a characteristic length of the order of the microstructural features giving rise to them. Thus, they will not be revealed by dissection methods. Examples can, of course, be found where the microstructural scale is on the order of the component dimensions, which blurs these definitions; however, they remain useful for most situations. ☆ Change History: July 2015. M. Rogante edited the Definitions section by removing an unnecessary link, extended the end of the Effects section and the References.
Reference Module in Materials Science and Materials Engineering
doi:10.1016/B978-0-12-803581-8.03038-1
1
2
Residual Stresses: Macro- and Microstresses
σ
〈ασII〉 σI 〈βσII〉
βσII
σIII
Figure 1 Illustration of residual stresses of types I, II, and III. Type I stresses, sI, are constant over the microstructural scale, while type II stresses, sII, vary from grain to grain. The type III stresses, sIII, vary on an even smaller scale and balance to zero within a single grain.
Residual macrostresses are occasionally referred to as RS of type I. Microstresses that balance over a scale comparable to the grain size are similarly referred to as RS of type II. Microstresses that originate from dislocations and point defects within grains, and will balance to zero over a characteristic length smaller than the grain dimensions, are then referred to as RS of type III. RS of type I will cause shifts in diffraction peaks. Those of type III will only cause diffraction peak broadening (Schwartz and Cohen, 1987), since their characteristic length is smaller than typical diffracting volumes. In single-phase materials, type II RS will cause peak broadening for the same reason. In two-phase materials, type II RS can also cause peak shifts, because the counterbalancing stresses for one phase can occur in the other phase whose effect does not appear in a diffraction peak for the first phase. Figure 1 illustrates the various types of RS across a two-phase microstructure.
2
Origins
Residual macrostresses originate from nonuniformities in plastic deformation, temperature, or composition. In all cases, they arise from shape misfits between different regions of the material. Many manufacturing operations will produce residual macrostresses in materials. Operations such as rolling, drawing, forging, bending, straightening, shot peening, and grinding will plastically deform selected regions of the material. After removal of the loads causing the plastic deformation, the regions subject only to elastic deformation will spring back, producing RS in the component. In shot peening, for example, the surface layers are stretched by the impinging shot but are constrained by the undeformed layers below, resulting in a compressive surface layer that can improve the fatigue life. Temperature gradients can produce RS by generating thermal stresses large enough to produce plastic deformation, inducing phase transformations at different times in different locations, or by developing stresses in solidification. When a material is quenched, the outside cools first, which can generate thermal stresses large enough to deform the softer interior. When the interior subsequently cools and contracts, it puts the surface layers into compression. This is the mechanism for tempering glass and for residual macrostresses developed during heat treatment. If, during quenching, solid-state phase transformations occur with an associated volume change, the situation is more complicated. On cooling, volume changes occur due to both thermal contraction and phase transformation. Welding is another of the most significant sources of RS, usually creating large tensile stresses whose maximum value is roughly equal to the yield strength of the materials being joined, balanced by lower
Residual Stresses: Macro- and Microstresses
3
compressive RS elsewhere in the component (Rogante et al., 2006). RS on the surface can be either tensile or compressive, depending on the size and sign of the volume change with transformation. When materials solidify in a temperature gradient, residual macrostresses can be generated, most notably during welding. As material solidifies, it cools and begins to contract. The fluid in unsolidified regions cannot support stresses and accommodates the contracting material around it. When this material later solidifies, it will try to contract more than the cooler areas around it, leading to residual stresses. In a similar way, castings can develop residual stresses as different portions solidify at different rates. Again, solid-state phase transformations can produce additional complications. Residual macrostresses can also be generated by composition variations in a material. Carburizing or nitriding steel will swell the surface layers as atoms diffuse in. These layers will develop compressive RS as they are constrained by the bulk of the material. Operations such as brazing and ion implantation can also lead to residual macrostresses due to composition variations. Residual microstresses arise from the inhomogeneity of properties in the microstructure. A common source of microstresses is the difference in the coefficients of thermal expansion in multiphase materials. A composite consolidated at elevated temperatures will develop stresses between the matrix and reinforcement as they contract at different rates. In the same way, second-phase particles will generate microstresses as they cool from a precipitation temperature. Plastic deformation will also generate residual microstresses. The individual phases will have different deformation behaviors leading to microstresses. Hard particles in a softer matrix, for example, will not plastically deform and the load will preferentially transfer to them. Plastic deformation can thus produce both residual macrostresses and microstresses at the same time. Another source of residual microstresses is elastic inhomogeneities. Different phases usually have different elastic moduli, and thus a load will distribute unevenly among the phases. The same is true of a single-phase polycrytalline material, as the anisotropic grains are oriented differently. Microstresses can also be generated in maintaining coherency in precipitates such as in the precipitation of G.P. zones.
3
Equilibrium Relations
Stress gradients must obey the equilibrium relation of elementary elasticity: sij;j ¼ 0
½2
This must hold for both macro and microstresses at any point. Macrostresses vary slowly enough that gradients within a component can be measured and seen to obey this relation. The steep gradients in stress are, by definition, microstresses. The gradients in microstresses vary too rapidly to be measured by most measurement techniques and thus, only the average microstress can be measured at a location in a component. To maintain equilibrium, the stresses must balance between phases. For any area somewhat larger than the microstructural dimensions, the average phase stresses must sum to zero when weighted by their volume fraction (Noyan, 1983). For a two-phase material we have: f 〈a sIIij 〉 þ ð1 f Þ〈b sIIij 〉 ¼ 0
½3
Here the superscript II is used to denote type II stresses, the stresses in the individual phases are represented by the superscripts a and b, and f represents the volume fraction of the a phase. The microstress terms have been enclosed in brackets to emphasize that they are an average. Figure 1 illustrates these average phase stresses in a two-phase material. Equation [3] holds for each component of stress.
4
Separation of Macro- and Microstresses
Diffraction methods offer a unique way to measure microstresses in crystalline materials, because each phase will have its own diffraction pattern giving information on the stresses in that phase . Using diffraction to measure interplanar spacings in different directions, the complete strain tensor may be determined (Noyan and Cohen, 1987; Hauk, 1997). With an appropriate elastic model, the stresses can be computed from the strains. The stress tensor measured by diffraction in a particular phase is the sum of the type I and II stresses. For a two-phase material, we may write (Noyan, 1983): a total sij
¼ sIij þ 〈a sIIij i
½4
b total sij
¼ sIij þ 〈b sIIij i
½5
Here the superscripts ‘total’, I, and II denote the total stress measured by diffraction for a phase, the macrostress, and the microstress in that phase, respectively. RS of type III will be recorded as peak broadening and do not contribute to peak shifts. Note
4
Residual Stresses: Macro- and Microstresses
that the macrostress is, by definition, the same in both phases and hence is not given a phase designation. The microstresses are enclosed in brackets to indicate that a diffraction average is taken, i.e., the average over the grains in the irradiated volume oriented to diffract. Equations [4] and [5] are also valid for each component of stress. By measuring the stresses in each phase astotal and ij b total sij , and using eqns [3], [4], and [5] we may determine the macrostress, sIij, and the average microstress in each phase, 〈asIIij 〉 and 〈bsIIij 〉. The macrostress is the weighted average of the individual phase stresses: sIij ¼ f a stotal þ ð1 f Þb stotal ij ij
½6
The microstress components are then the total phase stresses less the macrostresses from eqns [4] and [5]. These equations can be generalized to include more phases.
5
Effects
RS affect many aspects of materials’ properties and performance (Rogante and Rosta, 2005; Rogante, 2008). Residual macrostresses can improve or worsen a component’s fatigue performance (ASM, 1996; Suresh, 1991; Hauk, 1997). Compressive macrostresses may delay crack initiation and slow crack growth. Tensile stresses, on the other hand, can accelerate crack initiation and growth, leading to more rapid fatigue failure. Similarly, residual macrostresses can affect the fracture behavior of components, enhancing or diminishing the fracture resistance, or the propensity for stress corrosion cracking (ASM, 1987). Shot peening and autofrettage have long been used to improve components’ performance by imparting beneficial compressive stresses where cracks would form. The strength of tempered glass is greatly improved by the presence of residual compressive macrostresses at the surface, where tensile stresses from bending lead to failure (Almen and Black, 1963). Residual macrostresses also lead to problems of distortion and dimensional control in components, particularly in welding and heat-treating (Radaj, 1992; ASM, 1991; Rogante et al., 2006). Components can also distort when material containing residual stresses is removed by machining operations. Microstresses can also affect the properties of materials in beneficial or deleterious ways. Microstresses are responsible for the strength differential effect in metal matrix composites, where the yield strength is lower in tension than in compression because the matrix contains tensile thermal microstresses (Clyne and Withers, 1993). Bauchinger effects in metals can also arise due to microstresses arising from nondeforming phases (Clyne and Withers, 1993; Suresh, 1991). Microstresses can help improve the toughness of ceramics through transformation toughening or by promoting microcracking (Lee and Rainforth, 1994). The strengthening in some metal matrix composites is achieved, in part, by the generation of dislocations from the thermal microstresses developed on cooling from consolidation (Taya and Arsenault, 1989; Clyne and Withers, 1993). Dedicated procedures have been developed to evaluate macro- and microstresses by using neutron diffraction, especially in the case of industrial applications (Rogante, 2008). For these applications, as the assesment of RS is always related to the stress free material state, the availability of carefully measured zero-strain standards is fundamental to confirm the absence of systematic instrumental effects determining the diffraction profile at a chosen scattering angle. Several efforts are currently under way, thus, to introduce new methods allowing more and more precise and practical evaluations of the unstressed lattice parameters, hence of the residual strains and stresses (Rogante, 2000; Rogante et al., 2014).
References ASM International, 1987. Corrosion. In: ASM Handbook, tenth ed., vol. 13. Materials Park, OH: ASM International. ASM International, 1991. Heat Treating. In: ASM Handbook, tenth ed., vol. 4. Materials Park, OH: ASM International. ASM International, 1996. Fatigue and Fracture. In: ASM Handbook, tenth ed. vol. 19. Materials Park, OH: ASM International. Almen, J.O., Black, P.H., 1963. Residual Stresses and Fatigue in Metals. New York: McGraw-Hill. Clyne, T.W., Withers, P.J., 1993. An Introduction to Metal Matrix Composites. Cambridge, UK: Cambridge University Press. Hauk, V., 1997. Structural and Residual Stress Analysis by Nondestructive Methods. Amsterdam: Elsevier Science. Lee, W.E., Rainforth, W.M., 1994. Ceramic Microstructures: Property Control by Processing. London: Chapman & Hall. Lu, J., 1996. Handbook of Measurement of Residual Stresses. Society for Experimental Mechanics. Lilburn, GA: Fairmont Press. Noyan, I.C., 1983. Equilibrium conditions for the average stresses measured by X-rays. Metallurgical and Materials Transactions A 14A, 1907–1914. Noyan, I.C., Cohen, J.B., 1987. Residual Stress: Measurement by Diffraction and Interpretation. New York: Springer-Verlag. Radaj, D., 1992. Heat Effects of Welding: Temperature Field, Residual Stress, Distortion. Berlin: Springer-Verlag. Schwartz, L.H., Cohen, J.B., 1987. Diffraction from Materials. New York: Springer-Verlag. Rogante, M., 2000. Physica B: Condensed Matter 276−278, 202–203. Rogante, M., Rosta, L., 2005. Proceedings of SPIE 5824, 294–305. Rogante, M., Lebedev, V.T., Kralj, S., Rosta, L., To+ ro+ k, Gy., 2006. Multidiscipline Modeling in Materials and Structures 2 (4), 419–433. Rogante, M., 2008. Applicazioni Industriali delle Tecniche Neutroniche. Proc. 1st Italian Workshop for Industry "Industrial Applications of Neutron Techniques", Civitanova Marche, Italy, 12−14 June 2008. Rogante Engineering, pp. 40−120 Rogante, M., Mikula, P., Vrána, M., 2014. Key Engineering Materials 592−593, 465–468. Suresh, S., 1991. Fatigue of Materials. Cambridge, UK: Cambridge University Press. Taya, M., Arsenault, M.J., 1989. Metal Matrix Composites. Oxford: Pergamon.
Residual Stresses: Macro- and Microstresses
Further Reading Arsenault, R.J., Taya, M., 1987. Thermal residual stress in metal matrix composite. Acta Metallurgica 35, 651–659. ASM International ASM Handbook, 10th edn, Vol. 11: Failure Analysis and Prevention 1986. ASM International Metals Park, OH ASTM Residual Stress Effects in Fatigue. ASTM STP 776 1982. American Society for Testing and Materials Philadelphia, PA Cohen, J.B., 1986. The measurement of stresses in composites. Powder Diffraction 1 (2), 15–21. Hertzberg, R.W., 1996. Deformation and Fracture Mechanics of Engineering Materials, 4th edn New York: Wiley. Thelning, K.-E., 1984. Steel and Its Heat Treatment, 2nd edn London: Butterworths.
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