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Thermal and Residual Stresses

I. INTRODUCTION

Thermal stresses arise in components that are subject to temperature gradients, and the thermal transients associated with engine operations are of particular concern. Residual stresses arise because of nonuniform plastic deformation, and they are important in that thev affect resistance to fatigue and stress corrosion cracking, and also can cause distortion during machining and heat treating. This chapter briefly reviews how these types of stress are developed.

II. THERMAL STRESS, THERMAL STRAIN, AND THERMAL SHOCK A. Thermal Stresses

Consider the bimetallic strip shown in Fig. 7-1. In order to avoid bending during heating, equal amounts of material 1 are placed on either side of material 2. The initial length of the strip is L0, and the area of material 1 is designated as A1, and that of material 2 as A2. The corresponding coefficients of expansion are α1 and α2.

Fig. 7-1. A bimetallic strip.

If the strips were free to expand due to a temperature change of △T, the lengths of each material would be

LL(1 T), 1  0 1 (7-1) 1  L 2  L 0 (  2  T ). (7-2)

But the final lengths must be equal, or L1f = L2f, or

L0(11T)1E1L0L0(12T)2E2L0 (7-3) Also, for equilibrium, P1 + P2 = 0, or σ1A1+ σ2A2 = 0. There are thus two equations with two unknowns, and the stresses σ1 and σ2 can therefore be determined: 212TE1E2E1E2(A2/A1)12TE21(E2/E1)/(A2/A1),(74) A122.A1(75)As long as the stresses are in the elastic range, there will be no residual effects when the temperature returns to its initial value,and the above equations hold. However, if the stresses exceed the yield strength of the materials involved, a state of residual stress will develop when the temperature returns to its initial value. For example, if material 2 in the above example had a lower coefficient of expansion and a relatively low yield stress, it could undergo plastic extension as the temperature increased. Subsequently, when the temperature was lowered, material 2 would be left in a state of residual compression, and material 1 would be in residual tension.

B. Thermal-Mechanical Cyclic Strains

Transient thermal strains of a cyclic nature can arise in jet engine components such as disks, blades, and vanes. Figure 7-2 is a simplified thermal-mechanical cycle for blades in a gas turbine (1). Under cruise conditions, the blades are at 600°C. When engine power is increased, the surface temperature rises to 1100°C in 105 seconds. Fig. 7-2a. Engine power is then returned to the cruise condition, and the temperature falls to 600°C in another 105 seconds. The rate of heating and cooling is 4.76 °C/sec. The total strain is the sum of the strain due to thermal expansion plus the strain due to the thermal stresses developed. The mechanical strain versus time plot, shown in Fig. 7-2b, includes both elastic and plastic strains. On heating, the outer surface initially goes into compressive strain, but then the strain reduces to zero as the interior of the blade heats up. On cooling, the reverse procedure occurs. Figure 7-2c shows the counterclockwise diamond cycle that is the thermal-mechanical hysteresis loop for this transient thermal history. The corresponding stress versus mechanical strain hysteresis loops for single crystals in the [001] and [111] orientations are shown in Fig. 7-3. It is noted that the mean stress for the cycle is close to zero. For the [001] orientation, plastic deformation (or viscoplastic deformation) is pro-nounced at a strain of —0.8. For the [111] orientation, plastic deformation begins at a strain of —0.25. Such thermal mechanical histories are complex, but they are also obviously important in assessing the fatigue life of such components.

Fig. 7-2. Thermal-mechanical cycle, (a) Temperature versus time, (b) Strain versus time. (c) Thermal-mechanical hysteresis loop. (From Remy, 1. Reprinted from Low Cycle Fatigue and Elasto-Plastic Behaviour of Materials, edited by K.-T. Rie and P. D. Portella, pp. 119-130, Copyright 1998, with permission of Elsevier Science.)

Fig. 7-3. Stress versus mechanical strain hysteresis loops for [001] and [111] thermal mechanical fatigue (TMF) single crystal specimens of AM1 superalloy using the cycle depicted in Fig. 7-2. Comparison between a viscoplastic model (solid line) and experiment (symbols). (After Remy, 1. Reprinted from Low Cycle Fatigue and Elastic-Plastic Behaviour of Materials, edited by K.-T. Rie and P. D. Portella, pp. 119-130. Copyright 1998. with permission of Elsevier Science.)

C. Thermal Shock

Thermal shock denotes the rapid development of a steep temperature gradient and accompanying high stresses that can result in the fracture of brittle materials. It can occur either on heating or cooling. For example, the sudden shutdown of turbine engine can result in cracking of protective platinum-aluminide coatings due to the tensile stresses that develop as the surface cools and tries to shrink but is restrained by the interior.

An example of thermal shock that occurs during heating is as follows. Consider a glass bowl whose thermal conductivity is low.

1. A hot liquid is poured into the bowl.

2. The inside surface of the wall tries to expand because of the sudden rise in temperature.

3. A biaxial compressive stress develops on the inside surface because expansion of the is resisted by the surounding, still cool, wall material.

4. This compressive stress system sets up a balancing tensile stress system in the outer, still cool portion of the

wall.

5. Fracture can initiate in the outer region if the magnitude of the tensile stress developed is sufficient to nucleate a crack at a weak point.

III. RESIDUAL STRESSES CAUSED BY NONUNIFORM PLASTIC DEFORMATION

Residual stresses arise because of a gradient in plastic deformation caused either by mechanical deformation or by a thermal gradient caused during cooling of a metal or alloy from a high temperature to a low temperature. The sign of the residual stress is always opposite in sign to the sign of the applied stress that gave rise to the residual stress. Residual stresses are important in fatigue and stress corrosion cracking where they can be either beneficial or detrimental. If residual stresses are present prior to heat treatment or machining, they can be detrimental because they can result in distortion (warping).

A. An Example of Mechanically Induced Residual Stresses: Springback after Bending into the Plastic Range (2)

In sheet bending, the width w is much greater than the thickness /, and width changes are negligible. Therefore, bending can be considered to be a plane-strain operation with εv = 0, εz = — εx. Let z be the distance measured from the mid-plane of the sheet in the thickness direction, and let r be the radius of curvature of the mid-plane. The value of ev varies linearly from — t/2r at the inside of the bend (z = —t/2) to zero at the mid-plane (z = 0), to +t/2r at the outside of the bend (z = t/2). Figure 7-4 shows the stress through the cross section. The principal of superposition is used to show that unloading can be considered to be the reverse of loading, so that for purely elastic bending there is no residual stress after unloading.

Fig. 7-4. Strip bending, elastic strains.

Next, assume that the material is elastic ideally plastic, that is, there is no strain hardening in the plastic range. If the tensile yield stress is Y, the flow stress in plane strain o-0 will be 1. 15 Y. Figure 7-5 shows the stress distribution throughout the sheet for fully plastic behavior. Except for an elastic core at mid-plane (which will be neglected), the entire section will be at a stress, σx = ±σ0.

Fig. 7-5. Strip bending, plastic strains on loading, elastic on unloading.

To calculate the bending moment M needed to create this fully plastic bend, note that dFx = σxw dz, and that dM = z dFx = zσx wdz. Therefore, the fully plastic bending moment is

Mt/2t/2wxzdz2t/20t2wxzdzw04(76)

(Note that the elastic bending moment that is needed to have σx just equal to σ0 would be wσ0 (t2/6), so that the fully plastic bending moment is 50% higher.)

When the external moment is released, the internal moment must go to zero. As the material springs back elastically, the internal residual stress distribution must result in a zero bending moment. Since the unloading is elastic,

xE'Ex.x21v(77)The change in strain is given as △εx = z/r — zlr', where r' is the radius of curvature after springback. This causes a change in bending moment △M, where

M2wxzdz2w0t/2t/20311wE't11E' z2dz12rr'rr'(78)M — △M= 0 after springback, and therefore equating M and △M gives

Or

w0t2wE't311,412rr'1130,rr'tE'

(79) (710 )

The resulting residual stress

x/xx0E'x3113z0E'0E'z001.trr'tE'(711)

On the outside surface, z = t/2, and the residual stress cr'x equals —σ0/2. On the inside surface, the residual stress is +σ0/2.

The distribution of stresses is shown in Fig. 7-5.

Note that the sign of the residual stress is opposite to the sign of the stress that caused it. Also, plastic deformation is required, but the plastic deformation must be nonuniform. There is no residual stress associated with a tensile bar that has been uniformly stretched into the plastic region. On the other hand, residual stresses will develop at notches, and so on, if the material at the base of the notch is strained into the plastic range while the surroundings remain elastic.

An important state of residual stress is that formed at the tip of a growing fatigue crack due to an overload in plane specimens. During fatigue crack growth, if a 100% overload is applied, the fatigue crack will subsequently undergo a period of reduction in rate of crack growth as the crack penetrates the plastic zone created by the overload. This slowdown is related to an increased level of com-pressive residual stress brought about by the overload. This residual stress is largely a plane stress, surface regions, and is brought about by the lateral contraction of material at the surface that occurs in the overload plastic zone. In ductile materials, this lateral contraction leads to the formation of an obvious dimple immediately ahead of the crack tip. The compressive stress develops as the load is reduced from the overload level, and for now. because of the lateral contraction, there is more material in planes parallel to and just below the surface of the overload plastic zone than before the overload. As the crack grows through the overload zone, these residual stresses are released, giving rise to an increased level of crack closure in the wake of the crack tip (see Chapter 10), which results in a slowing down of the rate of fatigue crack propagation. The extent of the slowdown in crack rate is a function of the overload level and the thickness of the specimen, being much more pronounced in thin as compared to thick specimens. The reason for this difference is that, in thick specimens, plane-strain conditions prevail throughout most of the specimen's thickness, and the lateral contraction associated with plane stress is absent under plane-strain conditions. Hence, the level of enhanced residual compressive stress due to an overload is much less in thick specimens than in thin. Surface compressive residual stresses are beneficial in fatigue and in stress corrosion cracking. For this reason, they are often deliberately introduced, by shot-peening, for example. Any cracks that form are retarded in their growth rates, much as in the case of an overload.

B. Case Study: Shaft of a Golf Club

At a golf driving range in Connecticut the shafts (tapered hollow tubes) of the clubs were sometimes bent or otherwise damaged in use and had to be replaced. A supply of replacement shafts was kept in a storage shed, which was at times damp and humid. The shafts were made of a low-alloy steel that had been chrome plated, which protected the exterior of the shaft from corrosion. Whereas a new shaft is sealed to the club head as well as at the grip end. and thereby protected from corrosion on the inside of the shaft, the replacement shafts were not sealed, and as a result the interiors of these replacement shafts underwent corrosion over a period of time.

A golfer was driving balls with a club whose shaft was a replacement. As he struck a ball, the shaft fractured, and the end he was holding struck him in the eye. Fortunately, the damage to his eye was not serious, and he recovered completely. Upon examination of the shaft it was noted that the fracture origin was at a small dent in the shaft that preexisted the accident. The fracture had initiated at this dent on the inside of the shaft. The fracture origin, shown in Fig. 7-6, was more planar and brittle in appearance than was the fracture away from the origin, shown in Fig. 7-7, which exhibited the dimples characteristic of ductile fracture. It was concluded that the fracture was due to hydrogen embrittlement,

which was associated with corrosion and the seemingly innocuous dent which, because of springback, resulted in a residual tensile stress on the inside of the shaft at the periphery of the dent. When the dent was formed, material on the inside near the periphery of the dent went into compression, and material near the center of the dent went into tension. Upon springback, the signs of the stresses were reversed, and a residual tensile stress was left on the inside surface at the periphery of the dent.

Fig.7-6. Macroscopic view of the area of the fracture origin in a failed golf shaft. The rough area at bottom is the corroded inside surface of the shaft.(Reprinted from Material Characterization, Vol.26,A.J.McEvily and I. Le May,pp.253-268,Copyright 1991,with permission of Elsevier Science.)

Fig.7-7. Macroscopic view of fracture surface awwy from fracture origin in failed golf shaft. (Reprinted from Material Characterization, Vol.26,A.J.McEvily and I. Le May,pp.253-268,Copyright 1991,with permission of Elsevier Science.)

IV. RESIDUAL STRESSES DUE TO QUENCHING

On cooling a metal part from an elevated temperature, residual stresses may be developed. For example, if a massive piece of copper is cooled rapidly, the surface layers will cool before the interior, thus setting up tensile strains and stresses in the surface that are counterbalanced by compressive stresses in the interior. At elevated temperatures, these tensile and compressive stresses are relaxed due to the low yield strength of the material. However, at a later point in the cooling process, the already cooled surface layer will be subjected to compression as the interior finally cools and shrinks. In general, the resultant compressive stress in the surface layers is beneficial, except when machining follows and distortion results from the nonuniform removal of the surface.

When steel is quenched to form martensite at a low temperature, the transformation from austenite to martensite is associated with a volume expansion of the order of 1-3%. At the surface, this initially results in a compressive stress, as in the case of copper, but when the underlying layer finally cools and expands, the surface is put into tension. These tensile stresses can be reduced by using a steel of lower hardenability so that the interior does not got through a martensitic transformation, but instead transforms to bainite at a higher temperature.

B. Quench Cracking

In a steel, there is always the possibility of immediate quench cracking, due to the level of tensile residual stresses developed when untempered martensite is formed. However, cold cracking (or delayed cracking) is the more probable event, and for this reason, alloy steel parts that have been quenched are quickly transferred to tempering furnances or salt baths in order to mnimize the time available for cold cracking to develop. If cracks develop on quenching into water and the part is then transferred to a salt bath for tempering, an explosive reaction can occur as the water trapped in the cracks transforms into steam. Therfore, personnel carrying out this operation need to wear protective face masks and protective clothing.

It is also possible to develop quench cracks above the quench temperature if, during the quench of a material with low ductility, the tensile surface stresses that develop due to the temperture gradient in the material exceed the resistance to cracking of the material, as indicated in Fig. 7-8.

Fig. 7-8. A schematic of thermal stress versus resistance to cracking as a function of temperature.

To avoid the development of residual stresses, heat-treating procedures are used that minimize the temperature gradients responsible for the residual stresses. Such methods are:

(a) Martempering: The steel is rapidly cooled to a temperature above the Ms and held to allow a uniform temperature to develop before further cooling to martensite.

(b) Austempering: The steel is rapidly cooled to a temperature above the Ms and held there until the transformation to bainite is complete before further slow cooling to room temperature.

V. RESIDUAL STRESS TOUGHENING

If the outer surface of the bowl discussed above in the section on thermal shock had been treated to have a residual compressive stress in the surface region, the resistance to thermal shock would have increased. Glass can be toughened by diffusing large atoms into the surface to develop residual compressive stresses on cooling as, for example, in Corningware. Tempered glass is created by cooling rapidly from an elevated temperature to develop compressive surface residual stresses, much as in the case of a block of copper. This type of glass is used in the side windows of automobiles. When such glass breaks, it fractures into many small fragments. However, the glass used in windshields is not tempered. A windshield consists of a sandwich of two sheets of glass between which is bonded a sheet of plastic. In a crash, the windshield is intended to be flexible enough to reduce head injuries, but be resistant enough to overall fracture to prevent front seat occupants from being ejected through the windshield. In this case the glass generally breaks into large shards that remain attached to the plastic membrane.

VI. RESIDUAL STRESSES RESULTING FROM CARBURIZING, NITRIDING, AND INDUCTION HARDENING

A. Carburizing

Carburizing is carried out in the austenitic range and can lead to the development of a compressive residual stress at the surface in a low-alloy steel during quenching for the following reason (3, 4). Carbon is one of the elements that depresses the Ms temperature of a steel. The Ms temperature of the carburized surface layer, therefore, can be much lower than that of the interior because of the differential in carbon contents. On quenching, the interior, even though at a higher temperature than the surface, is the first to transform due to its higher Ms temperature. Later on, the surface transforms and tries to expand but is now restrained by the already transformed interior. As a result, the surface is left in a state of residual compression, as indicated in Fig. 7-9.

Fig. 7-9. Effect of carburization on the residual stresses in a quenched and tempered (1 hr at 180-200°C) 0.93 Cr-0.26 C steel (After Ebert, 3, and Krause, 4).

B. Nitriding

Nitriding is carried out at an elevated temperature below the eutectoid temperature for time periods of the order of 9-24 hours, and during the nitriding process any prior residual stresses are relaxed. The purpose of nitriding is to improve the surface wear and fatigue properties. Since the temperatures are lower than in carbur-izing and no phase transformation is involved, problems with distortion are minimized, an important consideration when heat treating carefully machined parts such as crankshafts. The formation of nitrides leads to a beneficial compressiveresidual stress in the surface even after the usual slow cooling because of a lower coefficient of expansion of the nitrides. C. Induction Hardening

Induction hardenmg is a surface-hardening process in which only the surface layer ot a suitable ferrous workpiece is heated by electrical induction to above the transformation temperature and immediately quenched. Compressive residual stresses develop as the surface layers transform from austenite on quenching.

VII. RESIDUAL STRESSES DEVELOPED IN WELDING

In welding operations, the parts being joined often provide a large heat sink, and therefore cooling rates are rapid. As a result, untempered martensite may form in a residual tensile stress field and lead to weld cracking, either through cold cracking or quench cracking. The susceptibility to weld cracking increases as the number of unfavorable welding

conditions increases. For example, a medium-carbon steel welded wuh an electrode not of the low-hydrogen type, and without preheat or postheat. may perform satisfactorily even though there may be a HAZ containing a region of

martensite about 1/16 inch thick if joint restraint is low and cyclic loading m service is limited. However, if the section thickness is doubled, the level of residual stress rises due to greater constraint, and the thickness of the martensitic zone is increased due to a higher cooling rate, and cracking may develop. Even the use of low-hydrogen electrodes may not prevent cracking of the thicker section but the use ot a 315℃ (600°F) preheat will prevent cracking by retarding the rate of cooling, and thereby reducing the level of the residual stresses and the amount of marten-site formed.

On cooling below the Ms temperature, two forms of martensite can form depending upon carbon content. In low-carbon steels, the martensite is made up laths that contain a high density of dislocations. In higher carbon steel, the martensite is made up of plates that also contain a high dislocation content, but in addition may be twined. Preheating is used to reduce the cooling rate in order to minimize the likelihood of the formation of brittle martensite, particularly the twinned martensite. The preheat temperature increases with carbon content and with the thickness Of the plates being welded, With recommended preheat temperatures ranging from 100°F for a 0.2 wt % C steel up to 600°F for a 0.6 wt % C steel

Postweld heating of a weldment serves two purposes. One is to relieve the residual stresses that may have been developed during the welding process. The second is to temper both the weld deposit and HAZ to improve their fracture toughness. It is now mandatory that weldments in all nuclear components and in most pressure vessels be postweld heat treated (PWHT). These postweld heat treatments are carried out at a temperature of the order of 650°C (1200°F) for one hour.

(a) Stresses parallel to weld (b) Stresses transverse to weld

Fig. 7-10. Transverse and longitudinal residual stresses developed at a butt weld. (From Gurney, 5. With permission of the Cambridge University Press.) The residual stresses that are developed both parallel to and transverse to a butt weld due to shrinkage of the weld metal on cooling are shown in Fig. 7-10 (5). To minimize distortion, weld passes were made from each edge of the plate toward the center, which accounts for the transverse residual stress distribution, since the last weld metal to solidify was in the central region of the butt weld. The residual stress developed during welding can sometimes be beneficial. For example, the fatigue strength of steel lap welded joints for automotive use was improved by using low transformation temperature (M.s

200°C, Mf20°C) welding wire (10Cr-10Ni), which induced compressive residual stresses in the surface layers. The fatigue

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strength at 10 cycles was increased from 300 MPa to 450 MPa by this procedure (6).

VIII. MEASUREMENT OF RESIDUAL STRESSES

The two main methods for the determination of the magnitude of residual stress, the x-ray method and the metal removal method, were described in Chapter 5. Another example of the metal removal method is known as the boring-out method. This method can be used to determine residual stress levels in cannon or other cylindrical bodies. In the case of a cannon barrel, as a last step in the manufacturing process, a pressure is developed within the barrel of sufficient magnitude to expand the interior of the barrel into the plastic range. When the pressure is removed, a beneficial state of residual compressive stress is developed on the inner surface, a process known as autofrettage. The determination of the magnitude of the residual stresses by the boring-out method involves the machining away of successive layers from the interior of the barrel. As each layer is removed, some of the residual stress is relieved. As a result, on the outer surface of the barrel, longitudinal and cir-cumferential strains develop which are measured by strain gauges. From these measurements the magnitude of the initial residual stress state can be deduced. X. SUMMARY

THe thermal stresses that arise in engineered items, such as the turbine disks of jet aircraft engines, are no longer a major problem because of improvements in materials as well as better design procedures. Nevertheless, their presence must be taken into account. Problems with thermal cracking during heat treatment continue. Residual stresses are still a problem because their presence may not be recognized until a cracking problem associated with them develops.

REFERENCES

(1) L. Remy, in Low Cycle Fatigue and Elasto-Plastic Behavior of Materials, ed. by K.-T. Rie and D. P. Portella. Elsevier, Oxford. UK.

1998, pp. 119-130.

(2) W. F. Hosford and R. M. Caddell. Metal Forming, 2nd ed., Prentice-Hall, Engelwood Cliffs. NJ, 1993.

(3) L. J. Ebert, The Role of Residual Stresses in the Mechanical Performance of Case Car-burized Steel. Met Trans. A. vol. 9A. 1978.

pp. 1537-I551. (4) G. Krauss, Principles of Heat Treatment of Steel. ASM. Materials Park. OH, 1980.

(5) T. R. Gurney, Fatigue of Welded Structures. Cambridge University Press, Cambridge, UK, 1968. p. 58. (6) A. Ohta, Y. Maeda. and N. Suzuki, in Proc. 25th Symp. on Fatigue, Japan Soc. Mats. Sci., 2000, pp. 284-287.