Residual Stresses

1.13

Residual Stresses

Aaron D. Krawitz, University of Missouri, Columbia, Missouri, USA Eric F. Drake, Rice University, Houston, Texas, USA Ó 2014 Elsevier Ltd. All rights reserved.

1.13.1 1.13.2 1.13.2.1 1.13.2.2 1.13.2.3 1.13.3 1.13.3.1 1.13.3.1.1 1.13.3.1.2 1.13.3.1.3 1.13.3.2 1.13.4 1.13.4.1 1.13.4.1.1 1.13.4.1.2 1.13.4.2 1.13.4.2.1 1.13.4.2.2 1.13.5 References

1.13.1

Introduction Method of Measurement The Diffraction Method Use of Neutrons Limitations of X-rays Bulk Thermal Residual Microstresses Magnitude Binder Volume Fraction Temperature Particle Size Distribution Interaction with External Loads Monotonic Loading Compression Tension Cyclic Loading Repeated Loading Stepped Loading Role of Residual Stresses in Mechanical Behavior

385 387 387 388 390 393 393 393 393 394 395 396 396 396 397 400 400 401 401 403

Introduction

The subject of residual stresses in cemented carbides is an interesting one. The thermal residual stresses that are established between the binder and hard phases upon cooling from liquid- or solid-phase sintering temperatures are large and complex. They arise due to the difference in thermal expansion between the metal binder and the refractory carbide. Such stresses are classified as thermal residual microstresses or Type II residual stresses. Type I residual stresses are macrostresses that equilibrate over the length scale of a part. They arise from thermal and/or deformation treatments that subject the part to differences in thermal or mechanical treatment, such as welding or shot-peening. Type III stresses are very short range and result from plastic deformation, usually the strain fields associated with dislocations and are generally not treated as residual stresses. Though Type II stresses in cemented carbide composites are created during cooling after liquid-phase or solid-state sintering and are ubiquitous in these materials, their presence has not been characterized and their role in material performance has not been understood. Interest in these stresses largely derives from their huge levels. The values are high in the metal binder phase due to the small volume fraction present and the constraint provided by the surrounding carbide particles. An estimate may be obtained from the misfit strain in a spherical WC particle in an infinite matrix whose properties are composed of the average composite values of thermal expansion and elastic constants. Using the values in Table 1, results are shown in Table 2 for WC–Co composites using Gurland’s approach (Liu & Gurland, 1965). The thermal stress in a sphere of WC surrounded by a sphere of material representing the average properties of the composite for a specified composition is given by
2EWC Ecomp acomp  aWC DT
sWC ¼ (1) 1 þ ncomp EWC þ 2ð1  2nWC ÞEcomp

Comprehensive Hard Materials, Volume 1

http://dx.doi.org/10.1016/B978-0-08-096527-7.00013-1

385

386

Residual Stresses Table 1

Some room-temperature properties of WC, Co and Ni

Properties E (GPa) K (GPa) G (GPa) n a ( C1)  106 r (Mg m3)

WC

Co

Ni

672 397 292 0.25 6.2 15.7

200 185 76 0.32 13.8 8.8

207 192 79 0.31 13.3 8.9

where s is the stress, E is the Young’s modulus, a is the linear coefficient of thermal expansion, n is the Poisson’s ratio, and the comp values for the composite are given by   fWC fCo 1 þ Ecomp ¼ EWC ECo acomp ¼ fWC aWC þ fCo aCo  ncomp ¼ Ecomp

fWC nWC fCo nCo þ EWC ECo

(2) 

where f is the volume fraction. Stresses for Co may be obtained by exchanging the WC subscripts for Co subscripts in Eqn (1). The values shown in Table 2 have been shown to be reasonable compared to those measured in typical materials. Thus, for a WC–10 wt% Co sample, values of þ2061 MPa for Co and 407 MPa for WC were obtained using neutrons (Livescu et al., 2005). The Co values cannot be directly measured because Co gains W and C in solid solution during liquid-phase sintering so that the reference (zero stress) cell parameter is not known. Rather, the Co stress values are obtained from the force–balance relation that applies for microstresses, namely that the force exerted by each phase must balance for equilibrium: fCo sCo þ fWC sWC ¼ 0

(3)

where fi and si are the volume fraction and average stress, respectively, of the ith phase (Hutchings, Withers, Holden, & Lorentzen, 2005; Noyan & Cohen, 1987). Though Gurland’s formulation is for an elastic system, the values in Table 2 are lower than those measured, though actually quite close. The actual values depend on more than just the relative amounts of the two phases and an effective set-up temperature. Factors such as cooling rate, carbide size, carbide shape, in situ binder yield strength, and the point-to-point distribution of local stress in both the particles and binder are not addressed by such analytical relations. As will be seen, carbide particle size can have a large effect on the thermal stress magnitude and distribution. The stress magnitude increases as the particle size decreases due to the greater constraint of the binder phase through reduced binder mean free path. Also, the distribution of stress within the carbide particles and matrix is substantial and important. It is due to the angular shape of the particles, the crystallography of the hexagonal WC, and the variable binder distances of the complex microstructure. The range of stress around the mean values in the binder and carbide is, we now know, significant. This has been investigated in a preliminary

Table 2 Thermal residual stresses using Gurland’s formulation (Liu & Gurland, 1965) wt% Co 5 10 20

vol% Co

sCo (MPa)

sWC (MPa)

8.6 16.5 30.8

þ2222 þ1840 þ1305

209 364 581

Residual Stresses

387

way using finite-element analysis and “real,” two-dimensional microstructure meshes, and documented through diffraction peak shape effects. Finally, the interaction of the residual stresses with applied load has been explored. The response to uniaxial tension and compression has been studied in detail, as well as effects due to cyclic loading. The results suggest that this interaction provides an important contribution to the unusual toughness of cemented carbide composites. The method of choice for these studies is neutron diffraction. Diffraction provides independent views of the binder and carbide phases, and their response to in situ loading. Neutrons are not subject to the extremely high absorption of X-rays in the presence of the heavy metal tungsten because they are uncharged particles, so that meaningful averages over many particles and freedom from surface effects may be obtained.

1.13.2

Method of Measurement

1.13.2.1

The Diffraction Method

The utility of diffraction to measure elastic strain can be seen using Bragg’s law: l ¼ 2dhkl sin qhkl

(4)

where l is the wavelength of the incident radiation, dhkl is the interplanar or d-spacing of the hkl planes, and 2qhkl is the diffraction angle for the hkl planes (Krawitz, 2001). The atomic plane spacings are used as “strain gauges” to measure changes due to elastic load. The idea is that (1) stress creates strain; (2) strain alters d-spacings; and (3) changes in d-spacings cause changes in diffraction peak positions. Cemented carbides are polycrystalline so that measurements can be made in arbitrary sample orientations, and individual diffraction peaks can be used to yield strain information in specific crystallographic orientations. Strain is given by ε ¼

d  do do

(5)

where d is the interplanar spacing and do is the stress-free interplanar spacing. Strains as low as 1  104 can be reproducibly measured. Obtaining stress-free interplanar spacings can be problematic. In cemented carbides with WC as the carbide, the WC is a reliable reference phase because it remains stoichiometric and does not take solute in solution. However, the metal binder does take W and C into solution during sintering, so that the starting binder powder cannot be used as a stress-free reference. In this case, as stated, Eqn (3) is used to obtain the mean binder stress. For applied stress (in situ) measurements, changes in binder stress can be readily measured relative to the startingdunstresseddvalue regardless of any uptake of W or C. In addition to the determination of existing residual stresses, in situ response of a material or component to mechanical and/or temperature loading can be observed. In situ capability can also be used for validation of finite-element modeling (FEM), analytical calculations, or vetting of other measurement methods (Krawitz, 2001). Recently, a multiaxial loading capability has been introduced, as well as a more sophisticated capability for making temporal measurements as a function of temperature or load (Liaw, Choo, Buchanan, Hubbard, & Wang, 2006). In principle, as mentioned above, in situ measurements are simpler with respect to stress-free reference values because the strains are reckoned relative to the initial values of interplanar spacing, regardless of the residual stress state: ε ¼

d  di di

(6)

where di is the initial d-spacing before load/temperature is applied, which may include microstresses and/or macrostresses. However, the situation becomes complicated if the preexisting stresses change during mechanical or thermal treatment (Paggett et al., 2007). High-temperature measurements may be particularly problematic as compositional changes can occur, in which case the initial values of the d-spacings are no longer meaningful. Thus, creep or other time-dependent changes with temperature (e.g. phase change, precipitation, and composition) are problematic with respect to the direct measurement of change in the binder phase (Mari, Clausen, Bourke, & Buss, 2009).

388

Residual Stresses

(a)

(b)

Diffracted beam

Plane normals

Incident beam

Figure 1 Schematic showing that different particles are selected to diffract at (a) low and (b) high diffraction angles.

The diffraction process involves a great deal of averaging, especially for powder diffraction. Figure 1 shows the schematic of a small portion of microstructure. Sketches 1(a) and 1(b) show that different grains are oriented to diffract for each hkl peak. This depicts a fixed wavelength, angular scan mode of data collection. There is also a fixed angle, variable wavelength (energy) mode in which case the grains satisfying the diffraction conditions vary with the wavelength and Bragg’s law becomes lhkl ¼ 2dhkl sin q

(7)

Due to the angularity of the WC particles, there is a considerable range of stress values, as discussed below. However, this range of values is averaged over the volume of each diffracting grain for the Bragg angle (or incident beam energy) being measured. Thus, grains in many different local environments contribute to a given diffraction peak, and another set contributes to the next diffraction peak, creating another layer of averaging. Finally, the information in each peak is averaged to create an average cell parameter for the structure. This is usually done by fitting the overall pattern using the Rietveld profile refinement method (Rietveld, 1969). Since there is no preferred orientation of the carbide grains, the same averaged diffraction pattern will be seen no matter how the sample is oriented in the beam because the statistical sample is so large (some 109 WC grains would be irradiated in a 3  3  3 mm3 volume). Thus, the average values of the cell parameters (peak position) and the range of those values (peak shape) do not change as a function of sample orientation in the beam even though there are significant variations from point to point within a grain, and from grain to grain. The stress state appears, after all the averaging is done, to be hydrostaticdthe same in all directions. The distinction between a mechanically hydrostatic and a diffraction hydrostatic stress state is in the shape of the diffraction peak. A particle under a true hydrostatic stress state, for example, a sphere surrounded by a matrix that has been thermally shrunk around it, would exhibit an instrumentally sharp peak, i.e. the peak breadth would represent the minimum breadth achievable due to beam size, beam divergence, monochromator mosaic, sample size, etc., because the stress in the sphere would be the same everywhere. In the diffraction hydrostatic case, the averaging process would result in a broadened peak due to the range of interplanar spacings diffracting. The situation is schematically illustrated in Figure 2. It shows (1) the sharpest peak obtainable for a given instrumental configuration, from an unstressed assemblage of annealed particles; (2) the peak obtained from a spherical particle under a uniform hydrostatic compression that is still instrumentally sharp but has shifted to higher angle due to a uniform hydrostatic compression; and, (3) the same peak but diffracting from angular particles that are under the same average particle compression as in (2). Since there are a range of stresses in the particles due to their angularity, the result is a peak whose mean position is the same but whose shape is broader.

1.13.2.2

Use of Neutrons

Neutrons enable study of the volumetric residual thermal microstresses in cemented carbides due to their much greater penetration power in most engineering materials because they are uncharged particles. Two types of neutron sources are now utilized. Reactor sources produce beams via nuclear fission from which fixed wavelengths are extracted with monochromators. The use of position-sensitive linear or area detectors enables whole

Residual Stresses

389

1

Intensity

0.8

Instrumental breadth no stress

Hydrostatic compression spherical particles

0.6 Hydrostatic compression angular particles

0.4

0.2

0 86

88

90

92

94

96

Diffraction angle (degrees)

Figure 2 Schematic showing the relative peak position and peak shape for (1) an instrumentally sharp peak from annealed powder particles of a carbide phase, for which the peak position is taken as 90 ; (2) spherical carbide particles under uniform hydrostatic compression; and (3) angular carbide particles under the same mean hydrostatic compression.

peaks to be collected without scanning, increasing throughput. Reactor spectrometers are similar in concept to X-ray instruments. Pulsed neutron sources, also called spallation sources, produce neutrons by the impact of accelerated particles on a heavy-metal target, rather than by fission. The resultant neutron beam has a range of velocity (energy) and, hence, wavelength. In this case, the diffraction angle is fixed and there is a range of wavelengths in the incident beam, typically from 0.05 to 0.50 nm. The energies are resolved using time-of-flight methods. This enables whole diffraction patterns to be recorded simultaneously, without the requirement of sample motion. The neutron energy, velocity and wavelength are related by   1 2 1 h 2 E ¼ mv ¼ (8) 2 2m l where E is the kinetic energy of the neutron, v is the neutron velocity, m is the neutron mass, h is the Planck’s constant and l is the wavelength (Bacon, 1975). The relation between time-of-flight t, distance traveled L, and wavelength l is given by t ¼

mLl h

(9)

More and more neutron measurements are being done at pulsed sources. Advantages include (1) the ability to record whole patterns from all phases simultaneously with no sample or detector movement; (2) the ability to measure in two orthogonal directions simultaneously, as shown in Figure 3; (3) superior data for lowsymmetry structures; and (4) greater ease of use for mechanical/temperature stages. A listing of both types of neutron facilities is given by Krawitz (2011). A few words about the flux (number of neutrons per unit area per unit time at a specified distance) of sources are in order, as the low flux in neutron beams relative even to laboratory X-rays is often cited as a limitation of the method. While it is true that incident beam flux is low relative to laboratory and, especially, synchrotron X-rays, this argument does not account for the “effective flux” represented by the measurement of whole diffraction patterns of all crystalline phases, and in two orthogonal directions simultaneously. These features represent a significant gain in effective flux over the traditional peak-scanning mode of recording used for many years on neutron powder instruments. Also, detector efficiency has improved. The performance of hundreds of measurements on engineering materials is testament to the practical utility of the method. Thus, the answer to the question “Can useful measurements be made in reasonable times?” is yes, within limits that allow many aspects to be studieddaspects that cannot be addressed as effectively in other ways.

390

Residual Stresses

Incident beam optics

Left bank measures in the transverse direction (Q⊥)

Right bank measures in the axial direction (Q )

Gauge volume Sample

Beam stop

Figure 3 The geometry often used for strain measurements at pulsed neutron sources, from Tanaka et al., 2002. The sample is at a 45 angle to the incident beam. There are two detector banks: the right bank measures in the axial direction (Qǁ) and the left bank measures in the transverse direction (Qt).

Neutrons measure in three dimensions, and therefore fundamentally differ from traditional X-ray stress measurements. A major implication of the three-dimensional character of neutron strain measurements is that stress-free interplanar spacings are required to convert measured d-spacings to strains. Useful discussions of stress-free reference values in neutron strain measurements can be found in the literature (Tanaka, Akinawa, & Hayashi, 2002; Withers, Preuss, Steuwer, & Pang, 2007). For WC-based cemented carbides, the procedure is to use loose WC powder as the stress-free reference material and to directly determine strain in WC. The powder should be contained in a can having a shape as close as possible to that of the samples. WC is an ideal standard material because it is stoichiometric and does not take binder elements into solution during sintering. To obtain thermal residual stress values, strain is converted to stress using pWC ¼ KWC DWC ¼ 3KWC εWC

(10)

where pWC is the average diffraction hydrostatic thermal stress of WC, KWC is the bulk modulus, and εWC is the strain measured in one direction by use of Eqn (10). The stress in the binder phase is obtained using Eqn (3). If the strain is not the same in all directions, then a stress state must be assigned that matches the observed data. Examples can be found in Krawitz (2001). A general diffraction text that discusses diffraction stress measurements (Krawitz, 2001) and more specialized neutron (Hutchings et al., 2005; Kisl & Howard, 2008), and neutron and synchrotron stress measurements (Reimers, Pyzalla, Schreyer, & Clemens, 2008) is also available. A standard source book on residual stresses and their measurement by diffraction is by Noyan and Cohen (1987).

1.13.2.3

Limitations of X-rays

The problem with using X-rays to study thermal residual stresses in cemented carbide composites is the high absorption of W, as indicated in Table 3. It leads to shallow beam penetration and the inability to properly measure the bulk, volumetric stress state. The absorption of a beam of intensity Io is given by I ¼ Io eml t

(11)

Residual Stresses

391

Table 3 Comparison of neutron and X-ray scattering and absorption. The scattering powers of neutrons (b) and X-rays (f) are given in scattering length units for direct comparison. ml is the linear absorption coefficient and t50% is the thickness to absorb 50% of the incident beam intensity at normal incidence Neutronsa

X-raysb

Z

b (1012 cm)

ml (cm1)

t50% (cm)

f (1012 cm)

ml (cm1)

t50% (mm)

C Al Co Ni W

0.6646 0.3449 0.250 1.03 0.477

0.98 0.10 3.89 2.10 1.45

0.71 6.93 0.18 0.33 0.48

1.00 2.58 5.94 6.24 17.62

15.8 133.9 2857 434 3251

439 51.8 2.43 16.0 2.13

a

Thermal neutrons with a wavelength of 0.1798 nm. Copper Ka X-rays with a wavelength of 0.154178 nm (average of Ka1 and Ka2).

b

where I is the intensity of the absorbed beam, ml is the linear absorption coefficient, and t is the thickness of the plate of material (or the path length). The scattering powers for the elements are shown in Table 3 as b for neutrons and f for X-rays. They have been represented as scattering cross-sections so they can be directly compared. It is seen that the absorption values for X-rays are, in general, much greater than for neutrons, typically three orders of magnitude greater. This is especially true for heavy elements. This is reflected in the t50% values in Table 3. These are the plate thicknesses required to reduce the intensity of a normally incident beam by 50%. In general, the values are in millimeters for neutrons and in micrometers for X-rays. The contrast is greatest for high-Z elements. Thus, it takes 4.8 mm of W to absorb half the intensity of a thermal neutron beam and 2.1 mm for Cu X-rays. This illustrates the difficulty of using X-rays to study heavy elements, a problem even with synchrotron radiation. The greater penetration of neutrons offers several advantages: (1) surface effects such as deformation due to grinding or polishing, or oxidation are avoided; (2) X-rays are largely confined to the relaxation region near the surface for microstresses whereas this is not an issue for neutrons, i.e. neutrons enable proper measurement of such stresses; (3) good volume sampling is achieved; and (4) the use of mechanical and/or thermal stages is greatly facilitated because neutrons can pass through enclosure materials and strain gauges. To emphasize potential problems with X-ray stress measurements in heavy-metal systems, the standard X-ray approach to the measurement of residual stresses is briefly considered. It assumes that the residual stress state is two-dimensional and lies in the plane of the surface of the sample. This is because of the shallow penetration of X-rays and the fact that stress components normal to the surface go to zero as the surface is approached. In addition, near-surface stress states are far greater in the plane parallel to the surface than normal to it. The assumption greatly simplifies the measurement and analysis. The surface is systematically tilted by the angle j in a surface direction given by 4 (Figure 4(a)). A view looking normal to the plane containing directions X3 and L3 is shown in Figure 4(b). The interplanar spacing d4j in the direction 4 at tilt j, which lies in the surface, is given by dfj ¼ do

1þn n sf sin2 j  do ðs11 þ s22 Þ þ do E E

(12)

where n is the Poisson’s ratio, E is the Young’s modulus, and s11 and s22 are the principal plane stresses (Krawitz, 2001). A plot of dfj versus sin2 j is linear and the slope gives the desired stress s4. The slope contains the term do, which is the stress-free interplanar spacing. The approximation do z dj¼0 is made, which introduces a very small error (equal to the elastic strain, i.e. a few tenths of a percent or less). It also alleviates the need for knowing do. For WC–10 wt% Co, depths from which 50 and 90% of the diffracted beam arises are given in Table 4 for laboratory X-rays and thermal neutrons (Krawitz, 1985). Let us view the situation through the classical d versus sin2 j plot. Suppose the WC is subject to an average thermal microstress of 400 MPa at room temperature. If the WC 201 peak is measured as a function of j-tilt and plotted as d versus sin2 j, the possibilities are shown in Figure 4(c). The d-spacing for the (201) planes is 0.11520 nm. If the WC phase is free of stress, the d versus sin2 j plot would be the horizontal line marked “stress free”. If it is under an average stress of 400 MPa, the result would be the horizontal line marked “hydrostatic”. If the beam penetration is so low that only the surface is

392

Residual Stresses

(a)

(b) Diffracting grain

Sample Ψ = 0°

Ψ = 20°

Ψ = 40°

Normal to diffracting grains

(c)

(d)

0.11522

62.6 Biaxial

WC

Stress free

WC in WC-Ni

0.1152 62.4 0.11519 °2θ

WC 201 d-spacing (nm)

0.11521

0.11518

WC powder 62.2

0.11517 Hydrostatic 0.11516 0.11515

62.0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

sin2 ψ

sin2 ψ

Figure 4 (a) Geometry of the sin2 j method of diffraction stress measurement. The angle 4 determines the direction in the surface plane that the stress is measured. Values of interplanar spacing are measured as a function of sample tilts given by j. (b) View of j tilts normal to the X3–L3 plane. Three j tilts are shown. The stress component that is horizontal and in the plane of the surface is being measured. (c) Schematic showing the variation in diffraction peak position with sample tilt j for a material that is in three different stress states: (i) stress free; (ii) hydrostatic; and (iii) a biaxial stress state in the plane of the sample surface. (d) Actual results for the WC phase in a WC–Ni cemented carbide with angular particles. The WC is under compression that is diffraction hydrostatic.

measured, a two-dimensional stress state results for which the limiting case is the line marked “biaxial”. If the penetration depth contains the beam within the relaxation region of the volumetric thermal residual stress, a sloped line would result somewhere between the “biaxial” and “hydrostatic” lines. If the partially relaxed line is processed using Eqn (12), a compressive stress of a significantly lower value than the true 400 MPa would result. Another way to say this is if a sloped line is obtained, the true thermal microstress is not being measured, provided no other sources of residual stress are present. An example of a d versus sin2 j measurement with neutrons is shown in Figure 4(d) (Krawitz, Reichel, & Hitterman, 1989). It was made at a reactor neutron source and is plotted as j-tilt versus diffraction angle. It shows the diffraction hydrostatic nature of the thermal residual Table 4 Depths of penetration in WC–10 wt% Co from which 50 and 90% of the diffracted intensity arises for four X-ray wavelengths and thermal neutrons Radiation

ml (cm1)

50%

90%

Mo Cu Co Cr 1.8 Å neutrons

1239 2594 3691 5981 1.28

2.7 1.3 0.9 0.6 2.6  103a

9.1 4.3 3.0 1.9 8.8  103a

a

0.26 cm and 0.88 cm, respectively.

Residual Stresses

393

stress state in cemented carbide composites: the d versus sin2 j line has no slope. The compressive stress in WC reduces the d-spacing, thereby increasing the Bragg angle relative to the unstressed state. An illustrative work on steel matrix–NbC particulate composites used both X-rays and neutrons (Pyzalla, Genzel, & Reimers, 1996). It grapples with the complexities of using X-rays to attempt measurements of bulk thermal microstresses. From about 1960 to 1980, a number of X-ray studies were conducted and have been reviewed by Krawitz (1985). Some dealt with surface grinding/polishing issues and not just thermal stresses. Near-surface stresses due to grinding, thermal spray coatings, or even diamond coatings can, in principle, be measured with X-rays. However, it is possible that, for the case of surface ground/polished cemented carbides, relaxation of the thermal stress state affects the results. If it is known that the surface treatment relieves the thermal stress, leaving a macrostress due to plastic deformation of the binder, then useful data can be collected. Coatings may be very appropriate for laboratory and/or synchrotron X-ray study, and indeed many such studies are being made because of the importance of coated products. It is not the purpose of this chapter to evaluate such studies.

1.13.3

Bulk Thermal Residual Microstresses

A number of neutron diffraction measurements have been made using model WC–Ni-cemented carbides. This is because Ni is a very good neutron scatterer (Table 3) and has a stable fcc structure over a wide range of temperature. Co, on the other hand, scatters neutrons poorly and tends to stay in the high-temperature fcc structure upon cooling rather than transforming to the room-temperature hcp phase due to the sluggish fcc-tohcp transformation. The reasoning was that WC–Ni would more clearly reveal the fundamental responses of the composites with respect to binder content, carbide particle size, temperature, stress distribution, and interaction with external loads. Studies have also been performed on WC–Co materials and are included here.

1.13.3.1

Magnitude

1.13.3.1.1 Binder Volume Fraction The role of binder fraction is shown in Figure 5, for both WC–Co (Coats & Krawitz, 2003; O’Quigley, Luyckx, & James, 1997) and WC–Ni (Paggett, 2005; Paggett et al., 2007). The square symbols are from samples of a different source than the triangle symbols. However, only WC particle sizes of about 1 mm were used. The “x” symbols are calculated values using Eqn (1) for WC and Eqn (3) for Ni and Co. A temperature drop of 800 K from the sintering temperature was used. This DT was chosen to best fit the data and is rather high even though Eqn (1) is an elastic formulation. The constraint imposed by the fine-scale three-dimensional microstructure is not part of analytical calculations and is complicated for finite-element formulations, where a realistic threedimensional microstructure is required for an optimal result. As a result, an artificially high set-up temperature is needed to account for the constraint of surrounding WC particles. However, the calculated values show the proper functional form for variation of binder content over a wide range (about 5–50%). Processing and binder (Co and Ni) do not make much difference. 1.13.3.1.2 Temperature Samples of WC–9.2 vol% Ni and WC–29 vol% Ni were measured in the temperature range 100–900 K (Seol, Krawitz, Richardson, & Weisbrook, 2005). The resultant thermal stresses are shown in Figure 6. The set-up temperature is about 900 K. The stresses are elastic so the curves are reversible with thermal cycling, though low-level damage likely accrues. This damage would eventually appear as a change in peak shape and position, particularly for the binder, and would indicate accumulating plastic damage in the binder and a concomitant relaxation in the elastic thermal stress state, which would shift the peaks of both phases. Presumably this would lead to binder embrittlement after which crack initiation and growth would proceed. These data also show the effect of composition discussed above. The residual stresses for these materials are very high because the WC particle size is 0.5 mm (Section 1.13.3.1.3). Mari et al. measured stress as a function of temperature in WC–11 wt% (17.8 vol%) Co, and followed two heating–cooling cycles from room temperature to 1273 K (Mari et al., 2009; Mari, Krawitz, Richardson, & Benoit, 1996). Above about 1000 K, the cell parameter of Co increases due to solubility of W and C, not to residual stress. The observed hysteresis between heating and cooling is attributed to a “difference in heating and cooling kinetics of solution-precipitation”. These important studies indicate the complexity of the systems at temperatures where diffusional changes can occur.

394

Residual Stresses 4000

Thermal residual stress (MPa)

3000

Binder

2000

1000

0

–1000

–2000

WC

0

10

20

30

40

50

60

Volume percent binder

Figure 5 Compilation of many WC-based cemented carbides, all with approximately 1 mm WC particles. The experimental values are for WC–Ni (upward triangles) (Paggett et al., 2007) and WC–Co (squares (Coats & Krawitz, 2003; O’Quigley et al., 1997) and downward triangles (Paggett, 2005)). The “x” symbols are calculated values using Eqn (1). Experimental binder stress values are calculated using Eqn (3).

1.13.3.1.3 Particle Size The variation of the thermal residual stress with binder content and carbide particle size for a series of WC–Co composites is shown in Figure 7 (Coats & Krawitz, 2003; O’Quigley, Luyckx, & James, 1997). Composites with 10, 20 and 40 wt% Co (16.4, 30.6 and 54.0 vol% Co, respectively) and four particle sizes were measured for thermal stress. The particle sizes are ultrafine (0.6 mm), fine (1.0 mm), medium (3 mm) and coarse (5 mm). Two effects are represented. First, as the amount of a phase increases, its average residual stress decreases to satisfy the force balance. This composition effect has been discussed above. Second, for a given composition, as the WC particle size decreases, the stress magnitude in both phases increases. For example, as the WC particle size is

Thermal residual stress (MPa)

4000 9.2 vol.% Ni

3000

Ni 2000 29 vol.% Ni

1000

0

9.2 vol.% Ni

–1000

29 vol.% Ni

WC –2000 0

200

400

600

800

1000

T (K)

Figure 6 Thermal residual stress from 100 to 900 K for samples of WC–9.2 vol% Ni and WC–29 vol% Ni (Seol et al., 2005). The stresses are elastic and reversible with heating and cooling.

Residual Stresses

395

3000 Coarse WC Medium WC Fine WC Ultrafine WC Coarse Co Medium Co Fine Co Ultrafine Co

Thermal residual stress (MPa)

Co 2000

1000

0

–1000

10

WC 20

30

40

50

60

Volume % Co

Figure 7 The strong effect of carbide particle size on the thermal residual stress for a matrix of WC–Co samples (Coats & Krawitz, 2003; O’Quigley et al., 1997).

reduced from 5 to 0.6 mm for WC–10 wt% Co, the thermal stress in the Co increases from þ1500 to þ2600 MPa, and, for WC–40 wt% Co, the change is from þ400 to þ1000 MPa. The mean free paths in the Co binder range from 6.3 mm for coarse WC (5 mm) and 54.0 vol% Co to 0.2 mm for ultrafine WC (0.6 mm) and 16.4 vol% WC (Coats & Krawitz, 2003; O’Quigley et al., 1997). This very strong particle size effect has not, to our knowledge, been analytically modeled.

1.13.3.2

Distribution

The residual thermal stress values are averages over the volume of the sample, as discussed in Section 1.13.2.1. Figure 8 shows how the breadths of diffraction peaks from each phase, the WC 201 peak and the Ni 311 peak, vary with T for the samples shown in Figure 6 (Seol et al., 2005). These values are the Gaussian component of the peak breadths and are a measure of the range of elastic stresses in the sample. They are directly compared with breadths of the same peaks from annealed, stress-free WC and Ni powders, which do not change with T and represent the instrumental breadth plus any (minor) broadening sources in the annealed powder material. The breadth values are in microseconds as the data was taken at a pulsed source; see Eqns (7) and (9). This breadth versus T response is elastic, that is, the broadening is due to the distribution of elastic strain that exists in the irradiated volume. As the mean stress increases, the distribution broadens, and this effect is essentially reversible over a small number of cycles. However, as discussed in Section 1.13.3.1.2, many thermal cycles would lead to irreversible changes, though such an experiment has yet to be done. Finally, it is noted that if the Ni content increases, the stress in WC decreases and vice versa. The elastic strain distribution cannot be directly converted to a stress distribution (Krawitz, Winholtz, & Weisbrook, 1996). However, bounds can be set between pure deviatoric and pure hydrostatic limits. It has been shown that the stress state is close to the deviatoric (lower) bound in similar WC–Ni material (Krawitz et al., 1996). For the cemented carbides shown in Figures 6 and 8, the room-temperature thermal stresses in the WC phase are about 300 and 800 MPa for the 9.2 vol% Ni and 29 vol% Ni composites, respectively. The deviatoric standard deviations are about 400 and 800 MPa, respectively. It seems clear that the range of stress in WC ranges from very high compression to significant tension. Conversely, the binder stress ranges from compression to very high tension. It is apparent from the foregoing that diffraction and the sampling capability of neutrons offer an unprecedented view of the micro- and macrobehavior of cemented carbides. However, the quantification and roles of microscale plasticity behavior in these composites have awaited the application of microstructural FEM. Such work allows independent assessment of both elastic and plastic strain components developing during thermal

396

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10 0

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T (K)

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Figure 8 Variation of peak breadth for WC–9.2 vol% Ni and WC–32 vol% Ni sample as a function of T. These are the same samples as shown in Figure 6. (a) WC 201 peak; (b) Ni 311 peak.

and mechanical loadings. Such modeling would enable both prediction of macroscopic response and interpretation/validation of diffraction results. Finite-element studies have been conducted to gain insight into diffraction results, using a series of WC–Ni cemented carbides that were both modeled and physically produced and measured (Weisbrook, Gopalaratnam, & Krawitz, 1995; Weisbrook & Krawitz, 1996). The meshes employed were two-dimensional plane stress elastic models based on real microstructures and looked schematically like that shown in Figure 1. These model results corroborated both the diffraction mean stresses in WC and Ni and the observed broad distributions, including regions of tension and extreme compression in WC and regions of compression and extreme tension in Ni. They also provided insights into how these distributions develop and resolve on the scale of the microstructures, for example, showing the highest tensile stresses in WC at corners and near WC/Ni interfaces. Compressive areas in Ni were less widespread as the WC content increases, occurring in narrow Ni bands between WC grains. This is because the mean binder stress becomes increasingly tensile for high carbide-fraction material, which is the usual case.

1.13.4

Interaction with External Loads

1.13.4.1

Monotonic Loading

1.13.4.1.1 Compression Applied stress interacts with the preexisting thermal residual stress in cemented carbide composites. Figure 9 shows the axial response of the Ni phase in WC–20 wt% Ni as well as the macroscopic stress–strain curve for uniaxial compression to 2000 MPa (Paggett et al., 2007). The Ni curve shows only the elastic strain measured by diffraction while the composite curve shows the sum of the elastic and plastic strain response. Both curves show nonlinearity that begins below 0.2% strain. The rate of strain accumulation increases for total macroscopic strain in the composite but decreases for elastic strain in the Ni. Upon unloading, the composite sample is shorter by more than 0.3% while the elastic tensile strain in the Ni has actually increased by almost 0.1%. Elastic strain changes in both the axial and transverse directions are shown in Figure 10 for the Ni phase. These are the responses of the Ni phase during uniaxial compression to 2000 MPa for (1) WC–5 wt% Ni, (2) WC–10 wt% Ni, and (3) WC–20 wt% Ni. The transverse response for WC–5 wt% Ni (Figure 10(a)) shows that accumulation of positive Poisson strain slows down and actually begins to reverse at the end of the load cycle. After unloading, there is a net reduction in transverse strain and a net increase in axial strain. The increase in axial strain is due to the Poisson effect that results from the (greater) decrease in transverse strain. This trend increases in the 10% Ni and 20% Ni composites, where plasticity in the Ni is greatly increased. As for the 5% Ni material, the transverse strain magnitude in the 10% Ni and 20% Ni composites initially increases due to the axial compression, then decreases. The decrease is so great that the strain goes below the initial value for both

Residual Stresses

397

0

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Macro σ – ε

–500

–1000

–1500 Ni

–2000 –8000

–6000

–4000

–2000

0

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Figure 9 The macroscopic (elastic–plastic) load–unload stress–strain curves and the Ni phase load–unload (elastic-only) stress–strain curves for WC–20 wt% Ni in the axial direction (Paggett et al., 2007).

compositions. It is emphasized that these changes are relative to the starting values of thermal residual strain, and are the same in all directions when averaged over all diffracting grains. The Poisson reaction in the axial direction leads to an increase in the mean Ni strain in that direction even though there is an overall reduction in the thermal residual stress. The mechanics of the thermal stress relaxation is shown schematically in Figure 11. The applied uniaxial compressive strain opposes the mean tensile residual stress in the axial direction (Figure 11(a)). However, in the transverse direction, the applied Poisson strain is tensile and adds to the thermal residual strain in the Ni. This leads to preferential flow of the Ni in the transverse direction and asymmetric relaxation of the thermal residual stress. The transverse relaxation, in turn, induces a Poisson expansion in the axial direction, as indicated in Figure 11(b). For the WC–20 wt% Ni composite, the result is reduction in the transverse and axial thermal residual stresses by 523 and 227 MPa, respectively. Thus, the relaxed residual stress state becomes cylindrical and, for the WC–20 wt% Ni sample, is þ1778 MPa in the axial direction and þ1482 MPa in the transverse direction. To summarize, upon uniaxial compressive loading/unloading, the thermal residual stress in the Ni phase decreases in both the axial and transverse directions, but does so asymmetrically; the decrease is greater in the transverse direction. The elastic strain in the Ni phase, however, decreases in the transverse direction but increases in the axial direction, due to the Poisson effect. Macroscopic load–unload stress–strain curves of WC–10 wt% Ni for 500 and 2000 MPa are shown in Figure 12(a). An enlarged plot of the 500 MPa load–unload sequence is shown in Figure 12(b). This is shown to emphasize that nonlinearity begins very early in the loading cycle. By 500 MPa, a clear hysteresis is present. This is due to the interaction between the applied stress and the thermal residual stress. It can be shown that composite density is not conserved when the thermal residual stress levels decrease in the WC and Ni phases. The forces remain balanced but the density actually increases upon relaxation of the initial thermal stress. This is because, upon relaxation of the thermal residual stresses in the Ni and WC, the decrease of the Ni phase volume is greater than the increase of the WC-phase volume so that the density of the composite increases. 1.13.4.1.2 Tension The application of uniaxial tension also creates an asymmetric relaxation of the thermal residual stress, but in the opposite sense of that for compression. This is shown schematically in Figure 13. In this case, the applied axial tension leads to preferential flow of the Ni in the axial direction because the applied strain adds to the positive residual strain in this direction. In the transverse direction, the applied Poisson compressive strain is now negative and opposes the mean Ni thermal residual strain, which is tensile (positive). The result is greater

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Figure 10 The Ni phase load–unload stress–strain curves in the axial and transverse directions due to uniaxial compression in (a) WC–5 wt% Ni, (b) WC–10 wt% Ni, and (c) WC–20 wt% Ni.

(a)

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ε <0 Figure 11 Schematic response of WC–Ni to uniaxial compression. (a) Applied strain to composite in the axial and transverse directions. (b) Anisotropic relaxation response of the thermal residual stress for the Ni phase.

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Figure 12 The WC–10 wt% Ni macroscopic load–unload stress–strain curves of (a) 500 and 2000 MPa loadings and (b) enlargement of 500 MPa load–unload. Note that the 0 load level is actually 10 MPa in order to keep tension on the sample. This is true in all plots but is visible here due to the expanded scale.

relaxation of the thermal residual stress in the axial direction than in the transverse direction. In this case, the change in elastic strain in the Ni is negative in the axial direction and positive in the transverse direction. The situation is illustrated for a WC–10 wt% Ni composite that was subjected to a þ1500 MPa tensile stress, the highest value that could be obtained without fracture (Figure 14) (Krawitz, Drake, & Clausen, 2010). The response of the Ni phase is shown in Figure 14(a). For comparison, Ni-phase curves are shown for loading to 1000 MPa (Figure 14(b)). The relaxation asymmetry is reversed for tensile versus compressive loading. The macroscopic load–unload stress–strain curves for WC–10 wt% Ni to þ1500 MPa are shown in Figure 15(a). The corresponding curve for WC–10 wt% Ni loaded to 1000 MPa is shown in Figure 15(b). (The macroscopic load–unload stress–strain curves to 500 and 2000 MPa are shown in Figure 12.) The reverse nature of the relaxation asymmetry is clearly shown. For the 1000 MPa case, the composite has already been subjected to three load–unload cycles to 500 MPa, which induced some plasticity. Thus, the effect should be somewhat stronger than it appears. To summarize, upon uniaxial tensile loading/unloading, the thermal residual stress in the Ni phase decreases in both the axial and transverse directions, but does so asymmetrically: the decrease is greater in the axial direction. The elastic strain in the Ni phase, however, decreases in the axial direction but increases in the transverse direction, due to the Poisson effect. In both the compressive and tensile cases, the applied plastic strain is opposed by the change in Ni strain due to relaxation of some of the thermal residual stress. However, for applied uniaxial compression, it is the tensile

(a)

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- ++ - - - +-

- - + - -- ++ -

Poisson compression

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ε >0 Figure 13 Schematic response of WC–Ni to uniaxial tension. (a) Applied strain to the composite in the axial and transverse directions. (b) Anisotropic relaxation response of the thermal residual strain for the Ni phase.

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Figure 14 The Ni phase load–unload stress–strain curves for WC–10 wt% Ni under (a) 1500 MPa tension and (b) 1000 MPa compression.

Poisson reaction of the Ni elastic strain that opposes the applied compression while, for applied uniaxial tension, it is the direct reduction of the Ni elastic strain that opposes the applied uniaxial tension. This suggests, as is observed, that the overall macroscopic length change will be greater for applied compression.

1.13.4.2

Cyclic Loading

1.13.4.2.1 Repeated Loading The effect of repeatedly loading a cemented carbide was studied using a WC–10 wt% Ni sample subjected to 100 cycles of uniaxial compression from 10 to 2500 MPa (Krawitz, Venter, Drake, Luyckx, & Clausen, 2009). Diffraction data were taken during load–unload cycles 1, 2, 3, 10, 25, 50 and 100. The macroscopic response of the Ni phase for cycles 1 and 100 is shown in Figure 16. The changes occurring in the Ni phase are shown in Figure 17. The relaxation process is best seen as a function of the number of cycles (Figure 18). Most of the change occurs during the first three cycles, with a stable state reached after about 10 cycles. Although the

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Figure 15 The macroscopic load–unload stress–strain curves for WC–10 wt% Ni for (a) þ1500 MPa tension and (b) 1000 MPa compression.

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Load cycle 100 Unload cycle 100

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Figure 16 Macroscopic load–unload stress–strain curves for WC–10 wt% Ni loaded to 2500 MPa for 100 cycles.

relaxation stabilizes rather early in the process, there is still a hysteresis present in both the axial and transverse directions. Most of the composite volume does not yield but rather is elastically strained under load. This creates a range of strain in the sample that contributes to a strain variance (peak broadening) that is reversed upon unloading. The peak breadth responses follow a similar pattern and, after 10 cycles, are about 75% of the starting values, that is, they narrow because the mean thermal residual stress decreases (Krawitz et al., 2009). 1.13.4.2.2 Stepped Loading The stepped loading response of WC–Ni helps explain the response of the composite in service, and in accounting for the unusual toughness of cemented carbides (Figure 19). The hysteresis closes significantly after three cycles but, if the load is subsequently increased, the process begins again, as shown by the first cycles to 1000 and 2000 MPa following the initial loading to 500 MPa. A component in service would be able to absorb additional energy through the plastic deformation/relaxation process in regions where it experienced an increase in load as well as through the ongoing hysteresis process.

1.13.5

Role of Residual Stresses in Mechanical Behavior

Long-recognized but unexplained mechanical behavior anomalies of cemented carbides are now attributable to the influences of residual stresses, including small-strain yielding and plasticity-induced relaxation. The absence of linear elastic load response of commercial WC–Co grades is perhaps the principal observation. Felgar and Lubahn (1957) measured samples in both tension and compression. Nonlinearity in tension was observed at strains as low as 0.1%, as well as nonlinear responses, that were termed “anelastic”, in both tension and compression. These behaviors, identical to those seen herein, are due to thermal stress relaxation and the volumetric, asymmetric nature of the response of the system to applied stress. Another example is that preloading in compression has been observed to increase density and cause asymmetric Palmquist crack lengths (Exner & Gurland, 1970). Both are due to the asymmetric stress relaxation that results from the interaction of nonuniform applied strains with the preexisting thermal residual stresses. In general, the longer Palmquist cracks are in the direction of higher residual stress while the shorter ones are in the direction that has experienced greater relaxation. Finally, variation in Poisson’s ratio was observed during uniaxial compressive loading (Drake, 1980). Specifically, the initial value of Poisson’s ratio in a WC–15.6 wt% Ni sample began decreasing with the onset of load for the two load cycles measured. This is due to the greater stress relaxation in the

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Figure 17 The Ni phase response in WC–10 wt% Ni after 1 and 100 cycles of repeated compression loading to 2500 MPa: (a) axial direction; (b) transverse direction.

directions normal to the compression axis. Poisson’s ratio n ¼ εx/εz and the transverse direction x experiences a higher degree of strain relaxation than the direction of compressive loading, z. The plasticity behavior appears to comprise a primary source of toughening characteristic of cemented carbides. This suggests the feasibility of sufficiently realistic and scaled-up microstructural modeling, which would provide predictive capability for mechanical response of cemented carbide composites in terms of composition, microstructure, and thermal and load history. Toughness, perhaps the most important attribute of cemented carbides, and the one that distinguishes them from other engineering materials at equivalent hardness levels, has been quantified by fracture toughness testing based on linear elastic fracture mechanics (Chermant & Osterstock, 1976; Igelstrom & Nordberg, 1974; Lueth, 1972; Murray, 1977). Much effort has been given to explaining and predicting observed toughness in terms of microstructural parameters and in situ elastic/plastic behavior of the binder and carbide phases (Chermant & Osterstock, 1976; Igelstrom & Nordberg, 1974; Murray, 1977; Pickens & Gurland, 1978). However, assumptions implicit in this approach including far-field linear elasticity and isotropy are not valid for cemented carbides. The emergent view of cemented carbide mechanics seems to require a new, nonlinear-elastic model of 1500 1000

Strain relaxation (με με)

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0 –500 –1000 Transverse

–1500 –2000 0

20

40

60

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Figure 18 Strain relaxation versus number of cycles for WC–10 wt% Ni under repeated uniaxial compression to 2500 MPa.

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Figure 19 Stepped loading of WC–10 wt% Ni: three cycles to 500 MPa, three cycles to 1000 MPa and then three cycles to 2000 MPa. The loading cycles are solid lines; the unloading are dashed.

toughness behavior in these materials, both in terms of bulk “continuum” response and response in the presence of defects. Moreover, such models, if they are to provide accuracy, must also take into account the documented anisotropic relaxation and plasticity effects and their sensitivity to load directionality and history. In conclusion, it is apparent from the foregoing that diffraction techniques in combination with the sampling capability of neutrons provide an unprecedented view of the micro- and macroelastic behaviors in cemented carbides, and also allow by deduction an emergent understanding of macroplasticity response. However, the quantification and roles of microscale plasticity behavior in these composites has awaited the application of microstructural FEM. Recent 3-D elastic–plastic FEM efforts in this vein utilizing idealized cemented carbide microstructures with increasing levels of similitude have semiquantitatively corroborated the measured micro and macro components of residual stresses and their interaction with applied loads (Livescu et al., 2005), demonstrating the development of elastic and plastic strains on a microstructural scale arising from both thermal and applied loadings.

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