WO2015109144A1

WO2015109144A1 - Method for determining stress using raman spectroscopy

Info

Publication number: WO2015109144A1
Application number: PCT/US2015/011695
Authority: WO
Inventors: Ghatu Subhash; Muhammad Shafiq
Original assignee: University Of Florida Research Foundation, Inc.
Priority date: 2014-01-16
Filing date: 2015-01-16
Publication date: 2015-07-23

Abstract

The stress at one or more sites on a Raman active material can be determined by a linear relationship between the stress at a site and the Raman peak-shift for Raman peaks of stressed to unstressed transverse optical (TO) and/or longitudinal optical (LO) Raman peak positions at that site. The method is carried out using a Raman Spectrometer, optionally with a laser excitation source, where a multiplicity of Raman peak-shift at a multiplicity of sites allows the construction of a stress map in and on the material.

Description

DESCRIPTION

METHOD FOR DETERMINING STRESS USING RAMAN SPECTROSCOPY CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application Serial No. 61/928,241 , filed January 16, 2014, the disclosure of which is hereby incorporated by reference in its entirety, including all figures, tables and drawings.

BACKGROUND OF INVENTION

Ceramics exhibit superior ballistic performance when confined by a lateral stress due to delayed onset of brittle cracking when external loads are applied. High confinement stress also induces ductile deformation mechanism in the ceramic, which allows high levels of energy absorption during a ballistic event. A typical confined compression test consists of shrink fitting a metallic sleeve on to a ceramic specimen, where, based on the interference fit between the ceramic specimen and the metal sleeve, one can calculate the level of confinement. Chen et ah , J. Am. Ceram. Soc. 1996 79, 579-84 and Chen et ah, J. Meek Phys. Solids 1997, 45, 1303-28, disclosed the use of shrink-fit metal sleeves to study deformation behavior of A1N and Macor under static and dynamic loading conditions. Nielsen et ah, Powder Technol. 2006, 161, 220-6, applied confinement using different metallic tubes on granular powder: where the radial pressure was estimated by correlating radial bulging of the tube with FE analysis of tube bulging. Paliwal et ah, J. Am. Ceram. Soc. 2008, 91, 3619-29 used planar confinement to partially confine AION, and directly observing failure processes during high strain rate compression tests. Chocron et ah, J Appl Mech 2008, 75, 021006-1-7 and Dannemann et ah , Advances in Ceramic Armor II: Ceramic Engineering and Science Proceedings., John Wiley & Sons, Inc., 2008, 119-30 have performed axial compression experiments on borosilicate glass using a modified confinement technique that enabled attainment of high lateral pressure of up to lGPa. While these approaches are well suited for fundamental investigation of deformation mechanisms in ceramics under multi-axial compression, there is no reliable non-destructive experimental method to determine the residual stress in a ceramic that is confined in a device. BRIEF SUMMARY

Embodiments of the invention are directed to a method for determining local residual stress in a material, such as a ceramic or semiconductor that is Raman active, where the peak position of the unstressed material's transverse optical (TO) Raman peak position and/or unstressed longitudinal optical (LO) Raman peak position is known and one or more Raman spectra of a material suspected to be under stress is acquired and the TO Raman peak position and/or LO Raman peak position of that material is observed. The peak-shift observed for the peak position of stressed material from that the peak position for the unstressed material is calculated from which the magnitude of the stress can be determined from a linear relationship between the Raman peak-shift and the stress. The Raman peak-shift is a positive value for a tensile stress and a negative number for a compressive stress. The Raman Spectrometer can employ a laser excitation source having a sub-micron cross-section to selectively excite sites on the material to create a stress field map for the material.

BRIEF DESCRIPTION OF DRAWINGS

Figure 1 shows Raman peak positions for SiC in annealed (stress-free) and as-sintered ZrB2-SiC composite revealing the presence of thermal residual stresses through Raman peak- shift in the sintered specimen.

Figure 2 shows an SEM image of polished ZrB₂-SiC surface revealing SiC (dark phase) embedded in ZrB₂ matrix (gray phase).

Figure 3 shows: (a) a photograph of cylindrical specimens of ZrB₂-SiC (along with blank) cut using EDM; (b) a photograph of shrink-fit specimen assembly with various metal sleeves; and (c) a schematic of an axisymmetric boundary value problem at the cross-section of an elastic ceramic specimen and elastic-perfectly plastic hollow cylindrical metal sleeve.

Figure 4 shows: (a) a photograph of an uncoated surface (for Raman measurements); (b) a photograph of a coated surface (for DIC) of shaft-collar ring and ZrB₂-SiC assembly; and (c) DIC setup comprised of cameras, extension tubes, lenses and shaft-collar ZrB₂-SiC specimen assembly.

Figure 5 shows a Raman spectrum of stress free SiC powder particles.

Figure 6 shows a plot of measured Raman peak-shifts and the estimated lateral pressures from Equation (18), according to an embodiment of the invention, along the radius of ZrB₂-SiC specimens. Figure 7 shows a plot of estimated lateral pressures obtained using Equations (20) and (18), according to an embodiment of the invention, along the radius and as a function of Raman peak-shift.

Figure 8 shows: (a) a plot of DIC measured on the surface of ZrB₂-SiC; (b) a plot of average Von-mises strain on the surface of ZrB₂-SiC at different load steps measured using DIC; and (c) a composite of Raman spectra showing peak positions in SiC particles from ZrB₂-SiC specimens in annealed and sintered conditions as well as at three different confinement stresses using shaft-collar ring.

Figure 9 shows a plot of the confinement pressures measured from thermal shrink-fit metal sleeves using the classical formulation from Equation (20) and the Raman peak-shift relation from Equation (18), according to an embodiment of the invention, on shrink fitted sleeves and DIC measurement on shaft-collar ring specimen.

DETAILED DISCLOSURE

Micro Raman spectroscopy is a widely used technique for estimating local residual stress in semiconductor microstructures and to investigate crystal defects in SiC. In semiconductor structures, the difference in lattice constants and mismatch in properties between the substrate and the film material lead to generation of thermal residual stresses. When a perturbation is applied to the crystal lattice, either internally, a residual stress, or externally, an applied stress, the resulting strain deforms the sub-lattices, thereby altering effective harmonic force constants. Consequently, Raman wave numbers are shifted that correspond to phonon frequencies of the deformed crystal. The shifting of phonon frequencies to lower and higher wave numbers is related to the induced tensile or compressive nature of stress, respectively, within the crystal. Micro Raman spectroscopy has the ability to resolve the stress over a relatively large volume with high accuracy due to its high spatial resolution (Ι μηϊ). Using a focused laser beam, with a diameter as small as 150 nm, and a near field microscope, it is possible to analyze nano-to micro-scale features lying within tens of nanometers to several millimeters in penetration depths, depending on material and laser excitation wavelength. In recent years, Raman spectroscopy has been effectively applied to characterize 3D amorphization zones in ceramics, such as Boron Carbide.

Embodiments of the invention are directed to a method for detecting and analyzing for bi-axial residual stress in a non-destructive manner. Analytical expressions for bi-axial residual stress of a composite specimen are based on measured micro Raman peak positions of a semiconductor material, for example, on SiC particles in a ZrB₂ matrix. A determination of residual stresses using Raman microscopy can be carried out while designing and testing of semiconductor devices and circuits. The method, according to an embodiment of the invention, can be used in place of, or to augment, Raman microscopy methods presently used to determine wafer stress or layer stress within thin films, single crystals, and textured materials. Current Raman techniques are typically limited to determination of internal residual stresses developed as a result of mismatch in coefficient of thermal expansion between substrates and thin films.

To illustrate the use of Raman peak positions to analyze confinement stress, an expression and its derivation is presented below that relates Raman peak-shift to the applied radial confinement stress within SiC particles. The transverse optical (TO) peak-shift will be considered although the longitudinal optical (LO) peak can also provide similar results.

The method, according to an embodiment of the invention, has numerous advantages, as it can be applied to any Raman active material to determine the local confinement stress at any point when the material is stressed by an unknown load. The method is easy to use because it provides a simple linear relationship between peak-shift and confinement stress, allowing one to evaluate the unknown confinement stress developed in a material, which permits one to determine the safety or operational reliability of a structure, as long as the stress-free position of the material is known. Because of the high resolution possible, Raman peak-shift based stress analysis provides an opportunity to map a stress field with pin-point accuracy without the need for complex FE analysis of the structure.

The materials that can be analyzed by the method for determining bi-axial stress must be Raman active. To this end, a large number of ceramics are Raman active but metals are not Raman active. To use the method, a material's stress-strain relation, compliance terms, phonon deformation potentials, and mode Griineisen parameters for uniaxial and hydrostatic stresses must be known to permit the derivation of the relation between the peak-shift and subsequent stress state. The relationship, as will be derived and given in Equation (18), below, is valid for diamond and zinc-blend crystal structures, but not other structures.

Raman peak- shifts are related to external applied stresses using a calibration procedure where Raman peak positions are calibrated to known stress magnitudes using like curves to determine unknown stresses by changes in Raman peak position for a given specified material. Because these calibration curves have a dependence on the state of stress, the formulation does not guarantee a unique relationship between the Raman peak-shift and applied stress, but generally requires the determination of experimental constants for a specific material under specific conditions. Explicit expression between Raman peak-shift and residual stresses allow assignment of simple stress states such as uniaxial, equi-biaxial and hydrostatic states.

The frame-work to develop the relationship used to asses stress, according to embodiments of the invention, is developed from fundamentals such as phonon deformation potentials and compliance constants. This framework is applied to matrix-particulate systems and adopted with only minor modifications for determining stresses in materials with single or multiple polycrystalline phases. Although thermal residual stresses (internal loading) constitute a high proportion of generated stresses in processed composites, stress generated within the composite due to external loading is determined by Raman spectroscopy.

For exemplary purposes, a combined SiC and ZrB₂ ceramic is disclosed rather than the separate species. Zirconium diboride (ZrB₂) is a promising ultra-high temperature ceramic (UHTC) which is characterized by its high melting temperature (-3200) and relatively high mechanical strength (>800 MPa). ZrB₂ is a brittle ceramic with an electrical conductivity almost as a metal, which allows machining into cylindrical specimens using electric discharge machining (EDM). However, it is difficult to process this ceramic from powder particles. Therefore, silicon carbide (SiC) is added to enhance sinterability and toughness to a composite. The introduction of SiC also improves the oxidation resistance of ZrB₂ at elevated temperatures due to the formation of Si0₂ layer on the surface. SiC is a high strength brittle ceramic widely used in many industrial applications as well as in armor. However, due to its high strength and brittleness, the exemplary cylindrical test specimens are not readily feasible or are prohibitively expensive from bulk material, because traditional machining methods have the potential to introduce surface flaws and non-cylindricity. ZrB₂ is poorly Raman active, whereas all SiC polytypes (cubic and hexagonal) are highly Raman active, and the characteristic Raman peak positions are highly strain sensitive. For these reasons, in exemplary materials, ZrB₂-SiC composites are formed where ZrB₂ is the machineable matrix material and SiC is employed as a second particulate phase for probing Raman peak-shift as a function of applied confinement pressure. Compressive stresses induced as a result of lateral confinement on a ZrB₂-SiC specimen are estimated by analytical expressions relating Raman peak-shift to the induced radial stress using phonon deformation potentials for 3C-SiC. Exemplary ZrB₂-SiC cylindrical specimens are slirink-fit with different metallic sleeves having various levels of confinement stress and SiC particles are probed for Raman peak-shift due to the applied confinement. The resulting confinement stress values are compared using a classical formulation for a thick walled cylinder interference fit problem. The derived equations are consistent with independent measurements, where Digital Image Correlation (DIC) is used to measure the induced strain field on one surface of a ZrB₂-SiC cylindrical specimen confined in a shaft-collar ring with unknown pressure. The derived relation is employed on another surface where Raman peak-shift was measured on SiC particles and the two measurements are compared. The method, according to an embodiment of the invention, though developed for SiC phase, can be generalized for any Raman active material with known deformation potentials and residual stresses induced as a result of processing can be quantified.

Typical Raman spectra on a SiC particle in as-processed (sintered) and annealed

(stress-free) virgin ZrB₂-SiC specimens are shown in Figure 1. Both spectra consist of a transverse optical (TO) phonon peak at 796.98 cm^"1 and a longitudinal optical (LO) phonon peak at 971.1 cm^'1. The stress-free condition is achieved by annealing the ZrB₂-SiC specimen at 1200 °C and then cooling over 4 hr in a furnace to room temperature. Literature values of the TO and LO peak positions for stress free 3C-SiC thin films are reported as 796 cm^"1 and 972 cm^"1, respectively. TO and LO peak positions for sintered ZrB₂-SiC are shifted to higher wave numbers than that observed for annealed sample. The shift to higher wave numbers indicates residual thermal compressive stresses within the composite. The mismatch in lattice constant (a), Young's modulus (E) and coefficient of thermal expansion (a) between Hexagonal-ZrB₂ and Cubic-SiC, as shown in Table 1 , below, causes introduction of compressive residual stresses on SiC particles in the consolidated compact, after cooling from sintering temperature of 1750 °C to a room temperature of 25 °C.

Table 1 Properties of ZnB₂ matrix and SiC particles.

Material E (GPa) A C*^"1) V Vol Frac Grain Size μιη if)

ZnB₂ 523 6.66x10^"b 0.135 -91 5

SiC 485 3.5xl0^"6 0.17 9 1 When no confinement is applied, the SiC particles located within the as-sintered ZrB₂-SiC composite have only the thermal component of residual stress i.e. σ'^ι = σ" . When the lateral confinement is applied on the specimen, additional radial (^) and tangential (σ_θ) stresses are generated and the residual stresses are the sum of processing (σ* j and pressure- induced (σ* ) stresses. σ =σ_τ + σ_ρ (1)

Processing-induced residual stresses are calculated based on the three-phase model disclosed in Ling et al., J Mater. Sci. 2007, 42, 759-62, where the two phase composite sphere assemblage (CSA) is given as disclosed in Hashin, J. Appl. Mech. 1962, 29, 143-50 and Chawla, Composite Materials 1998, 212-51, are combined with an effective model to account for increase in fracture toughness of the matrix. The ZrB₂-SiC composite is modeled as a CSA where the SiC particles are assumed equiaxed and spherical and uniformly distributed in ZrB₂ matrix and considered as a single phase in an infinite homogenous medium of the effective phase. This results in an axially symmetric stress distribution around the SiC particles when subjected to a temperature difference, AT = T_r -T_s , where, T_r and T_s denote the room temperature and sintering temperature, respectively. The hydrostatic pressure P developed around a SiC is given by:

12f_mK_pK_mG_m (a_p - a_m )AT

(2)

3K_pK_M + 4f_pK_pG_p + 4f_mK_mG, where K and G are the bulk and shear moduli, / is volume fraction of particles, and subscript m and p denote matrix and particle. For a particle with effective radius b , and a surrounding matrix with radius c , the radial (a_r ) and tangential stresses (σ_θ) in the particle and matrix are given by:

σπ = σ_θ = -Ρ for r≤b

From Equation (2), cooling of ZrB₂-SiC composite from the sintering temperature of 1750 °C to room temperature of 25 °C results in thermal compressive residual stresses within SiC due to mismatch in CTE as _m > a_p. The radial and tangential stresses within ZrB₂ matrix are compressive and tensile, respectively, and attain their maximum at the matrix- particle interface at r = b .

As taught by Ganesan et al , Annals of Physics 1970, 56(2), 556-94, when a material is deformed either internally (due to residual stress) or externally (due to applied stress), the frequencies of three optical phonon modes can be calculated by determining the Eigen values, λ , = 1 -3 , of the following secular equation:

R R

Where p, q, and r are phonon deformation potentials and are material constants, ε and γ are residual normal and shear strain tensor components, respectively. For small deformation, the frequency w (y = 1-3) of each mode of the deformed crystal in the presence of strain is related to the unstrained frequency w through the relationship:

Aw, = W; - w, «— ~ (5) j j J° 2w .

The polarization direction of each mode can be obtained by considering the Eigen vectors of Equation (4). Considering the compressive nature of radial and tangential stresses due to lateral pressure, the relationship between the residual strain tensor ε in Equation (4) and the residual stress tensor for zinc blende crystal structures of 3C-SiC is given by Hooke 's law as:

where, S_tJ represents compliance, and c^and T^R are the residual normal and shear stress components, respectively. For cubic structures the following relationship holds:

¾ ⁼^13 ⁼¾1 ⁼¾ ⁼¾1 ⁼¾ (7) ^4 ⁼^55 ⁼^66

Since pressure is applied on the lateral surface of a cylindrical specimen, the magnitude of radial and tangential stresses are significant and the axial stress « 0. To simplify calculations, all shear components of stress and strain tensor of Equation (6) are neglected, i.e.,

Thus solving Equation (6) reduces to:

Substituting Equation (9) in Equation (4), the Eigenvalues can be obtained utilizing Equation (6) as: λ, = {pS_n + (S_u+S_n)q}a?+{pS_n+2qS_l2}a« (10) λ, = {S_np + 2S_nq) σ + [S_up + (S_n + S_l2)q} σ_θ ^κ

Substituting λ₁₅ λ₂ and λ₃ into Equation (5), the relation between the Raman peak-shift and the residual stress are derived as:

= { _PS_n +(S_U+ S_l2)q}(a? + σ* ) (11)

^2 =T^J-[{ ¾+(^,+¾)?}^+{^„+2^₁₂}^] (12)

Δ^=-^ί-[{5₁₁ + 2¾^_σ {5_η 7 + (5₁₁+¾)^_σ ]. (13) The above peak-shift equations is used to calculate residual stresses within SiC particles from the Raman spectrum collected from individual particles. In a strained zinc- blende crystal structure such as 3C-SiC, a maximum of two transverse optical (TO) and one longitudinal optical (LO) Raman modes are possible. For the above equations (1 1)-(13), the first two Raman modes Au and Aw₂ represent TO peaks, while the third Raman mode Aw₃ corresponds to LO peak. Therefore, Equation (11) and Equation (12) are used to calculate the residual stresses from TO peak-shift, whereas Equation(13) is used for residual stress calculation from LO peak-shift.

Values of phonon deformation potentials p, q and r are not readily available in the literature for zinc-blende crystal structure. Therefore, mode Griineisen parameters for hydrostatic stress { ₀ ) and uniaxial stress (y_s) are used to yield values for p and q , where and (^) are defined as:

where w_v is the Raman peak in the absence of strain. The more recent value for mode

Griineisen parameter, i.e. for TO-phonon (/J° ) in 3C-SiC, is reported as 1.10 in Zhuravlev et al, J. Appl. Phys. , 2013, 113, 1 13503. For simplicity, γ™ = γ ° = l .lO is assumed. From

Fig. 1, w₀ = 796.98 cm^"1 and for TO peak from Equation (14):

p ^O =-0.4658 x \0⁶cm-²

q^w = -1.8632 x lO ²

The compliance constants were reported by Toplygo, Sov. Phys. Solid State 1961, 2, 2367-76 as:

7 x l0^~13 cm² 1 dyne

1.05 l0^"13 cm² 1 dyne

Substituting Equation (15) and Equation (16) into Equation (1 1), the final expression for residual stress calculation based on the TO peak position is obtained as: σ + σ = -35832(A_Wl ) MPa (17) where Aw, is the Raman peak-shift for TO peak position. Equation (17) represents a realistic biaxial stress state that may exist within SiC particles, which are usually not circular in shape and hence may not have axisymmetric stress state around them. When the matrix-particle system resembles a composite sphere assemblage model of Ling et al, J. Mater. Sci. 2007, 42, 759-62, as above with the assumptions recited therein, the ideal case persists, i.e. σ'' = σ* , and Equation (17) can be further reduced to give the magnitude of radial confinement by a sleeve or ring as:

The above equation relates the Raman peak-shift (Aw) to the magnitude of confinement stress a_r for SiC particles. It is evident from Equation (18) that Raman peak will be shifted to a higher or lower wave number as a result of compressive or tensile residual stress, respectively. METHODS AND MATERIALS

A ZrB₂-5wt%SiC composite consolidated using spark plasma sintering (SPS) technique was prepared in the manner of Ghosh et al, Acta Materialia 2008 56, 301 1-22. The sintered composite is comprised of two crystalline phases, namely, hexagonal (H) ZrB₂ matrix and cubic (3C) SiC particles. The as-processed microstructure of metallographically polished ZrB₂-SiC composite surface is shown in Figure 2. The continuous grey phase is the ZrB₂ matrix which surrounds the dark SiC particles that are distributed uniformly throughout the matrix. The average grain size for the ZrB₂ matrix was estimated to be 5 μιη and the SiC particle size is approximately 1 μηι. Ceramics, in general, are difficult to machine into desired shapes due to their high brittleness, but the unusual high electrical conductivity (~10 S/m) of ZrB₂ makes it possible to cut cylindrical specimens of ZrB₂-SiC composite by electric discharge machining (EDM) without causing significant damage to the surface, as shown in Figure 3(a). These specimens were metallographically polished using colloidal silica on the measurement surface and by SiC paper on cylindrical surface to remove EDM damage. Confinement pressurized cylindrical specimens were achieved through installing a metal sleeve on the lateral surface by shrink fitting, as shown in Figure 3(b), or by mechanically loading a shaft-collar ring, as shown in Figures 4(a) and (b).

The metal sleeves were machined to have an inside diameter slightly smaller than the ceramic specimen diameter, as indicated in Figure 3(c). Shrink-fit assembly was accomplished by heating the sleeve to a predetermined temperature such that the inside diameter was slightly larger than the specimen diameter and then forcing the specimen into the hot sleeve by applying axial force in a vice. Upon cooling, the metal sleeve shrinkage provided the desired lateral confinement on the cylindrical surface. The magnitude of confinement can be varied using different sleeve materials and the level of interference fit (<5).

The lateral pressure achieved through the confinement sleeve can be approximated by treating the sleeve as a thick pressure vessel and solving an axisymmetric boundary value problem on the cross-section of a circular elastic specimen having elastic constants (Ei,vj) inside an elastic-perfectly plastic hollow metal sleeve with (E₂,v₂) as its elastic constants with a misfit δ⁵ at the interface, as shown in Figure 3(c). The solution leads to an expressions for misfit strain and radial pressure, respectively:

(19)

and

(20)

where Ei , E₂ and vi, ₂ are Young's moduli and Poisson's ratios of the ceramic and metal sleeve, respectively. P is the induced confining pressure on the lateral surface of the ceramic specimen, ^ is the yield strength of the confining sleeve material, n and r₂ are the specimen radius and outer radius of the sleeve material, respectively, δ is the misfit between the specimen diameter and the sleeve inner diameter, and R is the radius of the elastic-plastic boundary in the sleeve cross-section. Three different sleeve materials were used to generate three different confinement pressures, as listed in Table 2, below.

Table 2 Sleeve materials and estimated confinement pressure using Equations (19) and (20)

Sleeve Material δ (mm) r;(mm) r₂( m) i?(mm) E(GPa) V P

(MPa)

Aluminum-2024 0.02 1.99 3.92 2.64 73 0.33 180

Brass-385 0.01 1 1.99 3.99 2.98 98 0.32 195

Stainless Steel-303 0.014 1.91 3.99 2.52 196 0.25 284 The confinement stress was first calculated by measuring the Raman peak-shift from individual SiC particles in the ZrB₂ matrix before and after the confinement and then using the derived relation between the peak-shift and radial confinement stress of Equation (18). The shrink-fit assembly results, using Equations (19) and (20), provided the values of confinement stress induced by a hollow metallic sleeve for a given misfit radius, as indicated in Table 2, above. By comparing the values of confinement stress from these two approaches, the validity of the derived equation between Raman peak-shift and confinement stress is verified.

DIC on Shaft-Collar Confined ZrB₂-SiC Specimens

Determination of an unknown level of confinement stress using a shaft-collar ring were carried out to measure the average stress on one planar surface of a cylindrical specimen using DIC, as shown in Figure 4(b) and collecting Raman peak-shift from SiC particles on the other planar surface Figure 4(a). DIC is a full-field, non-contact optical technique for measurement of surface displacements based on specimen's digital images taken before and after the deformation of the specimen. Stereo imaging from two cameras was used to measure in-plane as well as out-of-plane displacements, as shown in Figure 4(c). A full 3D displacement field was constructed and strains evaluated from the displacements using standard elasticity relations.

To conduct DIC and Raman probing of SiC particles on a single assembly of shaft- collar ring and ZrB_2-SiC specimen, one planar surface was first coated with a flat white Valspar^® paint, followed by a speckle pattern of Valspar^® satin black as shown in Figure 4(b). The collar-ceramic assembly was then placed in a vice and secured tightly to avoid any rigid body motion. The locations of the vice and specimen were marked at several points on an optical table so as to relocate the specimen exactly at its original position after the Raman measurement and then apply the next increment of the load step. The DIC cameras were fixed via bolts on the table to avoid any movement during the experiment, as indicated in Figure 4(c). The other planar side of the collar-ceramic assembly was left un-coated to collect Raman measurements at pre-determined points on SiC particles. Point Grey Research Cameras (GRAS-50S5M-C) and Fujinon l : 1.8/75mm lenses were utilized to capture all images. To acquire clear images and reduce loss of data, lighting was provided using a Visual Instrumentation Corporation (VIC) LED light array (Model 900445) powered by a VIC Video Flood Controller. Correlated Solutions VIC-Snap 2007 was used to capture the images and VIC3D 2009 was used to post-process the images.

The average displacement on the painted specimen surface was first measured using the DIC and then the strain was calculated. The average stress on the specimen surface was calculated by multiplying the strain with the Young's modulus for ZrB₂-SiC. On the opposite side of the same specimen, the Raman peak-shift from selected SiC particles was measured and then the confinement stress was calculated using Equation (18). This stress was then compared to the one measured using DIC on the other side of the specimen. Thus, the entire stress measurement process and the derived relationship between Raman peak-shift and the level of confinement stress were further verified.

Raman Probing on SiC Particles

A Renishaw^® in Via Raman Microscope Spectrometer was used to measure ZrB₂-SiC

Raman peak-shift. The spectrometer uses a Si laser (532 nm) of spot size ~1 μ^η to excite the specimen and an optically connected Leica^® microscope with a motorized XYZ stage. The Raman probe measurements were conducted on selected circular SiC particles before and after the confinement to yield peak-shift data for each SiC particle. The procedure was repeated on 50-100 particles for each specimen subjected to different confinement pressures so as to obtain the peak-shift before and after the confinement. For each specimen, the SiC particles were identified under 100X objective lens using point acquisition feature in the software and then the laser beam was focused and Raman data was collected automatically. The point acquisition feature allows for Raman peaks to be probed on individual points that have pre-determined Cartesian coordinates. Extended Raman scans at room temperature with a maximum laser power of 25mW and a 10 sec exposure time was used on each spectrum. Values derived from Equations (2) and (3) were compared with those by Equation (18). Polycrystalline ZrB₂ and single crystal SiC elastic constants, as listed in Table 1, above, were used in Equations (2) and (3) to calculate thermal residual stresses. Assuming a temperature difference of 1725 °C, it was determined that SiC particles in the as-sintered composite were under a uniform residual compressive stress of 2.04 GPa by (Equation (2); whereas the compressive radial and tensile tangential stresses by Equation (3) at the ZrB₂-SiC interface were 2.04 GPa and 1.32 GPa, respectively.

The characteristic Raman wave number for β-SiC powder was recorded as 789

as indicated in Figure 5, which is consistent with literature values. The maximum Raman wave number on cylindrical shaped SiC particle in the as-processed ZrB₂-SiC sample was recorded as ~ 800 cm^"1, revealing a Raman wave shift of 11 cm^'1 from powder (stress-free) to processed condition. By Equation (18), this peak-shift value gives a thermal compressive residual stress of 1.97 GPa, which is almost same as that obtained by Equations (2) and (3), i.e., 2.04 GPa. The minor difference may be due to the assumption a_r = a_s, which assumes circular particles in the derivation of Equation (18).

Compressive Stress Due to Confinement by a Metallic Sleeve

Figure 6 presents the measured peak-shift data for several SiC particles and the calculated confinement stress for these particles using Equation (18), according to an embodiment of the invention. These SiC particles lie at various distances from the center of the specimen. The data is plotted for three different metallic sleeves. The data reveal that confinement stress on a ZrB₂-SiC composite changes with the type of sleeve material as shown in Figures 6 and 7, but remain relatively constant for each sleeve material.

The replotted data of Figure 7 has the confinement stress calculated using Equation (20) and that given in Table 1, above. These two values match reasonably well, which supports the relationship derived for biaxial stress (Equation (18)) as a function of the Raman peak-shift.

Validation Using DIC experiments

The relationship between confinement stress and Raman peak-shift using DIC was independently verify. First, an unknown level of confinement was applied using the shaft- collar ring, as shown in Figure 4, and the average strain on one planar surface of the specimen was determined using the DIC. On the other planar side of the specimen, a total of 16 circular SiC particles were identified on the virgin specimen and the Raman peak-shift was measured on these particles after the confinement pressure was applied by a shaft-collar ring. Both DIC and the Raman scans of the same particles were repeated for three load increments. Table 3, below, compares average confinement pressures obtained from DIC measurements on one planar surface and the Raman spectroscopy of individual SiC particles on the other planar surface.

Table 3 Shaft-collar ring confinement pressure measurements using DIC and Equation

(18)

Confinement Pressure

Load Step Aw cm ¹ Equation (18) DIC (MPa) _ _ _ _

1 0.93 168 189

2 2.01 361 379

3 2.75 493 511

Figure 8(a) shows uniformity of DIC-measured strain on one surface, where the spot is an area of stress concentration caused by a rather large dot of Valspar paint. Figure 8(b) shows the measured average Von-mises strain (e_vm) at three levels of confinement using the shaft-collar ring. Figure 8(c) shows the corresponding TO Raman peaks for these three stress levels, where the Raman peak-shifts to the right with increasing confinement stress.

The Raman peak-shift to higher wave numbers, as shown in Figure 8(c), is attributed to the macroscopic strain induced within SiC particles due to progressively increasing confinement pressure. The applied bi-axial stress on bulk ZrB₂-SiC specimen governs the magnitude of macroscopic strain induced within SiC particles, and, hence, the degree of distortion of crystal lattice. The lower Raman wave numbers corresponds to lower crystal strain. The Raman peak width or peak broadening with confinement pressure is primarily due to photon damping by lattice imperfection and grain boundaries. The sharp Raman peaks in the as-sintered composite is an indication of perfect crystallinity as compared to blunt peaks in the strained composite. Peak broadening and peak asymmetry indicate the extent of structural disorder or crystal imperfection within SiC particles. Stacking faults are the known primary defects in SiC whiskers and in polycrystalline SiC. Rhomfeld et al, Physica Status Solidi (b), 2008, 215, 115-9, gives simulated Raman intensity profiles of one-dimensionally disordered SiC structures based on bond polarizability model; observing that as the density of stacking faults increased (or the average stacking fault distance decreased), the structural disorder increased, causing peak broadening. Lu et al, Thin Solid Films 200, 377-378, 389- 93, showed evidence of crystal structure defects, such as dislocations, stacking faults, and dislocation forests, on TEM micrographs of SiC grain due to internal stress. Based on the observation of Figure 8(c), it is inferred that the application of bi-axial compressive stresses on SiC particles may result in an increase in the density of stacking faults, thus increasing the structural disorder within SiC. Consequently, Raman peaks are shifted to higher wave numbers.

Summary of Test Results

Figure 9 summarizes all the confinement pressures measured from the DIC on one planar surface of a ZrB₂-SiC specimen and the Raman peak-shift measured stress, according to an embodiment of the invention, on selected SiC particles on the other planar surface of the shaft-collar ring specimen assembly. The stress values for shrink-fit sleeve assembly are also superposed on the same plot. All three methods predict the same linear relationship between confinement stress and peak-shift. The confinement stress due to processing-induced thermal residual stress is also shown to follow the same linear relationship.

The experimental results clearly validate the derived relationship (Equations (18)) between Raman peak-shift and the level of confinement stress. By using the mechanistic formulation for shrink-fit sleeve assembly using Equations (19) and (20) and the relation between Raman peak-shift and confinement stress of Equation (18), the derived equation is verified. This equation was further verified to provide a good agreement with the process induced residual stress based on established relationships of Equations (2) and (3). Validation of the method, according to an embodiment of the invention, was also shown by measuring the unknown confinement stress induced by a shaft-collar ring but measuring the stresses using another independent experimental method (DIC) and comparing these values with Raman peak-shift vs. stress relation (Equation (18)). The similar values for applied confinement provided by these methods further validate the derived relationship. The Raman spectroscopy method, according to an embodiment of the invention, provides an experimental tool for rapidly and accurately determining stress in a Raman active material.

All publications referred to or cited herein are incorporated by reference in their entirety, including all figures and tables, to the extent they are not inconsistent with the explicit teachings of this specification. It should be understood that the examples and embodiments described herein are for illustrative purposes only and that various modifications or changes in light thereof will be suggested to persons skilled in the art and are to be included within the spirit and purview of this application.

Claims

CLAIMS We claim:

1. A method of determining local residual stress in a material, comprising:

providing a material that is Raman active;

providing a Raman Spectrometer;

providing an unstressed transverse optical (TO) Raman peak position and/or unstressed longitudinal optical (LO) Raman peak position for the material in an un-stressed state;

acquiring at least one Raman spectrum displaying a stressed TO Raman peak position and/or stressed LO Raman peak position of at least one site on the material in a stressed state;

calculate at least one Raman peak-shift between the stressed TO Raman peak position and the unstressed TO Raman peak position and/or the stressed LO Raman peak position and the unstressed LO Raman peak position; and

determining at least one magnitude of stress from a linear relationship between the Raman peak-shift and the stress.

2. The method according to claim 1 , wherein the Raman peak-shift is a positive value for a tensile stress and a negative number for a compressive stress.

3. The method according to claim 1 , wherein the Raman active is a ceramic or a semiconductor.

4. The method according to claim 1, wherein the Raman Spectrometer is a Raman Microscope Spectrometer.

5. The method according to claim 1, wherein the Raman Spectrometer employs a laser excitation source having a sub-micron cross-section to selectively excite the at least one site.

6. The method according to claim 1, further comprising creating a stress field map, wherein the at least one Raman spectrum is a multiplicity of Raman Spectra acquired at a multiplicity of sites on the material, wherein the multiplicity of magnitude of stress is plotted against coordinates for the multiplicity of sites.