Keywords

5.1 Introduction

The amount of residual stresses present into a part is very important information in most engineering applications. When combined with loading stresses, residual stresses can bring the material close, or even beyond, the acceptable limits and may result in part failure. Several residual measurement methods are already established: X-ray or neutral diffraction, relaxation, slitting, contouring are among them [1]. Perhaps the most used for in-site applications is the hole-drilling method [2], which is a particular case of relaxation methods.

The hole drilling method consists of progressively drilling a blind hole in the part with residual stresses. As the hole is drilled, a new free surface is built, which progressively releases the amount of residual stresses present in the part. The stress relaxation produces a local deformation around the drilled hole. The deformation is measured and fitted to a model to evaluate residual stresses. The ASTM E837-13a standardizes a procedure to measure residual stresses using a special strain gauge rosette [2].

The hole-drilling method using strain gauges is a time consuming approach, which can reach over 1 h for each measurement point. The authors have developed an integrated device to measure residual stresses in a very practical and faster way using a special optical interferometer in combination with a high speed drilling unit [3–5]. A brief description of the device is presented in this paper. The authors emphasize here the residual stresses calculation approach using the interferometer data.

5.2 Radial In-Plane DSPI Interferometer

The integrated portable device for residuals stresses measurement is described in details in [3–5]. The device uses a special binary diffractive optical element (DOE) to double illuminate a circular area of about 10 mm in diameter. Figure 5.1 helps to describe the working principle of the DSPI interferometer.

Fig. 5.1
figure 1

Double illumination speckle interferometer with radial in-plane sensitivity. (a) Optical circuit configurations. (b) Double illumination by an axis-symmetrical diffractive optical element (DOE)

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A plane concave lens (E) expands the light coming from a diode laser (L). It passes through the elliptical hole of the mirror M 1 , which forms a 45° angle with the optical axis of the device, illuminating mirrors M 2 and M 3 and being reflected back to the mirror M 1 . Mirror M 1 directs the expanding light to the collimating lens (CL) in order to obtain an annular collimated beam. Finally, the light is diffracted by the DOE mainly in the first diffraction order towards the specimen surface. Residual non-diffracted light or light from other diffraction orders would not be considered troublesome since they are not directed to the central measuring area on the specimen surface. Mirrors M 2 and M 3 are two special circular mirrors. The former is joined to a piezoelectric actuator (PZT) and the later has a circular hole with a diameter slightly larger than diameter of M 2 . Furthermore, mirror M 3 is fixed while M 2 is mobile. The PZT actuator moves the mirror M 2 along its axial direction generating a relative phase difference between the beam reflected by M 2 (central beam) and the one reflected by M 3 (external annular beam). The boundary between both beams is indicated in Fig. 5.1a with dashed lines. According to this figure, it is possible to see that every point over the illuminated area receives only one ray coming from M 2 and only other one from M 3 . Thus, PZT enables the introduction of a relative phase shift to calculate the optical phase distribution by means of phase shifting algorithms. Thus, the central hole placed at M 1 a has several functions: (a) to allows that the light coming from the laser source reaches mirrors M 2 and M 3 , (b) to prevent that the laser light reaches directly the specimen surface having triple illumination and (c) to provide a viewing window for the camera (CCD).

Part (b) of Fig. 5.1 shows only one of the diffraction order reaching the double illuminated circular area. The special DOE is a binary circular diffraction grid with a pitch of 1.32 μm, what result in a diffraction angle of about 30° at a λ = 660 nm wavelength laser. Since each point on the illuminated area is double illuminated in this special scheme, the sensitivity direction is radial. Therefore, only the displacement component pointing to the circular area center is measured. It is possible to demonstrate and verify that this configuration results in an achromatic interferometer [3].

The authors integrated the radial in-plane DSPI interferometer with a high-speed drilling unit in a portable residual stresses measurement device. Figure 5.2a shows an actual view of the portable device [3–5]. The device has a mechanical base that is firmly attached to the specimen surface by rare earth magnets and three legs with sharp conical tips to avoid relative motions. A motorized mechanical stage is used to automatically exchange the measuring unit and the drilling unit. A measurement software automatically control the test and acquire and process all necessary images to compute the combined stresses profile as a function of the surface depth. Figure 5.2b shows a typical phase difference image after drilling a blind hole in a part with compressive residual stresses with the principal compressive axis near horizontal.

Fig. 5.2
figure 2

(a) Portable radial in-plane DSPI interferometer combined with a high speed drilling unit to measure combined stresses. (b) A typical radial displacement field due to blind hole drilling measurement

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5.3 Residual Stress Calculation

The relationship between the principal residual stresses σ 1 , σ 2 , the principal angle β and the radial displacement field u r (r, θ) expressed in polar coordinates (r and θ) is in the form of (5.1):

$$ {u}_r\left(r,\theta \right)=A(r)\left({\sigma}_1+{\sigma}_2\right)+B(r)\left({\sigma}_1-{\sigma}_2\right) \cos \left(2\theta -2\beta \right) $$
(5.1)

A(r) and B(r) are functions of: the radius r, the drilled hole radius r 0 , the hole depth h, and material properties E and ν. There is a closed form for A(r) and B(r) only for a thin infinity plate with uniform stresses and a through hole. For all other cases, only numerical solutions are available. Note that the residual stress sum component does not depend on the polar angle θ. Therefore, it is axially symmetrical. On the other hand, the residual stress difference component depends on the cosine of 2θ and the principal stress direction β.

To quantify the amount for residual stresses in an experiment, two vectors A(r k ) and B(r k ) are extracted from a phase difference map for N discrete radius values (r k ) ranging from k = 1 to N and compared with numerically determined reference values. After phase unwrapping, the extraction of A(r k ) and B(r k ) is done through Fourier analysis. The mean radial displacement value for a given radius r k all way around 2π rad corresponds to the zero order harmonic and to A(r k ). The second harmonic components in cosine C 2 (r k ) and the second harmonic components in sine S 2 (r k ) are used to compute B(r k ) by B(r k ) = √[(C 2 (r k ))2 + (S 2 (r k ))2] and β k  = Atan(S 2 (r k )/C 2 (r k ))/2.

Let a(r k ) and b(r k ) be the radial displacement reference values for A(r k ) and B(r k ) respectively. They can be determined through numerical simulations by the finite element method by loading the hole cylindrical surface with an unitary stress and by setting a zero stress at the distant boundary of the object. The finite element used in this analysis belongs to a family of elements of axi-symmetric solids. For the a(r k ) coefficients a uniform biaxial stress field with σ 1  = σ 2  = 1.0 MPa was simulated. For the b(r k ) coefficients calculation the simulation used σ 1  = 1.0 MPa and σ 2  = −1.0 MPa. The radius r, ranged from r 1  = 1.025 to r N  = 7.475 mm with radial increments of 0.025 mm.

The numerical solution of the problem was evaluated by using the ANSYS software. For this reason, two dimensional eight-node elements (PLANE 83) were used. The obtained mesh, shown in Fig. 5.3, had a total of 14,722 elements and 44,601 nodes. Simulations were performed in the same way as an incremental hole-drilling test is executed by following loading recommendations presented in [2]. Hole increments were simulated by drastically reducing the modulus of elasticity at the layers removed by the hole. Twenty steps of 0.05 mm of depth were simulated reaching a final hole depth of 1.00 mm. Displacement coefficients a ij (r k ) and b ij (r k ) were exported as vectors for a future calculations, where i denotes the hole depth increment and j the stress layer count [2, 6]. As an example, Fig. 5.4 shows a plot of these coefficients for the 15th hole step.

Fig. 5.3
figure 3

Mesh used during simulation. Right part shows a detailed view of the hole region (a) Complete mesh view and (b) Enlarged view of the drilled hole region

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Fig. 5.4
figure 4

Example of a ij (r k ) and b ij (r k ) coefficients

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To quantify the amount of residual stresses from the measured quantities A(r k ) and B(r k ) a linear behavior was assumed. Therefore, the stress values are proportional to the amount of radial displacements. As an example, let’s suppose that A(r k ) is P times bigger than a(r k ). In this case, the sum of residual stresses will be P times bigger than 1.0 + 1.0 MPa. Since random errors are expected in the experimental values A(r k ), as well as an unexpected bias C may exists, the relationship between A (r k ) and a(r k ) is in the form of (5.2), where e(r k ) is a random error component. The ij sub indexes were omitted here for simplicity. A similar model is used to relate B(r k ) and b(r k ), but due to space limitation, it will not be described here.

$$ A\left({r}_k\right)=P\cdot a\left({r}_k\right)+C+e\left({r}_k\right) $$
(5.2)

A least square approach is used to determine P and C. To do that, define the square error function SE by (5.3):

$$ SE={\displaystyle \sum_{all\;{r}_k}{\left[P\cdot a(r)+C-A(r)\right]}^2} $$
(5.3)

The minimum of SE can be found by differentiation of (5.3) to P and C and solving the resulting system of equation. The values for P and C are given by (5.4):

$$ C=\frac{{\displaystyle \sum_{all\;{r}_k}A(r)}\cdot {\displaystyle \sum_{all\;{r}_k}{\left[a(r)\right]}^2}-{\displaystyle \sum_{all\;{r}_k}\left[A(r)\cdot a(r)\right]}\cdot {\displaystyle \sum_{all\;{r}_k}a(r)}}{N{\displaystyle \sum_{all\;{r}_k}{\left[a(r)\right]}^2}-{\left[{\displaystyle \sum_{all\;{r}_k}a(r)}\right]}^2}\ \mathrm{and}\ P=\frac{N{\displaystyle \sum_{all\;{r}_k}\left[A(r)\cdot a(r)\right]}-{\displaystyle \sum_{all\;{r}_k}A(r)}\cdot {\displaystyle \sum_{all\;{r}_k}a(r)}}{N{\displaystyle \sum_{all\;{r}_k}{\left[a(r)\right]}^2}-{\left[{\displaystyle \sum_{all\;{r}_k}a(r)}\right]}^2} $$
(5.4)

To apply the integral method for residual stresses measurement by the incremental hole-drilling technique, the accumulated experimental data for each drilling increment have to be simultaneously fitted to a series of numerical signals. Each signal is related to the hole depth (i) and the stress layer (j). Let a ij (r) denotes the numerical signal computed for the i-depth and j-stress layer [6]. For the first hole depth, the relationship between the experimental data and numerical signal is given by (5.5). The k sub index was omitted for simplicity.

$$ {A}_1(r)={P}_1\cdot {a}_{11}(r)+{C}_1+{e}_1(r) $$
(5.5)

For the second incremental hole depth, the integral method expects to relate the experimental signal to the numerical ones according to (5.6). Since (5.5) alone can determine the value of P 1 , (5.6) involves two unknowns: P 2 the stress value for depth 2 and the additive constant C 2 .

$$ {A}_2(r)={P}_1\cdot {a}_{21}(r)+{P}_2\cdot {a}_{22}(r)+{C}_2+{e}_2(r) $$
(5.6)

For the third hole increment, (5.6) becomes:

$$ {A}_3(r)={P}_1\cdot {a}_{31}(r)+{P}_2\cdot {a}_{32}(r)+{P}_3\cdot {a}_{33}(r)+{C}_3+{e}_3(r) $$
(5.7)

The same idea can be extended for any number of S hole-drilling increments. A general least square solution for the simultaneous system of equations can be developed starting from the sum of the square error given by:

$$ \begin{array}{l}SE={\displaystyle \sum_{all\;r}{\left[{P}_1\cdot {a}_{11}(r)+{C}_1-{A}_1(r)\right]}^2}+{\displaystyle \sum_{all\;r}{\left[{P}_1\cdot {a}_{21}(r)+{P}_2\cdot {a}_{22}(r)+{C}_2-{A}_2(r)\right]}^2}+\\ {}{\displaystyle \sum_{all\;r}{\left[{P}_1\cdot {a}_{31}(r)+{P}_2\cdot {a}_{32}(r)+{P}_3\cdot {a}_{33}(r)+{C}_3-{A}_3(r)\right]}^2}+\dots +{\displaystyle \sum_{all\;r}{\left[{P}_1\cdot {a}_{S1}(r)+\dots +{P}_S\cdot {a}_{SS}(r)+{C}_S-{A}_S(r)\right]}^2}\end{array} $$
(5.8)

The least squares solution for the simultaneous 2S x 2S system can be formed by differentiation of (5.8) to P 1 to P S and C 1 to C S and setting all to zero. The 2S equations can be easily rearranged to solve the values of the additive constants C 1 to C S . Since their values have no practical interest, they can be substituted in the equation system that becomes a S x S system with the S unknowns P 1 to P S . The resulting matrix system can be written as (5.9) where the elements of the matrix [X] and vectors {P} and {Y} are given in (5.10).

$$ \left[X\right]\left\{P\right\}=\left\{Y\right\} $$
(5.9)

Where:

$$ \left\{P\right\}=\left\{\begin{array}{c}\hfill {P}_1\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {P}_S\hfill \end{array}\right\} $$
$$ {X}_{m,n}={\displaystyle \sum_{i=Q}^S\left[{\displaystyle \sum_{all\kern0.2em r}\left({a}_{im}\cdot {a}_{in}\right)}-\frac{1}{N}{\displaystyle \sum_{all\kern0.2em r}{a}_{im}}\cdot {\displaystyle \sum_{all\kern0.2em r}{a}_{in}}\right]} $$
(5.10)
$$ {Y}_m={\displaystyle \sum_{i=Q}^S\left[{\displaystyle \sum_{all\kern0.2em r}\left({a}_{im}\cdot {A}_i\right)}-\frac{1}{N}{\displaystyle \sum_{all\kern0.2em r}{a}_{im}}\cdot {\displaystyle \sum_{all\kern0.2em r}{A}_i}\right]} $$

The quantity Q is defined as the minimum integer between m or n: Q = min(m, n).

Usually (5.9) becomes ill-conditioned when S increases. The system (9) can be solved by Tikhonov regularization by (5.11), where α is the regularization factor and [D] is a tri-diagonal second derivative S x S matrix. The first and last rows of [D] contain zeros; all other rows have [–1 2 –1] centered along the diagonal [2, 6].

$$ \left(\left[X\right]-\alpha {\left[D\right]}^T\left[D\right]\right)\left\{P\right\}=\left\{Y\right\} $$
(5.11)

Solution of (5.11) result in the principal residual stresses sum for each S drilling increment. The same approach can be applied for b ij (r k ) and B ij (r k ) to determine the principal residual stresses differences as functions of the depth. A combination of both equations allows the determination of both principal residual stresses values individually and for each drilling increment. More details about the derivation of those equations, as well as, a set of difference equations are the focus of a detailed paper that will be submitted to Experimental Mechanics late 2015.

5.4 Experimental Verification

A stress free specimen was bent along a well-defined curved surface as a way to generate reference values for stresses along the specimen depth. All material treatment and geometry were carefully prepared according to recommendations of reference [7]. Figure 5.5 shows a view of the specimen. It is made by: (a) a thick aluminum base with is upper surface machined with a radius of 508 mm, and (b) a thin sheet of aluminum with a thickness of 1.6 mm. The latter part was prepared from a thicker sheet. The final thickness was achieved by machining the material of the surface. Thus, residual stresses generated by the rolling manufacturing process of the original plate were ideally removed resulting in a free-stressed material. The thin sheet was glued to the curved surface introducing a well-defined bending stress profile into the material.

Fig. 5.5
figure 5

View of the specimen bent over a curved surface and four holes drilled on it

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A 1.6 mm diameter drill was used for incremental hole-drilling. A total of S = 20 drilling steps of 0.050 mm each were used. For each one a set of four phase shifted images were acquired and used to compute phase values. As a result, 20 phase differences images were obtained always involving the phase differences between the reference state (without the hole) and the actual hole depth. After phase unwrapping, the vectors A i (r k ) and B i (r k ) were extracted from each phase difference map of i = 1 to 20 and the residual stresses computed using (5.11) for the stresses sum and later for stresses differences. Additional residual stresses measurement were carried out by strain gauge rosettes using the conventional hole-drilling method also with 20 depth increments of 0.050 mm each. Figure 5.6 shows plots of measured residual stresses along the material thickness. S1_DSPI and S2_DSPI are standing for the optically measured principal stresses σ 1 and σ 2 respectively. S1_SG and S2_SG are the plots obtained by strain gauge rosette measurements for σ 1 and σ 2 respectively. All stress profiles were computed after using appropriate Tikhonov regularization coefficients. Figure 5.6 shows a good agreement between measurements obtained by both techniques. The total measurement time for the optical device was about 20 min. For strain gauge measurement the surface was prepared, the strain gauge cemented, wired and the measurement was done by. The total time for strain gauge measurement was about 60 min.

Fig. 5.6
figure 6

Principal stresses profile results as functions of the hole depth for optical measurement (S1_DSPI and S2_DSPI) and for strain gauge measurement (S1_SG and S2_SG)

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5.5 Conclusions

This paper presented a residual stresses calculation model using the incremental method and finite element coefficients expressed in terms of radial displacement along the radius. The measured radial displacement signal is split in two components: the zero and the second order harmonics. The residuals stresses sum is computed from the zero order harmonic using a least squares fitting algorithm with Tikhonov regularizations. The second harmonics are used to compute the residual stresses differences using the same approach, as well as, the principal directions. A bending controlled specimen was used for model validation. The principal stresses profile along the material depth were also measured using strain gauges rosettes. Both signals present a very good agreement and encourage further development of this approach. The total measurement time for the optical device was about 30 % of the time needed for strain gauge measurement.

The use of difference equations as well, as predefined residual stresses profiles, are currently under development and will be present in a paper that has been prepared to be submitted to Experimental Mechanics late 2015.