Analytical solutions for determining residual stresses in two-dimensional domains using the contour method

Abstract

The contour method is one of the most prevalent destructive techniques for residual stress measurement. Up to now, the method has involved the use of the finite-element (FE) method to determine the residual stresses from the experimental measurements. This paper presents analytical solutions, obtained for a semi-infinite strip and a finite rectangle, which can be used to calculate the residual stresses directly from the measured data; thereby, eliminating the need for an FE approach. The technique is then used to determine the residual stresses in a variable-polarity plasma-arc welded plate and the results show good agreement with independent neutron diffraction measurements.

1. Introduction

Residual stresses exist in many engineering materials and structures. They arise within a structure when a material undergoes virtually any manufacturing process that causes thermal or compositional gradients or that involve plastic deformations. Residual stresses play a critical role for structural integrity of materials in terms of service performance, strength, fatigue life and dimensional stability. Therefore, the ability to accurately quantify residual stresses by means of measurements is an important engineering task. Residual stresses are present across multi-scales, and suitable measurement tools have to be used for their evaluation. Although there are a number of destructive methods available for measuring residual stresses at the macroscale (hole drilling [1], slitting [2], sectioning [3], curvature [4] and conventional and multi-axial contour methods [5–7]), with the rapid advances of nanotechnology in recent decades, a few destructive techniques, such as cantilever [8], microslotting [9,10], hole/core drilling [11,12] and the inverse problem of eigenstrain [13] have become applicable down to the microscopic scale and allow determination of residual stresses in a variety of amorphous or crystalline materials.

In 2001, Prime [5] introduced a new residual stress measurement method called the contour method, and it has become one of the most powerful techniques to measure residual stresses as it can determine an arbitrary cross-sectional area map of residual stress. As for all destructive methods, this technique is based on elastic stress relaxation after material removal. The contour method consists of three main steps and they are outlined in the following paragraphs.

Consider a two-dimensional body possessing the arbitrary residual stress components σ_xx(x,y), σ_yy(x,y) and σ_xy(x,y) as shown in figure 1. In step A, the two-dimensional body prior to the cutting process has both initial tension and compressive residual stresses. When the body is sectioned into two halves along the line x=0, the residual stresses on the cut plane are fully released and new traction-free surfaces are created. The boundary conditions (BCs) of the new traction-free surfaces require that the normal and shear stress components along the line x=0 are zero. Therefore, this condition owing to relaxation of the residual stresses results in deforming of the body on both cut surfaces (step B). Deformations caused by the residual stress relaxation can be assessed by considering an equivalent surface traction on the cut plane. As the traction-free BCs certify that the normal and shear stress components along the cut surface are zero, the displacement vector on the cut surface is a result of the residual stress relaxation (by assuming elastic relaxation), and hence the initial residual stresses can be calculated by applying a displacement vector in the opposite direction to a half part as BCs. As the stresses in step B are unknown, we cannot measure the original stresses within the entire body. However, the stresses normal to the free surfaces in step B are zero. Hence, step C by itself enables calculation of the correct stresses along the plane of the cut.

Figure 1.

Contour method principle: (step A) a two-dimensional body possessing both tension and compressive residual stresses. (Step B) cut surface deforms due to residual stress relaxation. (Step C) applying displacement vector in the opposite direction to a half part as boundary conditions.

As an experimental procedure, only the x component of the displacement vector u_x can be measured on the relaxed surface. In other words, the initial y coordinate of each point is unknown. However, it is still possible to determine the initial normal component of the residual stresses σ_xx(0,y), if additional computation effort is taken. In the plane of the cut, the displacements owing to the normal stress relaxation are always symmetric, whereas the displacements owing to release of the shear stress are antisymmetric and in the half plane, the left-hand side is equal to opposite sign of the right-hand side. Therefore, if the measured displacements on both surfaces are averaged, the displacements owing to shear stress relaxation will cancel out, and the average displacements will be equal to the normal stress relaxation [5].

One additional essential step is data smoothing of the averaged displacements. This is vital as the averaged displacements consist of both deformations resulting from residual stress release and some noise owing to cutting-induced surface roughness and measurement errors. As the noise is not the result of residual stress relaxation, it is important to remove it from the data before proceeding, otherwise it produces an amplified effect during the calculation of residual stresses. The noise is typically removed from the data by fitting the averaged displacements to a smooth analytical function by using Fourier series or polynomials.

Hence, the implementation of the contour method reduces to modelling a half part with traction-free BCs at each edge except for the cutting plane where averaged and smoothed displacements are prescribed in the normal direction to the cut plane and the transverse displacements are unconstrained as the shear stresses are zero.

There have been numerous articles in the literature in which the contour method was used for measuring residual stresses in engineering components. Applications of the contour method include welded joints [14], quenched materials [15], railroad rails [16], impacted plates [17], etc. Being a relatively new technique, the experimental validations of the contour method have been attempted with the help of different non-destructive and destructive techniques such as neutron diffraction [18,19], synchrotron diffraction [20], X-ray diffraction [21] and slitting [22]. Overall, the contour method results agree well with the majority of the independent validations. However, the contour method has so far used the finite-element (FE) method to calculate the original stress state from experimental data. Hence, the method has not been validated by itself.

This work develops analytical solutions based on semi-infinite strips and finite rectangles for determining the residual stress state using the contour method which removes the present dependence on the FE method for this purpose. Good examples of this type of residually stressed samples are welded or peened plates [23,24]. In the following section, a formulation for determining residual stresses from measured displacements using the contour method is described. The method is then implemented to calculate the residual stress in a thin variable-polarity plasma-arc (VPPA) welded sample. The residual stresses obtained by the analytical solutions are compared with the conventional contour method (which is FE based) and the neutron diffraction results.

2. Problem formulation

As mentioned earlier, a half part of the model for the implementation of the contour method is adequate owing to symmetry about the plane of the cut. The geometries of the semi-infinite strip and the finite rectangle to be studied here are depicted in figure 2a,b, respectively. The origin of the coordinate system is the centre of the finite rectangle and on the right edge of the semi-infinite strip. It is assumed that the cut line for both geometries is on the right-hand side. In the case of the finite rectangle, the direction of the surface displacements is taken in the opposite sense to the x component of its coordinate system. However, for the semi-infinite strip, the displacements are considered in the same sense.

Figure 2.

Geometries: (a) the semi-infinite strip (b) the finite rectangle. (Online version in colour.)

(a). The semi-infinite strip

The solution for the semi-infinite strip is formed from an Airy stress function ϕ(x,y), which satisfies the biharmonic equation [25] and stresses are defined by

2.1

The solution is constructed as a series of Papkovich–Fadle eigenfunctions, each of which is chosen so as to satisfy the traction-free conditions on the edges y=±a. Considering Saint–Venant's theorem, we anticipate that the stresses in the solution will decay exponentially with x, so we start from the separation of variables in the form

2.2

substituting (2.2) into the biharmonic equation (∇⁴ϕ(x,y)=0), which reduces to a fourth-order ordinary differential equation

2.3

The most general form satisfying equation (2.3) is

2.4

where C₁, C₂, C₃ and C₄ are four arbitrary constants. The terms , are symmetric (even) and , are antisymmetric (odd) in equation (2.4). It should be noted that measured normal displacements on the cut line for the implementation of the contour method mostly consist of both symmetric and antisymmetric parts in y. Hence, we expand attention to both odd and even functions Y (y). The arbitrary constants will be chosen so as to satisfy the traction-free BCs, i.e.

2.5

which lead to the four homogeneous equations

2.6

2.7

2.8

and

2.9

The set of equations (2.6)–(2.9) has two sets of eigenvalues. One set corresponds to symmetric and the other to antisymmetric. If each set is partitioned to yield the two independent matrix equations, we get

2.10

and

2.11

The symmetric terms (C₁,C₄) have non-trivial solutions if and only if the determinant of the coefficient matrix in (2.10) is zero, hence λ satisfies the characteristic equation such that

2.12

while the antisymmetric eigenvalues are obtained in the same way with the result

2.13

where superscripts ‘S’ and ‘A’ denote symmetric and antisymmetric eigenvalues, respectively. The only wholly real roots of these equations are zero, but there are an infinite number of complex ones, which are found numerically, as described in appendix A. Note that conjugate pairs of the symmetric eigenvalues and antisymmetric eigenvalues are also roots of equations (2.12) and (2.13), respectively, but only those eigenvalues with positive real parts are used in order to satisfy the requirement of a decaying solution when . The corresponding eigenvectors, from equations (2.6) and (2.8) are

2.14

and

2.15

A sufficiently general solution for each of the symmetric and antisymmetric solutions can be constructed as an eigenfunction series

2.16

and

2.17

where A_n and B_n are unknown complex coefficients; and are the eigenfunctions of the symmetric and antisymmetric solutions corresponding to the eigenvalues and , respectively, and they are found to be

2.18

and

2.19

General solutions can then be obtained by superimposing the symmetric and antisymmetric solutions as

2.20

2.21

and

2.22

The required stress components can then be written as

2.23

2.24

and

2.25

Using the stress–strain and the strain–displacement relations, we obtain the expressions for the displacements

2.26

and

2.27

in which μ is the modulus of rigidity and the terms , , , , and are given in appendix B. We now look at conditions along the line, x=0, where, first, transverse displacements are unconstrained therefore the shear stresses are zero, so that

2.28

and

2.29

Secondly, normal displacements are prescribed on the boundary as

2.30

in which f^S(y) and f^A(y) are symmetric and antisymmetric smooth functions in y, respectively, and they are obtained by fitting the averaged normal displacement component on the cut line obtained from experimental measurements. Hence, normal displacements on the boundary for both antisymmetric and symmetric parts can be written as

2.31

and

2.32

where the quantities u₀ and u₁ represent displacements of the strip as a rigid body and they are analytically determined as shown in appendix C. These rigid body terms always arise in the integration of strains to determine displacements. These coefficients reflect the fact that the known stress state and thus strain in the body is sufficient to determine its deformed shape but not its location in space. Rigid body translation arises in the normal displacement component for the symmetric solution as it is the zeroth order of the polynomial u₀ (even). On the other hand, rigid body rotation arises in the normal displacement component for the antisymmetric solution as it is the first order of the polynomial u₁y (odd).

(i). Biorthogonality condition

The complex constants (A_n and B_n) can be determined by using the BCs at x=0. There are three essentially different methods appearing in the literature for formulating infinite sets of equations. The first one is the direct method in which the equations are satisfied identically at a suitable set of collocation points [26]. The second approach is the Galerkin method where sets of weighting functions are introduced and mean equations are obtained and solved simultaneously [27]. Both direct and Galerkin methods result in a general eigenvalue problem with square matrix size of N×N and N unknown complex constants. The more elegant techniques are to use a set of biorthogonality approaches exploited by Johnson & Little [28] or Spence [29] and developed by Gregory [30]. By means of the biorthogonality relations, the complex expansion coefficients can be determined analytically. In the terminology of this study, the biorthogonality condition for the two-dimensional elastic problem of the strip by Gregory [30] takes the form:

2.33

where

2.34

can be obtained by integration using equations (B1), (B3), (B7) and (B8). In order to apply the relation in equation (2.33) to the case where σ_xy and u_x are prescribed as BCs (2.28) and (2.31) at the line x=0, equation (2.33) is multiplied by A_l and summed on l in the range of , obtaining

2.35

The summation on the left-hand side of this expression is taken under the integral sign and summation is performed as

2.36

substituting equations (2.28) and (2.31) into this equation, we get

2.37

hence

2.38

Unknown complex coefficients of the series expansion for the symmetric solution are calculated as

2.39

In a similar manner, unknown complex coefficients of the series expansion for the antisymmetric solution are calculated as

2.40

in which

2.41

For prescribed symmetric and antisymmetric displacements f^S(y) and f^A(y), unknown complex coefficients A_m and B_m are determined using equations (2.39) and (2.40) and substituting these complex coefficients into equation (2.23), residual stress fields are obtained. It should be noted that the solutions are given for plane stress. In order to get plane strain solutions, equations (B11)–(B14) are used in replacement of equations (B7)–(B10) given in appendix B.

(b). The finite rectangle

The technique for residual stress calculations used in §2a is particularly appropriate for rectangular domains possessing large aspect ratio. In other cases, if rectangles do not have large aspect ratio, we lose the advantage of the solution obtained in §2a, and hence an alternative approach must be developed to represent the solution of the entire problem as the sum of two independent systems of complex biharmonic functions for both symmetric and antisymmetric parts in the form:

2.42

and

2.43

where C_n, D_n, F_n and G_n are unknown complex coefficients. Here, the symmetric (2.42) and antisymmetric (2.43) solutions can be expressed in the form of

2.44

and

2.45

where the details of the algebra are given in appendix D. The stress and displacement components can then be expressed as

2.46

2.47

and

2.48

under plane stress

2.49

under plane strain

2.50

where primes denote differentiation with respect to y. Note that components for the symmetric part are denoted here; those for the antisymmetric part can be expressed in a similar manner. We now look at conditions along the lines, x=±b, where, first, the left edge is traction free, i.e.

2.51

and

2.52

Secondly, the displacements in the y-direction are unconstrained hence the shear stresses at the right edge are zero, so that

2.53

and

2.54

while the normal displacements are prescribed at x=b, so we set

2.55

and

2.56

where the constants u₀ and u₁ are rigid body terms which are given in appendix C. Similar to the semi-infinite strip solutions, as a result of there being two sets of eigenvalues and eigenfunctions, the general solutions for the finite rectangle are made up of symmetric and antisymmetric solutions each of which must satisfy the BCs. Hence, measured displacements are decomposed into symmetric and antisymmetric parts and they are used for the corresponding solutions.

Most existing solutions in the literature [31] to problems involving finite elastic rectangles work well only when the boundary conditions to be imposed on the edges are of displacement or traction type where their magnitudes are equal at the opposite edges. Here, we wish to impose a set of BCs which are of mixed type at one edge but of traction type at another and this requires solving multiple subproblems to obtain a general solution to the present problem possessing different types of BCs using the approach exploited by Meleshko & Gomilko [31].

The Galerkin method seems the most suitable method for the BCs we need to impose. Here, sets of weighting functions are introduced and mean equations are solved simultaneously. Fourier weight functions worked well for the semi-infinite strip problem in the work of Kartal et al. [27] in which BCs were mixed over one region and traction-free type over another. Fourier weight functions

2.57

and

2.58

are introduced, in which m is a positive integer, . Equations (2.51)–(2.56) are now enforced in a weak sense, giving

2.59

2.60

2.61

and

2.62

which impose y-direction BCs along the opposite edges and

2.63

2.64

2.65

and

2.66

which enforce x-direction BC along the opposite edges. Evaluation of the integrals in equations (2.59), (2.61), (2.63) and (2.65) provides an infinite set of algebraic equations for the symmetric solution,

2.67

Similarly, an infinite set of algebraic equations for the antisymmetric solution is obtained by evaluating the integrals in equations (2.60), (2.62), (2.64) and (2.66) as

2.68

in which, M^ij, N^ij,H^j and J^j are given in appendix E. Equations (2.67) and (2.68) may be truncated to a finite value, and the coefficients of series expansion, C_n, D_n, F_n and G_n are found. Finally, by superposing the symmetric and antisymmetric solutions, the general solutions are obtained. Here, details of the algebra are given for plane stress. In order to study the plane strain condition, equation (2.50) is used instead of equation (2.49).

3. Experimental procedure

The specimen used in this study for a practical implementation of the analytical contour method was the aluminium alloy 2024-T351, which was previously used to measure residual stress components with the help of the inverse problem of eigenstrain and neutron diffraction by Kartal et al. [32]. Hence, only a brief description of the experimental procedure is given here. The 2024-T351 aluminium alloy was welded using the technique of variable polarity plasma arc welding (VPPA) by Cranfield University [33]. It was assumed that the specimen was isotropic and linear elastic with Young's modulus 72.4 GPa and Poisson's ratio 0.34, which are typical values for aluminium alloys.

The ‘as received’ specimen was supplied as a ‘dogbone’ specimen with 160 mm gauge length, 350 mm full length, 80 mm gauge width and 7 mm thickness. The welding direction was in the longitudinal direction. The ‘dogbone’ specimen was used for the neutron diffraction measurements by Ganguly [34] to determine the residual stresses across the width of the specimen at the ISIS facility of the Rutherford Appleton Laboratory.

Neutron diffraction is a well-established non-destructive technique to determine strains within metallic structures. The interplanar atomic lattice distance can be determined from the position of the diffraction peaks using Bragg's law [35]. Measurements in the welded sample were carried out in three directions (longitudinal, transverse and normal to the weld) to be able to compute the normal stress components. For the longitudinal strain measurements, the gauge volume was 2×2×2 mm³. The gauge volume was increased to 2×20×2 mm³ for measurements in the transverse and normal directions to reduce the overall measurement time.

After the neutron diffraction measurement, the specimen was machined to produce a 160×80×7 mm³ block to perform the contour method. A wire electro-discharge machine (EDM) with a 0.1 mm diameter brass wire was used to cut the specimen into two halves. In order to minimize any movement during the cutting processes, the specimen was clamped firmly by using a steel plate. The cutting process was performed by submerging the specimen. Figure 3 shows the dimensions of the sample with the position of the EDM cut.

Figure 3.

Welded aluminium alloy 2024-T351 specimen used for a practical implementation of the analytical contour method. (Online version in colour.)

The next step was to accurately measure the surface profile (topography) of the two cut surfaces. In this way, the mutual displacements normal to the cut plane owing to the stress release were measured. Previous experiments have shown that the use of a coordinate measurement machine (CMM) provides adequately accurate measurement for a variety of specimens [36,37]. Both surface displacements were then averaged to remove any shear stress relief effects and cut path wandering, as explained earlier. Assuming that the state of plane stress persists in the thin welded plate, the displacements were also averaged across the thickness. Fourier series was then used to fit the experimental data to an analytical function. Fourier series are in the form of

3.1

where, c and d are coefficients of series expansions, ω is the fundamental frequency and j is order of the series. The smooth displacement function was then decomposed into symmetric and antisymmetric parts as

3.2

and

3.3

4. Results and discussion

The average experimental displacements and data fit for the welded aluminium alloy 2024-T351 specimen are depicted in figure 4a. By choosing a 15th-order Fourier series to fit the experimental data, a good agreement could be obtained to the measured values and the resulting fitted displacements were indeed smooth. In order to investigate the best fit for the experiment, the order of the Fourier series was increased up to 20. It was found, however, that there was no significant change between the 15th and 20th orders. Figure 4b shows the symmetric and antisymmetric smooth displacements obtained by decomposing the Fourier series into two parts as defined in equations (3.2) and (3.3). As can be seen from the figure, deformations owing to residual stress relaxation on the cut surface are predominantly symmetric in y and antisymmetric displacements are very small. Hence, resulting symmetric residual stresses are expected to be much higher than the antisymmetric case.

Figure 4.

(a) Experimental displacements and fit for contour measurements on the thin welded sample and (b) symmetric and antisymmetric displacement components in y.

The numerical scheme described for the analytical contour method was implemented within commercial code ‘MATLAB’ but with, of course, the infinite set of simultaneous equations truncated to a finite value, N. It is found that, when N is set to 300 the change in the calculated coefficients of the series expansion is negligible. Figure 5 shows the contour plots of the symmetric and antisymmetric parts for residual stress component σ_xx determined by using the solutions of the analytical contour method for the finite rectangle (§2b). As we have traction-free BCs at the left-hand side, both solutions, as expected, decay remote from the cut plane. In figure 5, the antisymmetric solution rapidly decays, whereas decay rate for the symmetric solution is slower. We may estimate the decay rate noting the distance between zeros in y in the corresponding BCs [25], where if the distance between zeros reduces, it will result in decaying the solution more rapidly. In addition, the magnitude of the eigenvalues for the symmetric solution is smaller than those for the antisymmetric solution in equations (A3) and (A4); hence the end effects in symmetric problems decay more slowly owing to exponential terms.

Figure 5.

Contour plots of the (a) symmetric and (b) antisymmetric parts obtained by solutions of the analytical contour method for the finite rectangle for the residual stress component σ_xx. (Online version in colour.)

Figure 6 denotes the normal residual stress component σ_xx for both symmetric and antisymmetric parts on the line x=40 mm. As can be seen from the figure, the magnitude of the symmetric component of the residual stress is much bigger than the antisymmetric one. This is because of the fact that welding processes mostly affect the surrounded material around the weld region equally. During the welding procedure, the region surrounded by the weld is subjected to high temperature which results in a volume expansion. This volume expansion is restricted by the cooler material and causes thermal stress. After the thermal stresses reach the yield point, the material locally yields. When the heated weld metal shrinks during solidification, it is restrained by the colder material around the weld region. As a result, this brings about a tensile residual stress around the weld region, balanced by compressive residual stress in a region further away from the weld region. Notwithstanding this, we can still obtain antisymmetric residual stress in y. This can be attributed to precise welding conditions, the cooling condition of the weld and the constraint of the weld material.

Figure 6.

Plots of the residual stress component σ_xx on the line x=40 mm for both symmetric and antisymmetric solutions of the analytical contour method for the finite rectangle.

In order to illustrate the agreement level between the analytical contour method and the conventional contour method, an 80×80 mm² rectangle was modelled under the plane stress condition using the FE method. The element size was chosen as 0.1 mm. The averaged and smoothed displacements were applied as displacement BCs to the nodes on the surface representing the cut. Figure 7 shows a comparison of the total (symmetric and antisymmetric) residual stress component σ_xx. Three contour plots are shown: the conventional contour method is based on the FE model; that labelled ‘analytical CM for the rectangle’ was obtained by the solutions of the finite rectangle (§2b); finally, that labelled ‘analytical CM for the semi-infinite strip’ employs the solutions of the semi-infinite strip (§2a). For the semi-infinite strip, the first 80 mm length from the cut line was plotted. As can be seen from the figure, the three different solutions show very good agreement and the component of normal stress persists for a distance x=6a/4 from the cut plane. It is found that residual stress has fallen to around 2 MPa at x=2a for the solutions of the semi-infinite strip. It should be noted that although there is very good agreement found between the solutions for the semi-infinite strip and the finite rectangle in this example, some other applications of these two solutions for the analytical contour method on different finite-size samples might be significantly different from each other depending upon the measured displacement profiles affecting the decay rate and also dimensions of samples.

Figure 7.

Field plots obtained from (a) analytical contour method (CM) for the rectangle, (b) conventional CM and (c) analytical CM for the semi-infinite strip for the total normal stress component σ_xx. (Online version in colour.)

To compare the results for the two different independent techniques, the total residual stress variation through the line at x=40 mm obtained from the solutions of the analytical contour method for the finite rectangle was compared with the neutron diffraction measurements in figure 8. Calculation of the normal stresses by the conventional contour method is also presented for comparison. The error bars on the neutron data were calculated from the reported uncertainties in the refinement fitting of the diffraction spectra according to Pawley [38]: other uncertainties such as those in the elastic modulus and possible systematic errors in d₀ are not included, and the total uncertainty may therefore be greater than that shown by the error bars. The agreement between the two different techniques is very good, although there is a slight spatial shift in the data towards the right of the plot. This difference might be owing to the fact that during specimen cutting from one side, there may be local deformation ahead of the cutting wire as a result of the local stress relaxation, which alters the width of the cut and introduces some error into the final stress values. In addition, the maximum difference between the analytical and conventional contour method was found to be 14 MPa at the right corner.

Figure 8.

Line plots of the total residual stress component σ_xx on the line x= 40 mm.

Even though a fine mesh size (0.1 mm) has been used for the conventional contour method in this study, we obtained a considerably big difference between analytical and conventional contour methods at the corners y=±40 mm. For example, the maximum difference was 42 MPa at the corners when the element size for the FE model was chosen as 0.5 mm, which is a typical mesh size for the implementation of the conventional contour method in three-dimensional applications. These results also suggest that errors induced by the use of the FE method must be taken into account when error estimations are carried out for the conventional contour method.

It should be noted that a change in the geometry of a body alters the residual stress. Hence, it would not be a fair comparison between the contour and neutron diffraction stresses if the length of the ‘dogbone’ specimen had been reduced to significantly small size. However, in this study, the results obtained from semi-infinite strip solutions show that the maximum stress value reduces to 2 MPa at line x=80 mm, away from the contour cut plane. In addition, previous experimental results in the literature show that the cutting-related errors and uncertainties for neutron diffraction measurements are much bigger than this effect.

5. Conclusion

This paper presents analytical solutions, obtained for a semi-infinite strip and a finite rectangle, which can be used to determine the residual stress field using the contour method. Previously, FE software was required for this task. This new analytical technique has been applied to the determination of residual stresses in a VPPA-welded plate and the results were in excellent agreement with the conventional FE-based approach and also with independent neutron diffraction measurements. It is anticipated that the analytical solutions described here will find wide application within the residual stress community as a replacement or supplementary technique for the FE approach.

Appendix A. Eigenvalue evaluation

An initial estimate of the complex roots of the characteristic equations in the asymptotic form is provided by Buchwald [39].

A1

and

A2

where n is an integer. With the help of equations (A1) and (A2), the complex roots of equations (2.12) and (2.13) are solved numerically by an iterative approach such as by using FindRoot[eqns,vars] in MATHEMATICA. The first few terms of and are found to be

A3

and

A4

Appendix B. Eigenfunctions

B1

B2

B3

B4

B5

and

B6

Under plane stress, expressions for displacement components are

B7

B8

B9

and

B10

Equations (B7)–(B10) are replaced with the following equations for plane strain:

B11

B12

B13

and

B14

Appendix C. Rigid body displacements

Equations (2.28) and (2.31) can be expressed in terms of the symmetric eigenfunction

C1

and

C2

in which primes denote differentiation with respect to y. Multiplying equation (C1) by y and integrating both equations with respect to y, over the interval [0,a], we obtain

C3

remembering that

C4

In a similar manner, equations (2.29) and (2.32) are written in terms of the antisymmetric eigenfunction

C5

and

C6

multiplying equation (C5) by y² and equation (C6) by y, then integrating both results with respect to y over the interval [0,a] and integrating by parts will yield

C7

noting that

C8

Appendix D. Potential function evaluation

The symmetric potential function (2.42) can be written as

D1

Noting that first and second summations in this equation are conjugates to each other and the evaluation of (D1) is given in equation (2.44).The antisymmetric potential function (2.43) can be evaluated in a similar manner.

Appendix E. Coefficients of systems

The coefficients of systems in equations (2.67) and (2.68) are determined as

E1

E2

E3

E4

E5

E6

E7

and

E8

Funding statement

The author acknowledges the financial support of the Engineering and Physical Sciences Research Council in the UK (EPSRC) under grant reference EP/G004676/1, ‘Micromechanical Modelling and Experimentation’.