*3.6.4 Cartesian coordinates of plane stress/plane strain solutions for a single hole with internal pressure*

The polar coordinate solutions of Section 3.5 have been transformed to Cartesian coordinates, using the appropriate coordinate transition equations of Section 3.6.1:

$$u\_{\mathbf{x}} = P\left(\frac{a^2}{\mathbf{x}^2 + \mathbf{y}^2}\right)\left(\frac{\mathbf{1} + \nu}{E}\right)\mathbf{x} \tag{27}$$

$$
\mu\_{\mathcal{Y}} = P \left( \frac{a^2}{\varkappa^2 + \mathcal{Y}^2} \right) \left( \frac{\mathbf{1} + \nu}{E} \right) \mathcal{Y} \tag{28}
$$

#### **3.7 Constitutive equations**

From the displacement field equations, once converted to Cartesian coordinates, one may compute the displacement gradients to obtain the strain tensor components in every location of the elastic medium:

$$
\varepsilon\_{\infty} = \frac{\partial u\_{\infty}}{\partial \mathbf{x}}\tag{29}
$$

$$
\varepsilon\_{\mathcal{Y}} = \frac{\partial u\_{\mathcal{Y}}}{\partial \mathbf{y}} \tag{30}
$$

$$\varepsilon\_{\rm xy} = \frac{1}{2} \left[ \frac{\partial u\_{\rm x}}{\partial \mathbf{y}} + \frac{\partial u\_{\rm y}}{\partial \mathbf{x}} \right] \tag{31}$$

In the present study, we follow the mechanical engineering convention where extension and tension are positive. Once the strain components have been identified for our specific problem, the stresses in each point of the elastic medium can be resolved using the constitutive equations. The following equations are valid for either plane strain or plane stress, depending on the value assigned to *κ* with a linear elasticity assumption [17]:

$$
\varepsilon\_{\text{xx}} = \frac{1}{8G} \left[ (\kappa + \mathbf{1}) \sigma\_{\text{xx}} + (\kappa - \mathbf{3}) \sigma\_{\text{yy}} \right] \tag{32}
$$

$$
\varepsilon\_{\mathcal{Y}} = \frac{1}{8G} \left[ (\kappa - \mathfrak{Z}) \sigma\_{\text{xx}} + (\kappa + \mathfrak{1}) \sigma\_{\mathcal{Y}} \right] \tag{33}
$$

$$
\varepsilon\_{\text{xy}} = \frac{\sigma\_{\text{xy}}}{\text{2G}} \tag{34}
$$

The constitutive equation for plane strain is given by substituting *κ* ¼ 3 � 4 *ν*; for plane stress, one should use *κ* ¼ ð Þ 3 � *ν =*ð Þ 1 þ *v* . Separately, when *G* is used in the equations, the solutions for plane stress may be converted to plane strain by replacing *ν* with *ν=*ð Þ 1 � *ν* , which means replacing 3ð Þ � *ν =*ð Þ 1 þ *ν* by 3ð Þ � 4*ν* . Likewise, solutions for plane strain may be converted to plane stress by replacing *ν* with *ν=*ð Þ 1 þ *ν* , which means replacing 3ð Þ � 4*ν* by 3ð Þ � *ν =*ð Þ 1 þ *ν* . The strain in the *z*-direction, *εzz*, vanishes for plane strain but does not necessarily vanish for plane stress (*σzz* ¼ *σxz* ¼ *σyz* ¼ 0) where it is given by:

$$
\varepsilon\_{\rm xx} = -\frac{v}{E} \left( \sigma\_{\rm xx} + \sigma\_{\rm yy} \right) = -\frac{v}{\mathbf{1} - v} \left( \varepsilon\_{\rm xx} + \varepsilon\_{\rm yy} \right) \tag{35}
$$

The principal strain magnitude can now be obtained as follows:

$$\varepsilon\_1, \varepsilon\_2 = \frac{\varepsilon\_{\infty} + \varepsilon\_{\mathcal{py}}}{2} \pm \sqrt{\left(\frac{\varepsilon\_{\infty} - \varepsilon\_{\mathcal{py}}}{2}\right)^2 + \varepsilon\_{\infty}^2} \tag{36}$$

## **4. Results**

A MATLAB code was written to evaluate—for specific hole arrangements—the delta between the plane strain and the plane stress solutions based on the algorithms developed in Section 3. We consider holes of equal and different radii, with and without farfield stress, with and without internal pressures, and the pressure of individual holes may be varied. All solutions given are for static conditions, in the sense that timedependent changes are not considered in the present study. Two types of borehole problems are addressed: single hole (Section 4.1) and multi-hole (Section 4.2).

#### **4.1 Single-hole problems**

The principal stress distributions *σ*<sup>1</sup> and *σ*<sup>2</sup> are computed using

$$
\sigma\_1 = \frac{1}{2} \left( \sigma\_{\text{xx}} + \sigma\_{\text{\mathcal{W}}} \right) + \left[ \sigma\_{\text{xy}}^2 + \frac{1}{4} \left( \sigma\_{\text{xx}} - \sigma\_{\text{\mathcal{W}}} \right)^2 \right]^{\frac{1}{2}} \tag{37}
$$

$$
\sigma\_2 = \frac{1}{2} \left( \sigma\_{\text{xx}} + \sigma\_{\text{yy}} \right) - \left[ \sigma\_{\text{xy}}^2 + \frac{1}{4} \left( \sigma\_{\text{xx}} - \sigma\_{\text{yy}} \right)^2 \right]^{\frac{1}{2}} \tag{38}
$$

*Asymptotic Solutions for Multi-Hole Problems: Plane Strain versus Plane Stress Boundary… DOI: http://dx.doi.org/10.5772/intechopen.105048*


#### **Table 1.**

*Model inputs used for the single-hole problem.*

The first case considers a single hole subject to either. Case 1–1: a far-field stress only, Case 1–2: an internal pressure only, Case 1–3: the superposed Cases 1–1 and 1–2.

Model inputs are given in **Table 1**. A comprehensive comparison of the vector displacement fields, strain tensor components, principal strains, stress tensor components, and principal stresses for all the above cases is given in Appendix C.

### *4.1.1 Case 1–1: single hole subject to a far-field stress*

**Figure 1** quantifies the delta of the principal stress distributions *σ*<sup>1</sup> and *σ*<sup>2</sup> for the plane strain and plane stress boundary conditions in the case of applying far-field

#### **Figure 1.**

*Principle stress distributions σ*<sup>1</sup> *(top row) and σ*<sup>2</sup> *(bottom row) for single hole subject to far-field stress. The first column is for plane strain boundary conditions, the second column is for plane stress boundary conditions, and the third column quantifies the difference (delta) between the first and second cases.*

**Figure 2.**

*Principle stress distributions σ*<sup>1</sup> *(top row) and σ*<sup>2</sup> *(bottom row) for single hole subject to internal pressure. The first column is for the plane strain boundary condition; the second column is for the plane stress boundary condition.*

stress only. The first column in **Figure 1** is for plane strain boundary conditions, the second column is for plane stress boundary conditions, and the third column represents the residual of the principal stress magnitude.

#### *4.1.2 Case 1-2: single hole subject to internal pressure*

For the hole using only internal pressure, the principal stress solutions for plane strain and plane stress are identical (**Figure 2**) due to the same displacement fields (see the reasoning in Section 3.5).

### *4.1.3 Case 1-3: superposed cases 1-1 and 1-2*

The superposed Cases 1–1 and 1–2 seem trivial, but the deltas in **Figure 3** differ from those in **Figure 1**. The explanation is that the displacements due to the internal pressure on the hole add lateral uniform volumetric displacements that shift the locations where the deltas occur. When the internal pressure on the hole increases, the overall delta remains limited.

*Asymptotic Solutions for Multi-Hole Problems: Plane Strain versus Plane Stress Boundary… DOI: http://dx.doi.org/10.5772/intechopen.105048*

**Figure 3.**

*The residual of the principle stress distributions σ*<sup>1</sup> *(top row) and σ*<sup>2</sup> *(bottom row) for a single hole subject to farfield stress (10 MPa) and different internal pressure loads (first column:* P *= 1 MPa, second column:* P *= 5 MPa, and third column:* P *= 10 MPa).*

#### **4.2 Multi-hole problems**

For multi-hole modeling, the elastic displacements due to the individual contributions are superposed, then converted to the overall stress state via the constitutive equations. The procedure has been previously applied and was coined the Linear Superposition Method (LSM) in prior work. For perfect analytical accuracy of LSM multi-hole solutions, the superposition patterns would require perfect symmetry patterns for hole arrangements and endless repetitions as in the method of images. This symmetrical superposition principle also lies at the heart of earlier analytical multihole stress interference solutions [24–27].

A previous multi-hole solution departing from symmetric superposition by instead using randomly placed holes was assumed a valid approach [28]. The 11-hole problem in ref. [28], solved by them with a system of linear algebraic equations using a complex boundary integral method based on truncated Fourier series, was closely matched with an LSM solution [29]. We are well aware that LSM solutions for arbitrarily placed holes would be only asymptotically correct, due to hole patterns lacking symmetry. However, based on close matches of LSM-based solutions with photoelastic patterns in our prior studies [4], as well as a comparison against Abaqus solutions in [30] our conclusion was that LSM gives very reliable results even for randomly placed hole arrangements.

To further support the practical reliability of LSM solutions for randomly placed holes, several new comparisons of stress field solutions with LSM for multihole problems with those obtained with other methods are given below. These comparisons are for a photo-elastic prototype strain and stress visualization (Case 2–1) and a prototype solution based on a finite element solution method (Case 2–2).

### *4.2.1 Case 2–1: photo-elastic prototype*

For multi-hole analysis, we first consider a traditional example of photo-elastic visualization of displacement and strain components. The 5-hole photo-elastic prototype (**Figure 4**) has a total thickness of 5 mm (3 mm aluminum and 2 mm photo-elastic coating). The aluminum strips are 100 mm wide and 450 mm long. The long dimension may be assumed well suited for an infinite plate solution. However, the lateral width of 100 mm leaves only 30 mm between the boundaries of the outer holes (all have radii of 10 mm) and the left and right boundaries of the elastic plate.

A point that has been little elaborated is whether photo-elastic experiments typically represent thin- or thick-plate solutions, that is, represent plane stress or plane strain solutions. Theoretically, solutions for a plane stress boundary condition would apply to holes in very thin strips (for which *σzz* will be negligibly small). However, when a plate is "thicker" instead of *σzz* ! 0, we will have the *ϵzz* ! 0, and the boundary condition approaches a plane strain case. For exactly what "finite thickness" of an elastic strip with holes, the plane stress solutions would need to be replaced by a plane strain solution has never been made explicit.

The accurate LSM solutions for either an infinite plate [with thin plate *<sup>a</sup> <sup>h</sup>* ! <sup>∞</sup> or a thick plate *<sup>a</sup> <sup>h</sup>* ! <sup>0</sup> solutions] will not be able to perfectly match the photo-elastic prototype with finite width, finite length, and for 0 < *<sup>a</sup> <sup>h</sup>* < ∞. Nonetheless, we can still use LSM to investigate which solution (plane strain or plane stress) gives the best approximation for a particular case. We tested for both, following [14] reasoning (summarized in Section 2.2 of the present study), for *<sup>a</sup> <sup>h</sup>* ! ∞ we have a thin-plate problem (plane stress); for *<sup>a</sup> <sup>h</sup>* ! 0 we have a thick-plate problem (plane strain).

#### *4.2.2 Case 2–1: results*

Model inputs are given in **Table 2**. Match attempts of **Figure 4 b-d** contour patterns with plane strain and plane stress LSM codes are given in **Figures 5**–**7**,

#### **Figure 4.**

*The 5-hole photo-elastic prototype. (a) Plate dimensions, (b) isochromatic pattern for strain state, (c) u x*ð Þ , *y displacement magnitude contours, (d) v x*ð Þ , *y displacement magnitude contours (after [10]).*

*Asymptotic Solutions for Multi-Hole Problems: Plane Strain versus Plane Stress Boundary… DOI: http://dx.doi.org/10.5772/intechopen.105048*


#### **Table 2.**

*Model inputs used for 5-hole problem.*

respectively. Any mismatches near the right and left margins of the sample may be due to differences in lateral boundary conditions: the photo-elastic strip has a finite width, while our solutions are for an infinite plate. The lateral boundary may be simulated by a mirror-image approach, but was not pursued in the present study. Separately, comprehensive comparisons of the vector displacement fields, strain tensor components, principal strains, stress tensor components, and principal stresses are given in Appendix D.

For the elastic prototype of **Figure 4**, the model scaling used was *<sup>a</sup> <sup>h</sup>* = 2, which means the elastic displacement field (**Figures 6** and **7**) and resulting strain state (**Figure 5**) in the plane of view represent the plane stress solution. The LSM method is used in this example for both plane stress and plane strain solution to validate this theoretical result. Clearly, the LSM plane stress solutions (**Figures 5c**–**7c**) are closer (but not "exactly") to the contour patterns in **Figures 5a**-**7a**, respectively, than the LSM plane strain solutions (**Figures 5b**–**7b**).

### *4.2.3 Case 2-2: Numerical benchmark; solution paths*

The accuracy of the LSM-based solutions was benchmarked in prior studies [3, 29], against results from independent solution methods (e.g., [28]), with excellent matches. Here we benchmark LSM in a multi-hole solution against the independent numerical solution for the tangential stress concentrations in the rim of a 5-hole problem by Yi et al. [11]. The 5-hole configuration studied is part of an infinite plate subject to a uniaxial far-field compression, with dimensions as shown in **Figure 8**. The numerical solution method (based on the finite element method) was validated by Yi et al. [11] against a prior analytical-numerical solution (based on a Laurent series method [31]).

The 5-hole problem of **Figure 8** has its holes positioned slightly different than those in **Figure 4**. We used the exact same 5-hole setup as in **Figure 8** to solve the tangential stresses with our LSM code. To quantify the radial and tangential stresses in a particular polar coordinate system ð Þ *r*, *θ* , one may follow two different computational paths.

**Path 1**: Use *x* ¼ *r* cos *θ* and *y* ¼ *r*sin *θ* as inputs for the Cartesian displacement equations. For specific locations ð Þ *r*, *θ* , such as at the rim of the central hole in

#### **Figure 5.**

*(a) Photo-elastic fringes near five equal holes due to far-field tension in the vertical direction of image view [10]. (b) LSM plane strain solution for ε*2*. (c) LSM plane stress solution for ε*2*. (d) the residual between the plane stress and plane strain solutions for ε*2*.*

#### **Figure 6.**

*(a) u x*ð Þ , *y displacement magnitude contours [10]. (b) LSM plane strain solution for uy (c) LSM plane stress solution foruy.*

*Asymptotic Solutions for Multi-Hole Problems: Plane Strain versus Plane Stress Boundary… DOI: http://dx.doi.org/10.5772/intechopen.105048*

**Figure 7.**

*(a) v x*ð Þ , *y displacement magnitude contours [10]. (b) LSM plane strain solution for ux. (c) LSM plane stress solution for ux.*

#### **Figure 8.**

*An infinite plate containing five equal circular holes under axial compressive stress (b* <sup>¼</sup> <sup>3</sup> ffiffiffiffiffi <sup>2</sup>*<sup>a</sup>* <sup>p</sup> *<sup>=</sup>*<sup>2</sup> *and <sup>σ</sup>* <sup>¼</sup> <sup>1</sup>Þ*: No internal pressure load.*

**Figure 8**, one can next compute the three polar strain tensor components using the following coordinate transformation for the strain tensor elements (e.g., Kelly Notes Solid Mechanics Part 2, Eq. 4.2.17):

$$
\varepsilon\_r = \varepsilon\_{\text{xx}} \cos^2 \theta + \varepsilon\_{\text{yy}} \sin^2 \theta + \varepsilon\_{\text{xy}} \sin 2\theta \tag{39}
$$

$$
\varepsilon\_{\theta} = \varepsilon\_{\text{xx}} \sin^{2}{\theta} + \varepsilon\_{\text{yy}} \cos^{2}{\theta} - \varepsilon\_{\text{xy}} \sin{2\theta} \tag{40}
$$

$$
\varepsilon\_{r\theta} = \left(\varepsilon\_{\mathcal{Y}} - \varepsilon\_{\infty}\right) \sin\theta \cos\theta + \varepsilon\_{\mathcal{X}} \cos 2\theta \tag{41}
$$

Please note that for the un-pressurized hole subjected to (only) far-field stress, the radial strain, *ε<sup>r</sup>* at the hole, the boundary will vanish (only at the hole boundary and not beyond).

Path 2: Revert to the original displacement equations in polar coordinates (Sections 2.1 to 2.5) and compute the displacement gradients in polar coordinates:

$$
\varepsilon\_r = \frac{\partial u\_r}{\partial r} \tag{42}
$$

$$
\varepsilon\_{\theta} = \frac{1}{r} \frac{\partial u\_{\theta}}{\partial \theta} + \frac{u\_{r}}{r} \tag{43}
$$

$$\varepsilon\_{r\theta} = \frac{1}{2} \left( \frac{1}{r} \frac{\partial u\_r}{\partial \theta} + \frac{\partial u\_\theta}{\partial r} - \frac{u\_\theta}{r} \right) \tag{44}$$

After having obtained the polar displacement gradients, one may compute the stresses for a plane stress (thin plate) problem from the following constitutive equations:

$$
\sigma\_r = \frac{E}{1 - \nu^2} (\varepsilon\_r + \varepsilon\_\theta \nu) \tag{45}
$$

$$
\sigma\_{\theta} = \frac{E}{1 - \nu^2} (\varepsilon\_{\theta} + \varepsilon\_r \nu) \tag{46}
$$

$$
\sigma\_{r\theta} = \frac{E}{2(1+\nu)} \varepsilon\_{r\theta} \tag{47}
$$

For plane strain (thick plate) problem, the corresponding constitutive equations are ([32], Eq. (5-38)):

$$
\sigma\_r = \frac{2G}{1 - 2\nu} \left[ (\mathbf{1} - \nu) + \varepsilon\_\theta \nu \right] = \frac{E}{(\mathbf{1} + \nu)(\mathbf{1} - 2\nu)} \left[ (\mathbf{1} - \nu) + \varepsilon\_\theta \nu \right] \tag{48}
$$

$$\sigma\_{\theta} = \frac{2G}{\mathbf{1} - 2\nu} \left[ \varepsilon\_{\theta} (\mathbf{1} - \nu) + \varepsilon\_{r} \nu \right] = \frac{E}{(\mathbf{1} + \nu)(\mathbf{1} - 2\nu)} \left[ \varepsilon\_{\theta} (\mathbf{1} - \nu) + \varepsilon\_{r} \nu \right] \tag{49}$$

$$
\sigma\_{r\theta} = G\varepsilon\_{r\theta} = \frac{E}{\Im(\mathbf{1} + \nu)} \varepsilon\_{r\theta} \tag{50}
$$

For completeness, polar strain tensor components can be computed from the stress tensor components as follows ([32], Eq. (5-37)):

$$\varepsilon\_r = \frac{1}{2G} [\sigma\_r(\mathbf{1} - \boldsymbol{\nu}) - \sigma\_\theta \boldsymbol{\nu}] \tag{51}$$

$$\varepsilon\_{\theta} = \frac{1}{2G} [\sigma\_{\theta}(\mathbf{1} - \nu) - \sigma\_{r}\nu] \tag{52}$$

$$
\varepsilon\_{r\theta} = \frac{\sigma\_{r\theta}}{G} \tag{53}
$$

The polar strain tensor components can be converted to the Cartesian components at any one time using the polar coordinate transformations [e.g. [12], Eq. (48)]:

$$
\varepsilon\_{\infty} = \varepsilon\_r \cos^2 \theta + \varepsilon\_\theta \sin^2 \theta \tag{54}
$$

*Asymptotic Solutions for Multi-Hole Problems: Plane Strain versus Plane Stress Boundary… DOI: http://dx.doi.org/10.5772/intechopen.105048*

$$
\varepsilon\_{\mathcal{V}} = \varepsilon\_r \sin^2 \theta + \varepsilon\_\theta \cos^2 \theta \tag{55}
$$

$$
\varepsilon\_{xy} = \frac{\varepsilon\_r - \varepsilon\_\theta}{2} \sin 2\theta \tag{56}
$$

#### *4.2.4 Case 2–2: results*

The results of our benchmark test for Case 2–2, as per the methodology outlined above, are given here. First, a baseline solution for the tangential stress around a single hole (Case 1–1) is given in **Figure 9**. An important finding is that the plane stress solution for tangential stress concentrations around the hole is less sensitive to the Poisson's ratio compared with the plane strain solution as shown in **Figure 9a, b**, and **Table 3** (for positive values of *ν*). This subtle difference has not been emphasized before. Overall, the delta between the plane strain and plane stress solutions becomes larger for larger Poisson's ratios, resulting in the stress concentration factor being 3 for the plane stress boundary condition, increasing to nearly 4 for the plane strain boundary condition.

The impacts of the Poison ratio and different boundary conditions were analyzed in more detail, based on the displacement fields quantified in Appendix C, which led to the following conclusions:


When the Poisson's ratio is 0, the stress concentrations, according to our LSM models, are the same for plane stress and plane strain boundary conditions (**Figure 9a**). This matching in the concentrations at *ν* ¼ 0 between the plane stress and plane strain solutions can also be seen in Eq. (46), for plane stress, and Eq. (49), for plane strain. For the higher Poisson's ratio *ν* ¼ 0*:*3, the plane strain

#### **Figure 9.**

*Tangential stress concentration variations around the rim of a single hole (case 1–1). (a) Poisson's ratio ν* ¼ 0*:*0*, and (b) ν* ¼ 0*:*3*.*

*Drilling Engineering and Technology - Recent Advances, New Perspectives and Applications*


#### **Table 3.**

*Maximum and minimum tangential stress (σθ*Þ *around the rim of the central hole in a single hole problem (case 1–1) corresponding to different values of Poisson's ratio ν. The max of σθ occurs at θ* ¼ 0 *and π. The minimum of σθ is at <sup>θ</sup>* <sup>¼</sup> *<sup>π</sup>* 2*.*

solution starts to show a higher stress concentration than the plane stress solution (**Figure 9b**).

Next, we show the 5-hole (Case 2–2) stress concentrations around the central hole (**Figure 10a, b**). For the small Poisson's ratio of *ν* ¼ 0, stress concentrations of LSM solutions under plane strain and plane stress boundary conditions are identical (**Figure 10a**). However, for *ν* ¼ 0*:*3, the plane strain solution shows higher stress concentrations at locations *θ* ¼ 0, π (**Figure 10b**) with a maximum value of 3.7728 corresponding to the maximum value for the plane stress solution 3.3481 (see **Table 4**). Overall, the minimum stress concentrations at *θ* ¼ π*=*2 approach �2 (due to stress interference effects between the central hole and its surrounding 4 holes). The maximum stress concentration at *θ* ¼ 0, π is decreased from 3 (for a single hole with *ν* ¼ 0, **Figure 9a**) to 2.9695 (for 5-holes with *ν* ¼ 0, **Figure 10a**).

#### **Figure 10.**

*Tangential stress concentration variations around the rim of the central hole in a 5-hole problem (case 2–2). (a) Poisson's ratio ν* ¼ 0*:*0*, and (b) ν* ¼ 0*:*3*.*

*Asymptotic Solutions for Multi-Hole Problems: Plane Strain versus Plane Stress Boundary… DOI: http://dx.doi.org/10.5772/intechopen.105048*


#### **Table 4.**

*Maximum and minimum tangential stress (σθ*Þ *around the rim of the central hole in a 5-hole problem (case 2–2) corresponding to different values of Poisson's ratio ν. The max of σθ occurs at θ* ¼ 0 *and π. The minimum of σθ is at <sup>θ</sup>* <sup>¼</sup> *<sup>π</sup>* 2*.*

**Figure 11a** and **b** include the prior solutions for the same 5-hole problem configuration using a numerical solution method in ref. [11] and Laurent series method in ref. [31]. Unfortunately, the Poisson's ratio is not specified by either [11] or [31]; neither was the boundary condition made explicit. However, **Figure 11a** shows that the plane strain solution is closer to [11, 31] results than the plane stress for *ν* ¼ 0*:*3, and for *ν* ¼ 0*:*4, the plane stress solution is closer than the plane strain. As for the photoelastic comparison of Case 2–1, the prototype used in Case 2–2 has finite lateral width, due to which the LSM for a similar domain but based on an infinite plate solution will be progressively mismatching.

#### **Figure 11.**

*Comparison of the stress concentrations around the rim of the central hole in a 5-hole problem (case 2–2) using (1) LSM solutions (red line for plane strain and blue line for plane stress boundary conditions) and (2) the prior solutions for the same 5-hole problem configuration using a numerical solution method [11] (solid black line) and Laurent series method [31] (black circles). (a) Poisson's ratio ν* ¼ 0*:*3*, and (b) ν* ¼ 0*:*4*.*
