1 Introduction

Many stiffening problems for the cracked components were proposed (Isida 1973; Chen 1994; Umamaheswar and Singh 1999; Duong and Yu 1997; Antipov et al. 1997). The tension problem of a long cracked strip with stiffened edges was investigated (Isida 1973). The tension problem of a finite cracked plate with stiffened edges was studied (Chen 1994). Two contact problems referring to the partially stiffened elastic half-plane were studied (Antipov et al. 1997).

The problem of assessing the effectiveness of a bonded repair to a cracked plate can be reduced to a one-dimensional integral equation for the special case when both the plate and the reinforcement are isotropic and have the same Poisson’s ratio (Wang and Rose 1998). Bonded composite repairs are efficient and cost effective means of repairing cracks and corrosion grind-out cavity in metallic structures (Duong and Wang 2007). Fundamental concept of crack patching was studied. The crack growth behavior of an aluminum plate cracked at the tip and repaired with a bonded boron/epoxy composite patch in the case of full-width disbond was investigated (Errouane et al. 2014). A numerical model for the optimization of composite patch repair of aluminum plate containing a central crack was developed (Errouane et al. 2014).

On the other hand, many researchers studied the antiplane problem for the elastic elliptic inclusion or layers (Gong 1995; Ru and Schiavone 1996; Chao and Young 1998; Shen et al. 2006; Chen and Wu 2007; Chen 2013). A generalized and unified treatment was presented for the antiplane problem of an elastic elliptic inclusion undergoing uniform eigenstrains and subjected to arbitrary loading in the surrounding matrix (Gong 1995). A novel efficient procedure to analyze the two-phase confocally elliptic inclusion embedded in an unbounded matrix under antiplane loadings was provided (Shen et al. 2006). Dual null-field integral equations was suggested for the multi-inclusion problem under antiplane shears (Chen and Wu 2007). A closed form solution for the Eshelby’s elliptic inclusion in antiplane elasticity was provided (Chen 2013). The interaction problem between a circular inclusion and a symmetrically branched crack embedded in an infinite elastic medium was solved (Lam et al. 1998). A crack problem for an array of collinear microcracks in composite matrix was investigated (Profant and Kotoul 2005). It is found that most previously published papers in the solution for confocally elliptic layers were devoted to a perfect inclusion without crack.

This paper provides a solution for a crack stiffened by elliptic layer in antiplane elasticity. The crack is embedded in an elliptic region and stiffened by a confocally elliptic layer. The whole medium is composed of three portions, the cracked elliptic plate, the confocally elliptic layer and the infinite matrix. The remote loading is denoted by \({\upsigma }_{\mathrm{yz}}^{\infty }\). The cracked elliptic plate and the infinite matrix have the same shear modulus of elasticity. The stiffening elliptic layer has a higher shear modulus of elasticity. By using the complex variable and the continuity conditions along interfaces, the problem can be solved. One numerical example with different sizes and properties of materials is given to show the effect of the stiffening layer.

2 Analysis

For convenience in derivation, we make a substitution \({\upphi } (\hbox {z})=-\hbox {i}{\uppsi }\,(\hbox {z})\) in (Chen et al. 2003) and obtain the following complex potential for antiplane elasticity

$$\begin{aligned} {\uppsi } (\hbox {z})=-\hbox {f}(\hbox {x},\hbox {y})+\hbox {iGw}(\hbox {x},\hbox {y}) \end{aligned}$$
(1)

where G is the shear modulus of elasticity, w(x, y) is the longitudinal displacement in antiplane elasticity. In addition, the result force function f(x, y) is defined by

$$\begin{aligned} \hbox {f}(\hbox {x}, \hbox {y})=\int _{\hbox {z}_\mathrm{o}}^\mathrm{z} { \left( {{\upsigma } _{\mathrm{xz}} \hbox {dy}-{\upsigma } _{\mathrm{yz}} \hbox {dx}} \right) } \end{aligned}$$
(2)

In Eq. (2), the integration is performed from a fixed point \(\hbox {z}_\mathrm{o}\) to a moving point “z”. In addition, \({\upsigma } _{\mathrm{xz}}\) and \({\upsigma }_{\mathrm{yz}}\) denote two stress components. Clearly, the displacement component w(x,y) and the resultant force function f(x,y) satisfy the following Laplace equation

$$\begin{aligned}&\nabla ^{2}\hbox {w(x, y)}=0,\;\nabla ^{2}\hbox {f(x, y)}=0\;\hbox {where}\nonumber \\&\quad \nabla ^{2}=\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}} \end{aligned}$$
(3)

From Eqs. (1) to (3), we can define the stress components by

$$\begin{aligned} {{\uppsi } }'\left( \hbox {z}\right)= & {} {\upsigma } _{\mathrm{yz}} +i{\upsigma } _{\mathrm{xz}} =-\frac{\partial \hbox {f}}{\partial \hbox {x}}+\hbox {iG}\frac{\partial \hbox {w}}{\partial \hbox {x}}\nonumber \\= & {} \hbox {G}\left( \frac{\partial \hbox {w}}{\partial \hbox {y}}+\hbox {i}\frac{\partial \hbox {w}}{\partial \hbox {x}}\right) \end{aligned}$$
(4)

In addition, from Eq. (1) we can get the following equations

$$\begin{aligned}&2\hbox {Gw}(\hbox {x},\hbox {y})=-\hbox {i}({\uppsi } (\hbox {z})-\overline{{\uppsi } (\hbox {z})})\end{aligned}$$
(5)
$$\begin{aligned}&\hbox {f(x, y)}=-\frac{1}{2}({\uppsi } (\hbox {z})+\overline{{\uppsi } (\hbox {z})}) \end{aligned}$$
(6)

For an edge crack shown in Fig. 1, the stress intensity factor at the crack tip can be defined by (Chen et al. 2003)

$$\begin{aligned} \hbox {K}_3 =\mathop {\hbox {Lim}}\limits _{\hbox {z}\rightarrow 0} \sqrt{2{\uppi } \hbox {z}} {{\uppsi } }'(\hbox {z}) \end{aligned}$$
(7)

It was proved that, the right hand term of Eq. (7) generally takes a real value in general (Chen et al. 2003).

Fig. 1
figure 1

An edge crack problem in antiplane elasticity

Full size image

If the stresses \({\upsigma } _{\mathrm{xz}}^\infty \) and \({\upsigma } _{\mathrm{yz}}^\infty \) are applied at infinity for an infinite medium, from Eq. (4) the relevant complex potential is as follows

$$\begin{aligned} {\uppsi } (\hbox {z})=({\upsigma } _{\mathrm{yz}}^\infty +\hbox {i}{\upsigma } _{\mathrm{xz}}^\infty )\hbox {z} \end{aligned}$$
(8)

For the crack embedded in an elliptic plate in antiplane elasticity, the following mapping function is used (Muskhelishvili 1953)

(9)

which maps the unit circle and its exterior region in the -plane into the crack configuration (\(-\)a, a) and its exterior region in the z-plane. The cracked medium is composed of (a) a cracked elliptic plate bounded by \(\Sigma _\mathrm{o}\) [crack along (\(-\)a a)] and \(\Sigma _1 \), (b) an elliptic layer bounded by \(\Sigma _1 \) and \(\Sigma _2 \) and (c) an infinite region exterior to the contour \(\Sigma _2 \) (Fig. 2).

Fig. 2
figure 2

Mapping relations for , (a) Ring region bounded by \(\Gamma _\mathrm{o}\) (with \({\uprho } _\mathrm{o} =1)\) and \(\Gamma _1 \) in the -plane into the corresponding elliptic region bounded by \(\Sigma _\mathrm{o}\)[crack along (\(-\)a a)] and \(\Sigma _1 \) in the z-plane, (b) Ring region bounded by \(\Gamma _1\) with \({\uprho } _1 \) and \(\Gamma _2\) with \({\uprho } _2 \) in the - plane into the corresponding elliptic layer bounded by \(\Sigma _1 \) and \(\Sigma _2 \) in the z-plane and (c) Infinite region exterior to \(\Gamma _2 \) with \({\uprho } _2\) in the -plane into the corresponding infinite region exterior to \(\Sigma _2 \) in the z-plane

Full size image

The mapping function also provides the following mappings: (a) it maps the circle \(\Gamma _\mathrm{o}\) (with and \({\uprho } _\mathrm{o} =1\)) in the -plane into a crack configuration \(\Sigma _\mathrm{o}\) [along the interval (\(-\)a, a)] in the z-plane, (b) it maps circles \(\Gamma _\mathrm{j}\) (with , j = 1, 2) in the -plane into contours \(\Sigma _\mathrm{j}\) (j = 1, 2) in the z-plane (Fig. 2).

In the formulation (Fig. 2), the layer bounded by \(\Sigma _1 \) and \(\Sigma _2 \) is thicker than other portions. In this case, we can assume the layer possesses a higher shear modulus of elasticity. However, if the layer bounded by \(\Sigma _1 \) and \(\Sigma _2 \) is a dissimilar material with a different modulus of elasticity, the derivation in this case is the same as in the studied case.

In addition, the shear modulus of elasticity for confocally elliptic layer bounded \(\Sigma _\mathrm{j} \) and \(\Sigma _{\mathrm{j}+1}\) is denoted by \(\hbox {G}_\mathrm{j}\) (j = 0, 1). The shear modulus of elasticity for the medium exterior to \(\Sigma _2 \) is denoted by \(G_2 \) (Fig. 2). In this case, we have \(\hbox {G}_\mathrm{o},\, \hbox {G}_1 ={\hbox {h}_1\, \hbox {G}_\mathrm{o} }/{\hbox {h}_\mathrm{o} }\) and \(\hbox {G}_2 =G_\mathrm{o}\) for different portions, respectively (Fig. 2).

The inverse of the mapping function of is denoted by

(10)

Clearly, for , or (j= \(-\) 1, 0, 1), from Eq. (9) we have

(11)
(12)

Eq. (12) reveals that along the boundary , or , the function can be converted into the form of an analytic function.

If the remote loading is \({\upsigma } _{\mathrm{xz}}^\infty \) only, the stress intensity factor \(\hbox {K}_3 \) always equal to zero. Therefore, we only study the case of the remote loading \({\upsigma } _{\mathrm{yz}}^\infty \).

The complex potentials defined on many confocally elliptic layers bounded by \(\Sigma _\mathrm{j} \)and \(\Sigma _{\mathrm{j}+1} \) (j  =  0, 1) are denoted by \({\uppsi } _\mathrm{j}^*(\hbox {z})\) (j  =  0, 1). The complex potential defined exterior to the \(\Sigma _2 \) is denoted by \({\uppsi } _2^*(\hbox {z})\) (Fig. 2).

Based on those complex potentials \({\uppsi } _\mathrm{j}^*(\hbox {z})\) (j  =  0, 1, 2), we can define the following complex potentials in the -plane as follows

(13)

Clearly, the displacement and traction and along the interface should be continuous. Therefore, from Eqs. (1), (5) and (6) we can propose the continuity conditions along the interfaces as follows

(14)
(15)

Eq. (14) is derived Eq. (5), which represents the displacement continuity condition along interfaces. In addition, Eq. (15) is derived Eq. (6), which represents the traction continuity condition along interfaces. It is easy to verify that the traction continuity condition along interfaces is equivalent to the same condition for resultant force “f” shown in Eq. (1). Thus, the equality shown by Eq. (15) is obtained.

The complex potentials for the cracked elliptic plate, confocal layer and infinite matrix are expressed in the form

(16)

From Eqs. (14) to (16), we can link the two sets of two undetermined coefficients in the adjacent layers as follows

$$\begin{aligned} \hbox {c}_{\mathrm{j}+1}= & {} {\upalpha }_\mathrm{j}\hbox {c}_\mathrm{j}-\frac{{\upbeta } _j}{{\uprho }_\mathrm{j+1}^2}\hbox {d}_\mathrm{j},\nonumber \\ \hbox {d}_\mathrm{j+1}= & {} -{\uprho }_\mathrm{j+1}^{2} {\upbeta }_\mathrm{j}\hbox {c}_\mathrm{j}+{\upalpha }_\mathrm{j}\hbox {d}_\mathrm{j},\;\left( {\hbox {j}=0,\,1} \right) \end{aligned}$$
(17)

where

$$\begin{aligned} {\upalpha }_\mathrm{j} =\frac{\hbox {G}_\mathrm{j+1} +\hbox {G}_\mathrm{j}}{2\hbox {G}_\mathrm{j} }, \quad {\upbeta }_\mathrm{j} =\frac{\hbox {G}_{\mathrm{j+1}}-\hbox {G}_\mathrm{j}}{2\hbox {G}_\mathrm{j}},\, (\hbox {j}=0, 1) \end{aligned}$$
(18)

We prefer to write Eq. (17) in an explicit form

$$\begin{aligned} \hbox {c}_1={\upalpha }_{0}\hbox {c}_{0}-\frac{{\upbeta }_0}{{\uprho }_{1}^{2}}d_0, \quad \hbox {d}_1=-{\uprho }_1^2 {\upbeta }_0\hbox {c}_0+{\upalpha }_{0}\hbox {d}_{0}\end{aligned}$$
(19)
$$\begin{aligned} \hbox {c}_2={\upalpha }_{1}\hbox {c}_{1}-\frac{{\upbeta }_\mathrm{1}}{{\uprho }_{2}^{2}}\hbox {d}_1, \quad \hbox {d}_2=-{\uprho }_{2}^{2}{{\upbeta }_{1}}\hbox {c}_{1}+{{\upalpha }_{1}}{\hbox {d}_1} \end{aligned}$$
(20)

On the other hand, from Eqs. (1), (6) and (16), we can propose the traction free condition along the crack face

(21)

From Eq. (16) we can express the complex potential as follows

(22)

Substituting Eq. (22) into (21) yields

(23)

Since \({\uppsi }_2^{*} (\hbox {z})={\upsigma }_\mathrm{yz}^\infty \hbox {z}+\hbox {O}(1/\hbox {z})\) and at the remote place, from Eqs. (9) and (16) we will find

$$\begin{aligned} \hbox {c}_2=\frac{\hbox {a}}{2}{\upsigma }_\mathrm{yz}^{\infty } \end{aligned}$$
(24)

and

(25)

In the formulation, there are six unknowns, or \(\hbox {c}_\mathrm{o},\, \hbox {d}_\mathrm{o},\hbox {c}_1 ,\, \hbox {d}_1,\,\hbox {c}_2,\, \hbox {d}_2\). In the meantime, we also have six equations for six unknowns: (1) four equations from Eqs. (19) and (20), representing the continuity conditions along two interfaces \(\Sigma _1 \) and \(\Sigma _2 \), (2) \(\hbox {d}_o =-\hbox {c}_o \) from Eq. (23), representing the traction free condition along the crack face and (3) \(\hbox {c}_2=\frac{\hbox {a}}{2}{\upsigma }_\mathrm{yz}^\infty \) from Eq. (24), representing the remote loading condition. Finally, we can obtain six undetermined coefficients.

In addition, from Eq. (7) the stress intensity factor at the crack tip can be defined by

(26)

Subsisting Eq. (23) into (26) yields

$$\begin{aligned} \hbox {K}_3 =\frac{2\hbox {c}_\mathrm{o}}{\hbox {a}}\sqrt{{\uppi }\,\hbox {a}} \end{aligned}$$
(27)

3 Numerical example

In the case of the remote loading \({\upsigma }_\mathrm{yz}^\infty \), one numerical example is provided to show the influence to stress intensity factor \(\hbox {K}_3\) from (a) the area of the thicker portion and (b) the assumed ratio for the shear modulus \(\hbox {G}_1 /\hbox {G}_o \) under the condition \(\hbox {G}_2 =\hbox {G}_o \).

In the example, the mapping function shown by Eq. (9) is used (Fig.2). In addition, we define the following three parameters. The first parameter \(\updelta \) is defined by

$$\begin{aligned} {\updelta } =\frac{{\uprho }_1 }{{\uprho } _o },\, (\hbox {with}\, {\uprho } _\mathrm{o} =1,\, \hbox {or}\, {\updelta } ={\uprho }_1) \end{aligned}$$
(28)

The second parameter is used for defining the relation of \({\uprho }_2\) to \({\updelta } ={\uprho }_1\). The semi-axes corresponding to \({\uprho }_\mathrm{j} (\hbox {j}=1, 2)\) are denoted by \(\hbox {a}_\mathrm{j}\) and \(\hbox {b}_\mathrm{j} (\hbox {j}=1, 2)\). It is assumed that the area bounded by \(\Sigma _1 \) and \(\Sigma _2 \) (Fig. 2) keeps a definite value. In this case we can define second parameter \(\upalpha \) by

$$\begin{aligned} \upalpha =\frac{\uppi (\hbox {a}_2 \hbox {b}_2 -\hbox {a}_1 \hbox {b}_1 )}{{\uppi }\, \hbox {a}^{2}}=\frac{\hbox {a}_2 \hbox {b}_2 -\hbox {a}_1 \hbox {b}_1 }{\hbox {a}^{2}} \end{aligned}$$
(29)

Clearly, from the mapping function shown by Eq. (9), we have

$$\begin{aligned} {\begin{array}{l} {\hbox {a}_1 } \\ {\hbox {b}_1 } \\ \end{array} }=\frac{\hbox {a}}{2}\left( {{\uprho }_1 \pm {\uprho }_1^{-1} } \right) , \quad {\begin{array}{l} {\hbox {a}_2 } \\ {\hbox {b}_2 } \\ \end{array} }=\frac{\hbox {a}}{2}\left( {{\uprho }_2 \pm {\uprho } _2^{-1} } \right) \end{aligned}$$
(30)

Substituting (30) into (29) yields

$$\begin{aligned} {\upalpha }= & {} \frac{1}{4}\left( {({\uprho }_2^2 -{\uprho } _2^{-2} )-({\uprho } _1^{2} -{\uprho } _1^{-2} )} \right) ,\, \hbox {or}\, ({\uprho } _2^2 -{\uprho } _2^{-2} )\nonumber \\= & {} 4\upalpha +({\uprho } _1^2 -{\uprho } _1^{-2} ) \end{aligned}$$
(31)

Eq. (31) reveals that if \({\upalpha } \) and \({\uprho } _1\) are given beforehand, we can get \({\uprho } _2 \) from Eq.  (31) accordingly. In the example, we choose \({\upalpha } =0.25\) or 0.5.

Finally, the third parameter \({\upgamma } \) is defined by

$$\begin{aligned} {\upgamma } =\frac{\hbox {G}_1 }{\hbox {G}_o } \end{aligned}$$
(32)

In the example, we choose \(\hbox {G}_2 =\hbox {G}_\mathrm{o}\).

In the example, we assume (1) \({\updelta } ={\uprho } _1 /{\uprho } _\mathrm{o}\)= 1.01, 1.02, 1.05, 1.1. 1.2, 1.5, 2, 5 (with \({\uprho } _\mathrm{o} =1)\), (2) \({\upgamma } =\hbox {G}_1 /\hbox {G}_o =\)1, 2, 3, 4, 5, 8, 10 and (3) \({\upalpha } =0.25\), 0.5. After using Eq. (27), the computed results for \(\hbox {K}_3\) at the crack tip can be expressed as

$$\begin{aligned} \hbox {K}_3 =\hbox {h}({\updelta } ,{\upgamma } ,{\upalpha } ){\upsigma } _{\mathrm{yz}}^\infty \sqrt{{\uppi } \hbox {a}} \end{aligned}$$
(33)

The computed non-dimensional stress intensity factors \(\hbox {h}({\updelta } ,{\upgamma } ,{\upalpha } )\) are listed in Table 1.

Table 1 The non-dimensional stress intensity factors \(\hbox {h}({\updelta } ,{\upgamma } ,{\upalpha }) (=\hbox {K}_3 /({\upsigma } _\mathrm{yz}^\infty \sqrt{{\uppi } \hbox {a}}))\) at the crack tip with \({\updelta } ={\uprho } _1 /{\uprho } _o ({\uprho } _\mathrm{o} =1),\, {\upgamma } =\hbox {G}_1 /\hbox {G}_\mathrm{o},\, \hbox {G}_2 =\hbox {G}_\mathrm{o}\) [see Eq. (33) and Fig. 2]
Full size table

From Table 1 we see that, the \({\upgamma }\) value \(({\upgamma } =\hbox {G}_1 /\hbox {G}_\mathrm{o} )\) has a significant influence to the non-dimensional stress intensity factors (SIFs) \(\hbox {h}({\updelta } ,{\upgamma } ,{\upalpha }) (=\hbox {K}_3 /({\upsigma } _{\mathrm{yz}}^\infty \sqrt{{\uppi } \hbox {a}}))\). For example, in the case of \({\upalpha } =0.25\), we have \(h({\updelta } ,{\upgamma } ,{\upalpha } )\left. \right| _{{\updelta } =1.01\,{\upgamma } =1}=1.000\) and \(\hbox {h}({\updelta } ,{\upgamma } ,{\upalpha } )\left. \right| _{{\updelta } =1.01\,{\upgamma } =5}=0.571\). That is to say, if the thickness of elliptic layer is magnified by five time, the non-dimensional SIF is reduced from 1 to 0.571. Similarly, the \({\updelta } \)value \(({\updelta } ={\uprho } _1 /{\uprho } _\mathrm{o} )\) also has a significant influence to the non-dimensional stress intensity factors (SIFs) \(\hbox {h}({\updelta } ,{\upgamma } ,{\upalpha } ) (=\hbox {K}_3 /({\upsigma } _{yz}^\infty \sqrt{{\uppi } \hbox {a}}))\). For example, in the case of \({\upalpha } =0.25\), we have \(\hbox {h}({\updelta } ,{\upgamma } ,{\upalpha } )\left. \right| _{{\updelta } =2{\upgamma } =1}=1.000\), \(\hbox {h}({\updelta } ,{\upgamma } ,{\upalpha } )\left. \right| _{{\updelta } =2\,{\upgamma } =2}=0.960\) and \(h({\updelta } ,{\upgamma } ,{\upalpha } )\left. \right| _{{\updelta } =2\,{\upgamma } =5}=0.825\). That is to say, if the stiffening layer is far away (\({\updelta } =2\) case) from the crack tip, the reduction of the non-dimensional SIF is not significant.

In addition, in the case of \({\upalpha } =0.5\), we have \(\hbox {h}({\updelta } ,{\upgamma } ,{\upalpha } )\left. \right| _{{\updelta } =1.01\,{\upgamma } =1}=1.000\) and \(\hbox {h}({\updelta } ,{\upgamma } ,{\upalpha } )\left. \right| _{{\updelta } =1.01\,{\upgamma } =5}=0.464\). That is to say, the thickness of elliptic layer is magnified by five time, the non-dimensional SIF is reduced from 1 to 0.464. Similarly, in the case of \({\upalpha } =0.5\), we have \(\hbox {h}({\updelta } ,{\upgamma } ,{\upalpha } )\left. \right| _{{\updelta } =2\,{\upgamma } =1}=1.000,\, \hbox {h}({\updelta } ,{\upgamma } ,{\upalpha } )\left. \right| _{{\updelta } =2\,{\upgamma } =2}=0.934\) and \(\hbox {h}({\updelta } ,{\upgamma } ,{\upalpha } )\left. \right| _{{\updelta } =2\,{\upgamma } =5}=0.737\). That is to say, if the stiffening layer is far away (\({\updelta } =2\) case) from the crack tip, the reduction of the non-dimensional SIF is minor.

Except for the \({\upgamma } =\hbox {G}_1 /\hbox {G}_\mathrm{o} =1\) case, the non-dimensional SIFs in the case of \({\upalpha } =0.5\) are generally lower than those for the case of \({\upalpha } =0.2\).

4 Conclusions

Generally, the continuity conditions along interfaces in the problem is rather complicated. We found that we only need to express the complex potentials in the form of Eq. (16). Therefore, the two sets of two undetermined coefficients in the adjacent layers can be linked by Eq. (17).

This paper provides an effective solution for a crack stiffened by elliptic layer in antiplane elasticity. A lot of numerical results are provided in this paper. It is found that if the stiffening layer is placed near the crack tip, for example, \({\uprho } _1 =1.01\), the stiffening effect is significant. However, if the stiffening layer is placed far away from the crack tip, for example \({\uprho } _1 =5\), the stiffening effect is weaker.

This paper provides a closed form solution for the mentioned crack problem. In fact, after substituting the relations \(\hbox {d}_\mathrm{o} =-\hbox {c}_\mathrm{o}\) and \(\hbox {c}_2=\frac{\hbox {a}}{2}{\upsigma }_\mathrm{yz}^\infty \) shown by Eqs. (23) and (24) into Eqs. (19) and (20), we will obtain an algebraic equation for four unknowns \(\hbox {c}_\mathrm{o},\, \hbox {c}_1,\, \hbox {d}_1\) and \(\hbox {d}_2\). This algebraic equation is indeed a very simple one, and no error is actually involved in computation. Thus, the computed results achieved must be very high.