Uniformity of anti-plane stresses inside a nonlinear elastic elliptical or parabolic inhomogeneity

Abstract

We study the anti-plane strain problem associated with a p-Laplacian nonlinear elastic elliptical inhomogeneity embedded in an infinite linear elastic matrix subjected to uniform remote anti-plane stresses. A full-field exact solution is derived using complex variable techniques. It is proved that the stress field inside the elliptical inhomogeneity is nevertheless uniform. The uniformity of stresses is also observed inside a p-Laplacian nonlinear elastic parabolic inhomogeneity.

1. Introduction

The uniformity of stresses inside an elastic inhomogeneity under a uniform loading at infinity has been discussed by many investigators [1–8]. The majority of these studies are confined within the framework of linear isotropic or anisotropic elasticity. Ru et al. [7] considered an elastic inhomogeneity in a particular class of harmonic materials and showed that if the Piola stress within the inhomogeneity is uniform, the inhomogeneity is necessarily an ellipse except for a special case of uniform remote loading.

In this paper, using complex variable techniques, we show that even when the elliptical inhomogeneity is composed of a p-Laplacian nonlinear elastic material [9,10], the stress field inside the inhomogeneity is still uniform when the linear elastic matrix is subjected to uniform remote anti-plane stresses. In addition, we obtain a full-field solution for an elliptical inhomogeneity with internal uniform stresses. More specifically, the uniform stress field inside the nonlinear elastic elliptical inhomogeneity and the non-uniform stress field in the linear elastic matrix are presented explicitly. It is proved that the state of stress inside a nonlinear elastic inhomogeneity is uniform if and only if the closed curvilinear contour enclosing the inhomogeneity is an ellipse. We also prove that the stresses inside a p-Laplacian nonlinear elastic parabolic inhomogeneity are uniform when the linear elastic matrix is subjected to uniform remote anti-plane stresses. The uniform stress field inside the nonlinear elastic parabolic inhomogeneity and the non-uniform stress field in the linear elastic matrix are presented explicitly.

2. A nonlinear elastic elliptical inhomogeneity with an internal uniform stress field

We begin by considering a fibrous composite in which a nonlinear elastic elliptical inhomogeneity $S_1:\{(x_1^2/a^2)+(x_2^2/b^2)\le 1\}$ with $a\ge b>0$ is perfectly bonded to an infinite linear elastic matrix $S_2:\{(x_1^2/a^2)+(x_2^2/b^2)\ge 1\}$ under uniform remote anti-plane stresses ${\sigma }_{32}^{\infty }$ and ${\sigma }_{31}^{\infty }$ . The linear elastic material occupying the matrix has a shear modulus ${\mu }_2(>0)$ , while the nonlinear elastic material occupying the inhomogeneity is characterized by its nonlinear shear modulus ${\mu }_1|\nabla w{|}^{p-2}$ with ${\mu }_1>0,p>1$ and w the out-of-plane displacement. In what follows, subscripts 1 and 2 are used to identify the respective quantities in $S_1$ and $S_2$ . The stress–strain law for the nonlinear elastic material in the inhomogeneity is

$${\sigma }_{31}={\mu }_1{|\nabla w|}^{p-2}{w}_{,1},{\sigma }_{32}={\mu }_1{|\nabla w|}^{p-2}{w}_{,2},$$

(1)

where ${\sigma }_{31}$ and ${\sigma }_{32}$ are the two anti-plane shear stresses. The nonlinear constitutive relation in equation (1) has been adopted by several authors [9–17]. Considering the equilibrium equation ${\sigma }_{31,1}+{\sigma }_{32,2}=0$ , the out-of-plane displacement w in the inhomogeneity will satisfy a p-Laplacian equation: $\nabla ·(|\nabla w{|}^{p-2}\nabla w)=0$ [9,10].

Under anti-plane shear deformation of the linear isotropic elastic material in the matrix, the two shear stress components, the out-of-plane displacement and the stress function $\varphi$ can be expressed in terms of a single analytic function $f(z)$ of the complex variable $z=x_1+\mathrm{i}{x}_2$ as [4]

$${\sigma }_{32}+\mathrm{i}{\sigma }_{31}={\mu }_{2}f\prime (z),\varphi +\mathrm{i}{\mu }_{2}w={\mu }_{2}f(z),$$

(2)

and the two stress components can be expressed in terms of the stress function as follows [4]:

$${\sigma }_{32}={\varphi }_{,1},{\sigma }_{31}=-{\varphi }_{,2}$$

(3)

We now introduce the following conformal mapping function for the matrix:

$$z=\omega (\xi )=R(\xi +\frac{m}{\xi }),\left|\xi \right|\ge 1,$$

(4)

where

$$R=\frac{a+b}{2},m=\frac{a-b}{a+b}(0\le m<1).$$

(5)

Using the mapping function in equation (4), the region occupied by the matrix $S_2$ is mapped onto $|\xi |\ge 1$ , and the elliptical interface is mapped onto the unit circle in the ξ-plane. In order to ensure that the anti-plane stress field inside the elliptical inhomogeneity is uniform, the anti-plane displacement within the nonlinear elliptical inhomogeneity should take the following form

$$w=\mathrm{Im}\left\{\mathrm{kz}\right\},z\in S_1,$$

(6)

where k is a complex constant to be determined. As a result, the uniform anti-plane stress field inside the elliptical inhomogeneity is given as follows:

$${\sigma }_{32}+\mathrm{i}{\sigma }_{31}={\mu }_1{\left|k\right|}^{p-2}k,z\in S_1.$$

(7)

The boundary value problem can now be expressed in terms of the analytic function $f(\xi )=f(\omega (\xi ))=f(z)$ in the ξ-plane as follows:

$$\begin{array}{c}{\mu }_2[f(\xi )+\overline{f(\xi )}]={\mu }_1[{\left|k\right|}^{p-2}k\omega (\xi )+{\left|k\right|}^{p-2}\overline{k}\overline{\omega }\left(\frac{1}{\xi }\right)],\hfill \\ f(\xi )-\overline{f(\xi )}=k\omega (\xi )-\overline{k}\overline{\omega }\left(\frac{1}{\xi }\right),\left|\xi \right|=1;\hfill \end{array}$$

(8)

$$f(\xi )\cong \frac{R({\sigma }_{32}^{\infty }+\mathrm{i}{\sigma }_{31}^{\infty })}{{\mu }_2}\xi +O(1),\left|\xi \right|\to \infty .$$

(9)

Equation (8) describes the continuity of traction and displacement across the perfect elliptical interface between the inhomogeneity and the matrix, while equation (9) gives the remote asymptotic behavior of $f(\xi )$ due to the uniform anti-plane stresses at infinity. The analytic function $f(\xi )$ can be readily derived from equation (8) as follows:

$$f(\xi )=\frac{{\mu }_1{\left|k\right|}^{p-2}+{\mu }_2}{2{\mu }_2}k\omega (\xi )+\frac{{\mu }_1{\left|k\right|}^{p-2}-{\mu }_2}{2{\mu }_2}\overline{k}\overline{\omega }\left(\frac{1}{\xi }\right),\left|\xi \right|\ge 1.$$

(10)

Using equation (10) to impose the asymptotic behavior in equation (9), we arrive at

$$({\left|k\right|}^{p-2}+\mathrm{\Gamma })k+m({\left|k\right|}^{p-2}-\mathrm{\Gamma })\overline{k}=2\delta (1+\mathrm{i}\alpha ),$$

(11)

where the three dimensionless parameters $\mathrm{\Gamma },\delta$ and α are defined as follows:

$$\mathrm{\Gamma }=\frac{{\mu }_2}{{\mu }_1},\delta =\frac{{\sigma }_{32}^{\infty }}{{\mu }_1},\alpha =\frac{{\sigma }_{31}^{\infty }}{{\sigma }_{32}^{\infty }}.$$

(12)

From equation (11), we obtain that

$$\begin{array}{c}k\prime =\frac{2\delta }{{\left|k\right|}^{p-2}(1+m)+\mathrm{\Gamma }(1-m)},\hfill \\ k″=\frac{2\delta \alpha }{{\left|k\right|}^{p-2}(1-m)+\mathrm{\Gamma }(1+m)},\hfill \end{array}$$

(13)

where $k\prime$ and $k″$ are, respectively, the real and imaginary parts of k.

It is further derived from equation (13) that $|k|$ should satisfy the following nonlinear equation:

$$\begin{array}{c}\multicolumn{1}{c}{g\left(\left|k\right|\right)=(1-m^2{)}^2{\left|k\right|}^{2(p-1)}+4\mathrm{\Gamma }(1-m^4){\left|k\right|}^p}\\ +{\mathrm{\Gamma }}^2[{(1+m)}^4+4{(1-m^2)}^2+{(1-m)}^4]{\left|k\right|}^2-4{\delta }^2[{(1-m)}^2+{\alpha }^2{(1+m)}^2]\hfill \\ +4{\mathrm{\Gamma }}^3(1-m^4){\left|k\right|}^{4-p}-8{\delta }^2\mathrm{\Gamma }(1+{\alpha }^2)(1-m^2){\left|k\right|}^{2-p}+{\mathrm{\Gamma }}^4(1-m^2{)}^2{\left|k\right|}^{6-2p}\hfill \\ -4{\delta }^2{\mathrm{\Gamma }}^2[{(1+m)}^2+{\alpha }^2{(1-m)}^2]{\left|k\right|}^{4-2p}=0,\hfill \end{array}$$

(14)

or equivalently

$$\begin{array}{c}\multicolumn{1}{c}{h\left(\left|k\right|\right)=(1-m^2{)}^2{\left|k\right|}^{4p-6}+4\mathrm{\Gamma }(1-m^4){\left|k\right|}^{3p-4}}\\ +{\mathrm{\Gamma }}^2[{(1-m)}^4+4{(1-m^2)}^2+{(1+m)}^4]{\left|k\right|}^{2p-2}-4{\delta }^2[{(1-m)}^2+{\alpha }^2{(1+m)}^2]{\left|k\right|}^{2(p-2)}\hfill \\ +4{\mathrm{\Gamma }}^3(1-m^4){\left|k\right|}^p-8{\delta }^2\mathrm{\Gamma }(1+{\alpha }^2)(1-m^2){\left|k\right|}^{p-2}+{\mathrm{\Gamma }}^4(1-m^2{)}^2{\left|k\right|}^2\hfill \\ -4{\delta }^2{\mathrm{\Gamma }}^2[{(1+m)}^2+{\alpha }^2{(1-m)}^2]=0.\hfill \end{array}$$

(15)

When $1<p<2$ , there exists at least a suitable solution of $|k|(>0)$ to equation (14) in view of the fact that

$$\begin{array}{c}g(0)=-4{\delta }^2[{(1-m)}^2+{\alpha }^2{(1+m)}^2]<0,\hfill \\ g(+\infty )\cong {\mathrm{\Gamma }}^4(1-m^2{)}^2{|+\infty |}^{6-2p}>0.\hfill \end{array}$$

(16)

When $p=2$ for a linear elastic material, it follows from equation (13) that $|k|(>0)$ can be given exactly by

$$\left|k\right|=2\left|\delta \right|\sqrt{\frac{1}{{[1+m+\mathrm{\Gamma }(1-m)]}^2}+\frac{{\alpha }^2}{{[1-m+\mathrm{\Gamma }(1+m)]}^2}},$$

(17)

which is proportional to $\left|\delta \right|$ .

Similarly, when $p>2$ , there exists at least a suitable solution of $|k|(>0)$ to equation (15) in view of the fact that

$$\begin{array}{c}h(0)=-4{\delta }^2{\mathrm{\Gamma }}^2[{(1+m)}^2+{\alpha }^2{(1-m)}^2]<0,\hfill \\ h(+\infty )\cong (1-m^2{)}^2(+\infty {)}^{4p-6}>0.\hfill \end{array}$$

(18)

Consequently, at least a suitable solution of $|k|(>0)$ to equation (14) or (15) exists for $p>1$ . Our extensive calculations reveal that there exists only a single solution of $|k|(>0)$ for prescribed values of the parameters $p(>1),m(0\le m<1),\mathrm{\Gamma }(>0),\delta$ , and $\alpha$ in view of the fact that $g(|k|)$ or $h(|k|)$ is always an increasing function of $|k|$ . It follows from equation (7) that

$$\sqrt{{\sigma }_{31}^2+{\sigma }_{32}^2}={\mu }_1{\left|k\right|}^{p-1},z\in S_1,$$

(19)

which gives a relation between the constant effective stress $\sqrt{{\sigma }_{31}^2+{\sigma }_{32}^2}$ within the inhomogeneity and $|k|$ . Once $|k|$ has been determined by solving equation (14) or (15), $k\prime$ and $k″$ can be uniquely determined using equation (13). We illustrate in Figures 1–3 the variations of $|k|$ as a function of $p,m,\mathrm{\Gamma },\delta$ and $\alpha$ . It is seen from the three figures that $|k|$ can be considerably large when $p\to 1$ . From Figure 2, it can be seen quite clearly that $|k|$ is no longer proportional to $|\delta |$ when $p\ne 2$ for a nonlinear elastic material. From Figure 3, we see that $|k|=2.8358$ is almost unaffected by the specific value of m when $p=1.33$ .

Figure 1.

Variations of $|k|$ determined from equation (14) or (15) as a function of p for different values of Γ and α with $m=0.5,\delta =1$ .

Figure 2.

Variations of $|k|$ determined from equation (14) or (15) as a function of p for different values of δ with $m=0.5,\mathrm{\Gamma }=0.5,\alpha =1$ .

Figure 3.

Variations of $|k|$ determined from equation (14) or (15) as a function of p for different values of m with $\mathrm{\Gamma }=1,\delta =1,\alpha =1$ .

From the above analysis, it is rather clear that the assumption of a uniform stress field inside the elliptical inhomogeneity is indeed valid. In addition, the non-uniform stress field in the matrix is given as follows:

$${\sigma }_{32}+\mathrm{i}{\sigma }_{31}={\mu }_1\frac{[k({\left|k\right|}^{p-2}+\mathrm{\Gamma })+\overline{k}m({\left|k\right|}^{p-2}-\mathrm{\Gamma })]{\xi }^2-[\mathrm{km}({\left|k\right|}^{p-2}+\mathrm{\Gamma })+\overline{k}({\left|k\right|}^{p-2}-\mathrm{\Gamma })]}{2({\xi }^2-m)},\left|\xi \right|\ge 1.$$

(20)

Similar to the argument by Sendeckyj [3] and Ru and Schiavone [5], the ellipse is the only shape of inhomogeneity with a closed curvilinear contour for which uniform anti-plane stresses at infinity will induce a uniform stress field within the inhomogeneity.

3. A nonlinear elastic parabolic inhomogeneity with an internal uniform stress field

Next, we consider a p-Laplacian nonlinear elastic inhomogeneity $S_1:\{x_1\le H-x_2^2/(4H)\}$ with $H>0$ bonded to an infinite linear elastic matrix $S_2:\{x_1\ge H-x_2^2/(4H)\}$ through a perfect parabolic interface $L:\{x_1=H-x_2^2/(4H)\}$ . The matrix is subjected to uniform remote anti-plane stresses ${\sigma }_{32}^{\infty }$ and ${\sigma }_{31}^{\infty }$ . The out-of-plane displacement and uniform anti-plane stresses within the parabolic inhomogeneity continue to take the forms in equations (6) and (7). The boundary value problem can be expressed in terms of $f(z)$ in the physical z-plane as follows:

$$\begin{array}{c}{\mu }_2[f(z)+\overline{f(z)}]={\mu }_1[{\left|k\right|}^{p-2}\mathrm{kz}+{\left|k\right|}^{p-2}\overline{k}\overline{z}],\hfill \\ f(z)-\overline{f(z)}=\mathrm{kz}-\overline{k}\overline{z},z\in L;\hfill \end{array}$$

(21)

$$f(z)\cong \frac{{\sigma }_{32}^{\infty }+\mathrm{i}{\sigma }_{31}^{\infty }}{{\mu }_2}z+O\left(z^{\frac{1}{2}}\right),\left|z\right|\to \infty .$$

(22)

Using the following identity established along the parabolic interface [8]

$${\overline{z}}^{\frac{1}{2}}=2H^{\frac{1}{2}}-z^{\frac{1}{2}},\mathrm{for}z\in L,$$

(23)

the analytic function $f(z)$ can be readily derived from equation (21) as follows:

$$f(z)=\frac{{\mu }_1{\left|k\right|}^{p-2}+{\mu }_2}{2{\mu }_2}\mathrm{kz}+\frac{{\mu }_1{\left|k\right|}^{p-2}-{\mu }_2}{2{\mu }_2}\overline{k}(z-4H^{\frac{1}{2}}{z}^{\frac{1}{2}}+4H),z\in S_2.$$

(24)

Using equations (24) to impose the asymptotic behavior in equations (22), we arrive at

$$({\left|k\right|}^{p-2}+\mathrm{\Gamma })k+({\left|k\right|}^{p-2}-\mathrm{\Gamma })\overline{k}=2\delta (1+\mathrm{i}\alpha ),$$

(25)

where $\mathrm{\Gamma },\delta$ and α have been defined in equation (12).

It is readily obtained from equation (25) that

$$\begin{array}{c}k\prime =\frac{\delta }{{\left|k\right|}^{p-2}},\hfill \\ k″=\frac{\delta \alpha }{\mathrm{\Gamma }}.\hfill \end{array}$$

(26)

It is further derived from equation (26) that $|k|$ should satisfy the following nonlinear equation:

$$g(\left|k\right|)={\mathrm{\Gamma }}^2{\left|k\right|}^2-{\delta }^2{\alpha }^2-{\delta }^2{\mathrm{\Gamma }}^2{\left|k\right|}^{4-2p}=0,$$

(27)

or equivalently

$$h(\left|k\right|)={\mathrm{\Gamma }}^2{\left|k\right|}^{2p-2}-{\delta }^2{\alpha }^2{\left|k\right|}^{2p-4}-{\delta }^2{\mathrm{\Gamma }}^2=0.$$

(28)

When $1<p<2$ , there exists at least a suitable solution of $|k|(>0)$ to equation (27) in view of the fact that

$$\begin{array}{c}g(0)=-{\delta }^2{\alpha }^2<0,\hfill \\ g(+\infty )\cong {\mathrm{\Gamma }}^2{|+\infty |}^2>0.\hfill \end{array}$$

(29)

When $p=2$ for a linearly elastic material, it follows from equation (26) that $|k|(>0)$ can be exactly given by

$$\left|k\right|=\left|\delta \right|\sqrt{1+\frac{{\alpha }^2}{{\mathrm{\Gamma }}^2}},$$

(30)

which is proportional to $|\delta |$ .

When $p>2$ , there exists at least a suitable solution of $|k|(>0)$ to equation (28) in view of the fact that

$$\begin{array}{c}h(0)=-{\mathrm{\Gamma }}^2{\delta }^2<0,\hfill \\ h(+\infty )\cong {\mathrm{\Gamma }}^2{|+\infty |}^{2p-2}>0.\hfill \end{array}$$

(31)

Consequently, at least a suitable solution of $|k|(>0)$ to equation (27) or (28) exists for $p>1$ . Our extensive calculations reveal that there exists only a single solution of $|k|(>0)$ for prescribed values of the parameters $p(>1),\mathrm{\Gamma }(>0),\delta$ , and $\alpha$ . Once $|k|$ has been determined by solving equation (27) or (28), $k\prime$ and $k″$ can be uniquely determined using equation (26). We illustrate in Figures 4 and 5 the variations of $|k|$ as a function of $p,\mathrm{\Gamma },\delta$ , and $\alpha$ . It is clear from Figure 5 that $|k|$ is no longer proportional to $|\delta |$ when $p\ne 2$ for a nonlinear elastic material. It is seen from the analysis carried out in this section that the assumption of a uniform stress field inside the parabolic inhomogeneity is indeed valid. Furthermore, the stresses are non-uniformly distributed in the linearly elastic matrix as follows:

$${\sigma }_{32}+\mathrm{i}{\sigma }_{31}={\sigma }_{32}^{\infty }+\mathrm{i}{\sigma }_{31}^{\infty }+{\mu }_1\overline{k}{H}^{\frac{1}{2}}(\mathrm{\Gamma }-{\left|k\right|}^{p-2})z^{-\frac{1}{2}},z\in S_2.$$

(32)

Figure 4.

Variations of $|k|$ determined from equation (27) or (28) as a function of p for different values of Γ and α with $\delta =0.5$ .

Figure 5.

Variations of $|k|$ determined from equation (27) or (28) as a function of p for different values of δ with $\mathrm{\Gamma }=0.5,\alpha =1$ .

4. Conclusion

We have proved the uniformity of stresses inside a p-Laplacian nonlinear elastic elliptical inhomogeneity when the linear elastic matrix is subjected to uniform anti-plane stresses at infinity. The internal uniform stress field is given by equation (7) with $|k|$ numerically determined by solving the nonlinear equation in equation (14) or (15) and then with the real and imaginary parts $k\prime$ and $k″$ obtained from equation (13). The exterior non-uniform stress field in the linear elastic matrix is given explicitly by equation (20). We have also proved the uniformity of stresses inside a p-Laplacian nonlinear elastic parabolic inhomogeneity. The internal uniform stress field is again given by equation (7) with $|k|$ numerically determined by solving the nonlinear equation (27) or (28) and then with the real and imaginary parts $k\prime$ and $k″$ obtained from equation (26). The exterior non-uniform stress field in the linear elastic matrix is given explicitly by equation (32).