Some Basic Problems Of The Mathematical Theory Of Elasticity Pdf

1. Introduction

The problem of stress concentration is a typical application of two-dimensional elasticity. From the earliest investigations conducted by Kirsh [1], a broad classes of problems has been solved. The major contribution to the solution of the problem was given by Kolosov [2] and Muskhelishvili [3], who first introduced the use of complex functions. Various cases and results can be found in the monograph by Savin [4]. From these pioneering investigations, much research has been published and the literature on the subject is very extensive. The basic problem of stress concentration has also been treated in textbooks, see e.g. [5].

Subsequently the problem has been studied in the context of non-classical elasticity. The research has been extended to the case of solids with microstructure, chiral solids, porous materials, and other media [6–10]. The attention given to this topic is due to its practical and technological interest. The problem is significant in the analysis of structural fatigue, which is part of fracture mechanics. Its solution provides insight into the possibility of dangerous stresses around irregularities of structural members whereas the nominal stresses are at an otherwise safe level. This problem is of crucial importance in the design of structural members such as pile foundations and in soil improvement structures.

In this paper, we study the problem of a cylindrical rigid inclusion in an infinite porous elastic body that is uniformly stretched along one axis.

Since the fundamental works by Truesdell and Toupin [11] and Truesdell and Noll [12], a number of theories have been formulated to describe the mechanical properties of porous materials. The most well known of them are the Biot consolidation theory based on Darcy's law [13] and the Nunziato–Cowin theory based on the volume fraction concept [14, 15]. In this formulation, the porous solids are modeled as elastic material with interstices or voids of material. The pores are vacuous, containing nothing of mechanical significance. They can contract or stretch. The bulk density is explicitated as the product of two fields, the matrix material density field and the volume fraction field. Denoting the mass density in the deformed configuration by $\rho$ , we assume that $\rho$ has the decomposition $\rho =\nu \tilde{\rho }$ , where $\tilde{\rho }$ is the density of the matrix material and $\nu$ is the volume fraction field. In the undeformed state we have ${\rho }_0={\nu }_0{\tilde{\rho }}_0$ , where ${\rho }_0$ , ${\tilde{\rho }}_0$ , and ${\nu }_0$ are the mass density, the density of the matrix material, and the volume fraction field in the reference configuration, respectively. We introduce the notation $\psi =\nu -{\nu }_0$ . As consequence, with respect to the classical elasticity, the theory of material with voids is characterized by four independent kinematic variables, the component of the displacement $u_i(i=1,2,3)$ and the change in volume fraction $\psi$ .

The mathematical model is adapted to describe the behavior of solids with small distributed voids such as geological material, rocks and soils, biomechanical materials, and manufactured porous materials such as ceramics and pressed powders.

Basic results on the Nunziato–Cowin theory may be found in the books by Ciarletta and Iesan [16], Iesan [17], Straughan [18], and references therein. Applications in various domains are discussed in the review article by de Boer [19].

The theory has been subject to significant research activity and has lead to much progress with regards to the development of mathematical models [20–23] and applications. The deformation of porous elastic cylinders was studied by Ieşan [24] and Dell'Isola and Batra [25]. A thermoelastic theory of material with voids was established by Ieşan [26]. Further applications on thermal effects can be found in [27–29]. Recently, the theory has been extended to solids with multiple porosity [30–33] and to fluids [34].

The outline of the paper is as follows. In Section 2, we present the basic equations of the equilibrium theory of elastic materials with voids and derive the equations of the plane strain problem for homogeneous and isotropic bodies. In Section 3, the problem of a cylindrical rigid inclusion is investigated. The solution is presented in a closed form. In Section 4, displacements and stresses are expressed by mean of explicit formulas. The solution of the analogous problem in the classical elastostatics is derived as a particular case. The maximum tensile stress is calculated and the factor concentration stress is deduced.

2. Basic equations

We consider a regular region $B$ of three-dimensional Euclidean space. We assume $\partial B$ as the boundary of $B,$ and consider $\mathit{n}$ to be the outward unit normal of $\partial B$ . The region $B$ is considered as occupied by a linearly elastic material with voids. A system of Cartesian coordinate frame O $\{{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\}$ is also assumed for the analysis of the body. Throughout this analysis, Latin indices range between 1 and 3. Let $\mathit{u}$ be the displacement field over $B$ . The linear strain measure $E$ is expressed by

where $\nabla \mathbf{u}$ is the gradient of $\mathbf{u}$ and $\nabla {\mathbf{u}}^{\mathrm{T}}$ is the transpose of $\nabla \mathbf{u}$ .

Let $T$ be the stress tensor and let $\mathbf{h}$ represent the equilibrated stress vector. The surface traction $\mathbf{t}$ and the equilibrated surface force $h$ at a regular point of $\partial B$ are expressed by

where the dot denotes scalar product.

The equilibrium equations are

$$divT+\mathbf{f}=\mathbf{0},div\mathbf{h}-p+q=0,$$

(2.3)

where $\mathbf{f}$ is the body force, $p$ is the intrinsic equilibrated body force, and $q$ is the extrinsic equilibrated body force. The second equation of (2.3) is the corresponding equation in the equilibrium theory of the balance of equilibrated force describing dynamical changes in void volume. The equation was first suggested by Goodman and Cowin [35] and then derived from a variational argument by Cowin and Goodman [36]. Specific interpretations were formulated by Nunziato and Cowin [14] and by Jenkins [37]. A detailed discussion and a comparison with analogous equations arising in microstructural theories can be found in [14, 15]. A physical interpretation of the microstructural variables $\mathbf{h}$ , $p$ , and $q$ in Equation (2.3)₂ was suggested by Nunziato and Cowin in [14, 15]. Roughly $\mathbf{h}$ can be associated with a single double force system without moment equivalent to two oppositely directed forces at the same point, $p$ and $q$ can be associated with the center of dilation or center of compression. Jenkins [37] suggested that the force $q$ can be interpreted as an externally controlled pore pressure. Such systems have no net force and no resulting moment. The equilibrated forces can also be identified with the singularities discussed by Love [38] in the classical linear elasticity. Another possible interpretation of the microstructural variables $\mathbf{h}$ , $p$ , and $q$ follows from an equivalence idea suggested by MacKenzie [39].

In the case of centrosymmetric isotropic material the constitutive equations are

$$\begin{array}{c}T=2\mu E+\lambda \mathrm{trEI}+\beta \psi I,\hfill \\ \mathbf{h}=\alpha \nabla \psi ,\hfill \\ p=\beta \mathrm{div}\mathbf{u}+\zeta \psi ,\hfill \end{array}$$

(2.4)

where $\psi$ is the volume fraction function, $I$ is the identity tensor, and $\mu ,$ $\lambda ,$ $\beta ,$ $\alpha$ , and $\zeta$ are constitutive coefficients. The constitutive equations (2.4) relate the stress tensor $T$ , the equilibrated stress vector $\mathbf{h}$ , and the intrinsic equilibrated body force $p$ to the strain tensor $E$ , the change in volume fraction $\psi$ and the gradient of $\psi$ . For an extensive discussion on constitutive equations that characterize the response of a material with voids in nonlinear theory, the reader is referred to the work of Nunziato and Cowin [14]. In [15], they constructed constitutive equations for an anisotropic elastic solid and then these equations were particularized in the case of isotropy and centrosymmetry. We restrict our attention to homogeneous materials so that the constitutive coefficients are constants. Further, we assume that the internal energy density is a positive-definite form. This assumption implies that (see Cowin and Nunziato [15])

$$\begin{array}{c}\mu >0,\alpha >0,\zeta >0,\hfill \\ \multicolumn{1}{c}{2\mu +3\lambda >0,(2\mu +3\lambda )\zeta >3{\beta }^2.}\end{array}$$

(2.5)

We consider the region $B$ as referred to the interior of a right cylinder with open cross section $\Omega$ and smooth lateral boundary $\Gamma$ . The rectangular Cartesian axes O $x_i$ $(i=1,2,3)$ are assumed so that the $x_3$ -axis is parallel to the generators of $B$ . Let $L$ indicate the boundary of $\Omega$ . The state of plane strain of the cylinder $B$ , parallel to the plane $x_{1}O{x}_2,$ is characterized by

$$\begin{array}{c}{u}_{\alpha }=u_{\alpha }(x_1,x_2),u_3=0,\hfill \\ \multicolumn{1}{c}{\psi =\psi (x_1,x_2),(x_1,x_2)\in \Omega .}\end{array}$$

(2.6)

The above restrictions, together with the geometrical equations (2.1) and the constitutive equations (2.4), entail that $E,T,\mathbf{h}$ , and $p$ are independent of $x_3$ . It follows that the non-zero strain measures are expressed by

where the Greek indices assume the value 1 or 2.

The constitutive equations indicate that the non-zero components of the stress tensor $T$ and equilibrated stress vector $\mathbf{h}$ are $t_{\alpha \beta }$ and $h_{\alpha }$ , respectively, and the result is

$$\begin{array}{c}{t}_{\alpha \beta }=2\mu e_{\alpha \beta }+\lambda e_{\rho \rho }{\delta }_{\alpha \beta }+\beta \psi {\delta }_{\alpha \beta },\hfill \\ h_{\alpha }=\alpha \psi {,}_{\alpha },\hfill \\ g=\beta e_{\rho \rho }+\zeta \psi ,\hfill \end{array}$$

(2.8)

where ${\delta }_{\alpha \beta }$ is Kronecker's delta.

The equilibrium equations (2.3) in the absence of body loads become

$$t_{\beta \alpha ,\beta }=0,h_{\alpha ,\alpha }-p=0,\mathrm{on}\Omega .$$

(2.9)

We assume that the surface loads are prescribed on the boundary of the body. Given the surface traction $\tilde{\mathit{t}}$ and the surface microtraction $\tilde{h}$ on $\Gamma$ , with $\tilde{\mathit{t}}$ and $\tilde{h}$ independent of $x_3$ and ${\tilde{t}}_3=0,$ the boundary conditions on the lateral surface become

$${\tilde{t}}_{\alpha }=t_{\beta \alpha }{n}_{\beta },\tilde{h}=h_{\alpha }{n}_{\alpha },\mathrm{on}L,$$

(2.10)

where ${\tilde{t}}_{\alpha }$ and $\tilde{h}$ are prescribed functions.

In the following, we consider a plane strain problem with the displacement vector and the volume fraction function expressed in cylindrical coordinates $(r,\theta ,z)$ so that

$$\begin{array}{c}{u}_r=u(r,\theta ),u_{\theta }=v(r,\theta ),\hfill \\ \multicolumn{1}{c}{u_z=0,\psi =\phi (r,\theta ),(r,\theta )\in \Theta .}\end{array}$$

(2.11)

The axis $\mathrm{Oz}$ of the cylindrical coordinate system is taken along the axis of the cylinder. The geometrical equations (2.7) are expressed as

$$\begin{array}{c}{\epsilon }_{\mathrm{rr}}=u{,}_r\hfill \\ {\epsilon }_{\theta \theta }=\frac{1}{r}(v{,}_{\theta }+u),\hfill \\ {\epsilon }_{r\theta }=\frac{1}{2}(\frac{1}{r}u{,}_{\theta }+v{,}_r-\frac{1}{r}v).\hfill \end{array}$$

(2.12)

In (2.12) and in the following subscripts preceded by a comma are adopted to represent partial differentiation with respect to the corresponding coordinate.

The constitutive equations (2.8) can be written in the form

$$\begin{array}{c}{\tau }_{\mathrm{rr}}=(\lambda +2\mu ){\epsilon }_{\mathrm{rr}}+\lambda {\epsilon }_{\theta \theta }+\beta \phi ,\hfill \\ {\tau }_{\theta \theta }=\lambda {\epsilon }_{\mathrm{rr}}+(\lambda +2\mu ){\epsilon }_{\theta \theta }+\beta \phi ,\hfill \\ {\tau }_{r\theta }=2\mu {\epsilon }_{r\theta },\hfill \\ {\chi }_r=\alpha \phi {,}_r,\hfill \\ {\chi }_{\theta }=\frac{1}{r}\alpha \phi {,}_{\theta },\hfill \\ \gamma =\beta [\frac{1}{r}\left(\mathrm{ru}\right){,}_r+\frac{1}{r}v{,}_{\theta }]+\zeta \phi .\hfill \end{array}$$

(2.13)

The equilibrium equations (2.9) are therefore formulated as

$$\begin{array}{c}{\tau }_{\mathrm{rr}}{,}_r+\frac{1}{r}{\tau }_{r\theta }{,}_{\theta }+\frac{1}{r}({\tau }_{\mathrm{rr}}-{\tau }_{\theta \theta })=0,\hfill \\ {\tau }_{r\theta }{,}_r+\frac{1}{r}{\tau }_{\theta \theta }{,}_{\theta }+\frac{2}{r}{\tau }_{r\theta }=0,\hfill \\ \frac{1}{r}\left(r{\chi }_r\right){,}_r+\frac{1}{r}\left({\chi }_{\theta }\right){,}_{\theta }-\gamma =0.\hfill \end{array}$$

(2.14)

The plane strain problem consists of finding the functions $u,$ $v$ , and $\psi$ on $\Omega$ , which satisfy (2.12)–(2.14) and the boundary conditions (2.10).

3. Stress concentration problem

In this section, we investigate the problem of a rigid cylindrical inclusion in an infinite body. We consider the body uniformly stretched along the axis $O{x}_1$ . Let us assume that the elastic body inhabits the region $B=\{(x_1,x_2,x_3)\in R^3:x_1^2+x_2^2>a^2\}$ , where $a$ is a positive constant. Further, we consider that the region $\{(x_1,x_2,x_3)\in R^3:x_1^2+x_2^2<a^2\}$ is occupied by a rigid body. The following boundary conditions are also assumed:

$$u_r=0,u_{\theta }=0,\phi =0,\mathrm{on}r=a,$$

(3.1)

and the conditions at infinity

$$\begin{array}{c}{\tau }_{\mathrm{rr}}=\frac{1}{2}A(1+\cos 2\theta ),\hfill \\ {\tau }_{\theta \theta }=\frac{1}{2}A(1-\cos 2\theta ),\hfill \\ {\tau }_{r\theta }={\tau }_{\theta r}=-\frac{1}{2}A\sin 2\theta ,\hfill \\ {\chi }_r={\chi }_{\theta }=0,\hfill \end{array}$$

(3.2)

where $A$ is a given constant. The body $B$ is then in a state of plane strain parallel to the plane $x_{1}O{x}_2$ in the absence of body loads. The solution is sought in the form

$$\begin{array}{c}u=Q\left(r\right)\cos 2\theta +\Lambda \left(r\right),\hfill \\ v=W\left(r\right)\sin 2\theta ,\hfill \\ \phi =\Phi \left(r\right)\cos 2\theta +\Psi \left(r\right),\hfill \end{array}$$

(3.3)

in which $Q,$ $W,$ $\Lambda ,$ $\Phi$ , and $\Psi$ are function only on $r$ . It is assumed that the functions $u$ and $v$ will have the same general function form as the solution to the same problem in classical elasticity theory. As a consequence, the function $\phi$ will have the same form. From (2.12), (3.3), and (2.14) it follows that

$$\begin{array}{c}{\tau }_{\mathrm{rr}}=(\lambda +2\mu )\Lambda \prime +\frac{1}{r}\lambda \Lambda +\beta \Psi +[(\lambda +2\mu )Q\prime +\frac{1}{r}\lambda (Q+2W)+\beta \Phi ]\cos 2\theta \hfill \\ {\tau }_{\theta \theta }=\lambda \Lambda \prime +\frac{1}{r}(\lambda +2\mu )\Lambda +\beta \Psi +[\lambda Q\prime +\frac{1}{r}(\lambda +2\mu )(Q+2W)+\beta \Phi ]\cos 2\theta \hfill \\ {\tau }_{r\theta }={\tau }_{\theta r}=[\mu W\prime -\frac{1}{r}\mu (2Q+W)]\sin 2\theta \hfill \\ {\chi }_r=\alpha (\Psi \prime +\Phi \prime \cos 2\theta ),\hfill \\ {\chi }_{\theta }=-\frac{2}{r}\alpha \Phi \sin 2\theta \hfill \\ \gamma =\beta (\Lambda \prime +\frac{1}{r}\Lambda )+\zeta \Psi +\{\beta [Q\prime +\frac{1}{2}(Q+2W)]+\zeta \Phi \}\cos 2\theta \hfill \end{array}$$

(3.4)

in which the prime denotes derivation with respect to $r$ . By substituting $(3.4)$ into the equilibrium equations (2.13), the following equations are obtained

$$\begin{array}{c}(\lambda +2\mu )(\Lambda \prime +\frac{1}{r}\Lambda )\prime +\beta \Psi \prime =0\hfill \\ \alpha ({\Psi }^{″}+\frac{1}{r}\Psi \prime -\frac{\zeta }{\alpha }\Psi )-\beta (\Lambda \prime +\frac{1}{r}\Lambda )=0\hfill \\ r[(\lambda +2\mu )(rQ\prime )\prime +2(\mu +\lambda )W\prime +\beta r\Phi \prime ]-(\lambda +6\mu )Q-2(\lambda +3\mu )W=0\hfill \\ r[\mu (rW\prime )\prime -2(\lambda +\mu )Q\prime -2\beta \Phi ]-2(\lambda +3\mu )Q-(4\lambda +9\mu )W=0\hfill \\ \alpha r^2({\Phi }^{″}+\frac{1}{r}\Phi \prime -\frac{\zeta }{\alpha }\Phi )-4\alpha \Phi -\beta r(rQ\prime +Q+2W)=0.\hfill \end{array}$$

(3.5)

In the following, we introduce the notation

Accordingly, from Equation (2.5) it follows that ${\xi }^2>0$ . From the first equation of (3.5) we obtain

in which $B_1$ is an arbitrary constant.

Given Equation (3.8), the second equation of (3.5) can be expressed as

Equation (3.9) has the solution

$$\Psi =A_1{K}_0\left(\xi r\right)+A_1^{*}{I}_0\left(\xi r\right)-\frac{\beta }{{\xi }^2\alpha }{B}_1,$$

in which $I_n$ and $K_n$ are the modified Bessel functions of order $n$ , and $A_1$ and $A_1^{*}$ are arbitrary constants. The function $\Psi$ must be finite at infinity and, accordingly, it is $A_1^{*}=0$ . Thus, we obtain

From Equations (3.10) and (3.8) we obtain

$$\Lambda =\frac{\zeta }{2{\xi }^2\alpha }{B}_{1}r+\frac{1}{r}{B}_2+\frac{{\nu }_2}{\xi }{A}_1{K}_1\left(\xi r\right),$$

(3.11)

in which $B_2$ is an arbitrary constant.

Let us denote $D=\frac{d}{\mathrm{dt}}$ and if we consider the independent variable $t$ given by

then Equations (3.5)_3,4 can be expressed in the form

$$\begin{array}{c}[D^2-(1+4{\nu }_1)]Q+2[(1-{\nu }_1)D-(1+{\nu }_1)]W=-e^t{\nu }_{2}D\Phi ,\hfill \\ 2[(1-{\nu }_1)D+(1+{\nu }_1)]Q+[{\nu }_1{D}^2-(4+{\nu }_1)]W=2e^t{\nu }_2\Phi .\hfill \end{array}$$

(3.13)

The general solution of the homogeneous system (3.13) which corresponds to a finite stress field at infinity is supplied by

$$\begin{array}{c}{Q}_0=C_1{e}^{-t}+C_2{e}^{-3t}+C_3{e}^t,\hfill \\ W_0=-{\nu }_1{C}_1{e}^{-t}+C_2{e}^{-3t}-C_3{e}^t,\hfill \end{array}$$

(3.14)

in which $C_1,$ $C_2$ , and $C_3$ are arbitrary constants. A particular solution of the system (3.13) is

$$\begin{array}{c}{Q}^{*}=-\frac{1}{2}{\nu }_2(e^t{J}_1+e^{-3t}{J}_2),\hfill \\ W^{*}=\frac{1}{2}{\nu }_2(e^t{J}_1-e^{-3t}{J}_2),\hfill \end{array}$$

(3.15)

where

$$\begin{array}{c}{J}_1\left(t\right)=\int \Phi \left(s\right)\mathrm{ds},J_2\left(t\right)=\int e^{4s}\Phi \left(s\right)\mathrm{ds}.\hfill \end{array}$$

(3.16)

We introduce the notation

$$\begin{array}{c}{H}_1\left(r\right)=\int x^{-1}\Phi \left(x\right)\mathrm{dx},H_2\left(r\right)=\int x^3\Phi \left(x\right)\mathrm{dx}.\hfill \end{array}$$

(3.17)

By considering Equations (3.12), (3.14), and (3.15) we obtain

$$\begin{array}{c}Q=C_1{r}^{-1}+C_2{r}^{-3}+C_{3}r-\frac{1}{2}{\nu }_2[r{H}_1+r^{-3}{H}_2],\hfill \\ W=-{\nu }_1{C}_1{r}^{-1}+C_2{r}^{-3}-C_{3}r+\frac{1}{2}{\nu }_2[r{H}_1-r^{-3}{H}_2].\hfill \end{array}$$

(3.18)

By substituting $Q$ and $W$ from (3.18)

$$r(r\Phi \prime )\prime -(4+{\xi }^2{r}^2)\Phi =-\frac{2{\nu }_1\beta C_1}{\alpha }.$$

(3.19)

For $r\to \infty$ the solution of (3.19) that generates finite stresses is supplied by

$$\Phi =A_2{K}_2\left(\xi r\right)+\frac{2}{{\xi }^2\alpha }{\nu }_1\beta C_1{r}^{-2},$$

(3.20)

in which $A_2$ is an arbitrary constant. By substituting (3.20) into relations (3.18) we obtain

$$\begin{array}{c}Q=\frac{1}{r}{C}_1+\frac{1}{r^3}{C}_2+C_{3}r+\frac{1}{2\xi }{\nu }_2{A}_2[K_3\left(\xi r\right)+K_1\left(\xi r\right)],\hfill \\ W=-\frac{1}{r}{\nu }_1\delta C_1+\frac{1}{r^3}{C}_2-C_{3}r+\frac{1}{2\xi }{\nu }_2{A}_2[K_3\left(\xi r\right)-K_1\left(\xi r\right)],\hfill \end{array}$$

(3.21)

in which

Let us now consider the notation

$$q_1=\zeta (\lambda +\mu )-{\beta }^2,q_2=\frac{\zeta (\lambda +\mu )-{\beta }^2}{\zeta (\lambda +2\mu )-{\beta }^2}.$$

(3.23)

We observe that restrictions (2.5) imply

From (2.13), (3.9), (3.11), and (3.20)–(3.23), we can derive that

$$\begin{array}{c}\multicolumn{1}{c}{{\tau }_{\mathrm{rr}}=q_1\frac{1}{\alpha {\xi }^2}{B}_1-2\mu r^{-2}{B}_2-2\mu {\nu }_2\frac{1}{\xi r}{K}_1\left(\xi r\right)A_1+\mu \{-2(2q_2{r}^{-2}{C}_1+3r^{-4}{C}_2-C_3)}\\ +\frac{1}{\xi r}{\nu }_2{A}_2[K_1\left(\xi r\right)-3K_3\left(\xi r\right)]\}\cos 2\theta ,\hfill \end{array}$$

$$\begin{array}{c}{\tau }_{\theta \theta }=q_1\frac{1}{\alpha {\xi }^2}{B}_1+2\mu r^{-2}{B}_2+2\mu {\nu }_2{A}_1[K_0\left(\xi r\right)+\frac{1}{\xi r}{K}_1\left(\xi r\right)]\hfill \\ +2\mu \{3r^{-4}{C}_2-C_3+\frac{1}{4}{\nu }_2{A}_2[K_0\left(\xi r\right)+3K_2\left(\xi r\right)+6\frac{1}{\xi r}{K}_3\left(\xi r\right)]\}\cos 2\theta ,\hfill \\ {\tau }_{r\theta }=-2\mu \{q_2{C}_1{r}^{-2}+3r^{-4}{C}_2+C_3+\frac{1}{2\xi }{\nu }_2{A}_2{r}^{-1}[3K_3\left(\xi r\right)+K_1\left(\xi r\right)]\}\sin 2\theta ,\hfill \\ {\chi }_r=-\alpha \xi K_1\left(\xi r\right)A_1-\{\frac{1}{2}\alpha \xi A_2[K_0\left(\xi r\right)+K_3\left(\xi r\right)]+4C_1{r}^{-3}{\nu }_1\frac{\beta }{{\xi }^2}\}\cos 2\theta ,\hfill \\ {\chi }_{\theta }=-2\alpha r^{-1}[A_2{K}_2\left(\xi r\right)+2C_1{r}^{-2}{\nu }_1\frac{\beta }{{\xi }^2\alpha }]\sin 2\theta .\hfill \end{array}$$

(3.24)

For $r\to \infty$ , Equations (3.24) reduce to

$$\begin{array}{c}{\tau }_{\mathrm{rr}}=\frac{q_1}{\alpha {\xi }^2}{B}_1+2\mu C_3\cos 2\theta ,\hfill \\ {\tau }_{\theta \theta }=\frac{q_1}{\alpha {\xi }^2}{B}_1-2\mu C_3\cos 2\theta ,\hfill \\ {\tau }_{r\theta }={\tau }_{\theta r}=-2\mu C_3\sin 2\theta ,\hfill \\ {\chi }_r={\chi }_{\theta }=0.\hfill \end{array}$$

(3.25)

From the conditions at infinity (3.2), we obtain

$$B_1=\frac{\alpha {\xi }^2}{2q_1}A,C_3=\frac{1}{4\mu }A.$$

(3.26)

By considering Equations (3.3), the boundary conditions (3.1) can be rewritten in the form

and

By (3.27) we have

$$\begin{array}{c}{A}_1=\frac{\beta }{2q_1}A{\left[K_0\left(\xi a\right)\right]}^{-1},\hfill \\ B_2=-\frac{a^2}{2q_1}A\{\frac{1}{2}\zeta +\beta {\nu }_2\frac{1}{\xi a}{K}_1(\xi a){[K_0(\xi r)]}^{-1}\}.\hfill \end{array}$$

(3.29)

Conditions (3.28) imply that

$$\begin{array}{c}{A}_2=-\frac{2}{\alpha {\xi }^2{a}^2}{\nu }_1\beta C_1{\left[K_2\left(\xi a\right)\right]}^{-1},\hfill \\ [a^2-\eta \xi \mathrm{aL}(\xi a)]C_1+C_2=-\frac{a^4}{4\mu }A,\hfill \\ -({\nu }_1\delta a^2+4\eta )C_1+C_2=\frac{a^4}{4\mu }A,\hfill \end{array}$$

(3.30)

where

$$\begin{array}{c}L\left(\xi r\right)=[K_3\left(\xi r\right)+K_1\left(\xi r\right)]{\left[K_2\left(\xi a\right)\right]}^{-1},\hfill \\ \eta =\frac{{\nu }_1{\nu }_2\beta }{\alpha {\xi }^4}.\hfill \end{array}$$

(3.31)

If we use the notation

$$\begin{array}{c}\multicolumn{1}{c}{{\kappa }_1=a^2(1+{\nu }_1\delta )+\eta [4-\xi \mathrm{aL}(\xi a)]}\\ \multicolumn{1}{c}{{\kappa }_2=a^2(1-{\nu }_1\delta )-\eta [4+\xi \mathrm{aL}(\xi a)]}\end{array}$$

(3.32)

by solving the system (3.30) we obtain

$$\begin{array}{c}{C}_1=-\frac{A{a}^4}{2\mu {\kappa }_1},\hfill \\ C_2=\frac{A{a}^4}{4\mu {\kappa }_1}{\kappa }_2,\hfill \\ A_2=\frac{A{a}^2}{\mu {\kappa }_1}\frac{{\nu }_1\beta }{\alpha {\xi }^2}{[K_2(\xi a)]}^{-1}.\hfill \end{array}$$

(3.33)

The solution of the problem is given by (3.3) in which the constants $A_{\alpha }$ , $B_{\alpha }$ , and $C_i$ are expressed by (3.26), (3.29), and (3.33).

4. Explicit formulas

In the following, it will be useful to express the solution in explicit form. After some calculations, we obtain explicit formulas for the displacements, the volume fraction function, the stress tensor, and the equilibrated stress vector:

$$\begin{array}{c}u=\frac{A}{4q_1}\{\zeta (r-\frac{a^2}{r})+\frac{2\beta {\nu }_2}{\xi }[\frac{K_1(\xi r)}{K_0(\xi a)}-\frac{a}{r}\frac{K_1(\xi a)}{K_0(\xi r)}]\}\hfill \\ +\frac{A{a}^4}{2\mu {\kappa }_1}[-\frac{1}{r}+\frac{1}{2}(\frac{{\kappa }_2}{r^3}+\frac{{\kappa }_{1}r}{a^4})+\frac{1}{a^2}\xi \eta L(\xi r)]\cos 2\theta ,\hfill \\ v=\frac{A{a}^4}{2\mu {\kappa }_1}[{\nu }_1\delta \frac{1}{r}+\frac{1}{2}(\frac{{\kappa }_2}{r^3}-\frac{{\kappa }_{1}r}{a^4})+4\eta \frac{1}{a^{2}r}\frac{K_2(\xi r)}{K_2(\xi a)}]\sin 2\theta ,\hfill \\ \phi =\frac{A\beta }{2q_1}[\frac{K_0(\xi r)}{K_0(\xi a)}-1]+\frac{A{a}^4}{\mu {\kappa }_1}\frac{{\nu }_1\beta }{{\xi }^2\alpha }[\frac{1}{r^2}-\frac{1}{a^2}\frac{K_2(\xi r)}{K_2(\xi a)}]\cos 2\theta \hfill \end{array}$$

(4.1)

and

$$\begin{array}{c}{\tau }_{\mathrm{rr}}=\frac{1}{2}A+\frac{A\mu }{q_1}\{\frac{1}{2}\zeta \frac{a^2}{r^2}+\beta {\nu }_2[\frac{1}{\xi a}\frac{a^2}{r^2}\frac{K_1(\xi a)}{K_0(\xi r)}-\frac{1}{\xi r}\frac{K_1(\xi r)}{K_0(\xi a)}]\}\hfill \\ -\frac{A{a}^4}{{\kappa }_1}[-2q_2\frac{1}{r^2}+\frac{1}{2}(\frac{3{\kappa }_2}{r^4}-\frac{{\kappa }_1}{a^4})-\eta \xi \frac{1}{a^{2}r}\frac{K_1(\xi r)-3K_3(\xi r)}{K_2(\xi a)}]\cos 2\theta ,\hfill \\ {\tau }_{\theta \theta }=\frac{1}{2}A-\frac{A\mu }{q_1}\{\frac{1}{2}\zeta \frac{a^2}{r^2}+\beta {\nu }_2[\frac{a^2}{r^2}\frac{1}{\xi a}\frac{K_1(\xi a)}{K_0(\xi r)}-\frac{1}{\xi r}\frac{K_1(\xi r)}{K_0(\xi a)}-\frac{K_0(\xi r)}{K_0(\xi a)}]\}\hfill \\ +\frac{1}{2}\frac{A}{{\kappa }_1}\{3{\kappa }_2\frac{a^4}{r^4}-{\kappa }_1+\eta {\xi }^2{a}^2[K_0(\xi r)+3K_2(\xi r)+6\frac{1}{\xi r}{K}_3(\xi r)]{[K_2(\xi a)]}^{-1}\}\cos 2\theta ,\hfill \\ {\tau }_{r\theta }=\{-\frac{1}{2}A+\frac{A{a}^2}{{\kappa }_1}[\frac{a^2}{r^2}(q_2-\frac{3}{2}{\kappa }_2\frac{1}{r^2})-\eta \xi \frac{1}{r}\frac{3K_3(\xi r)+K_1(\xi r)}{K_2(\xi r)}]\}\sin 2\theta \hfill \end{array}$$

$$\begin{array}{c}{\chi }_r=-\frac{1}{2}A\{\frac{\alpha \xi \beta }{q_1}\frac{K_1(\xi r)}{K_0(\xi a)}+\frac{a^2}{\mu {\kappa }_1}\frac{{\nu }_1\beta }{\xi }[\frac{K_0(\xi r)+K_3(\xi r)}{K_2(\xi a)}-\frac{4}{\xi }\frac{a^2}{r^3}]\cos 2\theta \}\hfill \\ {\chi }_{\theta }=\frac{2A{a}^2}{\mu {\kappa }_1}\frac{{\nu }_1\beta }{{\xi }^{2}r}[\frac{a^2}{r^2}-\frac{K_2(\xi r)}{K_2(\xi a)}]\sin 2\theta .\hfill \end{array}$$

(4.2)

If in Equations (4.1) and (4.2) we set $\alpha$ , $\beta$ , and $\eta$ equal to zero, we obtain as a particular case the solution of the problem of a rigid inclusion in a half-space made of classical elastic material (see Muskhelishvili [3, pp. 218–219])

$$\begin{array}{c}{u}^{*}=\frac{A}{4(\mu +\lambda )}(r-\frac{a^2}{r})+\frac{A{a}^2}{2\mu (1+{\nu }_1)}[-\frac{1}{r}+\frac{a^2}{2}(\frac{1-{\nu }_1}{r^3}+\frac{1+{\nu }_1}{a^4}r)]\cos 2\theta ,\hfill \\ v^{*}=\frac{A{a}^2}{2\mu (1+{\nu }_1)}[{\nu }_1\frac{1}{r}+\frac{a^2}{2}(\frac{1-{\nu }_1}{r^3}-\frac{1+{\nu }_1}{a^4}r)]\sin 2\theta ,\hfill \\ {\phi }^{*}=0\hfill \end{array}$$

(4.3)

and

$$\begin{array}{c}{\tau }_{\mathrm{rr}}^{*}=\frac{1}{2}A(1+\frac{\mu }{\mu +\lambda }\frac{a^2}{r^2})+\frac{1}{2}A[1+\frac{\lambda +\mu }{\lambda +3\mu }(\frac{4a^2}{r^2}-\frac{3a^4}{r^4})]\cos 2\theta \hfill \\ {\tau }_{\theta \theta }^{*}=\frac{1}{2}A(1-\frac{\mu }{\mu +\lambda }\frac{a^2}{r^2})-\frac{1}{2}A[1-\frac{3(\lambda +\mu )}{\lambda +3\mu }\frac{a^4}{r^4}]\cos 2\theta \hfill \\ {\tau }_{r\theta }^{*}=-\frac{1}{2}A[1+\frac{\mu +\lambda }{3\mu +\lambda }(\frac{3a^4}{r^4}-\frac{2a^2}{r^2})]\sin 2\theta \hfill \\ {\chi }_r^{*}=0,{\chi }_{\theta }^{*}=0.\hfill \end{array}$$

(4.4)

The stresses around the boundary of the cylinder will be determined from (4.2) putting $r=a$ . With the aim of deriving the next equations, it is useful to adopt the following relations:

$$\begin{array}{c}2q_2-1+2{\nu }_1\delta =1\hfill \\ 1-\frac{\zeta \mu }{q_1}+2\beta {\nu }_2\frac{\mu }{q_1}=\frac{1-2{\nu }_1}{q_2}\hfill \\ [K_0(\xi a)+3K_2(\xi a)+6\frac{1}{\xi a}{K}_3(\xi a)]{[K_2(\xi a)]}^{-1}=4+\frac{2}{\xi a}[3K_3(\xi a)-K_1(\xi a)]{[K_2(\xi a)]}^{-1}\hfill \\ 3{\kappa }_2-{\kappa }_1=2a^2(1-2{\nu }_1\delta )-16\eta -2\eta \xi a[K_1(\xi a)+K_3(\xi a)][K_2(\xi a)].\hfill \end{array}$$

(4.5)

By virtue of (4.5), Equations (4.2) become

$$\begin{array}{c}{\tau }_{\mathrm{rr}}=A(\frac{1}{2q_2}+\frac{a^2}{{\kappa }_1}\cos 2\theta )\hfill \\ {\tau }_{\theta \theta }=A(1-2{\nu }_1)(\frac{1}{2q_2}+\frac{a^2}{{\kappa }_1}\cos 2\theta )\hfill \\ {\tau }_{r\theta }=-\frac{1}{2}A\{1+\frac{1}{{\kappa }_1}[q_2{a}^2+2\eta \xi a\frac{K_1(\xi a)}{K_2(\xi a)}]\sin 2\theta \}\hfill \\ {\chi }_r=-\frac{1}{2}A\frac{{\nu }_2}{\xi }[\frac{1}{q_2}\frac{K_1(\xi a)}{K_0(\xi a)}+\frac{a^2}{{\kappa }_1}\frac{K_0(\xi a)+K_1(\xi a)}{K_2(\xi a)}\cos 2\theta ]\hfill \\ {\chi }_{\theta }=0\hfill \end{array}$$

(4.6)

In the classical theory of elasticity the stress concentration factor $K^{*}$ for the analogous problem under investigation occurs for $\theta =0$ and $r=a$ . In this case ${\tau }_{\mathrm{rr}}$ and ${\tau }_{\theta \theta }$ can be expressed in the form

$$\begin{array}{c}{\tau }_{\mathrm{rr}}=A(1+\frac{1-2\nu }{2}+\frac{1}{2(3-4\nu )})\hfill \\ \multicolumn{1}{c}{{\tau }_{\theta \theta }=A(\frac{3}{3-4\nu }-\frac{1-2\nu }{2})}\end{array}$$

(4.7)

where $\nu$ is the Poisson ratio. Figure 1 contains plots of the stress concentration factor for $0\le \nu <\frac{1}{2}$ . We note that

Figure 1. Stress concentration factor $K^{*}$ .

In the problem under consideration, the stress concentration factor $K$ is equal to maximum value between the ratios ${\tau }_{\mathrm{rr}}/A$ and ${\tau }_{\theta \theta }/A$ :

$$\begin{array}{c}\frac{{\tau }_{\mathrm{rr}}}{A}=\frac{1}{2q_2}+\frac{a^2}{{\kappa }_1}\hfill \\ \multicolumn{1}{c}{\frac{{\tau }_{\theta \theta }}{A}=(1-2{\nu }_1)(\frac{1}{2q_2}+\frac{a^2}{{\kappa }_1}).}\end{array}$$

(4.8)

In order to calculate the quantities $q_2$ and ${\kappa }_1$ in (4.8), the following relation is useful

that can be obtained from (2.5)₅ and where the positive parameter $\sigma$ verifies the inequality

Moreover, it appears to be realistic to put

Figures 2 and 3 contain plots of the ratio ${\tau }_{\mathrm{rr}}/A$ and ${\tau }_{\theta \theta }/A$ , for $\sigma =0.25$ , $\sigma =0.5$ , and $\sigma =0.75$ . The limiting value of the stress concentration factor is $K=2.2$ .

5. Conclusions

In this work, the equilibrium theory of elastic materials with voids has been analyzed and a plane strain problem has been solved. In the plane strain analysis, we have investigated the problem of a cylindrical rigid inclusion in an infinite porous elastic body that is uniformly stretched along one axis. The results are given in closed form and the connection with respect to previous works in classical elasticity can be easily deduced. The solution of the analogous problem in the classical theory can be obtained as a particular case. In fact, if in the solution we put $\alpha =\beta =\zeta =0$ , we obtain as a particular case the corresponding solution of the classic theory of elastic solids, see, e.g., Muskhelishvili [3]. Further applications of the present theory are problems of stress concentration, which can be studied in a similar way. Finally, we present explicit formulas of displacement field and stresses. We have determined the maximum stress and the stress concentration factor. An interesting feature of the solution is that the limiting value of the stress concentration factor is smaller than the value calculated in classical elasticity in the same situation.

Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article.

ORCID iDs
Simona De Cicco https://orcid.org/0000-0001-7635-8015

References

[1]	Kirsh, G . Die theorie der Elastizitat und die Bedurfuisse der Festigkeitslebre, Vol. 42. V.D.I. Berlin, 1898, pp. 797–807. Google Scholar
[2]	Kolosov, GV . On a application of complex function theory to a plane problem of the mathematical theory of elasticity (in Russian). Dissertation, Dorpat (Yuriev) University, 1909. Google Scholar
[3]	Muskhelishvili, NI . Some Basic Problems of the Mathematical Theory of Elasticity. 3rd revised and augmented ed. Moscow, 1949. Groningen: P. Noordhoff, N. V., 1953. Google Scholar
[4]	Savin, SN . Stress Concentration Around Holes. New York: Pergamon, 1961. Google Scholar
[5]	Hetnarski, RB, Ignaczak, J. The Mathematical Theory of Elasticity. Boca Raton, FL: CRC Press, 2011. Google Scholar
[6]	Ariman, T . On the stress around a circular hole in micropolar elasticity. Acta Mech 1967; 4: 216–229. Google Scholar | Crossref
[7]	Kim, BS, Eringen, AC. Stress distribution around an elliptical hole in an infinite micropolar plate. Lett Appl Eng Sci 1973; 1: 381–390. Google Scholar
[8]	De Cicco, S . Stress concentration effects in microstretch elastic bodies. Int J Eng Sci 2003; 41(2): 187–199. Google Scholar | Crossref
[9]	De Cicco, S . Problem of stress concentration in elastostatics of bodies with microstructure. Encyclopedia of Thermal Stresses 2014; Letter P: 4037–4046. Google Scholar
[10]	De Cicco, S, Ieşan, D. A theory of chiral Cosserat elastic plates. J Elasticity 2013; 111(2): 245–263. Google Scholar | Crossref
[11]	Truesdell, C, Toupin, R. The classical field theories. In: Flugge, S (ed.) Handbuch der Physik, Band III/1. Berlin: Springer, 1960. Google Scholar | Crossref
[12]	Truesdell, C, Noll, W. The non-linear field theories of mechanics. In: Flugge, S (ed.) Handbuch der Physik, Band III/3. Berlin: Springer, 1965. Google Scholar | Crossref
[13]	Biot, MA . General theory of three-dimensional consolidation. J Appl Phys 1941; 12: 155–164. Google Scholar | Crossref
[14]	Nunziato, JW, Cowin, SC. A nonlinear elastic materials with voids. Arch Rat Mech Anal 1979; 72: 175–201. Google Scholar | Crossref | ISI
[15]	Cowin, SC, Nunziato, JW. Linear elastic materials with voids. J Elasticity 1983; 13: 125–147. Google Scholar | Crossref | ISI
[16]	Ciarletta, M, Iesan, D. Non-Classical Elastic Solids. New York: John Wiley & Sons, Inc., 1993. Google Scholar
[17]	Iesan, D . Thermoelastic Models of Continua. Dordrecht: Springer Science and Business Media, 2004. Google Scholar | Crossref
[18]	Straughan, B . Stability and Wave Motion in Porous MediaNew York: Springer, 2011. Google Scholar
[19]	de Boer, R . Contemporary progress in porous media theory. Appl Mech Rev 2000; 53(12): 323–370. Google Scholar | Crossref
[20]	Day, WA . Time-reversal and the symmetry of the relaxation function of a linear viscoelastic material. Arch Rat Mech Anal 1971; 41: 132–162. Google Scholar
[21]	Nunziato, JW, Walsh, EK. Small-amplitude wave behaviour in one-dimensional granular materials. J Appl Mech 1977; 44: 559–564. Google Scholar | Crossref
[22]	Miled, K, Sab, K, Le Roy, R. Effective elastic properties of porous materials: Homogenization schemes vs experimental data. Mech Res Commun 2011; 38(2): 131–135. Google Scholar | Crossref
[23]	Trillat, M, Pastor, J, Francescato, P. Yield criterion for porous media with spherical voids. Mech Res Commun 2006; 33(3): 320–328. Google Scholar | Crossref
[24]	Ieşan, D . Classical and Generalized Models of Elastic Rods. Boca Raton, FL: CRC Press, 2009. Google Scholar
[25]	Dell'Isola, F, Batra, RC. Saint-Venant's problem for porous linear elastic materials. J Elasticity 1971; 47: 73–81. Google Scholar | Crossref
[26]	Ieşan, D . A theory of thermoelastic materials with voids. Acta Mech 1986; 60: 67–81. Google Scholar | Crossref | ISI
[27]	Ieşan, D, Nappa, L. Thermal stresses in plain strain of porous elastic solids. Meccanica 2004; 39: 125–138. Google Scholar | Crossref
[28]	De Cicco, S, Nappa, L. Singular surfaces in thermoviscoelastic materials with voids. J Elasticity 2003; 73: 191–210. Google Scholar | Crossref
[29]	De Cicco, S, Iesan, D. Thermal effects in anisotropic porous elastic rods. J Therm Stress 2013; 36(4): 364–377. Google Scholar | Crossref
[30]	Svanadze, M . Steady vibration problems in the theory of elasticity for materials with double voids. Acta Mech 2018; 229(4): 1517–1536. Google Scholar | Crossref
[31]	Svanadze, M . Fundamental solutions in the theory of elasticity for triple porosity materials. Meccanica 2016; 51(8): 1825–1837. Google Scholar | Crossref
[32]	Svanadze, M . Uniqueness theorems in the theory of thermoelasticity for solids with double porosity. Meccanica 2014; 49(9): 2099–2108. Google Scholar | Crossref
[33]	Svanadze, M . Potential method in the theory of elasticity for triple porosity materials. J Elasticity 2018; 130(1), DOI: 10.1007/s10659-017-9629-2. Google Scholar | Crossref
[34]	Nappa, L . Fundamental solutions in the theory of microfluids. Ann Univ Ferrara 2008; 54(1): 95–106. Google Scholar | Crossref
[35]	Goodman, MA, Cowin, SC. A continuum theory for granular materials. Arch Rat Mech Anal 1972; 44: 249–266. Google Scholar | Crossref | ISI
[36]	Cowin, SC, Goodman, MA. A variational principle for granular materials,. ZAMM 1976; 56: 281–286. Google Scholar | Crossref
[37]	Jenkins, JT . Static equilibrium of granular materials. J Appl Mech 1975; 42: 603–606. Google Scholar | Crossref
[38]	Love, AEH . A Treatise on Mathematical Theory of Elasticity. Cambridge: Dover, 1927. Google Scholar
[39]	MacKenzie, JK . The elastic constants of a solid containing spherical holes. Proc Phys Soc B 1950; 63: 2–11. Google Scholar | Crossref

Some Basic Problems Of The Mathematical Theory Of Elasticity Pdf

Source: https://journals.sagepub.com/doi/abs/10.1177/1081286519867109

Posted by: crowellniae1979.blogspot.com

Share this post