Complex potentials solutions for isotropic Cosserat bodies with voids

Abstract

This paper aims to obtain solutions in terms of the complex potential structure for the plain strain problem of an elastic micropolar and isotropic body with pores. The constitutive equations on which the method is applied and is useful are the well-known equations of the elasticity theory for the above-mentioned body. We intend to solve the Kirch problem using the procedure of complex variables.

1 Introduction

The Cosserat brothers laid the foundations of a theory of the mechanics of continuous media in which, for each material point, we have the freedom degree of a rigid. Later, the theory of micropolar media was introduced and studied by Eringen [1–3] for material studies. Unlike the generalized continuous media theory (Cosserat brothers’ theory) [4], which contains a conservation law for the microrotational inertia tensor, this theory considers three deformation directors. In this framework, the forces acting on the surface element are represented by the classic stress tensor and an additional couple tensor.

In this paper, we investigate plane strain within the equilibrium theory of micropolar, homogeneous, isotropic, and porous bodies. Using the constitutive equations (1)–(3), the geometric equations (4), and the equilibrium equations without body forces (5)–(7) from [5], we focus on addressing the fundamental boundary value problems of plane strain theory. Subsequently, we derive a depiction of the displacement of microrotations and pores using complex analytical functions and two real functions based on the homogeneous Helmholtz equations as described in [6]. In the fifth section, the structure of the potential functions for several domains of interest is studied, and in the sixth section, we apply the method of complex variables without introducing stress functions to solve the Kirch problem. The last section is dedicated to the numerical study, where we obtained the corresponding complex potential plots and stress and displacement distributions in a porous micropolar isotropic material. More studies on complex potentials can be found in [7, 8]. There also were countless studies aimed at the theory of micropolar media, among which we highlight [9–20].

2 Basic equations

We consider B a bounded domain from the three-dimensional Euclidean space, with ∂B as its boundary and n as the outer normal of the ∂B boundary. Assuming that we have a porous micropolar elastic medium occupying B, we associate the body with an orthogonal axis system $Ox_{i} (i=1,2,3)$.

The basic equations describing the evolution of an isotropic Cosserat medium with voids are as follows.

The constitutive equations:

$$\begin{aligned} & t_{ij}=\lambda u_{k,k}\delta _{ij}+\mu (u_{i,j}+u_{j,i})+k(u_{i,j}+ \varepsilon _{ijk}\phi _{k})+\xi \varphi \delta _{ij}, \end{aligned}$$

(1)

$$\begin{aligned} &m_{ij}=\alpha \phi _{k,k}\delta _{ij}+\gamma \phi _{j,i}+\psi \phi _{i,j}+ \zeta \varepsilon _{sji}\varphi _{,s}, \end{aligned}$$

(2)

$$\begin{aligned} &h_{i}=d\varphi _{,i}. \end{aligned}$$

(3)

The geometrical relations:

$$ e_{ij}=u_{i,j}+\varepsilon _{ijk}\phi _{k}, \ \ \ \ \ \psi _{ij}= \phi _{i,j}. $$

(4)

Equilibrium equations (body loads are absent):

$$\begin{aligned} & t_{ji,j}=0, \end{aligned}$$

(5)

$$\begin{aligned} & m_{ji,j}+\varepsilon _{irs}t_{rs}=0, \end{aligned}$$

(6)

$$\begin{aligned} & h_{i,i}+g=0. \end{aligned}$$

(7)

We consider:

$$ t_{i}=t_{ji}n_{j};\ m_{i}=m_{ji}n_{j}; \ N_{i}=h_{ji}n_{j}, $$

(8)

in which $N_{i}$ is the generalized surface force at a regular point on ∂B, $t_{i}$ is the surface force, and $m_{i}$ is the surface moment.

We will further presume a positive quadratic form for the internal energy density from where we have

$$\begin{aligned} &\kappa >0,\ \ \kappa + 2\mu >0, \ \ \kappa + 2\mu + 3 \lambda >0, \ \ \ d>0, \\ & \gamma -\beta >0,\ \ \gamma +\beta >0, \ \ \gamma +\beta + 3 \alpha >0. \ \end{aligned}$$

(9)

3 Plane strain problem

In this part of the paper, we will consider B to be the interior of a right cylinder whose cross-section is Σ and whose lateral boundary is Π. This setup is related to an orthogonal coordinate system so that its generators are parallel to the $x_{3}$ axis. We denote by L the boundary corresponding to the cross-section. The plane strain is considered to be parallel to the $x_{1}$, $x_{2}$-plane. Therefore,

$$\begin{aligned} & u_{\alpha}=u_{\alpha}(x_{1}, x_{2}),\ u_{3}=0, \ \phi _{\alpha}=0;\ \phi _{3}=\phi (x_{1},x_{2}), \\ & \varphi =\varphi (x_{1},x_{2}), \ \ \ (x_{1},x_{2}) \in \Sigma . \end{aligned}$$

(10)

Taking these restrictions into account, from the constitutive equations and the geometric equations, we deduce that $e_{ij}$, $\psi _{ij}$, $t_{ij}$, $m_{ij}$, $h_{i}$ are independent of $x_{3}$. So, taking into account that $t_{\alpha \beta}$, $m_{\alpha 3}$, $t_{33}$, $m_{3 \alpha}$ and $h_{\alpha}$ are nonzero, the constitutive equations become:

$$\begin{aligned} & t_{\alpha \beta}=\lambda u_{\rho ,\rho}\delta _{\alpha \beta}+\mu (u_{ \alpha ,\beta}+u_{\beta ,\alpha})+\kappa (u_{\alpha ,\beta}+\epsilon _{3 \alpha \beta}\phi _{\kappa})+\xi \varphi \delta _{\alpha \beta}, \end{aligned}$$

(11)

$$\begin{aligned} & m_{\alpha 3}=\psi \phi _{,\alpha}+\zeta \varepsilon _{3\beta \alpha} \varphi , \end{aligned}$$

(12)

$$\begin{aligned} & h_{i}=d\varphi _{,\alpha}. \end{aligned}$$

(13)

From (4) and (10) it follows that the nonzero measures of plane deformation are:

$$ e_{\alpha \beta}=u_{\beta ,\alpha}+\varepsilon _{\beta \alpha 3}\phi , \ \ \ \ \ \psi _{\alpha 3}=\phi _{,\alpha}. $$

(14)

Moreover, the equilibrium equations take the following form:

$$\begin{aligned} & t_{\beta \alpha ,\beta}=0, \end{aligned}$$

(15)

$$\begin{aligned} & m_{\alpha 3,\alpha}+\varepsilon _{3 \alpha \beta}t_{\alpha \beta}=0, \end{aligned}$$

(16)

$$\begin{aligned} & h_{\alpha , \alpha}+g=0 \ \text{on} \ \Sigma . \end{aligned}$$

(17)

For the vector $t_{i}$ for the surface force, the vector $m_{i}$ for the surface force couple, and the vector $N_{i}$ for the void evolution at an ordinary point on L, we have:

$$ t_{\alpha}= t_{\beta \alpha}n_{\beta}, \ \ m= m_{\rho 3}n_{\rho}, \ N=h_{ \alpha} n_{\alpha}. $$

(18)

We will add the boundary conditions to the basic equations so that the first boundary problem is characterized by

$$ u_{\alpha}= \tilde{u}_{\alpha},\ \phi =\tilde{\phi}, \ \varphi = \tilde{\varphi}\ \text{on} \ L $$

(19)

and the second by

$$ t_{\beta \alpha}n_{\beta}= \tilde{t}_{\alpha}, m_{\alpha 3}n_{\alpha}= \tilde{m}, h_{\alpha}n_{\alpha}=\tilde{N} \ \text{on} \ L, $$

(20)

where $\tilde{u}_{\alpha}$, ϕ̃, φ̃ are prescribed functions, and $\tilde{t}_{\alpha}$, m̃, Ñ are given.

From (11)–(17), we get the system below, in terms of displacement, microrotation and pores. (Δ is the Laplacian)

$$\begin{aligned} & (\lambda + \mu ) u_{\rho , \rho \alpha}+(\mu +\kappa ) \Delta u_{ \alpha}+ \xi \varphi _{,\alpha}+\kappa \varepsilon _{3 \alpha \beta} \phi _{, beta}=0, \\ &\psi \Delta \varphi + \kappa \varepsilon _{3 \alpha \beta } u_{ \beta , \alpha}- 2 k \phi =0, \\ &d \Delta \varphi - \xi u_{\rho , \rho}-a \varphi =0. \end{aligned}$$

(21)

More detailed steps to obtain the system can be found in [7].

4 Complex potentials

In this section, we will work in system (21) whose relations will be written in complex coordinates and integrated directly. In other words, we will determine the displacement using a pair of complex analytical functions, the microrotation, and the change in volume fraction in real functions verifying the homogeneous Helmholtz equations [6]. First, we introduce the complex coordinates

$$ z= x_{1}+ i x_{2}, \overline{z} = x_{1}- i x_{2}, $$

(22)

and complex displacement

$$ U= u_{1}+ i u_{2}. $$

(23)

So, we get

$$ \Delta = 4 \frac{\partial ^{2}}{\partial z\partial \overline{z}}; \ u_{ \rho , \rho}=\frac{\partial U}{\partial z}+ \frac{\partial \overline{U}}{\partial \overline{z}}; \ \varepsilon _{3 \alpha \beta} u_{\beta , \alpha}= i\Big( \frac{\partial \overline{U}}{\partial \overline{z}}- \frac{\partial U}{\partial z}\Big). $$

(24)

Taking into account (24), we will rewrite the relations of system (21) in the following form:

$$\begin{aligned} & 2(\kappa +\mu )\frac{\partial ^{2}U}{\partial z\partial \overline{z}}+( \mu +\lambda )\frac{\partial}{\partial \overline{z}}\Big( \frac{\partial U}{\partial z}+ \frac{\partial \overline{U}}{\partial \overline{z}}\Big)+\xi \frac{\partial \varphi}{\partial \overline{z}}- i\kappa \frac{\partial \phi}{\partial \overline{z}}=0, \\ &4\psi \frac{\partial ^{2}\phi}{\partial z\partial \overline{z}}-i \kappa \Big( \frac{\partial U}{\partial z}- \frac{\partial \overline{U}}{\partial \overline{z}}\Big) - 2 \kappa \phi =0, \\ &4 d\frac{ \partial ^{2} \varphi}{\partial z \partial \overline{z}}- \xi \Big( \frac{\partial U}{\partial z}+ \frac{\partial \overline{U}}{\partial \overline{z}}\Big) - a \varphi =0. \end{aligned}$$

(25)

We integrate the first relation of this system to obtain

$$ 2(\mu +\kappa ) \frac{\partial U}{\partial z}+(\lambda +\mu ) \Big( \frac{\partial U}{\partial z}+ \frac{\partial \overline{U}}{\partial \overline{z}}\Big)+ \xi \varphi -i \kappa \phi =\Gamma '(z), $$

(26)

where Γ is a complex analytic function on z, and $d\Gamma '(z)= \frac{d \Gamma (z)}{dz}$.

By conjugation of relation (26), we get

$$ 2(\mu +\kappa ) \frac{\partial \overline{U}}{\partial z}+(\lambda + \mu ) \Big( \frac{\partial U}{\partial z}+ \frac{\partial \overline{U}}{\partial \overline{z}}\Big)+ \xi \varphi +i \kappa \phi =\overline{\Gamma}'(z). $$

(27)

Next, using the relations (26) and (27), we get by summing the relation

$$ \frac{\partial U}{\partial z}+ \frac{\partial \overline{U}}{\partial \overline{z}}= \frac{1}{2(2\mu +\kappa +\lambda )}[\Gamma '(z)+\overline{\Gamma}'( \overline{z})-2\xi \varphi ],$$

(28)

and by subtraction the relation

$$ \frac{\partial U}{\partial z}- \frac{\partial \overline{U}}{\partial \overline{z}}= \frac{1}{2(\mu +\kappa )}[\Gamma '(z)-\overline{\Gamma}'(\overline{z})+ 2 i\kappa \phi (z, \overline{z})]. $$

(29)

From the third relation of system (25) and from (28), we deduce that

$$ \Big( 4\frac{\partial ^{2}}{\partial z \partial \overline{z}}- m^{2} \Big)\varphi =\frac{\xi}{2d(2\mu +\kappa +\lambda )}[\Gamma (z)+ \overline{\Gamma}'(\overline{z})], $$

(30)

where

$$ m= \frac{a(2\mu +\kappa +\lambda )-\xi ^{2}}{d(2\mu +\kappa +\lambda )},$$

with $m^{2}>0$, according to (9). Therefore, we deduce that the function φ can be written in the form

$$ \varphi = M- \frac{\xi}{2 dm^{2}(2\mu +\kappa +\lambda )}[\Gamma (z)+ \overline{\Gamma}'(\overline{z})], $$

(31)

where M is a real function that satisfies

$$ \Big(4 \frac{\partial ^{2}}{\partial z \partial \overline{z}}-m^{2} \Big)M=0.$$

(32)

Taking into account (28) and (31), we find:

$$ \frac{\partial U}{\partial z}+\frac{\partial \overline{U}}{\partial z}=R [\Gamma '(z)+\overline{\Gamma}'(\overline{z})] - \frac{\xi}{2\mu +\kappa +\lambda}M, $$

(33)

where

$$ R= \frac{\xi +dm^{2}(2\mu +\kappa +\lambda )}{2dm^{2}(2\mu +\kappa +\lambda )}.$$

Next, considering the second equation of system (25) and relation (29), we obtain

$$ \Big( 4 \frac{\partial ^{2}}{\partial z \partial \overline{z}}- p^{2} \Big) \phi = \frac{i \kappa}{2 \gamma (\mu +\kappa )}[\Gamma '(z)- \overline{\Gamma}'(\overline{z})],$$

(34)

where

$$ p^{2}=\frac{\kappa (2\mu +\kappa )}{\psi (\mu +\kappa )}.$$

The ϕ function can be rewritten as follows:

$$ \phi =P- \frac{i \kappa}{2 \gamma p^{2}(\mu +\kappa )}[\Gamma '(z)- \overline{\Gamma}'(\overline{z})], $$

(35)

where the real function P satisfies

$$ \Big( 4 \frac{\partial ^{2}}{\partial z \partial \overline{z}}-p^{2} \Big) P=0. $$

(36)

From (27), (31)–(33), and (36), we deduce that

$$ \frac{\partial U}{\partial z}= \eta _{1} \Gamma '(z)- \eta _{2} \overline{\Gamma}'(\overline{z})+ 4 i q_{1} \frac{\partial ^{2} P}{\partial z \partial \overline{z}}- 4 q_{2} \frac{\partial ^{2} M}{\partial z \partial \overline{z}}, $$

(37)

where we denoted by $\eta _{1}$, $\eta _{2}$, $q_{1}$ and $q_{2}$ the following:

$$\begin{aligned}& \eta _{1}= \frac{3 \mu + \lambda + 2 \kappa}{4(\mu +k)(2\mu +\lambda +\kappa )}+ \frac{\kappa}{4(2\mu +\kappa )(\mu +\kappa )}+ \frac{\xi ^{2}}{4 d m^{2}(2\mu +\lambda +\kappa )^{2}},\\& \eta _{2}=\frac{\mu + \lambda}{4(\mu +k)(2\mu +\lambda +\kappa )}+ \frac{\kappa}{4(2\mu +\kappa )(\mu +\kappa )}- \frac{\xi ^{2}}{4 d m^{2}(2\mu +\lambda +\kappa )^{2}},\\& q_{1}=\frac{\kappa}{2 p^{2}(\mu +\kappa )},\ q_{2}= \frac{\xi}{2m^{2}(2\mu +\lambda +\kappa )}. \end{aligned}$$

Let ω be a complex analytic function on z. By integrating equation (37), we get

$$ U= \eta _{1} \Gamma (z)- \eta _{2} \overline{\Gamma}(\overline{z}) z- \overline{\omega}(\overline{z})+ 4 i q_{1} \frac{\partial P}{\partial z}-4 q_{2} \frac{\partial M}{\partial \overline{z}}. $$

(38)

Previously, we were able to obtain in relations (31), (35), and (38) a representation of the functions φ, ϕ and U in terms of the analytic complex functions Γ, ω and the real functions M and P.

Next, following some simple calculations, we deduce the equalities below:

$$\begin{aligned} &t_{11}+t_{22}=2\xi \varphi +(2\lambda +2\mu +\kappa )u_{\rho ,\rho}, \\ &t_{11}+it_{12}-t_{22}+it_{21}=(\kappa +2\mu )[u_{1,1}+iu_{1,2}-u_{2,2}+iu_{2,1}]=2(2 \mu +\kappa )\frac{\partial U}{\partial \overline{z}}, \\ &t_{21}-t_{12}=(u_{2,1}-u_{1,2}-2 \phi )\kappa , \\ &m_{13}-i m_{23}=2\psi \frac{\partial \phi}{\partial z}- i \zeta \frac{\partial \varphi}{\partial z}, \\ &h_{1}-ih_{2}= 2d\frac{\partial \varphi}{\partial z}. \end{aligned}$$

(39)

Using (24), (34), (36), and (39), we obtain the following form of the relations presented in the previous system:

$$\begin{aligned} & t_{11}+t_{22}= \frac{a(2\mu +2\lambda +\kappa )-2\xi ^{2}}{2dm^{2}(2\mu +\kappa +\lambda )}[ \Gamma '(x)+\overline{\Gamma}'(\overline{z})]+ \frac{\xi (2\mu +\kappa )}{2\mu +\kappa +\lambda}M, \\ &t_{11}+it_{12}-t_{22}+it_{21}=-2(2\mu +k)\Big[\eta _{2} \overline{\Gamma}''(\overline{z})z+\overline{\omega}(\overline{z})- 4i q_{1} \frac{\partial ^{2} P}{\partial \overline{z}^{2}}+4q_{2} \frac{\partial ^{2} M}{\partial \overline{z}^{2}}\Big], \\ & t_{21}-t_{12}=\gamma p^{2} P, \\ & m_{13}-i m_{23}= 2\gamma \frac{\partial P}{\partial z}+2i\zeta \frac{\partial M}{\partial z}-i \Big[ \frac{\kappa}{p^{2}(\mu +\kappa )}+ \frac{\zeta \xi}{dm^{2}(2\mu +\lambda +\kappa )}\Big] \Gamma ''(z) , \\ & h_{1}-ih_{2}= 2d\frac{\partial M}{\partial z}- \frac{\xi}{m^{2}(2\mu +\lambda +\kappa )}\Gamma ''(z). \end{aligned}$$

(40)

Next, we will use the following relations:

$$ n_{1}=-\frac{1}{2}i\Big(\frac{dz}{ds}-\frac{d\overline{z}}{ds}\Big), \ \ \ \ n_{2}=-\frac{1}{2}\Big(\frac{dz}{ds}+\frac{d\overline{z}}{ds} \Big),$$

considering that L is a smooth piecewise arclength-parameterized curve. From (20), we have

$$\begin{aligned} & t_{1}+it_{2}=\frac{1}{2}[t_{12}-it_{11}-t_{21}-it_{22}] \frac{dz}{ds}+i\frac{1}{2}[t_{11}+it_{12}-t_{22}+it_{21}] \frac{d \overline{z}}{ds}, \\ & m=Im\{(m_{13}-im_{23}\frac{dz}{ds}\}, \\ & h=Im\{(h_{1}-ih_{2})\frac{dz}{ds}\} , \end{aligned}$$

(41)

where $Im\{\}$ represents the imaginary part of {}. We note that

$$ t_{11}+t_{22}=2(2\mu +k)\{\eta _{2}[\Gamma '(z)+\overline{\Gamma}'( \overline{z})]+q_{2} m^{2} M\}. $$

(42)

Using (40)–(42), we get

$$\begin{aligned} & t_{1}+it_{2}=-(2\mu +\kappa )i\frac{d}{ds}\{\eta _{2}[\Gamma (z)+z \overline{\Gamma}'(\overline{z})]+\overline{\omega}(\overline{z})-4iq_{1} \frac{\partial P}{\partial \overline{z}}+4 q_{2} \frac{\partial M}{\partial \overline{z}}\}, \\ & m=Im \{2 \gamma \frac{\partial P}{\partial z}+2 i\zeta \frac{\partial M}{\partial z}-i w_{1} \Gamma ''(z)]\frac{dz}{ds}\}, \\ & h=Im\{[2d \frac{\partial M}{\partial z}- w_{2} \Gamma ''(z)] \frac{dz}{ds}\}, \end{aligned}$$

(43)

where we used the following notations:

$$ w_{1}=\frac{\kappa}{p^{2}(\mu +\kappa )}+ \frac{\zeta \xi}{dm^{2}(2\mu +\lambda +\kappa )}; \ \ w_{2}= \frac{\xi}{m^{2}(2\mu +\lambda +\kappa )}. $$

We use the notations $S_{1}$ and $S_{2}$ for the vector components resulting from the application of external stresses to the contour L. Hence, we get

$$ S_{1}+i S_{2}=\int _{L}(t_{1}+it_{2})ds=-(2\mu +\kappa )i\{\eta _{2}[ \Gamma (z)+z\overline{\Gamma}'(\overline{z})]+\overline{\omega}( \overline{z})-4i q_{1}\frac{\partial P}{\partial \overline{z}}+4q_{2} \frac{\partial M}{\partial \overline{z}}\}_{A}^{A}. $$

(44)

A function F changes its value when moving around the contour L in the conventional positive direction. For one round, we denote these changes by $\{F\}_{A}^{A}$.

Now, we can express the boundary conditions (19) in the following form:

$$\begin{aligned} &\eta _{1} \Gamma (z)-\eta _{2} x \overline{\Gamma}'(\overline{z})- \overline{\omega}(\overline{z})+4iq_{1} \frac{\partial P}{\partial \overline{z}}-4q_{2} \frac{\partial M}{\partial \overline{z}}=\tilde{u}(s), \\ & P(z, \overline{z})-\frac{i\kappa}{2\gamma p^{2}(\kappa +\mu )}[\Gamma '(z)- \overline{\Gamma}'(\overline{z})]=\phi (s), \\ & M(z, \overline{z})-\frac{\xi}{2dm^{2}(2\mu +\lambda +\kappa )}[ \Gamma (z)+\overline{\Gamma}'(\overline{z})]=\varphi (s), \ z\in L, \end{aligned}$$

(45)

where $\tilde{u}=\tilde{u}_{1}+ i\tilde{u}_{2}$. Moreover, from (43), the boundary conditions (20) can take the following form:

$$\begin{aligned} & (2\mu +\kappa )\frac{d}{ds}\{\eta _{2} [ \Gamma (z)+z \overline{\Gamma}'(\overline{z})]+\overline{\omega}(\overline{z})-4iq_{1} \frac{\partial P}{\partial \overline{z}}+4q_{2} \frac{\partial M}{\partial \overline{z}}\}=\tilde{t}(s), \\ & Im\{[2\gamma \frac{\partial P}{\partial z}+2i\zeta \frac{\partial M}{\partial z}-iw_{1}\Gamma ''(z)]\frac{dz}{ds}\}= \tilde{m}(s), \\ & Im\{[2d\frac{\partial M}{\partial z}-w_{2}\Gamma ''(z)] \frac{dz}{ds}\}=\tilde{N}(s), z\in L, \end{aligned}$$

(46)

where $\tilde{t}(s)=i(\tilde{t}_{1}+i\tilde{t}_{2})$.

5 Construction of potentials

In this section, we aim to derive the structure of the potentials Γ, ω, P, and M and to explore their arbitrariness across various domains of interest. We analyze the differences between the configurations of the following sets of potentials $(\Gamma , \omega , P, M)$ and $(\Gamma ^{*}, \omega ^{*}, P^{*}, M^{*})$, corresponding to the same functions $t_{\alpha \beta}$, $m_{\alpha 3}$ and $h_{\alpha}$.

According to (40), it is required that

$$\begin{aligned} & Re[\Gamma '(z)]=Re[\Gamma ^{*'}(z)], \ \ \ \ M=M^{*}, \ \ \ P=P^{*}, \\ &\eta _{2} z\overline{\Gamma}^{\prime \prime }(\overline{z})+\overline{\omega}^{\prime }( \overline{z})=\eta _{2}z\overline{\Gamma}^{*''}(\overline{z})+ \overline{\omega}^{*'}(\overline{z}), \end{aligned}$$

where $Re[]$ denotes the real part of []. Therefore, we deduce that

$$\begin{aligned} &\Gamma (z)=\Gamma ^{*}(z)+i X_{z} +\rho _{1}, \\ & \omega (z)=\omega ^{*}(z)+\rho _{2}, \\ &P=P^{*}, \\ &M=M^{*}, \end{aligned}$$

(47)

where X is a real constant, and $\rho _{1}$, $\rho _{2}$ are complex constants.

We can set the origin of the coordinates in Σ such that X, $\rho _{1}$, $\rho _{2}$ satisfy the conditions

$$ \Gamma (0)=0, \ Im\{\Gamma '(0)\}=0, \omega (0)=0, $$

(48)

which ensure the unique determination of Γ and ω.

We consider that $(\Gamma , \omega , P, M)$ and $(\Gamma ^{*}, \omega ^{*}, P^{*}, M^{*})$ are correlated with $u_{\alpha}$, $\phi _{\alpha}$ and φ, which indicates that we cannot have a greater arbitrariness than in (46). Referring to relation (39), the displacement imposes $X=0$ and $\eta _{1} \rho _{1}=\overline{\rho}_{2}$. Consequently, we can select $\rho _{1}$ such that $\Gamma (0)=0$.

Given that Γ and ω are single-valued and analytic functions within a bounded, simply connected region, we focus on the situation where the cross section is bounded and multiple connected. We will consider the boundary L as comprising $(m+1)$ simple and closed $L_{j}$ contours, ensuring $L_{m+1}$ encompasses all $L_{k}$ contours (where ($k = 1,2, \ldots,m$)).

Assuming that $u_{\alpha}$, $\phi _{\alpha}$, φ and the stress functions $t_{\alpha , \beta}$, $m_{\alpha , \beta}$ and $h_{\alpha}$ have a unique value and from (40), we deduce that M, P and their second-order derivatives must also be single-valued. From this, we derive the following form of the complex potentials:

$$ \begin{aligned} &\Gamma (z)=\sum _{k=1}^{m}(zX_{k}+Y_{k})log(z-z_{k})+\Gamma _{1}(z), \\ &\omega (z)=\sum _{k=1}^{m} Z_{k} log(z-z_{k})+\omega _{1}(z), \end{aligned} $$

(49)

where $z_{k}$ represents a point in the simply connected region $\Sigma _{k}$ bounded by $L_{k}$, $\Gamma _{1}$, and $\omega _{1}$ are single-valued analytic functions on Σ, $Y_{k}$ and $Z_{k}$ are complex constants, and $A_{k}$ are real constants. Therefore, from (31), (35), (38), and (48), we deduce that

$$\begin{aligned} & [U]_{L_{k}}=2 \pi [(\eta _{1}+\eta _{2})z X_{k}+\eta _{1} Y_{k}+ \overline{Z}_{k}], \\ & [\phi ]_{L_{k}}=2 \pi (\eta _{1}+\eta _{2})z X_{k}, \\ & [\varphi ]_{L_{k}}=0. \end{aligned}$$

Here, the notation $[]_{L_{k}}$ represents the change in the value of the function when surrounding the contour $L_{k}$ once in the conventional positive sense. Taking into account the fact that $u_{\alpha}$ and $\phi _{\alpha}$ must have unique values, we deduce the following conditions:

$$ X_{k}=0, \ \ \eta _{1} Y_{k}+\overline{Z}_{k}=0. $$

(50)

We denote the resultant of the stress vector applied to the contour as $S_{1}^{(k)}$, $S_{2}^{(k)}$. Based on equations (43), (44), and (48), we derive

$$ S_{1}^{(k)}+iS_{2}^{(k)}= - 2\pi ( 2\mu +\kappa )(\eta _{2}Y_{k}- \overline{Z}_{k}). $$

(51)

From (50) and (51), it follows that

$$ Y_{k}=-\frac{1}{2\pi}(S_{1}^{(k)}+i S_{2}^{(k)}); \ \ \ Z_{k}=-\eta _{1} \overline{Y}_{k}. $$

(52)

Therefore, from (49), we get

$$ \begin{aligned} &\Gamma (z)=-\frac{1}{2\pi}\sum _{k=1}^{n}(S_{1}^{(k)}+iS_{2}^{(k)})log(z-z_{k})+ \Gamma _{1}(z), \\ & \omega (z)=\frac{1}{2\pi}\eta _{1}\sum _{k=1}^{m}(S_{1}^{(k)}-iS_{2}^{(k)})log(z-z_{k})+ \omega _{1}(z). \end{aligned} $$

(53)

In the study of complex analysis and potential theory, unbounded domains and the behavior of functions at infinity play a crucial role. Theorem 1, stated below, provides significant insights into the behavior of functions defined on such domains, particularly in terms of their integral representations and asymptotic properties.

Theorem 1

Let Σ be an unbounded domain with the outlines $\delta _{1}$, $\delta _{2}$, ..., $\delta _{m}$ as internal bounded regions. Assuming the origin $z=0$ is exterior to Σ and $h_{\alpha}$, $t_{\alpha \beta}$, and $m_{\alpha \beta}$ are delimited in the vicinity of the limit point and for $|z|=\chi$ sufficiently large, we have

$$\begin{aligned} &\Gamma (z)= -\frac{1}{2\pi}(R_{1}+iR_{2})log z+(a_{1}+ia_{2})z+ \Gamma _{0}(z), \\ & \omega (z)=\frac{1}{2\pi}\eta _{1}(R_{1}-iR_{2})log z+(b_{1}+ib_{2}) z+ \omega _{0}(z), \\ & P(z,\overline{z})=\sum _{n=0}^{\infty }(P_{n} e^{in\theta}+ \overline{P}_{n} e^{-i n\theta})K_{n}(\tau \chi ), \\ & M(z, \overline{z})=\sum _{n=0}^{\infty} ( M_{n} e^{ i n \theta}+ \overline{M}_{n} e^{-n \theta}) K_{n}(w \chi ). \end{aligned}$$

(54)

In the previous theorem, $\Gamma _{0}$ and $\omega _{0}$ represent single-valued analytic functions on Σ including the limit at infinity, and we used the notations $a_{\alpha}$ and $b_{\alpha}$ for real constants, $P_{n}$ and $M_{n}$ for complex constants, $K_{n}$ for modified Bessel functions of order n and

$$ R_{\alpha}=\sum _{k=1}^{m} S_{\alpha}^{(k)}; \ \ \ z=r e^{i \theta}. $$

(55)

For sufficiently large $|z|$, we can express the functions $\Gamma _{0}$ and $\omega _{0}$ in the following form:

$$ \Gamma _{0}=\sum _{n=0}^{\infty} D_{n} z^{-n}, \ \ \ \ \omega _{0}(z)= \sum _{n=0}^{\infty} E_{n} z^{-n}.$$

(56)

We consider that

$$ \lim _{P \to \infty} t_{\alpha \beta} (P)=t_{\alpha \beta}^{*}. $$

From (40), (42), and (53), it follows that

$$\begin{aligned} &t_{11}^{*}=(2\mu +\kappa )(2\eta _{2} a_{1}-b_{1}), \\ &t_{22}^{*}=(2\mu +\kappa )(2\eta _{2} a_{1}+b_{1}), \\ & t_{12}^{*}=t_{21}^{*}=(2\mu +\kappa )b_{2}. \end{aligned}$$

(57)

The constant $b_{2}$ depends on the rigid rotation at infinity ϵ via

$$ a_{2}=(2\mu +\kappa )\varepsilon . $$

(58)

6 The stress that occurs around the hole

In this part of the paper, we will use the results obtained in the previous sections. Using boundary conditions and complex potentials, the following theorem allows the analysis of stress and deformation of materials with circular inclusions under external load conditions.

Theorem 2

Let $\Sigma _{1}=\{(x_{1}, x_{2})\in \mathbb{R}^{2}, x_{1}^{2}+x_{2}^{2}> \chi ^{2}\}$ be an unbounded domain with a circular hole centered at the origin and with radius χ. Assuming that an axial uniform stress acts on the body in the $x_{1}$ direction, at infinity, we have:

$$ t_{11}^{*}=Q, \ \ t_{12}^{*}=t_{21}^{*}=t_{22}^{*}=0, \ \ m_{\alpha 3 }^{*}=0, h_{\alpha}^{*}=0, $$

(59)

where Q is a given constant.

At the boundary of the hole, we have

$$\begin{aligned} & \eta _{2}[\Gamma (z)+z\overline{\Gamma}'(\overline{z})]+ \overline{\omega}(\overline{z})- 4i q_{1} \frac{\partial P}{\partial \overline{z}}+4 q_{2} \frac{\partial M}{\partial \overline{z}}=0, \\ & Im\{[2\gamma \frac{\partial P}{\partial z}-iw_{1}\Gamma ''(z)] \frac{dz}{ds}\}=0, \\ & Im\{[2d\frac{\partial M}{\partial z}-w_{2} \Gamma ''(z)] \frac{dz}{ds}\}=0, \ \ \textit{for} \ \ |z|=\chi , \end{aligned}$$

(60)

where the components of the complex potentials are as follows:

$$\begin{aligned} &\Gamma (z)=\frac{1}{4\eta _{2}(2\mu +\kappa )}Qz+\frac{1}{z}D_{1}, \\ &\omega (z)=-\frac{1}{2(2\mu +\kappa )}Qz+\frac{1}{z}E_{1}+ \frac{1}{z^{3}}E_{3}, \\ &P(z, \overline{z})=iH_{1}\Big(\frac{z}{\overline{z}}- \frac{\overline{z}}{z}\Big)K_{2}(\tau \chi ), \\ & M(z, \overline{z})=H_{2} \Big(\frac{z}{\overline{z}}+ \frac{\overline{z}}{z}\Big)K_{2}(w \chi ), \ \ \ \chi =(z \overline{z})^{{1}/{2}}, \end{aligned}$$

(61)

where

$$\begin{aligned} & D_{1}=\frac{1}{2 (2\mu +\kappa )F}Q \xi ^{2}, \ \ \ E_{1}=- \frac{1}{2(2\mu +\kappa )}Q\chi ^{2}, \\ & E_{3}=\frac{1}{2(2\mu +\kappa )F}[\eta _{2}+1q_{1}\chi \tau T K_{3}( \tau \chi )+2 q_{2}\chi m H K_{3}(m \chi ), \\ & H_{1}=TD_{1}, \ \ H_{2}= HD_{1}, \ \ F= \eta _{2}+2q_{1} \tau a_{2} T K_{1}(\tau \chi )+ 2q_{2}m\chi K_{3}(m \chi ), \\ & T=\frac{4}{2\chi ^{4} \omega}\{8 d K_{2}(m\chi )+ 2m \chi \xi [K_{1}(m \chi )+K_{3}(\tau \chi )]\}, \\ & H=\frac{4}{3\chi ^{4} \omega}\{8 \gamma K_{2}(\tau \chi )-2\tau \chi \xi [K_{1}(\tau \chi )+K_{3}(\tau \chi )]\}, \\ & \Omega =\frac{16 d\gamma}{\chi ^{2}}K_{2}(\tau \chi )K_{2}(m \chi )+ 4m \tau \xi ^{2}[K_{1}(m \chi )+K_{3}(m\chi )][K_{1}(\tau \chi )+K_{3}( \tau \chi )]. \end{aligned}$$

(62)

We note that $R_{1}$ and $R_{2}$ are 0, and in the case of stress analysis, we can consider that $a_{2}$, $D_{0}$, $E_{0}$. From (57) and (59), we find that

$$ a_{1}=\frac{1}{4\eta _{2}(2\mu +\kappa )}Q, \ \ \ b_{1}=- \frac{1}{2(2\mu +\kappa )}Q, \ \ \ b_{2}=0. $$

(63)

In this case, the solution has the forms (54) and (56). Using them and (60), we get

$$ e^{i\theta}U= u_{\chi}+iu_{\theta}, $$

where the components $u_{\chi }$ and $u_{\theta}$ are in polar coordinates. From (31), (35), (38), and (63), it follows that

$$\begin{aligned} & u_{\chi}+iu_{\theta}= \frac{\eta _{1}-\eta _{2}}{4\eta _{2}(2\mu +\kappa )}Q\chi - \frac{1}{\chi}(E_{1}-\eta _{1}D_{1})+u cos2\theta +ivsin2\theta , \\ &\phi =-2[H_{1}K_{2}(\tau \chi )- \frac{\kappa}{2\gamma \tau ^{2}(\mu +\kappa )\chi ^{2}}D_{1}]sin2 \theta , \\ &\varphi =- \frac{\xi Q}{4dm^{2}\eta _{2}(2\mu +\lambda +\kappa )(2\mu +\kappa )}+2[H_{2}K_{2}(m \chi )+\frac{\xi}{2dm^{2}(2\mu +\lambda +\kappa )\chi ^{2}}]cos2 \theta , \end{aligned}$$

where

$$\begin{aligned} & u=\frac{1}{\chi} \eta _{2} D_{1}+\frac{Q \chi}{2(2\mu +\kappa )}- \frac{1}{\chi ^{3}}E_{3}+\frac{2}{\chi}q_{1}H_{1}K_{2}(\tau \chi )+2m q_{2}H_{2}[K_{1}(m \chi )+K_{3}(m\chi )], \\ & v=\frac{1}{\chi} \eta _{2}D_{1}-\frac{Q\chi}{2(2\mu +\kappa )}- \frac{1}{r^{3}}E_{3}+2q_{1}H_{1}[K_{1}(\tau \chi )+K_{3}(\tau \chi )]+ \frac{8}{rq_{2}}H_{2}K_{2}(m\chi ). \end{aligned}$$

Similarly, using (40) and (62), we obtain the stresses.

7 Numerical simulation

The graphs below (Figs. 1–4) correspond to an isotropic magnesium crystal with pores and are obtained using the “Wolfram Mathematica” computing system. The values used can be found in [9]. The acquired plots represent the real and imaginary parts of the $\Gamma (z)$ and $\omega (z)$ potentials, which can be found in Sect. 5. These allow us to visualize the variation of potentials in the complex. More precisely, these graphs help to understand the stress and displacement fields in the material, identify critical regions, and predict the behavior of the material under various conditions.

Figure 1

The behavior of $Re(\Gamma (z))$ and $Im(\Gamma (z))$

Figure 2

The behavior of $Re(\omega (z))$ and $Im(\omega (z))$

Figure 3

The behavior of $Re(u)$ and $Im(u)$

Figure 4

The behavior of $Re(v)$ and $Im(v)$

In the representation, $Im(\Gamma (z))$, a peak located in the center can be observed, which indicates a region where the imaginary part has a significant value. This may be due to stress concentration or a singularity in the material. In both representations, flat regions suggest slow changes in those areas. There is a visible central valley in the $Re(\Gamma (z))$ representation, which indicates a negative value and may suggest comprehensive stresses or regions of low potential.

Regarding the corresponding plots of $Re(\omega (z))$ and $Im(\omega (z))$, peaks and valleys can also be observed, indicating peaks of maximum and minimum displacement. Microrotations affect the displacement and strain fields, leading to more complex patterns in the graphics. Sharp peaks suggest the presence of singularities or displacement concentration points, possibly near pores or defects in the material. These plots help to understand the displacement and strain fields in the material, identify critical regions, and predict the behavior of the material under various conditions.

The following four graphs correspond to the imaginary and real parts of the radial displacement component “u” and the tangential displacement component “v” of Sect. 6. The axis corresponding to χ represents the radial distance from the center of the circular hole, and the corresponding axis of θ represents the angular coordinates around the hole. The plot of $\text{Re}[u(\chi , \theta )]$ for a porous isotropic magnesium crystal helps visualize the material’s response, highlighting areas of stress concentration and weakened stiffness due to porosity, essential for informed material design and application. The imaginary part of the function $u(\chi , \theta )$ is uniform, indicating consistent damping or phase effects throughout the material. This uniformity can be crucial for applications where stable phase characteristics are desired. The plot for the real part of $v(\chi , \theta )$ shows how the tangential displacement varies with χ and θ, highlighting areas of stress concentration and informing material design strategies. The plot for the imaginary part of $v(\chi , \theta )$ is similar to $Im(u(\chi , \theta ) )$.

8 Conclusion

Graphs representing complex potentials and stress and displacement distributions in a material have many practical applications in various fields of engineering and materials science. They are powerful tools in the analysis and design of materials and structures. They allow a deep understanding of the mechanical behavior of materials, identifying critical points and optimizing the design for superior performance and safety. These techniques are essential in a wide range of industries, from structural and aerospace engineering to bioengineering and scientific research.