In term of displacements (**u**) Navier equation is written as:

$$\begin{array}{}\mu {\mathrm{\nabla}}^{2}\mathbf{u}+\left(\lambda +\mu \right)\mathrm{\nabla}\left(\mathrm{\nabla}\cdot \mathbf{u}\right)=\mathbf{0}\end{array}$$(1)

where*λ* and *μ* are the Lamé’s constants [1]. Using PN formulation, Eq. (1) is represented by four potential functions, named displacement potentials, in the form

$$\begin{array}{}{\displaystyle 2\mu \mathbf{u}=\varphi -\mathrm{\nabla}\left({\varphi}_{0}+\frac{\mathbf{R}\cdot \varphi}{4(1-\nu )}\right),}\end{array}$$(2a)

$$\begin{array}{}{\mathrm{\nabla}}^{2}\varphi =\mathbf{0},\end{array}$$(2b)

$$\begin{array}{}{\mathrm{\nabla}}^{2}{\varphi}_{0}=0\end{array}$$(2c)

where *φ* is a harmonic vector potential, *φ*_{0} a harmonic scalar potential, **R -** the position vector and *ν* - the Poisson ratio. Eqs. (2a, 2b, 2c) are a general solution of Eq. (1) and have proved to be complete for the general case [22].

In rectangular coordinates, the completed PN solutions for the 2D-elasticity plane strain case reduce the displacement field from Eq. (2a) to

$$\begin{array}{}{\displaystyle 2\mu {u}_{x}=-\frac{\mathrm{\partial}{\varphi}_{0}}{\mathrm{\partial}x}+\frac{3-4\nu}{4\left(1-\nu \right)}{\varphi}_{x}-\frac{1}{4\left(1-\nu \right)}\left(x\frac{\mathrm{\partial}{\varphi}_{x}}{\mathrm{\partial}x}+y\frac{\mathrm{\partial}{\varphi}_{y}}{\mathrm{\partial}x}\right)}\end{array}$$(3a)

$$\begin{array}{}{\displaystyle 2\mu {u}_{y}=-\frac{\mathrm{\partial}{\varphi}_{0}}{\mathrm{\partial}y}+\frac{3-4\nu}{4\left(1-\nu \right)}{\varphi}_{y}-\frac{1}{4\left(1-\nu \right)}\left(x\frac{\mathrm{\partial}{\varphi}_{x}}{\mathrm{\partial}y}+y\frac{\mathrm{\partial}{\varphi}_{y}}{\mathrm{\partial}y}\right)}\end{array}$$(3b)

with *u*_{z} = *∂/∂z* = 0, *u*_{x} = *u*_{x}(*x*,*y*) and *u*_{y} = *u*_{y}(*x*,*y*).

The corresponding governing Eq. (2b, 2c) can be written as

$$\begin{array}{}{\displaystyle \frac{{\mathrm{\partial}}^{2}{\varphi}_{0}}{\mathrm{\partial}{x}^{2}}+\frac{{\mathrm{\partial}}^{2}{\varphi}_{0}}{\mathrm{\partial}{y}^{2}}=0}\end{array}$$(4a)

$$\begin{array}{}{\displaystyle \frac{{\mathrm{\partial}}^{2}{\varphi}_{x}}{\mathrm{\partial}{x}^{2}}+\frac{{\mathrm{\partial}}^{2}{\varphi}_{x}}{\mathrm{\partial}{y}^{2}}=0}\end{array}$$(4b)

$$\begin{array}{}{\displaystyle \frac{{\mathrm{\partial}}^{2}{\varphi}_{y}}{\mathrm{\partial}{x}^{2}}+\frac{{\mathrm{\partial}}^{2}{\varphi}_{y}}{\mathrm{\partial}{y}^{2}}=0}\end{array}$$(4c)

and finally, for the more general case, the mixed physical boundary conditions can be expressed in the form:

$$\begin{array}{}{u}_{i}={u}_{i}^{b}\phantom{\rule{thickmathspace}{0ex}}\mathrm{o}\mathrm{n}\phantom{\rule{thickmathspace}{0ex}}{S}_{u}\phantom{\rule{thickmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thickmathspace}{0ex}}{\sigma}_{ij}{n}_{j}={t}_{i}^{b}\phantom{\rule{thickmathspace}{0ex}}\mathrm{o}\mathrm{n}\phantom{\rule{thickmathspace}{0ex}}{S}_{t}\end{array}$$(5)

*S*_{u} denotes the boundary surface points where the displacement,
$\begin{array}{}{u}_{i}^{b}\end{array}$, is prescribed, while *S*_{t} refers to points where the traction values,
$\begin{array}{}{t}_{i}^{b}\end{array}$, are imposed. *σ*_{ij} is the stress field and *n*_{j} the outer normal vector on *S*_{t}. Note that *S* = *S*_{t}+ *S*_{u} represents the complete boundary surface, Figure 1.

Figure 1 Scheme of the boundary conditions: tractions and displacements [14]

The expressions that relate stress and potential functions, which are required for implementing the boundary condition (5), are [1]:

$$\begin{array}{}{\sigma}_{xx}=\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& {\displaystyle -\frac{{\mathrm{\partial}}^{2}{\varphi}_{0}}{\mathrm{\partial}{x}^{2}}-\left(x\frac{{\mathrm{\partial}}^{2}{\varphi}_{x}}{\mathrm{\partial}{x}^{2}}+y\frac{{\mathrm{\partial}}^{2}{\varphi}_{y}}{\mathrm{\partial}{x}^{2}}\right)\frac{1}{4\left(1-\nu \right)}}\\ & {\displaystyle +\frac{\mathrm{\partial}{\varphi}_{x}}{\mathrm{\partial}x}\frac{1}{2}+\frac{\mathrm{\partial}{\varphi}_{y}}{\mathrm{\partial}y}\frac{\nu}{2\left(1-\nu \right)}}\end{array}$$(6a)

$$\begin{array}{}{\sigma}_{yy}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& {\displaystyle -\frac{{\mathrm{\partial}}^{2}{\varphi}_{0}}{\mathrm{\partial}{y}^{2}}-\left(x\frac{{\mathrm{\partial}}^{2}{\varphi}_{x}}{\mathrm{\partial}{y}^{2}}+y\frac{{\mathrm{\partial}}^{2}{\varphi}_{y}}{\mathrm{\partial}{y}^{2}}\right)\frac{1}{4\left(1-\nu \right)}}\\ & {\displaystyle +\frac{\mathrm{\partial}{\varphi}_{y}}{\mathrm{\partial}y}\frac{1}{2}+\frac{\mathrm{\partial}{\varphi}_{x}}{\mathrm{\partial}x}\frac{\nu}{2\left(1-\nu \right)}}\end{array}$$(6b)

$$\begin{array}{}{\sigma}_{xy}=\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& {\displaystyle -\frac{{\mathrm{\partial}}^{2}{\varphi}_{0}}{\mathrm{\partial}x\mathrm{\partial}y}-\left(x\frac{{\mathrm{\partial}}^{2}{\varphi}_{x}}{\mathrm{\partial}x\mathrm{\partial}y}+y\frac{{\mathrm{\partial}}^{2}{\varphi}_{y}}{\mathrm{\partial}x\mathrm{\partial}y}\right)\frac{1}{4\left(1-\nu \right)}}\\ & {\displaystyle +\left(\frac{\mathrm{\partial}{\varphi}_{x}}{\mathrm{\partial}y}+\frac{\mathrm{\partial}{\varphi}_{y}}{\mathrm{\partial}x}\right)\frac{1-2\nu}{4\left(1-\nu \right)}}\end{array}$$(6c)

For each of the derived solutions, the resulting governing equations and boundary conditions are obtained by omitting one of the rectangular components, *φ*_{x} or *φ*_{y}, in Eqs. (3-6). Note that, since the physical boundary conditions for a given problem are unique, the problem can be formulated by different sets of PDEs, two governing equations for each derived solution or three governing equations for the complete PN solution.

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