Solid Mechanics

Basic equations of elasticity

To determine the distribution of static stresses and displacements in a solid body we must obtain a solution (either analytically or numerically) to the basic equations of the theory of elasticity, satisfying the boundary conditions on forces and/or displacements. For a general three dimensional solid object these equations are:

• strain-displacement equations (6)
• stress-strain equations (6)
• equations of equilibrium (3)

where the number in brackets indicates the number of equations. We see there are 15 equations for 15 unknown variables (6 strains, 6 stresses and 3 displacements). In two dimensions this simplifies to 8 equations with 2 displacements, 3 stresses, and 3 strains.

Strain-displacement equations

Under the action of applied forces, a point in the solid originally at $(x,y,z)$ moves to position $(X,Y,Z)$. This movement can be described completely by the displacement vector \begin{equation} \boldsymbol{u(\boldsymbol{x})} = \{u_x(x,y,z) \quad u_y(x,y,z) \quad u_z(x,y,z)\}, \end{equation} the brackets $\{\}$ indicating that this is a column vector.

Stress-strain equations

Two-dimensional stress distributions

Many problems in elasticity can be simplified as two-dimensional problems described by plane theory of elasticity. In general there are two types of problems we may encounter in plane analysis: plane stress and plane strain. The first problem arises in analysis of thin plates loaded in the plane of the plate, while the second is used for elongated bodies of constant cross section.

Plane stress

Plane stress distributions build on the assumption that the normal stress and shear stresses directed perpendicular to the $x$-$y$ plane are assumed zero: \begin{equation}\label{eq:pstress_assump} \sigma_{zz} = \sigma_{zx} = \sigma_{zy} = 0. \end{equation} It is also assumed that the stress components do not vary through the thickness of the plate (the assumptions do violate some compatibility conditions, but are still sufficiently accurate for practical applications if the plate is thin).

Using (\ref{eq:pstress_assump}) the three-dimensional Hooke's law can be reduced to: \begin{equation} \begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy} \end{bmatrix} = \frac{E}{1-\nu^2} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1-\nu}{2} \end{bmatrix} \begin{bmatrix} \epsilon_{xx} \\ \epsilon_{yy} \\ \epsilon_{xy} \end{bmatrix} \end{equation}

Plane strain

The plane strain problem arises in analysis of walls, dams, tunnels where one dimension of the structure is very large in comparison to the other two dimensions ($x$- and $y$- coordinates). It is also appropriate for small-scale problems such as bars and rollers compressed by forces normal to their cross section. In all such problems the body may be imagined as a prismatic cylinder with one dimension much larger that the other two. The applied forces act in the $x$-$y$ plane and do not vary in the $z$ direction, leading to the assumption \begin{equation} \frac{\partial}{\partial z} = u_z = 0. \end{equation}

With the above assumption it follows immediately that \begin{equation} \varepsilon_{zz} = \varepsilon_{zx} = \varepsilon_{zy} = 0. \end{equation} The three dimensional Hooke's law can now be reduced to \begin{equation} \sigma \end{equation}

Point contact on a 2D half-plane

A starting point to solve problems in contact mechanics is to understand the effect of a point-load applied to a homogeneous, linear elastic, isotropic half-plane. This problem may be defined either as plane stress or plain strain (for solution with FreeFem++ we have choosen the latter). The traction boundary conditions for this problem are: \begin{equation}\label{eq:bc} \sigma_{xy}(x,0) = 0, \quad \sigma_{yy}(x,y) = -P\delta(x,y) \end{equation} where $\delta(x,y)$ is the Dirac delta function. Together these boundary conditions state that there is a singular normal force $P$ applied at $(x,y) = (0,0)$ and there are no shear stresses on the surface of the elastic half-plane.

The analytical relations for the stresses can be found from the Flamant solution (stress distributions in a linear elastic wedge loaded by point forces a the tip. When the "wedge" is flat we get a half-plane. The derivation uses polar coordinates.) and are given as: \begin{equation} \sigma_{xx} = -\frac{2P}{\pi} \frac{x^2y}{\left(x^2+y^2\right)^2}, \end{equation} \begin{equation} \sigma_{xy} = -\frac{2P}{\pi} \frac{y^3}{\left(x^2+y^2\right)^2}, \end{equation} \begin{equation} \sigma_{yy} = -\frac{2P}{\pi} \frac{xy^2}{\left(x^2+y^2\right)^2}, \end{equation} for some point $(x,y)$ in the half-plane. From this stress field the strain components and thus the displacements $(u,v)$ can be determined. The displacements are given by \begin{align} u &= -\frac{P}{4\pi\mu}\left((\kappa-1)\theta - \frac{2xy}{r^2}\right), \label{eq:dispx}\\ v &= -\frac{P}{4\pi\mu}\left((\kappa+1)\log r - \frac{2y^2}{r^2}\right), \label{eq:dispy} \end{align} where $$r = \sqrt{x^2+y^2}$$ and $$\tan \theta = \frac{x}{y}.$$ The symbol $\kappa$ is known as Dundars constant and is defined as \[ \kappa = \begin{cases} 3 - 4\nu & \qquad \text{plane strain}, \\ \cfrac{3 - \nu}{1+\nu} & \qquad \text{plane stress}. \end{cases} \] The last remaining symbol is $\mu$ which represents the shear modulus (sometimes also denoted with $G$).

Numerical solution with FreeFem++

Due to the known analytical solution the point-contact problem can be used for benchmarking numerical PDE solvers in terms of accuracy (as well as computational efficiency). The purpose of this section is to compare the numerical solution obtained by FreeFem++ with the analytical solution, as well as provide a reference numerical solution for the C++ library developed in our laboratory.

For purposes of simplicity we limit ourselves to the domain $(x,y) \in \Omega = [-1,1] \times[-1,-0.1]$ and prescribe Dirichlet displacement on the boundaries $\Gamma_D$ from the known analytical solution (\ref{eq:dispx}, \ref{eq:dispy}). This way we avoid having to deal with the Dirac delta traction boundary condition (\ref{eq:bc}).

To solve the point-contact problem in FreeFem++ we must first provide the weak form of the balance equation: \begin{equation} \boldsymbol{\nabla}\cdot\boldsymbol{\sigma} + \boldsymbol{b} = 0. \end{equation} The corresponding weak formulation is \begin{equation} \int_\Omega \boldsymbol{\sigma} : \boldsymbol{\varepsilon}(\boldsymbol{v}) \, d\Omega - \int_\Omega \boldsymbol{b}\cdot\boldsymbol{v}\,d\Omega = 0 \end{equation} where $:$ denotes the tensor scalar product (tensor contraction), i.e. $\boldsymbol{A}:\boldsymbol{B} =\sum_{i,j} A_{ij}B_{ij}$. The vector $\boldsymbol{v}$ is the test function or so-called "virtual displacement".

Convergence studies

File:Convergence.png

Figure 1: Convergence results for the point contact problem. The different colours represent different finite elements.

For the convergence between analytical and numerical solution we vary the number of nodes by increasing the grid size in both $x$- and $y$- directions simultaneously by powers of two from $2^2$ (16 nodes all together) to $2^7$ (16384 nodes all together). The $L^2$ error norm is used to measure the "difference" between solutions. Since the displacements are the variable that obtain from FreeFem++, we use the displacement magnitude $|\boldsymbol{u}| = \sqrt{u_x^2+u_y^2}$ to define our $L^2$-error norm. The exact equation we have used is \begin{equation} L^2\text{-norm} = \sqrt{\frac{\int_\Omega (|\boldsymbol{u_{\text{numerical}}}|-|\boldsymbol{u_{\text{analytical}}}|)^2d\Omega}{\int_\Omega|\boldsymbol{u_{\text{analytical}}}|^2d\Omega}}. \end{equation}