Asymptotic Analysis for Plane Stress Problems

Abstract

In this classroom note, the old and well-known plane-stress elastic class of problems is revisited, using an analysis technique which is different than that commonly found in the literature, and with a pedagogical benefit. An asymptotic analysis is applied to problems of thin linear elastic plates, made of a homogeneous and rather general anisotropic material, under the plane stress assumption. It is assumed that there are no body forces, that the boundary conditions are uniform over the thickness, and that the material (hence also the solution) is symmetric about the middle plane. The small parameter in this analysis is $\epsilon =t/D$ where $t$ is the (uniform) thickness of the plate and $D$ is a measure of its overall size. The goal of this analysis is to show how the three-dimensional (3D) problem of this type is reduced asymptotically to a sequence of essentially two-dimensional (2D) problems for a small $\epsilon$. As expected, the leading problem in this sequence is shown to be the classical plane-stress problem. The solutions of the higher-order problems are corrections to the plane-stress solution. The analysis also shows that all six 3D compatibility equations are satisfied as $\epsilon$ goes to zero, and that the error incurred by the plane stress assumption is $O(\epsilon ^{2})$. For the special case of an isotropic in-the-plane material, the second-order solution is shown to be the exact solution of the 3D problem, up to an $O(\epsilon ^{2})$ error in the close vicinity of the edge (which agrees with a well-known result for an isotropic material).

Appendices

Appendix A: $\sigma ^{0}_{{{\alpha \alpha }},{\beta \beta }}=0$ for an Isotropic-in-the-Plane Material

The stresses $\sigma ^{0}_{{{\alpha \beta }}}$ satisfy $\mbox{E}^{0}_{\alpha }$ and $\mbox{C}^{0}_{1}$. The latter is

$$ {\mathrm {C}}^{0}_{1}:\quad \sigma ^{0}_{{11},{22}} + \sigma ^{0}_{{22},{11}} - \nu _{p} (\sigma ^{0}_{{11},{11}} + \sigma ^{0}_{{22},{22}}) - 2(1+ \nu _{p})\sigma ^{0}_{{12},{12}} = 0 $$

(50)

for a material that is isotropic in the plane $(x_{1},x_{2})$. From $\mbox{E}^{0}_{\alpha }$ we have

$$ \sigma ^{0}_{{12},{2}}=-\sigma ^{0}_{{11},{1}} \ ,\quad \sigma ^{0}_{{12},{1}}=- \sigma ^{0}_{{22},{2}} \ .$$

(51)

Differentiating the two equations by $x_{1}$ and $x_{2}$, respectively, we have

$$ \sigma ^{0}_{{12},{12}}=-\sigma ^{0}_{{11},{11}}=-\sigma ^{0}_{{22},{22}} \ ,$$

(52)

from which we also have

$$ \sigma ^{0}_{{12},{12}}=-\frac{1}{2}\ (\sigma ^{0}_{{11},{11}}+\sigma ^{0}_{{22},{22}}) \ .$$

(53)

Substituting (53) in (50), we find that the terms with $\nu _{p}$ cancel out, and we are left with

$$ \sigma ^{0}_{{11},{22}} + \sigma ^{0}_{{11},{11}} + \sigma ^{0}_{{22},{22}} + \sigma ^{0}_{{22},{11}} = 0\ ,$$

(54)

which can be written as $\sigma ^{0}_{{{\alpha \alpha }},{\beta \beta }}=0$.

Appendix B: 3D Numerical Calculation

It may be useful for students to see a demonstration of the theory via a solution to a specific problem. To this end, a 3D model of a thin plate of an isotropic material under planar loading was analyzed by the Finite Element (FE) software Ansys® [16]. See Fig. 2(a) for an illustration of the model. The dimensions of the plate are $200 \times 40 \times 4$, namely here $\epsilon =4/40=0.1$. (A thinner plate would have served as a better demonstration of the theory, but would have been more difficult to solve numerically with good accuracy.) A tensile traction $\sigma _{11}$ is applied on the left and right sides of the plate, and is piecewise linear (a “hat” function) in the width direction $x_{2}$. The load is distributed uniformly over the thickness. The FE model exploits the triple symmetry of the problem and includes only one eighth of the plate, using symmetry boundary conditions over the three symmetry planes. The model included 20,000 trilinear hexahedral elements.

Fig. 2

Three-dimensional finite element analysis of a thin isotropic plate with planar loading: (a) illustration of the model, (b) numerical result for the longitudinal stress $\sigma _{11}$ at the left part of the strip EFCD

Consider the (half) strip EFCD, of thickness $t/2$. The longitudinal stress field $\sigma _{11}$ in the left part of this strip, i.e., in the vicinity of the loaded side DE, is shown in Fig. 2(b). The stress is seen to remain uniformly distributed away from the loaded side (DE) up to a distance of about $5t$, where a non-uniform distribution develops. Taking into account the symmetry of this field over the midline EF, a parabola-like distribution is observed, in accordance with the theory.