Fundamentals

Expansion and Stiffness of Thin Adhesive or Rubber Layers


Thin layers of adhesive, plastic, or rubber are often employed in precision machines for joining, shimming, and sealing. These layers are often the most compliant and most dimensionally unstable elements of an assembly, so it is important to understand their behavior.

thinlayer

Consider a thin layer of relatively compliant material sandwiched between two rigid parts. If the layer is thin compared to its width, we can think of the strains \epsilon_x and \epsilon_y as being restrained to zero except near the edges. This is a state of \emph{uniaxial strain} in the z direction. As we show below, the stiffness and thermal expansion in uniaxial strain are often considerably larger than they are for uniaxial stress.

Hooke’s law for an isotropic solid is written as

    \begin{eqnarray*} E (\epsilon_{x} - \epsilon_T ) & = & \sigma_{x} - \nu ( \sigma_{y} + \sigma_{z} ) \\ E (\epsilon_{y} - \epsilon_T ) & = & \sigma_{y} - \nu ( \sigma_{x} + \sigma_{z} ) \\ E (\epsilon_{z} - \epsilon_T ) & = & \sigma_{z} - \nu ( \sigma_{x} + \sigma_{y} ) \end{eqnarray*}

The longitudinal strains \epsilon_x, \epsilon_y, and \epsilon_z are related to the longitudinal stresses \sigma_x, \sigma_y, and \sigma_z by the Young’s modulus E and Poisson’s ratio \nu. The strain \epsilon_T could arise from temperature changes, absorption or desorption of humidity, or changes in molecular structure of an adhesive during curing.

If \epsilon_T arises only from temperature change, it is common to replace \epsilon_T by a linear coefficient of thermal expansion (CTE) times a temperature change. But many polymers as well as low-expansion materials (for example Invar, ULE, or Zerodur) have nonlinear thermal expansion relationships, so we prefer to retain the more general \epsilon_T in our models.

Setting \epsilon_x and \epsilon_y to zero in Hooke’s law and combining the equations to eliminate \sigma_x and \sigma_y, we obtain Hooke’s law for uniaxial strain in the form

(1)   \begin{equation*} \frac{E(1-\nu)}{1-\nu-2\nu^2} \left( \epsilon_z - \frac{1+\nu}{1-\nu} \epsilon_T \right) = \sigma_z \end{equation*}

We can think of the leading coefficient as an effective modulus for uniaxial strain:

(2)   \begin{equation*} \hat{E} = \frac{E(1-\nu)}{1-\nu-2\nu^2} = \frac{E(1-\nu^2)}{1 - 3\nu^2 - 2\nu^3} \end{equation*}

The difference between the stiffnesses for uniaxial strain and uniaxial stress is usually significant. A Poisson’s ratio of 0.4 is common for plastics and adhesives, for which \hat{E} \approx 2.1 E.

Elastomers (rubbers) are nearly incompressible, so that \nu is nearly 0.5. If \nu is exactly 0.5, \hat{E} goes to infinity, but we can get a more realistic estimate by setting \nu = 0.5-\nu^* and expanding for small \nu^* to obtain

    \[ \hat{E} \approx \frac{E}{6 \nu^*} \]

For example, if \nu = 0.49, then \nu^* =0.01 and \hat{E} \approx 17 E. Thus a thin sheet of rubber is much stiffer in the normal direction than it is in shear, and this can be used to advantage in designing a class of flexural bearings. (For such applications, the value of \nu^* is very important, and quoted values of \nu may not be accurate enough. It is better for such cases to work from the bulk modulus if it is available.)

The effects of material expansion \epsilon_T are also amplified by the Poisson’s ratio. For example if, \nu =0.4, the coefficient of the strain \epsilon_T in Eq.~(1) is about 2.3, so that thermal expansion in the thin layer constrained in x and y results in more than twice as much thickness change than would be obtained if the thin layer were not restrained.