## PHY454H1S Continuum Mechanics. Lecture 5: Constitutive relationship. Taught by Prof. K. Das.

Posted by peeterjoot on January 28, 2012

# Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

# Review: Cauchy Tetrahedron.

Referring to figure (\ref{fig:continuumL5:continuumL5fig1})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL5fig1}

\caption{Cauchy tetrahedron direction cosines.}

\end{figure}

recall that we can decompose our force into components that refer to our direction cosines

Or in tensor form

We call this the traction vector and denote it in vector form as

# Constitutive relation.

Reading: section 2, section 4 and section 5 from the text [1].

We can find the relationship between stress and strain, both analytically and experimentally, and call this the Constitutive relation. We prefer to deal with ranges of distortion that are small enough that we can make a linear approximation for this relation. In general such a linear relationship takes the form

Consider the number of components that we are talking about for various rank tensors

latex 0^\text{th}$ rank tensor} & \mbox{ components} \\ \mbox{ rank tensor} & \mbox{ components} \\ \mbox{ rank tensor} & \mbox{ components} \\ \mbox{ rank tensor} & \mbox{ components}\end{array}\end{aligned} \hspace{\stretch{1}}(3.7)$

We have a lot of components, even for a linear relation between stress and strain. For isotropic materials we model the constitutive relation instead as

For such a modeling of the material the (measured) values and (shear modulus or modulus of rigidity) are called the Lam\’e parameters.

It will be useful to compute the trace of the stress tensor in the form of the constitutive relation for the isotropic model. We find

or

We can now also invert this, to find the trace of the strain tensor in terms of the stress tensor

Substituting back into our original relationship 3.8, and find

which finally provides an inverted expression with the strain tensor expressed in terms of the stress tensor

## Special cases.

### Hydrostatic compression

Hydrostatic compression is when we have no shear stress, only normal components of the stress matrix is nonzero. Strictly speaking we define Hydrostatic compression as

i.e. not only diagonal, but with all the components of the stress tensor equal.

We can write the trace of the stress tensor as

Now, from our discussion of the strain tensor recall that we found in the limit

allowing us to express the change in volume relative to the original volume in terms of the strain trace

Writing that relative volume difference as we find

or

where is called the Bulk modulus.

### Uniaxial stress

Again illustrated in the plane as in figure (\ref{fig:continuumL5:continuumL5fig2})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL5fig2}

\caption{Uniaxial stress.}

\end{figure}

Expanding out 3.12 we have for the element of the strain tensor

or

where is Young’s modulus. Young’s modulus in the text (5.3) is given in terms of the bulk modulus . Using we find

FIXME: figure (\ref{fig:continuumL5:continuumL5fig3}) reference?

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL5fig3}

\caption{stress associated with Young’s modulus}

\end{figure}

We define Poisson’s ratio as the quantity

Note that we are still talking about uniaxial stress here. Referring back to 3.12 we have

Recall (3.20) that we had

Inserting this gives us

so

We can also relate the Poisson’s ratio to the shear modulus

These ones are (5.14) in the text, and are easy enough to verify (not done here).

### Appendix. Computing the relation between Poisson’s ratio and shear modulus.

Young’s modulus is given in 3.21 (equation (43) in the Professor’s notes) as

and for Poisson’s ratio 3.24 (equation (46) in the Professor’s notes) we have

Let’s derive the other stated relationships (equation (47) in the Professor’s notes). I get

or

For substitution into the Young’s modulus equation calculate

and

Putting these together we find

Rearranging we have

This matches (5.9) in the text (where is used instead of ).

We also find

# References

[1] L.D. Landau, EM Lifshitz, JB Sykes, WH Reid, and E.H. Dill. Theory of elasticity: Vol. 7 of course of theoretical physics. 1960.

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