## PHY454H1S Continuum Mechanics. Lecture 6: Compatibility condition and elastostatics. Taught by Prof. K. Das.

Posted by peeterjoot on January 29, 2012

# Disclaimer.

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

# Review: Elastostatics

We’ve defined the strain tensor, where assuming the second order terms are ignored, was

We’ve also defined a stress tensor defined implicitly as a divergence relationship using the force per unit volume in direction

We’ve also discussed the constitutive relation, relating stress and strain .

We’ve also discussed linear constitutive relationships (Hooke’s law).

# 2D strain.

From 2.1 we see that we have

We have a relationship between these displacements (called the compatibility relationship), which is

We find this by straight computation

and

Now, looking at the cross term we find

This is called the compatibility condition, and ensures that we don’t have a disjoint deformation of the form in figure (\ref{fig:continuumL6:continuumL6fig1})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL6fig1}

\caption{disjoint deformation illustrated.}

\end{figure}

# 3D strain.

While we have 9 components in the tensor, not all of these are independent. The sets above and below the diagonal can be related, as illustrated in figure (\ref{fig:continuumL6:continuumL6fig2}).

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{continuumL6fig2}

\caption{continuumL6fig2}

\end{figure}

Here we have 6 relationships between the components of the strain tensor . Deriving these will be assigned in the homework.

# Elastodynamics. Elastic waves.

Reading: Chapter III (section 22 – section 26) of the text [1].

Example: sound or water waves (i.e. waves in a solid or liquid material that comes back to its original position.)

\begin{definition}

\emph{(Elastic Wave)}

An elastic wave is a type of mechanical wave that propagates through or on the surface of a medium. The elasticity of the material provides the restoring force (that returns the material to its original state). The displacement and the restoring force are assumed to be linearly related.

\end{definition}

In symbols we say

and specifically

This is just Newton’s second law, , but expressed in terms of a unit volume.

Should we have an external body force (per unit volume) acting on the body then we must modify this, writing

Note that we are separating out the “original” forces that produced the stress and strain on the object from any constant external forces that act on the body (i.e. a gravitational field).

With

we can expand the stress divergence, for the case of homogeneous deformation, in terms of the Lam\’e parameters

We compute

We find, for homogeneous deformations, that the force per unit volume on our element of mass, in the absence of external forces (the body forces), takes the form

This can be seen to be equivalent to the vector relationship

TODO: What form do the stress and strain tensors take in vector form?

# 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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