Energy at a discontinuity surface

As before, we are free to rewrite the above integral relationships in a form which incorporates the boundary conditions occurring at a material discontinuity surface:

$$H = \oint_S - {u}_{} \cdot {F}_{} ds + \oint_{\Sigma} \lbra... ... {\sigma}_{T} )_1 + ( {u}_{T} \cdot {\sigma}_{T} )_2 \rbrace ds$$ (9)

$\oint_{\Sigma}$ stands for an integral over all points in the medium which exhibit material and thus stress and strain discontinuities. u_N and u_T are displacements at the discontinuity interface, generated by the normal and tangential strains in effect on an infinitesimal surface element. For convenience we look at a small piece of such a discontinuity surface. The symbol < > means the addition of the two opposite surface effects. At a small surface region $\Delta s$, we obtain then  

$$h = < {u}_{N} \cdot {\sigma}_{N} \gt \Delta s + < {u}_{T} \cdot {\sigma}_{T} \gt \Delta s$$ (10)

Since the boundary conditions on welded interfaces require that ${\sigma}_{N}$ and u (${\epsilon}_{T}$) are continuous we can write for the change in surface energy:  

$$dh = {\sigma}_{N} \cdot < {\epsilon}_{N} \gt ~ \Delta s + {\epsilon}_{T} \cdot < {\sigma}_{T} \gt ~ \Delta s$$ (11)

We note that only the discontinuous components are in <   > . Expressing these discontinuous components in terms of continuous ones using the constitutive equation  

$$dh = {\sigma}_{N} < {\bf X}_{NT} \cdot {\epsilon}_{T} + {\b... ...\epsilon}_{T} + {\bf X}_{TN} \cdot {\sigma}_{N} \gt ~\Delta s$$ (12)

Separating out the continuous quantities and rewriting it as a matrix equation gives:

$$dh = \pmatrix{ {\epsilon}_{T} & {\sigma}_{N} \cr} <\pmatrix{... ...\pmatrix{ {\epsilon}_{T} \cr {\sigma}_{N} \cr} ~ dl ~ \Delta s$$ (13)

This takes care of one boundary point. Choosing an infinitesimal boundary region ds we can integrate the above expression over all discontinuity surfaces within the volume  

$$dH = \oint_{\Sigma} \pmatrix{ {\epsilon}_{T} & {\sigma}_{N} ... ...} \cr} \gt \pmatrix{ {\epsilon}_{T} \cr {\sigma}_{N} \cr} ~ ds$$ (14)

Consequently we obtain the elastic energy stored by deforming the medium. Note that this expression is an exact formula for calculating the energy; however all the quantities including the stresses and strains at each point inside the medium must be known. If we are willing to sacrifice exactness, we can approximate H by assuming an average stress and strain in the medium. Then the composite medium property depends only on properties of its composites by:  

$$H = \pmatrix{ {\epsilon}_{T} & {\sigma}_{N} \cr} \oint_{\Si... ...r} \gt ~ dl ~ ds \pmatrix{ {\epsilon}_{T} \cr {\sigma}_{N} \cr}$$ (15)

It is remarkable that in the layered geometry (14) and (15) are identical and thus exact. For an arbitrary heterogeneous medium the scale of heterogeneity determines the accuracy of approximation (15) to (14). The stress and strain field in the medium is not distorted much by small scale heterogeneities. Thus on an average much larger than the scale of the heterogeneities the stress and strain field is assumed to be constant, so that we can apply relation (15).