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Relations between E, G and K
The largest maximum shear strain and shear stress can be given by
Considering now the hydrostatic state of stress and strain we may write
Substituting in terms of we may write
So far the state of strain at a point was considered entirely due to applied forces. Changes in temperature may also cause stresses if a thermal gradient or some external constraints exist. Provided that the materials remain linearly elastic, stress pattern due to thermal effect may be superimposed upon that due to applied forces and we may write
It is important to note that the shear strains are not affected directly by temperature changes. It is sometimes convenient to express stresses in terms of strains. This may be done using the relation Substituting the above expressions for εx , εy and εz we have.
and substituting we have
Combining this with we have
Substituting we may write the normal and shear stresses as
These equations are considered to be suitable in thermoelastic situations.