Science Fair Project Encyclopedia
- the diagonal coefficients εii are the relative change in length in the direction of the i direction (along the xi-axis) ;
- the other terms εij (i ≠ j) are the γ, half the variation of the right angle (assuming a small cube of matter before deformation).
The deformation of an object is defined by a tensor field, i.e. this strain tensor is defined for every point of the object. This field is linked to the field of stress tensor by the generalized Hooke's law.
In case of small deformations, the strain tensor is the Green tensor, defined by the equation:
Where u represents the displacement field of the object's configuration (i.e. the difference between the object's configuration and its natural state). This is the 'symmetric part' of the Jacobian matrix.
Demonstration in simple cases
When the [AB] segment, parallel to the x1-axis, is deformed to become the [A'B' ] segment, the deformation being also parallel to x1
the ε11 strain is (expressed in algebraic length):
the strain is
The series expansion of u1 is
And in general
Pure shear strain
The tangent of the γ angle is:
for small deformations,
and u2(A) = 0. Thus,
Considering now the [AD] segment:
It is interesting to use the average because the formula is still valid when the rhombus rotates; in such a case, there are two different angles et .
Relative variation of the volume
The relative variation of the volume ΔV/V0 is the trace of the tensor :
Actually, if we consider a cube with an edge length a, it is a quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions and V0 = a3, thus
as we consider small deformations,
therefore the formula.
Real variation of volume (top) and the approximated one (bottom): the green drawing shows the estimated volume and the orange drawing the neglected volume
In case of pure shear, we can see that there is no change of the volume.
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