(
)
is the displacement of the sample from its unstressed state, at each point
in space. identifies
the position vector of the point in space.Go directly to derivation of several forms of strain.
A more formal treatment emphasizes the similarity between tensile and
shear stress, and resolves the rotational difficulty of the simple shear.
The treatment begins with a vector field: ()
is the displacement of the sample from its unstressed state, at each point
in space. identifies
the position vector of the point in space.
We will use
to identify the bit of material whose "home" position is at that
vector location in space. When the sample is distorted by a stress, the
strain causes this bit of material to move to a new position, 
.
We switch now to subscript notation for vectors. In this notation, a tensile strain is defined as a two-index mathematical object:
where 
can
be x, or y, or z and 
.
The is the
x coordinate of a position in space, and not part of the name of a bit
of material. The derivative is zero if the entire object is displaced,
so that all the displacement vectors,
are the same. The derivative is non-zero if neighboring bits are displaced
a different amount, e.g., stretched apart from each other.
If the tensile stress is applied in the x direction, and we are interested in the "stretch" in that direction, we calculate
We know that the side effect of this stress also produces a y component of distortion, and we may also calculate
For shear strains, we take a step to eliminate rotations from consideration. We define a shear strain symmetrically:
Inspection shows that this definition can also be used for tensile strains, since there the interchange of x and x (for example) makes no change in the result.
We define stress in a compatible manner, using two directions: The direction of the force applied to the face of a sample, and the direction of the unit vector normal to that face. We define tensile stress as
where 
has
the magnitude of the area of the face to which the force is applied, and
the direction perpendicular to that face. (For convenience we have chosen
one of our coordinate axes perpendicular to that face.) We agree that the
area vector points outward from the sample.
If we pull on the right and left faces of our cubic sample, the stress is the same (in magnitude and sign) whether we calculate it on the left or on the right face.
Shear stress is more complex because it is here that we take steps to prevent rotation. If we have a pair of forces; one pushing to the right on top and the other to the left on the bottom, we produce both a shear and a rotation. To cancel the rotation, we add two more forces: pushing up on the right face and down on the left face. For a cubical sample these forces have the same magnitude as the force on the top (and the force on the bottom).
Shear stress is defined as
We get the same answer no matter which of the four faces under stress we use for the calculation. We still have chosen our faces to be perpendicular to the x,y,z coordinate axes, for convenience. The notation ensures that for shear the force is parallel to the surface to which it is applied.
If we drop the restriction 
,
we have a general definition of stress that encompasses both tensile and
shear stress:
We continue to restrict our study and insist that strain be proportional
to stress. We do allow, however, that stress of one sort can produce strains
of several kinds. Our example is the fact that a tensile 
stress produces not only the 
strain, but also the side effect 
and 
strains.
Our general form for the elastic relation between stress and strain is
with the Einstein Summation Convention that repeated indices
are summed. Thus in this equation, the right side is summed over all possible
combinations of all possible x,y,z values of 
,
and of 
.
For an isotropic
material many of these elastic constants, 
,
are zero, and many more are equal to each other in groups. In the end,
there are only three independent elastic constants for an isotropic material.
This is consistent with a result for the four simpler elastic constants, bulk modulus, Young's modulus, Poisson's ratio, and simple shear modulus. For isotropic materials here is a relation among Young's modulus, the bulk modulus and Poisson's ratio. Thus in the simple description also, there are only three independent elastic constants for an isotropic material.
In only one case is there a simple relation between the 
's
and the simple elastic constants. This is for the shear modulus. For an
isotropic material, all 's
of the form 
are the same. Since they are the same, we replace each of these constants
with the symbol, 
.
It turns out that the "pure shear modulus,"
is related to the simple shear modulus by
The reason that
is called the "pure" shear modulus is historical.
For the other elastic constants, in an isotropic material,
A surprising result comes from these interrelations among elastic constants: A shear strain can be represented by a sum of two tensile strains. In the cube sheared top and bottom; left and right, one tensile strain would represent pull at the upper right and lower left corners, while the other would represent compression at the upper left and lower right corners.
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