We call the ratio of force on a surface to area of surface the stress. It has units of Newtons per square meter, or Pascal; just the same as pressure. We agree that tensile stress is positive if it tends to increase the length of the sample.

Positive tensile stress causes positive strain (positive ). If we follow the usual convention of assigning a vector to an area (with the direction of the vector perpendicular to the area, and pointing outward from the sample), then positive tensile stress occurs when the force on a surface is in the same direction as the area vector for the surface.

 


Two tensile stresses applied to a sample

The method of calculating the stress is illustrated in the figure. The sample is made of plastic, which is relatively easy to deform. Glued to each face is a stiff steel plate, which distributes the applied force uniformly across the plastic face.

The forces shown produce two stresses; one (B) that tends to compress the sample parallel to the x axis, and another (A) that tends to stretch the sample parallel to the y axis.

 

The tensile stresses are positive when the force pair acts parallel to the area vectors; negative when the force pair acts anti-parallel to the area vectors. For the (B) forces, the stress is

This calculation may be done using either the left or the right face, so the subscripts 1 and 2 are omitted. Since the x component of either (B) force points opposite to its area vector, this stress is negative. It will produce a negative strain (compression).

For the (A) forces, the stress is

which is positive, causing a positive strain (stretch).


Positive shear stress on a sample

The stress that produces a shear strain is sketched in the figure. As noted before, four forces are necessary to maintain equilibrium. Each force must be paired with an opposing force, to make the net force zero. The forces must be paired again, to make the net torque zero. On a cube the four forces have the same magnitude.

The shear stress is

The shear stress is positive in this case, since each area vector component has the same sign as its force component (even though the components are perpendicular). The positive shear stress produces a positive shear strain, with positive x displacement increasing with increasing y position.

The division of vector components by other vector components is a new idea. The pair of subscripts on the stress parameter represent the components involved in the ratio. The same notation applies for both tensile and shear stresses. Because the torques must balance for the shear stress, all four forces on a cube have the same magnitude. This means that the stress tensor is symmetric under interchange of indices:

ELASTIC CONSTANTS

For small strains, the stress and strain are proportional. This simple idea is complicated by the large number of possible stresses and strains. For example, the discussion of Poisson's ratio reminds us that a stress produces not only a strain but also (via the side effect) strains

Thus, if the perturbation is a uniaxial stress, we know there are at least three responses, , , and , each strain proportional to the stress. In the most general case we must anticipate that all possible strains might be induced by a single stress. For example, we must anticipate that

Note that strain is symmetric under interchange of indices. For example

This means that there are only 6 independent terms above, rather than 9.

There is an equation like this for each different stress, , ..etc., for a total of 9 stresses.

Because the stress is symmetric, only 6 can be different. But if we invoke no symmetry, there are 9x9=81 possible elastic constants, K. With the symmetry under interchange of indices, there are "only" 36. To keep track of this mess, we change notation to a more compact form:

We write a general stress as

where i can be x,y, or z. Also j can be x,y, or z.

In a similar change, we write a general strain as

where k and l can also be x, or y, or z.

Then the proportionality between stress and strain is written

For every particular stress, such as , there are 9 terms in the sum over k and l on the right. This "elastic equation" is a general form of Hooke's law, which we know as the relation between force and length change in a spring. Hooke's law is introduce in beginning physics in extensive form:

We will simplify this general form in order to get an idea of how it works.

The symmetry of stress and strain under interchange of indices means that there are "only" 36 independent elastic constants, . Further reductions come from experimental observation that, a uniaxial tensile stress does not cause a shear strain so that for example a stress does not cause a strain, implying that . Also, there is no shear "side effect," so, for example, a stress does not cause a strain, giving us .

These (and similar) arguments reduce the number of independent elastic constants to 21. See B.A. Auld, Acoustic Fields and Waves in Solids, Volume 1, Chapter 3, Wiley, New York, 1973.

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