1. Concepts about the science of "Strength of Materials" tasks, consistency, uniformity, priority, brief history

The first, second and third classical theories of strength.
One of the equally dangerous stress states is the well-studied linear stress by experiment, and the other is the stress state whose dangerous state needs to be determined.
The first theory of stability, one of the oldest theories, is based on the hypothesis that "the maximum normal stress causes the state of ultimate stress to appear." This theory is called the maximum normal stress theory.
According to the accepted hypothesis, the following condition must be fulfilled
(11.1.)
Here: - is the largest of the principal stresses for the stress case that checked.
- experimentally derived ultimate stress for linear tension.
The disadvantage of this theory of stability is that it does not take into account the main stresses , , other than .This theory gives good results only in tensile testing of brittle materials.
At present, the theory of the first continuity is not used, it is only of historical importance.
The second theory of strength is based on the hypothesis that the greatest tension causes the emergence of the state of limit stress in the material.This theory is called the greatest tension theory. The condition corresponding to this hypothesis is written as follows:
(11.2.)
Here: - is calculated value of the largest tension for the tested stress state;
- the limiting value of the relative tension obtained from the uniaxial tension test.
The formula representing Hooke's law is used to determine deformations and .
(11.3 )
(11.4)
Putting these expressions into expression (11.2), we get the following:
(11.5)
This expression is valid when the left side is positive. The results obtained according to this theory are not sufficiently confirmed by experimental results.
The third theory of strength is based on the hypothesis that the ultimate stress state is caused by the greatest shear stresses. The general condition of this theory is written as follows.
(11.6)
Here: - is calculated value of the maximum test stress for the stress condition that checked.
- the calculated value of the shear stress determined from the simple tension experiment.
The maximum shear stress in case of volumetric stress is found by the followingformula:
(11.7)
In the case of linear stress, the maximum shear stress is found from the following formula.
(11.8)
The general condition that takes into account these expressions is written as follows.
(11.9)

Condition (9) for the case of stress is written as follows:

(11.10)
R- calculative resistance

In most cases it is . Taking this into account, formula (10) is written as follows.

(11.11)
The main drawback of the third theory is that it does not take into account the effect of the main stress in the case of volumetric stress.This theory is called the theory of maximum shear stress and is widely used in evaluating the strength of plastic materials.