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The fourth theory of strength
The fourth theory of strength,the energetic theory is based on the hypothesis that not all the specific energy, but a part of the specific energy accumulated as a result of the change in the shape of a cube with edges equal to one, causes the state of limit stress to appear.In this theory, the general condition is written in the following form.
Here: - the calculated value of the energy associated with the change of the cube shape in the case of the limit stress that checked;
- the limiting value of this energy obtained from the result of an experiment on simple tension.
Substituting their known expressions for relative energies, we obtain the following condition:
(11.13)
Here: - limit stress found experimentally in simple tension.
In the case of flat stress, the formula (11.13) is written as follows.
If we say for a private point with
(11.17)
Energetic theory also gives good results in experiments with plastic materials, so it is widely used in practice.
9.-55Deformations and displacements (linear, angular, elastic, plastic).
When a force is applied to any solid body, its geometric dimensions, volume and shape change, but the total amount and mass of its constituent substances remain unchanged. This change is called deformation. For example, a bar is stretched under the influence of forces directed along the core, that is, its length changes, it bends under the influence of a force placed on a beam lying on two supports, that is, its shape changes, a shaft with a circular cross-sectional surface is twisted under the influence of torques, that is, the mutual location of one cross-sectional surface relative to another changes without changing its volume dimensions and shape.
Deformations are divided into elastic and plastic deformations from a physical point of view, and linear and angular deformations from a geometric point of view.
When an external force is removed from a deformed body, the body tends to return to its original shape, the deformation disappears partially or completely.
The property of objects to deform under force and return to their original state after the action of force is removed is called elasticity .The part of the deformation that disappears when the force is taken away is called elastic deformation, and the part that remains is called residual or plastic deformation.
The nature of the deformations depends on the amount of force, if the amount of the force causing the deformation does not exceed a certain limit, only elastic deformation is formed in the body, otherwise, plastic deformation occurs in the body in addition to elastic deformation.
The four simple types of deformation caused by force in the strength of materials are tension and compression, shear, torsion and bending deformations. In addition, complex deformations resulting from the combination of the above types of deformations are also seen.
In order to find the deformation at a point A of the body, we pass a cross-section AB with a length equal to S from this point in an arbitrary direction. After the deformation, the points A and B occupy the new state A 1 and B 1 , and the section between these points is stretched by the amount ΔS and changes its direction (Fig. 1.9)
- the ratio AB is called the relative linear deformation of the section. By shortening the length of the section, i.e. by bringing point B infinitely closer to point A, we get the following
the quantity is called the relative linear deformation at point A along the direction AB. If a system of rectilinear coordinate axes is transferred from point A, the linear deformations directed along these axes are respectively will be equal to .
We check whether the direction of the section changes. For this, from point A, we pass the sections AB and AC, which form a right angle with each other. After deformation, the cross-sections go to states A 1 B 1 and A 1 C 1 , the angles between them change (Fig. 1.10). By shortening the length of the sections, that is, by bringing points B and C closer to point A, we get the following.
The change of right angles is called relative angular deformation at point A. Relative angular deformations lying in different planes through point A are different. Relative angular deformations in the planes of the coordinate axes are defined by xu , xz , yz , respectively.
Thus, at any point of the body there are three linear and three angular deformation components.
11 -57-58Internal forces. Cutting method. Stresses (primary internal forces, internal tension forces, the essence of the cutting method, normal and transverse forces, bending and twisting moments, average real, total, pascal).
Interaction forces (primary internal forces) act between the elementary particles that make up the body, and they ensure the integrity of the body. When an external force is applied to a body, additional internal forces of resistance to the primary internal forces are formed between the elementary particles.
These forces are called internal forces , or stress forces. They resist to body deformation, that is, they tend to restore the original shape and dimensions of the body.
Under the influence of an external force, the body is deformed, which in turn causes a change in the relative location of the particles that make up the body. As a result of this, interaction forces arise between particles. These forces resist deformation of the body and strive to return its changed size, shape, and geometric dimensions to their original state.
The value of these internal forces increases with increasing deformation until the internal and external forces balance. If this balance does not occur, the balance of the elementary particles that make up the solid body, that is, their interdependence, is disturbed. As a result, a microcrack is formed in the appropriate place of the body and the body is destroyed. Therefore, internal tension forces which value can reach the magnitude of intermolecular interaction forces can be shown as the direct cause of the decay of the body . Therefore, determining their size is an important issue.
The forces of interaction between the particles of the body are called internal forces. The cutting method is used to determine the internal forces in the strength of materials.
To determine the internal forces, six equilibrium equations are written for the section lying on one side of the shear plane:
From the first three of these equations (1.1), the forces N, Q x , Q y are determined, and from the last three, the bending moment and torque M x ,M y andM bs are found.
Using the cutting method, it is possible to determine the amount and direction of internal force components generated in the section under the influence of external forces.
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