DEFINITION
Deformations are any changes in the shape, size and volume of the body. Deformation determines the final result of the movement of body parts relative to each other.
DEFINITION
Elastic deformations are called deformations that completely disappear after the removal of external forces.
Plastic deformations are called deformations that remain fully or partially after the cessation of external forces.
The ability to elastic and plastic deformations depends on the nature of the substance of which the body is composed, the conditions in which it is located; methods of its manufacture. For example, if we take different varieties iron or steel, then they can exhibit completely different elastic and plastic properties. With normal room temperatures iron is a very soft, ductile material; hardened steel, on the contrary, is a hard, elastic material. The plasticity of many materials is a condition for their processing and for the manufacture of the necessary parts from them. Therefore, it is considered one of the most important technical properties solid matter.
When deformed solid there is a displacement of particles (atoms, molecules or ions) from their original equilibrium positions to new positions. In this case, the force interactions between individual particles of the body change. As a result, the deformed body develops internal forces, preventing its deformation.
There are tensile (compressive), shear, bending, and torsional deformations.
Elastic forces
DEFINITION
Elastic forces– these are the forces that arise in a body during its elastic deformation and are directed in the direction opposite to the displacement of particles during deformation.
Elastic forces are of an electromagnetic nature. They prevent deformations and are directed perpendicular to the contact surface of interacting bodies, and if bodies such as springs or threads interact, then the elastic forces are directed along their axis.
The elastic force acting on the body from the support is often called the support reaction force.
DEFINITION
Tensile strain (linear strain) is a deformation in which only one change occurs linear size bodies. Its quantitative characteristics are absolute and relative elongation.
Absolute elongation:
where and is the length of the body in the deformed and undeformed state, respectively.
Relative extension:
Hooke's law
Small and short-term deformations with a sufficient degree of accuracy can be considered as elastic. For such deformations, Hooke’s law is valid:
where is the projection of force onto the rigidity axis of the body, depending on the size of the body and the material from which it is made, the unit of rigidity in the SI system is N/m.
Examples of problem solving
EXAMPLE 1
| Exercise | A spring with stiffness N/m in an unloaded state has a length of 25 cm. What will be the length of the spring if a load weighing 2 kg is suspended from it? |
| Solution | Let's make a drawing.
An elastic force also acts on a load suspended on a spring. Projecting this vector equality onto the coordinate axis, we obtain: According to Hooke's law, elastic force: so we can write: where does the length of the deformed spring come from: Let us convert the length of the undeformed spring, cm, to the SI system. Substituting the numerical values of physical quantities into the formula, we calculate: |
| Answer | The length of the deformed spring will be 29 cm. |
EXAMPLE 2
| Exercise | A body weighing 3 kg is moved along a horizontal surface using a spring with stiffness N/m. How much will the spring lengthen if under its action at uniformly accelerated motion in 10 s the speed of the body changed from 0 to 20 m/s? Ignore friction. |
| Solution | Let's make a drawing.
The body is acted upon by the reaction force of the support and the elastic force of the spring. |
CONTROL QUESTIONS
1) What is called deformation? What types of deformations do you know?
Deformation- a change in the relative position of body particles associated with their movement. Deformation is the result of changes in interatomic distances and rearrangement of blocks of atoms. Typically, deformation is accompanied by a change in the magnitude of interatomic forces, the measure of which is elastic stress.
Types of deformations:
Tension-compression- in the resistance of materials - type longitudinal deformation a rod or beam, which occurs when a load is applied to it along its longitudinal axis (the resultant of the forces acting on it is normal to the cross section of the rod and passes through its center of mass).
Tension causes elongation of the rod (rupture and residual deformation are also possible), compression causes shortening of the rod (loss of stability and longitudinal bending are possible).
Bend- a type of deformation in which there is a curvature of the axes of straight bars or a change in the curvature of the axes of curved bars. Bending is associated with the occurrence of bending moments in the cross sections of the beam. Direct bending occurs when the bending moment in a given cross-section of a beam acts in a plane passing through one of the main central axes of inertia of this section. In the case when the plane of action of the bending moment in a given cross section of the beam does not pass through any of the main axes of inertia of this section, it is called oblique.
If, during direct or oblique bending, only a bending moment acts in the cross section of the beam, then, accordingly, there is a pure straight or pure oblique bend. If a transverse force also acts in the cross section, then there is a transverse straight or transverse oblique bend.
Torsion- one of the types of body deformation. Occurs when a load is applied to a body in the form of a pair of forces (moment) in its transverse plane. In this case, only one internal force factor appears in the cross sections of the body - torque. Tension-compression springs and shafts work for torsion.
Types of deformation of a solid body. Deformation is elastic and plastic.
Deformation solid body may be a consequence of phase transformations associated with changes in volume, thermal expansion, magnetization (magnetostrictive effect), appearance electric charge(piezoelectric effect) or as a result of external forces.
A deformation is called elastic if it disappears after the load that caused it is removed, and plastic if it does not disappear (at least completely) after the load is removed. All real solids, when deformed, have plastic properties to a greater or lesser extent. Under certain conditions, the plastic properties of bodies can be neglected, as is done in the theory of elasticity. With sufficient accuracy, a solid body can be considered elastic, that is, it does not exhibit noticeable plastic deformations until the load exceeds a certain limit.
The nature of plastic deformation can vary depending on temperature, duration of load or strain rate. With a constant load applied to the body, the deformation changes with time; this phenomenon is called creep. As temperature increases, the creep rate increases. Special cases of creep are relaxation and elastic aftereffect. One of the theories explaining the mechanism of plastic deformation is the theory of dislocations in crystals.
Derivation of Hooke's law for various types deformation.
Net shift:
Pure torsion: 
4) What is called the shear modulus and torsional modulus, what are they? physical meaning?
Shear modulus or stiffness modulus (G or μ) characterizes the ability of a material to resist changes in shape while maintaining its volume; it is defined as the ratio of shear stress to shear strain, defined as the change right angle between planes along which shear stresses act). The shear modulus is one of the components of the viscosity phenomenon.
Shear modulus:
Torsion modulus: 
5) What is the mathematical expression of Hooke's law? In what units are elastic modulus and stress measured?
Measured in Pa, - Hooke's law
This force arises as a result of deformation (change in the initial state of the substance). For example, when we stretch a spring, we increase the distance between the molecules of the spring material. When we compress a spring, we decrease it. When we twist or shift. In all these examples, a force arises that prevents deformation - the elastic force.
Hooke's law
The elastic force is directed opposite to the deformation.
Since the body is represented as a material point, force can be represented from the center
When connecting springs in series, for example, the stiffness is calculated using the formula
When connected in parallel, the stiffness
Sample stiffness. Young's modulus.
Young's modulus characterizes the elastic properties of a substance. This is a constant value that depends only on the material and its physical state. Characterizes the ability of a material to resist tensile or compressive deformation. The value of Young's modulus is tabular.
Body weight
Body weight is the force with which an object acts on a support. You say, this is the force of gravity! The confusion occurs in the following: indeed, often the weight of a body is equal to the force of gravity, but these forces are completely different. Gravity is a force that arises as a result of interaction with the Earth. Weight is the result of interaction with support. The force of gravity is applied at the center of gravity of the object, while weight is the force that is applied to the support (not to the object)!
There is no formula for determining weight. This force is designated by the letter.
The support reaction force or elastic force arises in response to the impact of an object on the suspension or support, therefore the weight of the body is always numerically the same as the elastic force, but has the opposite direction.
The support reaction force and weight are forces of the same nature; according to Newton’s 3rd law, they are equal and oppositely directed. Weight is a force that acts on the support, not on the body. The force of gravity acts on the body.
Body weight may not be equal to gravity. It may be more or less, or it may be that the weight is zero. This condition is called weightlessness. Weightlessness is a state when an object does not interact with a support, for example, the state of flight: there is gravity, but the weight is zero!
It is possible to determine the direction of acceleration if you determine where the resultant force is directed.
Please note that weight is force, measured in Newtons. How to correctly answer the question: “How much do you weigh”? We answer 50 kg, not naming our weight, but our mass! In this example, our weight is equal to gravity, that is, approximately 500N!
Overload- ratio of weight to gravity
Archimedes' force
Force arises as a result of the interaction of a body with a liquid (gas), when it is immersed in a liquid (or gas). This force pushes the body out of the water (gas). Therefore, it is directed vertically upward (pushes). Determined by the formula:
In the air we neglect the power of Archimedes.
If the Archimedes force is equal to the force of gravity, the body floats. If the Archimedes force is greater, then it rises to the surface of the liquid, if less, it sinks.
Electrical forces
There are forces of electrical origin. Occurs in the presence of an electrical charge. These forces, such as the Coulomb force, Ampere force, Lorentz force.
Newton's laws
Newton's first law
There are such reference systems, which are called inertial, relative to which bodies retain their speed unchanged if they are not acted upon by other bodies or the action of other forces is compensated.
Newton's II law
The acceleration of a body is directly proportional to the resultant forces applied to the body and inversely proportional to its mass:
Newton's III law
The forces with which two bodies act on each other are equal in magnitude and opposite in direction. 
Local reference frame - this is a reference system that can be considered inertial, but only in an infinitesimal neighborhood of one point in space-time, or only along one open world line.
Galileo's transformations. The principle of relativity in classical mechanics.
Galileo's transformations. Let's consider two reference systems moving relative to each other and with a constant speed v 0. We will denote one of these systems by the letter K. We will consider it stationary. Then the second system Kwill move rectilinearly and uniformly. Let's choose coordinates x,y,z axes systems K and x",y",z" systems K" so that the axes x and x" coincided, and the axes y and y" , z and z" were parallel to each other. Let us find the relationship between the coordinates x,y,z of some point P in system K and coordinates x",y",z" of the same point in system K". If we start counting time from the moment when the origin of the system coordinates coincided, then x=x"+v 0, in addition, it is obvious that y=y", z=z". Let us add to these relations the assumption accepted in classical mechanics that time flows in the same way in both systems, that is, t=t". We obtain a set of four equations: x=x"+v 0 t;y=y";z=z"; t=t", called Galilean transformations. Mechanical principle of relativity. The provision that everything mechanical phenomena in different inertial reference systems proceed in the same way, as a result of which it is impossible to establish by any mechanical experiments whether the system is at rest or moves uniformly and in a straight line; it is called the principle of Galilean relativity. Violation of the classical law of addition of velocities. Based general principle relativity (no physical experience can distinguish one inertial system from another), formulated by Albert Einstein, Lawrence changed Galileo's transformations and received: x"=(x-vt)/(1-v 2 /c 2); y"=y; z"=z; t"=(t-vx/c 2)/(1-v 2 /c 2). These transformations are called Lawrence transformations.
Hooke's law is formulated as follows: the elastic force that occurs when a body is deformed due to the application of external forces is proportional to its elongation. Deformation, in turn, is a change in the interatomic or intermolecular distance of a substance under the influence of external forces. The elastic force is the force that tends to return these atoms or molecules to a state of equilibrium.
Formula 1 - Hooke's Law.
F - Elastic force.
k - body rigidity (Proportionality coefficient, which depends on the material of the body and its shape).
x - Body deformation (elongation or compression of the body).
This law was discovered by Robert Hooke in 1660. He conducted an experiment, which consisted of the following. A thin steel string was fixed at one end, and varying amounts of force were applied to the other end. Simply put, a string was suspended from the ceiling and a load of varying mass was applied to it.
Figure 1 - String stretching under the influence of gravity.
As a result of the experiment, Hooke found out that in small aisles the dependence of the stretching of a body is linear with respect to the elastic force. That is, when a unit of force is applied, the body lengthens by one unit of length.
Figure 2 - Graph of the dependence of elastic force on body elongation.
Zero on the graph is the original length of the body. Everything on the right is an increase in body length. In this case, the elastic force has a negative value. That is, she strives to return the body to its original state. Accordingly, it is directed counter to the deforming force. Everything on the left is body compression. The elastic force is positive.
The stretching of the string depends not only on the external force, but also on the cross-section of the string. A thin string will somehow stretch due to its light weight. But if you take a string of the same length, but with a diameter of, say, 1 m, it is difficult to imagine how much weight will be required to stretch it.
To assess how a force acts on a body of a certain cross-section, the concept of normal mechanical stress is introduced.
Formula 2 - normal mechanical stress.
S-Area cross section.
This stress is ultimately proportional to the elongation of the body. Relative elongation is the ratio of the increment in the length of a body to its total length. And the proportionality coefficient is called Young's modulus. Modulus because the value of the elongation of the body is taken modulo, without taking into account the sign. It does not take into account whether the body is shortened or lengthened. It is important to change its length.
Formula 3 - Young's modulus.
|e| - Relative elongation of the body.
s is normal body tension.
Types of deformations
Deformation called a change in the shape, size or volume of the body. Deformation can be caused by external forces applied to the body. Deformations that completely disappear after the action of external forces on the body ceases are called elastic, and deformations that persist even after external forces have ceased to act on the body - plastic. Distinguish tensile strain or compression(unilateral or comprehensive), bending, torsion And shift.
Elastic forces
When a solid body is deformed, its particles (atoms, molecules, ions) located at the nodes of the crystal lattice are displaced from their equilibrium positions. This displacement is counteracted by the interaction forces between particles of a solid body, which keep these particles at a certain distance from each other. Therefore, with any type of elastic deformation, internal forces arise in the body that prevent its deformation.
The forces that arise in a body during its elastic deformation and are directed against the direction of displacement of the particles of the body caused by the deformation are called elastic forces. Elastic forces act in any section of a deformed body, as well as at the point of its contact with the body causing deformation. In the case of unilateral tension or compression, the elastic force is directed along the straight line along which the external force acts, causing deformation of the body, opposite to the direction of this force and perpendicular to the surface of the body. The nature of elastic forces is electrical.
We will consider the case of the occurrence of elastic forces during unilateral tension and compression of a solid body.
Hooke's law
The connection between the elastic force and the elastic deformation of a body (at small deformations) was experimentally established by Newton's contemporary, the English physicist Hooke. The mathematical expression of Hooke's law for unilateral tension (compression) deformation has the form:
where f is the elastic force; x - elongation (deformation) of the body; k is a proportionality coefficient depending on the size and material of the body, called rigidity. The SI unit of stiffness is newton per meter (N/m).
Hooke's law for one-sided tension (compression) is formulated as follows: The elastic force arising during deformation of a body is proportional to the elongation of this body.
Let's consider an experiment illustrating Hooke's law. Let the axis of symmetry of the cylindrical spring coincide with the straight line Ax (Fig. 20, a). One end of the spring is fixed in the support at point A, and the second is free and the body M is attached to it. When the spring is not deformed, its free end is located at point C. This point will be taken as the origin of the coordinate x, which determines the position of the free end of the spring.
Let's stretch the spring so that its free end is at point D, the coordinate of which is x > 0: At this point the spring acts on the body M with an elastic force
Let us now compress the spring so that its free end is at point B, whose coordinate is x
It can be seen from the figure that the projection of the elastic force of the spring onto the Ax axis always has a sign opposite to the sign of the x coordinate, since the elastic force is always directed towards the equilibrium position C. In Fig. 20, b shows a graph of Hooke's law. The values of elongation x of the spring are plotted on the abscissa axis, and the elastic force values are plotted on the ordinate axis. The dependence of fx on x is linear, so the graph is a straight line passing through the origin of coordinates.
Let's consider another experiment.
Let one end of a thin steel wire be fixed to a bracket, and a load suspended from the other end, the weight of which is an external tensile force F acting on the wire perpendicular to its cross section (Fig. 21).
The action of this force on the wire depends not only on the force modulus F, but also on the cross-sectional area of the wire S.
Under the influence of an external force applied to it, the wire is deformed and stretched. If the stretch is not too great, this deformation is elastic. In an elastically deformed wire, an elastic force f unit arises. According to Newton's third law, the elastic force is equal in magnitude and opposite in direction external force, acting on the body, i.e.
f up = -F (2.10)
The state of an elastically deformed body is characterized by the value s, called normal mechanical stress(or, for short, just normal voltage). Normal stress s is equal to the ratio of the modulus of the elastic force to the cross-sectional area of the body:
s = f up /S (2.11)
Let the initial length of the unstretched wire be L 0 . After applying force F, the wire stretched and its length became equal to L. The quantity DL = L - L 0 is called absolute wire elongation. The quantity e = DL/L 0 (2.12) is called relative body elongation. For tensile strain e>0, for compressive strain e< 0.
Observations show that for small deformations the normal stress s is proportional to the relative elongation e:
s = E|e|. (2.13)
Formula (2.13) is one of the types of writing Hooke’s law for unilateral tension (compression). In this formula, the relative elongation is taken modulo, since it can be both positive and negative. The proportionality coefficient E in Hooke's law is called the longitudinal modulus of elasticity (Young's modulus).
Let us establish the physical meaning of Young's modulus. As can be seen from formula (2.12), e = 1 and L = 2L 0 for DL = L 0 . From formula (2.13) it follows that in this case s = E. Consequently, Young’s modulus is numerically equal to this normal voltage, which should appear in the body when its length increases by 2 times. (if Hooke's law were true for such a large deformation). From formula (2.13) it is also clear that in the SI Young’s modulus is expressed in pascals (1 Pa = 1 N/m2).