Elastic stresses Hooke's law. Hooke's law. Formula. Description of the experience
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 negative meaning. 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-Cross-sectional area.
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.
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 - a type of longitudinal deformation of a rod or beam that occurs if 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 in cross sections beam bending moments. 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 can be a consequence of phase transformations associated with changes in volume, thermal expansion, magnetization (magnetostrictive effect), the appearance of an electric charge (piezoelectric effect) or the result of the action 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 is their 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
The law of proportionality between the elongation of a spring and the applied force was discovered by the English physicist Robert Hooke (1635-1703)
Scientific interests Hooke's ideas were so broad that he often did not have time to complete his research. This gave rise to heated disputes about the priority in the discovery of certain laws with the greatest scientists (Huygens, Newton, etc.). However, Hooke's law was so convincingly substantiated by numerous experiments that Hooke's priority was never disputed.
Robert Hooke's spring theory:
This is Hooke's law!
PROBLEM SOLVING
Determine the stiffness of a spring that, under the action of a force of 10 N, lengthens by 5 cm.
Given:
g = 10 N/kg
F=10H
X = 5cm = 0.05m
Find:
k = ?
The load is in balance.
Answer: spring stiffness k = 200N/m.
TASK FOR "5"
(hand in on a piece of paper).
Explain why it is safe for an acrobat to jump onto a trampoline net from a great height? (we call on Robert Hooke for help)
I'm looking forward to your answer!
LITTLE EXPERIENCE
Place the rubber tube vertically, on which a metal ring has previously been tightly placed, and stretch the tube. What will happen to the ring?
Dynamics - Cool physics
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 comes from the following: indeed, body weight is often equal to force 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.
Electric 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 select the coordinate axes x,y,z systems K and x",y",z" of the K" system so that the x and x" axes coincided, and the y and y" axes, z and z", were parallel to each other. Let us find the relationship between the x,y,z coordinates of a certain point P in the K system and the coordinates x", y", z" of the same point in the K system". , 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 on the general principle of relativity (no physical experience can distinguish one inertial system from another), formulated by Albert Einstein, Lawrence changed the Galilean 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 usually called linear relationships between strain components and stress components.
Let's take an elementary rectangular parallelepiped with faces parallel to the coordinate axes, loaded with normal stress σ x, evenly distributed over two opposite faces (Fig. 1). Wherein σy = σ z = τ x y = τ x z = τ yz = 0.
Up to the limit of proportionality, the relative elongation is given by the formula
Where E— tensile modulus of elasticity. For steel E = 2*10 5 MPa, therefore, the deformations are very small and are measured as a percentage or 1 * 10 5 (in strain gauge instruments that measure deformations).
Extending an element in the axis direction X accompanied by its narrowing in the transverse direction, determined by the deformation components
Where μ - a constant called the lateral compression ratio or Poisson's ratio. For steel μ usually taken to be 0.25-0.3.
If the element in question is loaded simultaneously with normal stresses σ x, σy, σ z, evenly distributed along its faces, then deformations are added
By superimposing the deformation components caused by each of the three stresses, we obtain the relations
These relationships are confirmed by numerous experiments. Applied overlay method or superpositions to find the total strains and stresses caused by several forces is legitimate as long as the strains and stresses are small and linearly dependent on the applied forces. In such cases, we neglect small changes in the dimensions of the deformed body and small movements of the points of application of external forces and base our calculations on the initial dimensions and initial form bodies.
It should be noted that the smallness of the displacements does not necessarily mean that the relationships between forces and deformations are linear. So, for example, in a compressed force Q rod loaded additionally with shear force R, even with small deflection δ an additional point arises M = Qδ, which makes the problem nonlinear. In such cases, the total deflections are not linear functions of the forces and cannot be obtained by simple superposition.
It has been experimentally established that if shear stresses act along all faces of the element, then the distortion of the corresponding angle depends only on the corresponding components of the shear stress.
Constant G called the shear modulus of elasticity or shear modulus.
The general case of deformation of an element due to the action of three normal and three tangential stress components on it can be obtained using superposition: three shear deformations, determined by relations (5.2b), are superimposed on three linear deformations determined by expressions (5.2a). Equations (5.2a) and (5.2b) determine the relationship between the components of strains and stresses and are called generalized Hooke's law. Let us now show that the shear modulus G expressed in terms of tensile modulus of elasticity E and Poisson's ratio μ . To do this, consider the special case when σ x = σ , σy = -σ And σ z = 0.
Let's cut out the element abcd planes parallel to the axis z and inclined at an angle of 45° to the axes X And at(Fig. 3). As follows from the equilibrium conditions of element 0 bс, normal stress σ v on all faces of the element abcd are equal to zero, and the shear stresses are equal
This state of tension is called pure shear. From equations (5.2a) it follows that
that is, the extension of the horizontal element is 0 c equal to the shortening of the vertical element 0 b: εy = -ε x.
Angle between faces ab And bc changes, and the corresponding shear strain value γ can be found from triangle 0 bс:
It follows that
