Observations show that for most elastic bodies, such as steel, bronze, wood, etc., the magnitude of the deformations is proportional to the magnitude of the acting forces. A typical example explaining this property is a spring balance, in which the elongation of the spring is proportional to the acting force. This can be seen from the fact that the division scale of such scales is uniform. As a general property of elastic bodies, the law of proportionality between force and deformation was first formulated by R. Hooke in 1660 and published in 1678 in the work “De potentia restitutiva”. In the modern formulation of this law, it is not forces and movements of the points of their application that are considered, but stress and deformation.
Thus, for pure tension it is assumed:
Here is the relative elongation of any segment taken in the stretching direction. For example, if the ribs shown in Fig. 11 the prisms before applying the load were a, b and c, as shown in the drawing, and after deformation they will be respectively, then .
The constant E, which has the dimension of stress, is called the elastic modulus, or Young's modulus.
Tension of elements parallel to the acting stresses o is accompanied by a contraction of perpendicular elements, that is, a decrease in the transverse dimensions of the rod (dimensions in the drawing). Relative transverse strain
will be a negative value. It turns out that longitudinal and transverse deformations in an elastic body are related by a constant ratio:
The dimensionless quantity v, constant for each material, is called the lateral compression ratio or Poisson's ratio. Poisson himself, proceeding from theoretical considerations that later turned out to be incorrect, believed that for all materials (1829). In fact, the values of this coefficient are different. Yes, for steel
Replacing the expression in the last formula we get:
Hooke's Law is not an exact law. For steel, deviations from proportionality between are insignificant, while cast iron or carving clearly do not obey this law. For them, and can be approximated by a linear function only in the roughest approximation.
For a long time, strength of materials was concerned only with materials that obey Hooke's law, and the application of strength of materials formulas to other bodies could only be done with great reserve. Currently, nonlinear elasticity laws are beginning to be studied and applied to solving specific problems.
Action external forces on a solid body leads to the occurrence of stresses and deformations at points in its volume. In this case, the stress state at a point, the relationship between stresses on different areas passing through this point, are determined by the equations of statics and do not depend on physical properties material. The deformed state, the relationship between displacements and deformations, are established using geometric or kinematic considerations and also do not depend on the properties of the material. In order to establish a relationship between stresses and strains, it is necessary to take into account the actual properties of the material and loading conditions. Mathematical models describing the relationships between stresses and strains are developed on the basis of experimental data. These models must reflect the actual properties of materials and loading conditions with a sufficient degree of accuracy.
The most common models for structural materials are elasticity and plasticity. Elasticity is the property of a body to change shape and size under the influence of external loads and restore its original configuration when the load is removed. Mathematically, the property of elasticity is expressed in the establishment of a one-to-one functional relationship between the components of the stress tensor and the strain tensor. The property of elasticity reflects not only the properties of materials, but also loading conditions. For most structural materials, the property of elasticity manifests itself at moderate values of external forces leading to small deformations, and at low loading rates, when energy losses due to temperature effects are negligible. A material is called linearly elastic if the components of the stress tensor and strain tensor are related by linear relationships.
At high levels loading, when significant deformations occur in the body, the material partially loses its elastic properties: when unloaded, its original dimensions and shape are not completely restored, and when external loads are completely removed, residual deformations are recorded. In this case the relationship between stresses and strains ceases to be unambiguous. This material property is called plasticity. Residual deformations accumulated during plastic deformation are called plastic.
High load levels can cause destruction, i.e. division of the body into parts. Solids, made from various materials, are destroyed at different amounts of deformation. Fracture is brittle at small deformations and occurs, as a rule, without noticeable plastic deformations. Such destruction is typical for cast iron, alloy steels, concrete, glass, ceramics and some other structural materials. Low-carbon steels, non-ferrous metals, and plastics are characterized by a plastic type of failure in the presence of significant residual deformations. However, the division of materials into brittle and ductile according to the nature of their destruction is very arbitrary; it usually refers to some standard operating conditions. The same material can behave, depending on conditions (temperature, the nature of the load, manufacturing technology, etc.) as brittle or ductile. For example, plastic at normal temperature materials are destroyed as brittle when low temperatures. Therefore, it is more correct to speak not about brittle and ductile materials, but about the brittle or plastic state of the material.
Let the material be linearly elastic and isotropic. Let us consider an elementary volume under conditions of a uniaxial stress state (Fig. 1), so that the stress tensor has the form
With such a load, the dimensions increase in the direction of the axis Oh, characterized by linear deformation, which is proportional to the magnitude of the stress
Fig.1. Uniaxial stress state
This relation is a mathematical notation Hooke's law, establishing a proportional relationship between stress and the corresponding linear deformation in a uniaxial stress state. The proportionality coefficient E is called the longitudinal modulus of elasticity or Young's modulus. It has the dimension of stress.
Along with the increase in size in the direction of action; Under the same stress, the size decreases in two orthogonal directions (Fig. 1). We denote the corresponding deformations by and , and these deformations are negative while positive and are proportional to:
With the simultaneous action of stresses along three orthogonal axes, when there are no tangential stresses, the principle of superposition (superposition of solutions) is valid for a linearly elastic material:
Taking into account formulas (1 4) we obtain
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Tangential stresses cause angular deformations, and at small deformations they do not affect the change in linear dimensions, and therefore linear deformations. Therefore, they are also valid in the case of an arbitrary stress state and express the so-called generalized Hooke's law.
The angular deformation is caused by the tangential stress, and the deformation and , respectively, by the stresses and. There are proportional relationships between the corresponding tangential stresses and angular deformations for a linearly elastic isotropic body
which express the law Hooke's shear. The proportionality factor G is called shear module. It is important that normal stress does not affect angular deformations, since in this case only the linear dimensions of the segments change, and not the angles between them (Fig. 1).
A linear relationship also exists between the average stress (2.18), proportional to the first invariant of the stress tensor, and volumetric strain (2.32), coinciding with the first invariant of the strain tensor:
Fig.2. Plane shear strain
Corresponding proportionality factor TO called volumetric modulus of elasticity.
Formulas (1 7) include the elastic characteristics of the material E, , G And TO, determining its elastic properties. However, these characteristics are not independent. For an isotropic material, there are two independent elastic characteristics, which are usually chosen as the elastic modulus E and Poisson's ratio. To express the shear modulus G through E And , Let us consider plane shear deformation under the action of tangential stresses (Fig. 2). To simplify the calculations, we use a square element with a side A. Let's calculate the principal stresses , . These stresses act on areas located at an angle to the original areas. From Fig. 2 we will find the relationship between linear deformation in the direction of stress and angular deformation . The major diagonal of the rhombus, characterizing the deformation, is equal to
For small deformations
Taking these relations into account
Before deformation, this diagonal had the size . Then we will have
From the generalized Hooke's law (5) we obtain
Comparison of the resulting formula with the notation of Hooke's law for shift (6) gives
As a result we get
Comparing this expression with Hooke’s volumetric law (7), we arrive at the result
Mechanical characteristics E, , G And TO are found after processing experimental data from testing samples for different kinds loads From a physical point of view, all these characteristics cannot be negative. In addition, from the last expression it follows that Poisson's ratio for an isotropic material does not exceed 1/2. Thus, we obtain the following restrictions for the elastic constants of an isotropic material:
Limit value leads to limit value , which corresponds to an incompressible material (at). In conclusion, from elasticity relations (5) we express stress in terms of deformation. Let us write the first of relations (5) in the form
Using equality (9) we will have
Similar relationships can be derived for and . As a result we get
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Here we use relation (8) for the shear modulus. In addition, the designation
POTENTIAL ENERGY OF ELASTIC DEFORMATION
Let us first consider the elementary volume dV=dxdydz under uniaxial stress conditions (Fig. 1). Mentally fix the site x=0(Fig. 3). A force acts on the opposite surface .
This force does work on displacement .
When the voltage increases from zero level to the value
the corresponding deformation due to Hooke's law also increases from zero to the value ,
and the work is proportional to the shaded figure in Fig. 4 squares: 
. If you neglect kinetic energy and losses associated with thermal, electromagnetic and other phenomena, then, due to the law of conservation of energy, the work performed will turn into potential energy, accumulated during deformation: .
Value Ф= dU/dV called specific potential deformation energy,
meaningful potential energy accumulated per unit volume of the body. In the case of a uniaxial stress state
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 σx, 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 shape of the body.
It should be noted that the smallness of the displacements does not necessarily imply the linearity of the relationships between forces and deformations. So, for example, in a compressed force Q rod loaded additionally with shear force R, even with small deflection δ arises extra point 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 strains, 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
Hooke's law was discovered in the 17th century by the Englishman Robert Hooke. This discovery about the stretching of a spring is one of the laws of elasticity theory and plays an important role in science and technology.
Definition and formula of Hooke's law
The formulation of this law is as follows: the elastic force that appears at the moment of deformation of a body is proportional to the elongation of the body and is directed opposite to the movement of particles of this body relative to other particles during deformation.
The mathematical notation of the law looks like this:
Rice. 1. Formula of Hooke's law
Where Fupr– accordingly, the elastic force, x– elongation of the body (the distance by which the original length of the body changes), and k– proportionality coefficient, called body rigidity. Force is measured in Newtons, and elongation of a body is measured in meters.
To reveal the physical meaning of stiffness, you need to substitute the unit in which elongation is measured in the formula for Hooke’s law - 1 m, having previously obtained an expression for k.
Rice. 2. Body stiffness formula
This formula shows that the stiffness of a body is numerically equal to the elastic force that occurs in the body (spring) when it is deformed by 1 m. It is known that the stiffness of a spring depends on its shape, size and the material from which the body is made.
Elastic force
Now that we know what formula expresses Hooke’s law, it is necessary to understand its basic value. The main quantity is the elastic force. It appears at a certain moment when the body begins to deform, for example, when a spring is compressed or stretched. It is sent to reverse side from gravity. When the elastic force and the force of gravity acting on the body become equal, the support and the body stop.
Deformation is an irreversible change that occurs in the size of the body and its shape. They are associated with the movement of particles relative to each other. If a person sits in a soft chair, then deformation will occur to the chair, that is, its characteristics will change. It happens different types: bending, stretching, compression, shear, torsion.
Since the elastic force is related in origin to electromagnetic forces, you should know that it arises due to the fact that molecules and atoms - the smallest particles that make up all bodies - attract and repel each other. If the distance between the particles is very small, then they are affected by the repulsive force. If this distance is increased, then the force of attraction will act on them. Thus, the difference between attractive and repulsive forces manifests itself in elastic forces.
The elastic force includes the ground reaction force and body weight. The strength of the reaction is of particular interest. This is the force that acts on a body when it is placed on any surface. If the body is suspended, then the force acting on it is called the tension force of the thread.
Features of elastic forces
As we have already found out, the elastic force arises during deformation, and it is aimed at restoring the original shapes and sizes strictly perpendicular to the deformed surface. Elastic forces also have a number of features.
- they arise during deformation;
- they appear in two deformable bodies simultaneously;
- they are perpendicular to the surface in relation to which the body is deformed.
- they are opposite in direction to the displacement of body particles.
Application of the law in practice
Hooke's law is applied both in technical and high-tech devices, and in nature itself. For example, elastic forces are found in watch mechanisms, in shock absorbers in transport, in ropes, rubber bands, and even in human bones. The principle of Hooke's law underlies the dynamometer, a device used to measure force.
Ministry of Education of the Autonomous Republic of Crimea
Tauride National University named after. Vernadsky
Study of physical law
HOOKE'S LAW
Completed by: 1st year student
Faculty of Physics gr. F-111
Potapov Evgeniy
Simferopol-2010
Plan:
The connection between what phenomena or quantities is expressed by the law.
Statement of the law
Mathematical expression of the law.
How was the law discovered: based on experimental data or theoretically?
Experienced facts on the basis of which the law was formulated.
Experiments confirming the validity of the law formulated on the basis of the theory.
Examples of using the law and taking into account the effect of the law in practice.
Literature.
The relationship between what phenomena or quantities is expressed by the law:
Hooke's law relates phenomena such as stress and deformation of a solid, elastic modulus and elongation. The modulus of the elastic force arising during deformation of a body is proportional to its elongation. Elongation is a characteristic of the deformability of a material, assessed by the increase in the length of a sample of this material when stretched. Elastic force is a force that arises during deformation of a body and counteracts this deformation. Stress is a measure of internal forces that arise in a deformable body under the influence of external influences. Deformation is a change in the relative position of particles of a body associated with their movement relative to each other. These concepts are related by the so-called stiffness coefficient. It depends on the elastic properties of the material and the size of the body.
Statement of the law:
Hooke's law is an equation of the theory of elasticity that relates stress and deformation of an elastic medium.
The formulation of the law is that the elastic force is directly proportional to the deformation.
Mathematical expression of the law:
For a thin tensile rod, Hooke's law has the form:
Here F rod tension force, Δ l- its elongation (compression), and k called elasticity coefficient(or rigidity). The minus in the equation indicates that the tension force is always directed in the direction opposite to the deformation.
If you enter the relative elongation
and normal stress in the cross section
then Hooke's law will be written like this
In this form it is valid for any small volumes of matter.
In the general case, stress and strain are tensors of the second rank in three-dimensional space (they have 9 components each). The tensor of elastic constants connecting them is a tensor of the fourth rank C ijkl and contains 81 coefficients. Due to the symmetry of the tensor C ijkl, as well as stress and strain tensors, only 21 constants are independent. Hooke's law looks like this:
where σ ij- stress tensor, - strain tensor. For an isotropic material, the tensor C ijkl contains only two independent coefficients.
How was the law discovered: based on experimental data or theoretically:
The law was discovered in 1660 by the English scientist Robert Hooke (Hook) based on observations and experiments. The discovery, as stated by Hooke in his essay “De potentia restitutiva”, published in 1678, was made by him 18 years earlier, and in 1676 it was placed in another of his books under the guise of the anagram “ceiiinosssttuv”, meaning “Ut tensio sic vis” . According to the author's explanation, the above law of proportionality applies not only to metals, but also to wood, stones, horn, bones, glass, silk, hair, etc.
Experienced facts on the basis of which the law was formulated:
History is silent about this..
Experiments confirming the validity of the law formulated on the basis of the theory:
The law is formulated on the basis of experimental data. Indeed, when stretching a body (wire) with a certain stiffness coefficient k to a distance Δ l, then their product will be equal in magnitude to the force stretching the body (wire). This relationship will hold true, however, not for all deformations, but for small ones. With large deformations, Hooke's law ceases to apply and the body collapses.
Examples of using the law and taking into account the effect of the law in practice:
As follows from Hooke's law, the elongation of a spring can be used to judge the force acting on it. This fact is used to measure forces using a dynamometer - a spring with a linear scale calibrated for different force values.
Literature.
1. Internet resources: - Wikipedia website (http://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%93%D1%83 %D0%BA%D0%B0).
2. textbook on physics Peryshkin A.V. 9th grade
3. textbook on physics V.A. Kasyanov 10th grade
4. lectures on mechanics Ryabushkin D.S.
