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.
8.2. Basic laws used in strength of materials
Statics relations. They are written in the form of the following equilibrium equations.
Hooke's law ( 1678): the greater the force, the greater the deformation, and, moreover, is directly proportional to the force. Physically, this means that all bodies are springs, but with great rigidity.= When a beam is simply stretched by a longitudinal force N
F 
this law can be written as: Here longitudinal force, l- beam length, A- coefficient of elasticity of the first kind ( Young's modulus).
Taking into account the formulas for stresses and strains, Hooke’s law is written as follows: 
.
A similar relationship is observed in experiments between tangential stresses and shear angle:
.
G
calledshear modulus
, less often – elastic modulus of the second kind. Like any law, Hooke's law also has a limit of applicability. Voltage 
, up to which Hooke's law is valid, is called limit of proportionality(this is the most important characteristic in strength of materials).
Let's depict the dependence
from
graphically (Fig. 8.1). This picture is called stretch diagram
. After point B (i.e. at 
) this dependence ceases to be linear.
At 
after unloading, residual deformations appear in the body, therefore 
called elastic limit
.
When the voltage reaches the value σ = σ t, many metals begin to exhibit a property called fluidity. This means that even under constant load, the material continues to deform (that is, it behaves like a liquid). Graphically, this means that the diagram is parallel to the abscissa (section DL). The voltage σ t at which the material flows is called yield strength .
Some materials (St. 3 - construction steel) after a short flow begin to resist again. The resistance of the material continues up to a certain maximum value σ pr, and then gradual destruction begins. The quantity σ pr is called tensile strength (synonym for steel: tensile strength, for concrete - cubic or prismatic strength). The following designations are also used:
=R b
A similar relationship is observed in experiments between shear stresses and shears.
3) Duhamel–Neumann law (linear temperature expansion):
In the presence of a temperature difference, bodies change their size, and in direct proportion to this temperature difference.
Let there be a temperature difference 
. Then this law looks like:
Here α - coefficient of linear thermal expansion, Here - rod length, Δ Here- its lengthening.
4) Law of Creep .
Research has shown that all materials are highly heterogeneous in small areas. The schematic structure of steel is shown in Fig. 8.2.
Some of the components have the properties of a liquid, so many materials under load receive additional elongation over time 
(Fig. 8.3.) (metals at high temperatures, concrete, wood, plastics - at normal temperatures). This phenomenon is called creep material.
The law for liquids is: the greater the force, the greater the speed of movement of the body in the liquid. If this relationship is linear (i.e. force is proportional to speed), then it can be written as:
E 
If we move on to relative forces and relative elongations, we get
Here the index " cr
" means that the part of the elongation that is caused by the creep of the material is considered. Mechanical characteristics 
called the viscosity coefficient.
Law of energy conservation.
Consider a loaded beam
Let us introduce the concept of moving a point, for example,
- vertical movement of point B;
- horizontal displacement of point C.
Powers 
while doing some work U.
Considering that the forces 
begin to increase gradually and assuming that they increase in proportion to displacements, we obtain:
.
According to the conservation law: no work disappears, it is spent on doing other work or turns into another energy (energy- this is the work that the body can do.).
Work of forces 
, is spent on overcoming the resistance of elastic forces arising in our body. To calculate this work, we take into account that the body can be considered to consist of small elastic particles. Let's consider one of them:
It is subject to tension from neighboring particles 
. 
The resultant stress will be 
Under the influence
the particle will elongate. According to the definition, elongation is the elongation per unit length. Then: Let's calculate the work dW , which the force does dN , which the force does(here it is also taken into account that the forces
begin to increase gradually and they increase proportionally to the movements):
.
For the whole body we get: Job W 
which was committed , called
elastic deformation energy.
6)According to the law of conservation of energy: Principle .
possible movements
This is one of the options for writing the law of conservation of energy. When a beam is simply stretched by a longitudinal force 1
,
When a beam is simply stretched by a longitudinal force 2
,
…
Let the forces act on the beam 
. They cause points to move in the body 
and voltage . Let's give the body
additional small possible movements 
. In mechanics, a notation of the form means the phrase “possible value of the quantity A " These possible movements will cause the body
additional possible deformations . They will lead to the emergence of additional external forces 
,
δ.
and stresses
F 
Let us calculate the work of external forces on additional possible small displacements: When a beam is simply stretched by a longitudinal force 1
,
When a beam is simply stretched by a longitudinal force 2
,
…
- additional movements of those points at which forces are applied Consider again a small element with a cross section dA and length dz and length(see Fig. 8.5. and 8.6.). According to the definition, additional elongation
and length= of this element is calculated by the formula:
dz.
, which the force does = (+δ) Consider again a small element with a cross section ≈ Consider again a small element with a cross section..
The tensile force of the element will be:
The work of internal forces on additional displacements is calculated for a small element as follows: dW = dN dz = dW = dN dA
dV 
WITH
summing up the deformation energy of all small elements we obtain the total deformation energy: Job = U gives:
.
This ratio is called principle of possible movements(it is also called principle of virtual movements). Similarly, we can consider the case when tangential stresses also act. Then we can obtain that to the deformation energy Job the following term will be added:
Here is the shear stress, is the displacement of the small element. Then principle of possible movements will take the form:
Unlike the previous form of writing the law of conservation of energy, there is no assumption here that the forces begin to increase gradually, and they increase in proportion to the displacements
7) Poisson effect.
Let us consider the pattern of sample elongation:
The phenomenon of shortening a body element across the direction of elongation is called Poisson effect.
Let us find the longitudinal relative deformation.
The transverse relative deformation will be:
Poisson's ratio the quantity is called:
For isotropic materials (steel, cast iron, concrete) Poisson's ratio
This means that in the transverse direction the deformation less longitudinal
Note
: modern technologies can create composite materials with Poisson's ratio >1, that is, the transverse deformation will be greater than the longitudinal one. For example, this is the case for a material reinforced with rigid fibers at a low angle 
<<1
(см. рис.8.8.). Оказывается, что коэффициент
Пуассона при этом почти пропорционален
величине
, i.e. the less 
, the larger the Poisson's ratio.
Fig.8.8.
Fig.8.9
8) Even more surprising is the material shown in (Fig. 8.9.), and for such reinforcement there is a paradoxical result - longitudinal elongation leads to an increase in the size of the body in the transverse direction.
Generalized Hooke's law. 
Let's consider an element that stretches in the longitudinal and transverse directions. Let us find the deformation that occurs in these directions. 
Let's calculate the deformation 
:
arising from action 
Let's consider the deformation from the action
, which arises as a result of the Poisson effect:
The overall deformation will be: 
If valid and 
.
, then another shortening will be added in the direction of the x axis 
Hence: 
Likewise: These relations are called
generalized Hooke's law.
It is interesting that when writing Hooke’s law, an assumption is made about the independence of elongation strains from shear strains (about independence from shear stresses, which is the same thing) and vice versa. Experiments well confirm these assumptions. Looking ahead, we note that strength, on the contrary, strongly depends on the combination of tangential and normal stresses. The above laws and assumptions are confirmed by numerous direct and indirect experiments, but, like all other laws, they have a limited scope of applicability.
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.
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 When a beam is simply stretched by a longitudinal force rod tension force, Δ Here- 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 work “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.
Elasticity coefficient
Elasticity coefficient(sometimes called Hooke's coefficient, stiffness coefficient or spring stiffness) - a coefficient that in Hooke's law relates the elongation of an elastic body and the elastic force resulting from this elongation. It is used in solid mechanics in the section of elasticity. Denoted by the letter k, Sometimes D or c. It has the dimension N/m or kg/s2 (in SI), dyne/cm or g/s2 (in GHS).
The elasticity coefficient is numerically equal to the force that must be applied to the spring in order for its length to change per unit distance.
Definition and properties
The elasticity coefficient, by definition, is equal to the elastic force divided by the change in spring length: k = F e / Δ l. (\displaystyle k=F_(\mathrm (e) )/\Delta l.) The elasticity coefficient depends both on the properties of the material and on the dimensions of the elastic body. Thus, for an elastic rod, we can distinguish the dependence on the dimensions of the rod (cross-sectional area S (\displaystyle S) and length L (\displaystyle L)), writing the elasticity coefficient as k = E ⋅ S / L. (\displaystyle k=E\cdot S/L.) The quantity E (\displaystyle E) is called Young's modulus and, unlike the elasticity coefficient, depends only on the properties of the material of the rod.
Stiffness of deformable bodies when they are connected
Parallel connection of springs. 
Series connection of springs. When connecting several elastically deformable bodies (hereinafter referred to as springs for brevity), the overall rigidity of the system will change. With a parallel connection, the stiffness increases, with a series connection it decreases.
Parallel connection
With a parallel connection of n (\displaystyle n) springs with stiffnesses equal to k 1 , k 2 , k 3 , . . . , k n , (\displaystyle k_(1),k_(2),k_(3),...,k_(n),) the rigidity of the system is equal to the sum of the rigidities, that is, k = k 1 + k 2 + k 3 + . . . +kn. (\displaystyle k=k_(1)+k_(2)+k_(3)+...+k_(n).)
Proof
In a parallel connection there are n (\displaystyle n) springs with stiffnesses k 1 , k 2 , . . . ,kn. (\displaystyle k_(1),k_(2),...,k_(n).) From Newton's III law, F = F 1 + F 2 + . . . +Fn. (\displaystyle F=F_(1)+F_(2)+...+F_(n).) (A force F is applied to them (\displaystyle F). At the same time, a force F 1 is applied to spring 1, (\displaystyle F_(1),) to spring 2 force F 2 , (\displaystyle F_(2),) ... , to spring n (\displaystyle n) force F n (\displaystyle F_(n).))
Now from Hooke’s law (F = − k x (\displaystyle F=-kx), where x is the elongation) we derive: F = k x ; F 1 = k 1 x ; F 2 = k 2 x ; . . . ; F n = k n x . (\displaystyle F=kx;F_(1)=k_(1)x;F_(2)=k_(2)x;...;F_(n)=k_(n)x.) Substitute these expressions into the equality (1): k x = k 1 x + k 2 x + . . . + k n x ; (\displaystyle kx=k_(1)x+k_(2)x+...+k_(n)x;) reducing by x, (\displaystyle x,) we get: k = k 1 + k 2 + . . . + k n , (\displaystyle k=k_(1)+k_(2)+...+k_(n),) which is what needed to be proven.
Serial connection
With a series connection of n (\displaystyle n) springs with stiffnesses equal to k 1 , k 2 , k 3 , . . . , k n , (\displaystyle k_(1),k_(2),k_(3),...,k_(n),) the total stiffness is determined from the equation: 1 / k = (1 / k 1 + 1 / k 2 + 1 / k 3 + . (\displaystyle 1/k=(1/k_(1)+1/k_(2)+1/k_(3)+...+1/k_(n)).)
Proof
In a series connection there are n (\displaystyle n) springs with stiffnesses k 1 , k 2 , . . . ,kn. (\displaystyle k_(1),k_(2),...,k_(n).) From Hooke’s law (F = − k l (\displaystyle F=-kl) , where l is the elongation) it follows that F = k ⋅ l . (\displaystyle F=k\cdot l.) The sum of the elongations of each spring is equal to the total elongation of the entire connection l 1 + l 2 + . . . + l n = l . (\displaystyle l_(1)+l_(2)+...+l_(n)=l.)
Each spring is subject to the same force F. (\displaystyle F.) According to Hooke's law, F = l 1 ⋅ k 1 = l 2 ⋅ k 2 = . . . = l n ⋅ k n . (\displaystyle F=l_(1)\cdot k_(1)=l_(2)\cdot k_(2)=...=l_(n)\cdot k_(n).) From the previous expressions we deduce: l = F / k, l 1 = F / k 1, l 2 = F / k 2, . . . , l n = F / k n . (\displaystyle l=F/k,\quad l_(1)=F/k_(1),\quad l_(2)=F/k_(2),\quad ...,\quad l_(n)= F/k_(n).) Substituting these expressions into (2) and dividing by F, (\displaystyle F,) we get 1 / k = 1 / k 1 + 1 / k 2 + . . . + 1 / k n , (\displaystyle 1/k=1/k_(1)+1/k_(2)+...+1/k_(n),) which is what needed to be proven.
Stiffness of some deformable bodies
Constant cross-section rod
A homogeneous rod of constant cross-section, elastically deformed along the axis, has a stiffness coefficient
K = E S L 0 , (\displaystyle k=(\frac (E\,S)(L_(0))),) A- Young's modulus, which depends only on the material from which the rod is made; S- cross-sectional area; L 0 - length of the rod.
Cylindrical coil spring
Twisted cylindrical compression spring. A twisted cylindrical compression or tension spring, wound from a cylindrical wire and elastically deformed along the axis, has a stiffness coefficient
K = G ⋅ d D 4 8 ⋅ d F 3 ⋅ n , (\displaystyle k=(\frac (G\cdot d_(\mathrm (D) )^(4))(8\cdot d_(\mathrm (F ) )^(3)\cdot n)),) d- The diameter of the wire; d F - winding diameter (measured from the wire axis); n- number of turns; G- shear modulus (for ordinary steel G≈ 80 GPa, for spring steel G≈ 78.5 GPa, for copper ~ 45 GPa).
Sources and notes
- Elastic deformation (Russian). Archived June 30, 2012.
- Dieter Meschede, Christian Gerthsen. Physik. - Springer, 2004. - P. 181 ..
- Bruno Assmann. Technische Mechanik: Kinematik und Kinetik. - Oldenbourg, 2004. - P. 11 ..
- Dynamics, Elastic force (Russian). Archived June 30, 2012.
- Mechanical properties of bodies (Russian). Archived June 30, 2012.
10. Hooke's law in tension-compression. Modulus of elasticity (Young's modulus).
Under axial tension or compression to the limit of proportionality σ pr Hooke's law is valid, i.e. law on the directly proportional relationship between normal stresses
and longitudinal relative deformations 
:
(3.10)
or 
(3.11)
Here E - the proportionality coefficient in Hooke's law has the dimension of voltage and is called modulus of elasticity of the first kind, characterizing the elastic properties of the material, or Young's modulus.
Relative longitudinal strain is the ratio of the absolute longitudinal strain of the section 
rod to the length of this section 
before deformation: 
(3.12)
The relative transverse deformation will be equal to: " = = b/b, where b = b 1 – b.
The ratio of the relative transverse deformation " to the relative longitudinal deformation , taken modulo, is a constant value for each material and is called Poisson's ratio:
Determination of the absolute deformation of a section of timber
In formula (3.11) instead 
And 
Let's substitute expressions (3.1) and (3.12):
From here we obtain a formula for determining the absolute elongation (or shortening) of a section of a rod with length :
(3.13)
In formula (3.13) the product EA is called the rigidity of the beam in tension or compression, which is measured in kN, or MN.
This formula determines the absolute deformation if the longitudinal force is constant in the area. In the case where the longitudinal force is variable in the area, it is determined by the formula:
(3.14)
where N(x) is a function of the longitudinal force along the length of the section.
11. Transverse strain coefficient (Poisson's ratio
12.Determination of displacements during tension and compression. Hooke's law for a section of timber. Determination of displacements of beam sections
Let's determine the horizontal movement of the point means the phrase “possible value of the quantity axis of the beam (Fig. 3.5) – u a: it is equal to the absolute deformation of part of the beam means the phrase “possible value of the quantityd, enclosed between the embedment and the section drawn through the point, i.e. 
In turn, lengthening the section means the phrase “possible value of the quantityd consists of extensions of individual cargo sections 1, 2 and 3:
Longitudinal forces in the areas under consideration:
Hence,
Then 
Similarly, you can determine the movement of any section of a beam and formulate the following rule:
moving any section jof a rod under tension-compression is determined as the sum of absolute deformations ncargo areas enclosed between the considered and fixed (fixed) sections, i.e.
(3.16)
The condition for the rigidity of the beam will be written in the following form:
, (3.17)
Where 
– the greatest value of the section displacement, taken modulo from the displacement diagram; u – the permissible value of the section displacement for a given structure or its element, established in the standards.
13. Determination of mechanical characteristics of materials. Tensile test. Compression test.
To quantify the basic properties of materials, such as
As a rule, the tension diagram is experimentally determined in coordinates and (Fig. 2.9). Characteristic points are marked on the diagram. Let's define them.
The highest stress to which a material follows Hooke's law is called limit of proportionality P. Within the limits of Hooke's law, the tangent of the angle of inclination of the straight line = f() to the axis is determined by the value A.
The elastic properties of the material are maintained up to stress U called elastic limit. Below the elastic limit U is understood as the greatest stress up to which the material does not receive residual deformations, i.e. after complete unloading, the last point of the diagram coincides with the starting point 0.
Value T called yield strength material. The yield strength is understood as the stress at which strain increases without a noticeable increase in load. If it is necessary to distinguish between the yield strength in tension and compression T accordingly replaced by TR and TS. At voltages high T plastic deformations develop in the body of the structure P, which do not disappear when the load is removed.
The ratio of the maximum force that a sample can withstand to its initial cross-sectional area is called tensile strength, or tensile strength, and is denoted by VR(with compression Sun).
When performing practical calculations, the real diagram (Fig. 2.9) is simplified, and for this purpose various approximating diagrams are used. To solve problems taking into account elastically plastic properties of structural materials is most often used Prandtl diagram. According to this diagram, the stress changes from zero to the yield strength according to Hooke’s law = A, and then as increases, = T(Fig. 2.10).
The ability of materials to obtain residual deformations is called plasticity. In Fig. 2.9 presented a characteristic diagram for plastic materials.
Rice. 2.10 Fig. 2.11
The opposite of the property of plasticity is the property fragility, i.e. the ability of a material to collapse without the formation of noticeable residual deformations. A material with this property is called fragile. Brittle materials include cast iron, high-carbon steel, glass, brick, concrete, and natural stones. A typical diagram of the deformation of brittle materials is shown in Fig. 2.11.
1. What is body deformation called? How is Hooke's law formulated?
Vakhit Shavaliev
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.
Elastic deformations are deformations that completely disappear after the removal of external forces.
Plastic deformations are deformations that remain fully or partially after the action of external forces ceases.
Elastic forces are forces that arise in a body during its elastic deformation and are directed in the direction opposite to the displacement of particles during deformation.
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:
The elastic force that arises during deformation of a body is directly proportional to the absolute elongation of the body and is directed in the direction opposite to the displacement of the particles of the body:
\
where F_x is the projection of force on the x-axis, k is the rigidity 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 N/m.
http://ru.solverbook.com/spravochnik/mexanika/dinamika/deformacii-sily-uprugosti/
Varya Guseva
Deformation is a change in the shape or volume of a body. Types of deformation - stretching or compression (examples: stretching or squeezing an elastic band, accordion), bending (a board bent under a person, a sheet of paper bent), torsion (working with a screwdriver, squeezing out laundry by hand), shear (when a car brakes, the tires are deformed due to the friction force ) .
Hooke's law: The elastic force arising in a body during its deformation is directly proportional to the magnitude of this deformation
or
The elastic force that arises in a body during its deformation is directly proportional to the magnitude of this deformation.
Hooke's law formula: Fpr=kx
Hooke's law. Can it be expressed by the formula F= -khх or F= khх?
⚓ Otters ☸
Hooke's law is an equation of the theory of elasticity that relates stress and deformation of an elastic medium. Discovered in 1660 by the English scientist Robert Hooke. Since Hooke's law is written for small stresses and strains, it has the form of simple proportionality.
For a thin tensile rod, Hooke's law has the form:
Here F is the tension force of the rod, Δl is its elongation (compression), and k is called the elasticity coefficient (or rigidity). The minus in the equation indicates that the tension force is always directed in the direction opposite to the deformation.
The elasticity coefficient depends both on the properties of the material and on the dimensions of the rod. We can distinguish the dependence on the dimensions of the rod (cross-sectional area S and length L) explicitly by writing the elasticity coefficient as
The quantity E is called Young's modulus and depends only on the properties of the body.
If you enter the relative elongation
and normal stress in the cross section
then Hooke's law will be written as
In this form it is valid for any small volumes of matter.
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Generalized Hooke's law
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 Cijkl and contains 81 coefficients. Due to the symmetry of the Cijkl tensor, as well as the stress and strain tensors, only 21 constants are independent. Hooke's law looks like this:
For an isotropic material, the Cijkl tensor contains only two independent coefficients.
It should be borne in mind that Hooke's law is satisfied only for small deformations. When the proportionality limit is exceeded, the relationship between stress and strain becomes nonlinear. For many media, Hooke's law is not applicable even at small deformations.
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in short, you can do it this way or that, depending on what you want to indicate in the end: simply the modulus of the Hooke force or also the direction of this force. Strictly speaking, of course, -kx, since the Hooke force is directed against the positive increment in the coordinate of the end of the spring.
The action of external forces on a solid body leads to the occurrence of stresses and deformations at points in its volume. In this case, the stressed 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 the physical properties of the 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 of 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 of different materials fail 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, materials that are plastic at normal temperatures break down as brittle at 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, a decrease in size occurs 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 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 A and Poisson's ratio. To express the shear modulus G through A 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 under various types of 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 we 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 energy of deformation, having the meaning of potential energy accumulated in a unit volume of a body. In the case of a uniaxial stress state
