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 to the 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 the normal stress that should arise in the body if its length is doubled. (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).
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-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.
If a certain force is applied to a body, its size and (or) shape changes. This process is called body deformation. In bodies undergoing deformation, elastic forces arise that balance external forces.
Types of deformation
All deformations can be divided into two types: elastic deformation And plastic.
Definition
Elastic deformation is called if, after removing the load, the previous dimensions of the body and its shape are completely restored.
Definition
Plastic consider deformation in which changes in the size and shape of the body that appeared due to deformation are partially restored after removing the load.
The nature of the deformation depends on
- magnitude and time of exposure to external load;
- body material;
- body condition (temperature, processing methods, etc.).
There is no sharp boundary between elastic and plastic deformation. In a large number of cases, small and short-term deformations can be considered elastic.
Statements of Hooke's law
It has been empirically found that the greater the deformation necessary to obtain, the greater the deforming force should be applied to the body. By the magnitude of the deformation ($\Delta l$) one can judge the magnitude of the force:
\[\Delta l=\frac(F)(k)\left(1\right),\]
expression (1) means that the absolute value of elastic deformation is directly proportional to the applied force. This statement is the content of Hooke's law.
When deforming elongation (compression) of a body, the following equality holds:
where $F$ is the deforming force; $l_0$ - initial body length; $l$ is the length of the body after deformation; $k$ - elasticity coefficient (stiffness coefficient, stiffness), $ \left=\frac(N)(m)$. The elasticity coefficient depends on the material of the body, its size and shape.
Since elastic forces ($F_u$) arise in a deformed body, which tend to restore the previous size and shape of the body, Hooke’s law is often formulated in relation to elastic forces:
Hooke's law works well for deformations that occur in rods made of steel, cast iron, and other solid substances, in springs. Hooke's law is valid for tensile and compressive deformations.
Hooke's law for small deformations
The elastic force depends on the change in the distance between parts of the same body. It should be remembered that Hooke's law is valid only for small deformations. With large deformations, the elastic force is not proportional to the length measurement; with a further increase in the deforming effect, the body can collapse.
If the deformations of the body are small, then the elastic forces can be determined by the acceleration that these forces impart to the bodies. If the body is motionless, then the modulus of the elastic force is found from the equality to zero of the vector sum of the forces that act on the body.
Hooke's law can be written not only in relation to forces, but it is often formulated for such a quantity as stress ($\sigma =\frac(F)(S)$ is the force that acts on a unit cross-sectional area of a body), then for small deformations:
\[\sigma =E\frac(\Delta l)(l)\ \left(4\right),\]
where $E$ is Young's modulus;$\ \frac(\Delta l)(l)$ is the relative elongation of the body.
Examples of problems with solutions
Example 1
Exercise. A load of mass $m$ is suspended from a steel cable of length $l$ and diameter $d$. What is the tension in the cable ($\sigma $), as well as its absolute elongation ($\Delta l$)?
Solution. Let's make a drawing.
In order to find the elastic force, consider the forces that act on a body suspended from a cable, since the elastic force will be equal in magnitude to the tension force ($\overline(N)$). According to Newton's second law we have:
In the projection onto the Y axis of equation (1.1) we obtain:
According to Newton's third law, a body acts on a cable with a force equal in magnitude to the force $\overline(N)$, the cable acts on a body with a force $\overline(F)$ equal to $\overline(\N,)$ but opposite direction, so the cable deforming force ($\overline(F)$) is equal to:
\[\overline(F)=-\overline(N\ )\left(1.3\right).\]
Under the influence of a deforming force, an elastic force arises in the cable, which is equal in magnitude to:
We find the voltage in the cable ($\sigma $) as:
\[\sigma =\frac(F_u)(S)=\frac(mg)(S)\left(1.5\right).\]
Area S is the cross-sectional area of the cable:
\[\sigma =\frac(4mg\ )((\pi d)^2)\left(1.7\right).\]
According to Hooke's law:
\[\sigma =E\frac(\Delta l)(l)\left(1.8\right),\]
\[\frac(\Delta l)(l)=\frac(\sigma )(E)\to \Delta l=\frac(\sigma l)(E)\to \Delta l=\frac(4mgl\ ) ((\pi d)^2E).\]
Answer.$\sigma =\frac(4mg\ )((\pi d)^2);\ \Delta l=\frac(4mgl\ )((\pi d)^2E)$
Example 2
Exercise. What is the absolute deformation of the first spring of two springs connected in series (Fig. 2), if the spring stiffness coefficients are equal: $k_1\ and\ k_2$, and the elongation of the second spring is $\Delta x_2$?
Solution. If a system of series-connected springs is in a state of equilibrium, then the tension forces of these springs are the same:
According to Hooke's law:
According to (2.1) and (2.2) we have:
Let us express from (2.3) the elongation of the first spring:
\[\Delta x_1=\frac(k_2\Delta x_2)(k_1).\]
Answer.$\Delta x_1=\frac(k_2\Delta x_2)(k_1)$.
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 devices 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 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
How many of us have ever wondered how amazingly objects behave when acted upon?
For example, why can fabric, if we stretch it in different directions, stretch for a long time, and then suddenly tear at one moment? And why is the same experiment much more difficult to carry out with a pencil? What does the resistance of a material depend on? How can you determine to what extent it can be deformed or stretched?
An English researcher asked himself all these and many other questions more than 300 years ago and found the answers, now united under the general name “Hooke’s Law.”
According to his research, every material has a so-called elasticity coefficient. This is a property that allows a material to stretch within certain limits. The elasticity coefficient is a constant value. This means that each material can only withstand a certain level of resistance, after which it reaches a level of irreversible deformation.
In general, Hooke's Law can be expressed by the formula:
where F is the elastic force, k is the already mentioned elasticity coefficient, and /x/ is the change in the length of the material. What is meant by a change in this indicator? Under the influence of force, a certain object under study, be it a string, rubber or any other, changes, stretching or compressing. The change in length in this case is the difference between the initial and final length of the object being studied. That is, how much the spring (rubber, string, etc.) has stretched/compressed.
From here, knowing the length and constant coefficient of elasticity for a given material, you can find the force with which the material is tensioned, or elastic force, as Hooke's Law is often called.
There are also special cases in which this law in its standard form cannot be used. We are talking about measuring the force of deformation under shear conditions, that is, in situations where the deformation is produced by a certain force acting on the material at an angle. Hooke's law under shear can be expressed as follows:
where τ is the desired force, G is a constant coefficient known as the shear modulus of elasticity, y is the shear angle, the amount by which the angle of inclination of the object has changed.