Law of elastic deformation. Generalized Hooke's law

ELASTICITY, MODULE OF ELASTICITY, HOOKE'S LAW. Elasticity is the ability of a body to deform under load and restore its original shape and size after it is removed. The manifestation of elasticity is best observed by conducting a simple experiment with a spring balance - a dynamometer, the diagram of which is shown in Fig. 1.

With a load of 1 kg, the indicator arrow will move by 1 division, with 2 kg - by two divisions, and so on. If the loads are removed sequentially, the process proceeds reverse side. The dynamometer spring is an elastic body, its extension D l, firstly, proportional to the load P and, secondly, it completely disappears when the load is completely removed. If you build a graph, plot the load magnitude along the vertical axis, and the elongation of the spring along the horizontal axis, you get points lying on a straight line passing through the origin of coordinates, Fig. 2. This is true both for points depicting the loading process and for points corresponding to the load.

The angle of inclination of the straight line characterizes the ability of the spring to resist the action of the load: it is clear that the spring is “weak” (Fig. 3). These graphs are called spring characteristics.

The tangent of the slope of the characteristic is called the spring stiffness WITH. Now we can write the equation for the deformation of the spring D l = P/C

Spring stiffness WITH has a dimension of kg / cm\up122 and depends on the material of the spring (for example, steel or bronze) and its dimensions - the length of the spring, the diameter of its coil and the thickness of the wire from which it is made.

To one degree or another, all bodies that can be considered solid have the property of elasticity, but this circumstance cannot always be noticed: elastic deformations are usually very small and they can be observed without special instruments almost only when deforming plates, strings, springs, flexible rods .

A direct consequence of elastic deformations are elastic vibrations structures and natural objects. You can easily detect the shaking of the steel bridge on which the train passes; sometimes you can hear the clanging of dishes when a heavy truck passes on the street; all stringed musical instruments in one way or another convert elastic vibrations of the strings into vibrations of air particles; in percussion instruments, elastic vibrations (for example, drum membranes) are also converted into sound.

During an earthquake, elastic vibrations of the surface occur earth's crust; during a strong earthquake, in addition to elastic deformations, plastic deformations occur (which remain after the cataclysm as changes in the microrelief), and sometimes cracks appear. These phenomena do not relate to elasticity: we can say that in the process of deformation of a solid body, elastic deformations always appear first, then plastic deformations, and, finally, microcracks form. Elastic deformations are very small - no more than 1%, and plastic ones can reach 5-10% or more, so the usual idea of ​​​​deformations refers to plastic deformations - for example, plasticine or copper wire. However, despite their smallness, elastic deformations play a vital role in technology: strength calculations of airliners, submarines, tankers, bridges, tunnels, space rockets– this is, first of all, a scientific analysis of small elastic deformations that occur in the listed objects under the influence of operational loads.

Back in the Neolithic, our ancestors invented the first long-range weapon - a bow and arrow, using the elasticity of a curved tree branch; then catapults and ballistae, built for throwing large stones, used the elasticity of ropes twisted from plant fibers or even from women's long hair. These examples prove that the manifestation of elastic properties has long been known and used by people for a long time. But the understanding that any solid body under the influence of even small loads is necessarily deformed, albeit by a very small amount, first appeared in 1660 with Robert Hooke, a contemporary and colleague of the great Newton. Hooke was an outstanding scientist, engineer and architect. In 1676, he formulated his discovery very briefly, in the form of a Latin aphorism: “Ut tensio sic vis”, the meaning of which is that “as is the force, so is the elongation.” But Hooke did not publish this thesis, but only its anagram: “ceiiinosssttuu”. (In this way they ensured priority without disclosing the essence of the discovery.)

Probably, at this time, Hooke already understood that elasticity is a universal property of solids, but he considered it necessary to confirm his confidence experimentally. In 1678, Hooke’s book on elasticity was published, which described experiments from which it follows that elasticity is a property of “metals, wood, rocks, brick, hair, horn, silk, bone, muscle, glass, etc.” The anagram was also deciphered there. Robert Hooke's research led not only to the discovery of the fundamental law of elasticity, but also to the invention of spring chronometers (before that there were only pendulum ones). Studying various elastic bodies (springs, rods, bows), Hooke found that the “coefficient of proportionality” (in particular, the stiffness of the spring) strongly depends on the shape and size of the elastic body, although the material plays a decisive role.

More than a hundred years have passed, during which experiments with elastic materials were carried out by Boyle, Coulomb, Navier and some others, less famous physicists. One of the main experiments was stretching a test rod made of the material being studied. To compare results obtained in different laboratories, it was necessary either to always use the same samples, or to learn to eliminate the confluence of sample sizes. And in 1807, a book by Thomas Young appeared, in which the modulus of elasticity was introduced - a quantity that describes the elasticity property of a material, regardless of the shape and size of the sample used in the experiment. This requires strength P, attached to the sample, divided by the cross-sectional area F, and the resulting elongation D l divide by the original sample length l. The corresponding ratios are stress s and strain e.

Now Hooke's law of proportionality can be written as:

s = E e

Proportionality factor E called Young's modulus, has a dimension similar to stress (MPa), and its designation is the first letter of the Latin word elasticitat - elasticity.

Modulus of elasticity E is a characteristic of a material of the same type as its density or thermal conductivity.

Under normal conditions, significant force is required to deform a solid body. This means that the module E must be large compared to the ultimate stresses, after which elastic deformations are replaced by plastic ones and the shape of the body is noticeably distorted.

If we measure the modulus E in megapascals (MPa), the following average values ​​are obtained:

The physical nature of elasticity is related to electromagnetic interaction(including van der Waals forces in the crystal lattice). We can assume that elastic deformations are associated with changes in the distance between atoms.

An elastic rod has another fundamental property - it thins when stretched. The fact that ropes become thinner when stretched has been known for a long time, but special experiments have shown that when an elastic rod is stretched, a regularity always occurs: if you measure the transverse deformation e ", i.e., a decrease in the width of the rod d b, divided by the original width b, i.e.

and divide it by the longitudinal deformation e, then this ratio remains constant for all values ​​of the tensile force P, that is

(It is believed that e " < 0 ; therefore the absolute value is used). Constant v is called Poisson's ratio (named after the French mathematician and mechanic Simon Denis Poisson) and depends only on the material of the rod, but does not depend on its size and cross-sectional shape. The value of Poisson's ratio for different materials varies from 0 (for cork) to 0.5 (for rubber). In the latter case, the volume of the sample does not change during stretching (such materials are called incompressible). For metals the values ​​are different, but close to 0.3.

Modulus of elasticity E and Poisson's ratio together form a pair of quantities that fully characterize the elastic properties of any specific material (this refers to isotropic materials, i.e. those whose properties do not depend on direction; the example of wood shows that this is not always the case - its properties along fibers and across the fibers vary greatly. This is an anisotropic material. Anisotropic materials are single crystals, many composites such as fiberglass. Such materials also have elasticity within certain limits, but the phenomenon itself turns out to be much more complex).

The design of dynamometers - devices for determining forces - is based on the fact that elastic deformation is directly proportional to the force causing this deformation. An example of this is the well-known spring steelyard.

The connection between elastic deformations and internal forces in a material was first established by the English scientist R. Hooke. Currently, Hooke's law is formulated as follows: mechanical stress in an elastically deformed body is directly proportional to the relative deformation of this body

The value characterizing the dependence of mechanical stress in a material on the type of the latter and on external conditions is called the elastic modulus. The elastic modulus is measured by the mechanical stress that must arise in the material when the relative elastic deformation is equal to unity.

Note that relative elastic deformation is usually expressed by a number much less than unity. With rare exceptions, it is almost impossible to get equal to one, since the material is destroyed long before that. However, the elastic modulus can be found from experience as a ratio and at small values ​​in formula (11.5) it is a constant value.

The SI unit of elastic modulus is 1 Pa. (Prove it.)

Let's take an example of the application Hooke's law to deformation of one-sided tension or compression. Formula (11.5) for this case takes the form

where E denotes the modulus of elasticity for this type of deformation; it is called Young's modulus. Young's modulus is measured by the normal stress that must occur in a material

at a relative deformation equal to unity, i.e., when the length of the sample is doubled Numeric value Young's modulus is determined from experiments carried out within the limits of elastic deformation, and is taken from tables in calculations.

Since from (11.6) we obtain

Thus, the absolute deformation during longitudinal tension or compression is directly proportional to the force and length of the body acting on the body, and inversely proportional to the area cross section body and depends on the type of substance.

The greatest stress in a material, after the disappearance of which the shape and volume of the body are restored, is called the elastic limit. Formulas (11.5) and (11.7) are valid until the elastic limit is passed. When the elastic limit is reached, plastic deformations occur in the body. In this case, a moment may come when, under the same load, the deformation begins to increase and the material collapses. The load at which the greatest possible mechanical stress occurs in the material is called destructive.

When building machines and structures, a safety margin is always created. The safety factor is a value that shows how many times the actual maximum load in the most stressed place of the structure is less than the breaking load.

Tension (compression) the rod arises from the action of external forces directed along its axis. Tension (compression) is characterized by: - ​​absolute elongation (shortening) Δ l;

 relative longitudinal deformation ε= Δ l/l

 relative transverse deformation ε`= Δ a/ a= Δ b/ b

With elastic deformations between σ and ε there is a dependence described by Hooke’s law, ε=σ/E, where E is the elastic modulus of the first kind (Young’s modulus), Pa. The physical meaning of Young’s modulus: The elastic modulus is numerically equal to the stress at which the absolute elongation of the rod is equal to its original length, i.e. E= σ with ε=1.

14. Mechanical properties of structural materials. Tension diagram.

The mechanical properties of materials include strength indicators tensile strength σ in, yield strength σ t, and endurance limit σ -1; stiffness characteristic elastic modulus E and shear modulus G; contact stress resistance characteristic surface hardness NV, HRC; elasticity indicators relative elongation δ and relative transverse contraction φ; impact strength A.

Graphical representation of the relationship between the acting force F and elongation Δl called stretch diagram(compression) sample Δl= f(F).

X characteristic points and sections of the diagram: 0-1  section of the linear relationship between normal stress and relative elongation, which reflects Hooke’s law. Dot 1 corresponds to the proportionality limit σ pc =F pc /A 0, where F pc is the load corresponding to the proportionality limit. Dot 1` corresponds to the elastic limit σ y, i.e. the highest stress at which there are still no residual deformations in the material. IN point 2 diagram, the material enters the plasticity region - the phenomenon of material fluidity occurs . Section 2-3 parallel to the x-axis (yield area). On section 3-4 strengthening of the material is observed. IN point 4 local narrowing of the sample occurs. The ratio σ in =F in /A 0 is called tensile strength. IN point 5 the sample ruptures under a destructive load F size.

15. Permissible stresses. Calculations based on permissible stresses.

The stresses at which a sample of a given material fails or at which significant plastic deformations develop are called extreme. These stresses depend on the properties of the material and the type of deformation. The voltage, the value of which is regulated by technical specifications, is called acceptable. Allowable stresses are established taking into account the material of the structure and the variability of its mechanical properties during operation, the degree of responsibility of the structure, the accuracy of the loads, the service life of the structure, the accuracy of calculations for static and dynamic strength.

For plastic materials, the permissible stresses [σ] are chosen so that in case of any calculation inaccuracies or unforeseen operating conditions, residual deformations do not occur in the material, i.e. [σ] = σ 0.2 /[n] t, where [n] t is the safety factor in relation to σ t.

For brittle materials, permissible stresses are assigned based on the condition that the material does not collapse. In this case, [σ] = σ in /[n] in. Thus, the safety factor [n] has a complex structure and is intended to guarantee the strength of the structure against any accidents and inaccuracies that arise during the design and operation of the structure.

Action 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 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 plastic 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

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 modulus. 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:

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 various types 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

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

Hooke's law usually called linear relationships between strain components and stress components.

Let's take an elementary cuboid with edges parallel to the coordinate axes, loaded with normal stress σ x, evenly distributed over two opposite faces (Fig. 1). At the same time 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 direction of the axis 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 equal to 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 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 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 = -εx.

Angle between faces ab And bc changes, and the corresponding shear strain value γ can be found from triangle 0 bс:

It follows that