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- its cross-sectional area, E- 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, 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, l - rod length, Δ l- 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 (here it is also taken into account that the forces dN 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.

According to the law of conservation of energy:

6)Principle possible movements .

This is one of the options for writing the law of conservation of energy.

Let the forces act on the beam F 1 , F 2 , … . 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 appearance of additional external forces and stresses
, δ.

Let us calculate the work of external forces on additional possible small displacements:

Here
- additional movements of those points at which forces are applied F 1 , F 2 , …

Consider again a small element with a cross section dA and length dz (see Fig. 8.5. and 8.6.). According to the definition, additional elongation  dz of this element is calculated by the formula:

dz=  dz.

The tensile force of the element will be:

dN = (+δ) dA ≈  dA..

The work of internal forces on additional displacements is calculated for a small element as follows:

dW = dN  dz = dA dz =  dV

WITH
summing up the deformation energy of all small elements we obtain the total deformation energy:

Law of energy conservation W = 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 W 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

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.

8) 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 the 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.

Note: 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.

As you know, physics studies all the laws of nature: from the simplest to the most general principles of natural science. Even in those areas where it would seem that physics is not able to understand, it still plays a primary role, and every smallest law, every principle - nothing escapes it.

In contact with

It is physics that is the basis of the foundations; it is this that lies at the origins of all sciences.

Physics studies the interaction of all bodies, both paradoxically small and incredibly large. Modern physics is actively studying not just small, but hypothetical bodies, and even this sheds light on the essence of the universe.

Physics is divided into sections, this simplifies not only the science itself and its understanding, but also the study methodology. Mechanics deals with the movement of bodies and the interaction of moving bodies, thermodynamics deals with thermal processes, electrodynamics deals with electrical processes.

Why should mechanics study deformation?

When talking about compression or tension, you should ask yourself the question: which branch of physics should study this process? With strong distortions, heat can be released, perhaps thermodynamics should deal with these processes? Sometimes when liquids are compressed, it begins to boil, and when gases are compressed, liquids are formed? So, should hydrodynamics understand deformation? Or molecular kinetic theory?

It all depends on the force of deformation, on its degree. If the deformable medium (material that is compressed or stretched) allows, and the compression is small, it makes sense to consider this process as the movement of some points of the body relative to others.

And since the question is purely related, it means that the mechanics will deal with it.

Hooke's law and the condition for its fulfillment

In 1660, the famous English scientist Robert Hooke discovered a phenomenon that can be used to mechanically describe the process of deformation.

In order to understand under what conditions Hooke's law is satisfied, Let's limit ourselves to two parameters:

• Wednesday;
• force.

There are media (for example, gases, liquids, especially viscous liquids close to solid states or, conversely, very fluid liquids) for which it is impossible to describe the process mechanically. Conversely, there are environments in which, with sufficiently large forces, the mechanics stop “working.”

Important! To the question: “Under what conditions is Hooke’s law true?”, a definite answer can be given: “At small deformations.”

Hooke's Law, definition: The deformation that occurs in a body is directly proportional to the force that causes that deformation.

Naturally, this definition implies that:

• compression or stretching is small;
• elastic object;
• it consists of a material in which there are no nonlinear processes as a result of compression or tension.

Hooke's Law in Mathematical Form

Hooke's formulation, which we cited above, makes it possible to write it in the following form:

where is the change in the length of the body due to compression or stretching, F is the force applied to the body and causes deformation (elastic force), k is the elasticity coefficient, measured in N/m.

It should be remembered that Hooke's law valid only for small stretches.

We also note that it has the same appearance when stretched and compressed. Considering that force is a vector quantity and has a direction, then in the case of compression, the following formula will be more accurate:

But again, it all depends on where the axis relative to which you are measuring will be directed.

What is the fundamental difference between compression and extension? Nothing if it is insignificant.

The degree of applicability can be considered as follows:

Let's pay attention to the graph. As we can see, with small stretches (the first quarter of the coordinates), for a long time the force with the coordinate has a linear relationship (red line), but then the real relationship (dotted line) becomes nonlinear, and the law ceases to be true. In practice, this is reflected by such strong stretching that the spring stops returning to its original position and loses its properties. With even more stretching a fracture occurs and the structure collapses material.

With small compressions (third quarter of the coordinates), for a long time the force with the coordinate also has a linear relationship (red line), but then the real relationship (dotted line) becomes nonlinear, and everything stops working again. In practice, this results in such strong compression that heat begins to be released and the spring loses its properties. With even greater compression, the coils of the spring “stick together” and it begins to deform vertically and then completely melt.

As you can see, the formula expressing the law allows you to find the force, knowing the change in the length of the body, or, knowing the elastic force, measure the change in length:

Also, in some cases, you can find the elasticity coefficient. To understand how this is done, consider an example task:

A dynamometer is connected to the spring. It was stretched by applying a force of 20, due to which it became 1 meter long. Then they released her, waited until the vibrations stopped, and she returned to her normal state. In normal condition, its length was 87.5 centimeters. Let's try to find out what material the spring is made of.

Let's find the numerical value of the spring deformation:

From here we can express the value of the coefficient:

Looking at the table, we can find that this indicator corresponds to spring steel.

Trouble with elasticity coefficient

Physics, as we know, is a very precise science; moreover, it is so precise that it has created entire applied sciences that measure errors. A model of unwavering precision, she cannot afford to be clumsy.

Practice shows that the linear dependence we considered is nothing more than Hooke's law for a thin and tensile rod. Only as an exception can it be used for springs, but even this is undesirable.

It turns out that the coefficient k is a variable value that depends not only on what material the body is made of, but also on the diameter and its linear dimensions.

For this reason, our conclusions require clarification and development, because otherwise, the formula:

can be called nothing more than a dependence between three variables.

Young's modulus

Let's try to figure out the elasticity coefficient. This parameter, as we found out, depends on three quantities:

• material (which suits us quite well);
• length L (which indicates its dependence on);
• area S.

Important! Thus, if we manage to somehow “separate” the length L and area S from the coefficient, then we will obtain a coefficient that completely depends on the material.

What we know:

• the larger the cross-sectional area of ​​the body, the greater the coefficient k, and the dependence is linear;
• the greater the body length, the lower the coefficient k, and the dependence is inversely proportional.

This means that we can write the elasticity coefficient in this way:

where E is a new coefficient, which now precisely depends solely on the type of material.

Let us introduce the concept of “relative elongation”:

. 

Conclusion

Let us formulate Hooke's law for tension and compression: For small compressions, normal stress is directly proportional to elongation.

The coefficient E is called Young's modulus and depends solely on the material.

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.

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

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 based on 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 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, 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

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:

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 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

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