Deformations and displacements. Hooke's law
The action of external forces on a solid body leads to the appearance of stresses and strains at points in its volume. In this case, the stress state at a point, the relationship between stresses at different sites 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 connection 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 the loading conditions. Mathematical models describing the relationship between stresses and strains are developed on the basis of experimental data. These models should reflect the real properties of materials and loading conditions with a sufficient degree of accuracy.
The most common for structural materials are models of elasticity and plasticity. Elasticity is the property of a body to change shape and size under the action of external loads and restore its original configuration when the loads are 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 the loading conditions. For most structural materials, the elasticity property 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 the strain tensor are connected by linear relations.
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 fixed. In this case the relationship between stresses and strains ceases to be unambiguous. This material property is called plasticity. The residual deformations accumulated in the process of plastic deformation are called plastic.
A high level of stress can cause destruction, i.e., the division of the body into parts. Solid bodies made of different materials are destroyed at different amounts of deformation. Fracture is brittle at small strains 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. For low-carbon steels, non-ferrous metals, plastics, a plastic type of fracture is characteristic in the presence of significant residual deformations. However, the division of materials according to the nature of their destruction into brittle and ductile is very conditional; it usually refers to some standard operating conditions. One and the same material can behave, depending on the conditions (temperature, nature of the load, manufacturing technology, etc.), as brittle or as ductile. For example, materials that are plastic at normal temperatures are destroyed 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
Under such loading, there is an increase in dimensions 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 ratio is a mathematical notation Hooke's law, establishing a proportional relationship between stress and the corresponding linear deformation in a uniaxial stress state. The coefficient of proportionality E is called the modulus of longitudinal elasticity or Young's modulus. It has the dimension of stresses.
Along with the increase in size in the direction of action; under the same stress, the dimensions decrease in two orthogonal directions (Fig. 1). The corresponding deformations will be denoted by and , and these deformations are negative for positive ones 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 linear 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.
Angular deformation is due to shear stress , and deformations and , respectively, to stresses and . Between the corresponding shear stresses and angular deformations for a linearly elastic isotropic body, there are proportional relationships
which express the law Hook on shift. The proportionality factor G is called shear module. It is essential that the normal stress does not affect the angular deformations, since in this case only the linear dimensions of the segments change, and not the angles between them (Fig. 1).
A linear dependence also exists between the average stress (2.18), which is proportional to the first invariant of the stress tensor, and the volumetric strain (2.32), which coincides with the first invariant of the strain tensor:
Fig.2. Planar shear strain
Corresponding aspect ratio TO called bulk 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, two independent elastic characteristics are usually chosen as the elastic modulus E and Poisson's ratio. To express the shear modulus G through E And , Let us consider a plane shear deformation under the action of shear stresses (Fig. 2). To simplify the calculations, we use a square element with a side A. Calculate the principal stresses , . These stresses act on sites located at an angle to the original sites. From fig. 2 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
Given these ratios
Before deformation, this diagonal had the size . Then we will have
From the generalized Hooke's law (5) we obtain
Comparison of the obtained formula with the Hooke's law with 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 the experimental data of testing specimens for various types of loads. From the physical point of view, all these characteristics cannot be negative. In addition, it follows from the last expression 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, we express the stresses in terms of deformations from the elasticity relations (5). Let us write the first of relations (5) in the form
Using equality (9), we will have
Similar relations can be derived for and . As a result, we get
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Here relation (8) for the shear modulus is used. In addition, the designation
POTENTIAL ENERGY OF ELASTIC DEFORMATION
Consider first the elementary volume dV=dxdydz under conditions of uniaxial stress state (Fig. 1). Mentally fix the site x=0(Fig. 3). A force acts on the opposite side .
This force does work in displacement. .
As the voltage increases from zero to the value
the corresponding deformation, by virtue of Hooke's law, also increases from zero to the value ,
and the work is proportional to the shaded one in Fig. 4 squares: 
. If we neglect kinetic energy and losses associated with thermal, electromagnetic and other phenomena, then, by virtue of the law of conservation of energy, the work done will turn into potential energy accumulated during the deformation process: .
F= dU/dV called specific potential energy of deformation, meaningful potential energy accumulated per unit volume of the body. In the case of a uniaxial stress state
8.2. Basic Laws Used in Strength of Materials
Relationships of statics. 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. With a simple tension of the beam by the longitudinal force N= F this law can be written as:
Here 
longitudinal force, l- bar 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 shear stresses and shear angle:
.
G
calledshear modulus
, less often - the elastic modulus of the second kind. Like any law, it has a limit of applicability and Hooke's law. Voltage 
, up to which Hooke's law is valid, is called limit of proportionality(this is the most important characteristic in sopromat).
Let's depict the dependence
from
graphically (Fig. 8.1). This painting is called stretch diagram
. After point B (i.e. at 
), this dependence is no longer linear.
At 
after unloading, residual deformations appear in the body, therefore 
called elastic limit
.
When the stress 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 (i.e. behaves like a liquid). Graphically, this means that the diagram is parallel to the abscissa (DL plot). The stress σ t at which the material flows is called yield strength .
Some materials (Art. 3 - building 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 value 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 dependence is observed in experiments between tangential stresses and shears.
3) Dugamel–Neumann law (linear thermal expansion):
In the presence of a temperature difference, the body changes its size, and is directly proportional to this temperature difference.
Let there be a temperature difference 
. Then this law takes the form:
Here α - coefficient of linear thermal expansion, l - rod length, Δ l- its lengthening.
4) law of creep .
Studies have shown that all materials are highly inhomogeneous in the small. The schematic structure of steel is shown in Fig. 8.2.
Some of the components have fluid properties, so many materials under load gain additional elongation over time. 
(fig.8.3.) (metals at high temperatures, concrete, wood, plastics - at normal temperatures). This phenomenon is called creep material.
For a liquid, the law is true: how more power, the greater the speed of the body in the fluid. If this relationship is linear (i.e. force is proportional to speed), then it can be written as:
E 
If we go over 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 characteristic 
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 offset of point C.
Forces 
while doing some work U.
Considering that the forces 
begin to increase gradually and assuming that they increase in proportion to displacements, we get:
.
According to the conservation law: no work disappears, it is spent on doing other work or goes into another energy (energy is the work that the body can do.
The work of forces 
, is spent on overcoming the resistance of the elastic forces that arise in our body. To calculate this work, we take into account that the body can be considered as consisting of small elastic particles. Let's consider one of them:
From the side of neighboring particles, a stress acts on it 
. The resultant stress will be 
Under the influence 
the particle is elongated. By definition, elongation is the elongation per unit length. Then:
Let's calculate the work dW that the force does dN (here it is also taken into account that the forces dN begin to increase gradually and they increase in proportion to displacements):
For the whole body we get:
.
Job W committed 
, called elastic deformation energy.
According to the law of conservation of energy:
6)Principle possible movements .
This is one of the ways to write the law of conservation of energy.
Let forces act on the beam F 1
,
F 2
,
…
. They cause points to move in the body 
and stress 
. Let's give the body additional small possible displacements
. In mechanics, the record of the form 
means the phrase "possible value of the quantity A". These possible movements will cause in 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 displacements of those points where 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 the strain energy of all small elements, we obtain the total strain energy:
Law of energy conservation W = U gives:
.
This ratio is called principle of possible movements(also called principle of virtual movements). Similarly, we can consider the case when shear stresses also act. Then it can be obtained that the strain energy W add the following term:
Here - shear stress, - shear of a 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.
Consider the elongation pattern of the sample:
The phenomenon of shortening of a body element across the direction of lengthening is called Poisson effect.
Let us find the longitudinal relative deformation.
The transverse relative deformation will be:
Poisson's ratio 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 a Poisson ratio > 1, that is, the transverse deformation will be greater than the longitudinal one. For example, this is the case for material reinforced with hard fibers at a low angle. 
<<1
(см. рис.8.8.). Оказывается, что коэффициент
Пуассона при этом почти пропорционален
величине
, i.e. the less 
, the greater 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, a paradoxical result takes place - longitudinal elongation leads to an increase in the size of the body in the transverse direction.
8) Generalized Hooke's law.
Consider an element that stretches in the longitudinal and transverse directions. Let us find the deformation arising in these directions. 
Calculate the deformation 
arising from the action 
:
Consider the deformation from the action 
, which results from the Poisson effect:
The total deformation will be:
If it works and 
, then add one more shortening in the direction of the x-axis 
.
Hence: 
Similarly: 
These ratios are called generalized Hooke's law.
Interestingly, 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 the strength, on the contrary, strongly depends on the combination of shear 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 area of applicability.
Hooke's law usually referred to as linear relationships between strain components and stress components.
Take an elementary rectangular parallelepiped with faces parallel to the coordinate axes, loaded with normal stress σ x, uniformly distributed over two opposite faces (Fig. 1). Wherein y = σz = τ x y = τ x z = τ yz = 0.
Up to reaching the limit of proportionality, the relative elongation is given by the formula
Where E is the tensile modulus. For steel E = 2*10 5 MPa, therefore, the deformations are very small and are measured as a percentage or in 1 * 10 5 (in strain gauge instruments that measure deformations).
Extending an Element in the Axis Direction X is accompanied by its narrowing in the transverse direction, determined by the strain components
Where μ is a constant called the transverse compression ratio or Poisson's ratio. For steel μ usually taken equal to 0.25-0.3.
If the element under consideration is simultaneously loaded with normal stresses σ x, y, σz, uniformly distributed over its faces, then deformations are added
By superimposing the deformation components caused by each of the three stresses, we obtain the relations
These ratios are confirmed by numerous experiments. applied overlay method or superpositions to find the total strains and stresses caused by multiple 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 deformable body and small displacements 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 linearity of the relationships between forces and strains does not yet follow from the smallness of the displacements. So, for example, in a compressed Q rod loaded with an additional transverse force R, even with a small deflection δ there is an additional moment M = Qδ, which makes the problem non-linear. In such cases, the total deflections are not linear functions of the forces and cannot be obtained with a simple overlay (superposition).
It has been experimentally established that if shear stresses act on all faces of the element, then the distortion of the corresponding angle depends only on the corresponding shear stress components.
Constant G is called the shear modulus or shear modulus.
The general case of deformation of an element from the action of three normal and three tangential stress components on it can be obtained using superposition: three linear deformations determined by expressions (5.2a) are superimposed with three shear deformations determined by relations (5.2b). Equations (5.2a) and (5.2b) determine the relationship between the strain and stress components and are called generalized Hooke's law. Let us now show that the shear modulus G expressed in terms of tensile modulus E and Poisson's ratio μ . To do this, consider a special case where σ x = σ , y = -σ And σz = 0.
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 for the element 0 bс, normal stresses σ v on all faces of the element abcd are equal to zero, and shear stresses are equal
This stress state is called pure shift. Equations (5.2a) imply that
that is, the extension of the horizontal element 0 c equals the shortening of the vertical element 0 b: εy = -ε x.
Angle between faces ab And bc changes, and the corresponding amount of shear strain γ can be found from triangle 0 bс:
Hence it follows that
When a rod is stretched and compressed, its length and cross-sectional dimensions change. If we mentally select from the rod in the undeformed state an element of length dx, then after deformation its length will be equal to dx((Fig. 3.6). In this case, the absolute elongation in the direction of the axis Oh will be equal to
and relative linear deformation e x is defined by the equality
Since the axis Oh coincides with the axis of the rod, along which external loads act, we call the deformation e x longitudinal deformation, for which the index will be omitted below. Deformations in directions perpendicular to the axis are called transverse deformations. If denoted by b characteristic size of the cross section (Fig. 3.6), then the transverse deformation is determined by the relation 
Relative linear deformations are dimensionless quantities. It has been established that the transverse and longitudinal deformations during the central tension and compression of the rod are interconnected by the dependence
The quantity v included in this equality is called Poisson's ratio or coefficient of transverse strain. This coefficient is one of the main constants of elasticity of the material and characterizes its ability to transverse deformations. For each material, it is determined from a tensile or compression test (see § 3.5) and is calculated by the formula
As follows from equality (3.6), longitudinal and transverse strains always have opposite signs, which confirms the obvious fact that the cross-sectional dimensions decrease during tension, and increase during compression.
Poisson's ratio is different for different materials. For isotropic materials, it can take values ranging from 0 to 0.5. For example, for cork wood, Poisson's ratio is close to zero, while for rubber it is close to 0.5. For many metals at normal temperatures, the value of Poisson's ratio is in the range of 0.25 + 0.35.
As established in numerous experiments, for most structural materials at small strains, there is a linear relationship between stresses and strains
This law of proportionality was first established by the English scientist Robert Hooke and is called Hooke's law.
The constant included in Hooke's law E is called the modulus of elasticity. The modulus of elasticity is the second main constant of elasticity of a material and characterizes its rigidity. Since strains are dimensionless quantities, it follows from (3.7) that the modulus of elasticity has the dimension of stress.
In table. 3.1 shows the values of the modulus of elasticity and Poisson's ratio for various materials.
When designing and calculating structures, along with the calculation of stresses, it is also necessary to determine the displacements of individual points and nodes of structures. Consider a method for calculating displacements under central tension and compression of bars.
Absolute element extension length dx(Fig. 3.6) according to formula (3.5) is
Table 3.1
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Material name |
Modulus of elasticity, MPa |
Coefficient Poisson |
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Carbon steel |
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aluminum alloys |
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Titanium alloys |
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(1.15-s-1.6) 10 5 |
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along the fibers |
(0,1 ^ 0,12) 10 5 |
|
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across the fibers |
(0,0005 + 0,01)-10 5 |
|
|
(0,097 + 0,408) -10 5 |
||
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brickwork |
(0,027 +0,03)-10 5 |
|
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Fiberglass SVAM |
||
|
Textolite |
(0,07 + 0,13)-10 5 |
|
|
Rubber on rubber |
Integrating this expression in the range from 0 to x, we get
Where their) - axial displacement of an arbitrary section (Fig. 3.7), and C= and( 0) - axial displacement of the initial section x = 0. If this section is fixed, then u(0) = 0 and the displacement of an arbitrary section is
The elongation or shortening of the rod is equal to the axial displacement of its free end (Fig. 3.7), the value of which we obtain from (3.8), assuming x = 1:
Substituting into formula (3.8) the expression for the deformation? from Hooke's law (3.7), we obtain
For a rod made of a material with a constant modulus of elasticity E axial displacements are determined by the formula
The integral included in this equality can be calculated in two ways. The first way is to analytically write the function Oh) and subsequent integration. The second method is based on the fact that the integral under consideration is numerically equal to the plot area a in the section. Introducing the notation
Let's consider special cases. For a rod stretched by a concentrated force R(rice. 3.3, a), longitudinal force. / V is constant along the length and is equal to R. The stresses a according to (3.4) are also constant and equal to 
Then from (3.10) we obtain
It follows from this formula that if the stresses on a certain section of the rod are constant, then the displacements change according to a linear law. Substituting into the last formula x = 1, find the elongation of the rod:
Work EF called stiffness of the rod in tension and compression. The larger this value, the smaller the elongation or shortening of the rod.
Consider a rod under the action of a uniformly distributed load (Fig. 3.8). The longitudinal force in an arbitrary section, spaced at a distance x from the fastening, is equal to
Dividing N on F, we get the formula for stresses
Substituting this expression into (3.10) and integrating, we find
The largest displacement, equal to the elongation of the entire rod, is obtained by substituting x = / into (3.13):
From formulas (3.12) and (3.13) it can be seen that if the stresses depend linearly on x, then the displacements change according to the law of a square parabola. Plots N, oh and And shown in fig. 3.8.
General differential dependence linking functions their) and a(x), can be obtained from relation (3.5). Substituting e from Hooke's law (3.7) into this relation, we find
From this dependence follow, in particular, the patterns of change in the function noted in the above examples their).
In addition, it can be noted that if in any section the stresses a vanish, then on the diagram And there may be an extremum in this section.
As an example, let's build a diagram And for the rod shown in Fig. 3.2, putting E- 10 4 MPa. Calculating plot areas O for different areas, we find:
section x = 1 m:
section x = 3 m:
section x = 5 m:
On the upper section of the diagram bar And is a square parabola (Fig. 3.2, e). In this case, there is an extremum in the section x = 1 m. In the lower section, the character of the diagram is linear.
The total elongation of the rod, which in this case is equal to
can be calculated using formulas (3.11) and (3.14). Since the lower section of the rod (see Fig. 3.2, A) stretched by force R ( its lengthening according to (3.11) is equal to
Action of force R ( is also transmitted to the upper section of the rod. In addition, it is compressed by force R 2 and stretched by a uniformly distributed load q. In accordance with this, the change in its length is calculated by the formula
Summing up the values of A/, and A/ 2 , we get the same result as above.
In conclusion, it should be noted that, despite the small value of displacements and elongations (shortenings) of rods under tension and compression, they cannot be neglected. The ability to calculate these quantities is important in many technological problems (for example, when assembling structures), as well as for solving statically indeterminate problems.
