Stretch (compression) the rod arises from the action of external forces directed along its axis. Tension (compression) is characterized by: absolute lengthening (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 modulus of elasticity of the first kind (Young's modulus), Pa. The physical meaning of the Young's modulus: The modulus of elasticity is numerically equal to the stress at which the absolute elongation of the rod is equal to its original length, i.e. E= σ for ε=1.
14. Mechanical properties of structural materials. Stretch chart.
The mechanical properties of materials are 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 voltage resistance characteristic surface hardness HB, HRC; elasticity indicators relative elongation δ and relative transverse contraction φ; impact strength A.
Graphical representation of the relationship between 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 a straight-line relationship between normal stress and relative elongation, which reflects Hooke's law. Dot
1
corresponds to the limit of proportionality σ pts =F pts /A 0 , where F pts is the load corresponding to the proportionality limit. Dot
1`
corresponds to the elastic limit σy, i.e. the highest stress at which there is no permanent deformation in the material. IN point 2 diagram, the material enters the region of plasticity - the phenomenon of material flow occurs . Plot 2-3 parallel to the abscissa axis (yield area). On section 3-4 hardening of the material is observed. IN point 4 there is a local narrowing of the sample. The ratio σ in \u003d F in / A 0 is called the tensile strength. IN point 5 there is a rupture of the sample at the breaking load F res.
15. Permissible stresses. Calculations for allowable stresses.
The stresses at which a sample of a given material fails or at which significant plastic deformation, is called limiting. These stresses depend on the properties of the material and the type of deformation. Voltage, the value of which is regulated by technical conditions, is called allowed. Permissible stresses are set 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 setting loads, the service life of the structure, the accuracy of calculations for static and dynamic strength.
For ductile materials, the allowable stresses [σ] are chosen so that no residual deformations occur in the material under any inaccuracies in the calculation or unforeseen operating conditions, i.e. [σ] = σ 0.2 / [n] t, where [n] t is the safety factor in relation to σ t.
For brittle materials, allowable stresses are assigned from 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 occur during the design and operation of the structure.
DEFINITION
The deformation is elastic, in the event that it completely disappears when the deforming force stops.
Elastic deformation goes over inelastic (plastic), go over the elastic limit. Under elastic deformation, particles displaced to new equilibrium positions in crystal lattice, after removing the deforming force, take up the old places. The body completely restores its size and shape after the load is removed.
The law of elastic deformation
The English naturalist R. Hooke empirically obtained a direct relationship between the deforming force (F) and the elongation of the deformed spring (x). The external force generates the elastic forces of the body. These forces are equal in magnitude, the elastic force balances the action of the deformation force. Hooke's law is written as:
where is the projection of the force on the X axis; x - extension of the spring along the X axis; k - coefficient of elasticity of the spring (spring stiffness). When using such a quantity as the elastic force () for a deformed spring, then Hooke's law takes the form:
where is the projection of the elastic force on the X axis. The coefficient k is a value that depends on the material, the size of the coil of the spring and its length. Hooke's law is valid for small elongations and small loads.
The law of elastic deformation is valid for tension (compression) of an elastic rod. Usually, in this case, the elastic forces in the rod are described using stress.
At the same time, it is considered that the force is distributed evenly over the section and that it is perpendicular to the section surface. title="Rendered by QuickLaTeX.com" height="12" width="40" style="vertical-align: 0px;">, если происходит растяжение и при сжатии. Напряжение называют нормальным. При этом тангенциальное напряжение равно:!}
where is the elastic force that acts along the body layer; S is the area of the considered layer.
The change in the length of the rod () is equal to:
where E is Young's modulus; l is the length of the rod. Young's modulus characterizes the elastic properties of a material.
Law of elastic deformation in shear
A shear is a deformation in which flat layers solid body move parallel to each other. With this type of deformation, the layers do not change their shape and size. The measure of this deformation is the shear angle () or the amount of shear (). Hooke's law for elastic shear deformation is written as:
where G is the modulus of transverse elasticity (shear modulus), h is the thickness of the deformable layer; - shear angle.
All types of elastic deformation can be reduced to tensile or compressive deformations that occur simultaneously.
Examples of problem solving
EXAMPLE 1
| Exercise | The steel rod is heated from a temperature K to K. At the same time, in order for it not to increase its length, it is compressed with a force F. What is equal to given force, applied to both ends of the rod, if its cross-sectional area is equal to? |
| Solution | Based on the law of elastic deformation (Hooke's law), the rod should be compressed with force by the wound:
The elongation of the rod, which occurs when it is heated, we find as: We substitute the right side of expression (1.2) into Hooke's law, we have: Let's take Young's modulus for steel equal to Pa, coefficient of linear thermal expansion of steel |
| Answer | H |
. Let's do the calculations:Action external forces on a solid body leads to the occurrence 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 physical properties 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.
Most common for construction 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 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
|
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 samples for different kinds loads. From physical sense 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). We 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
|
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 platform 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 the 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
Hooke's law usually referred to as linear relationships between strain components and stress components.
Let's take elementary cuboid 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 . For this, consider special case, When 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 bc, 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 bc:
Hence it follows that
The device of dynamometers - devices for determining forces - is based on the fact that elastic deformation is directly proportional to the force that causes this deformation. An example of the above 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 the material on the type of the latter and on external conditions is called the modulus of elasticity. The modulus of elasticity is measured by the mechanical stress that must occur in the material during relative elastic deformation, equal to one.
Note that the relative elastic strain is usually expressed as a number much less than unity. With rare exceptions, it is practically impossible to get equal to unity, since the material is destroyed long before that. However, the modulus of elasticity can be found from experience as a ratio and for small since in formula (11.5) - a constant value.
The SI unit of modulus of elasticity is 1 Pa. (Prove it.)
Consider, as an example, the application of Hooke's law to the 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 the material
with a relative strain equal to one, i.e., with a doubling of the length of the sample. The numerical value of the Young's modulus is determined from experiments carried out within the limits of elastic deformation, and is taken from the tables in the calculations.
Since from (11.6) we get whence
Thus, the absolute deformation during longitudinal tension or compression is directly proportional to the force acting on the body and the length of the body, inversely proportional to the cross-sectional area of the body, and depends on the type of substance.
The greatest stress in the 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 is destroyed. The load at which the greatest possible mechanical stress occurs in the material is called destructive.
When building machines and structures, they always create a margin of safety. The margin of safety is a value showing how many times the actual maximum load in the most stressed part of the structure is less than the breaking load.
