Basic concepts of solid mechanics. General properties of solids
The mechanics of a deformable solid body is a science in which the laws of equilibrium and motion of solid bodies are studied under the conditions of their deformation under various influences. The deformation of a solid body is that its size and shape change. With this property of solids as elements of structures, structures and machines, the engineer constantly encounters in his practical activities. For example, a rod lengthens under the action of tensile forces, a beam loaded with a transverse load bends, etc.
Under the action of loads, as well as under thermal influences, internal forces arise in solids, which characterize the resistance of the body to deformation. Internal forces per unit area are called voltages.
The study of the stressed and deformed states of solids under various influences is the main problem of the mechanics of a deformable solid.
The resistance of materials, the theory of elasticity, the theory of plasticity, the theory of creep are sections of the mechanics of a deformable solid body. In technical, in particular construction, universities, these sections are of an applied nature and serve to develop and justify methods for calculating engineering structures and structures on strength, rigidity And sustainability. The correct solution of these problems is the basis for the calculation and design of structures, machines, mechanisms, etc., since it ensures their reliability throughout the entire period of operation.
Under strength usually understood as the ability of the safe operation of a structure, structure and their individual elements, which would exclude the possibility of their destruction. The loss (depletion) of strength is shown in fig. 1.1 on the example of the destruction of a beam under the action of a force R.
The process of strength exhaustion without changing the scheme of operation of the structure or the form of its equilibrium is usually accompanied by an increase characteristic phenomena such as the appearance and development of cracks.
Structural stability - it is its ability to maintain the original form of equilibrium until destruction. For example, for the rod in Fig. 1.2 but up to a certain value of the compressive force, the initial rectilinear form of equilibrium will be stable. If the force exceeds a certain critical value, then the bent state of the rod will be stable (Fig. 1.2, b). In this case, the rod will work not only in compression, but also in bending, which can lead to its rapid destruction due to loss of stability or to the appearance of unacceptably large deformations.
Loss of stability is very dangerous for structures and structures, since it can occur within a short period of time.
Structural rigidity characterizes its ability to prevent the development of deformations (elongations, deflections, twisting angles, etc.). Typically, the rigidity of structures and structures is regulated by design standards. For example, the maximum deflections of beams (Fig. 1.3) used in construction should be in the range /= (1/200 + 1/1000) /, the twist angles of the shafts usually do not exceed 2 ° per 1 meter of shaft length, etc.
Solving the problems of structural reliability is accompanied by the search for the most optimal options in terms of the efficiency of work or operation of structures, the consumption of materials, the manufacturability of erection or manufacture, aesthetic perception, etc.
Resistance of materials in technical universities is essentially the first engineering discipline in the learning process in the field of design and calculation of structures and machines. The course on the strength of materials mainly describes the methods for calculating the simplest structural elements - rods (beams, beams). At the same time, various simplifying hypotheses are introduced, with the help of which simple calculation formulas are derived.
In the strength of materials, the methods of theoretical mechanics and higher mathematics, as well as data experimental studies. As a basic discipline, the disciplines studied by undergraduate students, such as structural mechanics, building structures, testing of structures, dynamics and strength of machines, etc., largely rely on the strength of materials as a basic discipline.
The theory of elasticity, the theory of creep, the theory of plasticity are the most general sections of the mechanics of a deformable solid body. The hypotheses introduced in these sections are of a general nature and mainly concern the behavior of the material of the body during its deformation under the action of a load.
In the theories of elasticity, plasticity and creep, as accurate or sufficiently rigorous methods of analytical problem solving as possible are used, which requires the involvement of special branches of mathematics. The results obtained here make it possible to give methods for calculating more complex structural elements, such as plates and shells, to develop methods for solving special problems, such as, for example, the problem of stress concentration near holes, and also to establish the areas of application of solutions to the strength of materials.
In cases where the mechanics of a deformable solid body cannot provide methods for calculating structures that are sufficiently simple and accessible for engineering practice, various methods are used. experimental methods determination of stresses and strains in real structures or in their models (for example, strain gauge method, polarization-optical method, holography method, etc.).
The formation of the strength of materials as a science can be attributed to the middle of the last century, which was associated with the intensive development of industry and the construction of railways.
Requests for engineering practice gave impetus to research in the field of strength and reliability of structures, structures and machines. Scientists and engineers during this period developed enough simple methods calculation of structural elements and laid the foundation for the further development of the science of strength.
The theory of elasticity began to develop in early XIX centuries as a mathematical science that does not have an applied character. The theory of plasticity and the theory of creep as independent sections of the mechanics of a deformable solid body were formed in the 20th century.
The mechanics of a deformable solid body is a constantly developing science in all its branches. New methods are being developed for determining the stressed and deformed states of bodies. Various numerical methods for solving problems have been widely used, which is associated with the introduction and use of computers in almost all areas of science and engineering practice.
BASIC CONCEPTS OF MECHANICS
DEFORMABLE SOLID BODY
This chapter presents the basic concepts that were previously studied in the courses of physics, theoretical mechanics and strength of materials.
1.1. The subject of solid mechanics
The mechanics of a deformable solid body is the science of the balance and motion of solid bodies and their individual particles, taking into account changes in the distances between individual points of the body that arise as a result of external influences on the solid body. The mechanics of a deformable solid body is based on the laws of motion discovered by Newton, since the speeds of movement of real solid bodies and their individual particles relative to each other are significantly less than the speed of light. In contrast to theoretical mechanics, here we consider changes in the distances between individual particles of the body. The latter circumstance imposes certain restrictions on the principles of theoretical mechanics. In particular, in the mechanics of a deformable solid body, the transfer of points of application of external forces and moments is unacceptable.
Analysis of the behavior of deformable solids under the influence of external forces is based on mathematical models that reflect the most significant properties of deformable bodies and materials from which they are made. At the same time, the results of experimental studies are used to describe the properties of the material, which served as the basis for creating material models. Depending on the material model, the mechanics of a deformable solid body is divided into sections: the theory of elasticity, the theory of plasticity, the theory of creep, the theory of viscoelasticity. In turn, the mechanics of a deformable solid body is part of a more general part of mechanics - mechanics of continuous media. Continuum mechanics, being a branch of theoretical physics, studies the laws of motion of solid, liquid and gaseous media, as well as plasma and continuous physical fields.
The development of the mechanics of a deformable solid body is largely associated with the tasks of creating reliable structures and machines. The reliability of a structure and machine, as well as the reliability of all their elements, is ensured by strength, rigidity, stability and endurance throughout the entire service life. Strength is understood as the ability of a structure (machine) and all its (its) elements to maintain their integrity under external influences without being divided into parts that are not foreseen in advance. With insufficient strength, the structure or its individual elements are destroyed by dividing a single whole into parts. The rigidity of a structure is determined by the measure of the change in the shape and dimensions of the structure and its elements under external influences. If the changes in the shape and dimensions of the structure and its elements are not large and do not interfere with normal operation, then such a structure is considered sufficiently rigid. Otherwise, the rigidity is considered insufficient. The stability of a structure is characterized by the ability of a structure and its elements to maintain their form of equilibrium under the action of random forces not provided for by the operating conditions (disturbing forces). A structure is in a stable state if, after the removal of disturbing forces, it returns to its original form of equilibrium. Otherwise, there is a loss of stability of the original form of equilibrium, which, as a rule, is accompanied by the destruction of the structure. Endurance is understood as the ability of a structure to resist the influence of time-varying forces. Variable forces cause the growth of microscopic cracks inside the material of the structure, which can lead to the destruction of structural elements and the structure as a whole. Therefore, to prevent destruction, it is necessary to limit the magnitudes of the forces that are variable in time. In addition, the lowest frequencies of natural oscillations of the structure and its elements should not coincide (or be close to) the frequencies of oscillations of external forces. Otherwise, the structure or its individual elements enter into resonance, which can cause destruction and failure of the structure.
The vast majority of research in the field of solid mechanics is aimed at creating reliable structures and machines. This includes the design of structures and machines and problems technological processes material processing. But the scope of application of the mechanics of a deformable solid body is not limited to the technical sciences alone. Its methods are widely used in natural sciences such as geophysics, solid state physics, geology, biology. So in geophysics, with the help of the mechanics of a deformable solid body, the processes of propagation of seismic waves and the processes of formation of earth's crust, fundamental questions of the structure of the earth's crust are being studied, etc.
1.2. General properties of solids
All solids are made up of real materials with a huge variety of properties. Of these, only a few are of significant importance for the mechanics of a deformable solid body. Therefore, the material is endowed with only those properties that make it possible to study the behavior of solids at the lowest cost within the framework of the science under consideration.
Definition 1
Rigid body mechanics is an extensive branch of physics that studies the motion of a rigid body under the influence of external factors and strength.
Figure 1. Solid mechanics. Author24 - online exchange of student papers
Given scientific direction covers a very wide range of issues in physics - it studies various objects, as well as the smallest elementary particles substances. In these limiting cases, the conclusions of mechanics are of purely theoretical interest, the subject of which is also the design of many physical models and programs.
To date, there are 5 types of motion of a rigid body:
- progressive movement;
- plane-parallel movement;
- rotational movement around a fixed axis;
- rotational around a fixed point;
- free uniform movement.
Any complex movement of a material substance can be ultimately reduced to a set of rotational and forward movement. Fundamental and important for all this subject matter is the mechanics of motion of a rigid body, which involves a mathematical description of the probable changes in the environment and dynamics, which considers the movement of elements under the action of given forces.
Features of rigid body mechanics
A rigid body that systematically assumes various orientations in any space can be considered to be composed of huge amount material points. It's simple mathematical method, which helps to expand the applicability of theories of particle motion, but has nothing to do with the theory atomic structure real substance. Since the material points of the body under study will be directed in different directions with different velocities, it is necessary to apply the summation procedure.
In this case, it is not difficult to determine the kinetic energy of the cylinder, if the revolving around a fixed vector is known in advance with angular velocity parameter. The moment of inertia can be calculated by integration, and for a homogeneous object, the equilibrium of all forces is possible if the plate did not move, therefore, the components of the medium satisfy the condition of vector stability. As a result, the relation derived at the initial design stage is fulfilled. Both of these principles form the basis of the theory of structural mechanics and are necessary in the construction of bridges and buildings.
The foregoing can be generalized to the case when there are no fixed lines and the physical body freely rotates in any space. In such a process, there are three moments of inertia related to the "key axes". Conducted postulates in mechanics solid are simplified if we use the existing notation of mathematical analysis, which assumes the passage to the limit $(t → t0)$, so that there is no need to think all the time how to solve this problem.
Interestingly, Newton was the first to apply the principles of integral and differential calculus in solving complex physical problems, and the subsequent formation of mechanics as a complex science was the work of such eminent mathematicians, like J. Lagrange, L. Euler, P. Laplace and K. Jacobi. Each of these researchers found in Newton's teachings a source of inspiration for their universal mathematical research.
Moment of inertia
When studying the rotation of a rigid body, physicists often use the concept of moment of inertia.
Definition 2
The moment of inertia of the system (material body) about the axis of rotation is called physical quantity, which is equal to the sum of the products of the indicators of the points of the system and the squares of their distances to the considered vector.
The summation is made over all moving elementary masses into which the physical body is divided. If the moment of inertia of the object under study is initially known relative to the axis passing through its center of mass, then the whole process relative to any other parallel line is determined by the Steiner theorem.
Steiner's theorem states: the moment of inertia of a substance about the rotation vector is equal to the moment of its change about a parallel axis that passes through the center of mass of the system, obtained by multiplying the masses of the body by the square of the distance between the lines.
When an absolutely rigid body rotates around a fixed vector, each individual point moves along a circle of constant radius with a certain speed and the internal momentum is perpendicular to this radius.
Solid body deformation
Figure 2. Solid body deformation. Author24 - online exchange of student papers
Considering the mechanics of a rigid body, the concept of an absolutely rigid body is often used. However, such substances do not exist in nature, since all real objects under the influence of external forces change their size and shape, that is, they are deformed.
Definition 3
The deformation is called constant and elastic if, after the cessation of the influence of extraneous factors, the body assumes its original parameters.
Deformations that remain in the substance after the termination of the interaction of forces are called residual or plastic.
Deformations of an absolute real body in mechanics are always plastic, since they never completely disappear after the termination of the additional influence. However, if the residual changes are small, then they can be neglected and more elastic deformations can be investigated. All types of deformation (compression or tension, bending, torsion) can eventually be reduced to simultaneous transformations.
If the force moves strictly along the normal to a flat surface, the stress is called normal, but if it moves tangentially to the medium, it is called tangential.
A quantitative measure that characterizes the characterizing deformation experienced by a material body is its relative change.
Beyond the elastic limit, residual deformations appear in the solid, and the graph describing in detail the return of the substance to its original state after the final cessation of the force is depicted not on the curve, but parallel to it. The stress diagram for real physical bodies directly depends on various factors. One and the same object can, under short-term exposure to forces, manifest itself as completely fragile, and under long-term exposure - permanent and fluid.
The tasks of science
This is the science of strength and flexibility (rigidity) of engineering structure elements. Using the methods of mechanics of a deformable body, practical calculations are carried out and reliable (strong, stable) dimensions of machine parts and various building structures are determined. The introductory, initial part of the mechanics of a deformable body is a course called strength of materials. The main provisions of the strength of materials are based on the laws general mechanics solid body and, above all, the laws of statics, the knowledge of which is absolutely necessary for studying the mechanics of a deformable body. The mechanics of deformable bodies also includes other sections, such as the theory of elasticity, the theory of plasticity, the theory of creep, where the same issues are considered as in the resistance of materials, but in a more complete and rigorous formulation.
The resistance of materials, on the other hand, sets as its task the creation of practically acceptable and simple methods for calculating the strength and stiffness of typical, most frequently encountered structural elements. In this case, various approximate methods are widely used. The need to bring the solution of each practical problem to a numerical result makes it necessary in some cases to resort to simplifying hypotheses-assumptions, which are justified in the future by comparing the calculated data with the experiment.
General Approach
Many physical phenomena It is convenient to consider using the diagram shown in Figure 13:
Across X here some influence (control) applied to the input of the system is indicated BUT(machine, test sample of material, etc.), and through Y- reaction (response) of the system to this impact. We will assume that the reactions Y removed from the system output BUT.
Under managed system BUT Let us agree to understand any object capable of deterministically responding to some influence. This means that all copies of the system BUT under the same conditions, i.e. with the same impact x(t), behave in exactly the same way, i.e. issue the same y(t). Such an approach, of course, is only an approximation, since it is practically impossible to obtain either two completely identical systems, or two identical effects. Therefore, strictly speaking, one should consider not deterministic, but probabilistic systems. Nevertheless, for a number of phenomena it is convenient to ignore this obvious fact and consider the system to be deterministic, understanding all the quantitative relationships between the quantities under consideration in the sense of the relationships between their mathematical expectations.
The behavior of any deterministic controlled system can be determined by some relation connecting the output with the input, i.e. X from at. This relation will be called the equation states systems. Symbolically it is written as
where is the letter BUT, used earlier to denote the system, can be interpreted as some operator that allows you to determine y(t), if given x(t).
The introduced concept of a deterministic system with input and output is very general. Here are some examples of such systems: an ideal gas, the characteristics of which are related by the Mendeleev-Clapeyron equation, an electrical circuit that obeys one or another differential equation, a steam or gas turbine blade deforming in time, forces acting on it, etc. the type most suitable for modeling the behavior of a body deformed under load.
The analysis of any controlled system can in principle be carried out in two ways. The first one microscopic, is based on a detailed study of the structure of the system and the functioning of all its constituent elements. If all this can be done, then it becomes possible to write the equation of state of the entire system, since the behavior of each of its elements and the ways of their interaction are known. So, for example, the kinetic theory of gases allows us to write the Mendeleev-Clapeyron equation; knowledge of the structure of an electrical circuit and all its characteristics makes it possible to write its equations based on the laws of electrical engineering (Ohm's law, Kirchhoff's, etc.). Thus, the microscopic approach to the analysis of a controlled system is based on the consideration of the elementary processes that make up a given phenomenon, and, in principle, is capable of giving a direct, exhaustive description of the system under consideration.
However, the micro-approach cannot always be implemented due to the complex or not yet explored structure of the system. For example, at present it is not possible to write the equation of state of a deformable body, no matter how carefully it is studied. The same applies to more complex phenomena occurring in a living organism. In such cases, the so-called macroscopic phenomenological (functional) approach, in which they are not interested in the detailed structure of the system (for example, the microscopic structure of a deformable body) and its elements, but study the functioning of the system as a whole, which is considered as a connection between input and output. Generally speaking, this relationship can be arbitrary. However, for each specific class of systems, general restrictions are imposed on this connection, and a certain minimum of experiments may be sufficient to clarify this connection with the necessary details.
The use of the macroscopic approach is, as already noted, forced in many cases. Nevertheless, even the creation of a consistent microtheory of a phenomenon cannot completely devalue the corresponding macrotheory, since the latter is based on experiment and is therefore more reliable. Microtheory, on the other hand, when constructing a model of a system, is always forced to make some simplifying assumptions that lead to various kinds of inaccuracies. For example, all "microscopic" equations of state of an ideal gas (Mendeleev-Clapeyron, Van der Waals, etc.) have irreparable discrepancies with experimental data on real gases. The corresponding "macroscopic" equations, based on these experimental data, can describe the behavior of a real gas as accurately as desired. Moreover, the micro-approach is such only at a certain level - the level of the system under consideration. At the level of the elementary parts of the system, it is still a macro approach, so that the microanalysis of the system can be considered as a synthesis of its constituent parts analyzed macroscopically.
Since at present the micro-approach is not yet able to lead to the equation of state of a deformable body, it is natural to solve this problem macroscopically. We will adhere to this point of view in the future.
Displacements and deformations
A real rigid body, deprived of all degrees of freedom (the ability to move in space) and under the influence of external forces, deformed. By deformation we mean a change in the shape and size of the body, associated with the movement of individual points and elements of the body. Only such displacements are considered in the resistance of materials.
There are linear and angular displacements of individual points and elements of the body. These displacements correspond to linear and angular deformations (relative elongation and relative shear).
Deformations are divided into elastic, disappearing after the load is removed, and residual.
Hypotheses about the deformable body. Elastic deformations are usually (at least in structural materials such as metals, concrete, wood, etc.) insignificant, so the following simplifying provisions are accepted:
1. The principle of initial dimensions. In accordance with it, it is assumed that the equilibrium equations for a deformable body can be compiled without taking into account changes in the shape and size of the body, i.e. as for a perfectly rigid body.
2. The principle of independence of the action of forces. In accordance with it, if a system of forces (several forces) is applied to the body, then the action of each of them can be considered independently of the action of other forces.
Voltage
Under the action of external forces, internal forces arise in the body, which are distributed over the sections of the body. To determine the measure of internal forces at each point, the concept is introduced voltage. Stress is defined as an internal force per unit sectional area of a body. Let an elastically deformed body be in a state of equilibrium under the action of some system of external forces (Fig. 1). Through a dot (for example, k), in which we want to determine the stress, an arbitrary section is mentally drawn and part of the body is discarded (II). In order for the remaining part of the body to be in balance, internal forces must be applied instead of the discarded part. The interaction of two parts of the body occurs at all points of the section, and therefore the internal forces act over the entire section area. In the vicinity of the point under study, we select the area dA. We denote the resultant of internal forces on this site dF. Then the stress in the vicinity of the point will be (by definition)
N/m 2.
Voltage has the dimension of force divided by area, N/m 2 .
At a given point of the body, the stress has many values, depending on the direction of the sections, which can be drawn through a point through a set. Therefore, speaking of stress, it is necessary to indicate the cross section.
In the general case, the stress is directed at some angle to the section. This total voltage can be decomposed into two components:
1. Perpendicular plane sections - normal voltage s.
2. Lying in the plane of the section - shear stress t.
Determination of stresses. The problem is solved in three stages.
1. Through the point under consideration, a section is drawn in which they want to determine the stress. One part of the body is discarded and its action is replaced by internal forces. If the whole body is in balance, then the rest must also be in balance. Therefore, for the forces acting on the part of the body under consideration, it is possible to compose equilibrium equations. These equations will include both external and unknown internal forces (stresses). Therefore, we write them in the form
The first terms are the sums of the projections and the sums of the moments of all external forces acting on the part of the body remaining after the section, and the second terms are the sums of the projections and moments of all the internal forces acting in the section. As already noted, these equations include unknown internal forces (stresses). However, for their definition of the equations of statics not enough, since otherwise the difference between an absolutely rigid and deformable body disappears. Thus, the task of determining stresses is statically indeterminate.
2. To compile additional equations, the displacements and deformations of the body are considered, as a result of which the law of stress distribution over the section is obtained.
3. Solving jointly the equations of statics and the equations of deformations, it is possible to determine the stresses.
Power factors. We agree to call the sums of projections and the sums of moments of external or internal forces force factors. Consequently, the force factors in the considered section are defined as the sums of projections and the sums of the moments of all external forces located on one side of this section. In the same way, force factors can be determined by internal forces acting in the considered section. Force factors determined by external and internal forces are equal in magnitude and opposite in sign. Usually, external forces are known in problems, through which force factors are determined, and stresses are already determined from them.
Model of a deformable body
In the strength of materials, a model of a deformable body is considered. It is assumed that the body is deformable, solid and isotropic. In the resistance of materials, bodies are considered mainly in the form of rods (sometimes plates and shells). This is explained by the fact that in many practical problems the design scheme is reduced to a straight rod or to a system of such rods (trusses, frames).
The main types of the deformed state of the rods. Rod (beam) - a body in which two dimensions are small compared to the third (Fig. 15).
Consider a rod that is in equilibrium under the action of forces applied to it, arbitrarily located in space (Fig. 16).
We draw a section 1-1 and discard one part of the rod. Consider the balance of the remaining part. We use a rectangular coordinate system, for the beginning of which we take the center of gravity of the cross section. Axis X direct along the rod in the direction of the outer normal to the section, the axis Y And Z are the main central axes of the section. Using the equations of statics, we find the force factors
three forces
three moments or three pairs of forces
Thus, in the general case, six force factors arise in the cross section of the rod. Depending on the nature of the external forces acting on the rod, it is possible different kinds rod deformation. The main types of rod deformations are stretching, compression, shift, torsion, bend. Accordingly, the simplest loading schemes are as follows.
Stretch-compression. Forces are applied along the axis of the rod. Having discarded the right part of the rod, we select the force factors by the left external forces (Fig. 17)
We have one non-zero factor - the longitudinal force F.
We build a diagram of force factors (epure).
Rod torsion. In the planes of the end sections of the rod, two equal and opposite pairs of forces are applied with a moment M kr =T, called torque (Fig. 18).
As you can see, only one force factor acts in the cross section of the twisted rod - the moment T = F h.
Cross bend. It is caused by forces (concentrated and distributed) perpendicular to the axis of the beam and located in a plane passing through the axis of the beam, as well as pairs of forces acting in one of the main planes of the bar.
The beams have supports, i.e. are non-free bodies, a typical support is a hinged-movable support (Fig. 19).
Sometimes a beam with one embedded and the other free end is used - a cantilever beam (Fig. 20).
Consider the definition of force factors on the example of Fig.21a. First you need to find the support reactions R A and .
