Basic concepts of the mechanics of a deformable body. Basic concepts of solid mechanics. Internal forces and stresses
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 forces.
Figure 1. Solid mechanics. Author24 - online exchange of student papers
This scientific direction covers a very wide range of issues in physics - it studies various objects, as well as the smallest elementary particles of matter. 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 eventually be reduced to a set of rotational and translational movements. The mechanics of motion of a rigid body, which involves a mathematical description of probable changes in the environment, and dynamics, which considers the motion of elements under the action of given forces, is of fundamental and important importance for all this subject matter.
Features of rigid body mechanics
A rigid body that systematically assumes various orientations in any space can be considered as consisting of a huge number of material points. This is just a mathematical method to help expand the applicability of theories of particle motion, but has nothing to do with the theory of the atomic structure of real matter. 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 balance 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". The postulates that have been carried out in solid mechanics 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 outstanding mathematicians as J. Lagrange, L. Euler, P. Laplace and C. 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. Voltage 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.
Lecture #1
Strength of materials as a scientific discipline.
Schematization of structural elements and external loads.
Assumptions about the properties of the material of structural elements.
Internal forces and stresses
Section method
displacements and deformations.
The principle of superposition.
Basic concepts.
Strength of materials as a scientific discipline: strength, stiffness, stability. Calculation scheme, physical and mathematical model of the operation of an element or part of a structure.
Schematization of structural elements and external loads: timber, rod, beam, plate, shell, massive body.
External forces: volumetric, surface, distributed, concentrated; static and dynamic.
Assumptions about the properties of the material of structural elements: the material is solid, homogeneous, isotropic. Body deformation: elastic, residual. Material: linear elastic, non-linear elastic, elastic-plastic.
Internal forces and stresses: internal forces, normal and shear stresses, stress tensor. Expression of internal forces in the cross section of the rod in terms of stresses I.
Section method: determination of the components of internal forces in the section of the rod from the equilibrium equations of the separated part.
Displacements and deformations: displacement of a point and its components; linear and angular strains, strain tensor.
Superposition principle: geometrically linear and geometrically nonlinear systems.
Strength of materials as a scientific discipline.
Disciplines of the strength cycle: strength of materials, theory of elasticity, structural mechanics are united by the common name " Mechanics of a solid deformable body».
Strength of materials is the science of strength, rigidity and stability elements engineering structures.
by design It is customary to call a mechanical system of geometrically invariable elements, relative movement of points which is possible only as a result of its deformation.
Under the strength of structures understand their ability to resist destruction - separation into parts, as well as an irreversible change in shape under the action of external loads .
Deformation is a change relative position of body particles associated with their movement.
Rigidity is the ability of a body or structure to resist the occurrence of deformation.
Stability of an elastic system called its property to return to a state of equilibrium after small deviations from this state .
Elasticity - this is the property of the material to completely restore the geometric shape and dimensions of the body after removing the external load.
Plastic - this is the property of solids to change their shape and size under the action of external loads and retain it after the removal of these loads. Moreover, the change in the shape of the body (deformation) depends only on the applied external load and does not happen on its own over time.
Creep - this is the property of solids to deform under the influence of a constant load (deformations increase with time).
Building mechanics call science about calculation methods structures for strength, rigidity and stability .
1.2 Schematization of structural elements and external loads.
Design model It is customary to call an auxiliary object that replaces the real construction, presented in the most general form.
The strength of materials uses design schemes.
Design scheme - this is a simplified image of a real structure, which is freed from its non-essential, secondary features and which accepted for mathematical description and calculation.
Among the main types of elements, which in calculation scheme the whole structure is subdivided, include: timber, rod, plate, shell, massive body.
Rice. 1.1 Main types of structural elements
bar is a rigid body obtained by moving a flat figure along a guide so that its length is much greater than the other two dimensions.
rod called straight beam, which works in tension/compression (significantly exceeds the characteristic dimensions of the cross section h,b).
The locus of points that are the centers of gravity of cross sections will be called rod axis .
plate - a body whose thickness is much less than its dimensions a And b in respect of.
A naturally curved plate (curve before loading) is called shell .
massive body characteristic in that all its dimensions a ,b, And c have the same order.
Rice. 1.2 Examples of bar structures.
beam is called a bar that experiences bending as the main mode of loading.
Farm called a set of rods connected hingedly .
Frame – is a set of beams rigidly connected to each other.
External loads are divided on concentrated And distributed .
Fig 1.3 Schematization of the operation of the crane beam.
force or moment, which are conventionally considered to be attached at a point, are called concentrated .
Figure 1.4 Volumetric, surface and distributed loads.
A load that is constant or very slowly changing in time, when the speeds and accelerations of the resulting movement can be neglected, called static.
A rapidly changing load is called dynamic , calculation taking into account the resulting oscillatory motion - dynamic calculation.
Assumptions about the properties of the material of structural elements.
In the resistance of materials, a conditional material is used, endowed with certain idealized properties.
On fig. 1.5 shows three characteristic strain diagrams relating force values F and deformations at loading And unloading.
Rice. 1.5 Characteristic diagrams of material deformation
Total deformation consists of two components, elastic and plastic.
The part of the total deformation that disappears after the load is removed is called elastic .
The deformation remaining after unloading is called residual or plastic .
Elastic - plastic material is a material exhibiting elastic and plastic properties.
A material in which only elastic deformations occur is called perfectly elastic .
If the deformation diagram is expressed by a non-linear relationship, then the material is called nonlinear elastic, if linear dependence , then linearly elastic .
The material of structural elements will be further considered continuous, homogeneous, isotropic and linearly elastic.
Property continuity means that the material continuously fills the entire volume of the structural element.
Property homogeneity means that the entire volume of the material has the same mechanical properties.
The material is called isotropic if its mechanical properties are the same in all directions (otherwise anisotropic ).
The correspondence of the conditional material to real materials is achieved by the fact that experimentally obtained averaged quantitative characteristics of the mechanical properties of materials are introduced into the calculation of structural elements.
1.4 Internal forces and stresses
internal forces – increment of the forces of interaction between the particles of the body, arising when it is loaded .
Rice. 1.6 Normal and shear stresses at a point
The body is cut by a plane (Fig. 1.6 a) and in this section at the point under consideration M a small area is selected, its orientation in space is determined by the normal n. The resultant force on the site will be denoted by . middle the intensity on the site is determined by the formula . The intensity of internal forces at a point is defined as the limit
(1.1) The intensity of internal forces transmitted at a point through a selected area is called voltage at this site .
Voltage dimension .
The vector determines the total stress on a given site. We decompose it into components (Fig. 1.6 b) so that , where and - respectively normal And tangent stress on the site with the normal n.
When analyzing stresses in the vicinity of the considered point M(Fig. 1.6 c) select an infinitesimal element in the form of a parallelepiped with sides dx, dy, dz (carry out 6 sections). The total stresses acting on its faces are decomposed into normal and two tangential stresses. The set of stresses acting on the faces is presented in the form of a matrix (table), which is called stress tensor
The first index of the voltage, for example , shows that it acts on a site with a normal parallel to the x-axis, and the second shows that the stress vector is parallel to the y-axis. At normal voltage both indexes are the same, so one index is put.
Force factors in the cross section of the rod and their expression in terms of stresses.
Consider cross section rod of a loaded rod (Fig. 1.7, a). We reduce the internal forces distributed over the section to the main vector R, applied at the center of gravity of the section, and the main moment M. Next, we decompose them into six components: three forces N, Qy, Qz and three moments Mx, My, Mz, called internal forces in the cross section.
Rice. 1.7 Internal forces and stresses in the cross section of the rod.
The components of the main vector and the main moment of internal forces distributed over the section are called internal forces in the section ( N- longitudinal force ; Qy, Qz- transverse forces ,Mz,My- bending moments , Mx- torque) .
Let us express the internal forces in terms of the stresses acting in the cross section, assuming they are known at every point(Fig. 1.7, c)
Expression of internal forces through stresses I.
(1.3)
1.5 Section method
When external forces act on a body, it deforms. Consequently, the relative position of the particles of the body changes; as a result of this, additional forces of interaction between particles arise. These interaction forces in a deformed body are domestic efforts. Must be able to identify meanings and directions of internal efforts through external forces acting on the body. For this, it is used section method.
Rice. 1.8 Determination of internal forces by the method of sections.
Equilibrium equations for the rest of the rod.
From the equilibrium equations, we determine the internal forces in the section a-a.
1.6 Displacements and deformations.
Under the action of external forces, the body is deformed, i.e. changes its size and shape (Fig. 1.9). Some arbitrary point M moves to a new position M 1 . The total displacement MM 1 will be
decompose into components u, v, w parallel to the coordinate axes.
Fig 1.9 Full displacement of a point and its components.
But the displacement of a given point does not yet characterize the degree of deformation of the material element at this point ( example of beam bending with cantilever) .
We introduce the concept deformations at a point as a quantitative measure of material deformation in its vicinity . Let's single out an elementary parallelepiped in the vicinity of t.M (Fig. 1.10). Due to the deformation of the length of its ribs, they will receive an elongation.
Fig 1.10 Linear and angular deformation of a material element.
Linear relative deformations at a point defined like this():
In addition to linear deformations, there are angular deformations or shear angles, representing small changes in the original right angles of the parallelepiped(for example, in the xy plane it will be ). Shear angles are very small and are of the order of .
We reduce the introduced relative deformations at a point into the matrix
. (1.6)
Quantities (1.6) quantitatively determine the deformation of the material in the vicinity of the point and constitute the deformation tensor.
The principle of superposition.
A system in which internal forces, stresses, strains and displacements are directly proportional to the acting load is called linearly deformable (the material works as linearly elastic).
Bounded by two curved surfaces, the distance...
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. Correct solution these tasks is the basis for the calculation and design of structures, machines, mechanisms, etc., since it ensures their reliability during 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 in 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 A 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 within /= (1/200 + 1/1000) /, the twisting 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 best options from the point of view of the efficiency of work or operation of structures, consumption of materials, manufacturability of erection or manufacture, aesthetic perception, etc.
The strength 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 from experimental studies, are widely used. As a basic discipline, the disciplines studied by senior 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 experimental methods are used to determine stresses and strains in real structures or in their models (for example, the strain gauge method, the polarization-optical method, the method holography, 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 fairly simple methods for calculating structural elements and laid the foundations 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 motion 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 carried out on the basis of 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 the earth's crust are studied, fundamental questions of the structure of the earth's crust, etc. are studied.
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.
