7 SHORT DESCRIPTION OF STRENGTH OF MATERIALS
Note 7.18. It is assumed that a motion of the element of the rod occurs in the way that each cross-section rotates as a single item around the axis of the rod. To take
into account this fact in the exact theory, it is necessary to take into account the zeroth root. Then the solution is extended by introducing the additional summand of the form
, , o i kz t
u r z tM u reM Z (7.55)
This formula corresponds to the elementary theory. In practice, this mode is excited first of all.
Flexural waves
First, the horizontal plane should be chosen in the cylinder, which includes the axis of the rod and divides the cylinder into two equal parts. Just in this plane, the flexural vibrations occur. In the elementary theory, this corresponds to the neutral axis. The points of the
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that the radial and axial displacements are absent in this plane. The vertical plane, which also includes the axis of the rod and forms the right angle with the horizontal plane, must be also chosen. It is assumed that the angle displacements are absent in this plane and the points belonging to this plane vibrate only in this plane. The angle coordinate is chosen in the way that the angle reference starts with the vertical plane. Then the vertical plane corresponds to the value M 0 and the horizontal – to the value M S/ 2 .
In this form of describing the problem, the solution the system of Lame equations (7.29) is searched in the form
, , , cos i kz t ,
r r
u rM z t U r Me Z (7.56)
, , , sin i kz t ,
u rM M z t U rM Me Z (7.57)
, , , cos i kz t .
z z
u rM z t U r Me Z (7.58)
The depending on the radius functions U r U r U rr , M , z are expressed through the Bessel functions
1 , 1 1 1 ,
r L r T flex T
U r A Jª¬ N r º¼ B r J N r iCk J N r (7.59) 1 1 L 1 T ,r flex 1 T ,
U rM A r J N r B Jª¬ N r º¼ iCk J N r (7.60)
1 2 ,
z flex L T r
U r iAk J N r C J ª¬ N r º¼ , (7.61)
2 2 , 2 2 .
L kL kflex T kT kflex
N N
The substitution of representations (7.59)-(7.61) into the boundary conditions (7.22) gives a system of three linear algebraic homogeneous equations relative to constants A,B,C. When the determinant of this system is equated to zero, the frequency (characteristic) equation is obtained. This equation has a countable set of roots and the corresponding wavenumbers and modes.
The lowest flexural mode was investigated by Achenbach and Fang. For the small values of
flex o
k r (for the small wavenumbers of the lowest flexural mode under the fixed radius of the cylinder), they obtained the expression
1 2 3 2
2
o o
L flex
k r O P O P k r . (7.62)
They write also that as k rflex o increases the phase velocity of the lowest mode approaches the velocity of Rayleigh wave.
The shown example of the elementary and exact theory of propagation of waves in the circular rod (cylinder) demonstrates a deep difference between approaches of the strength of materials (elementary theory) and the linear theory of elasticity (exact theory). But it demonstrates also the role of the exact approaches in substantiation of the practicability of the elementary (simplified) approaches.
Comments
Comment 7.1. Perhaps, when analyzing the basics of the strength of materials as a science, the question of the adequacy of the studied mechanical phenomenon should be raised to the real phenomenon regar-ding the chosen system of its characteristics. This usually means a correct qualitative description of the pheno menon by selected characteristics and a correct quantitative description of the phenomenon accor- ding to the selected characteristics with some reasonable degree of accuracy.
If we talk about the adequacy of the model, it is convenient to introduce the concept of the degree of adequacy of the model. This degree may be understood as a grain of the model’s truth about selected chara- cteristics or as a correlation ratio of the model and the real mechanical phenomenon. The whole science of the strength of materials gives a lot of examples, where the adequacy of the engineering model plays a significant role and is emphasized in its description.
Comment 7.2. On the utility of the mathematical models of materials mechanics. It is believed that such models allow persons of a certain intellectual bent to understand the behavior of mechanical objects better than if it were outlined verbally. This occurs because mathematical language and, in particular, the language of differential equations has a high degree of commonality. A scientist who understands such a language immediately has many associations with similar, well-known to him, situations described by the same equations.
Comment 7.3. The terms of the exact theory and elementary theory should be defined.
Here the exact theory is understood as such, which is based on the model of the linear theory of elasticity and only on those assumptions in which this model is formulated. The accuracy of the theory, in this case, is determined not by the accuracy of the description of the elastic deformation of materials, but by the accuracy of the solutions of mathematical equations of the linear theory of elasticity. The elementary theory is understood as such,
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and does not allow further simplification. At the same time, this model corresponds to the exact model with additional simplifications and still describes the essence of the mechanical phenomenon in question.
Just in this sense, the linear theory of elasticity is understood as an exact theory and the strength of materials as an elementary theory, which imposes additional restrictions as compared to the linear theory of elasticity, allowing to simplify the mathematical formulations of this theory.
Further reading is presented in the Foreword Questions
7.1. Try to add some not shown in this chapter differences between the linear theory of elasticity and the strength of materials.
7.2. Which criteria of failure you would like to add to the list expounded in this chapter?
7.3. The frames and arches are the objects of the structural (building) mechanics. Should they be considered as the structural members or as the separate object of the strength of materials?
7.4. The stability of structures is nowadays the big area in the mechanics of materials. Do you think, the stability must still be described in the books on the strength of materials or the readers must use specialized books?
7.5. The same question can be formulated relative to the plates and shells.
7.6. The same question can be formulated relative to the theories of vibrations and waves.
7.7. What is better for studying – to systemize the strength of materials by the mechanical phenomena (tension, torsion, vibration, wave, etc) or by the object (rod, frame, plate, shell, etc)?