The condition of strength and stiffness Types of calculations
3) from the known dimensions of the part, the material and the loading scheme, determine the permissible load value
When stretched (or compressed), the stiffness condition has the following form:
adN x dx
l l
EF x
(4.15),
One of the main types of material testing is tensile testing, as this reveals their most
important properties. Special samples are made from the test material. Most often they are made cylindrical (Fig. 99, a); from sheet metal, flat samples are usually made (Fig. 99, b). In cylindrical samples, the relation between the calculated sample length lo and diameter d0 must be maintained.
Stretch Charts. For tensile tests, tensile testing machines are used, which make it possible to determine the forces and the corresponding deformations of the sample during the testing process. Based on these data, a primary tensile diagram is constructed in which the forces are plotted along the ordinate axis, and the elongations corresponding to them are plotted along the abscissa axis. The tensile diagram can also be obtained automatically with the help of special diagram devices. The nature of the tensile pattern depends on the properties of the test material. A typical view of such a diagram for mild steel is shown in Fig. 100.
From the beginning of loading to a certain value of the tensile force, there is a direct proportional relationship between the elongation of the sample and the force. This
dependence in the diagram is expressed as direct line OA. Hooke's law is valid at this stage of stretching.
We denote the force at which the law of proportionality ceases to be valid through the Pprop.
Point A corresponds to this value of the force in the diagram. The tension caused by the force of the Pprop is called the proportionality limit and is calculated by the formula (4.19)
(4.19) Thus, the limit of proportionality is called the stress
after which Hooke's law is broken
prop prop
o
P
F
A deformation is called elastic if it completely disappears after unloading. Assume that
gradually increasing the load P, we will carry out a complete unloading of the sample at each of its values. Until the force P reaches a certain value, the deformations caused by it will disappear during unloading. The unloading process will be depicted in the same line as the loading.
We denote by Pup the largest value of the force at which the sample still does not give residual deformation when unloading. Point B corresponds to this movement in the diagram, and the portion of the organic matter diagram corresponds to the elastic stage of sample extension – part OB.
The highest stress to which permanent deformation and unloading is not detected is called the elastic limit. Its tension is caused by the power of Pup and is determined by the formula
up up
o
P
F
Elastic limit is a characteristic not related to Hooke's law.
After point A, with further stretching of the sample, the curve becomes curved and smoothly rises to point C, where a transition to the horizontal section CD is observed, called fluidity area.
(4.20)
Fig. 99 Fig. 100
Pup - force-limit of elasticity , point B
Pprop - force-limit, point A, limit of proportionality before which Hooke’s law works Pfl - force-limit, points C-D, fluidity area
Pmax - force-limit , point E
up up
o
P
F
At this stage of elongation, the elongation of the sample increases at a constant value of the tensile force, denoted by PT. Such a deformation process, called fluidity of the material, is accompanied by residual (plastic) elongation that does not disappear after unloading.
Thus, the yield stress at is called the least stress, at which the deformation of the specimen occurs at constant tensile force f. The fluidity strength is calculated by the formula (4.21)
Fl PFl Fo
The onset of plastic deformation corresponds to the onset of a certain critical state of the metal, which can be detected not only by residual deformations, but also by other signs. With plastic deformation, the temperature of the sample rises; steel conductivity and magnetic properties change; on the polished surface of the samples, especially flat ones, tarnishing is noticeable, which is the result of the appearance of a dense grid of lines called Chernov- Luders lines. The latter are inclined to the sample axis approximately at an angle of 45 ° (Fig. 101) and are
microscopic irregular features arising due to shifts in those planes of crystals where the highest tangential stresses act. As a result of shifts along inclined planes, the sample receives residual deformations. The mechanism of their formation is simplified shown in Fig. 101.
After the stage of fluidity, the material again acquires the ability to increase the resistance to further deformation and accepts the force increasing to a certain limit. This corresponds to the ascending section DE (Fig. 100) of the tensile curve, called the hardening section. Point E
corresponds to the greatest Pmax force that the sample can perceive (withstand).
The tensile corresponding to the maximum force Pmax is called the temporary resistance It is calculated by the formula:
𝜎𝑡𝑒𝑚𝑝 (4.21)
max temp
o
P
F (4.22)
Up to this point, the elongations were distributed uniformly over the entire length of the sample, the cross-sectional areas of the calculated part of the sample changed
insignificantly and also uniformly in length. Therefore, to calculate the ; , the initial value of the area Fo was introduced into the calculation formulas.
After the Pmax force is reached with further stretching of the sample, deformation occurs mainly over a short length of the sample. This leads to the formation of local narrowing in the form of a neck (Fig. 102) and to a decrease in the force P, despite the fact that the tension in the neck cross section continuously increases.
prop; temp
T ; up
When loading by hanging loads, the destruction will occur at a constant load, but with an ever increasing strain rate.
Denoting by Pk the value of tensile force at the time of rupture, we obtain k k
o
P
F
The main characteristics of the elasticity and strength of the material used in practical calculations are the fluidity strength , yield strength and tensile strength
T 𝜎𝑢𝑝
tempFor mild steel with a yield strength, for example, St2 steel, these characteristics are as follows: = 200 MPa,
T = 220 ÷ 260 MPa, 𝜎𝑡𝑒𝑚𝑝= 340÷ 420 MPa.neck
Fig. 102
upUnloading and reloading. As already mentioned, if, with a tensile force causing a stress not exceeding the elastic limit, stop loading and then unload the specimen, the unloading process will be depicted on the diagram as a line that practically coincides with the load line.
After the final discharge of the sample, its elongation will completely disappear. The repeated loading in the diagram will follow the same line of organic matter obtained during the first loading of the sample.
It will be different if, at the beginning of unloading, the stress in the sample exceeds the elastic limit. After unloading, for example, after the force reaches the value depicted by the ordinate point M (Fig. 100), we note that the unloading process in the diagram is no longer described by a curve that coincides with the OABCDM loading curve, but by a straight line MN parallel to the straight section of the OA diagram.
The Δl’ elongation, obtained by the sample before unloading, does not completely disappear upon unloading. The disappeared part of the extension in the diagram is represented by the segment Δl’upand the remaining part by the segment Δl’0. Therefore, the total elongation of the sample beyond the elastic limit consists of two parts - elastic and plastic:
Δl’= Δl’up+ Δl’0
Elongation and contraction after rupture. The total elongation obtained by the sample before fracture will decrease after rupture, as elastic deformations disappear in parts of the sample.
Relative elongation after rupture δ is the ratio of the percentage increase in the estimated length of the sample after rupture to its original length:
0 0
l 100 l %
Elongation after rupture characterizes the ductility of the material. Depending on the magnitude of this elongation, the materials are divided into plastic and brittle.
For plastic materials when δ>5%, for the brittle ones – δ<5%.
The relative narrowing of the sample after rupture is determined by dividing the absolute decrease in the cross-sectional area in the neck by the initial area o100
o
F %
F
𝛹
The greater the relative narrowing after rupture, the more ductile the material. For example, for mild carbon steel grade St2: = 55 -65%.𝛹
Work strain. In addition to the characteristics of the mechanical properties of the material already mentioned, the tensile diagram makes it possible to determine its energy
characteristics as well.
The magnitude of the area of the tensile diagram in the coordinates P - characterizes the work spent on breaking the sample. This can be shown as follows.
Let the deformation λ of the specimen correspond to a certain tensile force P (Fig. 103). We give the force P an infinitely small increment dP, and the deformation will receive an
increment d λ. Obviously, the work of external forces on this movement dA = (P+ dP)d λ~ Pd λ.
The work spent on stretching the sample to elongation λ1 ,
l
As can be seen from fig. 103, the integral is the area of the OABCDMNO tensile diagram. The work spent on breaking the specimen will be equal to the entire area of the OABCDEFGO tensile diagram. Within elasticity, the total work of deformation is expressed by the area of the triangle (Fig. 104, a):
up 2 A P l
we get the specific work of deformation
2 2
up up
o o
A P l
a V F l
The specific work of deformation within the elastic range is expressed by the area of the
triangle in the diagram (Fig. 104, b). The specific work of deformation characterizes the ability of the material to resist the impact of the load: the greater the specific work of
deformation before breaking, the better the material resists impact.
𝜎−𝜀
static
Fig. 104, Fig. 103
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