RU2787307C1 - Method for determining the static strength of a complex stressed part outside the zones of contact forces acting on it (variants) - Google Patents

Abstract

FIELD: measuring technology.

SUBSTANCE: invention relates to methods for determining the strength of complex stressed parts loaded with a system of forces, individual components of which can change asynchronously with respect to each other, and the number of cycles of their loading does not exceed the limit of quasi-static fatigue. Essence: in the Oστ coordinate system, a series of limiting Mohr circles are constructed, which cover various tensors of the limiting plane-stressed states of the part material preceding its static strength limit or the beginning of plasticity. The diameters of these circles are drawn in straight lines parallel to the Oτ axis and a line is drawn around the intersection points of these diameters with the circles bounding the specified Mohr circles. This line is the boundary of the region of the limiting plane-stressed states of the part material. In the experimental construction of such a boundary, the limiting Mohr circles are constructed by creating various tensors of the plane-stressed state in samples from the part material, and in its calculated construction according to formulas accepted as strength hypotheses. Tensors of plane-stressed states that occur at various points of the surface layer of the part during loading are depicted in the coordinate system Oστ in the form of Mohr circles and monitored so that the diameters of these circles drawn in the vertical direction do not go beyond the boundary of the region of the limiting plane-stressed states of the part material. Additionally, the specified reserve of static strength n in the studied area of the surface of the part is determined.

EFFECT: increase in the accuracy of determining the static strength of parts on their surface areas located outside the impact zones of contact forces.

2 cl, 6 dwg, 1 tbl

Description

The invention relates to methods for determining the strength of complexly stressed parts loaded by a system of forces, the individual components of which can change out of sync with respect to each other, and the number of their loading cycles does not exceed the quasi-static fatigue limit.

The performance of a part with a number of loading cycles not exceeding 10 ³ -10 ⁴ is determined by its static strength [1]. The static strength of parts made of ductile metals is usually evaluated by means of a safety factor for the yield strength, and of brittle metals - by the tensile strength. Permissible stresses are assigned with the expectation that a certain safety factor n of static strength is provided before the onset of yield or fracture.

The greatest stresses outside the zones of action on the part of the contact forces usually occur on the surface of the parts. This is due to the fact that in any section of the part, as it approaches its surface, the stresses from bending and torsional moments reach the highest values. Therefore, destruction usually begins with the surface layer, and the static strength of a part outside the zones of external contact forces on it can usually be estimated by the magnitude of the stresses on the surface of the part.

Outside the impact zones of contact forces, there is no component of forces directed perpendicular to the surface of the part; therefore, the stress state in each of the sections of the surface layer of such a part is plane stressed. With an asynchronous change in the various components of the system of forces acting on the part, the individual components of the tensors of the plane stress state in each of the sections of the surface layer of the part change asynchronously with respect to each other. When analyzing the static strength of such sections, the assessment of the degree of danger of the stress states arising in them should be carried out taking into account the various combinations of the components of the tensors of their plane stress state.

There is a method of graphical recording of changes in the tensor of biaxial stresses at the considered point on the surface of the part [2]. In one of the variants of this method, the record of the change in the plane stress state tensor can be represented as Mohr circles sequentially displayed along the Oσ axis in the rectangular coordinate system Oστ. Such a record unambiguously determines the nature of the change in the stress tensor in the studied area of the part surface. The proposed method for recording the change in biaxial stresses can be implemented by processing the results of calculating the stresses in a part with sequentially changing schemes of its loading or by processing the readings experimentally taken in the investigated area of the surface of this part by a three-component socket of strain gauges. At the same time, Mohr circles, representing stress tensors, in which the tensile-compressive stresses acting in the plane of the part surface parallel to the direction initially chosen on it, have algebraically (taking into account the signs) higher values compared to the tensile-compressive stresses acting in the same direction. planes in the direction perpendicular to the direction initially selected on the surface of the part are shown by solid lines, and circles in which the tensile-compressive stresses are higher in the direction perpendicular to the direction initially selected are shown by dashed lines.

The disadvantages of this method include the fact that it does not indicate how to determine the static strength of the part in this zone by the nature of the recording of the sequential change in the stress tensor in the studied area of the surface of the part.

The following methods are known for determining the static strength of a part or its material in a complex (bi- and triaxial) stress state:

- a method using the Mohr method;

- methods using various strength hypotheses [3, 4].

A more accurate estimate of the static strength of a complexly stressed part is provided by the Mohr method [3, 4]. When it is used, first, under various stress states, the material of the part is tested. In this case, in the tested samples from the material of the part, certain relationships are created between the main stresses σ ₁ and σ ₃ acting in them. Further, without changing the proportion between the indicated principal stresses, they are increased until the appearance of fluidity or destruction of the sample. The limiting values of the main stresses σ ₁ and σ ₃ fixed at the same time are displayed in a rectangular coordinate system Оστ in the form of a Mohr circle. Performing similar operations with other proportions between σ ₁ and σ ₃ available for testing, in a rectangular coordinate system Oστ, a series of Mohr circles are constructed that correspond to the studied limit stress states at given ratios of the stress tensor components. Mohr's circles constructed in this way are enveloping their curve, which determines the boundary of the region of ultimate stress states (region of static strength) of the considered material of the part for various combinations of σ ₁ and σ ₃ .

After that, in the same rectangular coordinate system Oστ, Mohr circles are built, which, for each of the considered options for loading the studied part, display the stress tensors arising in it, acting in the plane of principal stresses σ ₁ and σ ₃ . Moreover, if each of these circles is within the area of permissible stress states, then the static strength of the part in the area under study is ensured.

The disadvantages of the method include its excessive complexity, due to the fact that mutually perpendicular stress vectors σ ₁ and σ ₃ can have very different directions in space. The values of σ ₁ and σ ₃ in the tested samples from the material of the part and in the part under study can often be established by calculation, and with inevitable calculation errors, and experimentally - only using frozen volumetric polarization optical models. Therefore, it is expedient to apply the specified classical Mohr method to study the static strength of parts in the zone of contact forces, where the maximum stress tensor has a triaxial character. When assessing the static strength of a complexly stressed part outside the zones of influence of contact forces on it, as already noted, it is sufficient to take into account the greatest stresses that arise on its surface and are of a plane stressed nature, therefore, they are easier to calculate and experimentally determine. In addition, the classical Mohr method does not take into account the influence on the static strength of the second main stress σ ₂ .

The disadvantages of the classical Mohr method also include the construction of the boundary of the area of limit stress states of the material under consideration (the boundary of the area of static strength) in the form of a line that envelops the circle of a series of Mohr limit circles. If, for example, it turns out that the loss of performance of the material of the part under triaxial tension occurs at a lower value of the first principal stress σ ₁ than with its uniaxial tension, then the Mohr limit circle corresponding to triaxial tension will be inside the Mohr circle corresponding to the stress state in uniaxial tension. stretching. In this case, the construction of the area of static strength in the form of an envelope of Mohr's limiting circles will be impossible.

There are also known methods for determining the static strength of a complexly stressed part using strength hypotheses that make it possible to bring the complex stress state of the material to a uniaxial stress state equivalent to it in terms of breaking capacity [3, 4]. These methods are convenient in that the static strength margin is calculated as the ratio of the equivalent stresses acting in the part under study to the equivalent stresses of the limit stress state of its material, which are expressed as numbers. The most common and frequently used at present are the Tresca-Saint-Venant tangential stresses and the Huber-Mises energy of shape change.

The disadvantages of methods for determining the static strength for each of these hypotheses include the fact that for a number of stress states, these hypotheses do not fully reflect the behavior of a real material. In particular, these hypotheses do not reflect that the yield or fracture strength of a real material during its tension and compression by stresses of the same modulus often do not coincide with each other. In addition, the calculation according to various strength hypotheses gives different results, and each of these hypotheses is more suitable for one type of material. At present, there has not been a visual comparison of the results obtained for different types of stressed state of the material of the part by experimentally determining the static strength and for each of the indicated strength hypotheses.

The technical result of the invention is to increase the accuracy of determining the static strength of parts in their surface areas located outside the zones of contact forces. The increase in accuracy is achieved due to:

- determining the margin of static strength n in the investigated area of the surface of the part by identifying the ratio of the maximum allowable stresses in the material of the part to the maximum stresses that occur in the specified area of the part with the same ratio of all components of the tensors of the compared plane stress states;

- determining, according to the data of experimental research on samples from the material of the part, the exact values of the components of the plane stress tensors corresponding to the yield strength or the achievement of the static strength limit of this material and constructing in a rectangular coordinate system Oστ the exact boundary of the region of limiting plane stress states of this material;

- selection for the calculation of one of the strength hypotheses used, according to which the boundary of the area of limiting plane stress states to the greatest extent repeats the boundary obtained experimentally;

- taking into account not two, but all three main stresses of the tensors of the stress states compared with each other (one of these main stresses is directed perpendicular to the surface of the part and is equal to zero).

The technical result according to the first variant is achieved by the fact that in the method for determining the static strength of a complexly stressed part outside the zones of influence of contact forces on it, including the construction in a rectangular coordinate system Оστ of a graphical record of a low-cycle change of biaxial stress tensors at the considered point of the part surface during its operation, performed in in the form of a series of Mohr circles constructed on the basis of data on biaxial stresses at this point at successively taken moments of time, with the centers of Mohr circles located on the Oσ axis, the direction of which is associated with the chosen direction on the surface of the part under study, which also includes construction in the same rectangular system coordinates Oστ of the boundary of the area of limit stress states of the material of the part, constructed by generalizing individual experimental data, which are displayed as a series of Mohr limit circles describing various tensors p limit stress states in samples from the material of the part, according to the experimental study conducted on samples from the material of the part, the exact values of various combinations of the components of the plane stress tensors are determined, at which the yield strength or static strength limit of the material of the part is reached, these combinations of components of the plane stress tensors reflect in a rectangular coordinate system Oστ in the form of a series of Mohr limit circles, then by drawing the diameters of the Mohr limit circles parallel to the Oτ axis and then constructing a line that envelops the point of intersection of these diameters with the circles bounding the indicated Mohr limit circles, the boundary of the area of limit plane stress states of the material of the part is constructed , while the criterion for ensuring the static strength of the investigated zone of the part is the location of the intersection points of the diameters, drawn parallel to the Oτ axis in all Mohr circles, which are displayed reflect individual plane-stressed states that occur in the investigated area of the surface layer of the part, with circles bounding these circles, inside the boundaries of the area of limiting plane-stressed states, additionally determine the refined static strength margin n in the studied area of the surface of the part by identifying the ratio of the maximum allowable shear stresses in the material of the part, identified according to the data of the experimental study, to the maximum tangential stresses arising in the investigated zone of the part with the same ratio of all components of the tensors of the plane stress states compared with each other.

The technical result according to the second variant is achieved by the fact that in the method for determining the static strength of a complexly stressed part outside the zones of influence of contact forces on it, including the construction in a rectangular coordinate system Оστ of a graphical record of a low-cycle change of biaxial stress tensors at the considered point of the part surface during its operation, performed in in the form of a series of Mohr circles constructed on the basis of data on biaxial stresses at this point at successively taken moments of time, with the centers of Mohr circles located on the Oσ axis, the direction of which is associated with the chosen direction on the surface of the part under study, which also includes construction in the same rectangular system coordinates Oστ of the boundary of the area of limit stress states of the material of the part, constructed by generalizing individual experimental data, which are displayed as a series of Mohr limit circles describing various tensors p limit stress states in samples from the material of the part, the construction of the boundary of the area of limit plane stress states of the material of the part in a rectangular coordinate system Оστ is performed by constructing for each of the hypotheses of strength at different stress tensors of a series of Mohr limit circles, the diameters of the indicated Mohr limit circles are drawn parallel to the Oτ axis and plotted the line enveloping the points of intersection of these diameters with the circles bounding the indicated limiting Mohr circles, choose the boundary of the area of limiting plane stress states that most closely repeats the boundary constructed experimentally, while the criterion for ensuring the static strength of the investigated zone of the part is the location of the intersection points of the diameters, drawn parallel to the Oτ axis, in all Mohr circles, displaying individual plane-stressed states that occur in the investigated area of the surface layer of the part, with circles bounding these circles inside the indicated th boundary of the area of limit plane stress states, additionally determine the margin of static strength n in the investigated area of the surface of the part by identifying the ratio of the maximum allowable shear stresses in the material of the part, obtained by the formulas of that hypothesis of strength, according to which the boundary of the area of limit plane stress states is constructed, which most closely matches with boundary, built experimentally, to the maximum tangential stresses arising in the investigated zone of the part with the same ratio of all components of the tensors of the plane stress states being compared with each other.

The proposed methods for determining the static strength of a complexly stressed part are illustrated by figures.

In FIG. 1 shows the boundaries of the region of limiting plane stress states of the material of the part, constructed by experimental and computational methods, and also shows an example of determining the static strength margin for one of the tensors of the plane stress state on the surface of the part under consideration.

In FIG. 2 shows the state of ultimate plane-stressed compression of the surface layer of the part.

In FIG. 3 - the state of limit plane stress tension - compression of the surface layer of the part.

In FIG. 4 - the state of limiting plane-stressed tension of the surface layer of the part.

In FIG. 5 shows the construction of the boundary of the area of limiting plane stress states of the material of the part by calculation according to the hypothesis of strength of the highest shear stresses Tresca - Saint-Venant.

In FIG. 6 shows the construction of the boundary of the area of limiting plane-stressed states of the material of the part by the calculation method according to the Huber-Mises strength hypothesis.

Lines 1, 2, 3 in Fig. 1 shows different boundaries of the area of limiting plane stress states of the material of the part, which determine the static strength of the surface layer of this part in areas where contact forces do not act on it. The boundaries are presented in a rectangular coordinate system Oστ, where σ and τ are, respectively, the normal and tangential stresses acting at the considered point of the surface layer. Solid line 1 denotes the boundary of the area of limiting plane stress states, built on the basis of experimental data on the study of test samples made from the material of the part. Dash-dotted line 2 denotes the boundary of the area of limiting plane stress states, constructed according to the hypothesis of the highest shear stresses. Dotted line 3 - the boundary of the area of limiting plane-stressed states of the material of the part, built on the hypothesis of the strength of the energy of deformation. Mohr's circle 4 corresponds to the maximum allowable uniaxial tension of the material under study. The Mohr circle 5 displays one of the stress tensors that sequentially appear at the considered point of the part surface during its operation.

In FIG. Figures 2-4 show three possible arrangements of the three Mohr circles, which characterize in a volumetric representation the plane stress state that occurs in the three main areas of the point under study, located on the surface of the part outside the zone of influence of external forces on it. In this case, position 6 denotes the Mohr circle corresponding to the tensor of limit stresses arising at the investigated point of the part surface in that main plane, which is located tangentially to the part surface. Positions 7 and 8 indicate the Mohr circles corresponding to the tensors of the limit stress states arising at the point under study in two other main planes located perpendicular to the directions of the two main stresses acting in the plane tangential to the surface of the part.

In FIG. 5 shows the construction in the Oστ coordinate system of the AVDED'B'A'C boundary of the area of static strength of the surface layer of the material of the part according to the hypothesis of strength of the largest shear stresses. Positions 9, 10, 11 denote individual Mohr circles corresponding to the tensors of the limiting plane stress states arising in the main plane coinciding with the plane tangent to the surface of the part (Mohr circles 9, 10, 11 in Fig. 5 correspond to Mohr circles 6 in Fig. 2 -3).

In FIG. 6 shows the construction in the Oστ coordinate system of the boundary of the static strength region of the surface layer of the material of the part according to the hypothesis of the strength of the forming energy. Positions 1', 2', ..., 18' mark the individual points of this boundary, corresponding to the tensors of the limiting plane stress states that occur in the main plane, which coincides with the plane tangent to the surface of the part.

The method for determining the static strength of a complexly stressed part outside the zones of influence of contact forces on it at the considered point on its surface is carried out as follows.

Test specimens are made from the material of the part, in the surface layer of which various predetermined proportions are created between the individual components of the stress tensors. Further, by gradually increasing the load applied to the samples while maintaining the specified proportion between the individual components of the stress tensor, the stress values are determined at which the normal operating conditions of these samples are violated due to the onset of fluidity or the achievement of the static strength limit of the material under study.

The tensors of limiting plane stress states that arise in the plane of the surface of the tested samples in the period preceding the onset of yield or fracture are depicted in the system of rectangular coordinates Оστ in the form of Mohr's limit circles. Further, parallel to the Oτ axis, the diameters of these circles are drawn. The points of intersection of these diameters with the circles bounding the Mohr circles correspond to the values of the maximum tangential stresses for the considered combination of components of the plane stress tensors. A line is drawn around these points, which is the boundary of the area of maximum allowable stresses for a given material, obtained on the basis of experimental data. A possible view of such a boundary is shown in Fig. 1 solid line 1.

The configuration of the boundary constructed in this way differs from that which is usually depicted in the presentation of Mohr's generalized theory of strength [3, 4]. This is due to the fact that in the classical generalized theory of Mohr's strength, the boundary of ultimate stress states is constructed from the stresses acting in the plane of the largest and smallest principal stresses σ ₁ and σ ₃ . In this case, the direction of the specified principal stresses in the space of the test sample is not taken into account. In Mohr's classical theory of strength, the value of the second principal stress σ ₂ in the Oστ coordinate system is between the values σ ₁ and σ ₃ . Negative values of σ ₁ and σ ₃ correspond to the state of three-sided compression. With a proportional increase in the absolute values of three compressive stresses close in modulus, it is impossible to destroy the material of the part. Therefore, the upper and lower branches of the boundary of the area of maximum allowable stresses of the material of the part in the classical generalized theory of Mohr's strength do not interlock with each other.

In FIG. 1, the left side of the boundary of the region of permissible plane stress states defines the Mohr limit circle, which is built at uniaxial or biaxial compressive stress with the main stresses acting in the plane of the surface layer of the part material, σ ₂ and σ ₃ , (Fig. 2). The maximum σ ₁ of the three main stresses in this case is directed perpendicular to the surface of the part and is equal to zero. Therefore, if at the point under study the value of σ ₃ modulo exceeds the allowable value, then the value of |σ ₁ - σ ₃ | will also exceed the allowable value, while the strength condition of the part at the point under study will be violated. In this regard, in FIG. 1, the left part of the boundary 1 of the area of permissible plane stress states is closed.

The configuration of the right part of the boundary 1 of the area of limit plane stress states in the material of the part, which is shown in Fig. 1 corresponds to such behavior of the material, in which, in the case of uniform biaxial tension of the surface layer, its destruction occurs at a lower value of the first two principal stresses than in the case of the limiting uniaxial tension shown in FIG. 1 by the Mohr circle 4. In this case, it is impossible to construct the boundary of the static strength region in the form of an envelope of a series of limiting Mohr circles, as is customary in the classical generalized Mohr theory of strength. Therefore, the boundary 1 of the area of limit plane stress states of the material of the part is proposed to be carried out in the Oστ coordinate system by connecting the upper and then the lower points of the vertically drawn diameters of the Mohr circles, which in the process of experimental studies display the limit plane stress states in this material.

With the indicated construction of the boundary 1 of the area of limiting plane-stressed states of the material of the part in the plane of its surface layer, an accurate account of all three principal stresses was made. (One of these principal stresses is directed perpendicular to the surface of the part and is equal to zero). Accounting for all three principal stresses increases the accuracy and reliability of the considered method for determining the static strength of a part in comparison with the method proposed in the classical Mohr theory of strength, where the effect on the static strength of the second principal stress is not taken into account.

After constructing according to the data of the experimental study of the boundary 1 of the static strength area (figure 1) in the same coordinate system Оστ, Mohr circles are built, which at different times display the stress tensors that occur at the considered point on the surface of the part during its operation, for example, the Mohr circle 5. Parallel to the Oτ axis, the radii or diameters of these circles are drawn. (In Mohr's circle 5 this is the radius AC). The points of intersection of the indicated radii or diameters with the circles bounding the Mohr circles (in the Mohr circle 5 this is point A) display the maximum shear stresses that sequentially occur in the specified zone of the part during its operation. The criterion for ensuring the static strength of the investigated zone of the part is the location of these points inside the boundary, built according to the data of the experimental study.

The static strength margin n in the investigated area of the part surface in each of its plane stressed states is determined by identifying the ratio of the maximum shear stresses at which a yield state occurs or the static strength limit of the part material is reached, to the maximum shear stresses that occur in the specified area of the part during its operation . In this case, the ratio of all components of the tensors of the plane stress states being compared with each other should be the same. To determine the margin of static strength n at the considered point of the surface of the part in its plane-stressed state, defined by the Mohr circle 5, through the point O of the origin of the coordinate system Oστ and point A of radius CA, drawn parallel to the axis Oτ, a straight line is drawn until it intersects at point B with boundary 1 considered area of static strength, constructed according to the data of the experimental study. Coefficient the static strength margin is determined by the ratio:

n \u003d VD / AS \u003d OB / OA \u003d OD / OS (1)

Expression (1) is based on the fact that with an increase in proportion to the static strength factor n of all components of the plane stress state tensor under consideration, displayed by the Mohr circle 5, the position of the center of the Mohr circle 5 on the Oσ axis and its diameter will increase in proportion to the static strength factor n . At the same time, at Mohr's circle 5, the upper point A of radius CA, drawn parallel to the axis Oτ, will coincide with point B (Fig. 1) and will be on the border 1 of the static strength region.

The boundary of the region of limiting plane stress states of the material of the part in the rectangular coordinate system Оστ can also be constructed according to various hypotheses of strength. In FIG. 1, positions 2 and 3 indicate the boundaries constructed, respectively, according to the Tresca-Saint-Venant hypothesis of the greatest tangential stresses and according to the Huber-Mises hypothesis of strength of the energy of shape change.

The principle of constructing the boundary according to the hypothesis of the greatest tangential stresses is illustrated in Fig. 2-4. Three Mohr circles are shown on each of these figures in the Oστ rectangular coordinate system, representing the full plane stress state in the part material. The Mohr circle 6 (Fig. 2-4) describes the plane stress state in the main area located in the plane tangent to the surface of the part, and the other two Mohr circles 7 and 8 describe the stress state in the main areas located in the planes perpendicular to this surface. Since the modulus of the main stress directed perpendicular to the surface of the part is equal to zero, the Mohr circles 7 and 8 touch the origin of the coordinate system Оστ at one of their points. The two main stresses acting in the plane of the part surface can have the same signs - biaxial compression (Fig. 2) or biaxial tension (Fig. 4) of the surface layer. The nature of the location of the three Mohr circles with different signs of the principal stresses acting in the plane of the part surface is shown in Fig. 3.

According to the hypothesis of strength of the highest tangential stresses Tresca - Saint-Venant, a dangerous stress state at the considered point of the material of the part occurs if the largest of the three maximum tangential stresses τ _max acting at this point in the main areas reaches the maximum allowable value τ _L . In turn, this maximum allowable value τ _L determines the equivalent stress σ _{ekv.kasat .} , according to the specified strength hypothesis [3, 4]. Thus, the strength condition is met if:

where τ _max - the largest of the three maximum shear stresses acting in the main areas at the point under consideration;

τ _L - maximum allowable shear stress;

σ 1 - maximum principal stress at the considered point;

σ 3 - the minimum principal stress at the considered point;

σ _{equiv.kasat .} - equivalent stress according to the hypothesis of strength of the largest tangential stresses.

According to (2), in order to comply with the Tresca-Saint-Venant hypothesis of strength of the greatest shear stresses, the diameter D of the largest of the three Mohr circles (Fig. 2-4), displaying the largest difference in principal stresses σ ₁ - σ ₃ , should not go beyond the area maximum shear stresses ±τ _L . Based on these conditions, by considering various combinations of two main stresses that act in the plane of the surface of the part and participate in the construction of the Mohr circle 6 with a diameter D ₁ , it is possible to construct the area of limit stress states in the surface layer of the part in the rectangular coordinate system Оστ, at which the complete the stress tensor in the investigated area of its surface takes the maximum allowable values. In this case, the maximum shear stresses τ _L can be achieved in the plane tangential to the surface layer of the part (Fig. 3) or in one of the main planes perpendicular to its surface layer (Fig. 2, 4).

The maximum tangential stresses τ _max occur in the plane tangential to the surface of the part, if biaxial tension-compression stresses act in its surface layer (Fig. 3). In this case, the static strength of the part is determined by the principal stresses σ ₁ and σ ₃ acting in the plane of its surface. According to the accepted condition (2), shear stresses τ _max should not exceed the maximum allowable stresses τ _L . Depending on the ratio between the absolute values of the principal stresses σ ₁ and σ ₃ Mohr's circle 6, which displays the tensor of the maximum allowable stresses in the surface layer of the material of the part, can move in Fig 3 to the left or right, and the boundary of the area of flat limiting plane stress states in the surface layer of the material of the part (the boundary of the area of static strength) in Fig 3 is located in the gap between the direct AED and A'B'D'. At stresses of biaxial tension-compression in the surface layer of the part, if ( σ ₁ - σ ₃ ) > 2 τ _L , the plane of the highest tangential stresses, along which the plastic shear of the material occurs, will be located perpendicular to the surface of the part.

Under various plane-stress states corresponding to biaxial compression or biaxial tension of the surface layer of the part (Figs. 2 and 4), the maximum shear stresses τ _max at the studied point of the part surface act in the main areas perpendicular to the surface layer. The maximum allowable values of the tensors of these stresses in Fig. 2 and 4 are depicted by Mohr circles 7. As noted above, the stress tensors that arise in this case in the plane of the part surface are displayed by Mohr circles 6. FIG. 5 stress tensors, which in the case of biaxial compression of the surface layer of the part, act in the plane of its surface and correspond to the appearance of limit stress tensors in planes perpendicular to the surface of the part, are shown by Mohr circles 9, 10 and 11. Similar circles can be shown on the right side of figure 5 in zone of biaxial tension of the surface layer. At stresses of biaxial tension or biaxial compression in the surface layer of the part, if ( σ ₁ - σ ₃ ) > 2 τ _L , the plane of the highest tangential stresses, along which the plastic shear of the material occurs, will be located approximately at an angle of 45 ° to the surface of the part.

In FIG. 5 it can be seen that it is impossible to represent the boundary of the region of maximum allowable stresses in the form of a line enveloping circles 9, 10 and 11 of Mohr's limit circles, which, under bilateral compression, display the limit stress tensors acting in the surface layer of the part, is impossible. This is due to the fact that in this case, in the Oστ coordinate system, the areas of maximum allowable stresses, which are defined by one Mohr circles, overlap with areas that are defined by other Mohr circles (Fig. 5), as a result of which the boundary of the static strength region will be determined ambiguously. A similar conclusion could be drawn when considering the Mohr circles corresponding to the biaxial tension of the surface layer (not shown in Fig. 5).

In connection with the foregoing, it is more convenient to build the boundary of the static strength region in the Oστ coordinate system, not in the form of an envelope of Mohr circles that display the maximum allowable stresses in the plane of the part surface, but in the form of an envelope of the upper and lower end points of the diameters of these circles drawn parallel to the Oτ axis, then there are points that express the magnitude of the maximum and minimum shear stresses defined by these circles. In FIG. 5, straight lines CA and CA', as well as DE and ED' are obtained in this way. This method of construction can be extended to the classical theory of Mohr's strength. With such a construction, each Mohr circle describing the limit stress states in the plane of the surface of the part is associated with two strictly defined points of the boundary of the static strength region. In FIG. 5, the boundary of the static strength region obtained in this way is represented by the hexagon SADED'A' represented by a dotted line.

No less common at present is the Huber-Mises strength hypothesis, which is based on the accepted assumption that a dangerous state of a material arises due to exceeding a certain threshold of the maximum allowable specific strain energy associated with a change in the shape of the elementary volumes of this material [3, 4] . According to this hypothesis, a complex stress state with principal stresses σ₁, σ₂ and σ₃ uniquely equivalent in terms of breaking capacity to uniaxial stress σ _{eq energy} , which can be determined by the formula:

where σ ₁ is the maximum principal stress at the point under consideration;

σ ₂ - the second main stress at the considered point;

σ ₃ is the minimum principal stress at the considered point.

In any of the strength hypotheses, various multiaxial stress states are reduced by means of certain formulas to equivalent ones in terms of breaking capacity, but more accessible for experimental studies. Usually, uniaxial tension of the material under study is chosen as such a more accessible stress state. The magnitude of stresses at which their maximum allowable value is reached during uniaxial tension σ _before , is determined experimentally and does not depend on the chosen strength hypothesis, therefore the Mohr circles corresponding to the stress state of uniaxial tension in different strength hypotheses have the same diameterD Mohr circle 4 (Fig. 1). Thus, when constructing in the Oστ coordinate system the boundary of the static strength region of the surface layer of the material, according to the Huber-Mises deformation energy hypothesis, the Mohr circle corresponding to the uniaxial tension of the material, in Fig. 6 coincides with what was chosen as the basis for constructing the boundary of the static strength region according to the Tresca-Saint-Venant hypothesis of the highest tangential stresses. The diameter of this circle, drawn parallel to the Oτ axis, also determines two points of the static strength region constructed on the basis of experimental data (Fig. 1 boundary 1).

. При этом на фиг. 4 точка, соответствующая напряжению

, переместится в начало координат, а диаметры кругов Мора 6 и 7, отображающих тензоры напряжений в двух главных плоскостях, проходящих через линию действия первого главного напряжения σ₁, будут иметь одинаковый диаметр D, отображающий указанную величину напряжений растяжения σ₁. Подставляя в формулу (3) указанные значения напряжений

, получим, что в этом случае σ _{экв энерг} =

. В тоже время первое главное напряжение

, является предельно допустимым напряжением:

=σ _пред .In the case of uniaxial tension of the material

. Meanwhile, in FIG. 4 point corresponding to voltage

, will move to the origin of coordinates, and the diameters of the Mohr circles 6 and 7, displaying the stress tensors in two main planes passing through the line of action of the first main stress σ ₁ , will have the same diameter D , displaying the specified value of tensile stresses σ ₁ . Substituting into formula (3) the indicated stress values

, we obtain that in this case σ _{equiv energy} =

. At the same time, the first principal stress

, is the maximum allowable voltage:

=σ _before .

As in the hypothesis of the greatest tangential stresses, in the hypothesis of the energy of deformation it is assumed that the ultimate stresses in uniaxial tension and uniaxial compression of the material of the structure are equal in absolute value, therefore the Mohr circles corresponding to these two stress states in Fig. 1 and 6 have the same diameter D . Said Mohr circles of diameter D are defined in FIG. 6 positions of four points of the border of the static strength area - points 7'.

The construction in the Oστ coordinate system of the entire contour of the boundary of the static strength region of the surface layer of the material of the part according to the hypothesis of the formation energy is performed according to the following algorithm:

,

и

в формуле (3) заменяют диаметрами D ₁, D ₂ и D ₃ трех главных кругов Мора (фиг. 2 - 4), которые в системе координат Оστ в выбранном масштабе отображают плосконапряженное состояние в рассматриваемой точке детали в его трехмерном представлении. Если рассматривается предельно допустимое плосконапряженное состояние, то величину σ_{экв энерг} заменяют диаметром D круга Мора, отображающего предельно допустимую величину напряжений в исследуемом материале при его одноосном растяжении (фиг.4), а диаметры D ₁, D ₂ и D ₃ определяют различные сочетания трех максимально допустимых главных напряжений, действующих в рассматриваемой точке детали. В результате указанных преобразований формула (3) может быть записана в виде:- difference in the values of the principal stresses

,

and

in formula (3) is replaced by the diameters D ₁ , D ₂ and D _{3 of} the three main Mohr circles (Fig. 2 - 4), which in the Oστ coordinate system on the selected scale display the plane stress state at the considered point of the part in its three-dimensional representation. If the maximum allowable plane stress state is considered, then the value of σ _{equiv energy is} replaced by the diameter D of the Mohr circle, which displays the maximum allowable stress in the material under study during its uniaxial tension (Fig.4), and the diameters D ₁ , D ₂ and D ₃ determine various combinations of three the maximum allowable principal stresses acting at the considered point of the part. As a result of these transformations, formula (3) can be written as:

whereD ₁ ,D ₂ , D ₃ - diameters of three Mohr circles, representing the maximum allowable plane stress state at the considered point of the part;

,

и

в формуле (3); D is the diameter of Mohr's circle, which reflects the maximum allowable stress in the material under study during its uniaxial tension. Depending on the location of the Mohr circles in Fig. 2 - 4, the sequence of the terms D ₁ , D ₂ and D ₃ in the formula (4) may not correspond to the sequence of the terms

,

and

in formula (3);

- on the selected scale, a Mohr circle with a diameter D is built (Fig. 6), which reflects the experimentally found value of the limiting plane stress state in the material under study during its uniaxial tension;

- set one of the possible proportions between the diameter of Mohr's circle D ₁ , representing the limiting plane stress state in the plane of the part surface, and the diameter of one of Mohr's circles, for example, D ₂ , representing the stress state in the plane of one of the main areas located perpendicular to the surface of the part. Given this proportion, the diameter D ₂ is expressed through D ₁ . Thereafter, taking into account the relationship between the diameters D ₁ , D ₂ and D _{3 of} the three Mohr circles shown in FIG. 2 - 4, according to which the extreme left and right points of two internal, contacting circles, go around the third circle, the diameter of the third circle D ₃ is also expressed through D ₁ and D ₂ . So in FIG. 2 and 4 diameter of the third circle: D ₃ = D ₂ - D ₁ and in FIG. 3: D ₃ \ u003d D ₁ - D ₂ ;

- diameter D ₁ , as well as the diameters of Mohr's circles D ₂ and D _{3 ,} expressed in terms of D ₁ , are substituted into formula (4). Further, using formula (4), through the diameter D of the Mohr circle (Fig. 2 - 4, 6), which corresponds to the yield strength or the achievement of the static strength limit in uniaxial tension of the material and is taken as the circle of maximum allowable equivalent stresses σ _{eq energy} = σ _before , determine the diameter D ₁ of Mohr's circle 6, which displays the maximum allowable stress in the plane of the surface of the part. After that, using the accepted proportions between the diameters of the three circles Mohr D ₁ , D ₂ and D ₃ determine the diameter of each of them;

- in accordance with the arrangement of circles D ₁ and D ₂ shown in FIG. 2 - 4 and the scale adopted when constructing the Mohr circle with a diameter D determine the position on the Oσ axis of the coordinate system Oστ (Fig. 6) of the center of the Mohr circle 6 with a diameter D ₁ . Through this center in the direction parallel to the axis Oτ, the diameter D _{1 of} the specified Mohr limit circle 6 is drawn, which reflects the maximum allowable plane-stressed state in the plane of the surface of the part, and the extreme points of this diameter are applied. Through these points, at the maximum allowable stress tensor, which is determined by the initially given proportion D ₁ / D ₂ , the boundary of the allowable stress area, built on the hypothesis of the energy of shape change, passes;

- by setting other proportions between D ₁ and D ₂ , it is possible to calculate the coordinates and construct in the Oστ coordinate system any other points of the boundary of the area of permissible plane stress states. With a limited number of points constructed in this way, an approximate construction of the boundary of the region of limiting plane stress states is performed by smoothly connecting the points constructed in this way (Fig. 6).

The proportions D ₁ /D ₂ that were set when calculating the diameters D _{1 of} Mohr's circles, which, according to the hypothesis of the strength of the energy of form change, display various maximum allowable stress tensors in the plane of the plane stress state acting on the surface of the part, as well as the values of D ₁ and point numbers obtained in this case, which in the plane of the plane stressed state define the boundary of the static strength region of the material of the part, are given in the table.

D1 _{/ D2} 2 3 four five 10 20 _D1 1.155D 1.134D 1.109D 1.091D 1.048D 1.025D No. of points
in fig. 6 1' 2' 3' four' five' 6' D1 _{/ D2} 1 19/20 9/10 4/5 3/4 2/3 _D1 D 0.973D 0.943D 0.873D 0.832D 0.756D No. of points
in fig. 6 7' 8' nine' 10' eleven' 12' D1 _{/ D2} 1/2 1/3 1/4 1/5 1/10 1/20 _D1 0.577D 0.378D 0.277D 0.218D 0.105D 0.051D No. of points
in fig. 6 13' fourteen' 15' sixteen' 17' eighteen'

The boundary of the region of limiting plane stress states of the material of the part, constructed from these points, is shown in Fig. 6.

Obtained in different ways, the boundaries 1, 2, 3 of the areas of limit plane stress states in the surface layer of the material of the part (Fig. 1) differ from each other. The most accurate representation of the limiting plane stress states in the surface layer of the material of the part is determined by boundary 1, since, unlike the other two, this boundary is not built according to strength hypotheses, but according to experimental data to determine the loss of the material's bearing capacity for various combinations of stress tensor components in test samples from the material of the part. Boundary 1, in particular, makes it possible to take into account the difference in static strength at the same modulus of tensile and compressive stresses.

Comparison of the static strength boundaries 2 and 3 (Fig. 1), constructed, respectively, according to the Tresca-Saint-Venant hypothesis of the greatest tangential stresses and according to the Huber-Mises shape change energy hypothesis, shows that more stringent, more conservative strength requirements are set by the hypothesis of the greatest shear stress Cod - Saint-Venant. The maximum difference in ultimate stresses according to the two indicated hypotheses reaches 13% and is observed when in formula (3) |σ ₁ - σ ₂ |= |σ ₂ - σ ₃ |, and in formula (4) the diameters of two inner Mohr circles equal to each other: D ₁ = D ₃ or D ₂ = D ₃ (see Fig. 2 - 4). The indicated plane stress states in Figs. 6 and in Table 1 are designated by point numbers 1' (pure shear) and 13' (uniform tension or compression of the surface layer of the material in two main directions).

At present, the static strength margin n in the studied plane stressed zone of the part is usually calculated by determining the ratio of the maximum allowable stresses in the material of the part to the maximum equivalent stresses that occur in the investigated zone of the part during its loading. In this case, the numerical value of the maximum allowable stresses is established on the basis of data from an experimental study on the destruction or the beginning of the yield of samples from the material of the part during their uniaxial tension. The numerical value of the equivalent stresses arising in the studied zone of the part can be determined by recalculating the values of the stress tensor components in this zone according to the corresponding formula (2 or 3) of the strength hypothesis used in this case.

Using boundaries 2 and 3, constructed in Fig. 1 by calculation, the margin of static strength n in the investigated area of the surface of the part can also be calculated by formula (1). In this case, the straight line OA is continued until it intersects with one of the specified boundaries 2 or 3. Point B is transferred to the point of intersection of the indicated boundaries with the straight line OA. The calculation of the static strength margin n according to formula (1) is carried out taking into account the changed distance from point O to point B.

To increase the accuracy of calculating the margin of static strength n in the studied area of the part surface using strength hypotheses, one of these hypotheses is used, according to which the boundary of the area of limiting plane stress states of the part material, constructed by calculation, to the greatest extent repeats the boundary of limiting plane stress states of the part material, constructed experimentally . In order to reduce the risks of obtaining, in this case, an underestimated static strength factor n, it is necessary that the boundary of the static strength region, constructed according to the formulas of the applied strength hypothesis, be inside boundary 1 (Fig. 1), constructed experimentally. In zones of a plane stressed state, where such a condition is not met, the determination of the static strength factor n according to the corresponding strength hypothesis will lead to overestimated results. Therefore, in such zones, the static strength factor must be specified by calculating it according to formula (1), while point B is the point of intersection of the straight line OA and the boundary constructed on the basis of experimental data.

Thus, a method is proposed for determining the static strength of a complexly stressed part outside the zones of external contact forces on it, in which the boundaries of the tensors of limiting plane stress states in the surface layer of the material of the part are depicted in the Oστ coordinate system, built experimentally and according to various strength hypotheses. When constructing the boundary of the static strength region of the surface layer of the material of the part, based on experimental data, the influence of not two, but three main stresses, one of which is always equal to zero, is taken into account. Compared to the method used in Mohr's classical theory of strength, this increases the accuracy of constructing such a boundary and the reliability of the results of calculating the static strength of a part using such a boundary.

It is easier to calculate the static strength of a complexly stressed part outside the zones of influence of contact forces on it by the proposed method than using Mohr's classical theory of strength for these purposes. This is due to the fact that when using the classical theory of Mohr's strength in the studied area of the part, it is necessary to determine experimentally or by calculation not two, but three main stresses σ ₁ , σ ₂ and σ ₃ , compare these stresses with each other, and then take into account only the maximum σ ₁ and minimum σ ₃ , while in the proposed method, only two principal stresses acting in the plane of the plane stress state are taken into account. The possibility in the proposed method of visual comparison of the boundaries of the static strength region, constructed according to various strength hypotheses, with the boundary, constructed on the basis of experimental data, allows you to select the most suitable strength hypothesis for each material of the part, according to which the boundary of the area of limiting allowable plane stress states of this material more accurately repeats the boundary constructed from the experimental data. The proposed method for constructing the boundary of the area of static strength of the material of the part, not in the form of an envelope of Mohr's limiting circles, but in the form of an envelope of points representing the maximum shear stresses on these circles, will make it possible to more accurately construct the specified boundary compared to the classical Mohr method.

List of sources used:

1. Vakhromeev, A.M. Determination of the cyclic durability of materials and structures of vehicles: guidelines - M.: MADI, 2015. - 64 p.

2. Patent for invention No. 2697022. Method for graphic recording of changes in biaxial stresses at the considered point of the part surface (options).

3. V. I. Feodos’ev Strength of materials. Seventh edition, revised. M. Nauka, 1974. - 560 p.

4. I.M. Belyaev Strength of materials. Twelfth edition. M.: State publishing house of physical and mathematical literature, 1959. - 856 p.

Claims (2)

1. A method for determining the static strength of a complexly stressed part outside the zones of contact forces on it, including the construction in a rectangular coordinate system Оστ of a graphical record of a low-cycle change of biaxial stress tensors at the considered point of the part surface during its operation, performed in the form of a series of Mohr circles, built on the basis of data on biaxial stresses at this point at successively taken moments of time, with the centers of the Mohr circles located on the Oσ axis, the direction of which is associated with the chosen direction on the surface of the part under study, also including the construction in the same rectangular coordinate system Oστ of the boundary of the region of limit stress states of the material part, constructed by summarizing individual experimental data, which are displayed as a series of Mohr's limit circles, describing various tensors of limit stress states available for obtaining during testing in samples from the material of the part , characterized in that, according to the data of an experimental study carried out on samples of the material of the part, the exact values of various combinations of the components of the plane stress tensors are determined, at which the yield strength or static strength of the material of the part is reached, these combinations of the components of the plane stress tensors are displayed in a rectangular coordinate system Оστ in the form of a series of Mohr limit circles, then by drawing the diameters of the Mohr limit circles parallel to the Oτ axis and then constructing a line that envelops the point of intersection of these diameters with the circles bounding the indicated Mohr limit circles, the boundary of the area of limit plane stress states of the material of the part is constructed, while the criterion for providing static strength of the studied zone of the part is the location of the intersection points of the diameters, drawn parallel to the Oτ axis in all Mohr circles, displaying individual plane stress states, in parts arising in the investigated area of the surface layer, with circles bounding these circles, inside the boundaries of the area of limit plane stress states, additionally determine the refined static strength margin n in the studied area of the surface of the part by identifying the ratio of the maximum allowable shear stresses in the material of the part, identified from the data of the experimental study , to the maximum tangential stresses arising in the investigated zone of the part with the same ratio of all components of the tensors of the plane stress states being compared with each other. 2. A method for determining the static strength of a complexly stressed part outside the zones of contact forces on it, including the construction in a rectangular coordinate system Оστ of a graphical record of a low-cycle change of biaxial stress tensors at the considered point of the part surface during its operation, performed in the form of a series of Mohr circles, built on the basis of data on biaxial stresses at this point at successively taken moments of time, with the centers of the Mohr circles located on the Oσ axis, the direction of which is associated with the chosen direction on the surface of the part under study, also including the construction in the same rectangular coordinate system Oστ of the boundary of the region of limit stress states of the material a part constructed by summarizing individual experimental data, which are displayed as a series of Mohr limit circles that describe various tensors of limit stress states available for testing in samples from the material of the part, characterized in that the construction of the boundary of the area of limiting plane-stressed states of the material of the part in a rectangular coordinate system Оστ is performed by constructing for each of the hypotheses of strength at different stress tensors of a series of Mohr limit circles, the diameters of the indicated Mohr limit circles are drawn parallel to the Oτ axis and a line is drawn that envelops the intersection points these diameters with circles bounding the indicated limiting Mohr circles, choose the boundary of the area of limiting plane stress states that most closely repeats the boundary constructed experimentally, while the criterion for ensuring the static strength of the investigated zone of the part is the location of the intersection points of the diameters drawn parallel to the Oτ axis, in all Mohr circles, which display individual plane stress states that occur in the investigated area of the surface layer of the part, with circles bounding these circles, inside the specified boundary of the area of limit plane stress states, additionally determine the margin of static strength n in the investigated area of the surface of the part by identifying the ratio of the maximum allowable shear stresses in the material of the part, obtained by the formulas of that hypothesis of strength, according to which the boundary of the area of limiting plane stress states is constructed, which most closely matches the boundary constructed by the experimental way, to the maximum tangential stresses arising in the investigated zone of the part, with the same ratio of all components of the tensors of the compared plane stress states.

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