CN111024520A - Calculation method, experimental method and experimental device for V-shaped grooving tip stress intensity factor under pure shear loading - Google Patents

Calculation method, experimental method and experimental device for V-shaped grooving tip stress intensity factor under pure shear loading Download PDF

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CN111024520A
CN111024520A CN201911326047.6A CN201911326047A CN111024520A CN 111024520 A CN111024520 A CN 111024520A CN 201911326047 A CN201911326047 A CN 201911326047A CN 111024520 A CN111024520 A CN 111024520A
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caustic
intensity factor
stress intensity
tip
formula
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杨立云
胡桓宁
宋烨
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China University of Mining and Technology Beijing CUMTB
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N3/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N3/24—Investigating strength properties of solid materials by application of mechanical stress by applying steady shearing forces
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N3/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N3/02—Details
    • G01N3/06—Special adaptations of indicating or recording means
    • G01N3/068—Special adaptations of indicating or recording means with optical indicating or recording means
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N2203/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N2203/02—Details not specific for a particular testing method
    • G01N2203/0202—Control of the test
    • G01N2203/0212—Theories, calculations
    • G01N2203/0218—Calculations based on experimental data
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N2203/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N2203/02—Details not specific for a particular testing method
    • G01N2203/026—Specifications of the specimen
    • G01N2203/0262—Shape of the specimen
    • G01N2203/027—Specimens with holes or notches
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N2203/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N2203/02—Details not specific for a particular testing method
    • G01N2203/06—Indicating or recording means; Sensing means
    • G01N2203/0641—Indicating or recording means; Sensing means using optical, X-ray, ultraviolet, infrared or similar detectors
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N2203/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N2203/02—Details not specific for a particular testing method
    • G01N2203/06—Indicating or recording means; Sensing means
    • G01N2203/067—Parameter measured for estimating the property
    • G01N2203/0676—Force, weight, load, energy, speed or acceleration

Abstract

The invention provides a calculation method, an experimental method and an experimental device for a V-shaped grooving tip stress intensity factor under pure shear loading, wherein the calculation method is based on a caustic method and comprises the following steps: the method comprises the following steps: selecting a planar object to be measured with a V-shaped cutting groove, and establishing an optical mapping relation between the plane of the object to be measured and a reference plane; step two: establishing a stress field expression of the V-shaped cutting groove tip; step three: establishing an initial curve radius r of the caustic0And stress intensity factor KIIThe relational expression of (1); step four: determining a caustic mapping equation of the V-shaped cutting groove under the pure shearing loading; step five: establishing a characteristic length DmaxInitial curve radius r from the caustic0The relational expression of (1); step six: determining the stress intensity factor K of the V-shaped notch tip under pure shear loadingII. The invention researches and obtains the stress intensity factor K of the V-shaped notch tip under pure shear loadingIIIs calculated byThe method provides a new way for acquiring the stress intensity factor of the V-shaped cutting groove tip. The method has good precision and numerical stability.

Description

Calculation method, experimental method and experimental device for V-shaped grooving tip stress intensity factor under pure shear loading
Technical Field
The invention belongs to the field of fracture mechanics, particularly relates to calculation of a stress intensity factor of a component containing a V-shaped groove, and relates to a calculation method, an experimental method and an experimental device of a stress intensity factor of a tip of the V-shaped groove under pure shear loading.
Background
V-shaped cutting grooves are widely formed in various engineering structures, such as large civil buildings and microelectronic structures. The presence of V-notches can lead to stress concentrations and even stress singularities, thereby reducing the load bearing capacity of the structural member. Therefore, it is important to study the stress distribution around the tip of the V-notch. The V-shaped cutting groove belongs to a kind of problem in fracture mechanics, and the key for researching the stress distribution of the tip of the V-shaped cutting groove is to determine the stress intensity factor of the corresponding cutting groove tip.
There are many methods for calculating the stress intensity factor of the tip of the V-notch, such as theoretical calculation, numerical calculation, and experiment. In terms of theory and numerical calculation methods, methods such as a boundary configuration method, a finite element method and an extended finite element method exist. However, the problems of complex theory, heavy modeling process and low calculation efficiency often exist by utilizing the theory or numerical calculation.
In terms of experimental methods, the caustic method is a photometric mechanical experimental method particularly suitable for solving the singularity problem, has the advantages of less measurement data, high precision and the like, and is widely used for measuring the stress intensity factor of the V-shaped cutting groove tip. The related scholars studied the fracture characteristics of the notch tips of the V-notch plates under type I loading and symmetric loading.
In the above background, the researchers have calculated the stress intensity factor of V-notch tip under tensile load by using the caustic method. However, for the V-shaped grooving under pure shear loading, the theory and application of solving the tip stress intensity factor by using the caustic method are still lack of research.
Disclosure of Invention
The invention provides a calculation method, an experimental method and an experimental device for a V-shaped grooving tip stress intensity factor under pure shear loading, aiming at the defects of the prior art and carrying out related research.
The invention provides a method for calculating a stress intensity factor of a V-shaped notch tip under pure shear loading, which is based on a caustic method and comprises the following steps:
the method comprises the following steps: selecting a planar object to be measured with a V-shaped cutting groove, and establishing an optical mapping relation between the plane of the object to be measured and a reference plane;
step two: establishing a stress field expression of the V-shaped cutting groove tip;
step three: establishing an initial curve radius r of the caustic0And stress intensity factor KIIThe relational expression of (1);
step four: determining a caustic mapping equation of the V-shaped cutting groove under the pure shearing loading;
step five: establishing a characteristic length DmaxInitial curve radius r from the caustic0The relational expression of (1);
step six: determining the stress intensity factor K of the V-shaped notch tip under pure shear loadingII。
As an improvement, the step two of establishing the expression of the stress field of the V-shaped notch tip includes:
establishing a polar coordinate system by taking the cutting groove tip of the object to be measured as the origin of coordinates, and establishing the generalized stress intensity factor K of the V-shaped cutting groove under pure shearing loadingIIIs defined as:
Figure BDA0002328402100000021
where r and theta are polar coordinates, taurθIs the shear stress of the tip of the grooving, and lambda is the stress singularity index of the tip of the grooving;
for a linear elastic, isotropic, homogeneous sheet, the stress field near the tip of the notch when subjected to a pure shear load is expressed as:
Figure BDA0002328402100000022
Figure BDA0002328402100000023
in the formula sigmarIs the radial stress, σθIs the tangential stress;
wherein the constant C is expressed as:
C＝(λ-1)sin(λ+1)γ-(λ+1)sin(λ-1)γ (3)
λ can be obtained from the following formula:
sin2λ(π-β)-λsin2(π-β)＝0 (4)。
as an improvement, step three establishes the initial curve radius r of the caustic line0And stress intensity factor KIIThe relational expression (A) includes an initial curve equation for determining a caustic line, specifically:
the corresponding relation between the reference plane and the point on the object plane to be measured is as follows:
Figure BDA0002328402100000024
the formula is a mapping equation, wherein z0Is the distance between the object plane to be measured and the reference plane, and Δ s is the optical path difference, λmIs the magnification of the optical system; when the light rays are parallel light, lambdam＝1；
The caustic is a singular curve formed by converging a plurality of light rays, and according to the diffraction theory, the sufficient necessary condition for generating the singularity is that the Jacobian determinant of the mapping equation (5) is zero, so that the initial curve equation for deriving the caustic is as follows:
Figure BDA0002328402100000031
as an improvement, step three establishes the initial curve radius r of the caustic line0And stress intensity factor KIIThe relationship of (a) further includes:
when a light beam passes through a transparent flat plate, the optical path difference of the isotropic material can be obtained by the following formula:
Δs＝cdeff(σr+σθ) (7)
wherein c is the caustic optical constant, deffIs the thickness of the flat plate;
substituting formula (2) for formula (7) to obtain:
Figure BDA0002328402100000032
the formula (8) may be substituted for the formula (6):
Figure BDA0002328402100000033
wherein A is:
Figure BDA0002328402100000034
from equation (9), when the load and the notch angle are constant, r is a constant, and is referred to as the initial curve radius of the caustic line, and is denoted as r0；
Equation (9) is the initial curve radius r of the caustic line0And stress intensity factor KIIThe relational expression (c) of (c).
As an improvement, the determination of the caustic mapping equation of the V-notch under the step four pure shear loading comprises:
by substituting equations (8) and (9) into equation (5), the equation for mapping the caustic of the V-notch under pure shear loading can be obtained:
Figure BDA0002328402100000035
the polar coordinates of the caustic at the reference plane are defined by the following relationship:
Figure BDA0002328402100000036
Figure BDA0002328402100000037
as an improvement, step five establishes the characteristic length DmaxInitial curve radius r from the caustic0The relationship of (1) includes:
to obtain a characteristic length DmaxLet WyThe derivative to the polar angle θ is zero, i.e.:
dWy/dθ＝0 (14)
since θ is within the range of the interval [ π - β, π + β ], it can be obtained from equation (14):
Figure BDA0002328402100000041
will theta1And theta2The value of (2) is substituted for the formula (11), the characteristic length D is obtainedmaxThe expression of (a) is as follows:
Figure BDA0002328402100000042
equation (16) can be written as follows:
r0＝Dmax/λmδ (17)
in the formula (I), the compound is shown in the specification,
Figure BDA0002328402100000043
the characteristic length D is represented by the formula (17)maxInitial curve radius r from the caustic0The relational expression (c) of (c).
As an improvement, step six determines the stress intensity factor K of the V-shaped notch tip under pure shear loadingIIThe method comprises the following steps:
establishing a characteristic length DmaxAnd stress intensity factor KIIBy substituting formula (17) for formula (9), the following can be obtained:
Figure BDA0002328402100000044
the characteristic length D is represented by formula (19)maxAnd stress intensity factor KIIThe relationship (c) of (a) to (b),
in the formula (I), the compound is shown in the specification,
Figure BDA0002328402100000045
only measuring the characteristic length D of the caustic line at the tip of the cutting groovemaxThat is, the stress intensity factor of the notch tip can be calculated by the formula (19)Seed KII。
The invention also provides an experimental method based on the calculation method, which comprises the following steps:
s11: selecting a pure shearing model test piece;
s12: carrying out a pure shear loading experiment on the pure shear model test piece by adopting a caustic wire experimental device;
s13: collecting the caustic line image under the action of corresponding load, and processing the caustic line image to obtain the characteristic length D of the caustic linemax；
S14: characteristic length D according to caustic linemaxCalculating the stress intensity factor K of the V-shaped notch tip under the pure shearing loadingII。
The invention also provides a caustic wire experimental device which is used for executing the experimental method.
The device comprises a laser transmitter, a beam expander, a field lens I, a model test piece, a reference plane, a field lens II, an image acquisition device and an image processing device which are sequentially arranged.
Has the advantages that: after the technical scheme is adopted, compared with the prior art, the invention has the following technical effects: the invention is based on the correlation theory of the caustic method and establishes the characteristic length DmaxAnd stress intensity factor KIIThe stress intensity factor K of the V-shaped cutting groove tip under pure shear loading is obtained through researchIIThe method of (3). The method enriches the theory and application of the caustic method, greatly improves the existing calculation method, avoids the problems of complex modeling and low calculation efficiency, facilitates the calculation, simplifies the complicated procedures of the experimental method, and only needs to obtain the characteristic length DmaxNamely, a new way is provided for acquiring the stress intensity factor of the V-shaped cutting groove tip. The method has good precision and numerical stability.
Drawings
FIG. 1 is a schematic view of a geometric model of a V-notch;
FIG. 2 is an optical diagram of the process of forming the caustic of the kerf tip (optical transformation diagram represented by the mapping);
FIG. 3 is a schematic diagram of theoretical caustic lines for different angle V-cuts;
FIG. 4 is a schematic diagram of defining the characteristic length of the caustic;
FIG. 5 is a schematic view of a pure shear loading model specimen;
FIG. 6 is a schematic view of a caustic experimental apparatus;
FIG. 7 is an experimental image of pure shear loading undercut tip caustic line (different loads);
FIG. 8 is a graph comparing experimental and theoretical caustic lines;
FIG. 9 is a graph comparing stress intensity factors calculated by the caustic method and the numerical method.
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this application, illustrate embodiment(s) of the invention and together with the description serve to explain the principles of the invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention more apparent, the embodiments of the present invention are described in further detail below with reference to the embodiments and the accompanying drawings. The exemplary embodiments and descriptions of the present invention are provided to explain the present invention, but not to limit the present invention.
In the description of the present invention, it is to be understood that the terms "upper", "lower", "front", "rear", "left", "right", "top", "bottom", "inner", "outer", and the like, indicate orientations or positional relationships based on the orientations or positional relationships shown in the drawings, and are only for convenience in describing the present invention and simplifying the description, but do not indicate or imply that the device, component, or structure referred to must have a particular orientation, be constructed or operated in a particular orientation, and should not be construed as limiting the present invention.
It will be further understood that the terms "comprises/comprising," "consists of … …," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a product, apparatus, process, or method that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such product, apparatus, process, or method if desired. Without further limitation, an element defined by the phrases "comprising/including … …," "consisting of … …," or "comprising" does not exclude the presence of other like elements in a product, device, process, or method that comprises the element.
The following will further explain the specific implementation method of the present invention with reference to the attached drawings.
The invention provides a method for calculating a stress intensity factor of a V-shaped cutting groove tip under pure shear loading, which comprises the following steps:
the method comprises the following steps: selecting a planar object to be measured with a V-shaped tip grooving, and establishing an optical mapping relation between the plane of the object to be measured and a reference plane;
considering a planar V-shaped notch geometry of an infinite homogeneous material, the notch angle is 2 β, γ ═ pi- β, taking the notch tip as the origin, and the bisector of the notch angle as the negative half axis of the x axis, a coordinate system is established, as shown in fig. 1, and according to the optical imaging principle, the optical mapping relationship between the object plane to be measured and the reference plane is shown in fig. 2.
Step two: establishing a stress field expression of the V-shaped cutting groove tip;
generalized stress intensity factor K of V-notch tip under pure shear loadingIICan be defined as:
Figure BDA0002328402100000061
where r and theta are polar coordinates, taurθIs the shear stress of the tip of the grooving, and lambda is the stress singularity index of the tip of the grooving;
for a linear elastic, isotropic, homogeneous sheet, the stress field near the tip of the notch when subjected to a pure shear load is expressed as:
Figure 3
in the formula sigmarIs the radial stress, σθIs the tangential stress;
wherein the constant C is expressed as:
C＝(λ-1)sin(λ+1)γ-(λ+1)sin(λ-1)γ (3)
λ can be obtained from the following formula:
sin2λ(π-β)-λsin2(π-β)＝0 (4)
from equation (4), the λ values corresponding to different slot angles 2 β can be calculated, and the λ values corresponding to common slot angles are shown in table 1 below:
table 1 table of correspondence between groove cutting angle 2 β and λ value
Figure BDA0002328402100000064
Figure BDA0002328402100000071
Step three: establishing an initial curve radius r of the caustic0And stress intensity factor KIIThe relational expression of (1);
first, the equation of the initial curve of the caustic is determined, and the principle of caustic formation at the tip of the V-notch is shown in FIG. 2. According to the transmission caustic method, a beam of parallel light maps an object plane to a reference plane, and when a test piece is loaded, the thickness and the material refractive index of the test piece are non-uniformly changed, so that the emergent light of the test piece is deflected. Therefore, the light intensity distribution is no longer uniform in any plane (reference plane) behind the specimen (i.e., the right side of the specimen). Some areas that are not illuminated by light are darkened, while other areas are multiplied in brightness due to doubling of light intensity. This results in a dark spot enveloped by a bright line, called a caustic line. Here, the correspondence between the reference plane and the point on the surface of the object is as follows:
Figure BDA0002328402100000072
the formula is a mapping equation, wherein z0Is the distance between the object plane to be measured and the reference plane, and Δ s is the optical path difference, λmIs a magnification of the optical systemCounting; when the light rays are parallel light, lambdam1 is ═ 1; where lambda ismIs 1;
the caustic is a singular curve formed by converging a plurality of light rays, and according to the diffraction theory, the sufficient necessary condition for generating the singularity is that the Jacobian determinant of the mapping equation (5) is zero, so that the initial curve equation for deriving the caustic is as follows:
Figure BDA0002328402100000073
on the basis of the above calculation results, further establishing the initial curve radius r of the caustic line0And stress intensity factor KIIIncludes:
when a light beam passes through a transparent flat plate, the optical path difference of the isotropic material can be obtained by the following formula:
Δs＝cdeff(σr+σθ) (7)
wherein c is the caustic optical constant, deffIs the thickness of the flat plate;
substituting formula (2) for formula (7) to obtain:
Figure BDA0002328402100000074
the formula (8) may be substituted for the formula (6):
Figure BDA0002328402100000075
wherein A is:
Figure BDA0002328402100000081
from equation (9), when the load and the notch angle are constant, r is a constant, and is referred to as the initial curve radius of the caustic line, and is denoted as r0；
Equation (9) is the initial curve radius r of the caustic line0And stress intensity factor KIIThe relational expression (c) of (c).
Step four: determining a caustic mapping equation for the V-notch under pure shear loading, comprising:
by substituting equations (8) and (9) into equation (5), the equation for mapping the caustic of the V-notch under pure shear loading can be obtained:
Figure BDA0002328402100000082
the polar coordinates of the caustic at the reference plane are defined by the following relationship:
Figure BDA0002328402100000083
Figure BDA0002328402100000084
the theoretical caustic curves for different slot angles under pure shear loading can be plotted from equation (11). At an initial curve radius r0The theoretical caustic curve was plotted with the slot angles of 0 °, 30 °, 60 °, 90 ° and 102.5 °, respectively, as 1, and the results are shown in fig. 3.
Step five: establishing a characteristic length DmaxInitial curve radius r from the caustic0The relational expression of (1);
as shown in FIG. 4, the tangent of the caustic line is taken along the direction of the bisector of the groove angle, i.e., the direction of the x-axis, and the distance between the two tangents is defined as the characteristic length Dmax。
To obtain a characteristic length DmaxLet WyThe derivative to the polar angle θ is zero, i.e.:
dWy/dθ＝0 (14)
since θ is within the range of the interval [ π - β, π + β ], it can be obtained from equation (14):
Figure BDA0002328402100000085
will theta1And theta2The value of (2) is substituted for the formula (11), the characteristic length D is obtainedmaxExpression (2)The following were used:
Figure BDA0002328402100000086
as shown in fig. 4, as long as α is determined on the reference plane1、α2The values of two angles, i.e. D can be directly measured out on the graphmaxα for different slot angles1And α2Can be determined by taking the coordinates (W) of two extreme pointsx1,Wy1)，(Wx2,Wy2) α corresponding to common notch angle is calculated by substituting formula (13)1、α2The values are shown in table 2 below:
TABLE 2 common grooving angles 2 β and α1、α2Table of corresponding relationship of values
Figure BDA0002328402100000091
Equation (16) can be written as follows:
r0＝Dmax/λmδ (17)
in the formula (I), the compound is shown in the specification,
Figure BDA0002328402100000092
the characteristic length D is represented by the formula (17)maxInitial curve radius r from the caustic0The relational expression (c) of (c).
Step six: determining the stress intensity factor K of the V-shaped notch tip under pure shear loadingIIThe method comprises the following steps:
from step 2 and step 4, a characteristic length D can be establishedmaxAnd stress intensity factor KIIThe relational expression (c) of (c). By substituting formula (17) for formula (9), it is possible to obtain:
Figure BDA0002328402100000093
the characteristic length D is represented by formula (19)maxAnd stress intensity factor KIIThe relationship (c) of (a) to (b),
in the formula (I), the compound is shown in the specification,
Figure BDA0002328402100000094
therefore, only the characteristic length D of the caustic line at the tip of the notch is measuredmaxThat is, the stress intensity factor K of the notch tip can be calculated by the equation (19)II。
Through the explanation, the invention fully discloses a method for calculating the stress intensity factor of the V-shaped cutting groove tip under pure shear loading, and establishes the characteristic length D based on the relevant theory and formula of the caustic methodmaxAnd stress intensity factor KIIThe relation of (A) successfully deduces the stress intensity factor K of the V-shaped notch tip under the pure shear loadingIIThe method of (3). The method enriches the theory and application of the caustic method and provides a new way for obtaining the stress intensity factor of the V-shaped cutting groove tip.
The invention further provides an experimental method based on the calculation method, which comprises the following steps:
s11: selecting a pure shearing model test piece;
experiments were performed using pure shear model specimens as shown in fig. 5. The organic glass Plate (PMMA) is selected as the material of the experimental test piece, and has the characteristics of homogeneity and isotropy. The thickness of the plate is 10mm, the depth of the cutting groove is 60mm, the opening angle of the cutting groove is 30 degrees, and the stress optical constant c is 85 × 10-6MPa, and the size of the test piece is shown in figure 5.
S12: carrying out a pure shear loading experiment on a pure shear model test piece by adopting a caustic wire experimental device;
the caustic experimental device is shown in fig. 6 and comprises a laser transmitter, a beam expander, a field lens I, a model test piece, a reference plane, a field lens II, an image acquisition device and an image processing device which are sequentially arranged. The first field lens and the second field lens adopt convex lenses, the image acquisition device can be a camera, and the image processing device can be a computer and is configured with image processing software.
S13: collecting a caustic line image under the action of corresponding load, and processing the caustic line image to obtain the characteristic length of a caustic line;
the experimental principle of the caustic wire experimental device is as follows: laser emitted by the laser emitter is diffused by the beam expander, then converted into parallel light by the convex lens, the light penetrates through the test piece and is received by the camera, finally a caustic line photo can be obtained through computer processing, and the distance z between the test piece and the reference plane01500 mm. The test pieces were pressed as shown in FIG. 5, and the loads p were 0MPa, 3.97MPa, 7.94MPa, 11.92MPa, 15.89MPa, 19.87MPa, 23.84MPa and 27.81MPa, respectively. The caustic line images under different loads are stored in a computer, and corresponding caustic line characteristic length D is obtained by image processing softwaremax。
S14: characteristic length D according to caustic linemaxCalculating the stress intensity factor K of the V-shaped notch tip under the pure shearing loadingII。
FIG. 7 is a photograph showing experimental results of a defocusing line at the tip of the notch under a load p of 0MPa, 11.92MPa, 19.87MPa, and 27.81MPa in a graded manner with increasing load. As can be seen from fig. 7, the slot tip caustic becomes progressively larger as the load increases.
Comparing the experimental chart of the caustic line with the load of 27.81MPa in the experiment with the deduced theoretical caustic line chart, as shown in FIG. 8, the shapes of the two are highly similar, and the correctness of the theoretical deduction of the invention is proved.
In order to prove the correctness of the stress intensity factor calculated by the method, the result of experimental calculation is compared with the result of numerical calculation.
For numerical calculation of the stress intensity factor at the tip of the kerf, it can be obtained using the finite element software ABAQUS. The specific process is as follows: firstly, establishing a V-shaped grooving model which is the same as the experiment by using ABAQUS, and calculating to obtain node coordinates and node stress. And then, according to the obtained node coordinates and the node stress, a stress intensity factor of the tip of the cutting groove is obtained by using a least square method.
According to the above analysis, the stress intensity factor K of the tip of the notch under the pure shear loading mode is obtained respectivelyIIThe specific comparison results are shown in table 3 below and fig. 9:
TABLE 3 stress Strength factor KIIExperiment (2)Comparison table of solution and numerical solution results
Figure BDA0002328402100000111
As can be seen from Table 3 and FIG. 9, the stress intensity factor K is obtained by the caustic method and the numerical methodIIThe difference is very small, which shows that the calculation method of the invention has good precision and numerical stability.
The invention also provides a caustic wire experimental device which is used for executing the experimental method.
Specifically, the caustic experimental apparatus is shown in fig. 6, and includes a laser emitter, a beam expander, a field lens i, a model test piece, a reference plane, a field lens ii, an image acquisition device, and an image processing device, which are sequentially arranged. The first field lens and the second field lens adopt convex lenses, the image acquisition device can be a camera, and the image processing device can be a computer and is configured with image processing software.
Thus, it should be understood by those skilled in the art that while exemplary embodiments of the present invention have been illustrated and described in detail herein, many other variations and modifications can be made, which are consistent with the principles of the invention, from the disclosure herein, without departing from the spirit and scope of the invention. Accordingly, the scope of the invention should be understood and interpreted to cover all such other variations or modifications.

Claims (10)

1. A method for calculating stress intensity factors of V-shaped cutting groove tips under pure shear loading is characterized in that the method is based on a caustic method and comprises the following steps:
the method comprises the following steps: selecting a planar object to be measured with a V-shaped cutting groove, and establishing an optical mapping relation between the plane of the object to be measured and a reference plane;
step two: establishing a stress field expression of the V-shaped cutting groove tip;
step three: establishing an initial curve radius r of the caustic0And stress intensity factor KIIThe relational expression of (1);
step four: determining a caustic mapping equation of the V-shaped cutting groove under the pure shearing loading;
step five: establishing a characteristic length DmaxInitial curve radius r from the caustic0The relational expression of (1);
step six: determining the stress intensity factor K of the V-shaped notch tip under pure shear loadingII。
2. The method of claim 1, wherein the step two of establishing the V-notch tip stress field expression comprises:
establishing a polar coordinate system by taking the cutting groove tip of the object to be measured as the origin of coordinates, and establishing the generalized stress intensity factor K of the V-shaped cutting groove under pure shearing loadingIIIs defined as:
Figure FDA0002328402090000011
where r and theta are polar coordinates, taurθIs the shear stress of the tip of the grooving, and lambda is the stress singularity index of the tip of the grooving;
for a linear elastic, isotropic, homogeneous sheet, the stress field near the tip of the notch when subjected to a pure shear load is expressed as:
Figure FDA0002328402090000012
Figure FDA0002328402090000013
in the formula sigmarIs the radial stress, σθIs the tangential stress;
wherein the constant C is expressed as:
C＝(λ-1)sin(λ+1)γ-(λ+1)sin(λ-1)γ (3)
λ can be obtained from the following formula:
sin2λ(π-β)-λsin2(π-β)＝0 (4)。
3. the computing method of claim 2The method is characterized in that the initial curve radius r of the caustic line is established in the third step0And stress intensity factor KIIThe relational expression (A) includes an initial curve equation for determining a caustic line, specifically:
the corresponding relation between the reference plane and the point on the object plane to be measured is as follows:
Figure FDA0002328402090000021
the formula is a mapping equation, wherein z0Is the distance between the object plane to be measured and the reference plane, and Δ s is the optical path difference, λmIs the magnification of the optical system; when the light rays are parallel light, lambdam＝1；
The caustic is a singular curve formed by converging a plurality of light rays, and according to the diffraction theory, the sufficient necessary condition for generating the singularity is that the Jacobian determinant of the mapping equation (5) is zero, so that the initial curve equation for deriving the caustic is as follows:
Figure FDA0002328402090000022
4. the calculation method according to claim 3, wherein step three establishes an initial curve radius r of the caustic line0And stress intensity factor KIIThe relationship of (a) further includes:
when a light beam passes through a transparent flat plate, the optical path difference of the isotropic material can be obtained by the following formula:
Δs＝cdeff(σr+σθ) (7)
wherein c is the caustic optical constant, deffIs the thickness of the flat plate;
substituting formula (2) for formula (7) to obtain:
Figure FDA0002328402090000023
the formula (8) may be substituted for the formula (6):
Figure FDA0002328402090000024
wherein A is:
Figure FDA0002328402090000025
from equation (9), when the load and the notch angle are constant, r is a constant, and is referred to as the initial curve radius of the caustic line, and is denoted as r0；
Equation (9) is the initial curve radius r of the caustic line0And stress intensity factor KIIThe relational expression (c) of (c).
5. The method of claim 4, wherein the step of determining the equation for the mapping of the caustic of the V-notch under pure shear loading comprises:
by substituting equations (8) and (9) into equation (5), the equation for mapping the caustic of the V-notch under pure shear loading can be obtained:
Figure FDA0002328402090000031
the polar coordinates of the caustic at the reference plane are defined by the following relationship:
Figure FDA0002328402090000032
Figure FDA0002328402090000033
。
6. the method of claim 5, wherein step five establishes a characteristic length DmaxInitial curve radius r from the caustic0The relationship of (1) includes:
to obtain a characteristic length DmaxLet WyThe derivative to the polar angle θ is zero, i.e.:
dWy/dθ＝0 (14)
since θ is within the range of the interval [ π - β, π + β ], it can be obtained from equation (14):
Figure FDA0002328402090000034
will theta1And theta2The value of (2) is substituted for the formula (11), the characteristic length D is obtainedmaxThe expression of (a) is as follows:
Figure FDA0002328402090000035
equation (16) can be written as follows:
r0＝Dmax/λmδ (17)
in the formula (I), the compound is shown in the specification,
Figure FDA0002328402090000036
the characteristic length D is represented by the formula (17)maxInitial curve radius r from the caustic0The relational expression (c) of (c).
7. The method of claim 6 wherein step six determines the V-notch tip stress intensity factor K at pure shear loadingIIThe method comprises the following steps:
establishing a characteristic length DmaxAnd stress intensity factor KIIBy substituting formula (17) for formula (9), the following can be obtained:
Figure FDA0002328402090000037
the characteristic length D is represented by formula (19)maxAnd stress intensity factor KIIThe relationship (c) of (a) to (b),
in the formula (I), the compound is shown in the specification,
Figure FDA0002328402090000038
only measuring the characteristic length D of the caustic line at the tip of the cutting groovemaxThat is, the stress intensity factor K of the notch tip can be calculated by the equation (19)II。
8. An experimental method based on the calculation method according to any one of claims 1 to 7, comprising the steps of:
s11: selecting a pure shearing model test piece;
s12: carrying out a pure shear loading experiment on the pure shear model test piece by adopting a caustic wire experimental device;
s13: collecting the caustic line image under the action of corresponding load, and processing the caustic line image to obtain the characteristic length D of the caustic linemax；
S14: characteristic length D according to caustic linemaxCalculating the stress intensity factor K of the V-shaped notch tip under the pure shearing loadingII。
9. A caustic experimental apparatus for performing the experimental method of claim 8.
10. The apparatus according to claim 9, comprising a laser emitter, a beam expander, a first field lens, a model test piece, a reference plane, a second field lens, an image acquisition device and an image processing device, which are arranged in sequence.
CN201911326047.6A 2019-12-20 2019-12-20 Calculation method, experimental method and experimental device for V-shaped grooving tip stress intensity factor under pure shear loading Pending CN111024520A (en)

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