KR20190137337A - A method for determining average radial stress and pressure vs. volume relationship of a compressible material - Google Patents

Abstract

According to one aspect of the present invention, relating to a method for determining an average radial stress of a compressible material by using a cylinder surrounding a side surface of the compressible material and a rod moving in an up/down direction in the same axis inside the cylinder, the present invention provides the method for determining a position of a strain gauge and a size of the cylinder according to a thick walled cylinder (TWC) theory and accurately measuring, determining, and investigating the average radial stress and the resulting pressure-volume relationship of the compressible material through a series of numerical analysis.

Description

A METHOD FOR DETERMINING AVERAGE RADIAL STRESS AND PRESSURE VS. VOLUME RELATIONSHIP OF A COMPRESSIBLE MATERIAL}

The present invention relates to a method of determining the average radial stress and pressure-volume relationship of a compressible material.

Unlike metallic materials that do not involve volume changes during plastic deformation, compressible materials, in particular compressible solids, involve volume changes during plastic deformation. These compressible solids have pressure-dependent yield conditions, and yield (plastic deformation) occurs even under the action of pure hydrostatic pressure. Therefore, almost all constitutive equations describing the behavior of compressible solids are described, including the relationship between the pressure and volume of the compressible solid, and for this reason it is very important to identify the pressure-volume (P-V) relationship of the compressible solid.

The most representative method for determining the pressure-volume relationship of a material is the conventional triaxial test (CTX), which is traditionally used to determine the pressure-volume relationship of soil, one of the representative compressible solids. come. In general, the measurable pressure level of the CTX test is limited to 1 MPa level, and the three-axis test apparatus capable of measuring pressures of 100 MPa or more is limited in high cost and accessibility. In addition, the CTX measurement method has a fundamental disadvantage of causing barreling of the specimen. Since the CTX method yields a pressure-volume relationship under the assumption that the specimen is in uniform deformation, the results measured from the specimens with barreling involve significant errors.

Another instrument that can measure P-V relationships up to about 100 MPa (and possibly 300 MPa) pressures is the instrumented die. In this method, a cylindrical specimen is placed inside a die with a sensor for measuring radial stresses, and a volumetric strain is generated in the specimen using upper and lower punches. The volume strain of a compressible solid using an instrumented die is determined by the stroke of the punch to which the axial load is applied. Instrumented dies are very expensive to design and build, just like high-pressure (100 MPa) CTX equipment. In addition, the reliability of the measurement results with the instrumented die is sensitively dependent on how precisely the instrument is calibrated, which is a very difficult (expensive) task. Calibration is performed by pushing a hydrostatic fluid of known pressure into the instrumented die, because introducing fluid into the die without limiting the free behavior of the die requires considerable effort.

For the same reason, it is not easy to measure the P-V relationship of compressible solids to the level of 100 MPa, despite its importance. Therefore, P-V relation data of compressible solids is very rarely provided in the literature, in spite of its importance, compared to other test data such as uniaxial test data. If a simpler and more economical way to measure the PV relationship of a compressible solid is developed, then the PV relationship, which is essential for describing the behavior of the compressible solid, can be provided in a much richer way than the present for various compressible solids. .

Accordingly, the present invention seeks to establish (develop) an alternative methodology that can more easily and economically measure the P-V relationship of compressible solids.

Summary of the Invention The present invention aims to solve the above-mentioned problems of the prior art, and an object of the present invention is to provide a method for more easily and economically determining the average radial stress and pressure-volume relationship of a compressible material.

)을 결정하는 단계; (d) 하기 식 2를 이용하여 보정함수(f(ε_v, μ))를 결정하는 단계; 및 (e) 하기 식 3을 이용하여 상기 압축성 소재의 평균 반경방향 응력(

)을 결정하는 단계;를 포함하는 방법을 제공한다.In one aspect of the present invention, a method for determining the average radial stress of the compressible material using a cylinder surrounding the side of the compressive material and a rod moving up and down coaxially in the interior of the cylinder, (a) the Determining a position of a strain gauge attached to an outer surface of the cylinder; (b) determining the wall thickness and inner diameter of the cylinder; (c) the radial stress of the compressible material using Equation 1

Determining); (d) determining a correction function f (ε _v , μ) using Equation 2 below; And (e) the average radial stress of the compressible material using Equation 3

It provides a method comprising;

<Equation 1>

<Equation 2>

<Equation 3>

In Formulas 1 to 3, a and b are the inner and outer diameters of the cylinder, E is the Young's modulus of the cylinder, ε _θ is the circumferential strain measured by the strain gauge, F _T and F _B are the loads measured at the top and bottom of the compressible material, μ is the coefficient of friction between the compressible material and the inner wall of the cylinder, ε _v is the volumetric strain of the compressible material, and A _c is The contact area between the compressive material and the inner wall of the cylinder.

In one embodiment, in step (a), when the ratio of the height of the compressive material to the height of the cylinder is 0.75, the strain gauge may be attached to a position that is less than 65% of the height of the cylinder.

In one embodiment, in step (a), when the ratio of the height of the compressible material to the height of the cylinder is 0.50, the strain gauge may be attached to a position that is 40% or less of the height of the cylinder.

In one embodiment, in step (a), when the ratio of the height of the compressive material to the height of the cylinder is 0.25, the strain gauge may be attached to a position that is less than 15% of the height of the cylinder.

In one embodiment, the correction function may be a quadratic function represented by Equation 4.

<Equation 4>

In Equation 4, z ₀ to z ₄ are real numbers obtained by non-linear curve fitting of ε _v versus σ _r ^avg / σ _r ^TWC at predetermined μs, respectively.

In one embodiment, the compressive material may be one selected from the group consisting of concrete, rock, porous ceramics, powder compacts, soil, hard plastic foam, and a combination of two or more thereof.

According to one aspect of the present invention, the position of the strain gauge and the cylinder size are determined according to the theory of thick walled cylinder (TWC), and through a series of numerical analysis, the average radial stress of the compressible material and the resulting pressure Can accurately measure, determine and identify volume relationships

It is to be understood that the effects of the present invention are not limited to the above effects, and include all effects deduced from the configuration of the invention described in the detailed description or claims of the present invention.

1 is a conceptual diagram of a compression test using TWC according to an embodiment of the present invention and its two-dimensional rotation axis symmetry model;
2 is a two-dimensional axis of rotation symmetry finite element model of TWC in accordance with one embodiment of the present invention;
3 and 4 are circumferential strain measured on the outer wall of the TWC according to an embodiment of the present invention;
5 is an equivalent stress calculated as a function of the radial stress applied to the inner wall of the TWC according to one embodiment of the invention;
6 is a circumferential strain measured at the outer wall of the TWC according to another embodiment of the present invention;
7 is an equivalent stress calculated as a function of radial stress applied to the inner wall of the TWC according to another embodiment of the present invention;
8 comprehensively illustrates design guidelines for a TWC in accordance with one embodiment of the present invention;
9 is a two-dimensional rotation axisymmetric finite element model of TWC in accordance with an embodiment of the present invention;
10 is a PV curve of a polyurethane foam according to one embodiment of the present invention;
11 is a PV curve of an alumina powder according to an embodiment of the present invention;
12 shows the (ratio) versus (volume strain) relationship of a polyurethane foam according to one embodiment of the present invention;
Figure 13 shows (rate) versus (volume strain) curve fitting capability according to the application of the correction function of the polyurethane foam according to one embodiment of the present invention;
14 shows (ratio) versus (volume strain) relationship of alumina powder according to an embodiment of the present invention;
Figure 15 shows (rate) versus (volume strain) curve fitting ability according to application of a correction function of alumina powder according to an embodiment of the present invention;
FIG. 16 shows the force equilibrium of a specimen located inside a TWC in accordance with one embodiment of the present invention; FIG.
Figure 17 illustrates a procedure for measuring the average radial stress of a compressive material in accordance with one embodiment of the present invention;
18 is a nonlinear curve fitting result for computational data sets of polyurethane foam according to one embodiment of the present invention;
19 is a nonlinear curve fitting result for computational numerical data sets of alumina powder according to an embodiment of the present invention;
20 is a numerical value analysis results of polyurethane foam and alumina powder according to an embodiment of the present invention;
21 shows radial stresses of polyurethane foam and alumina powder according to one embodiment of the present invention;
22 is an SEM image of a polyurethane foam according to one embodiment of the present invention;
23 is a conceptual diagram and preparation state of the experimental equipment for measuring the PV relationship of the polyurethane foam according to an embodiment of the present invention;
24 is a result of evaluation of the agreement between the left side and the right side of the formula 2 for the polyurethane foam according to an embodiment of the present invention;
25 illustrates an average radial stress-average axial force relationship of a polyurethane foam according to one embodiment of the invention;
26 shows a PV relationship of a polyurethane foam according to one embodiment of the invention;
27 is a compression test result according to ASTM D1621, ISO 3386 standard of the polyurethane foam according to an embodiment of the present invention;
FIG. 28 illustrates a superelastic constitutive equation model and parameters according to an embodiment of the present invention, the ability to fit the results of FIG. 27; FIG.
FIG. 29 illustrates the ability of the superelastic constitutive equation model and parameters to fit the results of FIG. 26 in accordance with an embodiment of the present invention; FIG.
FIG. 30 illustrates a PV curve derived as a result of performing computational numerical analysis according to an embodiment of the present invention, and an input physical property curve for computational numerical analysis.

Hereinafter, with reference to the accompanying drawings will be described the present invention. As those skilled in the art would realize, the described embodiments may be modified in various different ways, all without departing from the spirit or scope of the present invention. In the drawings, parts irrelevant to the description are omitted in order to clearly describe the present invention, and like reference numerals designate like parts throughout the specification.

Throughout the specification, when a part is "connected" to another part, it includes not only "directly connected" but also "indirectly connected" with another member in between. . In addition, when a part is said to "include" a certain component, this means that it may further include other components, without excluding the other components unless otherwise stated.

Hereinafter, embodiments of the present invention will be described in detail with reference to the accompanying drawings.

In the present invention, a thick walled cylinder (TWC) and its theory are considered as a means for measuring the radial stress of the compressible material. According to the TWC theory (open end condition), the radial stress exerted by the compressive material on the inner wall of the cylinder can be determined simply using the hoop strain value measured on the outer wall of the cylinder.

TWC theory assumes a situation where the radial stress (inner pressure) of the cylinder inner wall acts over the entire height of the cylinder, thereby causing the entire cylinder to expand radially. However, in the actual compression test, a portion of the cylinder which does not come into contact with the compressive material because of the shrinkage of the compressive material, that is, a portion which does not receive radial stress from the compressible material occurs. The portion of the cylinder not in contact with the compressive material limits the radial expansion of the portion of the cylinder in contact with the compressive material. For this reason, when measuring the radial stress of a compressible material using TWC theory without considering the part of the cylinder that does not come into contact with the compressible material, an incorrect radial stress measurement, i.e. the PV relationship of the wrong material, is determined. .

However, if the height of the compressible material is not too low than the height of the cylinder, i.e. compressing the compressible material within a limited punch stroke, it is possible to measure the radial stress of the compressible material using TWC theory. have. Starting from this possibility review, the present invention can use TWC and its theory to establish a methodology that can substantially determine the P-V relationship of compressible materials.

)을 결정하는 단계; (d) 하기 식 2를 이용하여 보정함수(f(ε_v, μ))를 결정하는 단계; 및 (e) 하기 식 3을 이용하여 상기 압축성 소재의 평균 반경방향 응력(

)을 결정하는 단계;를 포함하는 방법을 제공한다.In one aspect of the present invention, a method for determining the average radial stress of the compressible material using a cylinder surrounding the side of the compressive material and a rod moving up and down coaxially in the interior of the cylinder, (a) the Determining a position of a strain gauge attached to an outer surface of the cylinder; (b) determining the wall thickness and inner diameter of the cylinder; (c) the radial stress of the compressible material using Equation 1

Determining); (d) determining a correction function f (ε _v , μ) using Equation 2 below; And (e) the average radial stress of the compressible material using Equation 3

It provides a method comprising;

<Equation 1>

<Equation 2>

<Equation 3>

In Formulas 1 to 3, a and b are the inner and outer diameters of the cylinder, E is the Young's modulus of the cylinder, ε _θ is the circumferential strain measured by the strain gauge, F _T and F _B are the loads measured at the top and bottom of the compressible material, μ is the coefficient of friction between the compressible material and the inner wall of the cylinder, ε _v is the volumetric strain of the compressible material, and A _c is The contact area between the compressive material and the inner wall of the cylinder.

In the step (a) it is possible to determine the position of the strain gauge (strain gauge) attached to the outer surface of the cylinder using the TWC and its theory. TWC theory assumes a situation where the radial stress (inner pressure) of the inner wall is acting over the entire height of the cylinder, thereby causing the entire cylinder to expand radially. However, in the actual compression test, a portion of the cylinder which does not come into contact with the specimen, that is, a portion not subjected to radial stress from the specimen, may occur because the compressible material (sample) shrinks. The portion of the cylinder not in contact with the specimen may limit radial expansion of the portion of the cylinder in contact with the specimen. Therefore, there may be a limit in measuring the radial stress by applying the TWC theory (Equation 1) to the circumferential strain measured on the outer wall of the cylinder.

Nevertheless, if the height difference between the specimen and the cylinder is not too large (ie, the punch stroke is small) and the position of the strain gauge attached to the outer wall of the cylinder is close to the bottom surface of the cylinder, It may be possible to measure the radial stress by applying the TWC theory (Equation 1) to the circumferential strain. This assumption can be verified by a predetermined numerical analysis. In addition, it can be seen how deep it is possible to measure the radial stress by applying the TWC theory (Equation 1) up to the deep stroke (how large the volumetric strain of the specimen).

The conceptual diagram of the compression test using TWC and its two-dimensional symmetry model of axial rotation are considered in the numerical analysis.

As shown in FIG. 1 (b), numerical simulations were performed under a simplified assumption that the specimens located inside the cylinders were uniformly radially stressed across the specimen height. In FIG. 1, H is the current height of the specimen, H _o is the overall height of the cylinder, D _i is the inner diameter of the cylinder, and t is the wall thickness of the cylinder. In order to facilitate the representation of the height when the actual specimen is compressed, the current height of the specimen is expressed in a form normalized to the total cylinder height H _o , H / H _o . In addition, in order to consider the position of the strain gauge (strain gauge) attached to the outer wall of the cylinder, the position z in the cylinder height direction is normalized with respect to the cylinder height H _o and expressed as z / H _o .

To simulate the shrinkage of the specimen as the specimen was compressed in the actual test, a finite element model with various H / H _o was generated. An example of the generated finite element model is shown in FIG. 2. A radially biased mesh was applied to obtain numerical results that approximate the analytical solution (Equation 1). The inner diameter of the cylinder was considered 10 mm and the height was 1 mm, and a radial stress of 40 MPa was applied. The computational solver used a commercial finite element package (ABAQUS).

The material of the thick wall cylinder (TWC) was considered AISI 304L stainless steel, and only the elastic properties of the stainless steel (elastic modulus of 193 GPa and Poisson ratio 0.3) were used as the input physical properties of the numerical simulation.

Profiles showing how circumferential strains appear on the outer wall of the cylinder when the specimen has various current heights (H / H _o ) under a 40 MPa radial stress (inner pressure) applied to the inner wall of the cylinder. Shown in

Referring to FIG. 3, first, when the current height (H / H _o ) of the specimen is 1, the circumferential strain appearing on the outer wall shows a uniform value for the entire cylinder height, which is consistent with the analytical solution. This proves the reliability of the computational numerical analysis performed in the present invention. When the current height (H / H _o ) of the specimen is 0.75, the circumferential strain does not appear in the 80 to 90% z / H _o interval without contact with the specimen, but less than about 65%, preferably less than about 62% In the z / H _o interval, the value of the circumferential strain that corresponds to the solution is shown. H / H _o 0.50 indicates, preferably about 40%, the circumferential strain values that match the parsing in the z / H _o range less than about 37% is displayed. In addition, when H / H _o is 0.25, a circumferential strain value appears in the z / Ho interval of about 15% or less, preferably about 12% or less, which coincides with the solution.

From the above results, applying the position of the strain gauge (z / H _o ) attached to the outer wall of the cylinder close to the bottom of the cylinder, apply the TWC theory to a significantly lower specimen height H / H _o (i.e., a significantly deeper punch stroke). It can be seen that the radial stress of the specimen can be measured from the circumferential strain measured on the outer wall of the cylinder. For example, if the strain gage attachment position (z / H _o ) is less than 12% (z / Ho <0.12) of the specimen height from the bottom of the specimen, then the current height of the specimen is 25% (H / H _o = 0.25). The TWC theory can be applied to measure radial stresses.

In the step (b), it is possible to determine the wall thickness and the inner diameter of the cylinder from the following three aspects.

(i) the maximum measurable punch stroke (volume strain of the specimen); (ii) the maximum measurable radial stress point of view without yielding the cylinder; And (iii) a view for facilitating strain measurement in the early stages of compression of the specimen, ie when the measurable circumferential strain is very small. In order to determine the TWC shape (wall thickness and internal diameter), a predetermined numerical analysis was performed.

To simulate the shrinkage of the specimen as the specimen was compressed in the actual test, a finite element model with various H / H _o was generated. The generated finite element model is as shown in FIG. 2. The material of TWC is considered stainless steel, and the elastic-work hardening model which considers Ludwik hardening law is used to determine elastic behavior and yield. Ludwik hardening law applied is as shown in Equation 5.

<Equation 5>

In Equation 5, σ 'and ε' are equivalent stress and equivalent plastic strain, respectively, and A, B, and n are material parameters, respectively, 253.32 MPa, 685.1 MPa, and 0.3128. to be.

The specifications of the finite element models used in the present invention to investigate the effect of the shape (wall thickness and internal diameter) of the TWC on the above three aspects are shown in Table 1. Radial stress applied 1 MPa or 40 MPa to the inner wall of the cylinder.

D _i (mm) H / H _o t (mm) 10, 20 1, 0.75, 0.5, 0.25, 0.2, 0.1 1, 2, 5, 10

4 shows the profiles of the circumferential strain ε _θ according to the current height H / H _o of the specimen in a cylinder with various wall thicknesses. The results in FIG. 4 were obtained when the pressure (radial stress) applied to the inner wall of the cylinder having an inner diameter D _i of 10 mm was 1 MPa. In FIG. 4 (a) (t = 1 mm), the present height of the specimen ( If H / H _o ) is 0.20, the measured circumferential strain value is consistent with the value calculated by TWC theory when z / H _o is less than about 8%. In other words, if the strain gauge is located within 8% of the bottom of the cylinder (z / H _o <5%), the TWC can be up to about 1.61 (ln0.2) by volumetric strain using a cylinder with D _i = 10 mm and t = 1 mm. The theory can be applied to measure the radial stress of the specimen.

As the result of FIG. 4 increases the thickness of the cylinder wall, the magnitude of the maximum measurable volume strain ln (H / H _o ) decreases. For example, if the wall thickness of the cylinder is 5 mm (Fig. 4 (c)), when the current height (H / H _o ) of the specimen is 0.5, it is calculated by TWC theory in the interval z / H _o <10%. A circumferential strain that matches the value can be obtained. That is, the strain gauge may be a case to be located within _{10% (z / H o <} 10%) from the bottom of the cylinder to the volume strain of about 0.69 (ln0.5) by applying the TWC theory to measure the radial stress of the specimen . Therefore, the thinner the wall thickness of the cylinder is advantageous in terms of the maximum measurable volume strain of the specimen.

In order to examine the effect of wall thickness on the maximum measurable radial stress without the yielding of the cylinder, the variation of equivalent stress calculated as a function of the radial stress applied to the inner wall for various cylinder wall thicknesses is shown in FIG. Indicated.

5 is considered under cylinder conditions with an inner diameter D _i of 10 mm and 20 mm. The equivalent stress σ '' is calculated through the following equation.

<Equation 6>

The hoop stress (σ _θ ) in Equation 6 is calculated through Equation 7 below.

<Equation 7>

In Equation 7, a and b are the inner and outer diameters of the cylinder, respectively.

In FIG. 5, the horizontal line represents the plastic equivalent stress (parameter A of Equation 5) of 304L stainless steel. The slanted line of FIG. 5 has a physical meaning in the section before yield strength as a result calculated through Equation 6 under the assumption that the stress state of the cylinder is within the elastic limit.

In the condition that the cylinder does not yield as shown in FIG. 5, as the wall thickness of the cylinder increases, the value of the maximum measurable radial stress increases. Therefore, the thicker the wall thickness of the cylinder is advantageous when considering the maximum measurable radius " * stress.

In addition, the effect of the cylinder wall thickness on the circumferential strain magnitude that can be obtained when the radial stress is very low, for example 1 MPa, as in the initial stage of compression of the specimen, was investigated. When the inner diameter D _i of the cylinder is 10 mm using Equation 1, the circumferential strain values obtained from the outer wall of the cylinder for the wall thickness t 1, 2, 5, and 10 mm are 23.6, 10.8, 3.5, and 1.3, respectively. The micro-strain was calculated, and the circumferential strain values obtained for the same wall thickness for the inner diameter of 20 mm were calculated as 49.3, 23.6, 8.3, and 3.4 micro-strain, respectively. Therefore, when the radial stress applied by the specimen is small, the thinner the wall thickness of the cylinder is advantageous from the viewpoint of the circumferential strain measurement.

Ε _θ profile appearing on the outer wall of the cylinder by the current height (H / H _o ) of various specimens when a radial stress of 40 MPa is applied to the inner wall of the cylinder with an inner diameter (D _i ) and a wall thickness (t) of 1 mm. Is shown in FIG. 6.

When the 6 and compare Fig. _{3 (D i = 10mm, t} = 1mm), the strain gauge is a, if located within _{5% (z / H o <} 5%) smaller inner diameter (D _i) the cylinder from the bottom of the cylinder If used, the radial stress of the specimen can be measured by applying the TWC theory (Equation 1) up to a larger volumetric strain (ln (H / H _o )). For example, if the strain gauge is located on the z / _o H <5%, of the circumferential strain at the inner diameter of 10mm cylinder (3) to a volume of about 1.61 Strain (ln0.2) matches the value calculated through the TWC theory However, the maximum allowable volumetric strain of the specimen to obtain a circumferential strain consistent with the value calculated by TWC theory in a cylinder with an inner diameter of 20 mm (FIG. 6) is about 1.39 (ln0.25). Therefore, the smaller the inner diameter D _i of the cylinder, in view of the maximum allowable volume strain of the specimen to obtain a circumferential strain consistent with the value calculated by TWC theory, the more advantageous it is.

In order to examine the effect of the inner diameter on the maximum measurable radial stress without the yielding of the cylinder, the variation of the equivalent stress calculated as a function of the radial stress applied to the inner wall for various cylinder inner diameters is shown in FIG. 7. . The horizontal line in FIG. 7 represents the yield point (parameter A of Equation 5) of 304L stainless steel. Comparing the results of Figs. 7 (a) and 7 (b), the maximum measurable radial stress value without the yielding of the cylinder decreases as the inner diameter of the cylinder increases. Thus, the smaller the inner diameter of the cylinder is advantageous in terms of the maximum measurable radial stress.

Finally, the effect of the inner diameter of the cylinder on the magnitude of the circumferential strain that can be obtained when the radial stress is very low, for example 1 MPa, as in the initial stage of compression of the specimen, was investigated. When the wall thickness t of the cylinder is 1 mm using Equation 1, the circumferential strain values obtained from the outer wall of the cylinder for the inner diameters (D _i ) 10, 20, 30, and 40 mm are 23.6, 49.3, 75.2, and 101.1, respectively. Calculated in micro-strains, the circumferential strain values obtained for the same inner diameter at 2 mm wall thickness are calculated as 10.8, 23.6, 36.4, and 49.3 micro-strain, respectively. Therefore, when the radial stress applied by the specimen is small, the larger the inner diameter of the cylinder is advantageous from the viewpoint of the circumferential strain measurement.

8 shows the maximum allowable volume strain of the specimen, the maximum measurable radial stress without yielding TWC, and finally the measurable hoop strain when the radial stress is small. The influence of the design parameters of the TWC on the cylinder, ie the inner diameter of the cylinder and the wall thickness, is presented.

), 보정함수(f(ε_v, μ)) 및 상기 압축성 소재의 평균 반경방향 응력(

)을 순차적으로 결정할 수 있다.In the steps (c) to (e), the radial stress (

), The correction function f (ε _v , μ) and the average radial stress of the compressible material (

) Can be determined sequentially.

)만이 물리적 의미를 갖는다.In the presence of friction, the specimens apply non-uniform radial stresses in the height direction of the cylinder. Therefore, the radial stress distributed in the height direction of the specimen is

) Only has a physical meaning.

)과 실제 시편의 평균 반경방향 응력(

)을 정량화하였으며 이들의 상대적인 비율(

/

)을 구하였다.As described above, when (1) the height deviation of the specimen and the cylinder is not too large (i.e., the punch stroke is small) and (2) the position of the strain gauge on the outer wall of the cylinder is close to the bottom of the cylinder, The radial stress can be measured by applying the TWC theory (Equation 1) to the measured circumferential strain. In the following, it is examined whether the radial stress can be measured by applying the TWC and the theory even in the presence of friction between the actual specimen and the cylinder wall. If measurement is not possible, an appropriate correction equation (correction function) can be established to establish a methodology for measuring radial stresses using TWC and its theory even in the presence of friction. To do this, first calculate the radial stress obtained by applying TWC and its theory through numerical analysis.

) And the average radial stress of the actual specimen (

) And their relative proportions (

Of

) Was obtained.

Based on the cylinder design guidelines established in step (b) above, (i) the cylinder behaves within the elastic limits, (ii) the TWC and its theory can be applied to punch strokes of considerable depth, and (iii) TWC shape that can measure the ε _{θ of} size was designed, and the specifications are shown in Table 2 below.

D _i (mm) H _o (mm) t (mm) 10 10 5

9 shows a two-dimensional axisymmetric finite element model (FE model) used for computational numerical analysis. Punch (rod) supporting the upper and lower parts of the specimen was modeled in a simplified form considering the rigid body. To obtain more accurate numerical results, meshes deflected in the radial direction of the specimen and cylinder and in the height direction of the specimen (with strain gauges at the bottom of the cylinder) were applied. Friction coefficients of 0, 0.1, 0.2, 0.3, 0.4, and 0.5 were applied between the specimen and the cylinder wall. The numerical solver used a commercial finite element package (ABAQUS). The specimens are two types of polyurethane foam, and their PV (pressure-volume) relationship is shown in FIG. These two polyurethane foams have different PV relationships. The Ogden model was applied as the physical properties of the specimen, and the parameters of the model are shown in Table 3 below. Although there are limits to using the incompressibility hypothesis when calculating α _i and μ _i , we consider that the volume compression behavior of the specimen is described by using the D _i value obtained using the PV relationship only.

division Polyurethane Foam (ρ = 0.09g / cm ³ ) Polyurethane Foam (ρ = 0.32g / cm ³ ) i μ _i α _i D _i μ _i α _i D _i One -2.752 1.056 0.013 0.412 0.757 0.021 2 3.263 1.377 0.015 17.761 16.167 3.603E-003 3 -0.220 -0.895 5.557E-003 -8.881 -8.083 4.190E-003

In addition, in the present invention, two kinds of alumina powder were considered. Their P-V relationship is shown in FIG. They also have different P-V relationships. The behavior of the specimens was numerically analyzed using a Modified Drucker-Prager cap (MDPC) model, and the parameters of the model are shown in Tables 4 and 5 below.

Properties in the elastic regime Parameters of the MDPC model in the plastic regime E (GPa) v d (MPa) β (°) R α Hardening Law (MPa) (x = ε _v ) 9.03 0.28 5.5 16.5 0.558 0.03 p = 5.466 + 0.034e ^8.49x

Properties in the elastic regime Parameters of the MDPC model in the plastic regime E (GPa) v d (MPa) β (°) R α Hardening Law (MPa) (x = ε _v ) 10 0.26 4 44 0.5 0 p = 3.919 + 0.482e ^8.818x

값은 시편과 실린더 사이에 존재하는 마찰의 영향 및 원주 변형률을 측정하는 변형률 게이지의 위치(z/H_o)에 영향을 받는다. 따라서, 다양한 변형률 게이지의 위치(z/H_o=0, 0.1, 0.2, 0.3, 0.4, 0.5)에서, 각 마찰계수에 대한 정규화된 반경방향 응력(

/

) 대 체적 변형률의 관계를 검토하였다.In order to simulate the behavior of the cylinder, an elastic-work hardening model was used, which considered the Ludwik hardening law. The parameters of the model used in the numerical analysis are the same as the parameters A, B and n of Equation 5.

The value is influenced by the friction present between the specimen and the cylinder, and by the position of the strain gauge (z / H _o ) measuring the circumferential strain. Thus, at various strain gauge positions (z / H _o = 0, 0.1, 0.2, 0.3, 0.4, 0.5), the normalized radial stresses for each coefficient of friction (

Of

), The relationship between the bulk strain and the strain was examined.

/

값을 체적 변형률(ε_v)의 함수로 나타낸 결과이다. 도 12(a) 내지 12(f)는 변형률 게이지의 부착 위치가 다른 상황을 나타낸다. 마찰계수가 증가함에 따라

/

값은 마찰계수가 0일 때의 값으로부터 점점 더 큰 편차를 보인다. 즉, 마찰계수가 증가함에 따라

값은 실제 시편의

값으로부터 더 멀어진다. 도 12에서 주목할 점은 2종 폴리우레탄 폼의

/

값은 다양한 마찰계수 값에 걸쳐 매우 근사하다는 점이다. 이는 도 12에서 얻어진 데이터를 ε_v와 μ의 함수로 피팅할 수 있는 적절한 보정식(보정함수)을 수립한다면, 실린더 외벽 ε_θ로부터(TWC 이론을 통해) 구한

로부터

값을 구할 수 있다는 의미이다. 본 발명에서는 상기 식 3과 같은 보정함수를 수립하였다.12 is

Of

The result is a function of the volumetric strain (ε _v ). 12 (a) to 12 (f) show situations where the attachment positions of the strain gauges are different. As the coefficient of friction increases

Of

The value shows a larger deviation from the value when the coefficient of friction is zero. That is, as the coefficient of friction increases

The value of the actual specimen

Further away from the value Note that in Figure 12 of the two types of polyurethane foam

Of

The value is very close to the various coefficients of friction. This is obtained from the cylinder outer wall ε _θ (via TWC theory), if an appropriate correction equation (correction function) is fitted to fit the data obtained in FIG. 12 as a function of ε _v and μ.

from

It means you can get the value. In the present invention, a correction function as in Equation 3 was established.

The phenomenological correction function may be any shape as long as the data obtained in FIG. 12 can be well fitted as a function of ε _v and μ. For example, the correction function may be an n-th order polynomial, and more preferably, it may be a quadratic function represented by Equation 4 below.

<Equation 4>

In Equation 4, z ₀ to z ₄ are real numbers obtained by non-linear curve fitting of ε _v versus σ _r ^avg / σ _r ^TWC at predetermined μs, respectively.

As a result of performing nonlinear curve fitting on the data of FIG. 12 using Equation 4, fitting parameters as shown in Table 6 below were obtained. The curve fitting capability of equation 4 with these parameters applied is shown in FIG. 13.

z / H _o z ₀ z ₁ z ₂ z ₃ z ₄ R ² 0 0.9670 -0.3059 -0.8717 1.1908 0.5803 0.9846 0.1 0.9712 -0.2870 -0.8714 1.2058 0.5476 0.9884 0.2 0.9742 -0.2330 -0.8496 1.2882 0.5027 0.9897 0.3 0.9792 -0.1326 -0.8161 1.4607 0.4401 0.9917 0.4 0.9860 -0.0252 -0.7703 1.8016 0.3698 0.9941 0.5 0.9925 -0.2583 -0.7110 2.5101 0.3429 0.9947

로부터 보정함수를 통해

값을 구하는 절차를 수립할 수 있는 이유는 2종 폴리우레탄 폼들의

/

값이 다양한 마찰계수 값에 걸쳐서 매우 근사하기 때문이다.폴리우레탄 폼과는 다른 재질인 알루미나 분말들도 다양한 마찰계수 값에 걸쳐서

/

값이 매우 근사하게 나타나는지 알아보기 위해, 도 14에는 2종 알루마나 분말들에 대한

/

값을 체적 변형률의 함수로 나타내었다. 도 14에서 알 수 있듯이 서로 다른 P-V 관계를 갖는 2종의 알루미나 분말들 또한 다양한 마찰계수 값에 걸쳐서

/

값이 매우 근사하게 나타난다. 따라서, 알루미나 분말들 또한 폴리우레탄 폼들의 경우와 마찬가지로 적절한 보정식을 수립한다면 실린더 외벽 ε_θ로부터(TWC 이론을 통해) 구한

로부터

값을 구할 수 있다.As described above, two kinds of polyurethane foams with different PV curves were obtained from the cylinder outer wall ε _θ (via TWC theory).

Through the correction function from

The reason why we can establish the procedure for getting the value is because

Of

This is because the values are very close to the various coefficients of friction. Alumina powders, which are different from polyurethane foams, are also used over various coefficients of friction.

Of

To see if the value is very close, Figure 14 shows two kinds of alumina powders.

Of

Values are expressed as a function of volumetric strain. As can be seen in FIG. 14, two kinds of alumina powders having different PV relations also have various friction coefficient values.

Of

The value is very close. Thus, alumina powders are also obtained from the cylinder outer wall ε _θ (via TWC theory), as is the case with polyurethane foams, if an appropriate calibration equation is established.

from

You can get the value.

In the present invention, as in the case of the polyurethane foams, nonlinear curve fitting was performed on the alumina powder data shown in FIG. 14 using the correction function of Equation 4, and the fitting parameters obtained through the above are shown in Table 7 below. FIG. 15 shows the correction function and its parameters fitting the data of FIG. 14.

z / H _o z ₀ z ₁ z ₂ z ₃ z ₄ R ² 0 0.9818 -0.5240 -0.6535 1.6736 0.3348 0.9919 0.1 0.9852 -0.5381 -0.6382 1.6610 0.3155 0.9919 0.2 0.9911 -0.5312 -0.6007 1.8690 0.2722 0.9917 0.3 1.0027 -0.5315 -0.5372 2.2974 0.2048 0.9918 0.4 1.0214 -0.5568 -0.4411 3.0904 0.1176 0.9924 0.5 1.0474 -0.6459 -0.2864 4.5604 0.0131 0.9931

/

데이터들을 성공적으로 기술하는 것으로 볼 때, 상기 식 4와 같은 보정함수는 다양한 압축성 고체들에 대해 적용이 가능할 것으로 판단된다. 물론 동일한 보정함수라도 주어진 소재별로 피팅 파라미터 값들은 달라질 수 있다.상기와 같이, 상기 식 4의 보정함수뿐만 아니라 어떤 형태의 보정함수라도 도 12 및 도 14에 나타낸

/

데이터들을 ε_v와 μ의 함수로 성공적으로 기술할 수 있다면 이 또한 다양한 압축성 고체들에 대해 적용될 수 있다.Of the same polyurethane foam and alumina powder

Of

In view of the successful description of the data, it is determined that a correction function such as Equation 4 above can be applied to various compressible solids. Of course, even if the same correction function, the fitting parameter values may be different for each given material. As described above, not only the correction function of Equation 4 but also any type of correction function shown in FIGS.

Of

This can also be applied for various compressible solids if the data can be successfully described as a function of ε _v and μ.

로부터

를 구할 수 있다. 이 때, 상기 식 4의 보정함수를 적용하기 위해서는 별도의 실험, 예를 들어, instrumented die 이용법을 통해 해당 폴리우레탄 폼과 die 간의 마찰계수를 측정해야 한다. 마찬가지로 어떤 알루미나 분말의 P-V 곡선이 도 11에 나타낸 두 P-V 곡선 사이 혹은 근처에 있다면, 보정함수 및 그 파라미터(표 7)를 이용하여

로부터

를 구할 수 있다. 물론 이 경우에도 상기 식 4의 보정함수를 적용하기 위해서는 별도의 실험, 예를 들어, instrumented die 이용법을 통해 해당 알루미나 분말과 die 간의 마찰계수를 측정해야 한다.If the PV curve of any polyurethane foam is between or near the two PV curves shown in Fig. 10, the correction function and its parameters (Table 6)

from

Can be obtained. In this case, in order to apply the correction function of Equation 4, the friction coefficient between the polyurethane foam and the die must be measured by a separate experiment, for example, using an instrumented die. Likewise, if the PV curve of any alumina powder is between or near the two PV curves shown in Fig. 11, using the correction function and its parameters (Table 7)

from

Can be obtained. Of course, even in this case, in order to apply the correction function of Equation 4, the friction coefficient between the alumina powder and the die must be measured through a separate experiment, for example, using an instrumented die.

)을 구할 수 있다. 이를 위해, 본 발명에서는 도 16에 나타낸 것과 같은 실린더 내부에 위치한 시편의 힘 평형 상태를 분석하였으며, 그 결과 하기 식 8의 힘 평형식을 수립할 수 있다.However, according to the present invention, the average radial stress of the specimen is determined by simultaneously determining the correction function and the friction coefficient through the TWC experiment without measuring the friction coefficient separately.

) Can be obtained. To this end, in the present invention, the force equilibrium state of the specimen located in the cylinder as shown in Figure 16 was analyzed, and as a result, it is possible to establish the force equilibrium of Equation 8.

<Equation 8>

In Equation 8, F _T and F _B are the loads measured at the top and the bottom of the specimen, respectively, μ is the coefficient of friction between the specimen and the inner wall of the cylinder, A _c between the specimen and the inner wall of the cylinder Contact area. Substituting Equation 3 into Equation 8 leads to Equation 2.

, ε_v, A_c 값들은 실제 실험에서 실시간으로 무수히 많은 데이터 값들이 생성된다. μ와 보정함수 파라미터들은 상수이다.F _T -F _B term on the left side in equation 2,

, ε _v , and A _c values produce a myriad of data values in real time in real experiments. μ and correction function parameters are constants.

, ε_v, A_c 값들을 독립변수로 한 후, 실시간으로 얻어진 수많은 [F_T - F_B,

, ε_v, A_c] 데이터 세트들에 대해 상기 식 2를 이용하여 비선형 곡선 맞춤(non-linear curve fitting)을 실시한다면, 상수인 μ와 보정함수 파라미터들을 동시에 결정할 수 있다. 즉, μ와 보정함수 f(ε_v, μ)를 동시에 구할 수 있다. 보정함수 f(ε_v, μ)를 구하고 난 후에는, 이를 이용하여 시편의 평균 반경방향 응력

를 구할 수 있다. 도 17은 이러한 절차를 정리하여 도식화한 것이다.If F _T -F _B is the dependent variable,

, ε _v , A _c values are independent variables, and then [F _T -F _B ,

, ε _v , A _c ] If non-linear curve fitting is performed using Equation 2 on the data sets, the constant μ and the correction function parameters can be determined simultaneously. In other words, μ and the correction function f (ε _v , μ) can be obtained simultaneously. After obtaining the correction function f (ε _v , μ), use it to calculate the average radial stress of the specimen.

Can be obtained. Figure 17 is a schematic of these procedures.

If only the correction function f (ε _v , μ) is obtained, the radial stress can be obtained according to the procedure of FIG. 17, that is, using Equation 3 above. Hereinafter, this process will be described using computational numerical analysis data.

, ε_v, A_c 값들을 구하였으고, 그 결과를 도 18에 나타내었다. 하기 표 8에는 실시간으로 얻어진 수많은 [F_T - F_B,

, ε_v, A_c] 데이터 세트의 예를 나타내었다. 상기 식 2를 이용하여 도 18 및 표 8에 나타낸 데이터들에 대한 비선형 곡선 맞춤을 실시하였고, 그 결과 모든 마찰계수들에 대해 표 6과 동일한 보정함수 계수들을 찾아낼 수 있었다.Computational numerical analysis of the polyurethane foam was carried out by F _T -F _B ,

, ε _v , A _c values were obtained, and the results are shown in FIG. 18. Table 8 shows a number of [F _T -F _B , obtained in real time,

, ε _v , A _c ] shows an example of a data set. Nonlinear curve fitting was performed on the data shown in FIGS. 18 and 8 using Equation 2, and as a result, the same correction function coefficients as those of Table 6 could be found for all friction coefficients.

Time (s)

ε _v A _c F _T -F _B 0 0 0 314.159 0 0.01 0.483 0.004 312.903 13.363 0.02 0.971 0.008 311.646 27.313 0.03 1.464 0.012 310.389 41.336 0.04 1.962 0.016 309.133 55.343 0.05 2.466 0.020 307.876 69.328 0.06 2.974 0.024 306.619 83.408

In addition, for the alumina powder, the same kind of data as in FIG. 19 and Table 9 were obtained, and nonlinear curve fitting was performed on the data. As a result, the same correction function coefficients were found for all the friction coefficients considered.

Time (s)

ε _v A _c F _T -F _B 0 0 0 314.159 0 0.078 1.831 0.048 299.451 35.617 0.228 1.843 0.136 271.123 35.788 0.443 1.886 0.266 230.657 42.424 0.535 2.095 0.321 213.362 45.620 0.794 5.842 0.476 164.528 116.970 0.958 23.382 0.575 133.626 436.990

As a result of performing numerical analysis on the polyurethane foam, the obtained F _T -F _B value is shown by a solid line in FIG. 20 (a). By applying the determined f (ε _v , μ) to obtain a value of μσ _r ^TWC f (ε _v , μ) A _c , the results are shown in dashed lines in Fig. 20 (a). The left and right sides of 2 are very close. Therefore, the correction function adopted and its parameters are correct values, and the average radial stress obtained by following the procedure shown in Fig. 17, i. If the correction function and the parameters determined are inappropriate, the left and right values can never match. In actual experiments, by observing how well the left and right sides of Equation 2 coincide with each other, it is possible to determine how reliable the correction function and its parameters are.

For the alumina powder, the left and right sides of Equation 2 were obtained through the same procedure, and the results are shown in Fig. 20 (b). The left side value and the right side value of the alumina powder are also approximately equal.

, ε_v, A_c] 데이터 세트를 이용하여 본 발명에서 수립한 방법대로 반경방향 응력을 구하였으며, 그 결과를 도 21(a)에 대시선으로 나타내었다. 또한 전산수치해석을 실시할 때 시편 element들이 경험하는 방경방향 응력 값을 평균 처리한 결과(σ_r ^element)를 도 21(a)에 실선으로 나타내었다.Computed numerical analysis of polyurethane foam [F _T -F _B ,

, ε _v , A _c ] The radial stresses were calculated according to the method established in the present invention using the data set, and the results are shown by dashed lines in FIG. 21 (a). In addition, the results of averaging the radial stress values experienced by the specimen elements in the numerical analysis (σ _r ^element ) are shown by solid lines in FIG. 21 (a).

In FIG. 21 (a), the dashed line is obtained by numerically simulating the same procedure as that performed in the actual experiment. Solid lines, on the other hand, are not possible in experiments and can only be obtained from computational numerical analysis. From the agreement of the two curves, the method established in the study in the present invention can be verified numerically.

, ε_v, A_c] 데이터 세트를 이용하여 위와 동일한 방법으로 구한 곡선들을 도 21(b)에 나타내었다. 두 곡선의 일치로부터 본 발명에서 수립한 방법은 알루미나 분말에 대해서도 전산수치해석적으로 검증될 수 있다.Computed numerical analysis of alumina powder [F _T -F _B ,

, ε _v , A _c ] The curves obtained by the same method using the data set are shown in FIG. 21 (b). From the coincidence of the two curves, the method established in the present invention can be numerically verified even for alumina powder.

Through the methodology established above, in the following, the P-V relationship of the rigid polyurethane foam (foam) which is a kind of compressible material is measured through experiments. When the TWC measurement process is numerically simulated using the measured physical properties as the numerical input input properties, it is examined whether the PV curves derived from numerical simulations are consistent with the measured PV curves (calculated numerical input properties). Thus, the reliability of the measured physical properties can be verified.

Since the rigid polyurethane foam used is a material exhibiting mechanical behavior such as rubber rather than the characteristics of general foamed material, microstructure observation of the specimen was performed through SEM to verify whether the specimen is a foamed material.

Ultra-precision milling was performed to minimize pore damage on the surface of the rigid polyurethane foam material specimens. After polishing the viewing surface, platinum (Pt) coating was applied to the specimen surface in an ion coater vacuum chamber to impart electrical conductivity to the surface.

The microstructure of the rigid polyurethane foam was investigated using a scanning electron microscope (SEM, scanning electron microscope, Strate 400 STEM). The porosity of the rigid polyurethane foam specimens was measured using the Archimedes principle. Pentane (Sigma-Aldrich, Korea) having a density of 0.626 g / ml was used as a fluid for grading the catcher and the suspension weight of the specimen. The microstructure of the observed rigid polyurethane foam is shown in FIG. 22. From the results observed at 50 times the magnification, it can be confirmed that an average of 50 to 60 μm of pores exist on the surface of the rigid polyurethane foam.

Table 10 below shows the density and porosity of the rigid polyurethane foam. The total porosity (P) of the rigid polyurethane foam specimens measured using the Archimedes principle was about 38%, the open porosity (P _op ) was about 1.33%, and the porosity (P _cp ) was about 37.48%.

division Psalm 1 Psalm 2 Psalm 3 medium m _D (g) 0.7446 0.7140 0.7441 0.7342 m _SS (g) 0.1308 0.1027 0.1312 0.1216 m _D -m _SS (g) 0.6138 0.6113 0.6129 0.6127 ρ _a (g / ml) 0.7594 0.7312 0.7600 0.7502 ρ _b (g / ml) 0.9446 0.7140 0.7441 0.7342 ρ _t (g / ml) 1.2000 1.2000 1.2000 1.2000 P (%) 37.9508 40.4973 37.9932 38.8138 P _op (%) 1.2356 1.4275 1.3293 1.3308 P _cp (%) 36.7152 39.0697 36.6639 37.4829

-m _D , m _SS mean dry weight and suspended weight, respectively-ρ _a , ρ _b , ρ _t mean apparent density, bulk density and true density, respectively (sample volume = 1cm ³ )

True density of pure polyurethane = 1.2 g / cm ³

23 shows a conceptual diagram and a preparation state of the experimental equipment for measuring the P-V relationship of the rigid polyurethane foam using the methodology established above. Referring to Figure 23 (a), the device for measuring the P-V relationship consists of a thick wall cylinder (TWC), an upper cap, a lower support, an upper punch (upper rod) and a lower punch (lower rod).

In order to measure the pressure of 100MPa level, a thick wall cylinder (TWC) was designed in consideration of the design guidelines established in step (b). The dimensions of the designed cylinder are 10 mm inside diameter (D _i ), 5 mm wall thickness (t) and 11 mm height (H _o ). In the actual test, when the specimen was compressed to some extent, the specimen was pushed out of the TWC with increasing pressure on the inner wall of the TWC. If this occurs, an incorrect circumferential strain is measured as the pressure on the specimen's inner wall is drastically reduced. For this reason, the specimen was stably positioned inside the TWC using the upper cap and the lower support. In addition, in order to accurately measure the load deviation caused by the friction between the specimen and the inner wall of the TWC, the lower support was designed so that only the pure specimen load could be transmitted.

Machining tolerances between the TWC and the upper and lower punches apply general tolerances, not the rigid wall mold grade precision tolerances used in instrumented dies. In addition, no additional lubricant was applied in the TWC compression experiments to account for the pure frictional effects between the specimen and the TWC inner wall.

/

값을 피팅하는 능력을 변형률 게이지의 위치가 시편의 바닥으로부터 30% 이상(z/H_o > 0.3)일 때 우수하였다. 이러한 결과를 근거로 변형률 게이지의 부착 위치는 시편 바닥으로부터30%(z/H_o=0.3)로 선정하였다.Looking at the result of the obtained parameter of Equation 4, the correction function

Of

More than 30% the ability to fit the values from the bottom of the specimen position of the strain gauge (z / H _o> 0.3) was excellent when. Based on these results, the attachment position of the strain gauge was selected as 30% (z / H _o = 0.3) from the bottom of the specimen.

The foil strain gauge (N11-FA-5, SHOWA) used for measuring the circumferential strain (ε _θ ) in the TWC compression test was 2.12 and the resistance was 350Ω. The change in the resistance of the strain gauge was converted into strain by a signal processing device (SDL-350R, Radian) with a Wheatstone bridge. The load on the top and bottom of the specimen was measured using load cells located on the top and bottom of the TWC apparatus. In addition, the stroke of the upper punch was measured using an extensometer mounted to measure the displacement of the upper platen.

를 구하였다. TWC 압축 실험을 통해 결정한 ε_v,

, A_c, F_T - F_B 데이터들에 대해 상기 식 8의 힘 평형식 및 상기 식 2를 적용하여 비선형 곡선 맞춤을 실시함으로써 f(ε_v, μ) 값을 구하였다. 이 때, F_T - F_B 값을 종속변수로 두고, 실시간으로 계측되는 다른 변수들인 ε_v,

, A_c 값들을 독립변수로 설정하였으며, 보정함수 파라미터들과 마찰계수는 상수로 설정하였다. 여기서 사용한 보정함수는 상기 식 4이다. 비선형 곡선 맞춤을 실시함으로써 모든 상수 값들(보정함수 파라미터 및 마찰계수)을 결정하였다. 참고로, 상기 식 2에 적용된 보정함수가 얼마나 신뢰성이 있는지에 대해서는, 비선형 곡선 맞춤이 끝난 후 구해진 계수 값들을 상기 식 2에 적용하였을 때 좌변과 우변이 얼마나 잘 일치하는지를 관찰함으로써 알 수 있다. 상기 비선형 곡선 맞춤으로부터 상기 식 2의 모든 계수 값들을 구한 후(즉, f(ε_v, μ)를 구한 후), f(ε_v, μ)를

에 곱함으로써

값을 결정하였다.The data measured in real time in the TWC compression experiments are the cross-head displacements (from which ε _v and A _{c are} calculated), F _T , F _B , and ε _θ . The measured circumferential strain value is applied to Equation 1 above.

Was obtained. Ε _v , determined by TWC compression experiment,

F (ε _v , μ) values were obtained by applying the nonlinear curve fitting by applying the force equilibrium of Equation 8 and Equation 2 to the, A _c , F _T -F _B data. At this time, the value of F _T -F _B as a dependent variable, ε _v , other variables measured in real time

, A _c values were set as independent variables, and correction function parameters and friction coefficients were set as constants. The correction function used here is Equation 4 above. All constant values (correction function parameters and coefficient of friction) were determined by performing nonlinear curve fitting. For reference, how reliable the correction function applied to Equation 2 can be obtained by observing how well the left and right sides correspond to the coefficient values obtained after the nonlinear curve fitting is applied to Equation 2. After obtaining all the coefficients of the equation 2 from the non-linear curve fit (that is, after obtaining a f (ε _v, μ)), the f (ε _v, μ)

By multiplying by

The value was determined.

On the other hand, the average axial stress σ _a ^{avg on the} specimen was taken into account. The average axial force is (F _T + F _B ) / 2, and this value was divided by the cross-sectional area of the specimen to calculate the average axial stress σ _a ^avg of the specimen.

Finally, the pressure of the specimen was obtained using the following equation 9.

<Equation 9>

, A_c, F_T - F_B를 얻었으며, 상기 가공된 데이터들에 상기 식 2를 적용하여 비선형 곡선 맞춤(non-linear curve fitting)을 실시함으로써, f(ε_v, μ)를 결정하였다. 그 결과 얻어진 보정함수(식 4)의 파라미터 및 마찰계수를 하기 표 11에 나타내었다.By processing the cross-head displacement, F _T , F _B , ε _θ , ε _v data which are raw data as above, ε _v ,

, A _c , F _T -F _B were obtained, and f (ε _v , μ) was determined by performing non-linear curve fitting by applying Equation 2 to the processed data. The parameters and friction coefficients of the resulting correction function (Equation 4) are shown in Table 11 below.

μ z ₀ z ₁ z ₂ z ₃ z ₄ 0.4245 1.6589 -0.3197 -0.7149 3.1183 0.4922

/

비율을 ε_v와 μ의 함수로 제대로 기술해내지 못한다면, 상기 식 2의 좌변과 우변은 결코 일치할 수 없다. 본 발명에서 사용한 보정함수인 상기 식 4가 얼마나 신뢰성있게 경질 폴리우레탄 폼의

/

비율을 기술하고 있는지를 알아보기 위해, 표 11의 계수값들을 적용한 후 상기 식 2의 좌변과 우변이 얼마나 잘 일치하는지를 조사하였으며, 그 결과를 도 24에 나타내었다.도 24를 참고하면, 식 2의 좌변과 우변 결과 사이의 피팅 오차는 약 8.476%로 나타났다. 이러한 결과로부터 본 발명에서 사용한 보정함수인 상기 식 4는 상기 식 2의 좌변과 우변이 합리적으로 일치하도록 만들어준다는 것을 확인하였다. 이는 본 발명에서 사용한 보정함수의 신뢰성을 나타낸다. 또한, 이 사실은 본 발명에서 보정함수를 사용하여 측정한 경질 폴리우레탄 폼의 P-V 곡선의 신뢰성을 나타내는 것이기도 하다.If Equation 4 is used for the compressible material

Of

If the ratio is not properly described as a function of ε _v and μ, the left and right sides of Equation 2 can never coincide. Equation 4, which is a correction function used in the present invention, reliably shows how hard polyurethane foam

Of

In order to determine whether the ratio is described, after applying the coefficient values of Table 11, it was examined how well the left side and the right side of Equation 2 correspond, and the result is shown in FIG. 24. Referring to FIG. 24, Equation 2 The fitting error between the left and right results of is about 8.476%. From these results, it was confirmed that Equation 4, the correction function used in the present invention, makes the left side and the right side of Equation 2 reasonably coincide. This represents the reliability of the correction function used in the present invention. This fact also indicates the reliability of the PV curve of the rigid polyurethane foam measured using the calibration function in the present invention.

및 σ_a ^avg를 구하였고, 이를 도 25에 나타내었다. 여기서

는 상기 식 3을 통해 구했고, σ_a ^avg는 측정된 평균 축방향 힘((F_T+F_B)/2)을 시편의 단면적으로 나누어 계산하였다.Based on the above results, using the TWC compression test results

And σ _a ^avg were obtained, which are shown in FIG. 25. here

Was obtained through Equation 3, and σ _a ^avg was calculated by dividing the measured average axial force ((F _T + F _B ) / 2) by the cross-sectional area of the specimen.

및 σ_a ^avg 결과를 상기 식 9에 대입하여 구하였다. 측정된 시편의 P-V 결과를 도 26에 나타내었다.Finally, the pressure of the specimen is

And sigma _a ^avg results were obtained by substituting Equation 9 above. The PV results of the measured specimens are shown in FIG. 26.

Thus, in order to confirm the reliability of the PV relationship of the measured rigid polyurethane foam, the measured physical properties were used as numerical input values to simulate the TWC measurement process numerically. A hyperelastic constitutive equation was used to describe the deformation behavior of the specimen. This construction equation has the limitation of using the incompressibility hypothesis when calculating the C _ij (α _i , μ _i ) parameters, but considers that the volumetric compression behavior of the specimen is described by the D _i value obtained using only the PV relationship. Used in the invention. In addition, the reliability of the measured properties was verified by examining how the PV curve derived through numerical simulation matches the PV curve (computed numerical input properties) measured through the experiment.

Compression tests were performed with reference to ASTM D1621, ISO 3386 standards to obtain data for C _ij (α _i , μ _i ) calibration of rigid polyurethane foams. The strain rate of the specimen was 0.001 s ⁻¹ . 27 shows the experimental results.

Using the ABAQUS CAE program, the parameter calibration of the Polynomial, Yeoh, and Ogden models of the superelastic constitutive equations was performed from the results of FIG. 27, and the obtained parameters are shown in Tables 12 to 14, respectively. FIG. 28 shows each superelastic constitutive equation model and the parameters of Tables 12-14 fit the results of FIG. 27.

C ₁₀ [MPa] C ₀₁ [MPa] C ₂₀ [MPa] C ₁₁ [MPa] C ₀₂ [MPa] 0.5479 0.0396 -0.3400 0.0787 0.0619

C ₁₀ [MPa] C ₂₀ [MPa] C ₃₀ [MPa] 0.5890 -0.1431 0.0841

i μ _i [MPa] α _i [MPa] One 6.3123 2.0000 2 -4.2919 4.0000 3 0.8510 6.0000 4 -2.5510 -2.0000 5 0.8943 -4.0000 6 -0.0369 -6.0000

Referring to FIG. 28, the compressive stress-strain results (FIG. 27) of the actual rigid polyurethane foam are shown in solid lines, and the compressive stress-strain results fitted to each superelastic constitutive model to which the parameter values of Tables 12 to 14 are applied. Is indicated by a dotted line. The fitted results (dotted lines) and the compression test results (solid line) of the rigid polyurethane foams are almost identical for each superelastic constitutive equation model. The error between the results fitted with the Polynomial, Yeoh, and Ogden models is about 0.022%, about 0.015%, and about 0.014%, respectively. From the results, each superelastic constitutive model is a reliable level for the compression test results of rigid polyurethane foams. To fit.

Volumetric parameter calibration of the superelastic constitutive equations (Polynomial, Yeoh, Ogden model) was performed using the P-V curve of FIG. 26, and the parameters obtained through the P-V curves are shown in Tables 15 to 17 below. FIG. 29 shows the ability to fit the P-V curve results of FIG. 26 where each superelastic constitutive equation model and the parameters of Tables 15-17 were measured.

D ₁ [MPa ^-1 ] D ₂ [MPa ^-1 ] -2.1927 0.0119

D ₁ [MPa ^-1 ] D ₂ [MPa ^-1 ] D ₃ [MPa ^-1 ] 0.0893 -0.0059 8.2029E-04

D ₁ [MPa ^-1 ] 0.1363 D ₂ [MPa ^-1 ] -7.9976E-03 D ₃ [MPa ^-1 ] 2.9173E-04 D ₄ [MPa ^-1 ] -2.5696E-05 D ₅ [MPa ^-1 ] 5.3424E-06 D ₆ [MPa- ¹ ] -4.5160E-06

In FIG. 29, the solid line represents the P-V curve measurement result of the actual rigid polyurethane foam (FIG. 26), and the dotted line represents the P-V result fitted to each superelastic constitutive equation model to which the parameter values of Tables 15 to 17 are applied. The ability to fit the compression test results (FIG. 28) was excellent for all models examined, but only the Ogden model was reliable for fitting the measured P-V results (solid line). Here, the error between the measured PV curve and the results fitted with the Polynomial, Yeoh, and Ogden models is about 49.79%, about 25.29%, and about 4.77%, respectively. The Ogden model has the best ability to fit experimental results. Therefore, the P-V curve of the rigid polyurethane foam measured using the Ogden model's calibrated parameter values (Table 14 and Table 17) was used as the numerical input properties.

In order to simulate the P-V measurement experiment process using the TWC, the two-dimensional axisymmetric finite element model (FE model) of FIG. 9 was used. In this model, the coefficient of friction between the specimen and the inner wall of the TWC was calculated using the results of the coefficient of friction (Table 11) obtained through the established methodology.The input properties of the specimen were the calibrated parameter values of the Ogden model (Tables 14 and 17). It was. For the elastic properties to simulate the behavior of the cylinder, refer to the above-described elastic properties and material parameter values of the AISI 304L stainless steel TWC.

In FIG. 30, the P-V curve obtained as a result of performing the numerical value analysis is represented by a solid line, and the input property curve (the P-V curve measured through the experiment) for the numerical value analysis is represented by a dashed line. The P-V curve derived from the P-V curve numerical simulation used as an input property in FIG. 30 showed about 91% agreement. These results show that the correction function, Equation 4 and Equation 2, which is the equilibrium function established in the present invention, makes it possible to obtain the P-V curve and the friction coefficient of the specimen at a reasonable level.

The P-V curve (dashed line) derived from the numerical simulation is the result of uniform friction coefficient between the specimen and the inner wall of the TWC. However, the P-V curves (solid line) measured in the experiments were measured under conditions where the friction coefficient between the specimen and the inner wall of the TWC changed as the specimen was compressed. In addition to the coefficient of friction, the error between the two curves in FIG. 30 is due to various factors that may exist in the actual test (interference of specimens that may occur between TWC and punch as TWC expands, machining precision of specimen and TWC surface, etc.) It is analyzed as one.

The above description of the present invention is intended for illustration, and it will be understood by those skilled in the art that the present invention may be easily modified in other specific forms without changing the technical spirit or essential features of the present invention. will be. Therefore, it should be understood that the embodiments described above are exemplary in all respects and not restrictive. For example, each component described as a single type may be implemented in a distributed manner, and similarly, components described as distributed may be implemented in a combined form.

The scope of the invention is indicated by the following claims, and it should be construed that all changes or modifications derived from the meaning and scope of the claims and their equivalents are included in the scope of the invention.

Claims (6)

In the method for determining the average radial stress of the compressible material using a cylinder surrounding the side of the compressive material and a rod moving up and down coaxially in the interior of the cylinder,
(a) determining a position of a strain gauge attached to an outer surface of the cylinder;
(b) determining the wall thickness and inner diameter of the cylinder;
(c) the radial stress of the compressible material using Equation 1

Determining);
<Equation 1>

(d) determining a correction function f (ε _v , μ) using Equation 2 below; And
<Equation 2>

(e) Average radial stress of the compressible material using Equation 3

Determining a method comprising;
<Equation 3>

In Formulas 1 to 3,
a and b are the inner and outer diameters of the cylinder, respectively,
E is the Young's modulus of the cylinder,
ε _θ is the hoop strain measured by the strain gauge,
F _T and F _B are the loads measured at the top and bottom of the compressible material, respectively,
μ is the coefficient of friction between the compressive material and the inner wall of the cylinder,
ε _v is the volumetric strain of the compressible material,
A _c is the contact area between the compressive material and the inner wall of the cylinder. The method of claim 1,
In the step (a),
When the ratio of the height of the compressive material to the height of the cylinder is 0.75,
And the strain gauge is attached at a position that is no greater than 65% of the height of the cylinder. The method of claim 1,
In the step (a),
When the ratio of the height of the compressive material to the height of the cylinder is 0.50,
And the strain gauge is attached at a position that is 40% or less of the height of the cylinder. The method of claim 1,
In the step (a),
When the ratio of the height of the compressive material to the height of the cylinder is 0.25,
And the strain gauge is attached at a position that is no greater than 15% of the height of the cylinder. The method of claim 1,
The correction function is a quadratic function represented by Equation 4:
<Equation 4>

In Equation 4,
z ₀ to z ₄ are real numbers obtained by non-linear curve fitting of ε _v vs σ _r ^avg / σ _r ^TWC at predetermined μs, respectively. The method of claim 1,
Wherein the compressible material is one selected from the group consisting of concrete, rock, porous ceramics, powder compacts, soil, rigid plastic foams, and combinations of two or more thereof.

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