CN102436520B - Computing method for equivalent elastic modulus of two-dimensional porous materials - Google Patents
Computing method for equivalent elastic modulus of two-dimensional porous materials Download PDFInfo
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- CN102436520B CN102436520B CN 201110280821 CN201110280821A CN102436520B CN 102436520 B CN102436520 B CN 102436520B CN 201110280821 CN201110280821 CN 201110280821 CN 201110280821 A CN201110280821 A CN 201110280821A CN 102436520 B CN102436520 B CN 102436520B
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Abstract
The invention discloses a computing method for an equivalent elastic modulus of two-dimensional porous materials. The computing method comprises the following steps of: firstly building a finite element computation model of the two-dimensional porous materials, then dividing a grid, exerting a displacement load, obtaining a resultant force of all nodes at fixed ends of loads along a direction of the equivalent elastic modulus, and finally obtaining the equivalent elastic modulus of the finite element computation model of the two-dimensional porous materials according to formula computing. The method disclosed by the invention is applicable to solving of the equivalent modulus of the two-dimensional porous materials with regular and irregular conformation in a covariance direction, moreover, size limit of a product is avoided, and the computing method has the advantages of simplicity and rapidness.
Description
Technical field
The invention belongs to two-dimensional porous material static mechanical performance calculation of parameter technical field, relate to a kind of computing method of two-dimensional porous material equivalent elastic modulus.
Background technology
Two-dimensional porous material is divided into two kinds of regular and irregular configurations, wherein, the common structure form of regular configuration two-dimensional porous material has: triangle, alternately triangle, rectangle, alternately rectangle, protruding hexagon, recessed hexagon, sparse X-shaped, intensive X-shaped, circle, ellipse, corrugated and sinusoidal waveform etc.No matter be the two-dimensional porous material of which kind of configuration, when the direction along coplanar or antarafacial applied the static compressive load of low speed class, stress-strain diagram all can comprise the deformation processes such as linear elasticity, plastic yielding, platform area and densification.Slope at linear elastic deformation stage stress-strain diagram is the equivalent elastic modulus of two-dimensional porous material on corresponding loading direction.Calculate rapidly and accurately the equivalent elastic modulus on a direction, have important actual application value for the static mechanical performance of estimating two-dimensional porous material.
At present, the method for determining equivalent elastic modulus on two-dimensional porous material a direction has experimental method, theory method and finite element method.Experimental method is to carry out the class static compression test according to the relevant criterion in the test standards such as ASTM (American Society of Testing Materials), directly calculates corresponding equivalent elastic modulus from the trial curve that obtains.Experimental method need to be tested according to relevant criterion, the recording responses curve, and operating process is more loaded down with trivial details.And in experimental method, test specimen used from actual production, because production technology is limit, can not obtain the abundant sample of size and test.Experimental method also can consume sample, and is economical not.Theory method is for the periodic regular two-dimensional porous material, intercepts its feature micro unit, and it is considered as certain thickness beam, tries to achieve equivalent force on feature unit according to the macro-stress on a direction, derives accordingly and calculates relevant equivalent elastic modulus.Although theory method can be calculated the equivalent elastic modulus of the two-dimensional porous material of various tactical rules, also without the consumption test sample, be not suitable for calculating the equivalent elastic modulus of irregular structure two-dimensional porous material.Existing finite element method is to intercept the feature unit of periodicity two-dimensional porous material as computation model, model is applied periodic boundary condition, along a direction imposed load, the equivalent force of the feature micro unit that obtains in this load and theory method conforms to, by calculating corresponding displacement, try to achieve relevant equivalent elastic modulus in conjunction with the last calculating of the original size of micro unit and (see Sun Deqiang, Zhang Weihong, Sun Yujin. the elastic modulus of Aluminum Honeycomb Cores and material efficiency analysis [J]. mechanics and practice, 2008,30 (1): 35～40.).The method is based on the finite element method of feature unit, although overcome the shortcomings such as uneconomical, the sample size of experimental method is few, but cyclic load applies complexity, even sometimes is difficult to apply, and is not suitable for calculating the equivalent elastic modulus of erose two-dimensional porous material.
Summary of the invention
The object of the invention is to provide the computing method of two-dimensional porous material equivalent elastic modulus, is applicable to altogether finding the solution of the concrete moduli on the antarafacial direction of regular and irregular configuration two-dimensional porous material, and not limited by sample size, has simplicity and agility.
The technical solution adopted in the present invention is that a kind of computing method of two-dimensional porous material equivalent elastic modulus is characterized in that, carry out according to following steps:
Described two-dimensional porous material limited element calculation model is defined as the load stiff end in the bottom surface of equivalent elastic modulus direction, to be defined as at the end face of equivalent elastic modulus direction the load applying end, displacement and the rotary freedom of all nodes on the load stiff end are constrained to zero, and all nodes apply displacement load Δ l along the equivalent elastic modulus direction on the load applying end, start ANSYS/Multiphysics software, calculate each node of load stiff end along the ∑ F that makes a concerted effort of equivalent elastic modulus direction;
The equivalent elastic modulus E of step 2, the described two-dimensional porous material limited element calculation model of calculating
*: E
*=l * ∑ F/ (s
LS
W* Δ l), wherein, l is that this two-dimensional porous material limited element calculation model is along the length on loading direction, s
LBe the length of this two-dimensional porous material limited element calculation model along xsect on loading direction, s
WBe the width of this two-dimensional porous material limited element calculation model along xsect on loading direction, ∑ F is this each node of two-dimensional porous material limited element calculation model load stiff end along the making a concerted effort of equivalent elastic modulus direction, the displacement load that Δ l applies along the equivalent elastic modulus direction for all nodes on this two-dimensional porous material limited element calculation model load applying end.
In step 1, the span of displacement load Δ l is 0.002～0.02mm.
The beneficial effect of the inventive method is, only need to be on the basis of setting up the two-dimensional porous material limited element calculation model, therefore imposed load can be found the solution, and is applicable to two kinds of configuration two-dimensional porous materials of regular and irregular finding the solution of the equivalent elastic modulus on the antarafacial direction altogether; Secondly, this method does not need to apply cyclic load by means of existing ANSYS/Multiphysics software, compare existing finite element method and implement simpler, so operand is little, has simplicity and agility; At last, it has overcome experimental method and has sought the problem of various sizes sample difficulty, and does not need to consume sample, and is more economical feasible.
Description of drawings
Fig. 1 is two-dimensional porous material equivalent elastic modulus limited element calculation model schematic diagram in the inventive method;
Fig. 2 is the structural representation of hexagon two-dimensional porous material;
Fig. 3 is the structural representation of hexagon two-dimensional porous material limited element calculation model, and wherein, Fig. 3 a is this hexagon two-dimensional porous material x
1The structural representation of equivalent elastic modulus on direction, Fig. 3 b are this hexagon two-dimensional porous material x
2The structural representation of equivalent elastic modulus on direction, Fig. 3 c are this hexagon two-dimensional porous material x
3The structural representation of equivalent elastic modulus on direction;
Fig. 4 is that the hexagon two-dimensional porous material is at x
1Equivalent elastic modulus E on direction
1 *Change curve, wherein, Fig. 4 a is E
1 *Finite-Element Solution and the graph of relation of theoretical value and side ratio, Fig. 4 b is E
1 *Finite-Element Solution and the graph of relation of theoretical value and extended corner, Fig. 4 c is E
1 *Finite-Element Solution and the graph of relation of theoretical value and wall thickness;
Fig. 5 is that the hexagon two-dimensional porous material is at x
2Equivalent elastic modulus E on direction
2 *Change curve, wherein, Fig. 5 a is E
2 *Finite-Element Solution and the graph of relation of theoretical value and side ratio, Fig. 5 b is E
2 *Finite-Element Solution and the graph of relation of theoretical value and extended corner, Fig. 5 c is E
2 *Finite-Element Solution and the graph of relation of theoretical value and wall thickness;
Fig. 6 is that the hexagon two-dimensional porous material is at x
3Equivalent elastic modulus E on direction
3 *Change curve, wherein, Fig. 6 a is E
3 *Finite-Element Solution and the graph of relation of theoretical value and side ratio, Fig. 6 b is E
3 *Finite-Element Solution and the graph of relation of theoretical value and extended corner, Fig. 6 c is E
3 *Finite-Element Solution and the graph of relation of theoretical value and wall thickness;
Fig. 7 is triangle two-dimensional porous material limited element calculation model schematic diagram;
Fig. 8 is quadrilateral two-dimensional porous material limited element calculation model schematic diagram.
Embodiment
The present invention is described in detail below in conjunction with the drawings and specific embodiments.
The computing method of two-dimensional porous material equivalent elastic modulus of the present invention, carry out according to following steps:
As shown in Figure 1, the two-dimensional porous material limited element calculation model is defined as the load stiff end in the bottom surface of equivalent elastic modulus direction, to be defined as at the end face of equivalent elastic modulus direction the load applying end, displacement and the rotary freedom of all nodes on the load stiff end are constrained to zero, and all nodes apply displacement load Δ l along the equivalent elastic modulus direction on the load applying end, start ANSYS/Multiphysics software, calculate each node of load stiff end along the ∑ F that makes a concerted effort of equivalent elastic modulus direction; The span of displacement load Δ l is 0.002mm～0.02mm.
The equivalent elastic modulus E of step 2, Two-dimensional porosint limited element calculation model
*: E
*=l * ∑ F/ (s
LS
W* Δ l), wherein, l is that this two-dimensional porous material limited element calculation model is along the length on loading direction, s
LBe the length of this two-dimensional porous material limited element calculation model along xsect on loading direction, s
WBe the width of this two-dimensional porous material limited element calculation model along xsect on loading direction, ∑ F is this each node of two-dimensional porous material limited element calculation model load stiff end along the making a concerted effort of equivalent elastic modulus direction, the displacement load that Δ l applies along the equivalent elastic modulus direction for all nodes on this two-dimensional porous material limited element calculation model load applying end.
The inventive method is suitable for the calculating of various configuration two-dimensional porous material equivalent elastic modulus.No matter be rule or irregular two-dimensional porous material, only need to build computation model by the material physical size and get final product.In order effectively to verify the reliability of the inventive method result of calculation, following related embodiment will describe as an example of regular two-dimensional porous material example.
The hexagon two-dimensional porous material is along x
1, x
2And x
3Equivalent elastic modulus on direction.
In the structure of hexagon two-dimensional porous material, each feature unit has the inclined hole wall of 4 long d, thick t and the vertical core wall of 2 long h, thick t as shown in Figure 2, and hole depth is b.x
1Axle is perpendicular to vertical hole wall wall, x
2Axle is parallel to vertical hole wall wall, and x
3Axle is along the hole depth direction.Hypotenuse hole wall wall and x
1-x
3The angle of face is θ, is defined as extended corner.In the present embodiment, base material is aluminium alloy, and the physical parameter that its property linear elastic deformation is relevant has: Young modulus E
s=68.97GPa, Poisson ratio v
s=0.35, density p
s=2700Kg/m
3
In the present embodiment, get d=5mm, h/d=1.5, θ=30 °, t=0.15mm.Adopt ANSYS/Multiphysics software according to the modeling of ASTM stretching experiment standard, the thick 220mm of being of a size of of two-dimensional porous material limited element calculation model length and width that obtains * 130mm * 15mm.In order to guarantee the size of this two-dimensional porous material limited element calculation model, select abundant element number.As shown in Fig. 3 a, for asking this two-dimensional porous material x
1The structural representation of equivalent elastic modulus on direction, its x
1And x
2On direction, the quantity of complete characterization unit should be respectively m
1=26 and n
1=13, this model x
1Length l on loading direction is at x
1Length l on direction
1=2m
1Dcos θ+t=225.27mm, this model is along x
1The length s of xsect on loading direction
LBe at x
2Length w on direction
1=n
1(sin θ+h/d)+dsin θ=132.5mm, this model is along x for d
1The width s of xsect on loading direction
WBe hole depth b
1=15mm.As shown in Fig. 3 b, for asking this two-dimensional porous material x
2The structural representation of equivalent elastic modulus on direction, its x
2And x
1On direction, the quantity of complete characterization unit should be respectively m
2=22 and n
2=15, this model x
2Length l on loading direction is at x
2Length l on direction
2=m
2(sin θ+h/d)+dsin θ=222.5mm, this model is along x for d
2The length s of xsect on loading direction
LBe at x
1Length w on direction
2=(2n
2+ 1) dcos θ+t=134.33mm, this model is along x
2The width s of xsect on loading direction
WBe hole depth b
2=15mm.According to the standard of national army mark " glueing joint aluminium honeycomb sandwich construction and the sub-method for testing performance general provisions of aluminium honeycomb core " (GJB 130.1-1986) and " glueing joint aluminium honeycomb sandwich construction and fuse flat compressed method for testing performance " (GJB 130.5-1986), utilize compression test to ask antarafacial (along x
3On direction) the thick 60mm * 60mm * 60mm that is respectively of length and width of the two-dimensional porous material limited element calculation model of elastic modulus.As shown in Fig. 3 c, for asking this two-dimensional porous material x
3The structural representation of equivalent elastic modulus on direction, its x
2And x
1On direction, the quantity of complete characterization unit should be respectively m
3=6 and n
3=7, this model is along x
3The length s of xsect on loading direction
LBe at x
2Length l on direction
3=m
3(sin θ+h/d)=60mm, this model is along x for d
3The width s of xsect on loading direction
WBe at x
1Width w on direction
3=2n
3Dcos θ+t=60.72mm.This model x
3Length l on loading direction is hole depth b
3=60mm.
Utilize linear elasticity multilayered shell cell S hell99 with above-mentioned two-dimensional porous material limited element calculation model grid division, make thickness of shell elements equal the thickness t=0.15mm of wall in this two-dimensional porous material limited element calculation model.
Calculate this two-dimensional porous material limited element calculation model at x
1Equivalent elastic modulus E on direction
1 *The time, make this two-dimensional porous material limited element calculation model at x
1The bottom surface of direction is defined as x
1The load stiff end of direction, with this two-dimensional porous material limited element calculation model at x
1The end face of direction is defined as x
1The load applying end of direction is with x
1On direction load stiff end, the displacement of all nodes and rotary freedom are constrained to zero, and at x
1On direction load applying end, all nodes are along x
1Direction applies displacement load Δ l
1=0.01mm starts ANSYS/Multiphysics software, reads the x that software calculates
1All nodes of direction load stiff end are along x
1The ∑ F that makes a concerted effort of direction
1=0.3047N; According to E
*=l * ∑ F/ (s
LS
W* Δ l), get E
1 *=l
1* ∑ F
1/ (w
1B
1* Δ l
1), calculate E
1 *=3.4534MPa.
Calculate this two-dimensional porous material limited element calculation model at x
2Equivalent elastic modulus E on direction
2 *The time, make this two-dimensional porous material limited element calculation model at x
2The bottom surface of direction is defined as x
2The load stiff end of direction, with this two-dimensional porous material limited element calculation model at x
2The end face of direction is defined as x
2The load applying end of direction is with x
2On the load stiff end of direction, the displacement of all nodes and rotary freedom are constrained to zero, and at x
2On direction load applying end, all nodes are along x
2Direction applies displacement load Δ l
2=0.01mm starts ANSYS/Multiphysics software, reads the x that software calculates
2All nodes of direction load stiff end are along x
2The ∑ F that makes a concerted effort of direction
2=0.4887N; According to E
*=l * ∑ F/ (s
LS
W* Δ l), get E
2 *=l
2* ∑ F
2/ (w
2B
2* Δ l
2), calculate E
2 *=5.3964MPa.
Calculate this two-dimensional porous material limited element calculation model at x
3Equivalent elastic modulus E on direction
3 *The time, make this two-dimensional porous material limited element calculation model at x
3The bottom surface of direction is defined as x
3The load stiff end of direction, with this two-dimensional porous material limited element calculation model at x
3The end face of direction is defined as x
3The load applying end of direction is with x
3On direction load stiff end, the displacement of all nodes and rotary freedom are constrained to zero, and at x
3On direction load applying end, all nodes are along x
3Direction applies displacement load Δ l
3=0.01mm starts ANSYS/Multiphysics software, reads the x that software calculates
3All nodes of direction load stiff end are along x
3The ∑ F that makes a concerted effort of direction
3=1.2876KN; According to E
*=l * ∑ F/ (s
LS
W* Δ l), get E
3 *=b
3* ∑ F
3/ (l
3W
3* Δ l
3), calculate E
3 *=2.1205GPa.
Document (Gibson LJ, Ashby MF.Cellular solids:structures and properties.2nded.Cambridge University Press:Cambridge; 1997.) in about the hexagon two-dimensional porous material along x
1, x
2And x
3The equivalent elastic modulus of the above-mentioned hexagon two-dimensional porous material that the equivalent elastic modulus theory on direction calculates is respectively: E
1 *=3.3194MPa, E
2 *=5.6968MPa, E
3 *=2.0905MPa.
In order further to verify the reliability of computing method of the present invention, fixedly d=5mm, change all the other structural parameters, and side ratio h/d changes between 0.5～2.5, and the scope of extended corner θ is 10 °～80 °, and the scope of wall thickness t is 0.05mm～0.3mm.Get θ=45 °, t=0.15mm, only change h/d, carry out 9 simulations, can get E
1 *, E
2 *And E
3 *Numerical value under different side ratios (seeing Table 1).Get t=0.15mm, h/d=1.5 only changes θ, carries out respectively 8 simulations, can get E
1 *, E
2 *And E
3 *Numerical value under different θ (seeing Table 2).Get h/d=1.5, θ=45 ° only change t, carry out respectively 11 simulations, can get E
1 *, E
2 *And E
3 *Numerical value under different t (seeing Table 3).Wherein, in table 1, table 2 and table 3, the numerical value of theoretical value for obtaining according to existing equivalent elastic modulus theoretical calculation formula, Finite-Element Solution is the numerical value that computing method calculate according to the present invention.
E under the different hld of table 1
1 *, E
2 *And E
3 *Value and error
E under the different θ of table 2
1 *, E
2 *And E
3 *Value and error
E under the different t of table 3
1 *, E
2 *And E
3 *Value and error
Corresponding table 1, the relation curve of the Finite-Element Solution of each elastic modulus and theoretical value and side ratio is seen respectively Fig. 4 a, 5a and 6a.Corresponding table 2, the relation curve of the Finite-Element Solution of each elastic modulus and theoretical value and extended corner is seen respectively in Fig. 4 b, 5b and 6b.Corresponding table 3, the relation curve of the Finite-Element Solution of each elastic modulus and theoretical value and wall thickness is seen respectively in Fig. 4 c, 5c and 6c.
Can find out, within the scope that error allows, the result of calculation that the hexagon two-dimensional porous material that the equivalent elastic modulus theory calculates is total to antarafacial elastic modulus and the inventive method matches, and has proved the reliability of computing method of the present invention.
The calculating of the coplanar equivalent elastic modulus of triangle two-dimensional porous material
In the present embodiment, base material is aluminium alloy, and its linear elasticity stage, relevant physical parameter had: Young modulus is E
s=68.97GPa, Poisson ratio is v
s=0.35, density is ρ
s=2700Kg/m
3
Adopt ANSYS/Multiphysics software to set up this two-dimensional porous material limited element calculation model, obtain model as shown in Figure 7, along l
1Direction is called coplanar, and coplanar equivalent elastic modulus is defined as E
1 *The periodic feature unit intermediate cam shape length of side is 5mm, and the base angle is 60 °, and wall thickness is 0.15mm.According to ASTM stretching experiment standard, on this two-dimensional porous material limited element calculation model length and width direction, the quantity of full unit should be respectively m
1=51 and n
1=26, the length l of this model on in-plane direction
1=220.9865mm, width w
1=130mm, hole depth b
1Be 15mm.
Utilize linear elasticity multilayered shell cell S hell99 with above-mentioned two-dimensional porous material limited element calculation model grid division, make the thickness of shell unit equal the thickness 0.15mm of this two-dimensional porous material wall.
This two-dimensional porous material limited element calculation model is defined as the load stiff end in the bottom surface of in-plane direction, this two-dimensional porous material limited element calculation model is defined as the load applying end at the end face of in-plane direction, displacement and the rotary freedom of all nodes on the load stiff end are constrained to zero, and all nodes apply displacement load Δ l=0.001mm along in-plane direction on the load applying end, start ANSYS/Multiphysics software, read each node of load stiff end that software calculates along the ∑ F that makes a concerted effort of in-plane direction
1=21.0354N; According to E
*=l * ∑ F/ (s
LS
W* Δ l), get E
1 *=l
1* ∑ F
1/ (w
1B
1* Δ l
1), calculate E
1 *=2.3719GPa.
According to document (Wang AJ, McDowell DL.In-plane stiffness and yield strength of periodic metal honeycombs.Journal of Engineering Materials and Technology, 2004,126 (2): the computing formula of equivalent elastic modulus on the triangle two-dimensional porous material in-plane direction that provides 137-156.), the equivalent elastic modulus theoretical value that can calculate this moment is 2.3892GPa, can find out, the result of calculation of the inventive method is in the scope that error allows.
Embodiment 3
The calculating of the coplanar equivalent elastic modulus of square two-dimensional porous material
In the present embodiment, base material is aluminium alloy, and its linear elasticity stage, relevant physical parameter had: Young modulus is E
s=68.97GPa, Poisson ratio is v
s=0.35, density is ρ
s=2700Kg/m
3
Adopt ANSYS/Multiphysics software to set up this two-dimensional porous material limited element calculation model, obtain model as shown in Figure 8, along l
1Direction is called coplanar, and coplanar equivalent elastic modulus is defined as E
1 *In the periodic feature unit, the square length of side is 5mm, and wall thickness is 0.15mm.According to ASTM stretching experiment standard, on this two-dimensional porous material limited element calculation model length and width direction, the quantity of full unit should be respectively m
1=44 and n
1=26, the length l of this model on in-plane direction
1=220.15mm, width w
1=130mm, hole depth b
1Be 15mm.
Utilize linear elasticity multilayered shell cell S hell99 with above-mentioned two-dimensional porous material limited element calculation model grid division, make the thickness of shell unit equal the wall thickness 0.15mm of this two-dimensional porous material.
This two-dimensional porous material limited element calculation model is defined as the load stiff end in the bottom surface of in-plane direction, the end face of this two-dimensional porous material limited element calculation model in-plane direction is defined as the load applying end, displacement and the rotary freedom of all nodes on the load stiff end are constrained to zero, and all nodes apply displacement load Δ l along in-plane direction on the load applying end
1=0.001mm starts ANSYS/Multiphysics software, reads all nodes of load stiff end that software calculates along the ∑ F that makes a concerted effort of in-plane direction
1=18.7294N; According to E
*=l * ∑ F/ (s
LS
W* Δ l), get E
1 *=l
1* ∑ F
1/ (w
1B
1* Δ l
1), calculate E
1 *=2.1145Gpa.
According to document (Wang AJ, McDowell DL.In-plane stiffness and yield strength of periodic metal honeycombs.Journal of Engineering Materials and Technology, 2004,126 (2): the computing formula of equivalent elastic modulus on the square two-dimensional porous material in-plane direction that provides 137-156.), the equivalent elastic modulus theoretical value that can calculate this moment is 2.0691GPa, can find out, the result of calculation of the inventive method is in the scope that error allows.
Claims (1)
1. the computing method of two-dimensional porous material equivalent elastic modulus, it is characterized in that, the method is pressed the material physical size and is built computation model, is applicable to regular and irregular configuration two-dimensional porous material finding the solution of the concrete moduli on the antarafacial direction altogether, carries out according to following steps:
Step 1, employing ANSYS/Multiphysics software, set up the two-dimensional porous material limited element calculation model, utilize linear elasticity multilayered shell cell S hell99 with described two-dimensional porous material limited element calculation model grid division, make the thickness of shell unit equal the thickness of wall in described two-dimensional porous material limited element calculation model;
Described two-dimensional porous material limited element calculation model is defined as the load stiff end in the bottom surface of equivalent elastic modulus direction, to be defined as at the end face of equivalent elastic modulus direction the load applying end, displacement and the rotary freedom of all nodes on the load stiff end are constrained to zero, and all nodes apply displacement load Δ l along the equivalent elastic modulus direction on the load applying end, the span of displacement load Δ l is 0.002～0.02mm, start ANSYS/Multiphysics software, calculate each node of load stiff end along the ∑ F that makes a concerted effort of equivalent elastic modulus direction;
The equivalent elastic modulus E of step 2, the described two-dimensional porous material limited element calculation model of calculating
*: E
*=l * ∑ F/ (s
LS
W* Δ l), wherein, l is that this two-dimensional porous material limited element calculation model is along the length on loading direction, s
LBe the length of this two-dimensional porous material limited element calculation model along xsect on loading direction, s
WBe the width of this two-dimensional porous material limited element calculation model along xsect on loading direction, ∑ F is this each node of two-dimensional porous material limited element calculation model load stiff end along the making a concerted effort of equivalent elastic modulus direction, the displacement load that Δ l applies along the equivalent elastic modulus direction for all nodes on this two-dimensional porous material limited element calculation model load applying end.
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