CN101566544B - Method for predicating periodic porous material equivalent young's modulus - Google Patents

Abstract

The invention discloses a method for forecasting an equivalent Young's modulus of a periodic porous material, which comprises the following steps: firstly, establishing an entity model of a periodic porous girder structure; carrying out pure bending loading on the entity model of the periodic porous girder structure; equalizing the periodic porous girder into the homogenized girder; loading the same equivalent bending moment on the homogenized girder; and forecasting the equivalent Young's modulus of the periodic porous material and the equivalent Young's modulus of the periodic porous material single cell. A bending strain energy equivalence method is adopted, to solve the phenomena that the prior art cannot reflect the dimension effect of the equivalent Young's modulus of the periodic porous material along the dimension change of a microstructure when the prior art forecasts the equivalent Young's modulus of the periodic porous material, provide an analytic function relation between the equivalent Young's modulus of the periodic porous material and a dimension scaling factor n, really reflect the dimension effect of the equivalent Young's modulus of the periodic porous material along the change of the dimension scaling factor n of the microstructure.

Description

The method of predetermined period porous material equivalent Young modulus

Technical field

The present invention relates to a kind of method of predicting the material equivalent Young's modulus, specifically is the method for predetermined period porous material equivalent Young modulus.

Background technology

Document 1 " B Hassani; E Hinton.A review of homogenization and topology optimizationII-analytical and numerical solution of homogenization equations.Computers and Structures 69; 1998; 719-738. " discloses a kind of homogenization method that physical layer condensation material body born of the same parents' concrete moduli is found the solution in asymptotic expansion based on small scale, has provided the numerical result of rectangle hole bill of materials born of the same parents concrete moduli simultaneously.This method is that the mechanical property and the material bodies proportion by subtraction of material equivalent performance under the macro-scale and the microstructure configuration under the material yardstick, component material set up the strict mathematical description relation.But equivalent Young's modulus that the method is calculated only depends on the volume fraction ratio and the elastic modulus of different materials phase, and irrelevant with body born of the same parents' size, and size that can't the antimer born of the same parents is to the influence of unit cell mechanical property.

Document 2 " L J Gibson; M F Ashby.Cellular solids:structure and properties (2nd ed) [M] .Cambridge University Press; 1997. " has proposed a kind of for the thin-walled unit cell, body born of the same parents deformation mechanism satisfies the mesomechanics method of Euler's beam hypothesis, and to the unit cell of thin-walled rule configuration, as square unit cell, the triangle unit cell, the Equivalent Elasticity parameter of hexagon unit cell is predicted.Yet when unit cell did not satisfy the thin-walled hypothesis, equivalent Young's modulus predicts the outcome error will be very big; And when the volume fraction ratio of forming the unit cell material remains unchanged, result of calculation can't antimer born of the same parents size to the influence of unit cell mechanical property.

Summary of the invention

In order to overcome the deficiency that art methods prediction material equivalent Young's modulus can't antimer born of the same parents change in size, the invention provides a kind of method of predetermined period porous material equivalent Young modulus, the method of bending strain energy equivalence, can be when predetermined period porous material equivalent Young modulus, accurately antimer born of the same parents change in size is to the influence of unit cell mechanical property.

The technical solution adopted for the present invention to solve the technical problems: a kind of method of predetermined period porous material equivalent Young modulus is characterized in comprising the steps:

(a) set up the periodically solid model of multi-hole beam structure;

(b) periodicity multi-hole beam structural solid model is carried out pure bending and load, according to periodicity and the symmetry of structure along the x direction, the bending strain on the single cycle l can U _bFor:

$U_b=\frac{{\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">M</figure-callout>}}^2}{2}{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}$

In the formula, M is a moment of flexure, and E (x) is the Young modulus of answering solid model that changes with x, and I (x) changes the moment of inertia of the structure of solid model with respect to neutral line with x, and l is the cycle along x direction solid model microstructure;

(c) periodically the multi-hole beam equivalence is the homogeneous beam; Equivalence homogeneous beam is L along the length of x direction, is h along the cantilever thickness of z direction;

(d) equivalent homogeneous beam is loaded identical moment M, then can U ' along the bending strain in the l length on the x direction _bFor:

$U_b^{\&prime;}=\frac{1}{2}\frac{M^{2}l}{E^H{I}^H}$

Wherein, E ^HExpression unit cell microstructure equivalent elastic modulus, I ^HThe expression equivalent structure is with respect to the moment of inertia of neutral line;

(e) owing to the forward and backward bending strain of periodicity porous structure equivalence can equate, then

U′b＝Ub

$\frac{1}{2}\frac{M^{2}l}{E^H{I}^H}=\frac{{\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">M</figure-callout>}}^2}{2}{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}$

(f) equivalent Young's modulus of periodic porous material is:

$E^H=\frac{l}{I^H}{\left[{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}\right]}^{-1}$

(g) periodic porous material unit cell equivalent Young's modulus is:

$E^H=\frac{l}{I^H}{\left[{\&Integral;}_0^l\frac{1}{E(x,n)I(x,n)}\mathrm{dx}\right]}^{-1};$

In the formula, n is the unit cell size zoom factor, n=1,2....

The invention has the beneficial effects as follows: owing to adopt the method for bending strain energy equivalence, the size effect phenomenon that can't property reaction time porous material equivalent Young modulus when having solved prior art predetermined period porous material equivalent Young modulus changes with microstructure size, provide the analytical function relation between equivalent Young's modulus of periodic porous material and scale factor of n, really reflected the size effect that equivalent Young's modulus of periodic porous material changes with microstructure scale factor of n.

Below in conjunction with drawings and Examples the present invention is elaborated.

Description of drawings

Fig. 1 is periodically porous array structure standard specimen figure of the inventive method one dimension.

Fig. 2 is the periodically pure bending loading standard specimen figure of porous structure of the inventive method.

Fig. 3 is the inventive method and the corresponding equivalent structure standard specimen of periodicity porous structure figure.

Fig. 4 is the periodically pure bending loading standard specimen figure of porous structure equivalent structure of the inventive method.

Fig. 5 is the inventive method size effect standard specimen figure.

Fig. 6 is the Honeycomb Beam standard specimen figure of the inventive method embodiment 1 square unit cell and formation thereof.

Fig. 7 is the Honeycomb Beam standard specimen figure of the inventive method embodiment 2 triangle unit cells and formation thereof.

Fig. 8 is the Honeycomb Beam standard specimen figure of the inventive method embodiment 3 regular hexagon unit cells and formation thereof.

Fig. 9 is the inventive method embodiment 1 square hole unit cell equivalent Young's modulus relative scale factor of n calibration function graph of a relation.

Figure 10 is the inventive method embodiment 2 equilateral triangle hole unit cell equivalent Young's modulus relative scale factor of n calibration function graphs of a relation.

Figure 11 is the inventive method embodiment 3 regular hexagon hole unit cell equivalent Young's modulus relative scale factor of n calibration function graphs of a relation.

Embodiment

Following examples are with reference to Fig. 1-11.

Embodiment 1: the equivalent Young's modulus of square hole unit cell.

1) set up for periodicity square hole multi-hole beam structure, its parameter has length L=60 of beam, along the Cycle Length l=1 of x direction microstructure, and along the thickness h=2l of z direction beam, square hole unit cell wall thickness t=0.1; The solid material attribute is: Young modulus E=70e9, Poisson ratio v=0.34, density p=2774; If scale factor of n=1 of this moment.

2) periodicity square hole multi-hole beam structure is carried out pure bending and load, according to periodicity and the symmetry of structure along the x direction, the bending strain on the single cycle l can U _bFor:

$U_b=\frac{{\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">M</figure-callout>}}^2}{2}{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}$

Wherein, E (x)=E, I (x) is:

$I\left(x\right)=\left\{\begin{array}{cc}{I}_1=\frac{1}{12}b{h}^3,& x\&Element;[0,\frac{t}{2}]\&cup;[l-\frac{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">t</figure-callout>}}{2},l]\\ {\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">I</figure-callout>}}_2=\frac{1}{4}b{h}^{2}t-\frac{1}{4}\mathrm{<figure-callout\; id="2"\; label="bh\; t"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">bh</figure-callout>}{t}^{}{}^2+\frac{1}{6}b{t}^3,& x\&Element;(\frac{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">t</figure-callout>}}{2},l-\frac{t}{2})\end{array}\right.$

3) set up equivalent homogeneous beam model.Equivalence homogeneous beam is L=60 along the length of x direction, is h=2 along the cantilever thickness of z direction.

4) equivalent average beam is loaded identical moment M, then can U ' along the bending strain in the l length on the x direction _bFor:

$U_b^{\&prime;}=\frac{1}{2}\frac{M^{2}l}{E^H{I}^H}$

Wherein, equivalent structure is with respect to the moment of inertia I of neutral line ^HFor:

$I^H=\frac{b{h}^3}{12}$

5) according to equivalence, periodically the bending strain of porous structure equivalence front and back can equate as can be known, then

U′ _b＝U _b

$\frac{1}{2}\frac{M^{2}l}{E^H{I}^H}=\frac{{\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">M</figure-callout>}}^2}{2}{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}$

6) equivalent Young's modulus of microstructure can be expressed as:

$E^H=\frac{l}{I^H}{\left[{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}\right]}^{-1}$

$=\frac{3}{2b{l}^3}\frac{E{I}_1{I}_2}{\frac{\mathrm{<figure-callout\; id="2"\; label="t\; l\; I"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">t</figure-callout>}}{l}{I}_{}{}_2+\frac{l-\mathrm{<figure-callout\; id="1"\; label="t\; l\; I"\; filenames="H2009100227160E00015.png,H2009100227160E00021.png"\; state="\{\{state\}\}">t</figure-callout>}}{l}{I}_{}{}_1},(h=2l)$

7) the microstructure unit cell size changes with the variation of scale factor of n, and the structure proportion by subtraction remains unchanged in this course; Therefore the equivalent Young's modulus of microstructure should be expressed as:

$E^H=\frac{l}{I^H}{\left[{\&Integral;}_0^l\frac{1}{E(x,n)I(x,n)}\mathrm{dx}\right]}^{-1}$

Wherein, E (x, n)=E, I (x n) is:

$I(x,n)=\left\{\begin{array}{cc}{I}_1=\frac{1}{12}b{h}^3,& x\&Element;[0,\frac{t}{2}]\&cup;[l-\frac{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">t</figure-callout>}}{2},l]\\ {\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">I</figure-callout>}}_2=\frac{1}{12}b{h}^3-\frac{2}{3}b\underset{m=1}{\overset{n}{\mathrm{\&Sigma;}}}[{\left(\frac{\mathrm{mh}-t}{2n}\right)}^3-{\left(\frac{(m-1)h+t}{2n}\right)}^3],& x\&Element;(\frac{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">t</figure-callout>}}{2},l-\frac{t}{2})\end{array}\right.$

Obtain square hole unit cell equivalent Young's modulus E thus ^HWith the function analytical expression of scale factor of n, this formula really reflects the scale effect that equivalent Young's modulus changes with unit cell size.

Embodiment 2: the equivalent Young's modulus of equilateral triangle hole unit cell.

1) set up for periodicity equilateral triangle hole multi-hole beam structure, its parameter has length L=60 of beam, along the Cycle Length l=1 of x direction microstructure, along the thickness of z direction beam $h=\frac{\sqrt{3}}{2}l,$ Equilateral triangle hole unit cell wall thickness t=0.1; The solid material attribute is: Young modulus E=70e9, Poisson ratio v=0.34, density p=2774; If scale factor of n=1 of this moment.

2) periodicity equilateral triangle hole multi-hole beam structure is carried out pure bending and load, according to periodicity and the symmetry of structure along the x direction, the bending strain on the single cycle l can U _bFor:

$U_b=\frac{{\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">M</figure-callout>}}^2}{2}{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}$

Wherein, E (x)=E, I (x) is:

$I\left(x\right)=\frac{1}{12}b(h^3+t^3)+\frac{3}{4}b{x}^{2}t,x\&Element;[0,l]$

3) set up equivalent homogeneous beam model.Equivalence homogeneous beam is L=60 along the length of x direction, along the cantilever thickness of z direction is $h=\frac{\sqrt{3}}{2}l.$

4) equivalent average beam is loaded identical moment M, then can U ' along the bending strain in the l length on the x direction _bFor:

$U_b^{\&prime;}=\frac{1}{2}\frac{M^{2}l}{E^H{I}^H}$

Wherein, equivalent structure is with respect to the moment of inertia I of neutral line ^HFor:

$I^H=\frac{b{h}^3}{12}$

5) according to equivalence, periodically the bending strain of porous structure equivalence front and back can equate as can be known, then

U′ _b＝U _b

$\frac{1}{2}\frac{M^{2}l}{E^H{I}^H}=\frac{{\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">M</figure-callout>}}^2}{2}{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}$

6) equivalent Young's modulus of microstructure can be expressed as:

$E^H=\frac{l}{I^H}{\left[{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}\right]}^{-1}$

$=\frac{3\mathrm{lE}\sqrt{(h^3+t^3)t}}{h^3\arctan \left(\frac{3\mathrm{lt}}{\sqrt{(h^3+t^3)t}}\right)}$

7) the microstructure unit cell size changes with the variation of scale factor of n, and the structure proportion by subtraction remains unchanged in this course; Therefore the equivalent Young's modulus of microstructure should be expressed as:

$E^H=\frac{l}{I^H}{\left[{\&Integral;}_0^l\frac{1}{E(x,n)I(x,n)}\mathrm{dx}\right]}^{-1}$

$=\frac{8\sqrt{3}E}{9l^2}\left\{\underset{m=1}{\overset{n}{\mathrm{\&Sigma;}}}\right[\frac{\sqrt{3}{n}^3}{18t^3}\ln \left[\frac{3}{8}\right[\left(\frac{\sqrt{3}m}{n}l\right){\left(\frac{1}{n}t\right)}^2-{\left(\frac{\sqrt{3}m}{n}l\right)}^2\left(\frac{1}{n}t\right)]+\frac{1}{8}{\left(\frac{1}{n}t\right)}^3+6\sqrt{3}l{\left(\frac{1}{n}t\right)}^3]$

${-\frac{\sqrt{3}{n}^3}{18{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">t</figure-callout>}}^2}\ln \left[\frac{3}{8}\right[\left(\frac{\sqrt{3}m}{n}l\right){\left(\frac{1}{n}t\right)}^2-{\left(\frac{\sqrt{3}m}{n}l\right)}^2\left(\frac{1}{n}t\right)]+\frac{1}{8}{\left(\frac{1}{n}t\right)}^3]\left]\right\}}^{-1}$

Obtain equilateral triangle hole unit cell equivalent Young's modulus E thus ^HWith the function analytical expression of scale factor of n, this formula really reflects the scale effect that equivalent Young's modulus changes with unit cell size.

Embodiment 3: the equivalent Young's modulus of regular hexagon hole unit cell.

1) set up for periodicity regular hexagon hole multi-hole beam structure, its parameter has length L=60 of beam, along the Cycle Length 3l=3 of x direction microstructure, along the thickness of z direction beam $h=\sqrt{3}l,$ Regular hexagon hole unit cell wall thickness t=0.1; The solid material attribute is:

Young modulus E=70e9, Poisson ratio v=0.34, density p=2774; If scale factor of n=1 of this moment.

2) periodicity regular hexagon hole multi-hole beam structure is carried out pure bending and load, according to periodicity and the symmetry of structure along the x direction, the bending strain on the single cycle l can U _bFor:

$U_b=\frac{{\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">M</figure-callout>}}^2}{2}{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}$

Wherein, E (x)=E, I (x) is:

$I\left(x\right)=\left\{\begin{array}{cc}{I}_1=\frac{1}{3}b{t}^3+b{h}^{2}t,& x\&Element;[0,\frac{l}{2})\&cup;[\frac{5\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">l</figure-callout>}}{2},3l]\\ I_2=\frac{16}{3}b{t}^3+b{h}^{2}l+12b{(x-\frac{l}{2})}^{2}t,& x\&Element;[\frac{1}{2},l)\&cup;[2l,\frac{5l}{2})\\ I_3=\frac{1}{6}b{t}^3-\frac{1}{2}{\mathrm{<figure-callout\; id="2"\; label="bht"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">bht</figure-callout>}}^2+b{h}^{2}t,& x\&Element;[l,2l)\end{array}\right.$

3) set up equivalent homogeneous beam model.Equivalence homogeneous beam is L=60 along the length of x direction, along the cantilever thickness of z direction is $h=\sqrt{3}l.$

4) equivalent average beam is loaded identical moment M, then can U ' along the bending strain in the l length on the x direction _bFor:

$U_b^{\&prime;}=\frac{1}{2}\frac{M^{2}l}{E^H{I}^H}$

Wherein, equivalent structure is with respect to the moment of inertia I of neutral line ^HFor:

$I^H=\frac{b{h}^3}{12}$

5) according to equivalence, periodically the bending strain of porous structure equivalence front and back can equate as can be known, then

U′ _b＝U _b

$\frac{1}{2}\frac{M^{2}l}{E^H{I}^H}=\frac{{\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">M</figure-callout>}}^2}{2}{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}$

6) equivalent Young's modulus of microstructure can be expressed as:

$E^H=\frac{l}{I^H}{\left[{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}\right]}^{-1}$

$=\frac{12\mathrm{Elt}\sqrt{70{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">t</figure-callout>}}^2+36h^2-6\mathrm{ht}}}{h^3\arctan \left(\frac{6l}{\sqrt{70{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">t</figure-callout>}}^2+36h^2-6\mathrm{ht}}}\right)+h^3\arctan \left(\frac{30l}{\sqrt{70{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="H2009100227160E00015.png"\; state="\{\{state\}\}">t</figure-callout>}}^2+36h^2-6\mathrm{ht}}}\right)}$

7) the microstructure unit cell size changes with the variation of scale factor of n, and the structure proportion by subtraction remains unchanged in this course; Therefore the equivalent Young's modulus of microstructure should be expressed as:

$E^H=\frac{l}{I^H}{\left[{\&Integral;}_0^l\frac{1}{E(x,n)I(x,n)}\mathrm{dx}\right]}^{-1}$

$=\frac{8\sqrt{3}E}{9l^2}\left\{\underset{m=1}{\overset{n}{\mathrm{\&Sigma;}}}\right[\frac{\frac{m}{n}l}{{\left(\frac{t}{2n}\right)}^3}+\frac{2\sqrt{3}}{9{\left(\frac{t}{n}\right)}^2}\left[\ln \right[\frac{1}{4}{\left(\frac{t}{n}\right)}^3-\frac{3\sqrt{3}}{2}{\left(\frac{t}{n}\right)}^3\frac{m}{n}l]-\ln [\frac{1}{4}{\left(\frac{t}{n}\right)}^3\left]\right]$

${+\frac{\frac{m}{n}l}{\frac{9}{4}{\left(\frac{m}{n}l\right)}^2\frac{t}{2n}+\frac{3\sqrt{3}}{2}\frac{m}{n}l{\left(\frac{t}{2n}\right)}^2+{\left(\frac{t}{2n}\right)}^3}\left]\right\}}^{-1}$

Obtain regular hexagon hole unit cell equivalent Young's modulus E thus ^HWith the function analytical expression of scale factor of n, this formula really reflects the scale effect that equivalent Young's modulus changes with unit cell size.

As can see from Figure 9, adopt the inventive method prediction square hole unit cell equivalent Young's modulus to reduce gradually along with the increase of scale factor of n, when l＞＞when t and scale factor of n → ∞, following formula is reduced to:

$E^H=\frac{t}{l}E=\mathrm{\&alpha;E}$

The result that this result and homogenization method and G-A mesomechanics method are tried to achieve is in full accord.Illustrate that result that homogenization method and G-A mesomechanics method are tried to achieve is the ultimate value of microstructure unit cell when infinitely small.

As can see from Figure 10, adopt the inventive method prediction equilateral triangle hole unit cell equivalent Young's modulus to subtract the solving result that increases and finally level off to homogenization method and G-A mesomechanics method gradually along with the increase of scale factor of n.

As can see from Figure 11, adopt the inventive method prediction regular hexagon hole unit cell equivalent Young's modulus to be non-monotone variation trend along with the variation of scale factor of n.When the scale factor of n increased, the equivalent Young's modulus of regular hexagon hole unit cell changed from small to big; When n=5, E ^HReach maximal value, after this reduce gradually and finally level off to the result of homogenising.

Claims (1)

1. the method for predetermined period porous material equivalent Young modulus is characterized in that comprising the steps:

(a) set up the periodically solid model of multi-hole beam structure;

(b) periodicity multi-hole beam structural solid model is carried out pure bending and load, according to periodicity and the symmetry of structure along the x direction, the bending strain on the single cycle l can U _bFor:

$U_b=\frac{M^2}{2}{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}$

In the formula, M is a moment of flexure, and E (x) is the Young modulus of answering solid model that changes with x, and I (x) changes the moment of inertia of the structure of solid model with respect to neutral line with x, and l is the cycle along x direction solid model microstructure;

(c) periodically the multi-hole beam equivalence is the homogeneous beam; Equivalence homogeneous beam is L along the length of x direction, is h along the cantilever thickness of z direction;

(d) equivalent homogeneous beam is loaded identical moment M, then can U ' along the bending strain in the l length on the x direction _bFor:

$U_b^{\&prime;}=\frac{1}{2}\frac{M^{2}l}{E^H{I}^H}$

Wherein, E ^HExpression unit cell microstructure equivalent elastic modulus, I ^HThe expression equivalent structure is with respect to the moment of inertia of neutral line;

(e) owing to the forward and backward bending strain of periodicity porous structure equivalence can equate, then

U′ _b＝U _b

$\frac{1}{2}\frac{M^{2}l}{E^H{I}^H}=\frac{M^2}{2}{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}$

(f) equivalent Young's modulus of periodic porous material is:

$E^H=\frac{l}{I^H}{\left[{\&Integral;}_0^l\frac{1}{E\left(x\right)I\left(x\right)}\mathrm{dx}\right]}^{-1}$

(g) periodic porous material unit cell equivalent Young's modulus is:

$E^H=\frac{l}{I^H}{\left[{\&Integral;}_0^l\frac{1}{E(x,n)I(x,n)}\mathrm{dx}\right]}^{-1};$

In the formula, n is the unit cell size zoom factor, n=1,2.......

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