COMPUTATION METHOD OF MECHANICAL MATERIAL CONSTANT OF COMPOSITE MATERIAL AND VOLUME FRACTION OF MATERIAL COMPONENT IN COMPOSITE MATERIAL, AND RECORDING MEDIUM

• US 20100262407A1
• Filed: 04/14/2009
• Published: 10/14/2010
• Est. Priority Date: 04/14/2009
• Status: Active Grant

• Current Assignee: Yokohama Rubber Company Limited
• Original Assignee: Yokohama Rubber Company Limited
• Inventors: KABE, Kazuyuki, NOMURA, Seiichi, SUZUKI, Nobuo
• Granted Patent: US 8,170,846 B2
• Time in Patent Office: Days
• US Class Current: 703/2
• CPC Class Codes: G06F 2113/26 Composites
• CPC Class Codes: G06F 30/23 using finite element methods [FEM] or finite difference methods [FDM]

Abstract

In a method of computing the overall material constant of a composite material, a virtual composite material is defined as the one that a first material component is dispersed in a form of inner spherical particles in a matrix phase and each of the inner spherical particles is enveloped by the second material component, in a form of outer shell layers, as a coating layer. Based on this, a nonlinear equation is prepared, which has the material constant of the virtual composite material as an unknown. Next, the material constant of the virtual composite material is computed by solving the equation. In the equation, the material constant in each of the surrounding areas of the outer shell layers coating the inner spherical particles is defined as the overall material constant of the virtual composite material to be computed. The volume fractions of the material components in the composite material are computed using the equation.

Claims

1. A method of computing an overall mechanical material constant of a composite material, the composite material including a first material component and a second material component in a matrix phase, mechanical material constants of the first material component, the second material component and the matrix phase being known, the method comprising the steps of:

preparing a nonlinear equation having the overall material constant of a virtual composite material as an unknown by defining, as a composite material, the virtual composite material in which the first material component is dispersed in a form of inner spherical particles in the matrix phase at a known volume fraction and the second material component is in a form of outer shell layers, each of which envelops each of the inner spherical particles as a coating layer of the predetermined thickness; and

computing the overall material constant of the virtual composite material as the overall material constant of the composite material by solving the prepared nonlinear equation, and

wherein the nonlinear equation is a recursive nonlinear equation which is obtained by defining, as the overall material constant of the virtual composite material to be computed, a material constant in each of surrounding areas of the outer shell layers enveloping the inner spherical particles in the virtual composite material.

2. The method according to claim 1, wherein the nonlinear equation is prepared by determining a displacement in the virtual composite material under a condition that both a surface traction and the displacement are continuous across each of boundaries between the inner spherical particles and the coating layers, and a condition that both a surface traction and the displacement are continuous across each of boundaries between the outer shell layers and the matrix phase.

3. The method according to claim 1,

wherein the nonlinear equation is expressed by “

C*=C^A+v_B·

(C^B−

C^A)·

A^B(C^B, C^C, C*)+v_C·

(C^C−

C^A)·

A^C(C^B, C^C, C*)”

when the material constant of the matrix phase, the material constant of the first material component, the material constant of the second material component, the volume fraction of the first material component, a volume fraction of the second material component and the overall material constant of the virtual composite material are represented as C^A, C^B, C^C, v_B, v_C, and C*, respectively, and

wherein each of “

A^B(C^B, C^C, C*)” and

“

A^C(C^B, C^C, C*)”

in the equation is obtained by defining a material constant of the outer shell layers enveloping the inner spherical particles in the virtual composite material as the material constant C^C of the second material component and by defining a material constant in each of the surrounding areas of the outer shell layers enveloping the inner spherical particles as the overall material constant of the virtual composite material to be computed, and each of “

A^B(C^B, C^C, C*)” and

“

A^C(C^B, C^C, C*)”

is a proportional constant to be defined by a nonlinear expression having the material constant C* as an unknown.

4. The method according to the claim 1, wherein,

when each of stress and strain is decomposed into a shear term and a hydrostatic pressure term, and a displacement of the virtual composite material corresponding to the hydrostatic pressure term is defined as “

u_i (i=1, 2 or

     3),”

the nonlinear equation is prepared by defining;

the displacement u_i inside each of the pairs of the inner spherical particles and the outer shell layers to be proportional to a position “

x_i (i=1, 2 or

     3)”

with an origin at a center of each of the pairs of the inner spherical particles and the outer shell layers; and

the displacement u_i outside each of the pairs of the inner spherical particles and the outer shell layers to have;

a term proportional to the position x_i (i=1, 2 or

     3) with the origin at the center of each of the pairs of the inner spherical particles and the outer shell layers; and

a term proportional to the position x_i (i=1, 2 or

     3) with the origin at the center and inversely proportional to third-power of a distance from the center.

5. The method according to claim 1, wherein,

when each of stress and strain is decomposed into a shear term and a hydrostatic pressure term, and a displacement of the virtual composite material corresponding to the sheer term is defined as “

u_i (i=1, 2 or

     3),”

the nonlinear equation is prepared by defining the displacement u, inside and outside pairs of the inner spherical particles and the outer shell layers with an addition of terms proportional to the zeroth-power, second-power, inverse third-power and fifth-power of a distance from a center of the pairs of the inner spherical particles and the outer shell layers, respectively,

6. The method according to claim 1, wherein the nonlinear equation is solved using the Newton-Raphson method for a solution to reach convergence.

7. A method of computing a volume fraction of a first material component or a second material component in a composite material, the composite material including the first and second material components in a matrix phase, mechanical material constants of the first and the second material components and the matrix phase being known, the method comprising the steps of:

determining an overall material constant of the composite material based on an experimental result;

preparing a nonlinear equation having the volume fraction of the first material component or the second material component as an unknown by defining, as the composite material, a virtual composite material in which the first material component is dispersed in a form of inner spherical particles in the matrix phase and the second material component is in a form of outer shell layers, each of which envelops each of the inner spherical particles as a coating layer of the predetermined thickness; and

computing the volume fraction of the first material component or the second material component by solving the prepared nonlinear equation, and

wherein the nonlinear equation is a recursive nonlinear equation which is obtained by defining, as the predetermined overall material constant of the composite material, a material constant in each of surrounding areas of the outer shell layers enveloping the inner spherical particles in the virtual composite material.

8. The method according to claim 7, wherein the nonlinear equation is prepared by determining a displacement in the virtual composite material under a condition that a surface traction and the displacement are continuous across each of the boundaries between the inner spherical particles and the coating layers, and a condition that a surface traction and the displacement are continuous across each of boundaries between the outer shell layers and the matrix phase.

9. The method according to claim 7,

wherein the nonlinear equation is expressed by “

C*=C^A+v_B·

(C^B−

C^A)·

A^B(C^B, C^C, C*)+v_C·

(C^C−

C^A)·

A^C(C^B, C^C, C*)”

when the material constant of the matrix phase, the material constant of the first material component, the material constant of the second material component, the volume fraction of the first material component, the volume fraction of the second material component and the overall material constant of the virtual composite material are represented as C^A, C^B, C^C, v_B, v_C, and C*, respectively, and,

wherein each of “

A^B(C^B, C^C, C*)” and

“

A^C(C^B, C^C, C*)”

in the equation is obtained by defining a material constant of the outer shell layers enveloping the inner spherical particles in the virtual composite material as the material constant C^C of the second material component and by defining the material constant in each of the surrounding areas of the outer shell layers enveloping the inner spherical particles as the predetermined overall material constant of the composite material, and each of “

A^B(C^B, C^C, C*)” and

“

A^C(C^B, C^C, C*)”

is a proportional constant to be defined by a nonlinear expression having the material constant C* as an unknown.

10. The method according to claim 7, wherein,

when each of stress and strain is decomposed into a shear term and a hydrostatic pressure term, and a displacement of the virtual composite material corresponding to the hydrostatic pressure term is defined as “

u_i (i=1, 2 or

     3),”

the nonlinear equation is prepared by defining;

the displacement u_i inside each of the pairs of the inner spherical particles and the outer shell layers to be proportional to a position “

x_i (i=1, 2 or

     3)”

with an origin at a center of each of the pairs of the inner spherical particles and the outer shell layers; and

the displacement u_i outside each of the pairs of the inner spherical particles and the outer shell layers to have;

a term proportional to the position xi (i=1, 2 or

     3) with the origin at the center of each of the pairs of the inner spherical particles and the outer shell layers; and

a term proportional to the position x_i (i=1, 2 or

     3) with the origin at the center and inversely proportional to the third-power of the distance from the center.

11. The method according to claim 7, wherein,

when each of stress and strain is decomposed into a shear term and a hydrostatic pressure term, and a displacement of the virtual composite material corresponding to the shear term as “

u_i (i=1, 2 or

     3)”

,

the nonlinear equation is prepared by determining the displacement u, inside and outside pairs of the inner spherical particles and the outer shell layers with addition of terms proportional to the zeroth-power, second-power, inverse third-power and fifth-power of a distance from a center of the pairs of the inner spherical particles and the outer shell layers, respectively,

12. The method according to claim 7, wherein the nonlinear equation is solved using the bisection method.

13. A recording medium storing a computer-executable program for computing an overall mechanical material constant of a composite material, the composite material including a first material component and a second material component in a matrix phase, mechanical material constants of the first and second material components and the matrix phase being known,

wherein the program causes a computer to execute the steps of:

preparing a nonlinear equation having the overall material constant of a virtual composite material as an unknown by defining, as a composite material, the virtual composite material in which the first material component is dispersed in a form of inner spherical particles in the matrix phase at a known volume fraction and the second material component is in a form of outer shell layers, each of which envelops each of the inner spherical particles as a coating layer of the predetermined thickness; and

computing the overall material constant of the virtual composite material as the overall material constant of the composite material by solving the prepared nonlinear equation, and

wherein the nonlinear equation is a recursive nonlinear equation which is obtained by defining, as the overall material constant of the virtual composite material to be computed, a material constant in each of surrounding areas of the outer shell layers enveloping the inner spherical particles in the virtual composite material.

14. The recording media according to claim 13, wherein the nonlinear equation is prepared by defining a displacement in the virtual composite material under a condition that the displacement and a surface traction are continuous across each of boundaries between the inner spherical particles and the outer shell layers and a condition that the displacement and a surface traction are continuous across each of boundaries between the outer shell layers and the matrix phase.

15. The recording medium according to claim 13,

wherein the nonlinear equation is expressed by “

C*=C^A+v_B·

(C^B−

C^A)·

A^B(C^B, C^C, C*)+v_C·

(C^C−

C^A)·

A^C(C^B, C^C, C*)”

when the material constant of the matrix phase, the material constant of the first material component, the material constant of the second material component, the volume fraction of the first material component, a volume fraction of the second material component and the overall material constant of the virtual composite material are represented as C^A, C^B, C^C, v_B, v_C, and C*, respectively, and

wherein each of “

A^B(C^B, C^C, C*)” and

“

A^C(C^B, C^C, C*)”

in the equation is obtained by defining a material constant of the outer shell layers enveloping the inner spherical particles in the virtual composite material as the material constant C^C of the second material component and by defining, as the overall material constant of the virtual composite material to be computed, the material constant in each of surrounding areas of the outer shell layers enveloping the inner spherical particles, and each of “

A^B(C^B, C^C, C*)” and

“

A^C(C^B, C^C, C*)”

is a proportional constant determined by a nonlinear expression having the material constant C* as an unknown.

16. A recording medium storing a computer-executable program for computing a volume fraction of a first material component or a volume fraction of a second material component in a composite material, the composite material including the first and second composite material in a matrix phase, mechanical material constants of the first and second material components and the matrix phase being known,

wherein the program causes a computer to execute the steps of:

determining an overall material constant of the composite material based on an experimental result;

preparing a nonlinear equation having the volume fraction of the first material component or the volume fraction of the second material component as an unknown by defining, as the composite material, a virtual composite material in which the first material component is dispersed in a form of inner spherical particles in the matrix phase and the second material component is in a form of outer shell layers, each of which envelops each of the inner spherical particles as a coating layer of the predetermined thickness; and

computing the volume fraction of the first material component or the second material component by solving the prepared nonlinear equation, and

wherein the nonlinear equation is a recursive nonlinear equation obtained by defining, as the predetermined overall material constant of the composite material, a material constant in each of the surrounding areas of the outer shell layers enveloping the inner spherical particles in the virtual composite material.

17. The recording media according to claim 16, wherein the nonlinear equation is prepared by defining a displacement in the virtual composite material under a condition that the displacement and a surface traction are continuous across each of boundaries between the inner spherical particles and the coating layers and a condition that the displacement and a surface traction are continuous across each of boundaries between the outer shell layers and the matrix phase.

18. The recording medium according to claim 16,

wherein the nonlinear equation is expressed by “

C*=C^A+v_B·

(C^B−

C^A)·

A^B(C^B, C^C, C*)+v_C·

(C^C−

C^A)·

A^C(C^B, C^C, C*)”

when the material constant of the matrix phase, the material constant of the first material component, the material constant of the second material component, the volume fraction of the first material component, the volume fraction of the second material component and the overall material constant of the virtual composite material are represented as C^A, C^B, C^C, v_B, v_C, and C*, respectively, and,

wherein each of “

A^B(C^B, C^C, C*)” and

“

A^C(C^B, C^C, C*)”

in the equation is obtained by defining a material constant of the outer shell layers enveloping the inner spherical particles in the virtual composite material as the material constant C^C of the second material component, and by defining the material constant in each of surrounding areas of the outer shell layers enveloping the inner spherical particles as the predetermined overall material constant of the composite material, and each of “

A^B(C^B, C^C, C*)” and

“

A^C(C^B, C^C, C*)”

is a proportional constant determined by a nonlinear expression having the material constant C* as an unknown.