Micromechanics with Mathematica by Seiichi Nomura

Author:Seiichi Nomura [Nomura, Seiichi]
Language: eng
Format: epub
Published: 2016-02-18T03:33:47+00:00

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Micromechanics with Mathematica

The strain field in the matrix is lengthy and part of the expressed is listed as

3.2.6

Exact Solution for Four-Phase Materials

The analysis for a four-phase material where a spherical inclusion is surrounded by two extra

coating layers embedded in an infinite matrix as shown in Figure 3.12 is straightforward following the procedure described earlier. As the solution procedure is spelled out in the previous

subsection, this problem is an exercise of the employed method. As expected, the analytical

solution of the elastic field for such a material is expected to be extremely large, which is

verified by an output from Mathematica. This inevitably poses a fundamental question of how

k

useful an analytical solution is over a numerical solution if the length of the analytical solution

k

is too large, which is only traceable by computer.

Nevertheless, analytical solutions are desirable when used in parametric study that requires

all the relevant parameters to be present. In this subsection, instead of printing the output from

each process, only the logic and the corresponding Mathematica code are shown. The complete

program and its output can be downloaded from the companion web page.

Xij

a 2

X

a

ij

1

( μ 1, λ 1) ( μ 2, λ 2)

( μ 3, λ 3)

( μ , λ )

m

m

Xij

Figure 3.12

Four-phase material

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Inclusions in Infinite Media

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3.2.6.1

Exact Solution for X

The general form of the displacement and traction when a body is subject to X at the far field

can be expressed as

Using the aforementioned expression, the displacement and the traction at each phase can

be expressed as

k

k

where the numbers “1–3” refer to the inclusion and the two layers and “m” refers to the matrix.

The unknown coefficients are denoted as c1[1]–c1[4], c2[1]–c2[4], c3[1]–c3[4], and

cm[1]–cm[4]. From the requirement that ui must remain finite at r = 0, it follows that c1[1]

= 0. From the requirement that 𝜖 → ̂

ij

Xij as r → ∞, it follows that cm[2] = 1. Therefore,

the continuity condition for the displacement and the traction at r = a 1, r = a 2, and r = a 3 are entered as

The displacement, strain, stress, and traction for each phase are now computed from the fol-

lowing code:

k

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Micromechanics with Mathematica

The strain inside the inclusion, 𝜖 1, can be expressed as

ij

k

k

The strains in the second and third layers are expressed as

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Inclusions in Infinite Media

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The strain in the matrix is expressed as

3.2.6.2

Exact Solution for X′ ij

The Mathematica code is shown next but without output in this subsection, as the typical output

is too long. The displacement (Disp1–Dispm) and the traction (Tract1–Tractm) in each of

the four phases are set up first. Here, the numbers 1–3 refer to the material properties of the

inclusion and the surrounding two coating materials, and “m” refers to the matrix. They are

assumed to contain unknown coefficients (c1[i]–cm[i]) for each phase.

k

k

The following code sets up the equations for the unknowns (c1[i]–c4[i]) by applying the con-

tinuity conditions across each phase for the displacement and the traction. As

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