Following traditions of mechanics of heterogeneous media, the parameter is named partial if it characterizes one phase only

MODEL OF ELASTIC MIXTURES

Note 9.3. Following traditions of mechanics of heterogeneous media, the parameter is named partial if it characterizes one phase only

For the well-posed description of a mixture, three hypotheses also are formulated:

H1. All properties of a mixture should be the mathematical consequence of mixture phases properties.

H2. To describe the motion of a separately taken component, this component should be mentally isolated from the rest of a mixture with the condition that interaction between phases is taken into account.

H3. The motion of a mixture is described by the same balance equations as the motion of separately taken phase.

The basic equations of the theory of mixtures are derived, assuming the laws of mass, momentum, angular momentum, and energy conservation are valid for the mixture.

The first three laws are always written for each component separately. The theories of any mixtures are divided into two parts, depending on the law of energy conservation written either for each component separately or for a mixture as a whole. This division is essential because of its effects.

In the case “separately” the theory used the physical constants of each phase, whereas in the case “as a whole” the phase constants aren’t be used. In the first variant, minus consists in that theoretical constants may differ very markedly from the real one, plus in return consists in that these constants were long ago written in handbooks.

In the second variant, minus consists in that the complete set of physical constants have to be determined from special experiments, plus consists in that the constants will more or less precisely correspond to the concrete composite material.

Writing the basic equations needs an introduction of some classical notions and symbols.

Let the mixture fills the body with a volume

Focus on composite materials

V. The motion of each phase of the mixture will be described relative to the fixed orthogonal Cartesian coordinates. The basic equation is assumed as the equation of energy balance. It is considered as a consequence of previous balance equations (mass, momentum, angular momentum), if the energy scattering isn’t taken into account in the mixture. Since it is usually assumed that the mixture is elastic deformed only, hence it is not taken into account.

The assumption about the elastic character of deformation led to the theory of elastic mixtures. In turn, the assumption about the linearity of deformation and all other processes simplifies the basic system of equations.

In the linear elastic model, the mixture as a thermodynamical system is described by two kinematical parameters: the partial strain tensors

Focus on composite materials

( ) ikD

H and the relative displacements vectors

Focus on composite materials

FOUNDATIONS OF MECHANICS OF MATERIALS PART 1

FOCUS ON COMPOSITE MATERIALS. STRUCTURAL MODEL OF ELASTIC MIXTURES Focus on composite materials

U=U

H Hik^(1), ik^(2),u⁽¹⁾k uk⁽²⁾

. (9.1) From an analysis of internal energy (9.1), a few important formulas can be obtained, which are consequences of some thermodynamic hypotheses

Focus on composite materials

d U=

^wU/wHik^{( )D}

dHik^{( )D w} U/w

u⁽¹⁾k uk⁽²⁾

d u

k⁽¹⁾u⁽²⁾k

_,

ik( )D

V w_U/wHik^{( )}D

, Rk⁽¹⁾ w U/w

uk⁽¹⁾uk⁽²⁾

d u

k⁽¹⁾uk⁽²⁾

_,

U ²

^

²

, 1

1 2D E

ªw

¦

¬ ^{U 0 /}wHik^{( )}DwHnm^{( )}E

º¼H Hik^{( ) ( )}D nmE 2ª 2

w¬ U ^{0 /}wHik^{( )D}w

uk⁽¹⁾uk⁽²⁾

^º¼Hik^{( )D}

u⁽¹⁾k uk⁽²⁾

2ª 2

w¬ U ^{0 /w}

^{uk(1)uk(2) w u(1)mum(2)}

^º¼

^{u(1)k u(2)k}

^u(1)mum(2)

`

_3!^1d3U+"

These formulas permit to write the constitutive equations for a linear anisotropic mixture can be written as

Focus on composite materials

( )( , ) ( ) ( )( , ) (3) ( )( , ) ( 3)

ik^D x t ciklm lm^{D D} x t ciklm lm^G x t

V H H D G . (9.2)

In the linear theory, an interaction between mixture phases is described by three mechanisms.

They are introduced into the linear theory using phenomenological considerations. The interaction force is represented as a sum of two forces, which characterize a change of kinetic and internal energies owing to the phase interaction, respectively From the kinetic energy consideration follows that the interaction of the phases is displayed by the presence of a new additive term in the kinetic energy, which has the form of added mass energy. This mechanism is called the inertial mechanism. The second mechanism consists in the cross in-fluence of one phase strains on stresses of another phase. It is displayed in constitutive equations (9.2). The third mechanism was initially offered for the one-dimensional model.

Here, the interaction force is directly proportional to the relative displacement of phases at each point. This force in the layer composite, along layers of which the shear wave propagates, is the shear force over the boundary between layers. This mechanism is therefore called the shear mechanism. Owing to Bedford’s works the theory with such a type of interaction mechanism is called the Bedford theory.

The mass in the solid mixtures is assumed usually invariable. Therefore, the balance equation for mass is trivial. The balance equations for momentum and momentum of momentum are written for each phase separately.

Finally, the basic system of equations of the linear theory of elastic mixtures is constituted from the coup-led system of six equations of the motion

FOUNDATIONS OF MECHANICS OF MATERIALS PART 1

FOCUS ON COMPOSITE MATERIALS. STRUCTURAL MODEL OF ELASTIC MIXTURES

( ) 2 (1) 2 (1) 2 ( )

( ) (1) (2)

12 2 2 2

( 1)

ik k k k

k k k k

i

u u u

F u u

x t t t

D D

D D

DD

V ^ªE U ^{§ ·º} U

w w w w

« ¨ ¸»

w ¬ © w w ¹¼ w

and six linear constitutive equations (1.58), thus giving six coupled hyperbolic differential equations of the 2^nd order

^{2 (1) 2 (1) 2 ( )}

( ) ( ) (3) ( ) ( ) (1) (2)

, , ( 1) 12 2^k 2^{k k}2

iklm lm i iklm lm i k k k k u u u

c c F u u

t t t

D D G D D D

H H ^ª«¬E U ^§¨©^ww ^ww ^·¸¹^º»¼ UDD^ww .. (9.3) Consider now the simplest case of the isotropic mixture. Then the basic equations (9.3) are some generali-zation of classical Lame equations and have the form of six coupled hyperbolic partial differential equations of the 2^nd order

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( ) ( ) ( )

graddiv 3

u^D u^D u^G

D D D

P 'G O P G 'P G

Focus on composite materials

O P3 3graddivu^{( )G} E

u^{( )G} u^{( )D}

UDD U12u,^{( )}_tt^D U12 ,u^{( )}_tt^G .

G G G G G (9.4)

The next important fragment of this theory is the physical constants.

It seems to be worthy to repeat here the note from Chapter 8 devoted to the composite materials.

Note 9.4. It is considered that the constitutive equations and determination of