Composite Materials Design and Applications Part 8 pot

ANISOTROPIC ELASTIC MEDIA 9.1 REVIEW OF NOTATIONS9.1.1 Continuum Mechanics We consider the following classical notions and notations of the mechanics ofcontinuous media: State of stress

ANISOTROPIC ELASTIC MEDIA

9.1 REVIEW OF NOTATIONS9.1.1 Continuum Mechanics

We consider the following classical notions and notations of the mechanics ofcontinuous media:

State of stress at a point: This is defined by a second order tensor with

s11; s22; s33; s23; s13; s12

State of strain at a point: This is defined as a second order tensor eeee.The

3 by 3 matrix for this tensor is symmetric It consists of six distinct termsdenoted as eij:

e11; e22; e33; e23; e13; e12

Linear elastic medium:The strains are linear and homogeneous functions

of the stresses The corresponding relations are:

1

Homogeneous medium: In this case, the matrix terms jijklcharacterizingthe elastic behavior of the medium are not point functions They are thesame at all points in the considered medium

1

For example:

e 11 = j 1111 s 11 + j 1112 s 12 + j 1113 s 13 + j 1121 s 21 + j 1122 s 22 + j 1123 s 23 + j 1131 s 31 + j 1132 s 32 + j 1133 s 33

eij = jijk
¥sk

9.1.2 Number of Distinct

jjjjijk Terms

The above stress–strain relation can be written in matrix form as:

distinct coefficients is:

2

Consider two simple stress states:

• State No 1: One single stress, ( sk
) 1 , which causes the strain:

( eij)1= jijk
( sk
)1

• State No 2: One single stress, (spq)2, which causes the strain:

(emn)2 = jmnpq (spq)2

One can write that the work of the stress in state No 1 on the strain in state No 2 is equal

to the work of the stress in state No 2 on the strain in state No 1, as:

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In summary:

The previous stress–strain relation can then be written as:

The matrix above does not have the general symmetry as in the general form

around this inconvenience by doubling the terms e23, e13, e12, introducing the shearstrains:

from which the stress–strain behavior can then be written in a symmetric form as:

(9.2)

9.2 ORTHOTROPIC MATERIALSDefinition: An orthotropic material is a homogeneous linear elastic material having

two planes of symmetry at every point in terms of mechanical properties, these twoplanes being perpendicular to each other

Then one can show that3 the number of independent elastic constants is nine.

The constitutive relation expressed in the so-called “orthotropic” axes, defined

by three axes constructed on the two orthogonal planes and their intersection

line, can be written in the following form, called the engineering notation

because it utilizes the elastic modulus and Poisson ratios:

(9.3)

where

E1, E2, E3 are the longitudinal elastic moduli

G23, G13, G12 are the shear moduli

n12, n13, n23, n21, n31, n32 are the Poisson ratios

In addition, the symmetry of the stress–strain matrix above leads to the followingrelations:

perpendicular to the fibers characterize the transverse direction t.

ELASTIC CONSTANTS OF UNIDIRECTIONAL COMPOSITES

In this chapter we examine a distinct combination of two materials (matrix andfiber), with simple geometry and loading conditions, in order to estimate theelastic properties of the equivalent material, i.e., of the composite

10.1 LONGITUDINAL MODULUS E

f stands for fiber

Hypothesis: The two materials are bonded together More precisely, onemakes the following assumptions:

two materials to be different

and the corresponding state of strains:

Expressing the stresses in terms of the strains for each material yields

then:

(10.2)

Note: Among the real phenomena that are not taken into account in the

Example: Unidirectional layers with 60% fiber volume fraction (V f= 0.60) with

epoxy matrix:

10.2 POISSON COEFFICIENT

Considering again the loading defined in the previous paragraph, the transverse

The strain in the transverse direction can also be written as:

10.3 TRANSVERSE MODULUS Et

Hypothesis: At the interface between the two materials, assume the

following:

Freedom of movement in the l direction allows for different strains in

the two materials:

Freedom of movement in the z direction allows for different strains in

the two materials:

Then, the state of stress created by a load F (see Figure 10.2), can be reduced

for each material to the following:

The strains can be written as:

then:

(10.4)

Remarks:

Due to the above simplifications that allow the possibility for relative sliding

above may not be accurate

One finds in the technical literature many more complex formulae giving

Taking into consideration the load applied (see Figure 10.2),: the modulus

E f that appears in Equation 10.4 is the modulus of elasticity of the fiber in

a direction that is perpendicular to the fiber axis This modulus can bevery different from the modulus along the axis of the fiber, due to the

10.4 SHEAR MODULUS G
t

produced The state of stress, identical for both the matrix and fiber material, can

be written as:

The corresponding strains can be written as:

Using the constitutive equation, one has

then:

Also, from Figure 10.3, one has

which can be rewritten as:

(10.5)3

10.5 THERMOELASTIC PROPERTIES10.5.1 Isotropic Material: Recall

When the influence of temperature is taken into consideration, Hooke’s law forthe case of no temperature influence:

is replaced by the Hooke–Duhamel law:

DT = Change in temperature with respect to a reference temperature at

which the stresses and strains are nil

The coefficient of thermal expansion of the matrix is usually much larger (more

=

e 1 -+EnS n

E

- trace( )IS–

where, taking into account the equality of the strains:

V represents the volume fraction The longitudinal strain can then be written as:

It is also the longitudinal strain that is created only by the effect of temperature:

the above expressions to obtain:

(10.7)

10.5.2.2 Coefficient of Thermal Expansion along the Transverse Direction t

Using the expressions for stresses obtained before, one obtains:

The quantity between the brackets represent the coefficient of thermal

(10.8)

10.5.3 Thermomechanical Behavior of a Unidirectional Layer

Under the combined effect of the stresses and temperature, the global mechanical strains of a unidirectional layer can be obtained using the followingrelation:

=

ELASTIC CONSTANTS OF A PLY

ALONG AN ARBITRARY

DIRECTION

To study the behavior of a laminate made up of many plies with different orientations,

it is necessary to know the behavior of each of the plies in directions that aredifferent from the principal material directions of the ply We propose to determinethe elastic constants for this ply behavior using relatively simple calculations

11.1 COMPLIANCE COEFFICIENTS

an orthotropic material, to write the stress–strain relation in the plane
,t startingfrom Equation 9.3 or 9.5 in the form:

(11.1)

Problem: How can one transform this relation expressed in the coordinates
,t

First recall the following:

In a similar manner, the stresses acting on the surface with the normal are

written in the x,y axes as:

Therefore, the matrix of stresses in the x,y axes is:

in setting:

and observing that the matrix [P] is orthogonal ( t [P] = [P]–1), one has4

This expression can be developed to become

One can also rearrange the equation to be

This [T ] matrix is readily established if one knows the relation that allows one to express the

components of a tensor in one system in terms of the components of the same tensor in another system Here this relation is: sIJ= cosI smn with = cos( ); see Section 13.1.

m

cosJ m

cosI m

=

eg

where after substitution:

When all calculations are performed, one obtains the following constitutive relation,

moduli and Poisson coefficients appear in these relations One can also see the

E t

- 1

2G
t

–

––

11.2 STIFFNESS COEFFICIENTS

When one inverts Equation 11.1 written in the coordinate axes l,t of a ply, one obtains

where the “stiffness” coefficients appear, as opposed to the Equation 11.1 where the “compliance” coefficients appear To simplify the writing, one can denote

(11.6)

An identical procedure can be followed to arrive at the stress–strain relation:

(11.7)

axis z Substituting Equations 11.7 into 11.6, one obtains

which can be rewritten as:

Once the calculations are performed, one obtains the following expressions

for the stiffness coefficients , where c = cosq and s = sinq

(11.8)

sented in Figure 11.2 for a ply characterized by moduli E
and E t with very different

11.3 CASE OF THERMOMECHANICAL LOADING

11.3.1 Compliance Coefficients

Equation 11.1 with Equation 10.9:

Then, upon substituting,

One finds again in the first part of the second term a matrix of compliancecoefficients, the terms of which are described in details in Equation 11.5 Thesecond part of the second term is written as:

Therefore, the thermomechanical relation for a unidirectional layer written in the

axes x.y, different from the
,t coordinates, can be summarized as follows:

11.3.2 Stiffness Coefficients

Inverting Equation 10.9 gives

Following the procedure of Section 11.2, with the same notations, one can write:

where after substitution:

One finds again, in the first part of the second term, the matrix detailed in Equation11.8 The second part of the second term can be developed as follows:

E
nt
E
0

T1¢[ ]

Therefore, the thermomechanical behavior of a unidirectional layer in the

coor-dinate axes x,y can be written in the following form, in terms of the properties

MECHANICAL BEHAVIOR

OF THIN LAMINATED PLATES

practical calculation methods for the laminate was also described We propose here

to justify these methods, meaning to study the behavior of the laminate when it

is subjected to a combination of loadings This study is necessary if one wants

12.1 LAMINATE WITH MIDPLANE SYMMETRY12.1.1 Membrane Behavior

12.1.1.1 Loadings

The laminate is subjected to loadings in its plane The stress resultants are denoted

as N x, N y, T xy=T yx These are the membrane stress resultants.They are defined as:

N x: Stress resultant in the x direction over a unit width along the y direction

2 The problem of buckling of the laminates is not the scope of this chapter See Appendix 2 3

nthplyÂ

nthplyÂ

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inde-pendent of the stacking order of the plies.

distortions This coupling will disappear if the laminate is balanced This

means that apart from the midplane symmetry, there are as many and

nthplyÂ

nthplyÂ

with the x axis an angle -q.5 In effect, the coefficients 13 and 23 are

and, therefore, cancel each other out for the pairs of

and the stress–strain relation for the laminate is reduced to

One can also note that according to Equation 12.4 the terms of the matrix

[A] above can be written as:

the same orientation In case where these proportions have already been fixed—andtherefore their numerical values are known—it becomes possible to calculate the

(%) as the percentages of the pliesalong the different orientations, one has:

¥

k=1stply

nthplyÂ