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THEORETICAL CALCULATION OF ORTHOTROPIC MATERIAL PROPERTIES

3.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(tvid. 3a and 3b.)

in a Unidirectional and Continuous Fiber Reinforced Composites

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3.1 Our Purpose in This Section

While the elastic mechanical properties (E_f, ν_f, E_m, ν_m) of the isotropic components (matrix and fiber) that make up the composite are known;

It is to theoretically calculate the properties (E₁, E₂, ν₁₂, G₁₂) of a one-way reinforced orthotropic composite layer obtained by combining these. However, it should be noted that these calculations will also be valid for transversely isotropic composites.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

3.2 Local Axes in the Orthotropic Layer

1: Axis parallel to the fiber direction in the layer plane,

2: axis perpendicular to the fiber direction in the layer plane

3: Axis perpendicular to the layer plane and in the thickness direction

3.3 Representative Volume Element in an Orthotropic Layer

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• Calculations will be made on the representative volume element extracted from the unidirectional fiber reinforced composite plate.
• This element should be chosen in such a way that it represents the composite plate at a minimum level.

• Since a representative element is examined at the micro level, this subject is also called «micromechanics of composite materials».
• The mechanical properties of the composite will be determined by elementary strength calculations.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Unidirectional transverse isotropic composite structures (see: 2.11.3): The representative volume element is the same for these composites. This proves that the calculations made and the equations to be derived for composite plates reinforced with unidirectional continuous fibers are also valid for this type of composite.

3.4 Relationship Between Fiber and Matrix Volume Ratios�

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Fiber volumetric ratio

Matrix volumetric ratio

Theoretical Calculation of Composite Density:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(3.1)

(3.2)

Cross-sectional areas with normal in 1 direction :

A₁

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Attention: There are only matrix and fiber volumes in the structure. Also, there is no independent volume called Composite. The material or volume called composite is theoretical and represents the entire structure. (Or we can think that since it is the only orthotropic material, the entire structure is called composite.)

3.5 Theoretical Calculation of E₁ (Modulus of Elasticity in the Fiber Direction 1)

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We consider the representative volume element to which a pulling force P₁ is applied in direction 1. Since the fiber and matrix are completely adhered to each other, They affect each other and extend by the same ΔL in the 1 direction. Since their initial lengths are equal, their unit elongation (ΔL/L) will also be equal.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

From the static equilibrium of the left part of the Ι-Ι section,

(3.3)

(3.4)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Effect of Fiber Ratio on E₁ Value

Additionally, from the equation:

(3.5)

This equation 3.5 will appear in future calculations.

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

the relationship between the stresses in direction 1 is obtained as:

3.6. Theoretical Calculation of E₂ (Modulus of Elasticity Perpendicular to the Fibers)

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(3.6)

When we take Ι-Ι and II-II sections, respectively, in the representative volume element to which the P2 draft force is applied in the 2 direction; The internal forces and cross-sectional areas in the fiber and matrix are equal to those in the composite; Therefore, we can understand from the figures below that the stresses are the same as the stress in the composite.

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Total elongation in direction 2;

(3.9a)

(3.8)

(3.7a-c)

When the Poisson effect is neglected, the strains in the 2nd direction from Hooke's relations are:

or

(3.9b)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Effect of Fiber Ratio on E₂ Value

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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The general expression of Poisson's ratio for a layer showing orthotropic character is:

Major Poisson Ratio:

According to this;

Minor Poisson Ratio:

Transverse Poisson Ratio:

There are three other non-independent Poisson ratios:

There is also a general relationship between Poisson ratios and Elasticity Modules in an orthotropic material, as in equation 3.13:..>>

Notes: 1-) Equations 3.11, 3.12 and 3.13 can be used for all composite types with orthotropic properties.

2-) Poisson ratio cannot be negative except for some special materials. It cannot be greater than 0.5 in isotropic materials.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(3.11)

(3.12a-c)

(a)

(b)

(c)

(3.13)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

3.8 Theoretical Calculation of ν₁₂ (Major Poisson Ratio)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

In the representative volume element subjected to tension in direction 1, the total strain (ΔW) in direction 2 is equal to the sum of the strains in the fiber and matrix.

If we write the Poisson ratios of the matrix and fiber :

, Similarly for fiber..>>

(3.14)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Effect of Fiber Ratio on ν₁₂

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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3.9 Theoretical Calculation of ν₁₂ (Shear Modulus in plane 1-2)

shear strain angle in composite (γ₁₂);

For the whole composite material;

We can also write this relation for fiber and matrix :

Due to static equilbrium, the internal shear forces must be equal. Since the A₂ cross-sectional areas are also equal, the shear stresses in the fiber, matrix and composite are also equal.

or

,

Since the fiber and matrix are isotropic,

(3.15a)

(3.15b)

(3.16a-b)

,

(a)

(b)

(We thought of the entire structure as a single orthotropic material and named it composite.)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

15, Agust 2025

Shear stress (τ₁₂) occurring in 1-2 plane in representative volume element creates different deformation angles (γ_f , γ_m) in fiber and matrix.

Example 3.1

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Find the elastic properties of the Glass fiber-Epoxy composite to be obtained by combining the materials whose E, ν values are given in the table above.

(Fiber Volume Ratio = V_f = 0,3)

E₁ = ? , E₂= ?, G₁₂ = ? , ν₁₂=?

Solution:

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• Calculation of E₁ :

• Calculation of E₂ :

From equation (3.4):..>>

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

From equation (3.9):..>>

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• Calculation of G₁₂:

• Calculation of ν₁₂ (major poisson ratio) :

From equation (3.16a):..>>

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

From equation (3.16b):..>>

From equation (3.15b):..>>

From equation (3.14):..>>

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3.11 Mechanical properties of some fiber materials

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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3.12 Mechanical properties of some matrix materials

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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An aluminum sheet is reinforced unidirectionally with continuous boron fibers. Fibers constitute 36% of the total volume. The properties of the materials are given in the table below. A composite layer is produced by combining these materials. Calculate the following properties of this composite layer: a-) density, b-) Elasticity Modules in the 1 and 2 directions, c-) Poisson ratios in the 1-2 plane (major and minor), d-) Rigidity module in the 1-2 plane.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

d-)

Solution:

a-)

b-)

c-)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

ρ= 2,66 gr/cm³

Örnek 3.2

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3.13 ) Poisson Effects (p):

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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(3.17a)

(3.17b)

The stresses in 2 directions are equal:

(3.18a)

(3.18b)

(3.6)

If we substitute equation 3.6 into equations 3.17:

Strains in the fiber and matrix in the 1st direction: We substitute equation 3.6 into equations 2.17 (for isotropic materials).

Strains in direction 2:

(3.7a)

(3.19a)

(3.19b)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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(3.20)

(3.8)

(3.18b)

(3.18a)

(3.7a)

If the last equation is rearranged :

(3.5)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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(From equa. 3.19a):

(3.3)

(From equa. 3.12b and equa.3.7a):

(3.21)

(3.14)

(3.4 )

(3.13 )

(3.22)

(3.23)

(If equation 3.22 is substituted into 3.21 and rearranged,):

If we substitute equation 3.23 into equation 3.20:

..>>

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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The result after editing the last term is::

Poisson interaction term :

(3.24a)

• When the Poisson effect term (𝑝) is neglected, equations 3.24 and 3.9 will be the same.

• Although "p" can be at negligible levels in single-direction fiber-reinforced composites, "p" occurs at higher values ​​in bidirectionally woven composites and must be taken into account.

(3.25)

or

(3.24b)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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Example 3.3

What difference in E₂ value occurs when you take into account the Poisson's ratio for the composite layer in Example 3.2? Calculate.

Poisson effect term :

From equa. 3.24 :

When Poisson effect is neglected :

was found

The difference is:

Solution:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

If the temperature of the fiber-reinforced orthotropic composite plate is increased by ΔT while it is free, changes in the dimensions of the plate occur. These size changes are called thermal deformations. We calculate them as follows:

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Total Elongations:

Unit Elongatios (Strains):

(3.26)

(3.27)

Let's remember from equation 3.3 that the strains in the 1-direction in the fiber and matrix are the same as the strain of the composite (Tip 1):

No local shear deformation occurs due to ΔT :

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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3.14.1.a If we consider the structure as a single orthotropic material: Free deformation occurs in all directions. For this reason, the total internal forces, and therefore the stresses, that will arise in all axes within the material will be zero.

Now, we consider that the temperature of a unidirectionally reinforced orthotropic composite layer, which is not limited in any part, is increased by ΔT.

3.14.1 Thermal Stresses in Free State

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

representative volume element

We will examine the representative volume element for calculations.

(3.28)

(3.29)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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• from static equilibrium in direction 2:

• from static equilibrium in direction 1:

(Eq. (3.311) will be used in the α₂ calculation)

3.14.1.b If we consider the structure as 2 different isotropic materials (matrix and fiber):

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(3.31)

(3.30)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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Kompozit Malzeme Mekaniği-Ders Notları-Prof.Dr.Mehmet Zor

Eq. 3.26 :

(3.32)

(3.33)

3.14.2 Theoretical Calculation of α₁

We think that the temperature of a composite layer that is unconstrained on any surface (i.e, free layer) is changed by the amount ΔT.

Similarly:

Eq. 3.30 :

Eq. 3.3 :

(3.34)

(from eq. 3.4)

The total strain in the 1 direction caused by ΔT in the fibers. :

The total strain in the 1 direction caused by ΔT in the matrix. :

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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Kompozit Malzeme Mekaniği-Ders Notları-Prof.Dr.Mehmet Zor

3.14.3 Theoretical Calculation of α₂

Again, we consider that the temperature of an unconstrained composite plate is changed by ΔT.

If we substitute equation 3.26 into equations 3.32 and 3.33;

(3.35)

(3.36)

Hooke relations in Equation 2.15 for fiber and matrix;

(a)

(b)

,

(3.26)

(3.3)

(3.32)

(3.33)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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Similarly for the matrix :

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(3.37)

(3.38)

If we use equations 9.12 and 9.13 in equation 3.8:

If this equation is arranged..>>

(3.8)

In that case;

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

When the equations are arranged:

If we open the equation:

Then the equation takes the form:

When last edited:

(3.39)

(from eq. 3.14):

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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3.14.4 How are the theoretical calculations of α₁ and α₂ values of other types of composites made?

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• Particle reinforced, discontinuous fiber (whicker) reinforced, multidirectional continuous fiber reinforced composites are considered isotropic (called quasi-isotropic) and have 1 α value, and this value can only be found experimentally.

• The α₁ and α₂ values of double woven fabric reinforced composites or sandwich composites are equal to each other and can be calculated with one of the equations 3.34 or 3.39. (The same result should come from both equations.)

• Theoretical calculation methods of other mechanical properties (E₁,E₂,ν₁₂,G₁₂) of other types of composites are explained in section 3.10

• Equations 3.34 and 3.39 are used for unidirectional, continuous fiber reinforced composites. Other types of composites;

Discontinuous fiber (whicker)reinforced composite

Particle reinforced composite

Double woven fabric reinforced composite

Sandwich Composite

multidirectional continuous fiber reinforced composites

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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3.14.5 Thermal expansion coefficients of some materials

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

In the next section, we will further reinforce the subject with examples including formula deductions for a free or limited monolayer.

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties