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5

EXPERIMENTAL DETERMINATION OF ORTOTROPİC PROPERTIES

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(tvid: 5a and 5b)

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5.1 Purpose and Scope

Although any property of an isotropic material can only be determined by experimental measurements, the mechanical properties of an orthotropic composite material can be calculated theoretically (while the properties of its components are known) as explained in the 4th topic. However, the effects of factors such as internal material defects that may occur during composite production are ignored in theoretical calculations. In experimental measurements, the effects of these factors are reflected in the results, and therefore these results are much closer to reality. For this reason, as in other materials, the essential thing in composites is to determine the material properties experimentally

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

5. Experimental Determination of Ortotropic Properties

Our aim in this section

is to explain in detail how the mechanical properties (E₁, E₂, E₃, G₁₂ , ν₁₂ ) of a unidirectional, continuous fiber reinforced composite exhibiting orthotropic properties are obtained through experimental measurements.

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5.2 Let's Remember the Experimental Determination of Mechanical Properties in Isotropic Materials:

Thanks to tensile testing in isotropic materials,

Modulus of elasticity (E),

Poisson's ratio (ν),

Plastic zone curve,

Yield ((σ_a) and tensile (σ_T) strengths can be obtained experimentally. (After reaching the tensile strength, the sample elongates very quickly and breaks suddenly. Therefore, the stress σ_r at the moment of rupture does not matter.)

• The tensile sample can be removed from the material in any direction. Because the material is isotropic, the properties are the same in all directions.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• In the same experiment, measurements are taken for more than one moment in the elastic region.
• Measurements that are too different from the others are eliminated and not included in the calculation.
• The average of compatible values obtained from measurements in the same experiment is taken.
• Experiments are performed for different samples and the general average of the averages of all samples is taken.

5. Experimental Determination of Ortotropic Properties

(First, check out 1.11.5.)

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• At least 2 strain gauges should be attached to the tensile test sample.
• A strain gauge measures the unit elongation (ε) in its bonding direction.
• In this way, (ε_x, ε_y ) values are measured with these two strain gauges.
• The shearing strain ( angle γ) cannot be measured with strain gauges.

Extensometers can be used instead of strain gauges in tensile tests. In this case, the ∆𝐿 elongation amounts on the sample are measured for different instants and unit elongations can be obtained for those instants from the formula 𝜀=∆𝐿/𝐿_𝑜. Extonsometers provide significant convenience in this respect. Video extonsometers are also among the commonly used types.

Extensometer

Alternative to strain gauge :

• Since the elastic extension of the device jaws is included in the ∆𝐿 values read from the tensile test device's own indicator, these values mislead us. For this reason, extonmeters that take measurements directly on the sample are used.
• Strain gauges are also used to experimentally find the stresses at a point on the surface of an object under any loading condition.

5.2.1 Straingauges:

These are sensors used to measure strains.

Strain gauges

5. Experimental Determination of Ortotropic Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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While applying the tensile test in the y direction; for different instants such as b, c,… n, the following operations are performed respectively:

6) Yield Strength:

5.2.2 Testing Stages for Isotropic Materials:

5. Experimental Determination of Ortotropic Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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5.2.3 If the 3rd strain gauge is used in the tensile test;

• γ_xy is calculated from equation 5.1a.

• Because the Mohr circle is tangent to the y axis from the left. R= τ₄₅=σ_y / 2

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Strain

Transformation Formula

• Generally θ' = 45^o is chosen

(5.1b)

For Cartesian coordinates (x-y):

For local coordinates (1-2):

(5.1a)

5. Experimental Determination of Ortotropic Properties

Mohr Circle

• The G value found as a result of the tensile test should be equal to or close enough to the value to be obtained from this theoretical formula.

• During the test, G values are found for different moments and averaged.

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

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

3: Axis in the direction of layer thickness

5.3.1 Local Axes:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

In a unidirectional and continuous fiber reinforced composite layer

5.3.2. Global axes: These are the Cartesian (x, y, z) axes that represent the entire structure in layered composites.

5. Experimental Determination of Ortotropic Properties

Each layer of layered composites has its own specific local axes (1,2,3).

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

5.4.1- A sample is removed from the layer in direction 1.

5.4.2-) Strain gauges are glued in direction 1 and 2.

5.4.3-) The sample is connected to the tensile testing machine.

5.4.4-) Tensile Test is Performed, P-ΔL₁ Diagram is measured from the device

Tensile Test is performed in Direction 1. The following steps should be followed in order:

(The sample thickness should be equal to the layer thickness.)

5. Experimental Determination of Ortotropic Properties

step 5.4.5..>>

* P₁ force values read from the tensile device are correct for the sample, but ΔL₁ elongation values read from the device are not correct. Because the elastic extension effect of the device's jaws is also included in the ΔL_1a , ΔL_1b ,… values read from the device, and thus, it would be wrong to use the equation ε₁= ΔL₁ /L₁ . For this reason, strain-gauges are attached to the sample and (ε₁, ε₂) values are measured directly. Or ΔL_1a , ΔL_1b ,.. values should be read accurately on the sample by using extonsometers instead of strain gauges.

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

5.4.5) Stress – Strain Diagram and obtaining E₁, ν₁₂, X^T values from there

For different instant such as a, b,.. n, the following operations are performed respectively

5. Experimental Determination of Ortotropic Properties

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

5.5.1-) A sample is removed from the layer in direction 2.

5.5.2-) Strain gauges are glued in direction 1 and 2.

5.5.3-) The sample is connected to the tensile testing machine.

5.5.4-) Tensile Test is performed, P-ΔL₂ Diagram is measured from the device

Tensile Test is performed in 2 Direction Perpendicular to the Fibers. The following steps should be followed in order:

(The sample thickness should be equal to the layer thickness.)

5. Experimental Determination of Ortotropic Properties

Step 5.5.5..>>

* P₂ force values read from the tensile device are correct for the sample, but ΔL₂ elongation values read from the device are not correct. Because the elastic extension effect of the device's jaws is also included in the ΔL_2a , ΔL_2b ,.. values read from the device, and thus, it would be wrong to use the equation ε₂= ΔL₂ /L₂ . For this reason, strain-gauges are attached to the sample and (ε₁, ε₂) values are measured directly. Or ΔL_2a , ΔL_2b ,.. values should be read accurately on the sample by using extonsometers instead of strain gauges.

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

5.5.5-) Stress – Strain Diagram and obtaining E₂, ν₂₁, Y^T values from there

5. Experimental Determination of Ortotropic Properties

For different instant such as a, b,.. n, the following operations are performed respectively

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

5. Experimental Determination of Ortotropic Properties

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By taking i=1, j=2 in this formula, experimentally obtained E₁, E₂, ν₁₂, ν₂₁ values can be verified.

If the experiments have been done correctly, this equation should also satisfy, even if approximately. If this equation is not satisfied, errors may have been made in the measurements or calculations in the experiments and these should be checked again.

Equation (3.13) valid for an orthotropic composite layer:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

5.6 Verification of Experimental Measurements for Tensile Tests

5. Experimental Determination of Ortotropic Properties

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

5.7 Experimental Determination Methods for Shear Modulus (G₁₂) and Shear Strength (S)

5.7.1.1) A sample is removed from the layer at a 45° angle with the fibers.

5.7.1. Off – Axis Method (G₁₂ is found.):

5.7.1.2) A strain gauge is glued in the x direction.

5.7.1.3) The sample is connected to the tensile testing machine.

5.7.1.4) Tensile Test is performed, P_x -ΔL_x Diagram is measured from the device

(The sample thickness should be equal to the layer thickness.)

(5.2)

In this method, the S value is not calculated. G₁₂ is calculated from the transformation equation number 5.2 above. For this, the other values in equation 5 must have been determined beforehand. We accept that the values of E₁, E₂, ν₁₂ have been found experimentally before. We choose θ = 45^o. Elasticity Modulus (E_x) in the x direction is found by the experimental method whose steps are explained below.

5. Experimental Determination of Ortotropic Properties

As explained in section 5.4, since the jaws of the device also have an effect on the ΔL_x values read from the device, direct measurement is made on the sample using a strain gauge and ε_x values are obtained for different instant.

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

5.7.1.5) Stress – Strain Diagram and obtaining the E_x value from there.

*The explanations (important points) made in articles 5.4 and 5.5 also apply to this test.

5. G₁₂ is drawn from equation 5.2 and calculated

(5.3)

5. Experimental Determination of Ortotropic Properties

For different instant such as a, b,.. n, the following operations are performed respectively:

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

5.7.2.a Torsional Tube Test Method (G₁₂ and S are obtained)

5. By substituting the T and ε₄₅^o values measured for different instants into equation 5.4, the G₁₂ value is determined for each instant and the shear modulus (G₁₂) value of the material is obtained by taking the general average.

(5.4)

Average cross-sectional area:

6. By reading the torsional moment (Tult) at the time of damage from the device, the shear strength of the material is calculated from equation 5.5:

(5.5)

How these formulas are derived is explained on the next page...>>

1. In this method, first of all, a hollow, cylindrical (tube or pipe-shaped) sample with an average radius of "r" and a wall thickness of "t" must be prepared.
2. The sample must be reinforced with continuous fibers along the axis of the cylinder.
3. A strain gauge is placed on the outer surface of the cylinder at an angle of 45^o with the horizontal.
4. The sample is connected to the torsion tester.

5. Experimental Determination of Ortotropic Properties

Torsion Testing Machine

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

5.7.2.b Derivation of the formulas used in the torsion tube method:

I-I section – left part

B

I-I section

(avarage shear stress)

Strain Transformation equation from (5.1.b):

From the Hooke equations in equation (2.17);

(Average cross-sectional area)

5. Experimental Determination of Ortotropic Properties

(shear strength)

Shear Modulus in plane 1-2 :

(5.4)

(5.5)

Total internal moment in the section :

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5. Experimental Determination of Ortotropic Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

4. The apparatus and the sample are connected together to the tensile device and a compressive load P is applied. P compressive loads must be equidistant from the notch axis and in opposite directions. In this case, equal P1 forces will occur between the contact parts of the apparatus to the sample close to the notch, and P2 forces will occur in the distant contact parts.

Σ F_y =0

5.7.3 Iosipescu Method (obtains G12 and S):

1. In this method, a special sample with a V-notch is prepared in the direction of the fibers, as shown in the figure.

3. The sample is connected to the load fixture apparatus consisting of 2 parts. The most important point is that the V notch is right in the middle. Because only in this case, the bending moment in the notch section becomes zero and the shear stress in the notch section is found with the formula 𝜏₁₂=V/A. (V: shear force in section)

2. A strain-gauge is glued to the sample at an angle of 45° with the fiber axis, as shown in the figure.

load fixture apparatus

t:sample thickness

c:notch width

test sample

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5) If the balance of the left part of the a-a section of the sample is examined; It is understood that the cutting force in the middle region of the sample, including the notch section, is V = P.

From equation (2.17);

From equation (5.1.b) :

(5.6)

(5.7)

7. During the compression test performed on the tensile testing machine, P and 𝜀₄₅^o values are read for different instants.
8. For these recorded instants, S values are calculated from equation (5.6) and G₁₂ values are calculated from equation (5.7).
9. These instantaneous values are averaged and the S and G₁₂ values of the material are obtained.
10. Results much closer to reality can be obtained by taking the general average of the tests performed for different samples.

5. Experimental Determination of Ortotropic Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6) Calculations: Since the bending moment in the notch section is M=0, normal stresses will not occur r (σ₁=σ₂=0).

;

Situation at point D in the notch section:

D

D

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Iosipescu Apparatus and Test Setup

5. Experimental Determination of Ortotropic Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Example 5.1: A 2cm wide, 0.4cm thick sample is removed from a newly produced, unidirectional fiber reinforced layer as shown in the figure and is subjected to a tensile test. a and b strain gauges were glued on the sample in the directions shown in the figure. It is clear from the tensile diagram that the material is brittle and linear elastic. The values measured at different moments during the test are as shown in the table. Accordingly, by taking into account only the measurements in the table, determine the possible E₁,E₂,ν₁₂, ν₂₁, G₁₂, X^T, Y^T, S values of the material.

Solution:

Since a tensile test is performed perpendicular to the fibers, that is, in the 2 direction, E₂, ν₂₁ and Y^T values are obtained as a result of the test. (Explained in article 5.5)

a: direction 2 , b: direcition 1

From Equation ( 2.15b ) :

5. Experimental Determination of Ortotropic Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

From Equation ( 3.12.b ) :

rupture

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Measurements (given in the question)

Calculatios

• Measurements 4 are not taken into account as they are too different from the others.

5. Experimental Determination of Ortotropic Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• The tensile value at break is also equal to the breaking strength (Y^T) value.

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Example 5.2

In the experiment to be carried out with the Iosipescu Method, the notch spacing of the sample taken from a brittle, unidirectional continuous fiber reinforced layer is c = 1cm and the sample thickness is t = 1mm. The measurements taken for different instants from this sample during the experiment are given in the table below. Accordingly, determine the stiffness modulus (G₁₂) and shear strength (S) values of this sample.

Instant Measurements

Solution:

(Calculation)

5. Experimental Determination of Ortotropic Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(MPa)

(MPa)

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Example 5.3: With the off-axis experiment, the shear modulus (G₁₂) of a composite material in the 1-2 plane will be determined.

Previously found properties of the material : E₁ =26GPa, E₂ = 6GPa, ν₁₂ = 0.25

Values read during the test:

For this purpose, a tensile test in the x direction was applied to a test sample with a cross section of: A = 100mm², length: L = 10cm, and a fiber orientation of 45^o. Instant values read from the tensile testing machine during the test are given in the table below. Accepting that ΔL total elongation values belong only to the sample; Obtain the G₁₂ value from these measurements.

5. Experimental Determination of Ortotropic Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor