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Thermal Loading in Composites

9.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(tvids : 3a last part and 3 )

for Monolayer and Layered Structures )

(Thermal Stress and Deformation Calculations

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9.1 Our aim in this section

is to calculate the deformations and stresses caused by temperature changes in a composite structure with orthotropic properties.

• In the solved examples, problem-specific formulas will also be derived.

• Thermal expansion coefficients will be formulated in terms of known material properties.

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• The subject will be explained first for single layer and then for layered structures.

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):

• : Coefficient of Thermal Expansion ( CTE ). It is a material property defined as the change in unit volume per unit temperature change.
• For isotropic materials there is 1 a.
• For an orthotropic layer, there are a total of 2 coefficients of thermal expansion, in the 1 direction (α₁) and in the 2 direction(α₂)

9.2 Thermal Deformations in Free State

(9.1)

(9.2)

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 :

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9.3.1 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.

9.3 Thermal Stresses in Free State

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

representative volume element

We will examine the representative volume element for calculations.

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• from static equilibrium in direction 2:

,

• from static equilibrium in direction 1:

Although the structure is not limited anywhere, we see that stresses occur in the 1 direction in each of the matrix and fiber. Because the two materials are stuck to each other and their expansion coefficients are different. At the same DT temperature difference, they want to expand at different rates. The matrix wants to extend too much, pulling the fiber (Pf1 force occurs); The fiber wants to elongate less, preventing the matrix from elongating (Pm1 force occurs). In the 2 direction, the fiber and matrix do not interfere with each other. In addition to its own expansion, the matrix also drags (moves) the fiber with it in the 2 direction. Internal forces and therefore stresses in the matrix and fiber become zero in the 2 direction.

(This information will be used in the α₂ calculation)

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

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

9.4.1 Hooke's Relations Including the Temperature Effect

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

ΔT

Let's assume that in a composite layer, in plane stress case, while there are tensile forces in the 1 and 2 directions, we increase the temperature of the layer by the amount ΔT. In this case, we calculate the resulting strain values using the superposition method and Hooke's relations as follows:

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Hooke's equation including temperature for the plane stress case:

If we write it in matrix format:

(9.3.a)

(9.4.a)

For orthotropic or more specifically transversely isotropic materials; In local axes, α₁₂ = 0. In other words, temperature change has no effect on shear deformation and shear strain.

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(9.3.b)

(9.4.b)

Local strains:

Local stresses:

Global strains:

Global stresses:

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The transformation relation is also valid between local and global thermal expansion coefficients. Namely:

(9.5b)

We can write the global strains that occur only due to the temperature difference ΔT as follows:

9.4.2 Global Thermal expansion coefficients (α_x, α_y, α_xy) :

From equation 6.12 :

Recall the transformation matrix from Eq. 6.9:

From eq. 9.3.a

(While there is only ΔT)

(9.5a)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

9. Thermal Loading in Composites

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Eq.. 8.1 :

(9.6)

(9.7)

9.5 Theoretical Calculation of α₁

Unit elongation (strain) in fibers in direction 1:

Stress caused by the pull of the fibers by the matrix.

Elongation per unit length in the fiber due to temperature increase

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. If we apply the Hooke relations in Equation 2.17 for fiber and matrix:

Similarly:

Unit elongation (strain) in direction 1 in the matrix:

Eq. 3.5 :

Eq. 3.3 :

(9.8)

First of all, we will formulate the values of α₁ and α₂ in terms of known material properties and fiber ratio.

(from eq. 3.4)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

9.6 Theoretical Calculation of α₂

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

If we rearrange equations 9.6 and 9.7;

(9.9)

(9.10)

Hooke relations in Equation 2.15 for fiber and matrix;

(a)

(b)

(9.11)

,

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Similarly for the matrix :

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(9.12)

(9.13)

If we use equations 9.12 and 9.13 in equation 3.8:

If this equation is arranged..>>

(3.8)

In that case;

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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:

(9.14)

9. Thermal Loading in Composites

(from eq. 3.14):

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9.7 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 9.8 or 9.14. (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 9.8 and 9.14 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

9. Thermal Loading in Composites

multidirectional continuous fiber reinforced composites

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9.8 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.

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An aluminum plate with an initial temperature of 23 C is placed between two fixed walls as shown in the figure. According to this,

a-) Can the aluminum plate be used at 120 ^oC under these boundary conditions? Calculate.

Example 9.1* ^{(video 3)}

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

9. Thermal Loading in Composites

(* Attention: This example also includes formula inferences regarding thermal loads.)

b-)Can this composite structure be used at 120 ^oC?

c-) When the right wall is removed, to what temperature can the composite structure be heated within its strength limits?

d-) When the right wall is removed, what will be the stress value that will occur in the matrix component of the composite structure at the maximum allowable (safe) temperature?

The same aluminum plate will be unidirectionally reinforced with 25% steel fibers to create a composite structure and this structure will be placed between two fixed walls. Accordingly, for this composite structure, answer the following questions with calculations.

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a-) Can the aluminum plate be used at 120 ^oC ?

P

As a result, when the temperature exceeds 116.1 °C, the aluminum material will flow. This means that Aluminum plate cannot be used at 120 ^oC. (We can make this calculation with our strength information)

= 0

Aluminum can be heated to its yield limit. In the limit case,σ = σ_akma.

According to the superposition principle, we first lift the right wall and increase the temperature, then we apply the reaction force P coming from the wall. We start from the fact that the total extension (δ) is zero.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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b-) Can the plate be used at the same operating temperature (120 ^oC) if it is unidirectionally reinforced with 25% steel fibers?

(9.15)

In an orthotropic layer constrained in direction 1, the temperature difference at any moment is:

This time we will use the same solution as in part a for the orthotropic composite structure. Note that the only thing that changes are the material constants in Hooke's relations.

From the last equality …>>

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Then from the Equation 9.15:

:

The maximum temperature difference that can be applied to an orthotropic layer limited (constrained) in the 1 direction, within the strength limits:

(9.16)

The composite can be used up to this temperature in a constrained condition.

Note: In the constrained state, deformation is completely prevented. For this, it is necessary to have both walls.

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

9. Kompozitlerde Termal Yüklemeler

From equ.4.4..>>

From equ. 4.3..>>

(See: chapter 4)

From eq. 3.4..>>

From equ. 9.16...>>

From equ. 9.8..>>

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c-) When the right wall is removed, to what temperature can the composite structure be heated within its strength limits?

As explained in article 9.3.2, even if the composite is allowed to expand freely, stresses in the 1 direction will occur in the fiber and matrix. In addition to thermal elongation, the fibers also lengthen a little more due to the pull of the matrix itself.

Total elongation in composite

Total elongation in fiber:

Thermal elongation

Elongation caused by the matrix pulling the fiber

=

=

Temperature difference at any instant in free state: :

: Stress caused by the matrix pulling the fibers:

(9.17)

Maximum allowable temperature difference in free state:

(9.18)

From the above equation

For option c of the example we are examining, there is an unconstrained situation since the right wall is removed. Maximum temperature difference from equation 9.18:

(Maximum operating temperature that can be reached within endurance limits when the right wall is removed)

Note: If one or both of the right or left walls are removed, extension is allowed and the Free state is obtained.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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=

Thermal elongation

Shortening caused by fibers preventing the matrix from elongating

: Stress caused by the fibers working to prevent the matrix from elongating

(9.19)

From the above equation, the stress in the matrix for any ∆𝑇 :

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Total elongation in composite

Total elongation in matrix

=

d-) We will calculate the stress in the matrix at the maximum allowable temperature in the unconstrained case:

The maximum temperature difference was found in option c. At this instant the stress in the matrix:

(9.20)

or 2nd way

For free-state thermal loading:

From eq. 3.5:

If you pay attention, at maximum temperature, the yield strength of the matrix (150 MPa) is not exceeded and no damage occurs to the matrix.

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Example 9.2*

A composite layer with a fiber orientation angle of 0^o is placed in a rigid mold. The inner surfaces of the mold and the outer surfaces of the layer are in frictionless contact, and there is no compression between the surfaces. The required material properties of the layer are given in the table below. According to this,

a-) How much can the temperature of this layer be increased within the strength limits? Determine according to the Maximum Stress Criterion.

b-) Check for damage according to Tsai Hill and Modified Tsai-Hill criteria for fiber routing angle θ = 30⁰ and ΔT = 50 ⁰C

(This example includes formula inferences regarding thermal loads.)

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Solution…>>

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Stresses at any temperature difference ΔT :

(9.21)

(9.22)

a-)

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Solution:

There is no limitation in the thickness direction perpendicular to the plane (3 direction). The deformation is free and hence no stress occurs in this direction. Therefore, this problem is a plane problem.

If we first calculate the minor poisson ratio:

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If ΔT is found from equation 9.21 :

Review for direction 1:

(9.23)

If we examine it according to the Maximum Stress criterion:

Then, at the exact moment of damage :

No-damage condition for direction 1 (Eq. 7.1a):

Review for direction 2:

Similarly

Maximum temperature rise for direction 1, in the constrained layer in directions 1 and 2 :

Maximum temperature rise for direction 2, in the constrained layer in directions 1 and 2 :

Eq. 7.1.b:

at the time of damage:

From eq. 9.22 :

(9.24)

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• Since ΔT has no effect on shear stress and shear deformation, shear stress is zero and there is no need to examine inequality 7.1.c.

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When the given values ​​are substituted:

(Temperature difference that will cause yield in direction 1)

Or the condition that no damage occurs in this layer :

(Temperature difference that will cause yield in direction 2)

Therefore, the temperature increase limit value in this layer is :

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

b-) Check for damage according to Tsai Hill and Modified Tsai-Hill criteria for fiber routing angle θ = 30⁰ and ΔT = 50 ⁰C

According to Modified Tsai-Hill

no damage occur

According to for Tsai-Hill

damage occur

Answers:

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9.9 Thermal Loading in Layered (laminated) Composites

(9.25)

We think that a layered composite structure is placed in a cavity that will prevent deformation in both x and y directions, and its temperature is increased by the amount ΔT.

Eq. (9.4.b)

Eq. (8.14)

Eq. (8.18)

Internal forces per unit length:

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(9.26)

Internal Forces per Unit Length due to temperature difference:

Internal Forces Per Unit Length Resulting from Structural Loads:

Total Internal Forces Per Unit Length Resulting from

Temperature Difference + Structural Loads:

(9.28)

(9.29)

(9.27)

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Similarly, let's calculate the total internal moments :

Eq. (8.22)..>>

Internal moments per unit length:

If we substitute Equation (9.25) into Equation 8.22, we get

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(9.30)

Internal Moments per Unit Length due to temperature difference:

Internal Moments Per Unit Length Resulting from Structural Loads:

Total Internal Moments Per Unit Length Resulting from

Temperature Difference + Structural Loads:

(9.32)

(9.33)

(9.31)

(It is zero in symmetrical structures.)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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If Equations 9.29 and 9.33 are combined;

(9.34)

(9.35)

In this case, the strains and deformations of the middle plane are:

Equations 9.34 and 9.35 are general equations for layered composites, including the temperature effect.

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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9.9.1 Steps for calculating Stresses and Strains (including temperature effect):

Other Steps are the same as described in topic 8. Only instead of equation 8.32, the operations start with equation 9.35.

The only difference with the calculations in topic 8 is that instead of structural internal loads, total internal loads (including temperature) will be used.

Now we will try to understand the subject better by solving an example..>>

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Example 9.3

3 of the layers in Example 9.2 are glued on top of each other with a 30⁰ / 0⁰ / 30⁰ fiber arrangement and placed in the same mold. Each layer is 200mm x 200mm x 2mm in size. By tightening the bolts a little, a compression force of -20kN was created on the lateral surfaces. Additionally, the temperature of the system was increased by 50 ⁰C. Calculate the local stresses arising on the lower surface of the middle layer, ignoring all friction.

Solution:

Material properties will be taken from example 9.2.

a-) First of all, if we calculate the structural loads per unit length:

N_x = N_y = -20x10³ N / 200mm = -100N/mm

N_xy = 0 (Because frictions are neglected)

Step 1-) Total internal forces and moments per unit length are calculated.

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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b-) Calculation of internal forces and moments due to temperature difference:

For the 1st and 3rd layers with θ =30⁰

[Q] matrix terms for all layers:

(It was calculated in Example 9.2)

Since θ =0⁰ for the middle layer:

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Let's code the structure..>>

Calculation of global thermal expansion coefficients for each layer :

Eq. 9.5

Eq. 6.9

for the 2nd (middle) layer, θ =0⁰

for the 1st and 3rd layer, θ =30⁰

Values given in Example 9.2 :

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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We will calculate Thermal Stifness Matrices:

From eq. 9.2

Because the structure is symmetrical :

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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From eq. 9.30

Internal Forces

per Unit Length

due to temperature difference:

Internal Moments

per Unit Length

due to temperature difference:

From eq. 9.26

Total Internal Forces Per Unit Length Resulting from

Temperature Difference and Structural Loads:

Total Internal Moments Per Unit Length Resulting from

Temperature Difference and Structural Loads:

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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• Since the structure is symmetrical, it enters the 1st special case, therefore [B] =0

• Does the structure fall into the special situations in article 8.7?

Other Steps:

• The matrix [D] is non-zero. However, since the total internal moments per unit length are zero, the 2nd special case is achieved. For this reason, there is no need to calculate the [D] matrix as it will not enter into the calculations.

Then only matrix [A] will be calculated.

• Since the stresses in the mid-plane are not asked, the structure does not enter the 3rd special case.

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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- We will calculate the strains and curvatures in the midplane from equation 9.35

Or, since [B] = 0 and [D] have no effect in the calculations, the deformations of the middle plane can be reduced to a 3x3 matrix multiplication as follows:

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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- Global strains are calculated:

From eq. 8.13

Since the global strains of all points are independent of z, they are the same and equal to the strains of the middle plane.

- Local strains are calculated:

The local stresses on the lower surface of the middle layer, requested in the question are:

c:cos0^o , s:sin0^o

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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- Local stresses are calculated :

9. Thermal Loading in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor