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HOOKE AND TRANSFORMATION EQUATIONS IN COMPOSITES

(STRESS-STRAIN CALCULATIONS IN AN ORTHOTROPIC LAYER)

6.

(tvid : 6.)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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6.1 Our aim in this section is to obtain Hooke’s and transformation equations that help us calculate the stresses and strains in global axes in a composite plate with orthotropic properties. The examination will be carried out for the plane stress condition, and the mechanical properties of the composite layer according to the local axes (E1 , E2 , ν12 , G12 ) will be assumed to be known. Examples will also be solved within the subject.

1-2: local axes, x-y: global axes

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

θ: Fiber orientation angle,

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For an orthotropic layer, the relations between stresses and strains according to the 1-2 (local) axes (i.e. Hooke’s Laws) are rewritten on the side (It was explained in the second topic).

 

[𝑄] stiffness matrix and [𝑆] compliance matrix depend only on the material properties in 1-2 directions and are also seen in the equations on the side. (Theoretical calculations of material properties were explained in topic 3; experimental calculations were explained in topic 5.)

 

or

6.2 Let's Remember Hooke’s Equations in Local Coordinates:

 

 

 

 

 

 

 

 

 

 

 

,

,

,

,

,

,

,

From Equation 2.17;

Note: The [C] matrix mentioned in the 2nd topic is expressed as [Q] here. [C] = [Q]

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

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Hooke equations between stresses and strains according to the x-y global axis set can be expressed as follows:

 

 

 

 

6.3 Deriving Hooke Equations in Global Coordinates :

(6.1)

(6.2)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

Tip-4: Although Hooke equations want to be found in x-y global axes, material properties are known according to 1-2 local axes.

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Strain Transformation Equations:

 

Stress

Transformation Equaitions :

 

Now we continue our calculations to achieve our goal:

(These equations are obtained from static equilibrium and do not depend on material properties. Applies to all material types. It is shown in the Strength of Materials course.)

(6.3a-c)

(6.4a-c)

(a)

(b)

(c)

(a)

(b)

(c)

 

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

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where

In this case, if the stress transformation relations from equations 6.3 are written for directions 1-2:

Now we will write these equations in matrix format::

We can now write equations 6.5 in matrix format as follows:

(6.5a-c)

 

(6.6)

(6.7)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

 

We define a transformation matrix :

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or;

(6.8)

(6.9)

 

 

 

Similarly, the strain transformation equations can be written in the same format.

(6.10a-c)

 

 

(6.11)

(6.12)

or;

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

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From equation 6.11

 

,

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

 

 

 

If we consider Equation 6.8 again;

 

 

 

 

 

 

Reduced Stiffness Matrix :

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When the multiple matrix multiplication process on the previous page is performed, the reduced stiffness matrix and its terms are found as follows:

(6.13)

(6.14a-f)

(a)

(b)

(c)

(d)

(e)

(f)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

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(6.15)

(6.16a-f)

(a)

(b)

(c)

(d)

(e)

(f)

 

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

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6.5 ABSTRACT

Hooke’s Equations

Transformation Equations

Tip-5:

In single layer calculations, alternative solutions that do not use reduced matrices [𝑄 ̅ , 𝑆 ̅ ] should be preferred as much as possible, so that there are not too many operations.

Note:

It is inevitable to use reduced matrices in the calculations of layered composites that will be explained later.

In Local Coordinates :

In Global Coordinates:

For Stresses

For Strains

(6.1)

(6.2)

(6.7)

(6.8)

(6.11)

(6.12)

(2.16a)

(2.16b)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Example 6.1 At a point q with a fiber orientation angle of 45o, the plane stress condition shown in the figure emerges. Calculate the global strains accordingly.

 

σ1

= 30MPa

σ2

= -10MPa

τ12

= -30MPa

ε1

= 2,21x10-4

ε2

= -5,64x10-4

γ12

= -75x10-4

 

 

 

 

 

εx

= 35,7x10-4

εy

= -39,2x10-4

γxy

= 7,85x10-4

 

 

1st Solution Way: Using only [𝑸] 𝒗𝒆 [𝑺] Normal matrices:

First, let's find the local stresses from Equation 6.7:

 

Local strains from Equation 2.16b:

 

 

Global stresses from Equation 6.12:

 

 

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

q

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Global strains are calculated from Equation 6.2::

First, the terms of the Normal [S] matrix are calculated:

 

 

 

,

,

 

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

Similarly :

,

,

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E1

E2

ν12

G12

140 GPa

36 GPa

0,28

14 GPa

Mechanical Material Properties of the Composite Material

Example 6.2

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

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Let's first calculate the Minor Poisson Ratio :

 

 

When we look at the placement directions of strain gauges;

 

 

Local Strains

 

 

If we remember Equation 5.1b:

 

 

 

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

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Local Stresses

Let's calculate the local stress values from Equation 2.16a:

 

 

 

 

From equation 2.7d, the terms of the matrix [Q] are:

 

 

 

Global Stresses

Global strains from Equation 6.12 :

 

Global stresses from Equation 6.8 :

 

Global Strains

 

 

(Using reduced matrices)

or 2nd Solution

(From Equation 6.2 )

(From Equation 6.1 )

Due to the reduced matrices, Solution 2. The path is longer.

c: cos50o, s=sin50o

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)

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6.6 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:

ΔT

P2

P1

P2

P1

P1

P1

P2

P2

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

If we write it in matrix format:

 

 

 

 

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.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

 

 

 

 

 

 

Local strains:

Local stresses:

Global strains:

Global stresses:

(6.17.a)

(6.18.a)

(6.17.b)

(6.18.b)

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

 

 

 

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

9.4.2 Global Thermal expansion coefficients (αx, αy, αxy) :

 

 

From equation 6.12 :

 

 

Recall the transformation matrix from Eq. 6.9:

From eq. 6.17.a

(While there is only ΔT)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(6.19.b)

(6.19.a)

6. Hooke's and Transformatıon Equations In Composıtes (Stress-Strain Calculations In An Orthotropic Layer)