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SANDWİCH COMPOSİTES

11.

(Theoretical Calculations of Material Properties and Strength Limits and Examples)

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Sandwich Composites are composite structures obtained by gluing two more durable plates to the upper and lower surfaces of a core material. The components of sandwich composites (core and reinforcing top and bottom plates) can each be an isotropic material, as well as different composite structures within themselves. The core structure can be a solid block or it can be in forms with some areas emptied, such as honeycomb. For this reason, sandwich composites have many different alternatives.

11.1) Scope and Purpose

The equations will be derived for the most general case, that is, for sandwich composites with all components being orthotropic and having different thicknesses, and more specific cases can be passed from these general equations. For tension/compression or simple shear loading types, considering sandwich composites as a single orthotropic material allows us to produce more practical solutions. However, for different loadings such as bending and torsion, sandwich composites should be considered as layered structures and solutions should be produced.

Our aim in this section is to theoretically calculate the material and strength properties of sandwich composites.

Now let's start to derive the equations that give the theoretical calculations of the material properties for a sandwich structure in the most general case that we will examine:.>>

Figure11.1

(a)

(b)

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11.2) Theoretical Calculation of Modulus of Elasticity in 1 Direction (E1)

Let's apply a tensile force to the structure in direction 1. Let's examine the internal forces and stresses by cutting the structure from a plane such as c.

 

 

 

 

 

Volume Fractions:..>>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m

 

 

 

 

m:isotropic core

 

 

 

 

 

 

(11.1)

 

(11.2)

 

(11.3)

Sum of volumetric ratios :

Figure11.2

(a)

(b)

(c)

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From Hooke's relations:

 

Since the plates and the core volume are completely adhered to each other, their total elongation is equal. Since their lengths in the 1st direction are equal, the unit elongations (strains) in this direction will be equal.

(11.4a-d)

(11.5)

If we substitute equations 11.4a-d and 11.5 into equation 11.2:..>>

(11.6)

 

The modulus of elasticity in the 1st direction for a sandwich composite structure, each of whose components has a different volumetric ratio and is a different orthotropic material:

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(11.7)

* Equations 11.6 and 11.7 can be modified similarly for different special cases.

11.3) Theoretical Calculation of Modulus of Elasticity in 2 Direction (E2)

Since the components of the sandwich structure are completely adhered in the 2nd direction as well as in the 1st direction, it exhibits the same mechanical behavior in both directions. If the same approach and equations obtained for the direction 1 in topic 11.2 are applied for the direction 2 (similar to equation 11.6), the elastic modulus E2 is obtained as in equation 11.7.

 

 

 

(11.8)

The modulus of elasticity in the 2-direction for a sandwich composite structure, each of whose components has a different volumetric ratio and is a different orthotropic material:

11.4) Special case -1 : Sandwich Panels

Figure 11.3

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m

 

 

 

 

 

 

 

 

 

We apply a tensile load in direction 1 to the sandwich composite. There will be a narrowing in the structure in direction 2. Accordingly, the poisson ratio:..>>

 

 

 

m

 

 

 

 

 

The external load P1 will be distributed to the components u, m and a in certain proportions.

From static equilibrium:

 

 

 

(11.9)

(11.10)

(11.11)

 

 

 

 

 

 

 

 

 

 

 

 

m

 

 

 

 

 

Figure 11.4

(a)

(b)

(c)

(d)

(e)

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Let's write equation 11.11 in terms of stresses:

 

 

 

 

 

 

 

 

 

 

m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(11.12)

Since the components are completely bonded, the total elongations and strains in directions 1 and 2 will be equal:

 

 

(11.13)

(11.14)

If we expand equations 11.13 and 11.14 according to Hooke's Relations,

 

 

 

(11.15)

 

 

 

(11.16)

 

 

 

 

 

(11.17)

(11.18)

 

Figure 11.5

(a)

(b)

(c)

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Now we will arrange the equations on the previous page and get the result:

(11.16)

+

 

 

 

 

 

(11.19)

(11.20)

 

 

 

(11.21)

 

(11.22)

 

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(If we substitute equations 11.24 into equation 11.23)

(11.23a-c)

 

 

 

(a)

(b)

(c)

(a)

(b)

(c)

(11.24.a-c)

From equation 3.13 :

 

 

 

(11.25a-c)

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volume of u plate

 

 

 

m

 

 

 

 

 

 

 

 

 

 

Volume fractions:

 

If we rearrange equation 11.12:

Composite volume

volume of m core

volume of a plate

 

 

 

Let's rewrite equation 11.22 :

 

(11.22)

(11.26.b)

 

 

(11.27)

(11.26.a)

Figure 11.6

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If we substitute equation 11.27 into equation 11.19:

 

 

 

 

 

(11.28)

 

If we also use equations 11.24.a and b :

or

(11.29.a)

(11.29.b)

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From equations 11.14 and 11.16:

From equations 11.13 and 11.15 :

 

Substituting equation 11.28:

 

 

 

Let's substitute equations 11.28 and 11.29 into equation 11.9:

(11.30.a)

(11.30.b)

 

 

 

(11.31.a)

 

(11.31.b)

From equation 11.24.a:

 

or

Figure 11.7

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m

 

 

 

 

 

 

 

 

 

 

 

 

 

11.6) Theoretical calculation of shear modulus G12

The shear forces F coming to the entire composite in the 1-2 plane will be distributed to the components in certain proportions. Let's assume the structure is fixed to the ground.

Since the components are completely bonded, the shear angles will be equal:

 

 

m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hooke Relations

 

 

m

 

 

 

 

 

 

 

 

 

Volume Fractions:

 

 

 

 

 

 

 

 

Top view

(11.32)

(11.33)

Figure 11.8

(a)

(b)

(c)

(d)

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11.7) Strength Values ​​of Sandwich Composites

 

 

 

m

 

 

 

 

 

 

 

 

 

 

 

 

m

 

 

 

 

 

 

 

 

 

 

 

 

m

 

 

 

 

 

 

 

 

 

 

 

 

Volume Fractions:

 

 

 

When tensile load P1 is applied to the composite structure, this load will be distributed to the components at certain rates and different stresses will occur in each component in direction 1. Try to understand this situation from the figures below.

 

From Static Equations:

 

(11.34)

At any instant the stress in the composite in the direction 1:

 

 

 

Figure 11.9

(a)

(b)

(c)

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(11.35.a)

(11.35.b)

(11.35.c)

Tensile strength of the composite in direction 1:

Stresses in a and m components at the moment of tensile rupture in direction 1:

Warning: These equations were first derived assuming that the upper u plate would rupture or break. However, it is also possible that the lower "a" plate would rupture first. This can be understood by drawing the adjacent graph correctly at the beginning according to the given values. If it is understood from the drawn graph that the a plate will rupture first, the u and a indices in the upper formulas will only change places. It is also possible that the m component in the middle will rupture first and the situation should be evaluated similarly. However, having the m component break first is contrary to the purpose of the sandwich structure. Because the purpose is to strengthen the m middle core volume.

 

 

 

 

Elasticity modules

 

m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

​

​

 

 

 

 

 

koma

 

 

 

 

 

 

composite

 

Rupture

u

c

Rupture (breaking)

 

Figure 11.10

Figure 11.11

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Compressive Strength in 1 direction:

Sandwich structure shows the same mechanical behavior in direction 2. Therefore, strength limits in direction 2 are as follows:

Tensile Strength in 2 direction:

 

 

Stresses in components a and m at the time of compression damage in direction 1:

Stresses in components a and m at the time of tensile damage in direction 2::

 

 

Stresses in components a and m at the time of compression damage in direction 2:

 

 

Note that the «Warning» explanation on the previous page is also valid for the above strength limit values.

Again, assuming that upper plate u breaks first, other strength limit values ​​are obtained with a similar approach as follows.

(11.36.a)

(11.36.b)

(11.36.c)

(11.37.a)

(11.37.b)

(11.37.c)

(11.38.a)

(11.38.b)

(11.38.c)

11.7.2)

11.7.3)

11.7.4)

 

 

 

Compressive Strength in 2 direction:

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Shear Strength of Sandwich Composite (S):

11.7.5)

 

 

m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Volume Fractions:

 

 

 

As explained in the G12 calculation, the shear forces F in the composite as a whole and in the 1-2 plane are distributed to the components at different rates.

(11.39)

The shear stress in the composite at any instant:

 

m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 11.12

(a)

(b)

(c)

(d)

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(11.40.a)

(11.40.b)

(11.40.c)

Shear strength of the composite in plane 1-2:

Shear stresses in components a and m at the moment of shear failure in plane 1-2

It is possible that component a or m will be damaged before the u plate. This can be understood by drawing the graph correctly at the beginning. In this context, please note that the warning in 11.7.1 will also apply to shear strength.

 

 

 

 

Shear Modules

 

 

 

 

​

​

 

 

 

 

 

koma

 

 

 

 

 

 

composite

 

damage

u

c

Damage (tearing)

 

 

 

 

 

 

 

 

 

 

Figure 11.13

Figure 11.14

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11.8) Calculation Methods According to Loading Types in Sandwich Composites:

  • Sandwich composites are actually layered structures and the layered composite calculations explained in topic 8 for all loading types can also be done for sandwich composites.

​

  • If there are only tensile, compressive loadings or only simple shear loading in the 1-2 plane, then the sandwich structure can be considered as a single orthotropic volume. After the material properties and strength limit values ​​of this single orthotropic volume are calculated with the equations derived in this section 11, stress, strain calculations and strength checks can be carried out. The equations in Chapter 11 are also valid for cases where each of the components has a separate isotropic or orthotropic material and different volumetric ratios.
  • Now we will try to understand the subject better by solving some examples about sandwich composites..>>

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

11- Sandwich Composites