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The Zor Model (or Zor Approach), a new homogenization method for laminated structures, assumes perfect bonding between the layers, similar to the Voigt and CLT approaches.
Under any loading condition, the layers cannot deform independently of one another in any direction.
The model aims to represent the equivalent and overall macroscopic behavior of the entire structure.
In the general case, a layer may exhibit monoclinic behavior with respect to the global x–y coordinate system.
The reciprocity condition is not imposed as an initial assumption. Instead, it emerges naturally from the equivalent properties derived using the laws of static equilibrium and Hooke’s law.
The resulting equations are valid for both symmetric and asymmetric stacking sequences.

9.4.1 Fundamental Characteristics of the Zor Model:

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This condition and Eq. (9.4.1) are directly assumed in the Voigt approach. In the Reuss approach, however, an iso-stress assumption is adopted, and therefore the strains are generally different. In the CLT approach, this condition is satisfied for symmetric laminates, whereas different results may be obtained for asymmetric laminates.

(9.4.1)

9.4.2 Consequences of Perfect Bonding in the Zor Model: Perfect bonding implies that common points between adjacent layers continue to move together after loading and that no separation occurs at the interfaces. As a result of perfect bonding, the following three conditions arise, which form the basis of the governing equations of the Zor Model.

Figure 9.9

A comparison of the different approaches with respect to the interpretation of perfect bonding is presented separately in Section 9.4.9.4.

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Case 2, which represents the lateral interactions between the layers, is a distinctive assumption of the Zor Model and is not included in the Voigt, Reuss, or CLT approaches.

(9.4.2)

Figure 9.10

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9.4.2.3) Case 3: Preservation of the Rectangular Shape of the Layer Plane

Consider a tensile load applied in the x-direction. A layer plane that is rectangular before loading remains rectangular after loading. In other words, lines that are initially perpendicular to each other remain perpendicular after deformation. Therefore, no shear strain develops in either the layers or the equivalent volume.

Case 3 is not included in the Voigt and Reuss approaches. In the CLT approach, it is valid only for symmetric laminates, whereas in the Zor approach (model), it is applied to both symmetric and asymmetric laminates.

Note: These three conditions, which form the foundation of the Zor Model, are equally valid for tensile or compressive loading applied in either one or both directions within the x–y plane.

Since asymmetric laminates may produce different results in the CLT approach, this conclusion may appear unexpected at first glance. However, it should be remembered that the Zor Model is based on the macroscopic behavior of the equivalent volume and therefore follows a different framework from CLT.

(9.4.3)

Figure 9.11

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F_x

Since no external force is applied to the structure in the y-direction : F_y= 0

From static equilibrium,

(9.4.4)

(9.4.5)

Equation (9.4.4) is also present in the Voigt and CLT approaches. In contrast, Eq. (9.4.5) is a distinctive equation of the Zor Model.

Figure 9.12.a

Figure 9.12.b

Figure 9.12.c

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Expressing the force equilibrium equations, Eqs. (9.4.4) and (9.4.5), in terms of stresses gives

(9.4.4)

(9.4.6)

(9.4.7)

Equation (9.4.6) is also present in the Voigt and CLT approaches. In contrast, Eq. (9.4.7) is a distinctive result that emerges from the Zor Model.

Figure 9.13.a

Figure 9.13.b

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9.4.4 Stress Distribution in the Zor Model:

Similarly, from Eq. (9.4.5)

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9.4.5 Application of Hooke’s Relations in the Zor Model

(9.4.8)

(9.4.9)

9.4.5.2) For the Equivalent Volume : The equivalent volume of a structure composed of monoclinic layers also exhibits monoclinic behavior. Therefore, the Hooke relations for the equivalent volume can be written in a similar form as

(9.4.10)

(9.4.11)

Continuous fiber-reinforced layers are orthotropic with respect to their local 1–2 coordinate system, but generally exhibit monoclinic behavior with respect to the global x–y coordinate system.

Figure 9.14

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9.4.6 Calculation of the Equivalent Properties in the Zor Model

Using the equations derived so far, the elastic properties of the equivalent volume representing the mechanical behavior of symmetric or asymmetric laminated structures composed of layers that generally exhibit monoclinic behavior with respect to the global x–y coordinate system will now be determined. In the Zor Model, the order of calculation is important and should follow the sequence presented below.

Substituting Eqs. (9.4.8) and (9.4.10) into Eq. (9.4.1) yields

(9.4.12)

Substituting Eqs. (9.4.9) and (9.4.11) into Eq. (9.4.2) gives

(9.4.13)

Solving Eqs. (9.4.12) and (9.4.13) for the normal stresses in the i-th layer, we obtain

(9.4.14)

(9.4.15)

We consider only the equations derived for the case of tensile loading in the x-direction.

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Substituting Eq. (9.4.15) into Eq. (9.4.7):

(9.4.16)

(9.4.17)

From Eq. (9.4.17):

Let us rewrite Eq. (9.4.18) in terms of these coefficients

(9.4.19)

(9.4.18)

(9.4.22)

(9.4.20)

To observe the Poisson effect more clearly in Eq. (9.4.18), we define layer-specific coefficients for each layer:

(9.4.21)

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Substituting Eq. (9.4.14) into Eq. (9.4.6):

(9.4.23)

(9.4.24)

(9.4.25)

(9.4.26)

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F_x

F_y

..>>

Figure 9.15

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(9.4.26)

(9.4.27)

(9.4.28)

(9.4.29)

(9.4.30)

(9.4.31)

(9.4.32)

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(9.4.33)

As a result,

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9.4.7) Natural Satisfaction of the Reciprocity Condition in the Zor Model

(9.4.34.a)

(9.4.34.b)

(9.4.34.c)

(9.4.34.d)

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Let us express the final equations of the Zor Model in terms of these variables:

Eq.(9.4.18):

Eq.(9.4.26):

Expansion of Eq. (9.4.24):

Expansion of Eq. (9.4.30):

(9.4.35)

(9.4.36)

(9.4.37.a)

(9.4.37.b)

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We know that reciprocity is satisfied for each i-th layer:

Let us express the reciprocity terms for the entire structure (equivalent volume) in terms of these variables:

(9.4.38)

(9.4.39)

(9.4.40)

(9.4.41)

(9.4.42)

Eq. (9.4.42) shows that the reciprocity condition is satisfied for the equivalent volume in the Zor Model.

Eqs. (9.4.40) and (9.4.41) are found to be equal.

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In this section, we will obtain the general stiffness matrix that gives the stress–strain relations in the Zor Model.

(9.4.44)

In this case, the third row of the matrix equation numbered (9.4.43) is:

(9.4.43)

The stress state in the equivalent volume (at the macro level) is:

From Eq. (9.4.3), the shear strain is:

As a necessary consequence of Eq. (9.4.44):

(9.4.45)

(9.4.3)

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Similarly,

(9.4.46)

In this case, the third row of the matrix equation given in Eq. (9.4.43) becomes:

The stress state at the macro level in the equivalent volume is:

Eq. (9.4.3) is also valid for this loading condition:

As a necessary consequence of Eq. (9.4.46):

(9.4.47)

(9.4.3)

Thus, the equivalent compliance matrix of the Zor Model takes the following orthotropic form:

(9.4.48)

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The equivalent stiffness matrix is obtained by taking the inverse of the compliance matrix and again takes an orthotropic form:

(9.4.49)

It is stated in the literature how the terms of the stiffness matrix under plane stress conditions for an orthotropic material can be written in terms of the engineering constants. Accordingly, the equivalent stiffness matrix of the Zor Model can be expressed as follows:

(9.4.50)

As can be seen, the Zor Model enables a structure composed of layers that are generally monoclinic in the global axes to be represented by an orthotropic equivalent volume. This makes other mechanical calculations much easier.

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Example 9.1

θ₁

θ₃

θ₂

8mm

a-) Voigt

The layers are cut from a large continuous fiber-reinforced lamina, and the properties of this lamina with respect to the local 1–2 axes are as follows:

b-) Reuss

c-) CLT

d-) Zor Model

60⁰

0⁰

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Solution::

Step 2) Determination of the Material Properties of the Layers with Respect to the Global x–y Axes:

The cross Poisson’s ratios are calculated from Eq. (9.1):

Since the layers are made of the same material, their properties with respect to the local 1–2 axes are the same as the properties of the large lamina from which they were manufactured.

Properties of the second layer: >>..>>

Step 1) Determination of the Volume Fractions:

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(9.4.51.a)

(9.4.51.b)

(9.4.51.c)

(9.4.51.d)

Material Property Transformation Equations

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The cross Poisson’s ratio is calculated from Eq. (9.1):

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a-) According to the Voigt Approach

Step 3) Calculation of the Equivalent Volume Properties:

From Eq. (9.1.6):

From Eq. (9.1.7):

From Eq. (9.1.13):

From Eq. (9.1.14):

From Eq. (9.1.20):

From Eq. (9.1): Reciprocity Check

Reciprocity condition not satisfied

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b-) According to Reuss approach

From Eq. (9.2.5) :

From Eq. (9.2.6):

From Eq. (9.2.11):

From Eq. (9.2.12):

From Eq. (9.2.18):

From Eq. (9.1):

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Reciprocity condition is satisfied

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c-) According to the CLT Approach

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From Eq. (6.14) ;

where

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c₃) The laminated structure is coded:

8mm

c₄) The [A] matrix is calculated.

Eq. (8.23):

c₅) The [A]^-1 matrix is calculated.

(The result is given directly here.)

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From Eq. (9.3.1) :

From Eq. (9.3.2) :

From Eq. (9.3.3):

From Eq. (9.3.4):

From Eq. (9.3.5):

From (9.1) Reciprocity Check :

Reciprocity condition is satisfied

c₆) Calculation of the Equivalent Properties

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d ) According to the Zor Model

In the Zor Model, the order of calculation is also important and should be followed as given below:

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The Zor Model adopts the Voigt approach for the calculation of the shear modulus.

From Eq. (9.4.33):

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Equivalent Properties	Voigt	Reuss	CLT	Zor Model
E_x (GPa)	87,14	82,69	88,09	87,82
E_y(GPa)	75,51	74,94	77,73	76,79
ν_xy	0,26	0,29	0,26	0,26
ν_yx	0,25	0,26	0,23	0,23
G_xy (GPa)	23,63	22,66	24,31	23,63

0^o/ 60^o / 0^o (Symmetric)

Equivalent Properties	Voigt	Reuss	CLT	Zor Model
E_x (GPa)	82,45	79,12	85,28	82,96
E_y(GPa)	73,00	72,58	76,5	74,02
ν_xy	0,30	0,31	0,27	0,30
ν_yx	0,28	0,29	0,24	0,26
G_xy (GPa)	24,52	23,72	25,97	24,52

15^o/ 60^o / 0^o (Asymmetric)

(calculated in parts (a)–(d))

(Only final results are presented)

Results Tables and Comparisons

From the tables above, it can be seen that the equivalent properties vary depending on the homogenization approach, and that these differences become somewhat more pronounced for asymmetric stacking sequences. In addition, factors such as the use of different materials in the layers and the number of layers are also important parameters that can affect the differences among the results obtained by the various approaches.

Table 9.1.a

Table 9.1.b

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Example 9.2 :

15^o

8mm

60^o

0^o

The equivalent properties obtained for this structure as a result of the calculations were given previously in Table 9.1.b.

Calculate the stress and strain values at the mid-planes of the layers using the Zor Model and compare the results with those obtained from CLT. (The CLT results are given directly in the tables at the end of the solution.)

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Solution:

Equivalent volume properties calculated for the Zor Model from Table 9.1.b:

Stresses developed in the equivalent volume

Strains developed in the equivalent volume

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Global strains in each layer

According to the Zor Model, for plane tension/compression loading conditions, the strains are equal throughout the entire structure and in every layer; the shear strain is zero.

From Eq. (9.4.1):

From Eq. (9.4.2):

From Eq. (9.4.3):

Global properties of each layer:

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Global stresses in the 1st layer:

If the stresses are solved from Eqs. (9.4.8) and (9.4.9):

Eq. (9.4.8)

Eq. (9.4.9)

(9.4.52.a)

(9.4.52.b)

Global stresses in each layer::

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Global stresses of 2nd layer:

Global stresses of 3th layer:

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Layer Strains..>>
Layer Number (i)..>>	1	2	3	1	2	3	1	2	3
Zor	2,87	2,87	2,87	-2,16	-2,16	-2,16	0	0	0
CLT	3,021	2,825	2,629	-2,243	-2,067	-1,890	-0,057	-0,057	0,790

Layer Stresses..>>
Layer Number (i)..>>	1	2	3	1	2	3	1	2	3
Zor	21,62	14,38	26,08	-10,21	-8,25	-13,11	0	0	0
CLT	24,63	14,73	24,04	-11,43	-9,00	-11,28	4,21	-3,71	1,58

The CLT values for the mid-plane of each layer were obtained following the procedures described in Chapter 8 and were entered directly into the tables.

The differences between the Zor and CLT results may vary depending on the stacking sequence, layer material properties, and the applied loads.

While all layers share the same global strains in the Zor Model, the strains in CLT may vary from one layer to another.

Although failure assessment in CLT is performed on a layer-by-layer basis, in the Zor Model the failure assessment is performed on the equivalent volume. This subject will be discussed in greater detail in Chapter 10. In Example 10.1, the CLT and Zor Models are compared separately for a symmetric structure..

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Strength and Failure Analysis

of Laminated Structures Using the Zor Model

Chapter 10

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