MECHANICS OF COMPOSİTE MATERIALS

(Last Update: July 16, 2026)

Professor Mehmet Zor / Dokuz Eylul University

Lecture Notes

(Dowload pptx)

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For Communication and Feedback

• Faculty members who want to follow these lecture notes in their own department's course are sufficient to send me an information e-mail. Solutions to the questions marked (*) in the notes are sent to these faculty members, if they wish.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Statement for Faculty Members

mehmet.zor@deu.edu.tr , mehmetzor66@gmail.com

www.en.mehmetzor.com

You can access the following materials related to this course from my website:

General Information:

More comprehensive general information about types, classifications, matrix and fibers of Composite Materials

Turkish exam questions and examples:

Some Exams and Answer Keys Taken in the Department of Mechanical Eng. and examples

Supporting Documents :

Standard Formula Paper used in exams, Literature Review Sample and other documents

Lecture Notes:

You can freely download the Turkish and English pdf and pptx files of this lecture notes.

Note: You can access the same type of materials regarding Statics, Dynamics, Strength and CAE courses on my website.

My personal WebSite:

My Youtube Channel for all turkish course videos: https://www.youtube.com/@mehmetzor

(tvid : turkish video)

Preface

Dear Student and Researcher Friends

Composite materials are used in many different sectors today due to the advantages they provide in terms of strength, lightness and economy. In parallel with technological developments, classical isotropic structural elements can be replaced by alternatives made of composites over time, and this situation is becoming more widespread day by day.

Today, it is possible to encounter such alternative composite elements, parts or equipment in many mechanisms, machines, structures or systems used in industry. R&D and innovation studies based on composite material alternatives are frequently carried out in many universities or private institutions.

For all these reasons, you, as future engineers, can only take an active part in such activities related to composites if you have sufficient knowledge of the basic principles, theories and approaches of composite material mechanics. Otherwise, you will not be able to provide the required level of information, guidance and evaluations expected from you in the design and analysis of a composite alternative structure that is very likely to be encountered in an R&D unit. This will undoubtedly affect your career negatively.

With these lecture notes, we aimed to convey to you the basic issues that an engineer should know in terms of mechanical calculations and measurements of composite materials. There is no doubt that an engineer who thoroughly understands these course notes will gain a privilege and a strong reason for being preferred in terms of basic knowledge and skills.

At the beginning, general information about composites is summarized in these notes, and each topic is associated with course training videos, which are my own explanations and can be accessed on mehmetzor.com. Tips that play a key role in understanding the course topics are specifically stated in each chapter. Lecture notes are updated over time, new topics or examples may be added, and last update dates are specifically stated. You can follow these updates on my website.

I hope my notes will be useful to all students and researchers.

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Mechanics of Composite Materials- Lecture Notes

January 2024

Mehmet Zor

Contents

1. Summary General Information

2. Anisotropic Material Types and Elastic Mechanical Properties (Anistropic Elasticity)

A- LAMINA (Unidirectional continuous fiber reinforced)

3. Theoretical Calculation of Material Properties of Unidirectional Continuous Fiber Composites
4. Theoretical Strength and Efficiency Limits of Composites (Structural Loading Examples)
5. Experimental Determination of Mechanical Properties
6. Hooke and Transformation Equations in Composites (Stress-strain calculations for single layer)
7. Yield and Failure Criteria in Composites

B- LAMINATED COMPOSİTES

8. Classical Lamination Theory (CLT) : Stress-Strain Calculations

9. Homegenization in Laminated Structures (Theoretical Approaches)

9.1 -9.3 Voigt, Reuss and CLT Methods

9.4 Zor Model

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Summary of �General Information� About �Composite Materials

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

You can access more comprehensive and detailed general information in the "General Information" document on en.mehmetzor.com.

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1.1 What is Composite Material?

A new material created by combining at least two different materials at the macro level (in such a way that they do not dissolve in each other) is called composite material.

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The aim is to develop and bring together some features (lightness, strength, flexibility, etc.) that are not available in the components alone.

1-Summary of General Information About Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1.2 Basic Properties of Composite Materials

1. Although composite actually means mixture, it does not consist of soluble and dissolving components.
2. There is no exchange of atoms between the components.
3. Composite components do not chemically affect each other.
4. If the materials dissolve in each other and there is a mixture at the atomic level, such materials are not composites but alloys.
5. If the mixture is at the level of nanometer particles, these types of composites are called nano composites.

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1-Summary of General Information About Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

6. Composites are generally created from a main material called "matrix" and a more durable material called "reinforcement element (fiber)".
7. Of these two groups of materials, the reinforcement material increases the strength and load-bearing ability of the composite material.
8. The matrix material plays a role in preventing crack propagation that may occur during the transition to plastic deformation and delays the rupture of the composite material.

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1-Summary of General Information About Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

9. By placing continuous fibers, especially in directions where strength is important, the composite structure is provided with higher strength in those directions.
10. Since there is no need to place fibers in other directions, the composite structure is both lighter than classical metallic materials, has higher strength in the desired direction, and is more resistant to the same external loads. In fact, this is one of the most important purposes of composite manufacturing.

1.3 Application Areas of Composites

• Composite materials are used in a wide variety of areas thanks to their structure and properties. Since each sector has different needs and expectations, the product flexibility of composite materials appears as an important advantage.
• Composites are used as raw materials in different sectors as well as as auxiliary equipment in manufacturing.

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1-Summary of General Information About Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

The main sectors where composite materials are widely used and the product types used in these sectors are briefly summarized below:

• Space technology,
• Maritime industry,
• In the field of medicine (Manufacture of medical devices),
• Robotics technology,
• Chemical industry,
• Electrical-Electronic technology,
• Musical instruments industry,

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1.3.1 Some Areas Where Composites Are Used:

• Construction and building industry,
• Automotive industry,
• Defense Industry and Aviation Sector,
• Food and Agriculture Sector
• Manufacturing of Sports Equipment (high jump poles, tennis rackets, surfing, racing boats, skis, etc.).

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1-Summary of General Information About Composites

1.3.2 Composite Product Examples

• Building Industry

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• Exterior and interior coverings
• Decorative Applications
• Roofing Sheets
• Carrier profiles
• Rainwater Transport Systems
• Concrete molds
• prefabricated buildings
• Bridges, Water tanks,
• Food Section Coverings

• Automobile bodies
• Bus side panels
• Truck trailer side slats
• Truck dashboards
• Container manufacturing
• Highway signs
• Highway edge posts
• Pedals, Rear View Mirrors
• Air Intake Manifold

• Automotive industry

• Defense industry

• Aircraft and helicopter body parts,
• Airplane nose and wing parts
• Mortar hulls and chests
• Bulletproof panel manufacturing
• Helmets
• Mine and assault boat parts and hulls
• Shelters etc...

• Food and Agriculture Sector

• Silos
• food storage tanks
• brine tanks
• Aquaculture equipment
• Greenhouses, Grain warehouses
• Irrigation channels etc...

• Maritime industry

• Sailboats
• Motor boats,
• Lifeboats,
• Buoys,
• Pontoons-piers,
• Marine motorcycle,
• Canoes,
• Surfboards,
• Marina equipment etc...

• Chemical Industry

• Pipes for various purposes
• Chemical plant floor grates
• Acid tanks and coatings
• Purification equipment
• Industrial platform and railings
• Ventilation ducts etc...

• Energy Sector

• Insulators,
• Antennas,
• Circuit Breakers,
• Fuse-panel Boxes,
• Lighting Bodies,
• Insulated Platforms,
• Lighting Poles,
• Circuit Breaker Boxes,
• Cable Carriers,

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1-Summary of General Information About Composites

Various Composite Products

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1-Summary of General Information About Composites

Concrete columns are created by combining iron and concrete and are actually a composite structure.

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1-Summary of General Information About Composites

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1-Summary of General Information About Composites

Composites used in an aircraft fuselage and their proportions

1.4 Advantages of Composite Materials.

Composite materials, which have many advantages over other materials with their characteristic features, are preferred due to their many superior properties such as,

1- long life,

2- lightness,

3-high chemical and mechanical resistance.

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1-Summary of General Information About Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Other Advantages Achievable with Composites:

4. High Dielectric Resistance
5. Sealing
6. Color variety
7. U.V. resistance to rays

9. Repairability
10. Machinability
11. Ease of Assembly, Design and Molding
12. Can be applied to concrete, metal and wooden surfaces

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When the material properties in the table below are examined, it is seen that composite structures are both much lighter and much more durable than classical metals.

1-Summary of General Information About Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Isotropic Materials

Composites

1.5 Some Disadvantages of Composites

1. The properties of the manufactured composite may not always be ideal. The quality of the material depends on the quality of the production method, there is no standardized quality.

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1-Summary of General Information About Composites

Delaminaiton

2. Since laminated composites are sensitive to interlayer shear stresses, delaminations (separation between layers) may occur.
3. Since some composites are brittle, they are easily damaged, and their repair may create new problems.
4. They need to be cleaned very well and dried hot before they can be repaired. Some drying techniques can take a long time and be difficult.

1.6 Composites and Engineering Activities

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1-Summary of General Information About Composites

• As can be seen, composite materials can be used in many different sectors, they are constantly being developed and their usage rate is increasing day by day.
• The probability of encountering composites in any business is quite high.
• R&D activities for composite materials are frequently carried out not only in composite producing companies, but also in other companies depending on the usage situation.
• In these activities, solutions and development alternatives with composite materials are possible.

• For this reason, it is an important privilege and reason for preference for engineers to be able to carry out activities such as design, analysis, theoretical calculations, experimental measurements, developments and manufacturing for composite materials that they can do for isotropic (metal, ceramic, etc.) materials. This is only possible with a good basic knowledge of composite material mechanics.

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The fact that the mechanical behavior of composites is different from isotropic materials and can vary depending on the direction requires that they be examined with different mechanical approaches and criteria.

Sir, let's make the shaft material composite. Let's look at von-mises stresses again.

If we make the shaft composite, it would be more accurate to evaluate it according to the Tsai-Hill criterion, not Von-Mises. Yield-fracture criteria for composites are different.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1-Summary of General Information About Composites

The options for creating composite materials are almost endless. Therefore, they are very difficult to classify.

However, common classifications will be emphasized here.

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1.7 Classification of Composite Materials

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1-Summary of General Information About Composites

1.7.1 Classification According to Matrix Material Type :

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According to the type of matrix material, composites can be divided into 3 groups:

Classification Scheme by Matrix Material Type

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1-Summary of General Information About Composites

1a) Thermosets: They are thermosetting plastics. One of the most well-known is Epoxy.

1b) Thermoplastics: Plastics that soften with heat.

1c) Elastomers: These are plastics that can stretch a lot (show large elastic deformation).The most well-known elastomer is rubber.

3. Ceramic Matrix Composites: They are high temperature composites and ceramic materials are used as matrix.

1. Plastic (Polymer) Matrix Composites

2. Metal Matrix Composites: These are composites in which light metals such as Aluminum and Zinc are used as matrix.

Since approximately 90% of composites are produced from polymer (plastic) based matrices, composite materials are also called reinforced plastics. They are divided into 3 groups.

1.7.2 Classification According to Shape and Placement of Reinforcement Elements :

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1. Particulate Reinforced Composite
2. Discontinous Fibers or Whiskers Reinforced Composites
3. Continous Fibers Reinforced Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1-Summary of General Information About Composites

Since the composites in the first two groups are isotropic at the macro level, they are also called Quasi-Isotropic composites. , Group 3 composites show orthotropic properties.

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4 – Laminated (or Layered) Composites:

• Composite layers (lamina) are obtained by placing more durable rod or woven fiber (rod or woven) materials into the matrix material.
• Then, multiple layers are bonded on top of each other and a laminated composite plate is obtained.
• In these layered structures, the orientation angle (θ) of the fibers may differ from layer to layer.
• The orientation angle of the fibers is selected in directions where the stresses will be higher under operating conditions, and in this case, both lightness and strength are achieved at the same time.

1.7.2 Classification According to Shape and Placement of Reinforcement Elements -Continue:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1-Summary of General Information About Composites

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Sandwich composite structures fall into the laminated composite material class. Sandwich structures are obtained by gluing higher strength plates to the upper and lower surfaces of a low-density core material that does not carry load and has only insulation properties.

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5- Sandwich Composites

The bottom and top layers can each be an isotropic material or a fiber-reinforced layer.

Sandwich composite panels used in exterior cladding

6- Hibrid Composites

• It is possible to have two or more fiber types in the same composite structure.
• These types of composites are called hybrid composites.
• This field is very suitable for the development of new types of composites.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1-Summary of General Information About Composites

1.7.2 Classification According to Shape and Placement of Reinforcement Elements -Continue:

7- Natural Composites

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1-Summary of General Information About Composites

1.7.2 Classification According to Shape and Placement of Reinforcement Elements -Continue:

Composite structure of the tree

Composite structure of bone

Materials such as wood and bone are natural composite materials.

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1.8.1 Epoxy:

is an adhesive chemical resin from the thermoset group.

It is produced by polymerization of the epoxide group and its properties can be changed with different formulas. Depending on the type of hardener used, the properties of the composite material vary.

1.8.1.1 Some Superior Features of Epoxy :

1. Its resistance to water, acid, oil and chemicals is very good and does not lose its resistance over time.
2. Epoxies, which are generally two-component, change from liquid to solid after a certain period of time.

4. It has excellent mechanical durability. It can withstand temperatures up to 140^oC when wet and 220 ^oC when dry.

5. It creates surfaces that are resistant to friction and wear.

3. Low shrinkage occurs during hardening.

1. Their costs are high.
2. They are harmful to the skin.

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1.8.1.2 Some Disadvantages of Epoxy

1.8 Some Important Materials Used in Composites

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1-Summary of General Information About Composites

1.8.2 Glass Fiber:

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1-Summary of General Information About Composites

1. Glass fiber is produced from materials such as silica, colemanite, aluminum oxide and soda.
2. It is the most commonly used fiber type in the production of fiber reinforced composites.

3. Glass fiber is produced by passing molten glass under pressure through a specially designed furnace with small holes at its base.

4. There are types with different properties: A, C, E, S and R glass.

1.8.3 Carbon Fiber– Carbon Matrix (Carbon/carbon)

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1. Composites made of carbon matrix and carbon fiber can withstand up to 4000^oC.
2. These composites have very good thermal and mechanical properties at high temperatures.
3. Carbon fiber is lighter and has better mechanical properties than glass fiber. However, production costs are high.

1. The type of fiber that is stronger and also more expensive than carbon fiber is boron fiber.
2. Boron is the second lightest element that is solid at room temperature. It is manufactured by coating boron on a thin wire called the core (usually Tungsten/Wolfram). Therefore, boron fiber is a composite in itself.

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1.8.4 Aramid Fiber (Kevlar)

Kevlar (Aramid) is a polymer fiber and a lightweight reinforcement material that provides high strength and rigidity to the composite structure. Aramid is aromatic polyamide, a type of nylon.

1. Low density,
2. High strength and fatigue resistance,
3. High impact resistance and wear resistance,
4. High chemical resistance,
5. E-Compressive strength close to glass fiber,
6. Kevlar fiber composites are 35% lighter than glass fiber composites.

1.8.4.1 Some Outstanding Features

1.8.5 Boron Fiber :

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1.9.1 Mechanical and Thermal Properties of Thermoplastic Resins

1.9 Some Material Properties:

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1-Summary of General Information About Composites

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1.9.2 Some Properties of Reinforcement Elements (Fibers)

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1.9.3 Tensile Curves of Fiber Materials

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1-Summary of General Information About Composites

1. Hand Lay-Up
2. Spray-Up
3. Wet Filament Winding
4. Resin Transfer Molding (RTM)
5. Pultrusion
6. Compression molding
1. Sheet Moulding Composites (SMC)
2. Bulk Moulding Composites (BMC)
7. Vakum Bonding / Vakum Bagging
8. Autoclave bonding

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1.10.1 Major Composite Manufacturing Methods

Hand Lay-Up

Spray - up

Wet Filament Winding

RTM

Pultrusion

1.10 Manufacturing Technologies in Composites :

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1.10.2 General Features of Manufacturing Methods:

Compression Molding

Vakum Bonding / Vakum Bagging

1. Manufacturing methods of the composite may vary depending on the reinforcement and matrix material, part shape, and the properties targeted from the composite.
2. Raw materials, mold, heat and pressure are generally needed to produce a part.
3. During manufacturing, care should be taken to ensure
1. that the fiber has an evenly spaced and homogeneous distribution,
2. that the fiber materials are thoroughly wetted by the matrix since they are sensitive to mechanical contact,
3. and that a strong interface is created between the fiber and the matrix.

4. In addition to the resin and reinforcement material used, the manufacturing method also plays an important role in determining the final properties of a composite structure.

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1-Summary of General Information About Composites

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1.10.3 Grouping of Manufacturing Methods According to Composite Material Type:

Manufacturing Methods

Thermoset Materials:

Thermoplastic Materials:

Short fiber Composites

Continius fiber Composites

Short fiber Composites

Continius fiber Composites

1. SMC molding
2. SRIM
3. BMC molding
4. Spray -Up
5. Injection molding

1. Flament Winding
2. Pultrusion
3. Resin Trans. Molding
4. Hand Lay-Up
5. Autoclave
6. SCRIMP,RIFT,VARTM..

1. Injection Molding
2. Blow Molding

1. Thermal Shaping
2. Tape Wrapping
3. Press Molding
4. Autoclave

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1.11 Let's remember the important concepts in strength :

1.11.1 Isotropic Material:

1. They are materials that show the same mechanical behavior in all directions and directions against thermal or mechanical loading.
2. They show a homogeneous grain distribution in terms of internal structure distribution. Like all pure metals , alloys are isotropic materials.

3. In addition, particles whiskers or discontinuous fiber reinforced composites are considered isotropic materials at the macro level if they have a homogeneous distribution . Nano-composites also fall into this class. All mechanical calculations valid for isotropic materials also apply to them. ( They are also called quisa-isotropic materials because they contain more than one material . )

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1-Summary of General Information About Composites / important concepts in stregth

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1.11.2 Separation Principle and Concept of Stress:

If a system subject to the influence of external forces is in balance, each of its parts, which we have separated on an imaginary basis, is also in balance separately. This is called the separation principle.

• In addition to external forces, internal forces and internal moments are also applied to each part that we separate imaginary from the separation part. Internal and external forces provide static balance in each part.
• Internal forces and internal moments: These are the reactions that occur on the separation surface. It is the response of the system to external forces in that part.
• Stress: It can be defined as the force per unit area resulting from internal force and moment reactions.
• Stress is not applied, it occurs inside the object under the influence of external loads.

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1.11.3 Stress Types

1.11.3.1 Normal Stress : If the stress is parallel to the normal of the section plane, in other words, if it is perpendicular to the plane, it is called normal stress. It is denoted by σ . It occurs in tensile-compression and Bending loading.

1.11.3.2 Shear Stress : If the stress is perpendicular to the plane normal, in other words, if it is parallel to the plane, it is called shear stress. It is denoted by τ .

Meaning of shear stress indices:

plane normal

Stress Direction

τ_{i j} = τ_{j i}

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1.11.4 Deformation

• The most important concept in deformation is the magnitude we call strain.

• In its simplest definition, it is the amount of extension (or shortening) per unit length of an object.

• The subscript e also indicates its direction.( could be ε _{x ,} ε _y or ε _z ).

1.11.4.1 Unit Elongation : Strain (ε)

1.11.4.2 Shearing Strain: Angle (γ):

• At a point, in the most general case, there may be 3 different shear stresses (τ_xy , τ_yz , τ_xz )
• These stresses cause shear strain angles (γ_xy , γ_yz , γ_xz )

• It is the deformation angle caused by shear stress at a point. The angle γ is in the same plane as the shear stress and has the same indices. Try to understand this angle by carefully examining the figures below .

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1-Summary of General Information About Composites / important concepts in stregth

1.11.5 Tensile test and stress-strain diagram:

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that best characterizes the mechanical behavior of materials .

The part that connects to the chin

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Poisson's ratio:

Poisson's ratio: Varies between 0 and 0.5

Attention : Formula 1.1 is valid only in case of x-axis (uniaxial) loading . If there were forces in the y or z directions as well as P, this formula could not be used.

(Hooke's equation for uniaxial loading .)

Remember: Always,

Strain= Total elongation / initial length in that direction

If the rod in the figure is subjected to P load in the x direction, stress occurs in only x direction and no stress occurs in the y and z directions.

,

,

Poisson's ratio is the material property that gives the deformation effect of a load in other directions .

1.11.5.1 Poisson Ratio ( ν ):

(1.1 )

(1.2 )

(1.3ac )

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1.11.5.2 Modulus of Rigidity ( or Shear modulus) G

with The elastic relationship

(1.7 )

(1.4 )

(1.5 )

(1.6 )

Hooke’s equations between shear stresses and strains in other planes :

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

1-Summary of General Information About Composites / important concepts in stregth

1.11.6 Hooke’s Equations in the Most General Case:

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Temperature Effect

Shearing Strain

isotropic materials, these are the relations between stress-strain at any point Q:

(1.8.a )

(1.8.b )

(1.8.c )

(1.8.d )

(1.8.e)

(1.8.f )

Axial Strain

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ANISOTROPIC MATERIAL TYPES

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2.

(tvid - 2)

(tvid: turkish course video number)

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2- Anisotropic Material Types

2.1 What is an Anisotropic Material?

• Materials whose properties vary with direction are called anisotropic materials.
• Anisotropic materials show different mechanical behavior (stressing, deformation, etc.) depending on the direction, against thermal or mechanical loading.
• Composite materials are anisotropic materials and are more specialized versions of them.
• In order to understand the mechanics of composites, it is important to first know the more general mechanics of anisotropic materials.
• In fact, isotropic materials are the most special form of anisotropic material.

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2- Anisotropic Material Types

2.2 Our Aims in This Chapter:

1. To explain the types of anisotropic materials, starting from the most general to the specific,
2. for each type, it is to derive the stress-strain relations (Hooke’s equations) in the region where the loading is elastic and
3. the material constants that provide these equations.

In article 1.11.6, it was shown that there are 2 independent material constants (E and ν) in the elastic region for isotropic materials and the stress-strain relations (Hooke’s equations) were summarized.

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Stiffness matrix: [C]

(9x9= contains 81 independent material constants)

2.3 Elastic Material Constants for Most Generally Anisotropic Materials

1st subscript: indicates the plane normal.

2nd subscript: shows the direction of stress or deformation.

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2- Anisotropic Material Types

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Compliance Matrix: [S]

It is the inverse of the stiffness matrix.

[S]=[C]^-1

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Elastic Material Constants for Most Generally Anisotropic Materials

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If the same analysis is made for planes 1-3 and 2-3:

it is found as

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Elastic Material Constants for Most Generally Anisotropic Materials

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Abbreviated Notations

From this last equation it is found as

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In this case, in an anisotropic material, the independent elastic material constants in the stiffness or compliance matrices are reduced to 6x6 = 36.

Thus, using shortened notations for an anisotropic object, stress-strain or strain-stress relations can be expressed in 2 different matrix formats as follows:

Or with index notation:

• (i , j , k , l = 1, 2, 3,…,6)

or

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types / Elastic Material Constants for Most Generally Anisotropic Materials

​

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The energy density stored in a nonlinear elastic material is dW,

In this case, the energy density can also be written as:

Stress-strain relationship can be writen as:

(I)

(II)

Since equations (I) and (II) are equal;

In this case, the stiffness (C) and compliance ( S ) matrices must be symmetric about the diagonal.

The independent, elastic material constants in these matrices are 21. In terms of material properties, such an anisotropic material that does not have planes of symmetry is called a triclinic material.

2.5 Triclinic Material

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types

The reciprocity condition holds for all elastic materials for which Hooke’s law is valid. Namely:

Finaly,

Reciprocity condition:

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In the figure, monoclinic material that is symmetrical with respect to the (x-y) plane is symbolized. (Note that each little cube is symmetrical with respect to this plane.)

Potassium Feldspar

(It is an example of a monoclinic material found in nature.)

• Materials that, in addition to showing triclinic material properties, also have one plane of symmetry at the crystal or larger macro level are called monoclinic materials.
• Physical material properties are symmetrical with respect to this plane.
• In homogeneous structures, these symmetry planes at all points are parallel to each other.
• An example of these is Feldspar, which is the raw material of ceramic materials.
• Layered (laminated)composite materials can have planes of symmetry or planes of symmetry parallel to each other at much larger sizes than the crystal level (at the macro level that can be seen with the eye).

2.6 Monoclinic Material

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types

Symbolic form of monoclinic structure

plane of symmetry

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We think that there is material symmetry with respect to the 1-2 plane at some point in an anisotropic, elastic and monoclinic structure.

Now we will determine the material constant number of the monoclinic material:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Monoclinic Material

Similarly; In the following cases, there will be no distortion in the 1-3 and 2-3 directions:

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In this case, for a monoclinic and elastic material, the independent material constants in the compliance and stiffness matrices are reduced to 13.

,

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Monoclinic Material

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Below is shown the explicit expression of each term of the [S] compliance matrix for a monoclinic material.

(2.4)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Monoclinic Material

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2.6.1. Hooke’s Equations for Monoclinic Material – (3D Case)

(2.5)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(2.6.a-f)

(a)

(b)

(c)

(e)

(f)

(d)

Explicit versions of the stress-strain (Hooke) equations in 3D for a material showing monoclinic behavior:

2- Anisotropic Material Types/ Monoclinic Material

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2.7 Orthotropic Material

• In other words, in the orthotropic material type, there are 2 more symmetry planes in addition to the monoclinic material.

• At each point of the orthotropic material, there is symmetry with respect to all three planes 1-2, 2-3 and 1-3.

• Continuous fiber reinforced composite layers show orthotropic properties.

• In addition to the triclinic material feature, materials that have three planes of symmetry at the crystal or larger macro level are called orthotropic materials.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Ortotropic Material

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2- Anisotropic Material Types/ Ortotropic Material

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2- Anisotropic Material Types/ Ortotropic Material

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Compliance [S] matrix for orthotropic material:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

In 2D Case:

In 3D Case

As a result, there are 9 independent material constants for an orthotropic material:

(2.7a)

(2.7b)

2- Anisotropic Material Types/ Ortotropic Material

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Stiffness [C] matrix for orthotropic material :

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

In 2D Case

In 3D Case

Attention: The [C] matrix is symbolized as [Q] for orthotropic materials.

(2.7c)

(2.7d)

2- Anisotropic Material Types/ Ortotropic Material

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2.7.1 Industrial examples for orthotropic composites.

a-) Unidirectional continuous fiber reinforced composite layers: These are structures formed by arranging fibers, each of which is a single wire (without discontinuity), in a matrix in a way that they are parallel to each other in one direction.

(a)

(b)

Since the following composite types are symmetrical with respect to three planes, they exhibit orthotropic character and have widespread applications in industry.

b-) Bidirectional woven fabric composite layers: : These are structures formed by arranging fibers, each of which is a single wire (without discontinuity), in a matrix in a way that they are parallel and perpendicular to each other in two directions.

c-) Sandwich Composites: These are composite structures obtained by gluing two more durable plates to the upper and lower surfaces of a core material.

(c)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Ortotropic Material

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Example 2.1: The plane stress state at a point of the unidirectionally reinforced Graphite/Epoxy Composite is shown in the figure.

1. The Compliance [S] matrix

​

• The Minor Poisson’s ratio

​

• The Stiffness [C ] matrix

​

• The strain values in directions 1 and 2.

Mechanical Properties of a Unidirectional Layer for Different Materials

Accordingly, find the following values:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Ortotropic Material

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a-)

Solution:

From the table on the previous page, the material properties for graphite/epoxy read as follows:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Ortotropic Material

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b-)

c-)

d-)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Ortotropic Material

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2.7.1 Hooke’s Equations for Orthotropic Material – (3D Case)

(2.8)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(2.9.a-f)

(a)

(b)

(c)

(e)

(f)

(d)

Stress-strain relations (Hooke’s equations) in 3D for a material showing orthotropic behavior.(valid for elastic loading)

2- Anisotropic Material Types/ Ortotropic Material

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If there are isotropy planes parallel to each other at every point of the orthotropic material, these materials are called transversely isotropic materials.

Although the material properties are the same in all directions on that plane, they differ in the direction perpendicular to the plane.

The structures obtained by placing the fibers parallel to each other and irregularly in the matrix show transverse isotropic character. The properties are almost the same in every direction in the 2-3 plane. The properties in the 1 direction are different.

2.8 Transversely Isotropic Material

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

An Isotropy plane

There are infinite planes of symmetry, the normal of which is direction 1 and parallel to plane 2-3.

2- Anisotropic Material Types/ Transversely Isotropic Material

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In composite materials with rotational symmetry with respect to fiber axis 1, material properties can be considered the same in all directions in sections parallel to plane 2-3.

Thus, the independent material constant is reduced from 9 to 5:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(2.10.a-e)

(2.11)

Since the 2-3 plane is isotropic, equation 2.11 is also valid for this plane.

While E₂ and ν₂₃ are known G₂₃ can be calculated from equation 2.11 valid for isotropic materials:

2- Anisotropic Material Types/ Transversely Isotropic Material

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False

True

The tensile speciment transverse dimensions must be large enough and be representative of the entire structure.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Material constants matrices for transversely isotropic materials:

Test speciment

2- Anisotropic Material Types/ Transversely Isotropic Material

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2.8.1 Hooke’s Equations for Transversely Isotropic Materials

(2.12)

(3d Case)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(2.13.a-f)

(a)

(b)

(c)

(e)

(f)

(d)

Stress-strain relations (Hooke’s equations) in 3D for a material showing transverse isotropic behavior. (valid for elastic loading)

2- Anisotropic Material Types/ Transversely Isotropic Material

In fact, transversely isotropic materials are a special case of orthotropic materials, and the Hooke relations valid for orthotropic materials are also valid for transversely isotropic materials (See topic 2.7.1).

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a- Reinforced Concrete Columns: Reinforced concrete columns, which are mostly exposed to tension or compression, are transversely isotropic materials. Iron bars are placed in a mixed manner in the concrete matrix so that they are parallel to the column axis, which is the loading direction. The aim is to strengthen the matrix in the direction of the tension/compression load.

2.8.2 Natural and industrial examples of Transverse Isotropic composites :

b-) Sedimentary Rocks :The property of transverse isotropy is seen in nature in layered sedimentary rocks with long wavelengths. Each layer has approximately the same properties in its own plane (each layer is parallel to the 2-3 planes). However, there are different properties along the thickness (from bottom to top, as axis 1). The plane of each layer is the isotropy plane and the vertical axis is the axis of symmetry.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Transversely Isotropic Material

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2.9 Isotropic Material

• If there are an infinite number of isotropy planes parallel or angled to the axes in the structure, these types of materials are called isotropic materials.
• In other words, the properties are the same in all directions parallel or at an angle to the axes in the material, so there are infinite directions.
• In this case, the elastic material constants are reduced from 5 to 2 (E,ν):

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Isotropic Material

• All metallic materials such as steel, aluminum are isotropic.
• No matter which direction the sample is removed from the material, the tensile test curve and E, n values do not change.

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2.9.1 Hooke’s Equations for Isotropic Materials (3D Case)

(2.14b)

2- Anisotropic Material Types/ Isotropic Material

Stress-strain relations (Hooke’s Equations) for a material showing isotropic behavior. (valid for elastic loading)

(a)

(b)

(c)

(e)

(f)

(d)

(2.15.a-f)

As can be seen, there are only two independent elastic material constants (E, ν) in isotropic materials.

ve

(2.14a)

Shear Modulus

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Or if we calculate the stresses from equation 2.14 b, we obtain the following equation for isotropic materials:

2- Anisotropic Material Types/ Isotropic Material

(2.14.c)

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2.9.2 Composites Having Isotropic Character:

b-) Chopped Fiber (Whiskers) Reinforced Composites

c-) Multidirectional, Continuous Fiber Reinforced Composites

a-) Particle reinforced composites and nano composites

• These types of composites show isotropic properties at the macro level.
• Since they are isotropic materials consisting of at least two different materials, they are also called quasi-isotropic materials.
• If the size of the particles is at the nano level, these types of composites are called nano-composites.
• There are 2 independent material constants: elasticity modulus (E ) and poisson's ratio (n).
• The shear modulus (G) value depends on E and n and is calculated from equation 1.5.
• The values ​​of E and n cannot be found theoretically; they can only be obtained experimentally.
• Since they are isotropic, all calculations shown in the strength course are valid for these composites.
• Therefore, for calculations regarding these composites, it is necessary to look at the sources where strength topics are explained.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types/ Isotropic Material

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2.10 Hooke’s Equations in Plane Stress : (For Orthotropic, Transversely Isotropic and Isotropic Materials)

(2.16b)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

temperature effect *

(2.17)

Important points:

1. The temperature effect will be explained in more detail in the 9th topic. Its effect on strains has been added here.
2. These formulas are also valid for orthotropic, transversely isotropic and isotropic materials.
3. For isotropic materials; It is taken as (E₁=E₂=E, ν₁₂=ν₂₁=ν, G₁₂ =G, α₁=α₂=α )

For orthotropic materials, in the case of plane stress we can write the Hooke equations in matrix format as follows:

(a)

(b)

(c)

(2.16a)

2- Anisotropic Material Types

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Tablo 2.1 Classification of Composites According to Reinforcement Direction and Material, and Sections Explaining Mechanical Calculations

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

2- Anisotropic Material Types

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THEORETICAL CALCULATION OF ORTHOTROPIC MATERIAL PROPERTIES

3.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(tvid. 3a and 3b.)

in a Unidirectional and Continuous Fiber Reinforced Composites

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3.1 Our Purpose in This Section

While the elastic mechanical properties (E_f, ν_f, E_m, ν_m) of the isotropic components (matrix and fiber) that make up the composite are known;

It is to theoretically calculate the properties (E₁, E₂, ν₁₂, G₁₂) of a one-way reinforced orthotropic composite layer obtained by combining these. However, it should be noted that these calculations will also be valid for transversely isotropic composites.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

3.2 Local Axes in the Orthotropic Layer

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

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

3: Axis perpendicular to the layer plane and in the thickness direction

3.3 Representative Volume Element in an Orthotropic Layer

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• Calculations will be made on the representative volume element extracted from the unidirectional fiber reinforced composite plate.
• This element should be chosen in such a way that it represents the composite plate at a minimum level.

• Since a representative element is examined at the micro level, this subject is also called «micromechanics of composite materials».
• The mechanical properties of the composite will be determined by elementary strength calculations.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Unidirectional transverse isotropic composite structures (see: 2.11.3): The representative volume element is the same for these composites. This proves that the calculations made and the equations to be derived for composite plates reinforced with unidirectional continuous fibers are also valid for this type of composite.

3.4 Relationship Between Fiber and Matrix Volume Ratios�

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Fiber volumetric ratio

Matrix volumetric ratio

Theoretical Calculation of Composite Density:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(3.1)

(3.2)

Cross-sectional areas with normal in 1 direction :

A₁

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Attention: There are only matrix and fiber volumes in the structure. Also, there is no independent volume called Composite. The material or volume called composite is theoretical and represents the entire structure. (Or we can think that since it is the only orthotropic material, the entire structure is called composite.)

3.5 Theoretical Calculation of E₁ (Modulus of Elasticity in the Fiber Direction 1)

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We consider the representative volume element to which a pulling force P₁ is applied in direction 1. Since the fiber and matrix are completely adhered to each other, They affect each other and extend by the same ΔL in the 1 direction. Since their initial lengths are equal, their unit elongation (ΔL/L) will also be equal.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

From the static equilibrium of the left part of the Ι-Ι section,

(3.3)

(3.4)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Effect of Fiber Ratio on E₁ Value

Additionally, from the equation:

(3.5)

This equation 3.5 will appear in future calculations.

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

the relationship between the stresses in direction 1 is obtained as:

3.6. Theoretical Calculation of E₂ (Modulus of Elasticity Perpendicular to the Fibers)

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(3.6)

When we take Ι-Ι and II-II sections, respectively, in the representative volume element to which the P2 draft force is applied in the 2 direction; The internal forces and cross-sectional areas in the fiber and matrix are equal to those in the composite; Therefore, we can understand from the figures below that the stresses are the same as the stress in the composite.

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Total elongation in direction 2;

(3.9a)

(3.8)

(3.7a-c)

When the Poisson effect is neglected, the strains in the 2nd direction from Hooke's relations are:

or

(3.9b)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Effect of Fiber Ratio on E₂ Value

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The general expression of Poisson's ratio for a layer showing orthotropic character is:

Major Poisson Ratio:

According to this;

Minor Poisson Ratio:

Transverse Poisson Ratio:

There are three other non-independent Poisson ratios:

There is also a general relationship between Poisson ratios and Elasticity Modules in an orthotropic material, as in equation 3.13:..>>

Notes: 1-) Equations 3.11, 3.12 and 3.13 can be used for all composite types with orthotropic properties.

2-) Poisson ratio cannot be negative except for some special materials. It cannot be greater than 0.5 in isotropic materials.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(3.11)

(3.12a-c)

(a)

(b)

(c)

(3.13)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

3.8 Theoretical Calculation of ν₁₂ (Major Poisson Ratio)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

In the representative volume element subjected to tension in direction 1, the total strain (ΔW) in direction 2 is equal to the sum of the strains in the fiber and matrix.

If we write the Poisson ratios of the matrix and fiber :

, Similarly for fiber..>>

(3.14)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Effect of Fiber Ratio on ν₁₂

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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3.9 Theoretical Calculation of ν₁₂ (Shear Modulus in plane 1-2)

shear strain angle in composite (γ₁₂);

For the whole composite material;

We can also write this relation for fiber and matrix :

Due to static equilbrium, the internal shear forces must be equal. Since the A₂ cross-sectional areas are also equal, the shear stresses in the fiber, matrix and composite are also equal.

or

,

Since the fiber and matrix are isotropic,

(3.15a)

(3.15b)

(3.16a-b)

,

(a)

(b)

(We thought of the entire structure as a single orthotropic material and named it composite.)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

15, Agust 2025

Shear stress (τ₁₂) occurring in 1-2 plane in representative volume element creates different deformation angles (γ_f , γ_m) in fiber and matrix.

Example 3.1

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

Find the elastic properties of the Glass fiber-Epoxy composite to be obtained by combining the materials whose E, ν values are given in the table above.

(Fiber Volume Ratio = V_f = 0,3)

E₁ = ? , E₂= ?, G₁₂ = ? , ν₁₂=?

Solution:

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• Calculation of E₁ :

• Calculation of E₂ :

From equation (3.4):..>>

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

From equation (3.9):..>>

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• Calculation of G₁₂:

• Calculation of ν₁₂ (major poisson ratio) :

From equation (3.16a):..>>

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

From equation (3.16b):..>>

From equation (3.15b):..>>

From equation (3.14):..>>

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3.11 Mechanical properties of some fiber materials

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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3.12 Mechanical properties of some matrix materials

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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An aluminum sheet is reinforced unidirectionally with continuous boron fibers. Fibers constitute 36% of the total volume. The properties of the materials are given in the table below. A composite layer is produced by combining these materials. Calculate the following properties of this composite layer: a-) density, b-) Elasticity Modules in the 1 and 2 directions, c-) Poisson ratios in the 1-2 plane (major and minor), d-) Rigidity module in the 1-2 plane.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

d-)

Solution:

a-)

b-)

c-)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

ρ= 2,66 gr/cm³

Örnek 3.2

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3.13 ) Poisson Effects (p):

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(3.17a)

(3.17b)

The stresses in 2 directions are equal:

(3.18a)

(3.18b)

(3.6)

If we substitute equation 3.6 into equations 3.17:

Strains in the fiber and matrix in the 1st direction: We substitute equation 3.6 into equations 2.17 (for isotropic materials).

Strains in direction 2:

(3.7a)

(3.19a)

(3.19b)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(3.20)

(3.8)

(3.18b)

(3.18a)

(3.7a)

If the last equation is rearranged :

(3.5)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(From equa. 3.19a):

(3.3)

(From equa. 3.12b and equa.3.7a):

(3.21)

(3.14)

(3.4 )

(3.13 )

(3.22)

(3.23)

(If equation 3.22 is substituted into 3.21 and rearranged,):

If we substitute equation 3.23 into equation 3.20:

..>>

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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The result after editing the last term is::

Poisson interaction term :

(3.24a)

• When the Poisson effect term (𝑝) is neglected, equations 3.24 and 3.9 will be the same.

• Although "p" can be at negligible levels in single-direction fiber-reinforced composites, "p" occurs at higher values ​​in bidirectionally woven composites and must be taken into account.

(3.25)

or

(3.24b)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Example 3.3

What difference in E₂ value occurs when you take into account the Poisson's ratio for the composite layer in Example 3.2? Calculate.

Poisson effect term :

From equa. 3.24 :

When Poisson effect is neglected :

was found

The difference is:

Solution:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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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Total Elongations:

Unit Elongatios (Strains):

(3.26)

(3.27)

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 :

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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3.14.1.a 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.

3.14.1 Thermal Stresses in Free State

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

representative volume element

We will examine the representative volume element for calculations.

(3.28)

(3.29)

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• from static equilibrium in direction 2:

• from static equilibrium in direction 1:

(Eq. (3.311) will be used in the α₂ calculation)

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

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(3.31)

(3.30)

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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

Eq. 3.26 :

(3.32)

(3.33)

3.14.2 Theoretical Calculation of α₁

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.

Similarly:

Eq. 3.30 :

Eq. 3.3 :

(3.34)

(from eq. 3.4)

The total strain in the 1 direction caused by ΔT in the fibers. :

The total strain in the 1 direction caused by ΔT in the matrix. :

3. Unidirectional - Continuous Fiber Reinforced Composites / Theoretical Calculation of Orthotropic Material Properties

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

3.14.3 Theoretical Calculation of α₂

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

If we substitute equation 3.26 into equations 3.32 and 3.33;

(3.35)

(3.36)

Hooke relations in Equation 2.15 for fiber and matrix;

(a)

(b)

,

(3.26)

(3.3)

(3.32)

(3.33)

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Similarly for the matrix :

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(3.37)

(3.38)

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:

(3.39)

(from eq. 3.14):

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3.14.4 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 3.34 or 3.39. (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 3.34 and 3.39 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

multidirectional continuous fiber reinforced composites

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3.14.5 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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THEORETICAL STRENGTH AND EFFICIENCY LIMITS

OF COMPOSITE

4.

(STRESS-STRAIN CALCULATION EXAMPLES RESULTING FROM STRUCTURAL LOADING)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(tvids : 4a and 4b)

1. While the strength limit values (breaking stresses) of the fiber and matrix and the fiber volumetric fraction (ratio) are known, theoretically calculating the tensile and compressive strengths of the composite in the 1st and 2nd directions,
2. Explaining the stresses and deformations that occur as a result of structural loading with examples,

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4. Theoretical Strength And Efficiency Limits of Composite

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

4.1 Our Aims in This Section are:

3. To formulate, using examples, the temperature increases that will bring the strength limits to their peak in thermal loading.
4. and are to determine the lower limit values of the fiber volumetric fraction so that the fibers can carry loads and the composite can be efficient in terms of strength.

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4.2 Load Distribution fraction Calculation in Composite Structure:

Tip for direction 1: Unit elongations are equal to:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(4.1)

4. Theoretical Strength And Efficiency Limits of Composite

Total Force on composite

Force carried by fibers

Force carried by matrix

;

Representative volume element :

How much of the tensile load in direction 1 do the fibers carry? First we are looking for an answer to this question :

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Stage 1: Elastic deformation occurs in the fibers and matrix.

​

Stage 2: While elastic deformation continues to occur in the fibers, the matrix undergoes plastic deformation.

​

Stage 3: Plastic deformation occurs in both fibers and matrix.

​

Stage 4: First the fibers and then the matrix are damaged.

• Some of these stages may not occur depending on the properties of the composite components

(such as brittleness-ductility).

4.3 Deformation Stages of Composite Structure

When a unidirectionally reinforced composite layer with continuous fibers is subjected to tension in the 1-direction, it deforms in four stages as the load increases.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• Attention: Even if the fibers have higher strength, the stress on them will be higher and the fibers will break first. (For a correctly and efficiently designed lamina).

4. Theoretical Strength And Efficiency Limits of Composite

(It is accepted that the composite is damaged when the fibers break.)