Mechanics of Composite Materials, Faculty of Civil Engineering and Applied Mechanics Department of Structures,

C1 : Introduction to composite materials C2 : Mechanical behaviors of composite materials C3 : Elastic behavior of unidirectional composite 2 materials C4 : Elastic behavior of orthotropic composite C5 : Offaxis behavior of composite materials C6 : Fracture and damage of composite materials C7 : Modeling of mechanical behaviors of laminated plates C8 : Homogenization of composite materialsReferences Autar K. Kaw, Mechanics of Composite Materials, Taylor Francis, NewYork, 2006 JeanMarie Berthelot, Composite Materials – 3 Mechanical behavior and Structural analysis, Springer, 1999 J. N. Reddy, Mechanics of laminated composite plates and shells – Theory and Analysis, CRC Press, 2004. S. LI, Introduction to micromechanics and nanomechanics, Lecture notesContents C1 : Introduction to composite materials C2 : Mechanical behaviors of composite materials C3 : Elastic behavior of unidirectional composite 4 materials C4 : Elastic behavior of orthotropic composite C5 : Offaxis behavior of composite materials C6 : Fracture and damage of composite materials C7 : Modeling of mechanical behaviors of laminated plates C8 : Homogenization of composite materialsINTRODUCTION TO COMPOSITE MATERIALS Introduction Composite materials Matrix materials 5 o o Fibers o Architecture of composite materials o Study the mechanical behavior of composite materials Composite materials for civil engineering applicationsIntroduction Composite materials used more and more for primary structures in aerospace, marine, energy,… 6Introduction Composite materials used more and more for primary structures in civil engineering, etc 7Composite materials Definition: o “Composite” means made of two or more different parts Classification: 8 o Form of constituents Fiber composite Particle composite o Nature of Constituents Organic matrix composites Metallic matrix composites Mineral matrix compositesComposite materials Classification by class of constituents 9 Fiber Reinforcement Matrix Composite Particle Matrix Composite Mechanical properties of composites the nature of the constituents the proportions of the constituents the orientation of the fibersComposite materials Matrix comprises a resin (polyester, epoxide, etc.) and fillers which is to improve the characteristics of the resin: o Thermosetting Resins: 10 Polyester Resins Condensation Resins Epoxide Resins o Thermoplastic Resins: polyvinyl chloride (PVC), polyethylene, polypropylene, polystirene, polyamide, and polycarbonate o Thermostable Resins: o Bismaleimide Resins, Polyimide ResinsComposite materials Epoxide Resins: 11 Advantages of epoxide resins are the following: good mechanical properties (tension, bending, compression, shock, etc.) superior to those of polyesters good behavior at high temperatures: up to 150190°C in continuous use excellent chemical resistance low shrinkage in molding process and during cur

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Faculty of Civil Engineering and Applied Mechanics

Department of Structures

Mechanics of Composite Materials

PhD Nguyễn Trung Kiên

Email: ntkien@hcmute.edu.vn

Faculty of Civil Engineering and Applied Mechanics

1 Vo Van Ngan Street, Thu Duc District

Ho Chi Minh City, Viet Nam

Keywords:

- Mechanics of Composite Materials

- Laminated materials and Structures

- Homogenization

- Theory of plates and beams

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C1 : Introduction to composite materials

C2 : Mechanical behaviors of composite materials C3 : Elastic behavior of unidirectional composite

C3 : Elastic behavior of unidirectional composite

materials

C4 : Elastic behavior of orthotropic composite

C5 : Off-axis behavior of composite materials

C6 : Fracture and damage of composite materials

C7 : Modeling of mechanical behaviors of laminated plates

C8 : Homogenization of composite materials

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Jean-Marie Berthelot, Composite Materials –

Mechanical behavior and Structural analysis,

Springer, 1999

J N Reddy, Mechanics of laminated composite plates and shells – Theory and Analysis, CRC

Press, 2004

S LI, Introduction to micromechanics and

nanomechanics, Lecture notes

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C1 : Introduction to composite materials

materials

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INTRODUCTION TO COMPOSITE MATERIALS

o Architecture of composite materials

o Study the mechanical behavior of composite

materials

Composite materials for civil engineering applications

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Composite materials used more and more for primary structures in aerospace, marine, energy,…

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Composite materials used more and more for primary structures in civil engineering, etc

7

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o Nature of Constituents

Organic matrix composites Metallic matrix composites Mineral matrix composites

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Composite materials

Classification by class of constituents

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Mechanical properties of composites

the nature of the constituents

the proportions of the constituents

the orientation of the fibers

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Matrix comprises a resin (polyester, epoxide, etc.) and fillers which is to improve the characteristics of the resin:

o Thermosetting Resins:

Polyester Resins Condensation Resins Epoxide Resins

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Epoxide Resins:

11

Advantages of epoxide resins are the following:

good mechanical properties (tension, bending, compression, shock, etc.) superior to those of polyesters

good behavior at high temperatures: up to 150-190°C in continuous use excellent chemical resistance

low shrinkage in molding process and during cure (from 0.5-1 %)

very good wettability of reinforcements

excellent adhesion to metallic materials

Disadvantages:

High cost, manufacture, sensibility to cracking

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Polypropylene, polyamide:

Advantages of epoxide resins are the following:

low cost, fabrication

Disadvantages:

mechanical and thermomechanical properties : low

Limited development

Thermostable Resins: Bismaleimide Resins, Polyimide Resins

Thermal performance developed especially in the aviation and space

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Fillers and additives: function of improving the mechanical and

physical characteristics of the finished product or making their manufacture easier

Fillers: Reinforcing Fillers, Nonreinforcing Fillers

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Fillers: Reinforcing Fillers, Nonreinforcing Fillers

o Reinforcing Fillers : improve the mechanical properties of a resin

Spherical fillers: diameter usually lying between 10 and 150 µ m They can be glass, carbon, or organic (epoxide, phenolic, polystirene, etc.),

Nonspherical fillers: mica used most (dimension: 100-500 µ m, thickness: 1-20

µ m)

o Nonreinforcing Fillers: reducing the cost of resins, preserving their

performance carbonates, silicates

Additives: pigments and colorants, antishrinkage agents,

antiultraviolet agents

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Improve mechanical characteristics: stiffness,strength, hardness, etcImprove certain of the physical properties: thermal properties, fire

Improve certain of the physical properties: thermal properties, fire

resistance, resistance to abrasion, electrical properties

Reinforcements origins: vegetable, mineral, artificial, synthetic fibers

linear forms (strands, yarns, rovings, etc.)surfacing tissues (woven fabrics, mats, etc.)multidirectional forms (preforms, complex cloths, etc.)

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Specific mechanical characteristics of materials, made

in the form of fibers

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Architecture of composite materials

Laminates

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Architecture of composite materials

o Sandwich

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Study the mechanical behavior of composite materials

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Study the mechanical behavior of composite materials

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C2 : Mechanical behaviors of composite materials

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materials

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Linear elastic scheme

Elastic behavior of a unidirectional composite material Elastic behavior of an orthotropic composite material

Elastic behavior of an orthotropic composite material

Elastic behavior of composite materials outside of main axes

Strength failure theories

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Mechanical behaviors of composite materials

Linear elastic scheme

o Stiffness and compliance matrix

C, S 6x6-matrix is called the

stiffness matrix and compliance matrix having 21 independent

constants: S = C-1

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Change of coordinate system

Vector:

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where

Tensor:

Rotation of a ɵ angle of coordinate

system around 3-axis

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where :

Rotation of a ɵ angle of coordinate

system around 3-axis

1' = σ ε−

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26

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Engineering Matrix Notation

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Note that these constants can vary

from point to point if the material is

nonhomogeneous

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Monoclinic material

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1–2 symmetry plane of a monoclinic material

Note:

A monoclinic material is a material that has a symmetry plane

13 independent elastic constants

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Orthotropic material

Note:

Three mutually perpendicular planes of material symmetry

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Transverse isotropic material (Unidirectional material)

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Note:

Orthotropic material having one axis of revolution

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Isotropic material

Note:

Properties are independent of the choice of its reference axis

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Exercise 1: In the case of a monoclinic material with the symmetry

plane (1,2) show that the stiffness matrix has the form (a).

Exercise 2: The symmetry plane (1,3) is added to a monoclinic

material in order to obtain an orthotropic material Show that the stiffness matrix has the form (b)

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C2 – Mechanical behaviors of composite materials

Exercises 3: Consider a rotation through an angle e about the I-axis

of an orthotropic material Write the stiffness matrix in the new axes and deduce the form (a) of the stiffness matrix of a transverse

isotropic material.

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isotropic material.

Exercise 4: In case of isotropic material, show that the stiffness

matrix has the form (b)

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C2 : Mechanical behaviors of composite materials

C3 : Elastic behavior of unidirectional composite

C3 : Elastic behavior of unidirectional composite materials

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ELASTIC BEHAVIOR OF UNIDIRECTIONAL COMPOSITE MATERIALS

Homogenized problems of designing structures can be solved by considering theaverage properties measured on the scale δ

Macroscopic homogeneity or statistical homogeneity Homogenization.

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How to determine the homogenized properties

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Effective properties of composites

An element of volume V and size δ :

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Basic equations of the average strain and stress field:

C: Effective stiffness matrixS: Effective compliance matrix

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Homogenization method: 3 main stages

• Representative Volume Element (RVE)

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Periodic material media

Localization problem of periodic composites

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Strain energy:

Macroscopic constitutive equation:

Resolution method of the elastic problem on a unit cell: FEM, Fourier

(Gusev, Kanit et al., Suquet et al., Mishneavsky, etc)

Strain energy:

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Unidirectional composite material

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Engineering constants - Tests

Engineering constants: Young's moduli (E), Poisson

ratios ( ν ), shear moduli (G).

Longitudinal Tensile Test

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Longitudinal Tensile Test

Stress and strain:

Elastic moduli:

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Transverse tensile test

Stress and strain:

Elastic moduli:

(ν21)

(ν23)

Nota:

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Longitudinal shear test Transverse shear test

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Lateral hydrostatic compression

Stress and strain:

Lateral compression modulus:

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Moduli as functions of the stiffness

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Only 5 independent moduli, practically: E , E , νννν , G , G

Only 5 independent moduli, practically: EL, ET, ννννLT, GLT, GTT'

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Moduli as functions of the stiffness

Stiffness as functions of Moduli

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Engineering constants – Theoretical approach

Different approach to the problem

Find 5 independent constants as functions of the mechanical and geometric properties of the constituents (engineering constants of

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the matrix and fibers, volume fraction of the fibers),…

Periodic fiber arrangements:

Random fiber arrangements:

How to estimate elastic constants ?

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Estimation of elastic constants:

Bounds (upper and lower bounds) using energy variational

theorems (total potential energy theorem, Hashin-Shtrikman,…) : not accurate for high contrast of materials

Exact solutions: simple geometry

Numerical methods (FEM, Fourier)

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Bounds on engineering constants:

Total potential energy theorem (displacement approach): upper bounds

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Bounds on engineering constants:

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Simplified approach:

Longitudinal Young’s modulus:

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Transverse Young’s modulus:

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Longitudinal Poisson ratio:

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Longitudinal shear modulus:

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materials

C4 : Elastic behavior of orthotropic composite

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Elastic behavior of orthotropic composite

Orthotropic composite material

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Orthotropic composite material

C 11 , C 12 , C 13 , C 22 , C 23 , C 44 ,C 55 , C 66

S 11 , S 12 , S 13 , S 22 , S 23 , S 44 , S 55 , S 66

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Engineering constants: Young's moduli (E), Poisson

ratios ( ν ), shear moduli (G).

Tensile test in direction 1

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Shear test

Similarly,

Conclusion:

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Compliance constants:

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Stiffness constants:

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Exercise 1: Calculate the stiffness and compliance constants of an orthotropic

composite with the following characteristics:

Exercise 2: Calculate the stiffness and compliance constants of an orthotropic

composite with the following characteristics:

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materials

C5 : Off-axis behavior of composite materials

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Constitutive equations of off-axis layers

1, 2, 3: principal directions1’, 2’, 3’: reference system

Stiffness and compliance constants:

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Off-axis behavior of composite materials

Elastic constants

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Two dimensional stress state

Plane stress state:

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Plane stress state:

where

Q: reduced stiffness matrix

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Plane stress state in principal directions:

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Plane stress state in off-axis:

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C2 : Mechanical behaviors of composite materials

materials

C6 : Fracture and damage of composite materials

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Fracture and Damage of Composite Materials

Fracture Processes Induced in Composite Materials :

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Failure Criteria:

Maximum stress criterion

+ Xt, Xc: the tensile and compressive strengths in the longitudinal direction

+ Yt , Yc: the tensile and compressive strengths in the transverse direction

+ S: the in-plane shear strength of the layer

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Fracture and Damage of Composite Materials

Failure Criteria:

Maximum strain criterion

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+ Xεt, Xεc: the tensile and compressive strains in the longitudinal direction

+ Yεt , Yεc: the tensile and compressive strains in the transverse direction

+ S: the in-plane shear strain of the layer

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C3 : Elastic behavior of unidirectional composite materials

C7 : Modeling of mechanical behaviors of

laminated plates

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Basics of Laminate Theory :

Plate element Laminated element

Equivalent single-layer theories (2D)

Classical laminate theory

Shear deformation laminate theories

Three-dimensional elasticity theory

3D elasticity formulations

Layerwise theories

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Modeling the Mechanical Behavior of Laminated Plates

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Plate model of Love-Kirchhoff:

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Plate models

Plate model of Reissner-Mindlin: First-order shear deformation theory

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High-Order Shear Deformation Plate Model

In-plane displacements varied in the thickness:

TSDT :

SSDT :

Transverse displacement varied in the thickness:

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2D plate theories

Assumption: σzz=0

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Plate models

2D plate theories

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Resultants and Moments:

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Plate models

Resultants and Moments:

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Straight lines perpendicular to

the midsurface before

deformation remain straight

after deformation

Transverse normals are

inextensible

Transverse normals rotate

such that they remain

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