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
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 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 Contents C1 : Introduction to composite materials C2 : Mechanical behaviors of composite materials 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 References Autar K Kaw, Mechanics of Composite Materials, Taylor & Francis, NewYork, 2006 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 Contents C1 : Introduction to composite materials C2 : Mechanical behaviors of composite materials 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 INTRODUCTION TO COMPOSITE MATERIALS Introduction Composite materials o Matrix materials o Fibers o Architecture of composite materials o Study the mechanical behavior of composite materials Composite materials for civil engineering applications Introduction Composite materials used more and more for primary structures in aerospace, marine, energy,… Introduction Composite materials used more and more for primary structures in civil engineering, etc Composite materials Definition: o “Composite” means "made of two or more different parts Classification: o Form of constituents Fiber composite Particle composite o Nature of Constituents Organic matrix composites Metallic matrix composites Mineral matrix composites Composite materials Classification by class of constituents Fiber Reinforcement Particle Matrix Composite Matrix Composite Mechanical properties of composites the nature of the constituents the proportions of the constituents the orientation of the fibers Composite materials 10 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 o Thermoplastic Resins: polyvinyl chloride (PVC), polyethylene, polypropylene, polystirene, polyamide, and polycarbonate o Thermostable Resins: o Bismaleimide Resins, Polyimide Resins Contents 109 C1 : Introduction to composite materials C2 : Mechanical behaviors of composite materials 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 Homogenization of composite materials 110 Introduction: Microscopic ↔ Macroscopic Homogenization Material can be considered to be homogeneous or heterogeneous according to the scale at which it is observed Material can be described in the framework of continuum mechanics by two models: one at microscopic scale where the behavior is heterogeneous, the other at macroscopic scale where the behavior is homogeneous Objective of homogenization: study the relation of these two models, especially determination of behavior at macroscopic scale in terms of one at microscopic scale Homogenization of composite materials 111 Introduction: Phenomenological method Homogenization Determination of behavior at macroscopic scale in terms of one at microscopic scale: Representative volume element (RVE) whose boundary subjected homogeneous boundary conditions in strain and stress to Macroscopic behavior is the relation between volume average of stress and strain in RVE Size of RVE should be large enough at micro scale to well describe heterogeneity, and small enough at macro scale in which calculated mechanical fields are very few variables in RVE Homogenization of composite materials 112 Introduction: Heterogeneity scale RVE scale Scale of structure: macroscopic scale Scale of heterogeneity : microscopic scale Scale of RVE: mesoscopic scale Concept of RVE: linear elasticity, nonlinear elasticity, elastoplasticity, limit analysis Homogenization of composite materials 113 Objective: Definition of homogenized linear elastic properties of heterogeneous materials Presentation of bounds that enable to estimate the properties in terms of constituent properties Hypothesis: Small deformation Heterogeneous linear elastic materials No cavities, cracks Constituents are perfectly adherent Homogenization of composite materials 114 Average value on RVE: Definition: f D = V ∫ f ( x )dV x∈V Average of stress: σ ( x ) = {div σ ( x ) = in V , σ.n = σ n} ⇒ σ V = σ0 Homogenization of composite materials 115 Average of strain on RVE: Consider : ε ( x ) = ( ∇u + t ∇u ) , u ( x ) = u ( x ) = ε x on ∂V Average strain field of RVE: Homogenization of composite materials 116 Average value on RVE: Homogeneous boundary condition in stress: 1 σ ε V = σε dV = u ( σ.n )dS ∫ ∫ V x∈D V x∈∂V σ V = Σ ∀x ∈ ∂V , σ.n = Σ.n ⇒ σ ε V = σ V ε V Homogeneous boundary condition in strain: ε V = E ∀x ∈ ∂V , u ( x ) = E.x ⇒ σ ε V = σ V ε Hill – Mandel’s principle : σ ε V = σ V ε V V Homogenization of composite materials 117 Homogenization with strain approach: Direct method: u ( x ) = E.x x ∈ ∂V AV∈ V and BCs ∀E ⇒ Σ = σ E Heterogeneous : a(x) Homogeneous : A Elastic problem on RVE: solutions (σE, εE, uE) div σ ( x ) = σ ( x ) = a ( x ) ε ( x ) t ε x = ∇ u + ∇u ) ( ) ( u ( x ) = E.x x ∈ ∂V V = AV E A ? for a(x) Nota: ∀x ∈ V , if a ( x ) = A, σE εE V = σE V εE V = ΣE σ E ( x ) = AE ⇒ ∀x ∈ V , ε E ( x ) = E u E ( x ) = E.x E : uniform strain in RVE Homogenization of composite materials 118 Potential energy method W ( u E ) − Φ ( u E ) = Minu∈KA W ( u ) − Φ ( u ) u ∈ KA : u ( x ) = E.x x ∈ ∂V W (u ) = ε ( x ) a ( x ).ε ( x ) dV , Φ ( u ) = ∫ x∈D Minimum principle of potential energy: u ( x ) = uɶ ( x ) + E.x ⇒ uɶ ( x ) = ∀x ∈ ∂V ε ( x ) = ε ( uɶ ) + E ( ε ( uɶ ) + E ) a ( x ) ( ε ( uɶ ) + E ) = Min ( ε ( uɶ ) + E ) a ( x ) ( ε ( uɶ ) + E ) ε ( u ) = ε ( uɶ ) + E, σ = a ( x ) ( ε ( uɶ ) + E ) ( ε ( uɶ ) + E ) a ( x ) ( ε ( uɶ ) + E ) = σ ε = σ ε = EA E E E E E E u∈KA V E E E V E E V E V ⇒ ∀E, EAV E = Minu∈KA ( ε ( uɶ ) + E ) a ( x ) ( ε ( uɶ ) + E ) E V V V V Homogenization of composite materials 119 Complementary energy method ∀E,W ( σ E ) − Φ ( σ E ) = Minσ∈SA W ( σ ) − Φ ( σ ) σ ∈ SA : div σ = W (σ ) = σ ( x ) s ( x ).σ ( x ) dV , Φ ( u ) = ∫ ( E.x ) ( σ.n )dS = E ∫ σ ( x ) dV ∫ x∈D x∈∂V x∈V Minimum principle of complementary energy: σ E s ( x ) σ E V − 2E σ E σ E s ( x ) σ E V = σ Eε E V V = Minσ∈SA σ.s ( x ) σ = σE εE V ⇒ ∀E, EAV E = Maxσ∈SA 2E σ V V V − 2E σ V = EAV E − σ.s ( x ) σ V Conclusions: ∀E, Maxσ∈SA 2E σ V − σ.s ( x ) σ V = EAV E = Minu∈KA ( ε ( uɶ ) + E ) a ( x ) ( ε ( uɶ ) + E ) V Homogenization of composite materials 120 Homogenization with stress approach: Direct method: σ x n = Σn x ∈ ∂V ( ) Heterogeneous : s(x) Homogeneous : S Elastic problem on RVE: solutions (σ∑, ε ∑, u ∑): σ ∈ SA : div σ = 0, σ ( x ) n = Σn x ∈ ∂V div σ ( x ) = σ ( x ) = a ( x ) ε ( x ) ε x = ∇u + t ∇u ) ( ) ( σ ( x ) n = Σn x ∈ ∂V ∀ Σ , Maxu Σ ε ( u ) εΣ V − ε ( u ) a ( x ) ε ( u ) V V = SV Σ = ΣS V Σ = Minσ∈SA ( σɶ + Σ ) s ( x ) ( σɶ + Σ ) V Homogenization of composite materials 121 Homogenized properties of a heterogeneous medium: Homogenization of composite materials 122 Bounds on Ahom: Voigt and Reuss ∀E, Maxσ∈SA 2E σ V − σ.s ( x ) σ V = EAV E = Minu∈KA ( ε ( uɶ ) + E ) a ( x ) ( ε ( uɶ ) + E ) V ∀E, EAV E ≤ E a ( x ) V E = E ∑ f α aα E (Voigt’s bound) α ∀E, σ 2Eσ − σ s ( x ) V σ ≤ EAV E ⇒ ∀E, E s ( x ) −1 V E ≤ EAV E −1 ⇒ ∀E, E ∑ f α sα E ≤ EA hom E α (Reuss’s bound) −1 fα hom α α ≤ K ≤ f K ∑ ∑ α K α α −1 fα hom α α ≤ µ ≤ f µ ∑ ∑ α α α µ 9K µ 3K − µ E= ,ν = 3K + µ K + 2µ Homogenization of composite materials 123 Bounds on Ahom: Hashin-Shtrikman K − − K1 V2 = ; K K − K − K1 + − V ( 2) K1 + µ1 / µ − − µ1 V2 = µ2 − µ1 + − V µ2 − µ1 ( 2) µ1 + f1 K + − K1 V2 = ; K K − K − K1 + − V ( 2) K1 + µ2 / V2 µ + − µ1 = µ − µ1 + − V µ2 − µ1 ( 2) µ1 + f fα = µα ( Kα + 8µα ) / ( Kα + µα ) E= 9K µ 3K − µ ,ν = 3K + µ K + 2µ Nota : K ≥ K1 , µ ≥ µ1 ... of composite materials C3 : Elastic behavior of unidirectional composite materials C4 : Elastic behavior of orthotropic composite C5 : Off-axis behavior of composite materials C6 : Fracture and. .. characteristics of materials, made in the form of fibers Composite materials 16 Architecture of composite materials Laminates Composite materials 17 Architecture of composite materials o Sandwich Composite. .. of composite materials C3 : Elastic behavior of unidirectional composite materials C4 : Elastic behavior of orthotropic composite C5 : Off-axis behavior of composite materials C6 : Fracture and