Bài giảng Mechanics of Composite Material_9_2015

Mechanics of Composite Materials Assoc Prof Dr Nguyen Trung Kien Email: kiennt@hcmute.edu.vn, https://fceam.hcmute.edu.vn Faculty of Civil engineering and Applied mechanics HCMC University of Technology and Education Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Vietnam September 2015 Contents Introduction to composite materials Mechanical behaviors of composite materials Elastic behaviour of unidirectional composite materials Elastic behaviour of orthotropic composite Off-axis behaviour of composite materials Fracture and damage of composite materials Modeling of mechanical behaviours of laminated plates Homogenization of composite materials References 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 Autar K Kaw, Mechanics of Composite Materials, Taylor & Francis, NewYork, 2006 S LI, Introduction to micromechanics nanomechanics, Lecture notes and Significance & Objectives Investigation of characteristics of the constituent and composite materials Material optimization Development of analytical procedures for determining material properties Development of analytical procedures for determining material properties Development of analytical procedures for determining structural behavior Contents Introduction to composite materials Mechanical behaviors of composite materials Elastic behavior of unidirectional composite materials Elastic behavior of orthotropic composite Off-axis behavior of composite materials Fracture and damage of composite materials Modeling of mechanical behaviors of laminated plates 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 Application of composite materials Composite materials Old and new aspects of composites o Human body, plants o Early 1960s (fibrous composites) Definition: “Composite” means "made of two or more different parts heterogeneous All materials may be considered heterogeneous if the scale of interest is sufficiently small Fibrous composites (Fiber-Reinforced Composites) are materials in which one phase acts as a reinforcement of a second phase Composite materials Classification: o Form of constituents Fiber composite Particle composite o Nature of Constituents Organic matrix composites Metallic matrix composites Mineral matrix composites Particle composite interphase Continuous phase (matrix) Dispersed phase (reinforcement) Fiber composite Composite materials Classification by class of constituents Fiber Reinforcement Particle Matrix Composite Matrix Composite Mechanical properties of composites nature of the constituents proportions of the constituents orientation of the fibers Properties of composite materials 10 Low-medium performance composites The reinforcement provides stiffening and local strengthening of the materials (short fiber) The matrix is the main load bearing constituent governing the mechanical properties High performance composites The reinforcement is the backbone of the materials (continuous fiber) The matrix provides protection and support for the sensitive fiber The interphase controls the failure mechanisms Contents 139 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 140 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 141 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 142 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 143 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 144 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 145 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 146 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 147 Homogenization with strain approach: Direct method: u ( x ) = E.x x ∈ ∂V ∀E ⇒ Σ = σ E Chom ? Homogeneous : Chom Heterogeneous : C(x) V = Chom E Elastic problem on RVE: solutions (σE, εE, uE) Nota: div σ ( x ) =  σ ( x ) = C ( x ) ε ( x )   ε x = ∇u + t ∇u ) ( ) (   u ( x ) = E.x x ∈ ∂V ∀x ∈ V , if C ( x ) = Chom , σE εE V = σE V εE V = ΣE σ E ( x ) = Chom E  ⇒ ∀x ∈ V , ε E ( x ) = E  u E ( x ) = E.x ∂V E : uniform strain in RVE Homogenization of composite materials 148 Bounds based on potential energy method W ( u E ) = Minu∈KAW ( u ) with u ∈ KA : u ( x ) = E.x x ∈ ∂V W (u) = ε ( x ) C ( x )ε ( x ) dV ∫ x∈D Minimum principle of potential energy: u ( x ) = u% ( x ) + E.x ⇒ u% ( x ) = ∀x ∈ ∂V , ε ( x ) = ε ( u% ) + E ( ε ( u% ) + E ) C ( x ) ( ε ( u% ) + E ) = Min ( ε ( u% ) + E ) C ( x ) ( ε ( u% ) + E ) ε ( u ) = ε ( u% ) + E, σ = C ( x ) ( ε ( u% ) + E ) ( ε ( u% ) + E ) C ( x ) ( ε ( u% ) + E ) = σ ε = σ ε = EC E E E E E u∈KA V E E hom E E V E E V E V E V ⇒ ∀E, EChom E = Minu∈KA ( ε ( u% ) + E ) C ( x ) ( ε ( u% ) + E ) V V Homogenization of composite materials 149 Bounds based on 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 σ E s ( x ) σ E V − 2E σ E V = σ Eε E V V = Minσ∈SA  σ.s ( x ) σ = σE V ⇒ ∀E, EChom E = Maxσ∈SA  2E σ εE V V V − 2E σ V  = EChom E − σ.s ( x ) σ V  Conclusions: ∀E, Maxσ∈SA  2E σ V − σ.s ( x ) σ V  = EChom E = Minu∈KA ( ε ( u% ) + E ) C ( x ) ( ε ( u% ) + E ) V Homogenization of composite materials 150 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  ( σ% + Σ ) s ( x ) ( σ% + Σ ) V Homogenization of composite materials 151 Homogenized properties of a heterogeneous medium: Homogenization of composite materials 152 Bounds on Ahom: Voigt and Reuss ∀E, Maxσ∈SA  2E σ V − σ.s ( x ) σ V  = EAV E = Minu∈KA ( ε ( u% ) + E ) a ( x ) ( ε ( u% ) + 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 0 ⇒ ∀E, E s ( x ) 0 −1 V E ≤ EAV E −1   ⇒ ∀E, E  ∑ f α sα  E ≤ EA hom E  α  (Reuss’s bound) −1  fα  hom α α ∑ α  ≤ K ≤ ∑ f K α  α K  −1  fα  hom α α ∑ α  ≤ µ ≤ ∑ f µ α  α µ  9K µ 3K − µ ,ν = E= 3K + µ 6K + 2µ Homogenization of composite materials 153 Bounds on Ahom: Hashin-Shtrikman K − − K1 V2 = ; K − K1 K − K1 + − V ( 2) K1 + µ1 / µ − − µ1 V2 = µ2 − µ1 + − V µ2 − µ1 ( 2) µ1 + f1 K + − K1 V2 = ; K − K1 K − K1 + − V ( 2) K1 + µ2 / V2 µ + − µ1 = µ2 − 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 , µ2 ≥ µ1 [...]... etc.) Composite materials 21 Specific mechanical characteristics of materials, made in the form of fibers Composite materials 22 Specific mechanical characteristics of materials, made in the form of fibers Composite materials 23 Architecture of composite materials Laminates Composite materials 24 Architecture of composite materials o Sandwich Composite materials 25 Laminated composite materials Composite. .. and of high rigidity, the H-IIA rocket adopts CFRP parts Applications of Composite Materials 31 Transportation Composite wings & boot Automotive Composite materials can be used in a number of applications within the transportation market from exterior door panels, radiators and ignition components on passenger vehicles, to the side panels on light- and heavy - duty trucks The benefits of using composites... plane, and Boeing's Sonic Cruisers shall be the state of the art examples Use of CFRP for airplane parts will be developed further and further toward the future Aircraft flooring panels Cargo liners Helicopter rotor blades Radomes Leading and trailing edges of wings Stabilizers etc Applications of Composite Materials 29 Aircraft/Aerospace: Applications of Composite Materials 30 Aircraft/Aerospace: In aerospace... Study the mechanical behavior of composite materials Applications of Composite Materials Aircraft/Aerospace Transportation Marine Energy Electrical/Electronic Leisure/Sports Medical Construction 2015-09-09 Applications of Composite Materials 28 Aircraft/Aerospace: "Further Possibility to Use CFRP parts on Commercial Aircraft" Airplane manufacturers now plan to expand use of CFRP Airbus's A380s, the new... last decade utilization of wind power has expanded at the impressive rate of 20 percent per annum The Blade Factory in Nakskov, Denmark Turbine blades(32-39m) Applications of Composite Materials 35 Electrical/Electronic Composite applications for electrical and electronics markets include light poles, circuit boards, electrical junction boxes and ladder rails The benefits of using composites in these applications... equipping of powerboats, sailboats and other crafts Composites bring a myriad of benefits to marine applications, including high strength, reduced weight, corrosion resistance, dimensional stability and design flexibility Baja Marine Manufactures composite high-performance powerboats, with Sophisticated In-gel graphics that have become the company’s trade mark 2015-09-09 Applications of Composite Materials... Formula 3 bodywork in carbon fiber Applications of Composite Materials 32 Light Weight Interior & Exterior Panels for Rail Way Vehicles Glass/Phenolic Honeycomb Sandwich Panels Honeycomb sandwich panel has superior flexural stiffness, buckling resistance, and light weight Applications of Composite Materials 33 Marine Composite materials can be used in a variety of applications in the marine market, including... dimensional stability, design flexibility, cost performance and corrosion resistance 2015-09-09 Applications of Composite Materials 36 Leisure/Sports Increasingly, composites can be found in a variety of consumer goods, particularly recreational and sporting goods products The benefits of using composites for consumer goods include strength, low weight, resilience, flexibility and corrosion resistance... performance developed especially in the aviation and space Composite materials 19 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 o Reinforcing Fillers : improve the mechanical properties of a resin Spherical fillers: diameter usually lying between... B 2015-09-09 Definition of Composites 13 Nonhomogeneous anisotropic isotropic No conventional method Structural analysis tool for composites required orthotropic anisotropic 2015-09-09 MATRIX 14 Polymer Metal (higher use temperature) Ceramics (very high use temperature) MATRIX 15 Matrix: a resin (polyester, epoxide, etc.) and fillers which is to improve the characteristics of the resin: o Thermosetting ... behaviors of composite materials Elastic behavior of unidirectional composite materials Elastic behavior of orthotropic composite Off-axis behavior of composite materials Fracture and damage of composite. .. to composite materials Mechanical behaviors of composite materials Elastic behaviour of unidirectional composite materials Elastic behaviour of orthotropic composite Off-axis behaviour of composite. .. behavior of unidirectional composite materials Elastic behavior of orthotropic composite Off-axis behavior of composite materials Fracture and damage of composite materials Modeling of mechanical