Subject: Composite Materials Science and Engineering Subject code: 0210080060

1. Subject: Composite MaterialsScience and Engineering Subject code: 0210080060 Prof C. H. XU School of Materials Science and Engineering Henan University of Science and Technology Chapter 9: Mechanical Properties of Composites

2. Chapter 9: Mechanical Properties of Composites -Introduction The methods of calculating the stiffness and strength of composites • Based on the orientation of reinforcement, the properties of a composite can be • Isotropic (各向同性）: • compositeswith the random orientation of reinforcement, such as sort fibers or particles • Anisotropic (各向异性）: • continuous fiber composites, • Lamellar composites • Skin-Core-Skin pattern • Some composite with short fibers (pressure molded short fiber composites) • This chapter introduces thecalculation of stiffness and strength of the composite fiber-reinforced or lamellar composites at special directions.

3. Mechanical Property - Fiber-reinforced or lamellar composites Volume fraction of reinforcement Volume fraction of matrix

4. Mechanical Property - Fiber-reinforced or lamellar composites • Heterogeneity (异质) Composites’ properties and structures vary from point to point. • Property relationship The properties of a composite are determined largely by • properties of the constituents, • their relative concentration, • their geometric arrangement and • the nature of the interface between them. • Anisotropy the strength and stiffness are highest in the direction of fibre orientation, but are week in the direction of the transverse direction.

5. Deformation · The mechanical characteristics of a fiber-reinforced composite depend on the properties of the fiber and on the degree of load transmittance to the matrix phase. · Important to the degree of this load transmittance is the magnitude of the interfacial bond between the fiber and matrix phases. · Under an applied stress, this fiber- matrix bond ceases at the fiber ends.

7. Fiber-reinforced Composites

8. Fiber-reinforced Composites

9. Fiber-reinforced Composites Influence of Fiber Orientation and Concentration Two extremes: (a) a parallel alignment of the longitudinal axis of the fibers in a single direction; and (c) a totally random alignment.

10. Continuous and aligned fiber composites • Tensile stress-strain behavior depends on the fiber and matrix phases; the phase volume fractions; the direction of loading. • with longitudinal loading • in stage I, both fiber and matrix deform elastically. • in stage II, the fiber continues to deform elastically, but the matrix has yielded. • from stage I to II, the fiber picks up more load. • the onset of composite failure begins as the fibers start to fracture. • composite failure is not catastrophic.

11. Longitudinal Loading · The properties of a composite depend on the fibre direction. · Assuming the fiber-matrix interfacial bond is very good, such that deformation of both matrix and fibers is the same (Isostrain) . Fiber-reinforced Composites - Continuous and Aligned Fibre Composites ec = ef = em

12. Fiber-reinforced Composites

13. s = s + s V V c m m f f Fiber-reinforced Composites Using Volume Fraction: = : the volume V = V /V A /A m vol,m vol,c m c fractions of the matrix = : the volume fractions V = V /V A /A f vol,f vol,c f c of the fiber The composite stress becomes:

14. Fiber-reinforced Composites

15. The modulus of elasticity of a continuous and aligned fibrous composite in the direction of alignment (or longitudinal direction): Ecl = + E E V E V cl m m f f + = V V 1 m f = - + E E ( 1 V ) E V cl m f f f Rule of Mixture is equal to the volume- E cl fraction weighted average 加权平均值 of the moduli of elasticity of the fiber and matrix phases.

16. ·Other properties, including tensile strength, also have this dependence on volume fractions. Students cangive this equation. (TS)lc= ? It can also be shown, for longitudinal loading, that the ratio of the load carried by the fibers to that carried by the matrix is

17. Fiber-reinforced Composites

18. Rule of Mixture

19. Fiber-reinforced CompositesDiscontinuous and aligned fiber composites Moduli of elasticity and tensile strengths of short fiber composites are about 50 -90 % these of long fiber composites. • When l > lc, the longitudinal strength (TS)cd Where (TS)f is the fracture strength of the fiber and (TS)’m is the stress in the matrix when the composite fails. • When l < lc, the longitudinal strength is Where d is the fiber diameter and c is the shear yields strength of the matrix.

20. Fiber-reinforced CompositesDiscontinuous and randomly oriented composites The orientation of the short and discontinuous fibers is random in matrix. A ‘rule-of-mixtures’ expression for the elastic modulus Ecd where K is a fiber efficiency parameter depended on Vf and Ef/Em ratio (0.1 ~ 0.6)

21. Example Problem: A continuous and aligned glass-reinforced composite consists of 40 vol% of glass fibers having a modulus of elasticity of 69 GPa and 60 vol% of a polyester resin that, when hardened, display a modulus of 3.4 GPa. Compute the modulus of elasticity of this composite in the longitudinal direction. If the cross-sectional area is 250mm2 and a stress of 50 MPa is applied in this longitudinal direction, compute the magnitude of the load carries by each of the fiber and matrix phases. Determine the strain that is sustained by each phase when the stress in part b is applied. Compute the elastic modulus of the composite material, but assume that the stress is applied perpendicular to the direction of fiber alignment. Assuming tensile strengths of 3.5 GPa and 69 MPa, respectively, for glass fibers and polyester resin, determine the longitudinal tensile strength of this fiber composite.

22. Equations on the calculation for long fiber reinforced matrix composite • longitudinal direction: • Transverse direction

23. Solution We have Ef = 69 GPa Em = 3.4 GPa Vf = 0.4 Vm = 0.6 • The modulus of elasticity of the composite is calculated using equation

24. To solve this portion of the problem, first find the ration of fiber load to matrix load, using equation or Ff =13.5Fm In addition, the total force sustained by the composite Fc may be computed from the applied stress s and total composites cross section area Ac according to However, this total load is just the sum of the loads carried by fiber and matrix phases, that is Fc=Ff+Fm = 12500 N. Substitution for Ff from the above yields 13.5Fm + Fm=12500N, Fm=862N Whereas Ff=Fc- Fm=12500N-860N=11640N Thus, the fiber phase supports the majority of the applied load.

25. The stress for both fiber and matrix phases must be calculated. Then, by using the elastic modulus for each (from part a), the strain values my determined. For stress calculations, phase cross-sectional areas are necessary: Am= VmAc= (0.6)(250mm2)=150mm2 and Af =VfAc = (0.4)(250mm2)=100mm2 Thus, Finally, strains are commutated as Therefore, strains for both matrix and fiber phases are identical, which they should be, according to in the previous development.

26. d) According to equation This value for Ect is slightly greater than that of the matrix phase, but, only approximately one-fifth of the modulus of elasticity along the fiber direction (Ecl), which indicates the degree of anisotropy of continuous and oriented fiber composites. e) For the tensile strength TS, (TS)cl = (TS)mVm +(TS)f Vf (TS)cl=(69 MPa) (0.6) + (3.5 x 103MPa) (0.4)=1441 MPa