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ADVANCED MECHANICS OF COMPOSITE MATERIALS Episode 4

Tham khảo tài liệu 'advanced mechanics of composite materials episode 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 92 Advanced mechanics of composite materials The first of these equations specifies the apparent longitudinal modulus of the ply and corresponds to the so-called rule of mixtures according to which the property of a composite can be calculated as the sum of its constituent material properties multiplied by the corresponding volume fractions. Now consider Eq. 3.67 which can be written as 2 2 Vf e Vm Substituting strains e2 and em from Eqs. 3.72 stresses ơỊ and ơjm from Eqs. 3.74 and e1 from Eqs. 3.58 with due regard to Eqs. 3.76 and 3.77 we can express e2 in terms of Ơ1 and ơ2. Comparing this expression with the second constitutive equation in Eqs. 3.58 we get 1 Vf Vm _ Vf Vm Ef Vm Em Vf 2 E2 Ef Em Ef Em Ef Vf Em Vm 3.78 V21 Vf Vf VmVm E1 Ef Vf Em Vm 3.79 Using Eqs. 3.76 and 3.79 we have V21 Vf Vf Vm Vm 3.80 This result corresponds to the rule of mixtures. The second Poisson s ratio can be found from Eqs. 3.77 and 3.78 . Finally Eqs. 3.58 3.70 and 3.73 yield the apparent shear modulus 1 Vf Vm G12 Gf Gm 3.81 This expression can be derived from the rule of mixtures if we use compliance coefficients instead of stiffnesses as in Eq. 3.76 . Since the fiber modulus is typically many times greater than the matrix modulus Eqs. 3.76 3.78 and 3.81 can be simplified neglecting small terms and presented in the following approximate form Em Gm E1 Ef Vf E2 fm 2 G12 m Vmp mi Vm Only two of the foregoing expressions namely Eq. 3.76 for E1 and Eq. 3.80 for V21 both following from the rule of mixtures demonstrate good agreement with experimental results. Moreover expressions analogous to Eqs. 3.76 and 3.80 follow practically from the numerous studies based on different micromechanical models. Comparison of predicted and experimental results is presented in Figs. 3.35-3.37 where theoretical dependencies of normalized moduli on the fiber volume fraction are shown with lines. The dots correspond to the test data for epoxy composites reinforced with different fibers Chapter 3. Mechanics