O.N. Grigoriev et aL./Composites: Part B 37(2006)530-541 539 Table 7 The calculation results of first stage on the base a-e dependence Calculation results on the base of corrected a-e dependence F(H) w(MKM) GA(MIla) oB (MIla) D(HM) F(H) w(MKM) A (MIla) oB(MIla) D(HM) 844×100.75×10°4.79 842×100.75×10° 4.579 1.48×10° 166×10147×10° 248×10220×10p 175 46×10°2.16×10° 4.287 32.6×10°290×10° 2.82×1 40.3×10°3.58×10°4277 98×10° 45×1 3.982 47.8×10° 0°4147 47.1×10°405×1 3.825 52×10° 10°4017 542×10°462×1 2.5×10°5.54×10° 61.1×105.16×10°3.509 Stiffnesses calculated by this equation are summarized in Table 5. Maximum normal stresses oa and oB arising in the upper and lower layers, respectively, are found by Hooke's law 4. Let us equate the stiffnesses by Eqs.(6)and( 8); as a result o=Ee with account of calculated elastic moduli and strains get the equation for the elastic modulus of material A Stress values are also cited in Table 5 To plot a-E diagrams for the materials under study, EA Material A: 0A The elastic modulus of material B is determined by the equation EB=mEA Calculated elastic moduli(secant moduli) =1.83×1012g3-1.21×10°c2+5.35 are summarized in Table 5 To obtain longitudinal strains in the middle cross-section of X 10E(MPa) the beam, let us use the equation of applied mechanics 174×10g3-1.13×10c2+487 where z is the coordinate of the which is examined relative to the cross-sectional The ab X10*E(MPa) assumptions allow one to take the arbitrary z coordinate of the cross-section for the plotting of a-E diagrams. Let us take the These relations became the basis for determining the stress- coordinate za=1.569x10m(upper point of the cross- strain state section)for A layers and the coordinate zB=1.531X10 m (lower point of the cross-section) for B layers relative to the 3. 3. 4. Stress-strain state of the composite specimen center of stiffness. Strains EA and EB in those points are given in The procedure for determining the stress-strain state (second substage)taking into account the nonlinear properties of the layers(Eqs.(11)and(12)is as follows: Fig. 11. Experimental (points), nonlinear(solid line), and linear(dash line)F-w Fig. 12. Stress-strain curves for A and B layers and effective deformation diagram for the beam as a quasi-homogeneous material.Stiffnesses calculated by this equation are summarized in Table 5. Let us equate the stiffnesses by Eqs. (6) and (8); as a result we get the equation for the elastic modulus of material A: EA Z Fl3 128ðIA CmIBÞwe (9) The elastic modulus of material B is determined by the equation EBZmEA. Calculated elastic moduli (secant moduli) are summarized in Table 5. To obtain longitudinal strains in the middle cross-section of the beam, let us use the equation of applied mechanics 3 1 2 ;z  Z Fl 4D z; (10) where z is the coordinate of the point which is examined relative to the cross-sectional stiffness center. The above assumptions allow one to take the arbitrary z coordinate of the cross-section for the plotting of s–3 diagrams. Let us take the coordinate zAZ1.569!10K3 m (upper point of the cross￾section) for A layers and the coordinate zBZ1.531!10K3 m (lower point of the cross-section) for B layers relative to the center of stiffness. Strains 3A and 3B in those points are given in Table 5. Maximum normal stresses sA and sB arising in the upper and lower layers, respectively, are found by Hooke’s law sZE3 with account of calculated elastic moduli and strains. Stress values are also cited in Table 5. To plot s–3 diagrams for the materials under study, calculated data was fitted by cubic polynomials: Material A :sA Z1:83!1012 3 3K1:21!109 3 2 C5:35 !105 3 ðMPaÞ; (11) Material B :sB Z1:74!1011 3 3K1:13!108 3 2 C4:87 !104 3 ðMPaÞ: (12) These relations became the basis for determining the stress– strain state. 3.3.4. Stress–strain state of the composite specimen The procedure for determining the stress–strain state (second substage) taking into account the nonlinear properties of the layers (Eqs. (11) and (12)) is as follows: Table 6 The calculation results of first stage on the base s–3 dependence F (H) w (MKM) sA (MPa) sB (MPa) D (H M2 ) 0 0 0 0 5.060 5 0.52 8.44!106 0.75!106 4.796 10 1.07 16.7!106 1.48!106 4.668 15 1.65 24.8!106 2.20!106 4.538 20 2.27 32.6!106 2.90!106 4.408 25 2.92 40.3!106 3.58!106 4.277 30 3.62 47.8!106 4.24!106 4.147 35 4.36 55.2!106 4.90!106 4.017 40 5.14 62.5!106 5.54!106 3.891 45 5.97 69.8!106 6.19!106 3.770 50 6.84 77.2!106 6.84!106 3.656 Fig. 11. Experimental (points), nonlinear (solid line), and linear (dash line) F–w curves. Table 7 Calculation results on the base of corrected s–3 dependence F (H) w (MKM) sA (MPa) sB (MPa) D (H M2 ) 0 0 0 0 5.060 5 0.55 8.42!106 0.75!106 4.579 10 1.13 16.6!106 1.47!106 4.435 15 1.75 24.6!106 2.16!106 4.287 20 2.42 32.3!106 2.82!106 4.136 25 3.14 39.8!106 3.45!106 3.982 30 3.92 47.1!106 4.05!106 3.825 35 4.77 54.2!106 4.62!106 3.667 40 5.70 61.1!106 5.16!106 3.509 45 6.70 68.0!106 5.68!106 3.356 50 7.79 75.1!106 6.21!106 3.211 Fig. 12. Stress–strain curves for A and B layers and effective deformation diagram for the beam as a quasi-homogeneous material. O.N. Grigoriev et al. / Composites: Part B 37 (2006) 530–541 539