O.N. Grigoriev et al. / Composites: Part B 37(2006)530-54 Fig. 13. Distribution of normal stresses over the middle cross-section(a) and shear stresses(b) over cantilever portions across the specimen thickness 1. Stiffness D is derived from the initial elastic moduli by Eq. Material B OB 2. Longitudinal strains are determined by Eq(10), taking a 1.84×10g3-1.31×102+4.68 fixed load in a given point (for the first point, it is 5 N; as a X I0E(MPa) 3. Stresses in A and B layers are found by Eqs. (11)and(12) 4. Elastic moduli are determined and then recalculated for Results of calculations by these relations are given in each layer proceeding from the following condition: the Table 7. Calculated deflections w correspond practically to elastic modulus of a layer is the average of the elastic experimental ones(Table 5), which confirms the reliability of moduli for its upper Eu=O,Eu and lower E1=0E points, other results(stress, stiffness). i.e. E=(Eu+E1/2 Fig. 12 depicts the a-E curve for a quasi-homogeneous 5. Stiffness D is recalculated by Eg.(3)with the new elastic specimen and similar curves calculated by Eqs. (13)and(14) moduli; then(2)-(5)are repeated until elastic moduli values As is seen, the diagram for the laminated specimen as a quasi- in a given loading point are stabilized. homogeneous material is between the diagrams for A and B 6. The value of an applied load is increased by 5 N; then(1)- layers (6)are repeated, the final elastic moduli from the previous The distribution of normal and shear stresses by Eqs. (13) loading point are taken as the initial ones, and this process is and (14)across the specimen thickness is shown in Fig 13. The continued until the applied load values reach 50 N distribution of normal stresses(Fig. 13a)is presented in the two versions: nonlinear(solid line)according to Eqs. ( 13)and(14) Results of calculations by a-E diagrams as well and linear(dash line) based on initial elastic moduli values. deflections determined by Eq. (7)are summarized in Table 6. Observed stress steps correspond to the layer interface.These The calculated nonlinear F-w diagram in comparison with the curves represent the case of the largest experimental load experimental one and the diagram derived from the initial (50N). Maximum normal stresses by nonlinear approach are moduli by linear approach are illustrated in Fig. 11 equal to 75.1 MPa, while by linear approach, they are larger, In the final loading point, the difference between calculated 85.3 MPa Maximum shear stresses(Fig. 13b) take place on the and experimental deflections is about 13%. To reduce this neutral axis and equal 4.2 MPa. As is seen in the plot of shear difference, the coefficients of polynomials(11) and (12)were stresses, significant ditferences correspond to A layers and corrected by varying them in such a way that in each loading small ones to B layers point the ratio of elastic moduli EB/EA=m was maintained and stiffness D approached the value calculated by Eq.( 8). At 4. Conclusions the same time, for each loading point and for the variation of the coefficients of the above polynomials, the calculations The theory of heterogeneous layered systems was used to were performed in accordance with(1)(5)of the second develop a procedure for solving the inverse problem based on substage. Several iterations resulted in the following final o-E experimental data on deformation of a laminated composite relations beam. This approach allows for the evaluation of the mechanical characteristics of laminated composites: to plot the effective nonlinear a-e diagram for the case of the composite quasi-homogeneous material as well as a-E MaterialA diagrams for the materials of its layers. These diagrams =1.79×1012e3-1.28×102+5.13 provided the basis for theoretical investigation of the stress- strain state of the co mposite and evaluation of the refinements introduced to nonlinear calculations, in particular a decrease in X 10E(MPa) (13) the largest normal stresses compared to their linear values1. Stiffness D is derived from the initial elastic moduli by Eq. (3). 2. Longitudinal strains are determined by Eq. (10), taking a fixed load in a given point (for the first point, it is 5 N; as a whole it varies from 5 to 50 N with a step of 5 N). 3. Stresses in A and B layers are found by Eqs. (11) and (12). 4. Elastic moduli are determined and then recalculated for each layer proceeding from the following condition: the elastic modulus of a layer is the average of the elastic moduli for its upper EuZsu/3u and lower ElZsl/3l points, i.e. EZ(EuCEl)/2. 5. Stiffness D is recalculated by Eq. (3) with the new elastic moduli; then (2)–(5) are repeated until elastic moduli values in a given loading point are stabilized. 6. The value of an applied load is increased by 5 N; then (1)– (6) are repeated, the final elastic moduli from the previous loading point are taken as the initial ones, and this process is continued until the applied load values reach 50 N. Results of calculations by s–3 diagrams as well as deflections determined by Eq. (7) are summarized in Table 6. The calculated nonlinear F–w diagram in comparison with the experimental one and the diagram derived from the initial moduli by linear approach are illustrated in Fig. 11. In the final loading point, the difference between calculated and experimental deflections is about 13%. To reduce this difference, the coefficients of polynomials (11) and (12) were corrected by varying them in such a way that in each loading point the ratio of elastic moduli EB/EAZm was maintained and stiffness D approached the value calculated by Eq. (8). At the same time, for each loading point and for the variation of the coefficients of the above polynomials, the calculations were performed in accordance with (1)–(5) of the second substage. Several iterations resulted in the following final s–3 relations: Material A :sA Z1:79!1012 3 3K1:28!109 3 2 C5:13 !105 3 ðMPaÞ; (13) Material B :sB Z1:84!1011 3 3K1:31!108 3 2 C4:68 !104 3 ðMPaÞ: (14) Results of calculations by these relations are given in Table 7. Calculated deflections w correspond practically to experimental ones (Table 5), which confirms the reliability of other results (stress, stiffness). Fig. 12 depicts the s–3 curve for a quasi-homogeneous specimen and similar curves calculated by Eqs. (13) and (14). As is seen, the diagram for the laminated specimen as a quasi￾homogeneous material is between the diagrams for A and B layers. The distribution of normal and shear stresses by Eqs. (13) and (14) across the specimen thickness is shown in Fig. 13. The distribution of normal stresses (Fig. 13a) is presented in the two versions: nonlinear (solid line) according to Eqs. (13) and (14) and linear (dash line) based on initial elastic moduli values. Observed stress steps correspond to the layer interface. These curves represent the case of the largest experimental load (50 N). Maximum normal stresses by nonlinear approach are equal to 75.1 MPa, while by linear approach, they are larger, 85.3 MPa. Maximum shear stresses (Fig. 13b) take place on the neutral axis and equal 4.2 MPa. As is seen in the plot of shear stresses, significant differences correspond to A layers and small ones to B layers. 4. Conclusions The theory of heterogeneous layered systems was used to develop a procedure for solving the inverse problem based on experimental data on deformation of a laminated composite beam. This approach allows for the evaluation of the mechanical characteristics of laminated composites: to plot the effective nonlinear s–3 diagram for the case of the composite as a quasi-homogeneous material as well as s–3 diagrams for the materials of its layers. These diagrams provided the basis for theoretical investigation of the stress– strain state of the composite and evaluation of the refinements introduced to nonlinear calculations, in particular a decrease in the largest normal stresses compared to their linear values. Fig. 13. Distribution of normal stresses over the middle cross-section (a) and shear stresses (b) over cantilever portions across the specimen thickness. 540 O.N. Grigoriev et al. / Composites: Part B 37 (2006) 530–541