Z-M. Huang / Computers and Structures 80(2002)1159-1176 1161 materials. The latter are dependent on the fiber and L Overal stiffness matrix matrix current stress states. On the other hand, having explicitly known the internal stresses in the fiber and strain a analysis匚今ilne matrix materials, the effective properties as well as stress-strain response of the lamina can be completely Stress shared by eachlamina identified. The lamina failure status can be detected by m checking whether the fiber or the matrix has attained its ultimate stress state or not. if the lamina fails the cor- Check lamina tailure les responding overall applied stress on the laminate is de- ilure strengt fined as a progressive failure strength. The remaining Check laminate ultimate failureS aminate is analyzed by discounting the stiffness contri- bution from the failed lamina. In this way, the ultimate strength of the laminate can be determined incremen- tally Thus, a necessary and sufficient condition for un Fig. 2. A flow chart to show analysis procedure for a fibrous derstanding the inelastic and strength behavior of a fi- laminate brous composite is to explicitly determine the internal stresses in its constituent fiber and matrix materials. all nation theory. The purpose of the laminate analysis is to ness matrix. the internal stresses in the fiber. and the obtain the in-plane strain and curvature increments of internal stresses in the matrix, pertaining to the lamina the laminate, and further to determine the stresses analysis at any load level can be obtained by using the sheared by each lamina in the laminate, as indicated Fig. 3 for a multidirectional tape laminate. The most bridging micromechanics model, which is summarized in the next section important quantity involved in this step analysis is the instantaneous stiffness matrix of the lamina. which will be used to construct the laminate overall stiffness matrix Initially, all the laminae in the laminate are in linear and 3. Summary of the bridging model elastic deformation (provided that the laminate is not Similarly as other composite theories, the bridgir subjected to high thermal residual stresses), and their model is developed with respect to a UD composite iffness matrices can be defined in a usual way. How- This is because other fibrous composites can be dis- ever, with the increase of load level, some laminae may cretized into a number of UD composites, see subse- ergo inelastic deformation. Their initial (elastic) stifness matrices are no longer applicable. Thus, the element (RVE) of a ud composite is indicated in Fig 4 second step analysis, i.e lamina analysis, is concerned with determination of the lamina instantaneous stiffness There are two material elements in the rve, i.e., the matrix even if an inelastic deformation has occurred fiber and the matrix. No other element such as fiber- This instantaneous stiffness matrix cannot be determined matrix interface region has been distinctly considered in ithout an explicit knowledge of the internal stress the bridging model Let us deal with a plane problem first. The incre- states generated in the fiber and matrix materials of the mental stresses in the fiber and matrix can be correlated lamina. This is because the lamina instantaneous stiff- through a bridging matrix, [a. ness matrix is closely related with the instantaneous tiffness matrices of the constituent fiber and matrix {da}=4]{do where da)= don, doz, do1) with f and m referring to the fiber and matrix, respectively. Based on(1), the three basic quantities of the composite are found to be {dd}=(F+4]){d}={]{d} {d}=团l(+h4)-{da}=]l{d},(3) ]=(]+S"4)(h+V团4 Fig. 3. A schematic show for the analysis of a multidirectional (S]+VmSlADIB (4)nation theory. The purpose of the laminate analysis is to obtain the in-plane strain and curvature increments of the laminate, and further to determine the stresses sheared by each lamina in the laminate, as indicated in Fig. 3 for a multidirectional tape laminate. The most important quantity involved in this step analysis is the instantaneous stiffness matrix of the lamina, which will be used to construct the laminate overall stiffness matrix. Initially, all the laminae in the laminate are in linear and elastic deformation (provided that the laminate is not subjected to high thermal residual stresses), and their stiffness matrices can be defined in a usual way. How￾ever, with the increase of load level, some laminae may undergo inelastic deformation. Their initial (elastic) stiffness matrices are no longer applicable. Thus, the second step analysis, i.e., lamina analysis, is concerned with determination of the lamina instantaneous stiffness matrix even if an inelastic deformation has occurred. This instantaneous stiffness matrix cannot be determined without an explicit knowledge of the internal stress states generated in the fiber and matrix materials of the lamina. This is because the lamina instantaneous stiff- ness matrix is closely related with the instantaneous stiffness matrices of the constituent fiber and matrix materials. The latter are dependent on the fiber and matrix current stress states. On the other hand, having explicitly known the internal stresses in the fiber and matrix materials, the effective properties as well as stress–strain response of the lamina can be completely identified. The lamina failure status can be detected by checking whether the fiber or the matrix has attained its ultimate stress state or not. If the lamina fails, the cor￾responding overall applied stress on the laminate is de- fined as a progressive failure strength. The remaining laminate is analyzed by discounting the stiffness contri￾bution from the failed lamina. In this way, the ultimate strength of the laminate can be determined incremen￾tally. Thus, a necessary and sufficient condition for un￾derstanding the inelastic and strength behavior of a fi- brous composite is to explicitly determine the internal stresses in its constituent fiber and matrix materials. All the three quantities, i.e., the lamina instantaneous stiff- ness matrix, the internal stresses in the fiber, and the internal stresses in the matrix, pertaining to the lamina analysis at any load level can be obtained by using the bridging micromechanics model, which is summarized in the next section. 3. Summary of the bridging model Similarly as other composite theories, the bridging model is developed with respect to a UD composite. This is because other fibrous composites can be dis￾cretized into a number of UD composites, see subse￾quent sections for detail. A representative volume element (RVE) of a UD composite is indicated in Fig. 4. There are two material elements in the RVE, i.e., the fiber and the matrix. No other element such as fiber– matrix interface region has been distinctly considered in the bridging model. Let us deal with a plane problem first. The incre￾mental stresses in the fiber and matrix can be correlated through a bridging matrix, ½A, as fdrmg¼½Afdrf g; ð1Þ where fdrg¼fdr11; dr22; dr12gT with f and m referring to the fiber and matrix, respectively. Based on (1), the three basic quantities of the composite are found to be [3,7] fdrf g¼ðVf ½I þ Vm½AÞ 1 fdrg¼½Bfdrg; ð2Þ fdrmg¼½AðVf½I þ Vm½AÞ 1 fdrg¼½A½Bfdrg; ð3Þ and ½S¼ðVf½Sf þ Vm½Sm½AÞðVf½I þ Vm½AÞ 1 ¼ ðVf½Sf þ Vm½Sm½AÞ½B: ð4Þ Fig. 2. A flow chart to show analysis procedure for a fibrous laminate. Fig. 3. A schematic show for the analysis of a multidirectional tape laminate. Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176 1161