Z-M. Huang/Computers and Structures 80(2002)1159-1176 essentially fibrous composites. An expanding interest in fiber reinforced composite [3-7 until rupture. The ma- biological engineering attracts people's increasing atten terial parameters involved in the model are minimal and tion on those tissues, and on the development of nano- can be measured or determined independently. Appli- scale biocomposite substitutes cations of this model to a number of fibrous composites Apparently, any successful use/critical design of including various textile(woven, braided, and knitted synthetic composites or achievement in tissue engineer fabric reinforced composites have been successfully ing depends on a thorough understanding for the com- achieved [7-1l] However, the composites considered posite whole properties. However, according to the were mainly subjected to in-plane load conditions and critical survey organised by UK Science Engineering not enough attentions have been given to the responses Research council and uk institution of mechanica of laminated composites. For the convenience or an Engineers, the composite theories in current use are sti nite element method based structural analysis, it is useful less successful [2]. The main reason for this is that only to bring all the information, i.e. the systematic simula- the linear elastic constitutive equations of the composite tion procedure as well as its applications to a broad have been well established, and are essentially used for range of instructive examples, into a summary article. analyses in the current literature. Little is known about The purpose of these two parts of papers is to present a the composite inelastic behaviors. There is hardly any unified constitutive description for the thermo-mechan- commercial finite element analysis software package that ical response of a lamin has incorporated an efficient material constitutive model variety of application examples of which several are with which the response of a composite structure out of new. The present paper focuses on the theoretical de- a linear elastic deformation range can be directly simu velopment and an accompanied one deals with the ap- lated. However, without a good knowledge of the com- plications posite inelastic behavior, the composite load carrying The constitutive relationship is established through capacity cannot be well assessed, and hence, a critical combining the bridging model, for lamina analysis, with design of a composite material/structure cannot be made. the classical lamination theory, for laminate analysis As This is because most composites are in laminated struc- mentioned earlier, one purpose for the lamina analysis tures, and each lamina involved is statically indetermi is to obtain instantaneous stiffness/compliance matrix. nate(Fig. 1). The lamina load share depends on Considered in the paper are various laminae reinforced constitutive equations. Just before the laminate attains with different fiber preforms, including UD fiber pre- its failure status. some laminae in the laminate must form and woven braided and knitted fabric structures have undergone more or less inelastic deformations. The One of the most critical factors that influence the com- lamina load share cannot be correctly determined if only posite response is matrix behaviour. a different matrix its linear elastic constitutive relationship is used material may require a different theory for its constitu- throughout. In the case of a living tissue especially a soft tive description. In the present paper, an additional ssue. the inelastic deformation is even more distinct constitutive theory, the Bodner-Partom unified theory All these necessitate the development of a rational con- was employed to describe the response of titanium stitutive model for describing the composite inelastic matrix material at high temperature. However, the behavior up to failure bridging model is developed based on an incremental Very recently, the present author proposed one sucl odel, called the bridging model, which best applicable. As the Bodner-Partom theory uses a simulate the inelastic behaviour of a unidirectional (ud) total stress-total strain description in the differential form,a transformation between the Bodner-Partom description and the Prandth-Reuss description has been illustrated in the paper. The laminate analysis and the numerical examples will be presented in the subsequent 2. Simulation procedure Based on an incremental solution strategy which is best applicable to nonlinear problems, a detailed flow chart to show the simulation procedure for a gener laminated composite is indicated in Fig. 2. Essentially, two steps are involved in the simulation. Fig.i.compaRisonofanisolatedlaminawiththattakenfromInthefirststepalaminateanalysisisperformedThisis a laminate accomplished in this work by using the classical lami-essentially fibrous composites. An expanding interest in biological engineering attracts people’s increasing atten￾tion on those tissues, and on the development of nano￾scale biocomposite substitutes. Apparently, any successful use/critical design of synthetic composites or achievement in tissue engineer￾ing depends on a thorough understanding for the com￾posite whole properties. However, according to the critical survey organised by UK Science & Engineering Research Council and UK Institution of Mechanical Engineers, the composite theories in current use are still less successful [2]. The main reason for this is that only the linear elastic constitutive equations of the composite have been well established, and are essentially used for analyses in the current literature. Little is known about the composite inelastic behaviors. There is hardly any commercial finite element analysis software package that has incorporated an efficient material constitutive model with which the response of a composite structure out of a linear elastic deformation range can be directly simu￾lated. However, without a good knowledge of the com￾posite inelastic behavior, the composite load carrying capacity cannot be well assessed, and hence, a critical design of a composite material/structure cannot be made. This is because most composites are in laminated struc￾tures, and each lamina involved is statically indetermi￾nate (Fig. 1). The lamina load share depends on its constitutive equations. Just before the laminate attains its failure status, some laminae in the laminate must have undergone more or less inelastic deformations. The lamina load share cannot be correctly determined if only its linear elastic constitutive relationship is used throughout. In the case of a living tissue especially a soft tissue, the inelastic deformation is even more distinct. All these necessitate the development of a rational con￾stitutive model for describing the composite inelastic behavior up to failure. Very recently, the present author proposed one such model, called the Bridging Model, which can fairly well simulate the inelastic behaviour of a unidirectional (UD) fiber reinforced composite [3–7] until rupture. The ma￾terial parameters involved in the model are minimal and can be measured or determined independently. Appli￾cations of this model to a number of fibrous composites including various textile (woven, braided, and knitted) fabric reinforced composites have been successfully achieved [7–11]. However, the composites considered were mainly subjected to in-plane load conditions and not enough attentions have been given to the responses of laminated composites. For the convenience of an fi- nite element method based structural analysis, it is useful to bring all the information, i.e. the systematic simula￾tion procedure as well as its applications to a broad range of instructive examples, into a summary article. The purpose of these two parts of papers is to present a unified constitutive description for the thermo-mechan￾ical response of a laminated composite and to show a variety of application examples of which several are new. The present paper focuses on the theoretical de￾velopment and an accompanied one deals with the ap￾plications. The constitutive relationship is established through combining the bridging model, for lamina analysis, with the classical lamination theory, for laminate analysis. As mentioned earlier, one purpose for the lamina analysis is to obtain instantaneous stiffness/compliance matrix. Considered in the paper are various laminae reinforced with different fiber preforms, including UD fiber pre￾form and woven, braided, and knitted fabric structures. One of the most critical factors that influence the com￾posite response is matrix behaviour. A different matrix material may require a different theory for its constitu￾tive description. In the present paper, an additional constitutive theory, the Bodner–Partom unified theory, was employed to describe the response of a titanium matrix material at high temperature. However, the bridging model is developed based on an incremental solution strategy with which the Prandtl–Reuss theory is best applicable. As the Bodner–Partom theory uses a total stress–total strain description in the differential form, a transformation between the Bodner–Partom description and the Prandtl–Reuss description has been illustrated in the paper. The laminate analysis and the numerical examples will be presented in the subsequent paper. 2. Simulation procedure Based on an incremental solution strategy which is best applicable to nonlinear problems, a detailed flow chart to show the simulation procedure for a general laminated composite is indicated in Fig. 2. Essentially, two steps are involved in the simulation. In the first step, a laminate analysis is performed. This is accomplished in this work by using the classical lami￾Fig. 1. Comparison of an isolated lamina with that taken from a laminate. 1160 Z.-M. Huang / Computers andStructures 80 (2002) 1159–1176