12 Introduction to Damage Mechanics of Composite Materials 12.1 Basic Equations The objective of this final chapter is to introduce the reader to the subject of damage mechanics of composite materials.For further details,the reader is referred to the comprehensive book on this subject written by the authors: Advances in Damage Mechanics:Metals and Metal Matrix Composites.This chapter does not contain any MATLAB functions or examples. In this chapter,only elastic composites are considered.The fibers are as- sumed to be continuous and perfectly aligned.In addition,a perfect bond is assumed to exist at the matrix-fiber interface.A consistent mathematical for- mulation is presented in the next sections to derive the equations of damage mechanics for these composite materials using two different approaches:one overall and one local.The elastic stiffness matrix is derived using both these two approaches and is shown to be identical in both cases. For simplicity,the composite system is assumed to consist of a matrix re- inforced with continuous fibers.Both the matrix and fibers are linearly elastic with different material constants.Let C denote the configuration of the un- damaged composite system and let Cm and Cf denote the configurations of the undamaged matrix and fibers,respectively.Since the composite system as- sumes a perfect bond at the matrix-fiber interface,it is clear that Cmncf= and Cm U Cf=C.In the overall approach,the problem reduces to trans- forming the undamaged configuration C into the damaged configuration C. In contrast,two intermediate configurations Cm and Cf are considered in the local approach for the matrix and fibers,respectively.In the latter approach, the problem is reduced to transforming each of the undamaged configurations Cm and Cf into the damaged configurations cm and C/,respectively. In case of elastic fiber-reinforced composites,the following linear relation is used for each constituent in its respective undamaged configuration: kEk :k =m,f (12.1)12 Introduction to Damage Mechanics of Composite Materials 12.1 Basic Equations The objective of this final chapter is to introduce the reader to the subject of damage mechanics of composite materials. For further details, the reader is referred to the comprehensive book on this subject written by the authors: Advances in Damage Mechanics: Metals and Metal Matrix Composites. This chapter does not contain any MATLAB functions or examples. In this chapter, only elastic composites are considered. The fibers are as￾sumed to be continuous and perfectly aligned. In addition, a perfect bond is assumed to exist at the matrix-fiber interface. A consistent mathematical for￾mulation is presented in the next sections to derive the equations of damage mechanics for these composite materials using two different approaches: one overall and one local. The elastic stiffness matrix is derived using both these two approaches and is shown to be identical in both cases. For simplicity, the composite system is assumed to consist of a matrix re￾inforced with continuous fibers. Both the matrix and fibers are linearly elastic with different material constants. Let C¯ denote the configuration of the un￾damaged composite system and let C¯m and C¯f denote the configurations of the undamaged matrix and fibers, respectively. Since the composite system as￾sumes a perfect bond at the matrix-fiber interface, it is clear that C¯m ∩ C¯f = φ and C¯m ∪ C¯f = C¯. In the overall approach, the problem reduces to trans￾forming the undamaged configuration C¯ into the damaged configuration C. In contrast, two intermediate configurations Cm and Cf are considered in the local approach for the matrix and fibers, respectively. In the latter approach, the problem is reduced to transforming each of the undamaged configurations C¯m and C¯f into the damaged configurations Cm and Cf , respectively. In case of elastic fiber-reinforced composites, the following linear relation is used for each constituent in its respective undamaged configuration: σ¯k = E¯k : ¯εk, k = m, f (12.1)