544 R.Talreja and C.V.Singh a b c） d Fig.12.8.Distributed damage configurations in CMCs:(a)matrix cracking, (b)interfacial slip in conjunction with matrix cracking,(c)debonding,and (d)debonding in conjunction with matrix cracking [59] Matrix cracking A matrix crack can be viewed as a pair of internal surfaces in a composite that are able to perturb the stress state in a region around the surfaces by conducting displacement,i.e.,separation of surfaces,from the undeformed configuration.The surface separation per unit of applied external load depends on the size and shape of the surfaces as well as on the constraint, if any,imposed by the surroundings.For a matrix crack in a unidirectional CMC,the constraint comes from the bridging fibers as well as from the stiffening effect of fibers in the matrix surrounding the crack. The description for matrix cracks follows a second-order tensor char- acterization as suggested first by Vakulenko and Kachanov [64]and described in further detail by Kachanov [31].Talreja [58]used a diad an to characterize a damage entity of a finite volume bounded by a surface S.In this characterization,n is the unit outward normal to the surface at a point
Fig. 12.8. Distributed damage configurations in CMCs: (a) matrix cracking, Matrix cracking A matrix crack can be viewed as a pair of internal surfaces in a composite that are able to perturb the stress state in a region around the surfaces by conducting displacement, i.e., separation of surfaces, from the undeformed CMC, the constraint comes from the bridging fibers as well as from the stiffening effect of fibers in the matrix surrounding the crack. The description for matrix cracks follows a second-order tensor characterization as suggested first by Vakulenko and Kachanov [64] and described in further detail by Kachanov [31]. Talreja [58] used a diad an to characterize a damage entity of a finite volume bounded by a surface S. In this characterization, n is the unit outward normal to the surface at a point R. Talreja and C.V. Singh (d) debonding in conjunction with matrix cracking [59] (b) interfacial slip in conjunction with matrix cracking, (c) debonding, and depends on the size and shape of the surfaces as well as on the constraint, if any, imposed by the surroundings. For a matrix crack in a unidirectional configuration. The surface separation per unit of applied external load 544
Chapter 12:Multiscale Modeling for Damage Analysis 545 on the surface and a represents the extent and direction of some"influence," e.g.,the disturbance of the strain fields,due to presence of the damage entity,referred to the same point on the surface.A"damage entity tensor" is then defined as d,=an,ds, (12.5) with reference to a Cartesian coordinate system Xi.The influence vector can be resolved along the normal and tangential directions with respect to the crack surface.For the type of crack considered here,it is reasonable to assume that only the normal (crack opening)displacement matters, allowing a;to be expressed as a41=an:, (12.6) where the quantity a now represents a measure of the crack influence. From dimensional analysis,with di;taken to be dimensionless,a has dimensions of length.Drawing upon fracture mechanics,this length is in proportion to the crack length.For a fiber-bridged matrix crack,the crack length can be expressed in multiples of the average interfiber spacing. Thus, (12.7) 1=kd5 where k is a constant,d is the fiber diameter,and vr is the fiber volume fraction.The expression in (12.7)is based on a hexagonal fiber arrange- ment.Similar expression will result from another assumption of fiber distribution in the cross section.It can now be inferred that the micro- structural length scale for matrix microcracking is the fiber diameter.Note that,for an irregularly shaped crack surface,the interfiber spacing and, therefore,the fiber diameter will still be the length scale. The consequence of the presence of a matrix crack is generally in changing the composite's deformational response,which is defined and measured at a larger length scale,e.g.,the characteristic length of a volume containing a representative sample of the cracks.This volume is called a representative volume element (RVE).For the Stage II stress-strain res- ponse [53],the model proposed in [59]was used.Accordingly,assuming the influence vector magnitude a to be proportional to the crack length, a=kl, (12.8)
on the surface and a represents the extent and direction of some “influence,” e.g., the disturbance of the strain fields, due to presence of the damage entity, referred to the same point on the surface. A “damage entity tensor” is then defined as d , ij i j S d an S = ∫ (12.5) with reference to a Cartesian coordinate system Xi. The influence vector can be resolved along the normal and tangential directions with respect to allowing ai to be expressed as , i i a an = (12.6) where the quantity a now represents a measure of the crack influence. From dimensional analysis, with dij taken to be dimensionless, a has dimensions of length. Drawing upon fracture mechanics, this length is in proportion to the crack length. For a fiber-bridged matrix crack, the crack length l can be expressed in multiples of the average interfiber spacing. Thus, f f 1 , v l kd v − = (12.7) where k is a constant, d is the fiber diameter, and vf is the fiber volume fraction. The expression in (12.7) is based on a hexagonal fiber arrangement. Similar expression will result from another assumption of fiber distribution in the cross section. It can now be inferred that the microstructural length scale for matrix microcracking is the fiber diameter. Note that, for an irregularly shaped crack surface, the interfiber spacing and, therefore, the fiber diameter will still be the length scale. The consequence of the presence of a matrix crack is generally in changing the composite’s deformational response, which is defined and measured at a larger length scale, e.g., the characteristic length of a volume containing a representative sample of the cracks. This volume is called a representative volume element (RVE). For the Stage II stress–strain response [53], the model proposed in [59] was used. Accordingly, assuming the influence vector magnitude a to be proportional to the crack length, a l = κ , (12.8) Chapter 12: Multiscale Modeling for Damage Analysis to assume that only the normal (crack opening) displacement matters, the crack surface. For the type of crack considered here, it is reasonable 545