《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_fiber115

J Mater Sci(2006)41:6800-6812 DOI10.1007/s10853-0060210-9 Multi-scale modeling in damage mechanics of composite materials Ramesh talreja C Springer Science+Business Media, LLC 2006 Abstract This paper addresses the multi-scale modeling changes with nano-scale elements(clay particles, nanotubes, aspects of damage in composite materials. The multiplicity etc. ) In all cases, effective engineering necessitates mathe- of the scales of the operating mechanisms is discussed and matical modeling of mechanisms and the consequent clarified by taking examples of damage in a unidirectional responses. From the early days of the rule of mixture type ceramic matrix composite and in a cross ply polymer estimates to today's multi-scale modeling, composite matrix composite laminate. Two multi-scale modeling mechanics has been concerned with continuous refinement strategies-the hierarchical and the synergistic-are of methods to accomplish this goal. The computational tools reviewed in the context of deformational response. Finally, available today have motivated a renewed emphasis on the" big picture"as it relates to the cost-effective manu- multi-scale modeling facturing of composite structures intended for long-term This article will focus on the multi-scale modeling performance is outlined and desired future direction in aspects of damage in composites. In order to specify the multi-scale modeling is discussed scope of the treatment, we shall first define the key terms- and failure. The context of th will be structural integrity and durability of composites under mechanical loading. The role of damage mechanics in the ""big picture"of cost effective manufacturing will be Introduction discussed at the end. where directior opment of modeling for this purpose will be outlined. Composite materials can be viewed as material systems with a wide range of possibilities for engineering design. As fracture, damage and failure-Definitions engineered materials, composites can be made"advanced fracture is conventionally understood to be"" of e.g. by using constituents that have high(advanced) prop- material, or at a more fundamental level. breakage erties, or by use of fiber architecture to create combinations of atomic bonds which manifests itself in formation of and anisotropy of properties not possible in single(mono- internal surfaces. Examples of fracture in composites are lithic)materials,or both. Composites can also be engineered fiber fragmentation, cracks in matrix, fiber/matrix deb- ts that modify, or impart properties to meet specific needs. Examples of onding and separation of bonded plies(delamination).The field of fracture mechanics concerns itself with conditions enrichment range from optical property modification by for enlargement of the surfaces of material separation. mixing curing agents to molecular-level morphological Damage is a collective reference to irreversible changes ought about by energy dissipating mechanisms, of which atomic bond breakage is an example. Unless specified R. Talreja(<) Department of Aerospace Engineering, Texas A&M University differently, damage is understood to refer to distributed ollege Station, TX 77843-3141, USA changes. Examples of damage are multiple fiber-bridged e-mail: Talreja@aero. tamu. edu matrix cracking in a unidirectional composite, multiple 2 Springer

Multi-scale modeling in damage mechanics of composite materials Ramesh Talreja Published online: 12 August 2006
Springer Science+Business Media, LLC 2006 Abstract This paper addresses the multi-scale modeling aspects of damage in composite materials. The multiplicity of the scales of the operating mechanisms is discussed and clarified by taking examples of damage in a unidirectional ceramic matrix composite and in a cross ply polymer matrix composite laminate. Two multi-scale modeling strategies––the hierarchical and the synergistic––are reviewed in the context of deformational response. Finally, the ‘‘big picture’’ as it relates to the cost-effective manu￾facturing of composite structures intended for long-term performance is outlined and desired future direction in multi-scale modeling is discussed. Introduction Composite materials can be viewed as material systems with a wide range of possibilities for engineering design. As engineered materials, composites can be made ‘‘advanced’’, e.g. by using constituents that have high (advanced) prop￾erties, or by use of fiber architecture to create combinations and anisotropy of properties not possible in single (mono￾lithic) materials, or both. Composites can also be engineered by ‘‘enrichment’’, e.g. by adding elements that modify, alter or impart properties to meet specific needs. Examples of enrichment range from optical property modification by mixing curing agents to molecular-level morphological changes with nano-scale elements (clay particles, nanotubes, etc.). In all cases, effective engineering necessitates mathe￾matical modeling of mechanisms and the consequent responses. From the early days of the rule of mixture type estimates to today’s multi-scale modeling, composite mechanics has been concerned with continuous refinement of methods to accomplish this goal. The computational tools available today have motivated a renewed emphasis on multi-scale modeling. This article will focus on the multi-scale modeling aspects of damage in composites. In order to specify the scope of the treatment, we shall first define the key terms–– fracture, damage and failure. The context of the treatment will be structural integrity and durability of composites under mechanical loading. The role of damage mechanics in the ‘‘big picture’’ of cost effective manufacturing will be discussed at the end, where directions for further devel￾opment of modeling for this purpose will be outlined. Fracture, damage and failure––Definitions Fracture is conventionally understood to be ‘‘breakage’’ of material, or at a more fundamental level, breakage of atomic bonds, which manifests itself in formation of internal surfaces. Examples of fracture in composites are fiber fragmentation, cracks in matrix, fiber/matrix deb￾onding and separation of bonded plies (delamination). The field of fracture mechanics concerns itself with conditions for enlargement of the surfaces of material separation. Damage is a collective reference to irreversible changes brought about by energy dissipating mechanisms, of which atomic bond breakage is an example. Unless specified differently, damage is understood to refer to distributed changes. Examples of damage are multiple fiber-bridged matrix cracking in a unidirectional composite, multiple R. Talreja (&) Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA e-mail: Talreja@aero.tamu.edu J Mater Sci (2006) 41:6800–6812 DOI 10.1007/s10853-006-0210-9 123

J Mater Sci(2006)41:68006812 intralaminar cracking in a laminate, local delamination In composite materials, the scales of inhomogeneities distributed in an interlaminar plane, and fiber/matrix inter-(reinforcements, additives, second phases, etc. ) embedded facial slip associated with multiple matrix cracking. The in the baseline material (matrix) determine the character- field of damage mechanics is concerned with conditions for istic scales of operation of the mechanisms of energy initiation and progression of distributed changes as well as dissipation. Although energy dissipation may also be consequences of those changes on the response of a material occurring at other(smaller)scales, e.g. the scale of the (and by implication, a structure)to external loading matrix material's microstructure, the dissipative mecha Failure is defined as the inability of a given material nisms associated with the inhomogeneities has usually system(and consequently, a structure made from it) to overriding influence on the composite behavior. For perform its design function. Fracture is one example of a instance in short-fiber polymer matrix composites, the size possible failure, but generally, a material could fracture of fiber diameter manifests the scale at which matrix cracks (locally) and still perform its design function. On suffering form, although energy dissipation may also occur at the damage(e. g. in the form of multiple cracking) a composite matrix polymer's molecular scale. The complexity intro- material could continue to carry loads, and thereby meet its duced by inhomogeneities in composite damage is in the load-bearing requirement, but fail to deform in a manner form of multiple scales of dissipative mechanism needed for its other design requirements such as vibration depending on the geometrical features of the inhomoge characteristics and deflection limits deities. Fo or the case of short fibers for instance. the matrix cracking from the fiber ends and the fiber/matrix debonding occur at two length scales, determined by the fiber diameter The multi-scale nature of damage and fiber length, respectively. For composite laminates, the thickness of identically oriented plies sets the scale for In a purist view, the first(basic) scale at which dissipative development of intralaminar cracking, while for formation mechanisms occur is the lowest possible material size- of these cracks the appropriate scale is given by the fiber scale. In reality, however, identifying this scale is limited diameter. Thus in modeling of a composite material by our ability to observe as well as to model and analyze behavior one faces a complex situation concerning the the mechanisms at the observed scale. The so-called length scales, and taking a hierarchical approach may not micro"scale is a reference to the scale at which entities be the most efficient way, as we shall discuss later or features within a material are observable by a certain In the following the multi-scale nature of damage be a few micrometers, if an electron microscope is used to further by examining two particular casee nd elaborated ype of microscope. Thus, for example, the micro scale can composite materials will be illustrated an observe entities such as cracks or crystalline slip within grains or at grain boundaries. The scale reduces by an order Unidirectional ceramic fiber reinforced ceramic matrix of magnitude if one focuses on dislocations observed by a composites( CMCs) transmission electron microscope. Today, the use of nano- scale elements(particles, fibers, tubes, etc. has moved the The first case for consideration is a unidirectionally basic scale further down where it is necessary to revisit the reinforced CMC subjected to uniform, monotonically fundamentals of continuum mechanics and to develop increasing tension in the fiber direction. The set of four modeling tools that can bridge the discrete-level descrip- micrographs from Sorensen and Talreja [7] shown in Fig tions (quantum mechanics) to continuum type(smeared- illustrate the progressive matrix cracking in a Sic fiber out)descriptions reinforced glass-ceramic matrix. The axial strains at which In an engineering approach, the purpose at hand should these pictures are taken by a surface replication technique guide the choice of the basic scale. Thus if the overall are indicated in each The first picture at 0. 15% (effective) characteristics of inelastic response are of strain shows an early stage of the matrix cracks lying interest, it would suffice to incorporate the energy dissi- normal to the(horizontal) fiber axis and not fully spanning pating mechanisms in a model, directly or indirectly, in an the specimen cross-section. Progressively at higher strain appropriate average sense, while if, for instance, a partic- levels the cracks are fully fiber-bridged ular material failure characteristic is aimed, the analysis number and reaching a saturation density (maximum may need to be conducted at the local physical scale of the number of cracks per unit axial length). A schematic relevant details of the mechanisms On the other hand, if overview of the stages of damage corresponding to the the purpose is to design a material, i.e. to engineer its stress-strain response is depicted in Fig. 2. In the first stage response or to provide it with certain functionalities, then it of no cracking the response is linearly elastic, followed by would be necessary to address scales where the material Stage II where the multiple matrix cracking renders the (micro) structure can be modified, manipulated or intruded. stress-strain response inelastic. The unloading modul 2 Springer

intralaminar cracking in a laminate, local delamination distributed in an interlaminar plane, and fiber/matrix inter￾facial slip associated with multiple matrix cracking. The field of damage mechanics is concerned with conditions for initiation and progression of distributed changes as well as consequences of those changes on the response of a material (and by implication, a structure) to external loading. Failure is defined as the inability of a given material system (and consequently, a structure made from it) to perform its design function. Fracture is one example of a possible failure, but generally, a material could fracture (locally) and still perform its design function. On suffering damage (e.g. in the form of multiple cracking) a composite material could continue to carry loads, and thereby meet its load-bearing requirement, but fail to deform in a manner needed for its other design requirements such as vibration characteristics and deflection limits. The multi-scale nature of damage In a purist view, the first (basic) scale at which dissipative mechanisms occur is the lowest possible material size￾scale. In reality, however, identifying this scale is limited by our ability to observe as well as to model and analyze the mechanisms at the observed scale. The so-called ‘‘micro’’ scale is a reference to the scale at which entities or features within a material are observable by a certain type of microscope. Thus, for example, the micro scale can be a few micrometers, if an electron microscope is used to observe entities such as cracks or crystalline slip within grains or at grain boundaries. The scale reduces by an order of magnitude if one focuses on dislocations observed by a transmission electron microscope. Today, the use of nano￾scale elements (particles, fibers, tubes, etc.) has moved the basic scale further down where it is necessary to revisit the fundamentals of continuum mechanics and to develop modeling tools that can bridge the discrete-level descrip￾tions (quantum mechanics) to continuum type (smeared￾out) descriptions. In an engineering approach, the purpose at hand should guide the choice of the basic scale. Thus if the overall (effective) characteristics of inelastic response are of interest, it would suffice to incorporate the energy dissi￾pating mechanisms in a model, directly or indirectly, in an appropriate average sense, while if, for instance, a partic￾ular material failure characteristic is aimed, the analysis may need to be conducted at the local physical scale of the relevant details of the mechanisms. On the other hand, if the purpose is to design a material, i.e. to engineer its response or to provide it with certain functionalities, then it would be necessary to address scales where the material (micro) structure can be modified, manipulated or intruded. In composite materials, the scales of inhomogeneities (reinforcements, additives, second phases, etc.) embedded in the baseline material (matrix) determine the character￾istic scales of operation of the mechanisms of energy dissipation. Although energy dissipation may also be occurring at other (smaller) scales, e.g. the scale of the matrix material’s microstructure, the dissipative mecha￾nisms associated with the inhomogeneities has usually an overriding influence on the composite behavior. For instance in short-fiber polymer matrix composites, the size of fiber diameter manifests the scale at which matrix cracks form, although energy dissipation may also occur at the matrix polymer’s molecular scale. The complexity intro￾duced by inhomogeneities in composite damage is in the form of multiple scales of dissipative mechanisms depending on the geometrical features of the inhomoge￾neities. For the case of short fibers, for instance, the matrix cracking from the fiber ends and the fiber/matrix debonding occur at two length scales, determined by the fiber diameter and fiber length, respectively. For composite laminates, the thickness of identically oriented plies sets the scale for development of intralaminar cracking, while for formation of these cracks the appropriate scale is given by the fiber diameter. Thus in modeling of a composite material’s behavior one faces a complex situation concerning the length scales, and taking a hierarchical approach may not be the most efficient way, as we shall discuss later. In the following the multi-scale nature of damage in composite materials will be illustrated and elaborated further by examining two particular cases. Unidirectional ceramic fiber reinforced ceramic matrix composites (CMCs) The first case for consideration is a unidirectionally reinforced CMC subjected to uniform, monotonically increasing tension in the fiber direction. The set of four micrographs from Sørensen and Talreja [7] shown in Fig. 1 illustrate the progressive matrix cracking in a SiC fiber reinforced glass-ceramic matrix. The axial strains at which these pictures are taken by a surface replication technique are indicated in each picture. The first picture at 0.15% strain shows an early stage of the matrix cracks lying normal to the (horizontal) fiber axis and not fully spanning the specimen cross-section. Progressively at higher strain levels the cracks are fully fiber-bridged, increasing in number and reaching a saturation density (maximum number of cracks per unit axial length). A schematic overview of the stages of damage corresponding to the stress-strain response is depicted in Fig. 2. In the first stage of no cracking the response is linearly elastic, followed by Stage II where the multiple matrix cracking renders the stress-strain response inelastic. The unloading moduli 123 J Mater Sci (2006) 41:6800–6812 6801

6802 J Mater Sci(2006)41:6800-6812 a SiC fiber reinforced glass. ceramic composite at different axial strains. Tensile loading was in the (horizontal) fiber direction. From Sorensen and Talreja [71 c01%112 c=035% c=0.50% and Talreja[7] ch for the stage II matrix cracking described above. In the following we shall use this example to illustrate the multi-scale nature of and discuss how the scales can be incorporated in a mechanics framework. Figure 3 shows schematically a fiber-bridged matrix tage/ I crack typical of the Stage Il damage in CMCs described above. This type of crack can be viewed as having three components, each causing dissipation of energy by a sep- arate mechanism. One component is the crack surface formed by breakage of atomic bonds in the matrix. The second component consists of the fiber/matrix disbonds, ig. 2 A schematic overview of the three stages of stress-strain which occur by breakage of atomic bonds at the Sorensen and Talreja [7] Finally, the third energy-dissipating component frictional sliding at the fiber/matrix interface that (axial Youngs modulus and Poisson's ratio) show steady debonding. Each component has a characteristic geometry degradation with increasing crack number density in Stage and an associated""influenceto signify its presence. In Il, which initiates at 0. 13% strain and extends to 0.5% Talreja [9] the three mechanisms-matrix cracking, deb. strain. Beyond this strain Stage Ill occurs where the fric onding and sliding-were treated as individual damage modes and were characterized by symmetric second order tional sliding at the fiber/matrix interface becomes signi tensors, incorporating appropriate measures of influence icant. At a later part of this stage, beyond 0.7% strain progressive fiber breakage takes place leading to localiza- tion of damage and subsequent failure. The Stage II progressive cracking can be treated with amage mechanics, a field that concerns itself with char debonding acterization of damage, evolution of damage and relating damage to material response. Talreja [9] presented char acterization of damage modes in ceramic matrix com ites(matrix cracking, fiber/matrix debonding and fib matrix sliding) and derived for each mode expressions for he changes in moduli as functions of damage. Sorensen Fig 3 Schematic illustration of a fiber-bridged matrix crack 2 Springer

(axial Young’s modulus and Poisson’s ratio) show steady degradation with increasing crack number density in Stage II, which initiates at 0.13% strain and extends to 0.5% strain. Beyond this strain Stage III occurs where the fric￾tional sliding at the fiber/matrix interface becomes signif￾icant. At a later part of this stage, beyond 0.7% strain, progressive fiber breakage takes place leading to localiza￾tion of damage and subsequent failure. The Stage II progressive cracking can be treated with damage mechanics, a field that concerns itself with char￾acterization of damage, evolution of damage and relating damage to material response. Talreja [9] presented char￾acterization of damage modes in ceramic matrix compos￾ites (matrix cracking, fiber/matrix debonding and fiber/ matrix sliding) and derived for each mode expressions for the changes in moduli as functions of damage. Sørensen and Talreja [7] used that modeling approach for the Stage II matrix cracking described above. In the following we shall use this example to illustrate the multi-scale nature of damage and discuss how the scales can be incorporated in a damage mechanics framework. Figure 3 shows schematically a fiber-bridged matrix crack typical of the Stage II damage in CMCs described above. This type of crack can be viewed as having three components, each causing dissipation of energy by a sep￾arate mechanism. One component is the crack surface formed by breakage of atomic bonds in the matrix. The second component consists of the fiber/matrix disbonds, which occur by breakage of atomic bonds at the interface. Finally, the third energy-dissipating component is the frictional sliding at the fiber/matrix interface that follows debonding. Each component has a characteristic geometry and an associated ‘‘influence’’ to signify its presence. In Talreja [9] the three mechanisms––matrix cracking, deb￾onding and sliding––were treated as individual damage modes and were characterized by symmetric second order tensors, incorporating appropriate measures of influence Fig. 1 Surface micrographs of a SiC fiber reinforced glass￾ceramic composite at different axial strains. Tensile loading was in the (horizontal) fiber direction. From Sørensen and Talreja [7] εL σ Stage I Stage II Stage III εT Fig. 2 A schematic overview of the three stages of stress–strain response in a SiC fiber reinforced glass-ceramic composite. Based on Sørensen and Talreja [7] sliding debonding Fig. 3 Schematic illustration of a fiber-bridged matrix crack 123 6802 J Mater Sci (2006) 41:6800–6812

J Mater Sci(2006)41:68006812 for each mode. We shall discuss those damage modes here response, which is defined and measured at a larger length with a view to bringing out the multi-scale features. 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 Il stress-strain response [7] used the model proposed in A matrix crack can be viewed as a pair of internal surfaces Talreja [9]. Accordingly, assuming the influence vector in a composite that are able to perturb the stress state in a magnitude a to be proportional to the crack length region around the surfaces by conducting displacement (i.e. separation of surfaces) from the undeformed configu- a=al ation. The surface separation per unit of applied external load depends on the size and shape of the surfaces as well where a is a constant representing the constraint to the as on the constraint, if any, imposed by the surroundings. crack surface displacement. This constant equals zero when For a matrix crack in a unidirectional CMc the constraint the constraint allows no crack separation, while it increases comes from the bridging fibers as well as from the stiff- as the constraint reduces ening effect of fibers in the matrix surrounding the crack. From Eqs. (1),(2)and(4), the damage entity tensor for As described in Talreja [9] a single crack can be charac- ng Is terized by a"damage entity tensor, given by 山=/ands where t is the specimen thickness (or the through-thickness characteristic dimension of the crack) where ai are components of an"influence vector'placed The macro-level deformational response is derived from on a crack of surface area S at a point with outward unit n energy density function normal vector of components n; The influence vector can and damage states. The matrix-cracking damage state is be resolved along the crack surface normal and tangential characterized by 19) directions. For the type of crack considered here it is rea- onable to assume that only the normal( (crack opening)D=∑雩 displacement matters, allowing ai to be expressed as where V is the rve volume of a representative volume (2) element(RVE) over which the summation is conducted Substituting Eq (5)in Eq(6), one obtains where the quantity a now represents a measure of the crack influence From dimensional analysis, with d taken to be Dmc=and<fI> dimensionless, a has dimensions of length. Drawing upon fracture mechanics this length is in proportion to the crack where Dmc= Dm, the only surviving component of the length. For a fiber-bridged matrix crack the crack length I damage mode tensor, f is the fraction of RVE width can be expressed in multiples of the average inter-fiber spanned by a crack and n is the crack number density, i.e spacing. Thus, the number of cracks per unit volume, and A is the cross sectional area. The quantity within the brackets <>is (3) averaged over the RVE volume The matrix crack length, Eq. (3), thus appears in the damage descriptor, Eq (7). Also, as shown in [9). the crack where k is a constant, d is fiber diameter and vy is the fiber length also governs the elastic constants at a given crack volume fraction. The expression in Eq.(3)is based on a density n. For instance, the axial Youngs modulus can be hexagonal fiber arrangement. Similar expression will result written as from other assumption of fiber distribution in the on. We can now infer that the microstructural Eu=E(I-cnd) scale for matrix microcracking is the fiber diameter. that for an irregularly shaped crack surface the inter-fiber where c is a constant and the superscript 0 is for the initial spacing, and therefore the fiber diameter, will still be the value length scale In characterizing matrix cracks as a damage mode, Eq The consequence of the presence of a matrix crack is (7), no specific account is made of the associated fiber/ generally in changing the composite's deformational matrix debonding and sliding mechanisms. These can be

for each mode. We shall discuss those damage modes here with a view to bringing out the multi-scale features. 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 configu￾ration. 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 stiff￾ening effect of fibers in the matrix surrounding the crack. As described in Talreja [9] a single crack can be charac￾terized by a ‘‘damage entity tensor’’, given by dij ¼ Z S ainjdS ð1Þ where ai are components of an ‘‘influence vector’’ placed on a crack of surface area S at a point with outward unit normal vector of components nj. The influence vector can be resolved along the crack surface normal and tangential directions. For the type of crack considered here it is rea￾sonable to assume that only the normal (crack opening) displacement matters, allowing ai to be expressed as ai ¼ ani ð2Þ 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 inter-fiber spacing. Thus, l ¼ kd 1
ffiffiffiffi vf p ffiffiffiffi vf p ð3Þ where k is a constant, d is fiber diameter and vf is the fiber volume fraction. The expression in Eq. (3) is based on a hexagonal fiber arrangement. Similar expression will result from other assumption of fiber distribution in the cross￾section. We can now infer that the microstructural length scale for matrix microcracking is the fiber diameter. Note that for an irregularly shaped crack surface the inter-fiber 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 [7] used the model proposed in Talreja [9]. Accordingly, assuming the influence vector magnitude a to be proportional to the crack length, a ¼ al ð4Þ where a is a constant representing the constraint to the crack surface displacement. This constant equals zero when the constraint allows no crack separation, while it increases as the constraint reduces. From Eqs. (1), (2) and (4), the damage entity tensor for matrix cracking is dmc ij ¼ al 2 tninj ð5Þ where t is the specimen thickness (or the through-thickness characteristic dimension of the crack). The macro-level deformational response is derived from a strain energy density function that depends on the strain and damage states. The matrix-cracking damage state is characterized by [9], Dmc ij ¼ 1 V Xdmc ij ð6Þ where V is the RVE volume of a representative volume element (RVE) over which the summation is conducted. Substituting Eq. (5) in Eq. (6), one obtains Dmc ¼ agA\fl[ ð7Þ where Dmc ¼ Dmc 11 , the only surviving component of the damage mode tensor, f is the fraction of RVE width spanned by a crack and g is the crack number density, i.e. the number of cracks per unit volume, and A is the cross￾sectional area. The quantity within the brackets < > is averaged over the RVE volume. The matrix crack length, Eq. (3), thus appears in the damage descriptor, Eq. (7). Also, as shown in [9], the crack length also governs the elastic constants at a given crack density g. For instance, the axial Young’s modulus can be written as E11 ¼ E0 11ð1
cglÞ ð8Þ where c is a constant and the superscript 0 is for the initial value. In characterizing matrix cracks as a damage mode, Eq. (7), no specific account is made of the associated fiber/ matrix debonding and sliding mechanisms. These can be 123 J Mater Sci (2006) 41:6800–6812 6803

6804 J Mater Sci(2006)41:6800-6812 considered separately and then accounted for by their where interactions with the matrix cracks 9]. Discussions of these mechanisms follow c=c+kdn Interfacial debonding where d, is the ratio of the debond length to the crack length and k, is a constant. Here a fixed ratio of the number of The fiber/matrix interface can debond due to several debonds per unit of matrix crack length has been assumed causes. Essentially, a stress normal to fibers or a shear From Eqs.(9)and (10) it can be seen that the debond stress along fibers or a combination of the two, must exist length does not enter into the rve response directly but via for the bond to fail. These stresses can be generated by a its ratio to the crack length, suggesting that the governing fiber break or brought into play by an approaching matrix length for this response is the crack length crack. Alternatively, a preexisting flaw at the fiber surface or an imperfect fiber, or its misalignment, can produce Interfacial sliding those stresses. If debonds are produced without interaction with matrix cracks, then they can be characterized in a Interfacial sliding occurs when fibers and matrix remain in manner similar to matrix cracks. A characterization of such contact after debonding of the interface and undergo distributed debonding is given in Talreja [9] based on unequal displacements. Talreja [9] defined a measure of the certain simplifying assumptions. The only surviving dam- slip at the interface as the area swept off by the relative age mode tensor component for this case is D2 and its displacement of one constituent over the other and form is the same as that of Dmc in Eq (7). Thus the debond expressed this measure in terms of a slippage vector. A length and the debond number density enter into the slippage tensor was then constructed as a dyadic product of damage mode description. The debond length will depend the slippage vector with itself to account for the insensi on the characteristic flaw length, which in turn depends on tivity of the material response to the direction of slip. As in the manufacturing process. Unless the ability of the man- the case of debonding discussed above, when sliding ufacturing process to produce interfacial Alaws somehow occurs in conjunction with matrix cracking, the slip dam- depends on the composite microstructure, no microstruc- age tensor, which represents this damage mode averaged tural length scale can be identified for the debonding over the RVE, turns out to depend on the average matrix mechanism crack length. In fact it depends explicitly on the average For the case of a matrix crack initiating debonding and crack opening displacement, which in turn depends on the then merging with the debond crack, further driving force average crack length. The only surviving component of the to the advancement of the debond crack comes from the slip damage tensor can be written as [9] opening displacement of the matrix crack. The damage configuration of interest then is not the debond crack by Dsl (11 itself but a combined matrix-debond crack. The latter can be viewed as a fiber-bridged matrix crack. discussed above,with the constraint to its surface displacement where d is the fiber diameter, vy is the fiber volume fraction now modified by the presence of debonding. Then the and ca is the crack opening displacement and the quantity constant a in Eqs.(4),(5)and(7)may be changed to within the brackets <>is averaged over the rve volume another value, resulting in a change of the constant c in Assuming the crack opening displacement to be pro- Eq.(8) portional to the crack length we may rewrite Eq(11)as Thus for debonding that occurs in conjunction with with a modified influence. This length can still be where s is a constant depending on the fiber volume expressed by Eq.(3), giving the fiber diameter as the fraction and fiber stiffness. The fiber diameter is placed microstructural length scale within the brackets to allow for its variation. Equation(12) Specific treatments of debonding by itself and of indicates that this damage mode depends directly and debonding in conjunction with matrix cracking are given in trongly on the fiber diameter in addition to depending or Talreja [9]. Based on that work the axial modulus for th the matrix crack length, which in turn is expressible in latter case can be modified from Eq.( 8 )to be terms of the fiber diameter. as in microstructural length scale also in this case is the fiber E1=E1(1-cn) diameter. Note that the fiber length over which sliding 2 Springer

considered separately and then accounted for by their interactions with the matrix cracks [9]. Discussions of these mechanisms follow. Interfacial debonding The fiber/matrix interface can debond due to several causes. Essentially, a stress normal to fibers or a shear stress along fibers, or a combination of the two, must exist for the bond to fail. These stresses can be generated by a fiber break or brought into play by an approaching matrix crack. Alternatively, a preexisting flaw at the fiber surface or an imperfect fiber, or its misalignment, can produce those stresses. If debonds are produced without interaction with matrix cracks, then they can be characterized in a manner similar to matrix cracks. A characterization of such distributed debonding is given in Talreja [9] based on certain simplifying assumptions. The only surviving dam￾age mode tensor component for this case is D22 and its form is the same as that of Dmc in Eq. (7). Thus the debond length and the debond number density enter into the damage mode description. The debond length will depend on the characteristic flaw length, which in turn depends on the manufacturing process. Unless the ability of the man￾ufacturing process to produce interfacial flaws somehow depends on the composite microstructure, no microstruc￾tural length scale can be identified for the debonding mechanism. For the case of a matrix crack initiating debonding and then merging with the debond crack, further driving force to the advancement of the debond crack comes from the opening displacement of the matrix crack. The damage configuration of interest then is not the debond crack by itself but a combined matrix-debond crack. The latter can be viewed as a fiber-bridged matrix crack, discussed above, with the constraint to its surface displacement now modified by the presence of debonding. Then the constant a in Eqs. (4), (5) and (7) may be changed to another value, resulting in a change of the constant c in Eq. (8). Thus for debonding that occurs in conjunction with matrix cracking the determining length associated with the damage mode is still the matrix crack length l, although with a modified influence. This length can still be expressed by Eq. (3), giving the fiber diameter as the microstructural length scale. Specific treatments of debonding by itself and of debonding in conjunction with matrix cracking are given in Talreja [9]. Based on that work the axial modulus for the latter case can be modified from Eq. (8) to be E11 ¼ E0 11ð1
c0 glÞ ð9Þ where c0 ¼ c þ kldl ð10Þ where dl is the ratio of the debond length to the crack length and kl is a constant. Here a fixed ratio of the number of debonds per unit of matrix crack length has been assumed. From Eqs. (9) and (10) it can be seen that the debond length does not enter into the RVE response directly but via its ratio to the crack length, suggesting that the governing length for this response is the crack length. Interfacial sliding Interfacial sliding occurs when fibers and matrix remain in contact after debonding of the interface and undergo unequal displacements. Talreja [9] defined a measure of the slip at the interface as the area swept off by the relative displacement of one constituent over the other and expressed this measure in terms of a slippage vector. A slippage tensor was then constructed as a dyadic product of the slippage vector with itself to account for the insensi￾tivity of the material response to the direction of slip. As in the case of debonding discussed above, when sliding occurs in conjunction with matrix cracking, the slip dam￾age tensor, which represents this damage mode averaged over the RVE, turns out to depend on the average matrix crack length. In fact it depends explicitly on the average crack opening displacement, which in turn depends on the average crack length. The only surviving component of the slip damage tensor can be written as [9] Dsl ¼ p2d4g2 64v2 f \c2 d[ ð11Þ where d is the fiber diameter, vf is the fiber volume fraction and cd is the crack opening displacement and the quantity within the brackets < > is averaged over the RVE volume. Assuming the crack opening displacement to be pro￾portional to the crack length we may rewrite Eq. (11) as Dsl ¼ ng2 \d4 l 2 [ ð12Þ where n is a constant depending on the fiber volume fraction and fiber stiffness. The fiber diameter is placed within the brackets to allow for its variation. Equation (12) indicates that this damage mode depends directly and strongly on the fiber diameter in addition to depending on the matrix crack length, which in turn is expressible in terms of the fiber diameter, as in Eq. (3). Thus the microstructural length scale also in this case is the fiber diameter. Note that the fiber length over which sliding 123 6804 J Mater Sci (2006) 41:6800–6812