《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 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

J Mater Sci(2006)41:68006812 6805 occurs is not a characteristic dimension of the mechanism when it occurs in conjunction with matrix cracking. Ply cracking in laminates Figure 4 shows an X-ray radiograph of a carbon-epoxy cross-ply laminate after being subjected to tension-tension cycling. In this two-dimensional view the horizontal lines are images of cracks in the 90-plies, while the vertical lines indicate cracks (also called axial splits) that lie in the Fig. 5 A schematic illustration of the cracks and delaminations seen 0-plies [3]. The shaded areas are sites of interlaminar in the X-ray radiograph, Fig. 4 cracks(delaminations), which are depicted in Fig. 5. For the sake of our discussion on length scales of damage we shall primarily focus on ply cracking Figure 6 illustrates multiple matrix cracking in a ply of an arbitrary orientation 0 with respect to the 90-direction The cracks are shown at a mutual spacing s, which repre sents the average crack spacing in a rve. Using a second order tensor characterization for this mode of damage [8 glves D Fig. 6 Illustration of multiple cracking in a general off-axis ply of a where the superscript pc stands for ply cracking, K is a ply constraint parameter, te is the thickness of the cracked ply and t is the laminate thickness. The components n; of the unit vector normal on a crack surface are given by (cos 0, sin 0, 0) where 0 as shown in fig. 6 is the crack inclination The laminate stiffness matrix in the presence of a fixed tate of ply cracking is given by [8] and delaminations in a cross ply laminate after fatigue /3 r lal cracks where the indices p and g take values from 1 to 6 in Fig. 4 An X-ray radiograph showing transverse cracks accordance with the Voigt notation. The superscript 0 on

occurs is not a characteristic dimension of the mechanism when it occurs in conjunction with matrix cracking. Ply cracking in laminates Figure 4 shows an X-ray radiograph of a carbon-epoxy cross-ply laminate after being subjected to tension-tension cycling. In this two-dimensional view the horizontal lines are images of cracks in the 90-plies, while the vertical lines indicate cracks (also called axial splits) that lie in the 0-plies [3]. The shaded areas are sites of interlaminar cracks (delaminations), which are depicted in Fig. 5. For the sake of our discussion on length scales of damage we shall primarily focus on ply cracking. Figure 6 illustrates multiple matrix cracking in a ply of an arbitrary orientation h with respect to the 90-direction. The cracks are shown at a mutual spacing s, which repre￾sents the average crack spacing in a RVE. Using a second order tensor characterization for this mode of damage [8] gives Dpc ij ¼ jt 2 c st cos h ninj ð13Þ where the superscript pc stands for ply cracking, j is a ply constraint parameter, tc is the thickness of the cracked ply and t is the laminate thickness. The components ni of the unit vector normal on a crack surface are given by ni ¼ ðcos h;sin h; 0Þ ð14Þ where h as shown in Fig. 6 is the crack inclination. The laminate stiffness matrix in the presence of a fixed state of ply cracking is given by [8] Cpq ¼ C0 pq
CD pq ð15Þ where the indices p and q take values from 1 to 6 in accordance with the Voigt notation. The superscript 0 on Fig. 4 An X-ray radiograph showing transverse cracks, axial cracks and delaminations in a cross ply laminate after fatigue [3] Fig. 5 A schematic illustration of the cracks and delaminations seen in the X-ray radiograph, Fig. 4 Fig. 6 Illustration of multiple cracking in a general off-axis ply of a laminate 123 J Mater Sci (2006) 41:6800–6812 6805

6806 J Mater Sci(2006)41:6800-6812 the stiffness matrix indicates initial value while d indicate the contribution due to damage The stiffness matrix reduction due to damage can be expressed in a first approximation as where K' is a modified ply constraint parameter, which depends on the ply constraint, the crack orientation and ply 圈 The length scale variable entering the damage descrip tor, Eq.(13), and consequently, the elastic response change,Eq.(16), is the crack dimension, which for fully Fig. 7 Illustration of matrix cracks(left) and delamination (right)in developed ply cracks, as assumed here, is the ply thickness an angle ply laminate te. The other crack dimension along the fiber axis in the ply extends as far as the imposed stress acts and is therefore not scale is speculative at present. It appears, however, that the the characteristic length scale of the cracking mechanism. growth of the delamination is mainly along the two Expressed differently, the crack surface displacement, orthogonal ply crack directions, suggesting thereby two which is the cause of stress perturbations and thereby the length scales. These length scales may be described as the elastic response changes, depends on the crack dimension two principal directions of an ellipse, which may be taken through the ply thickness. to approximate the delamination geometry Although the ply cracks are assumed for simplicity to be sharp-tipped as illustrated in Fig. 6, in reality they must get blunted by merging with the local delamination, i. e. the Multi-scale modeling of damage for elastic response separation of plies at the interface, caused by the intense ress field carried by the approaching ply crack fronts. The Damage entity size and microstructural length scales extent of the delamination cracks along the ply interfaces must depend on the interfacial bond strength as well as on In the Section"The Multi-scale Nature of Damage"we the ply crack length te. In fact this situation is analogous to have examined the governing dimensions of the damage the fiber/matrix debonding in conjunction with matrix entities from the viewpoint of elastic response in the cracking in unidirectional CMCs, discussed in the section presence of damage. We have looked at two specific cases Interfacial Debonding"above. Drawing upon that anal- of damage in unidirectional CMCs and in PMC laminates ogy we deduce that the delamination length enters the The framework within which we have analyzed the issue of analysis not directly but via its ratio to the ply crack length characteristic lengths is the continuum damage mechanics te. Thus, once again the relevant length scale variable is the using characterization of damage with second order ten total cracking ply thickness te. ors.This particular representation of damage, in the form When cracking occurs in more than one ply orientation, used here, provides a consistent characterization of the multiple length scales result with each length scale variable basic damage entity involved in each case, e.g., a matrix equaling to the combined thickness of the set of consecu- crack in a CMC and a ply crack in a PMC laminate. The tive cracking plies of the corresponding orientation. Also, crux of the characterization is the "influence"vector, the delamination associated with each ply cracking con- which provides a relevant measure of the action induced by tributes to the effect on the elastic response ratio of the presence of the damage entity. By expressing the the associated delamination length to the ply crack length. magnitude of this vector in terms of the characteristic and gure 7 illustrates the ply(matrix)cracking and delani- governing dimension of the damage entity the"essence nation in an angle ply laminate of the damage entity is carried into the damage entity Finally, let us return to the delamination mode observed tensor. This dimension, when properly identified and in fatigue of cross ply laminates depicted in Figs. 4 and 5. related to the microstructural entities, provides the lengt As illustrated in Fig. 5 this delamination occurs locally at scale associated with the damage mechanism considered the intersection of cracks in the two orthogonal orientations The relevance of the length scale to determining the elastic in adjacent plies. The cause of this delamination and the response affected by that damage becomes clear effect of its presence have not been adequately analyzed. when we examine the response measured over the RVE. Therefore, any inference regarding its characteristic length For this a damage mode tensor, which acts as an internal 2 Springer

the stiffness matrix indicates initial value while D indicates the contribution due to damage. The stiffness matrix reduction due to damage can be expressed in a first approximation as CD pq ¼ j0 st t 2 c ð16Þ where j¢ is a modified ply constraint parameter, which depends on the ply constraint, the crack orientation and ply properties. The length scale variable entering the damage descrip￾tor, Eq. (13), and consequently, the elastic response change, Eq. (16), is the crack dimension, which for fully developed ply cracks, as assumed here, is the ply thickness tc. The other crack dimension along the fiber axis in the ply extends as far as the imposed stress acts and is therefore not the characteristic length scale of the cracking mechanism. Expressed differently, the crack surface displacement, which is the cause of stress perturbations and thereby the elastic response changes, depends on the crack dimension through the ply thickness. Although the ply cracks are assumed for simplicity to be sharp-tipped, as illustrated in Fig. 6, in reality they must get blunted by merging with the local delamination, i.e. the separation of plies at the interface, caused by the intense stress field carried by the approaching ply crack fronts. The extent of the delamination cracks along the ply interfaces must depend on the interfacial bond strength as well as on the ply crack length tc. In fact this situation is analogous to the fiber/matrix debonding in conjunction with matrix cracking in unidirectional CMCs, discussed in the section ‘‘Interfacial Debonding’’ above. Drawing upon that anal￾ogy we deduce that the delamination length enters the analysis not directly but via its ratio to the ply crack length tc. Thus, once again the relevant length scale variable is the total cracking ply thickness tc. When cracking occurs in more than one ply orientation, multiple length scales result with each length scale variable equaling to the combined thickness of the set of consecu￾tive cracking plies of the corresponding orientation. Also, the delamination associated with each ply cracking con￾tributes to the effect on the elastic response via the ratio of the associated delamination length to the ply crack length. Figure 7 illustrates the ply (matrix) cracking and delami￾nation in an angle ply laminate. Finally, let us return to the delamination mode observed in fatigue of cross ply laminates depicted in Figs. 4 and 5. As illustrated in Fig. 5 this delamination occurs locally at the intersection of cracks in the two orthogonal orientations in adjacent plies. The cause of this delamination and the effect of its presence have not been adequately analyzed. Therefore, any inference regarding its characteristic length scale is speculative at present. It appears, however, that the growth of the delamination is mainly along the two orthogonal ply crack directions, suggesting thereby two length scales. These length scales may be described as the two principal directions of an ellipse, which may be taken to approximate the delamination geometry. Multi-scale modeling of damage for elastic response Damage entity size and microstructural length scales In the Section ‘‘The Multi-scale Nature of Damage’’ we have examined the governing dimensions of the damage entities from the viewpoint of elastic response in the presence of damage. We have looked at two specific cases of damage in unidirectional CMCs and in PMC laminates. The framework within which we have analyzed the issue of characteristic lengths is the continuum damage mechanics using characterization of damage with second order ten￾sors. This particular representation of damage, in the form used here, provides a consistent characterization of the basic damage entity involved in each case, e.g., a matrix crack in a CMC and a ply crack in a PMC laminate. The crux of the characterization is the ‘‘influence’’ vector, which provides a relevant measure of the action induced by the presence of the damage entity. By expressing the magnitude of this vector in terms of the characteristic and governing dimension of the damage entity the ‘‘essence’’ of the damage entity is carried into the damage entity tensor. This dimension, when properly identified and related to the microstructural entities, provides the length scale associated with the damage mechanism considered. The relevance of the length scale to determining the elastic response affected by that damage mode becomes clear when we examine the response measured over the RVE. For this a damage mode tensor, which acts as an internal Matrix crack θ θ Delamination Fig. 7 Illustration of matrix cracks (left) and delamination (right) in an angle ply laminate 123 6806 J Mater Sci (2006) 41:6800–6812

J Mater Sci(2006)41:68006812 6807 state variable in a continuum damage framework, is Hierarchical vS Synergistic multi-scale approach considered. We have examined the damage mode tensor in a simple form, such as that in Eq (6), which is the volume It would be fair to say at the outset that the hierarchical average of the dama the RvE. This multi-scale approach is intuitively logical. For a complex does not account for the damage entity distribution and can composite architecture, which is quite often the case in therefore be used only for the RVE-averaged, i. e the meso- practical applications, one can think of starting with the scale, response, and not for de scribing damage evolution. smallest basic unit-a discrete fiber embedded in matrix Thus, we have examined the length scales of damage and proceed to the level of a representative unit of col within the context of the elastic response lective fiber arrangement. The basic unit can be analyzed as cance of the length scales of damage? The basic concept fiber and matrix are considered, or three, if an imlentao i The next question to address is: What is the signifi- a piecewise homogeneous continuum, with two regions, behind length scales appears to be the intuitive idea that layer is added. The result(stress, strain, temperature, etc. effects seen at a given observation"window"(e.g. RVe can then be averaged in some sense over a representative size)must be determined by events occurring at dimen- unit to get a description for the homogenized medium. sions smaller than the window size. Implicit is the Several models for doing this exist, e. g, the Mori-Tanaka assumption that those events are associated with certain method. These models aim at bridging the two scales-the discrete entities such as grains in a polycrystalline mate- scale of the basic unit and the rve scale. Generally, the rial, and that the action of those entities and interactions issue of uniqueness remains unresolved in the sense of between them when averaged over the window size pro- representation of the collective fiber effect. There is as yet vides the"response" variables applicable at that scale. no precise and rigorous definition of a representative unit Carrying this logic one stepbehind'would suggest that for a general case, which is the source of lack of unique he response at the scale of the discrete entities in a rve ness. A logical extension of the discrete-to-collectiv could be given by the sub-entities lying within those bridging of the fiber-matrix case to higher scales produces entities. Thus, if a grain is viewed as an entity, then the the hierarchical approach in multi-scale modeling. One can dislocations within the grain could be the sub-entities. argue if this approach is efficient, in spite of This move to smaller and smaller size-scales could in Historically, the hierarchical approach has not principle have no end other than the limit set by the tools other approaches. A structural analyst has worked with of observation and analysis available at a given time. In macro-level descriptions of material behavior, e. g, the engineering science, in contrast to"pure"sciences, one classical laminate plate theory, and has looked micro- takes a pragmatic approach driven by the application or level information as needed. A materials developer, on the need at hand. From this point of view, one must consider other hand, has focused on effects of constituents and their the purpose first and go as far down in scales as needed. microstructural arrangements on properties. In recent years Thus if the purpose is to determine the elastic response the seemingly abundant computer power has motivated the hanges induced by damage in a composite material, then hierarchical approach with the hope of integrating mater one must go as much down in length scales as necessary als design and structural analysis to determine the reversible deformation (or stress) related Our objective here is to examine approaches for multi effects. but no further scale analysis of damage in composites. A first thought The next question is whether a hierarchy of length would be to conduct damage initiation and progression scales can be identified. What we have illustrated by the analysis as a part of the hierarchical multi-scale approach. discussion of the two cases of composites with damage is It turns out not to that a simple hierarchy of length scales does not exist. confronting this approach will be discussed below, along Instead, a complex damage mode may involve more than with the merits of an alternative approach. one governing length, e.g., the matrix-crack length and Let us first take an overall view of the multi-scale the interfacial sliding length for a fiber-bridged crack. approach. Figure 8 illustrates, from left, an object of Also, multiple damage modes may operate simulta- structural integrity assessment within which a region of neously, and interactively, leading to multiple length potential criticality (failure) exists. This region (a scales, e.g., in the case of multiple off-axis plies in a substructure) is analyzed to determine the loading on its laminate. These considerations suggest that an alternative boundary. Next step is to examine how this loading is needed to the strategy of starting at the smallest length induces damage. This step requires analyzing heteroge scale and working up the scale hierarch ly. Talreja [10] neities(microstructure, generally), which govern initiation proposed one such strategy, the so-called"synergistic" of damage. Simple examples are debonding of fibers from damage mechanics. The following discussion will address matrix and matrix cracking from broken fiber ends the two strategies. The analysis of local stress/strain fields to determine 2 Springer

state variable in a continuum damage framework, is considered. We have examined the damage mode tensor in a simple form, such as that in Eq. (6), which is the volume average of the damage entity tensor over the RVE. This does not account for the damage entity distribution and can therefore be used only for the RVE-averaged, i.e. the meso￾scale, response, and not for describing damage evolution. Thus, we have examined the length scales of damage within the context of the elastic response. The next question to address is: What is the signifi- cance of the length scales of damage? The basic concept behind length scales appears to be the intuitive idea that effects seen at a given observation ‘‘window’’ (e.g. RVE size) must be determined by events occurring at dimen￾sions smaller than the window size. Implicit is the assumption that those events are associated with certain discrete entities such as grains in a polycrystalline mate￾rial, and that the action of those entities and interactions between them when averaged over the window size pro￾vides the ‘‘response’’ variables applicable at that scale. Carrying this logic one step ‘‘behind’’ would suggest that the response at the scale of the discrete entities in a RVE could be given by the sub-entities lying within those entities. Thus, if a grain is viewed as an entity, then the dislocations within the grain could be the sub-entities. This move to smaller and smaller size-scales could in principle have no end other than the limit set by the tools of observation and analysis available at a given time. In engineering science, in contrast to ‘‘pure’’ sciences, one takes a pragmatic approach driven by the application or need at hand. From this point of view, one must consider the purpose first and go as far down in scales as needed. Thus if the purpose is to determine the elastic response changes induced by damage in a composite material, then one must go as much down in length scales as necessary to determine the reversible deformation (or stress) related effects, but no further. The next question is whether a hierarchy of length scales can be identified. What we have illustrated by the discussion of the two cases of composites with damage is that a simple hierarchy of length scales does not exist. Instead, a complex damage mode may involve more than one governing length, e.g., the matrix-crack length and the interfacial sliding length for a fiber-bridged crack. Also, multiple damage modes may operate simulta￾neously, and interactively, leading to multiple length scales, e.g., in the case of multiple off-axis plies in a laminate. These considerations suggest that an alternative is needed to the strategy of starting at the smallest length scale and working up the scale hierarchy. Talreja [10] proposed one such strategy, the so-called ‘‘synergistic’’ damage mechanics. The following discussion will address the two strategies. Hierarchical vs. Synergistic multi-scale approach It would be fair to say at the outset that the hierarchical multi-scale approach is intuitively logical. For a complex composite architecture, which is quite often the case in practical applications, one can think of starting with the smallest basic unit––a discrete fiber embedded in matrix–– and proceed to the level of a representative unit of col￾lective fiber arrangement. The basic unit can be analyzed as a piecewise homogeneous continuum, with two regions, if fiber and matrix are considered, or three, if an interfacial layer is added. The result (stress, strain, temperature, etc.) can then be averaged in some sense over a representative unit to get a description for the homogenized medium. Several models for doing this exist, e.g., the Mori-Tanaka method. These models aim at bridging the two scales––the scale of the basic unit and the RVE scale. Generally, the issue of uniqueness remains unresolved in the sense of representation of the collective fiber effect. There is as yet no precise and rigorous definition of a representative unit for a general case, which is the source of lack of unique￾ness. A logical extension of the discrete-to-collective bridging of the fiber–matrix case to higher scales produces the hierarchical approach in multi-scale modeling. One can argue if this approach is efficient, in spite of its logic. Historically, the hierarchical approach has not preceded other approaches. A structural analyst has worked with macro-level descriptions of material behavior, e.g., the classical laminate plate theory, and has looked for micro￾level information as needed. A materials developer, on the other hand, has focused on effects of constituents and their microstructural arrangements on properties. In recent years the seemingly abundant computer power has motivated the hierarchical approach with the hope of integrating materi￾als design and structural analysis. Our objective here is to examine approaches for multi￾scale analysis of damage in composites. A first thought would be to conduct damage initiation and progression analysis as a part of the hierarchical multi-scale approach. It turns out not to be that straightforward. The issues confronting this approach will be discussed below, along with the merits of an alternative approach. Let us first take an overall view of the multi-scale approach. Figure 8 illustrates, from left, an object of structural integrity assessment within which a region of potential criticality (failure) exists. This region (a substructure) is analyzed to determine the loading on its boundary. Next step is to examine how this loading induces damage. This step requires analyzing heteroge￾neities (microstructure, generally), which govern initiation of damage. Simple examples are debonding of fibers from matrix and matrix cracking from broken fiber ends. The analysis of local stress/strain fields to determine 123 J Mater Sci (2006) 41:6800–6812 6807

6808 J Mater Sci(2006)41:6800-6812 CDM length scales of damage. Thus, length scales of damage and Micromechanics their hierarchy are not fixed but are subject to evolution. To llustrate this, consider ply cracking in a laminate. In the early stage, individual ply cracks initiate from debonding Structure RVE Unit cell of fibers, giving the damage length scale in terms of the fiber diameter. When the ply cracks are fully-grown Fig8 A multi-scale approach starting from the structural scale. through the ply thickness, the mechanism of interest is the moving down to lower scales multiplication of cracks. At this stage, the damage length cale is crack spacing, which in turn depends on the ply thickness as well as the constraint to surface displacement such micro-failures is commonly referred to as of the ply cracks. The two-stage behavior and the evolving micromechanics. The micromechanics itself could be nature of damage complicate any hierarchical scheme for onducted at multiple scales. An example is fiber/matrix prediction of response debonding at the fiber diameter scale and coalescence of Another issue in a hierarchical approach is the multi- the debond cracks at the scale of a representative number plicity of damage modes. If more than one damage mode operates at a time, and there is interaction between the In Fig. 8 the direction of the arrows indicates moving modes, then a hierarchy of length scales becomes ques rom structural(macro)scale downwards to decreasing tionable. Consider ply cracking in a commonly used, quasi length scales. Until the microstructural entities are explic- isotropic [0/+45/90]s laminate in axial tension. Multiple itly included in an analysis, the regime of analysis is matrix cracking occurs in 90-plies, followed by the same in characterized as""continuum, beyond which it is known plies of-45 and 45 orientations. The three ply cracking as micromechanics. In the context of damage the contin- modes progress interactively and at some stage concur uum regime is called Continuum Damage Mechanics rently. The length scales associated with the three damage CDM), while the micromechanics is typically not given an modes do not show hierarchy. Consequently bridging the additional characterization (except, perhaps, occasionally scales by some averaging scheme becomes irrelevant as micro-damage mechanics, see [2]). Historically, the Further complication to the hierarchical scale arrangement fields of CDM and micromechanics have developed inde- is given by the interlaminar cracking that results from the pendently, CDM going back to Kachanov (4), while cracking in individual off-axis plies micromechanics originated in various works, but its char- An alternative to the hierarchical approach is the syn acterization as a coherent field may be credited to eristic damage mechanics(SDM) approach proposed by Budiansky [1], who defined it as"the mechanics of very this author [10]. Conceptually the approach combines the small things". In recent years the upsurge of computational strengths of CDM and micro-damage mechanics(MDM) mechanics has also boosted micromechanics, adding the In CDM the material microstructure(e.g,distributed aspects of numerical simulation and length-scale based fibers)and the distributed damage, which may be called characterizations such as" nanomechanics". The increas- micro-damage structure, are treated as smeared-out fields. ing confidence in the power of computation has led to the This homogenization is illustrated in Fig. 9 as a two-step notion of the hierarchical approach, with the implicit process, where the material microstructure is viewed as assumption that"basic"laws, when placed into a simu- consisting of"stationary"entities(e. g, fibers and plies) lation scheme, will lead to physically correct results. Thus, and the micro-damage structure is considered as a family of once the microstructure, at any chosen level of length scale, evolving entities(.g, cracks and voids). A set of response has been codified in a simulation scheme, the results of functions are expressed in terms of the field variables computation will describe the behavior at the next higher (stress, strain, temperature)and internal variables, which level, the assumption goes. In the context of damage represent the smeared-out field of evolving damage enti mechanics this may raise a few issues that we shall discus ties. The internal variables, although being field quantit actually have an RVE associated with them at each mate The first issue in a hierarchical approach is the choice of rial point. Strictly speaking, there is another RVE associ length scales. As discussed in Section"The Multi-Scale ated with the stationary microstructure but it is customary Nature of Damage"above, the microstructural length in continuum treatments to bypass it by requiring that the scales are relatively straightforward, and consequently quantities such as the elastic moduli of virgin material be setting up a hierarchy of scales and procedures for bridging measured at a scale much larger than the scale of the between can accomplished relatively easily. individual stationary microstructure entities. There is a microstructural configuration and driving tendency to do that for the rve associated with damage as forces for damage initiation and progression determine the well, as evidenced in finite element based analyses where 2 Springer

such micro-failures is commonly referred to as micromechanics. The micromechanics itself could be conducted at multiple scales. An example is fiber/matrix debonding at the fiber diameter scale and coalescence of the debond cracks at the scale of a representative number of fibers. In Fig. 8 the direction of the arrows indicates moving from structural (macro) scale downwards to decreasing length scales. Until the microstructural entities are explic￾itly included in an analysis, the regime of analysis is characterized as ‘‘continuum’’, beyond which it is known as micromechanics. In the context of damage the contin￾uum regime is called Continuum Damage Mechanics (CDM), while the micromechanics is typically not given an additional characterization (except, perhaps, occasionally as micro-damage mechanics, see [2]). Historically, the fields of CDM and micromechanics have developed inde￾pendently, CDM going back to Kachanov [4], while micromechanics originated in various works, but its char￾acterization as a coherent field may be credited to Budiansky [1], who defined it as ‘‘the mechanics of very small things’’. In recent years the upsurge of computational mechanics has also boosted micromechanics, adding the aspects of numerical simulation and length-scale based characterizations such as ‘‘nanomechanics’’. The increas￾ing confidence in the power of computation has led to the notion of the hierarchical approach, with the implicit assumption that ‘‘basic’’ laws, when placed into a simu￾lation scheme, will lead to physically correct results. Thus, once the microstructure, at any chosen level of length scale, has been codified in a simulation scheme, the results of computation will describe the behavior at the next higher level, the assumption goes. In the context of damage mechanics this may raise a few issues that we shall discuss below. The first issue in a hierarchical approach is the choice of length scales. As discussed in Section ‘‘The Multi-Scale Nature of Damage’’ above, the microstructural length scales are relatively straightforward, and consequently setting up a hierarchy of scales and procedures for bridging between them can be accomplished relatively easily. However, the microstructural configuration and driving forces for damage initiation and progression determine the length scales of damage. Thus, length scales of damage and their hierarchy are not fixed but are subject to evolution. To illustrate this, consider ply cracking in a laminate. In the early stage, individual ply cracks initiate from debonding of fibers, giving the damage length scale in terms of the fiber diameter. When the ply cracks are fully-grown through the ply thickness, the mechanism of interest is the multiplication of cracks. At this stage, the damage length scale is crack spacing, which in turn depends on the ply thickness as well as the constraint to surface displacement of the ply cracks. The two-stage behavior and the evolving nature of damage complicate any hierarchical scheme for prediction of response. Another issue in a hierarchical approach is the multi￾plicity of damage modes. If more than one damage mode operates at a time, and there is interaction between the modes, then a hierarchy of length scales becomes ques￾tionable. Consider ply cracking in a commonly used, quasi￾isotropic [0/45/90]s laminate in axial tension. Multiple matrix cracking occurs in 90-plies, followed by the same in plies of –45 and 45 orientations. The three ply cracking modes progress interactively and at some stage concur￾rently. The length scales associated with the three damage modes do not show hierarchy. Consequently, bridging the scales by some averaging scheme becomes irrelevant. Further complication to the hierarchical scale arrangement is given by the interlaminar cracking that results from the cracking in individual off-axis plies. An alternative to the hierarchical approach is the syn￾ergistic damage mechanics (SDM) approach proposed by this author [10]. Conceptually the approach combines the strengths of CDM and micro-damage mechanics (MDM). In CDM the material microstructure (e.g., distributed fibers) and the distributed damage, which may be called micro-damage structure, are treated as smeared-out fields. This homogenization is illustrated in Fig. 9 as a two-step process, where the material microstructure is viewed as consisting of ‘‘stationary’’ entities (e.g., fibers and plies) and the micro-damage structure is considered as a family of evolving entities (e.g., cracks and voids). A set of response functions are expressed in terms of the field variables (stress, strain, temperature) and internal variables, which represent the smeared-out field of evolving damage enti￾ties. The internal variables, although being field quantities, actually have an RVE associated with them at each mate￾rial point. Strictly speaking, there is another RVE associ￾ated with the stationary microstructure, but it is customary in continuum treatments to bypass it by requiring that the quantities such as the elastic moduli of virgin material be measured at a scale much larger than the scale of the individual stationary microstructure entities. There is a tendency to do that for the RVE associated with damage as well, as evidenced in finite element based analyses where Structure Substructure RVE Unit cell CDM Micromechanics Fig. 8 A multi-scale approach starting from the structural scale, moving down to lower scales 123 6808 J Mater Sci (2006) 41:6800–6812

J Mater Sci(2006)41:68006812 6809 Fig. 9 Illustration of the two- Stationary microstructure composites with damage RVE microstru △ He P Homogenized continuum reduced(damage induced) properties are assigned at nodal number density, as noted above. The characterization of points. The reduced properties are meaningful only at the infuence is accomplished by assigning a two-vector RVE Scale, which depends on the length scales of damage representation to a damage entity, as illustrated in Fig 9 discussed above The vector a carries the influence through its magnitude To illustrate the structure of SDM let us consider the and direction. The magnitude of the vector represents a Helmholz free energy function for isothermal, mechanical measure of how much the damage entity is able to affect its surroundings, while the direction of the vector indicates the orientation in which this effect acts. For instance. if we are φ=f(,D) (17) concerned about deformational response of a composite, then clearly the surface of a given damage entity must where the strain tensor e and damage variable D, generally conduct a displacement in order to affect this response also a tensor, are independent variables representing the Imagine for instance a transverse crack in 90-plies of a material state. The variable D is viewed as an internal [0/+45/90]s laminate. The degree to which this crack opens variable, representing some measure of the collective under an imposed axial load increment will determine how presence of damage entities in an Rve at the considered much the axial elastic modulus of the composite will reduce point where the material response is sought In Fig. 9 the If the axial stiffness of the sublaminate [0/+45]is high, then RVE at a point is shown as a finite-sized cube of material the crack opening displacement will be low, and conse- containing a representative sample of damage entities quently, the modulus reduction will be small. In a passive The internal state(damage)in a general case may con- damage characterization, where only the crack number ain multiple modes, such as the ply cracking modes in a density enters, no distinction can be made between the [0/+45/90]s laminate. In the conventional CDM approach presence of cracks in different constraining environments the response function R, and any function derived from it, In the SDM approach the constraint to the damage entity is expressed in terms of D, which is formulated to represent influence is represented in a constraint parameter, such as ax a measure of the intensity of damage. Examples of such in Eq (4)for a fiber-bridged matrix crack. The determina- measures are void volume fraction and crack number tion of the constraint parameter, and generally any influence density. Such"passive measures end the CDM at the function, is accomplished by a micromechanics analysis RVE level, i.e., the role of CDM gets limited to generating levels warranted by the length scales of damage constitutive relationships at the RvE(meso)level that are The SDM approach has been illustrated in Varna et al hen used for analyzing structural(macro)response. In the [11] for the elastic response of [0/904]s laminates and ir SDM approach we proceed down from the RVE level to Varna et al. [12] for the linear viscoelastic response of one or more micro-levels as warranted by the situation at cross ply laminates of different 0/90 ply mix. In each case hand. This is accomplished by developing a characteriza- the transverse cracking in 90-plies was considered as the tion of damage entities that is"active"in the sense that the damage mode subjected to different constraints. Thus in the presence of damage entities is accounted for by including [+e/904]s laminates 0 is varied to vary the constraint, while he"influence"of damage entities. In contrast, the passive no cracking is considered in the +e plies. The objective in characterization is limited to only accounting for the both the works just cited was to demonstrate SDM for the presence'of damage entities by measures such as crack case of one damage mode with varying constraint and 2 Springer

reduced (damage induced) properties are assigned at nodal points. The reduced properties are meaningful only at the RVE scale, which depends on the length scales of damage discussed above. To illustrate the structure of SDM let us consider the Helmholz free energy function for isothermal, mechanical response, as / ¼ f ðe; DÞ ð17Þ where the strain tensor
and damage variable D, generally also a tensor, are independent variables representing the material state. The variable D is viewed as an internal variable, representing some measure of the collective presence of damage entities in an RVE at the considered point where the material response is sought. In Fig. 9 the RVE at a point is shown as a finite-sized cube of material containing a representative sample of damage entities. The internal state (damage) in a general case may con￾tain multiple modes, such as the ply cracking modes in a [0/45/90]s laminate. In the conventional CDM approach the response function R, and any function derived from it, is expressed in terms of D, which is formulated to represent a measure of the intensity of damage. Examples of such measures are void volume fraction and crack number density. Such ‘‘passive’’ measures end the CDM at the RVE level, i.e., the role of CDM gets limited to generating constitutive relationships at the RVE (meso) level that are then used for analyzing structural (macro) response. In the SDM approach we proceed down from the RVE level to one or more micro-levels as warranted by the situation at hand. This is accomplished by developing a characteriza￾tion of damage entities that is ‘‘active’’ in the sense that the presence of damage entities is accounted for by including the ‘‘influence’’ of damage entities. In contrast, the passive characterization is limited to only accounting for the ‘‘presence’’ of damage entities by measures such as crack number density, as noted above. The characterization of influence is accomplished by assigning a two-vector representation to a damage entity, as illustrated in Fig. 9. The vector a carries the influence through its magnitude and direction. The magnitude of the vector represents a measure of how much the damage entity is able to affect its surroundings, while the direction of the vector indicates the orientation in which this effect acts. For instance, if we are concerned about deformational response of a composite, then clearly the surface of a given damage entity must conduct a displacement in order to affect this response. Imagine for instance a transverse crack in 90-plies of a [0/45/90]s laminate. The degree to which this crack opens under an imposed axial load increment will determine how much the axial elastic modulus of the composite will reduce. If the axial stiffness of the sublaminate [0/45] is high, then the crack opening displacement will be low, and conse￾quently, the modulus reduction will be small. In a passive damage characterization, where only the crack number density enters, no distinction can be made between the presence of cracks in different constraining environments. In the SDM approach the constraint to the damage entity influence is represented in a constraint parameter, such as a in Eq. (4) for a fiber-bridged matrix crack. The determina￾tion of the constraint parameter, and generally any influence function, is accomplished by a micromechanics analysis at levels warranted by the length scales of damage. The SDM approach has been illustrated in Varna et al. [11] for the elastic response of [h/904]s laminates and in Varna et al. [12] for the linear viscoelastic response of cross ply laminates of different 0/90 ply mix. In each case the transverse cracking in 90-plies was considered as the damage mode subjected to different constraints. Thus in the [h/904]s laminates h is varied to vary the constraint, while no cracking is considered in the h plies. The objective in both the works just cited was to demonstrate SDM for the case of one damage mode with varying constraint and P n a Stationary microstructure Evolving microstructure Homogenization of stationary microstructure RVE Homogenized continuum with damage Damage entity Fig. 9 Illustration of the two￾step homogenization process for composites with damage 123 J Mater Sci (2006) 41:6800–6812 6809