Chapter 12:Multiscale Modeling for Damage Analysis 539 D a°ply failure % 1.00 Gr./Ep.(04.90ns Q.75 0.50 A 0.25 go°ply failure 0 0 2.0 4.0 6.0 8.0 2n,number of 90°plies Fig.12.4.The strain at first ply failure as a function of the number of transverse plies in [0/90ls laminate B.Multiple Fracture Low Constraint A.Single Fracture No Constraint EFPF EFPF e C.Multiple Fracture D.Multiple Fracture High Constraint Full Constraint (Crack suppression) EFPF Fig.12.5.Stress-strain response at different constraints to transverse cracking in crossply laminates
Fig. 12.4. The strain at first ply failure as a function of the number of transverse plies in [04/90n]s laminate Fig. 12.5. Stress–strain response at different constraints to transverse cracking in crossply laminates Chapter 12: Multiscale Modeling for Damage Analysis 539
540 R.Talreja and C.V.Singh low constraint,(C)high constraint,and (D)full constraint.In the case of [04/90ls laminate,PE varies with the number 2n of 90 plies.As this number increases,the constraint of the 0o plies becomes increasingly insignificant and &PF approaches the failure strain of the unconstrained 90 plies,i.e.,the failure strain normal to fibers.On the other extreme,as the constraint of the 0 plies becomes effective,this strain increases;and multiple cracking occurs.This process continues to higher constraint;and at some point,the &pr exceeds the fiber failure strain,at which point the constraining plies fail. Unconstrained CMC loaded in tension along fibers The stress-strain response of a unidirectional CMC loaded in axial tension is described in Fig.12.6.This response develops in stages as the matrix cracking progresses,as evidenced by the set of micrographs obtained by Sorensen and Talreja [53]shown in Fig.12.7.The micrograph taken at 0.15% axial strain shows the matrix cracks normal to the (horizontal)fiber axis that do not span the complete specimen cross section.As strain increases, more cracks form and quickly span the whole specimen width.Finally,the cracking saturates,i.e.,no more cracks form on increasing the load.This stage of progressive matrix cracking represents Stage II,extending from 0.13 to 0.5%axial strain in Fig.12.6.The preceding stage (Stage I)con- sists of linear elastic behavior before the onset of cracking.Beyond 0.5% Stage I Stage IⅡ Stage IⅢl Fig.12.6.The three stages of stress-strain response in a SiC fiber-reinforced glass-ceramic composite [53]
low constraint, (C) high constraint, and (D) full constraint. In the case of [04/90n]s laminate, εFPF varies with the number 2n of 90° plies. As this number increases, the constraint of the 0° plies becomes increasingly insignificant and εFPF approaches the failure strain of the unconstrained 90° plies, i.e., the failure strain normal to fibers. On the other extreme, as the constraint of the 0° plies becomes effective, this strain increases; and multiple cracking occurs. This process continues to higher constraint; and at some point, the εFPF exceeds the fiber failure strain, at which point the constraining plies fail. Unconstrained CMC loaded in tension along fibers The stress–strain response of a unidirectional CMC loaded in axial tension is described in Fig. 12.6. This response develops in stages as the matrix cracking progresses, as evidenced by the set of micrographs obtained by Sørensen and Talreja [53] shown in Fig. 12.7. The micrograph taken at 0.15% axial strain shows the matrix cracks normal to the (horizontal) fiber axis that do not span the complete specimen cross section. As strain increases, more cracks form and quickly span the whole specimen width. Finally, the cracking saturates, i.e., no more cracks form on increasing the load. This stage of progressive matrix cracking represents Stage II, extending from 0.13 to 0.5% axial strain in Fig. 12.6. The preceding stage (Stage I) consists of linear elastic behavior before the onset of cracking. Beyond 0.5% Fig. 12.6. The three stages of stress–strain response in a SiC fiber-reinforced glass-ceramic composite [53] 540 R. Talreja and C.V. Singh
Chapter 12:Multiscale Modeling for Damage Analysis 541 strain,the frictional sliding at the fiber/matrix interface becomes signi- ficant.Finally,beyond 0.7%strain,a progressive fiber breakage takes place leading to localization of damage and catastrophic failure.The Stage II progressive cracking was treated by Sorensen and Talreja [53].In Sect.12.3, two cases of damage will be used to illustrate the multiscale nature of damage and discuss how the scales can be incorporated into a damage mechanics framework. 200m c0.15% 200m 气=035% 2004网 无=0.50% 200m =080 Fig.12.7.Surface micrographs of a SiC fiber-reinforced glass-ceramic composite at different axial strains.Tensile loading was in the (horizontal)fiber direction [53] 12.3 Multiscale Nature of Damage As described in Sect.12.2,the damage in composites occurs due to a variety of dissipative mechanisms which cause permanent changes in the internal microstructure of the material and decrease the energy storing capacity of the material.The most basic scale at which these mechanisms occur depends upon the size of inhomogeneities in the microstructure of the material.As an example,nanocomposites may show dissipative mech- anisms at the nanometer scale.In reality,however,identifying this scale is limited by the ability to observe as well as to model and analyze the
strain, the frictional sliding at the fiber/matrix interface becomes significant. Finally, beyond 0.7% strain, a progressive fiber breakage takes place leading to localization of damage and catastrophic failure. The Stage II progressive cracking was treated by Sørensen and Talreja [53]. In Sect. 12.3, two cases of damage will be used to illustrate the multiscale nature of damage and discuss how the scales can be incorporated into a damage mechanics framework. Fig. 12.7. Surface micrographs of a SiC fiber-reinforced glass-ceramic composite at different axial strains. Tensile loading was in the (horizontal) fiber direction [53] 12.3 Multiscale Nature of Damage As described in Sect. 12.2, the damage in composites occurs due to a variety of dissipative mechanisms which cause permanent changes in the internal microstructure of the material and decrease the energy storing capacity of the material. The most basic scale at which these mechanisms occur depends upon the size of inhomogeneities in the microstructure of the material. As an example, nanocomposites may show dissipative mechanisms at the nanometer scale. In reality, however, identifying this scale is limited by the ability to observe as well as to model and analyze the Chapter 12: Multiscale Modeling for Damage Analysis 541
542 R.Talreja and C.V.Singh mechanisms at the observed scale.The so-called microscale 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 microscale 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 to the atomic scale.At this scale,the basic notions of continuum mechanics fail;and it is necessary to develop modeling tools that can bridge the discrete-level descriptions(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 dissipating mechanisms in a model,directly or indirectly,in an appropriate average sense;while if,for instance,the aim is a particular material failure characteristic,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 characteristic 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 mechanisms associated with the inhomogeneities have usually an overriding influence on the composite behavior.For instance,in short- fiber PMCs,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 introduced by inhomogeneities in composite damage is in the form of multiple scales of dissipative mechanisms depending on the geometrical features of the inhomogeneities.In the case of short fibers, for instance,the matrix cracking from the fiber ends and the fiber/matrix debonding occurs 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
mechanisms at the observed scale. The so-called microscale 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 microscale 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 nanoscale elements (particles, fibers, tubes, etc.) has moved the basic scale further down to the atomic scale. At this scale, the basic notions of continuum mechanics fail; and it is necessary to develop modeling tools that can bridge the discrete-level descriptions (quantum mechanics) to continuumtype (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 dissipating mechanisms in a model, directly or indirectly, in an appropriate average sense; while if, for instance, the aim is a particular material failure characteristic, 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 characteristic 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 mechanisms associated with the inhomogeneities have usually an overriding influence on the composite behavior. For instance, in shortfiber PMCs, 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 introduced by inhomogeneities in composite damage is in the form of multiple scales of dissipative mechanisms depending on the geometrical features of the inhomogeneities. In the case of short fibers, for instance, the matrix cracking from the fiber ends and the fiber/matrix debonding occurs 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, 542 R. Talreja and C.V. Singh
Chapter 12:Multiscale Modeling for Damage Analysis 543 one faces a complex situation concerning the length scales;and taking a hierarchical approach may not be efficient,as will be discussed later.In the following,the multiscale nature of damage in composite materials will be illustrated by examining the two cases pertaining to damage in uni- directional CMCs and ply cracking in laminates. 12.3.1 Unidirectional CMCs In the case of CMCs,the fiber/matrix interfacial region has a strong influence on the thermomechanical response.The properties of the inter- facial region determine whether a matrix crack front approaching a fiber advances into the fiber or bypasses it by causing interfacial slip and/or debonding.The damage configurations at the microscopic level thus gene- rated govern the macroscopic (overall)response of a composite.There are three basic mechanisms of damage in CMCs:matrix cracking,interfacial sliding,and interfacial debonding.They can occur independently or inter- actively.The experimental evidence indicates that the interfacial damage (slip and debonding)occurs primarily in conjunction with matrix cracking [41].Talreja [59]used these damage configurations to characterize damage, as shown in Fig.12.8a-d.Figure 12.8b,d shows interfacial slip and debonding,both in conjunction with matrix cracking.This situation (b) will result if the fibers are held in the matrix by frictional forces at the interfaces,while (d)is likely to result from a nonuniform interfacial bond strength [59]. The characterization of damage is done by regarding damage entities as internal structure of the homogeneous body.The internal structure changes with loading and causes changes of the overall response of the compo- site.The internal structure of a continuum is described using the so-called internal variables.These variables are some appropriately defined quantities representing the geometry,i.e.,size,shape,orientation,etc.,of the internal structure as well as the influence of the internal structure on the response considered.The quantities chosen depend on the geometrical characteristics of the entities involved in the internal structure constitution and the nature of the influence of these entities on the response of the composite.The ele- mentary damage entities present in the damage configurations treated here are cracks,debonds,and slipped surfaces.The characterization used for cracks and debonds is different from that used for slipped surfaces.These two types are,therefore,treated separately in the following
one faces a complex situation concerning the length scales; and taking a hierarchical approach may not be efficient, as will be discussed later. In the following, the multiscale nature of damage in composite materials will be illustrated by examining the two cases pertaining to damage in unidirectional CMCs and ply cracking in laminates. 12.3.1 Unidirectional CMCs In the case of CMCs, the fiber/matrix interfacial region has a strong influence on the thermomechanical response. The properties of the interfacial region determine whether a matrix crack front approaching a fiber advances into the fiber or bypasses it by causing interfacial slip and/or debonding. The damage configurations at the microscopic level thus generated govern the macroscopic (overall) response of a composite. There are three basic mechanisms of damage in CMCs: matrix cracking, interfacial sliding, and interfacial debonding. They can occur independently or interactively. The experimental evidence indicates that the interfacial damage (slip and debonding) occurs primarily in conjunction with matrix cracking [41]. Talreja [59] used these damage configurations to characterize damage, as shown in Fig. 12.8a–d. Figure 12.8b,d shows interfacial slip and debonding, both in conjunction with matrix cracking. This situation (b) will result if the fibers are held in the matrix by frictional forces at the interfaces, while (d) is likely to result from a nonuniform interfacial bond strength [59]. The characterization of damage is done by regarding damage entities as internal structure of the homogeneous body. The internal structure changes with loading and causes changes of the overall response of the composite. The internal structure of a continuum is described using the so-called internal variables. These variables are some appropriately defined quantities representing the geometry, i.e., size, shape, orientation, etc., of the internal structure as well as the influence of the internal structure on the response considered. The quantities chosen depend on the geometrical characteristics of the entities involved in the internal structure constitution and the nature of the influence of these entities on the response of the composite. The elementary damage entities present in the damage configurations treated here are cracks, debonds, and slipped surfaces. The characterization used for cracks and debonds is different from that used for slipped surfaces. These two types are, therefore, treated separately in the following. Chapter 12: Multiscale Modeling for Damage Analysis 543