500 D.H.Allen and R.F.Soares become quite popular in the engineering field.Furthermore,because the resulting body is elastic,the analyses on the smaller and larger scales can be performed independently of one another,so that no coupling between the two-length scales is necessary. In the case of inelastic media,this,unfortunately,is not the case.When materials undergo load-induced energy dissipation,such as that occurs in elastoplastic or viscoelastic media,the micromechanical description does not decouple from the analysis to be performed on the larger scale.In other words,the material properties become spatially variable and dependent on the load history,so that coupling between the macro-and microscale is unavoidable.Therefore,it becomes essential to develop modeling approaches that account for this fundamental increase in the level of complexity of the problem if there is to be any hope of achieving accuracy of prediction. For the better part of the last half of the twentieth century,efforts to account for this complexity in inelastic media centered on development of ever more complicated constitutive theories for the microscale,similar to that used successfully to model heterogeneous elastic media,as described above.This had the pragmatic basis that one could perform a finite element analysis on a single length scale,which was just about the limit that computers of that time could handle.However,as it became apparent that microscale cracking would have to be included in constitutive models of heterogeneous media at the macroscale,efforts began to bog down and become very complicated indeed.To account for observed behavior in test specimens with time-dependent microcracking,more and more (often un- explained)phenomenological parameters had to be introduced into models. This approach developed the name"continuum damage mechanics."It also inherited the unpalatable complication that sometimes many experi- mentally measured material parameters were required,especially when it became necessary to model evolving microcracks. Enter the twenty-first century and more and more powerful computers. What required a supercomputer 10 years ago now requires only a desktop computer.Therefore,it is now possible to conceive of algorithms that obviate the necessity to perform many complicated experiments at the microscale.Furthermore,these new algorithms have the added advantage that,by performing simultaneous computations on both the micro-and global scales,they possess the flexibility to include heretofore unmanage- able design variables at the microscale in the global design process,and without recourse to expensive constitutive testing
become quite popular in the engineering field. Furthermore, because the resulting body is elastic, the analyses on the smaller and larger scales can be performed independently of one another, so that no coupling between the two-length scales is necessary. In the case of inelastic media, this, unfortunately, is not the case. When materials undergo load-induced energy dissipation, such as that occurs in elastoplastic or viscoelastic media, the micromechanical description does not decouple from the analysis to be performed on the larger scale. In other words, the material properties become spatially variable and dependent on the load history, so that coupling between the macro- and microscale is unavoidable. Therefore, it becomes essential to develop modeling approaches that account for this fundamental increase in the level of complexity of the problem if there is to be any hope of achieving accuracy of prediction. For the better part of the last half of the twentieth century, efforts to account for this complexity in inelastic media centered on development of ever more complicated constitutive theories for the microscale, similar to that used successfully to model heterogeneous elastic media, as described above. This had the pragmatic basis that one could perform a finite element analysis on a single length scale, which was just about the limit that computers of that time could handle. However, as it became apparent that microscale cracking would have to be included in constitutive models of heterogeneous media at the macroscale, efforts began to bog down and become very complicated indeed. To account for observed behavior in test specimens with time-dependent microcracking, more and more (often unexplained) phenomenological parameters had to be introduced into models. This approach developed the name “continuum damage mechanics.” It also inherited the unpalatable complication that sometimes many experimentally measured material parameters were required, especially when it became necessary to model evolving microcracks. Enter the twenty-first century and more and more powerful computers. What required a supercomputer 10 years ago now requires only a desktop computer. Therefore, it is now possible to conceive of algorithms that obviate the necessity to perform many complicated experiments at the microscale. Furthermore, these new algorithms have the added advantage that, by performing simultaneous computations on both the micro- and global scales, they possess the flexibility to include heretofore unmanageable design variables at the microscale in the global design process, and without recourse to expensive constitutive testing. 500 D.H. Allen and R.F. Soares
Chapter 11:Multiscale Modeling of the Evolution of Damage 501 11.4 Multiscale Modeling in Inelastic Media with Damage In this section,a multiscale model is proposed for predicting the evolution of damage on multiple scales in inelastic media.The formulation is taken from [5]. 11.4.1 Microscale Model Consider an approach proposed herein that can be used on any number of length scales observed in a solid object.The number of scales n utilized is determined by the physics of the problem on the one hand and the amount of computational speed and size available on the other hand.To that end,consider a solid object with a region wherein microcracks are evolving on the smallest length scale considered /as shown in Fig.11.1. Macrocrack Macroscale RVE for Microscale Microcracks Fig.11.1.Scale problem with cracks on both length scales
11.4 Multiscale Modeling in Inelastic Media with Damage In this section, a multiscale model is proposed for predicting the evolution of damage on multiple scales in inelastic media. The formulation is taken from [5]. 11.4.1 Microscale Model Consider an approach proposed herein that can be used on any number of length scales lµ observed in a solid object. The number of scales n utilized is determined by the physics of the problem on the one hand and the amount of computational speed and size available on the other hand. To that end, consider a solid object with a region wherein microcracks are evolving on the smallest length scale considered l1, as shown in Fig. 11.1. Fig. 11.1. Scale problem with cracks on both length scales Chapter 11: Multiscale Modeling of the Evolution of Damage 501 Macroscale RVE for Microscale Macrocrack Microcracks l µ+1 l µ x2 µ+1 x2 µ x1 µ x3 µ x1 µ+1 x3 µ+1
502 D.H.Allen and R.F.Soares While it is not necessary (or even always correct)that a representative volume of the object on this length scale be accurately modeled by con- tinuum mechanics,it is assumed that this is the case in this chapter to simplify the discussion.Suppose that the object can be treated as linear viscoelastic,again for simplicity,so that the following initial-boundary value problem(IBVP)may be posed. Conservation of linear momentum i.6。+pf=0,re', (11.2) where is the Cauchy stress tensor defined on length scale u,p is the mass density,and f is the body force vector per unit mass.Note that inertial effects have been neglected,implying that the length scale of interest is small compared to the next larger length scale,thus neglecting the effects of waves at this scale on the next scale up.Ultimately,it will be convenient within this context to model waves only on the largest,or global,scale Strain-displacement equations ,=i。+a,)'门 (11.3) where is the strain tensor on the length scale u and i is the displacement vector on the length scale u.Note that the linearized form of the strain tensor has been taken for simplicity,although a nonlinear form may be employed without loss of generality. Constitutive equations G(4,)=2{(4,t)}, (11.4) where is the coordinate location in the object on the length scale 4, which has interior V and boundary oV The above description implies that the entire history of strain at any point in the body is mapped into the current stress,which is termed a viscoelastic material model.Because only the value of strain (the sym- metric part of the deformation gradient is used in this model)is required at
While it is not necessary (or even always correct) that a representative volume of the object on this length scale be accurately modeled by continuum mechanics, it is assumed that this is the case in this chapter to simplify the discussion. Suppose that the object can be treated as linear viscoelastic, again for simplicity, so that the following initial–boundary value problem (IBVP) may be posed. Conservation of linear momentum f xV 0, , ∇⋅ + = ∀ ∈ σ ρ µ µ µ G G G (11.2) where σ µ is the Cauchy stress tensor defined on length scale µ, ρ is the mass density, and f G is the body force vector per unit mass. Note that inertial effects have been neglected, implying that the length scale of interest is small compared to the next larger length scale, thus neglecting the effects of waves at this scale on the next scale up. Ultimately, it will be convenient within this context to model waves only on the largest, or global, scale. Strain–displacement equations 1 T [ ( ) ], 2 u u µ µµ ε ≡ ∇ +∇ G G G G (11.3) where µ ε is the strain tensor on the length scale µ and uµ G is the displacement vector on the length scale µ. Note that the linearized form of the strain tensor has been taken for simplicity, although a nonlinear form may be employed without loss of generality. Constitutive equations ( , ) { ( , )}, t xt x τ σ µµ τ µµ ε τ = = Ω =−∞ G G (11.4) where xµ G is the coordinate location in the object on the length scale µ, which has interior Vµ and boundary Vµ ∂ . The above description implies that the entire history of strain at any point in the body is mapped into the current stress, which is termed a viscoelastic material model. Because only the value of strain (the symmetric part of the deformation gradient is used in this model) is required at 502 D.H. Allen and R.F. Soares
Chapter 11:Multiscale Modeling of the Evolution of Damage 503 the point of interest,it is sometimes called a simple (or local)model [15]. Note that a local elastic material model,such as Hooke's law [321,is a special case of(11.4). Equations(11.2)(11.4)must apply in the body,together with appro- priate initial and boundary conditions.These are then adjoined with a frac- ture criterion that is capable of predicting the growth of new or existing cracks anywhere in the object.There are multiple possibilities but,for example,the Griffith criterion given by inequality (11.1)can be taken.The above then constitutes a well-posed boundary value problem,albeit non- linear due to the crack growth criterion(perhaps as well as the constitutive model (11.4)). Obtaining solutions for this problem,even for simple geometries,is in itself a difficult challenge,as anyone who has every attempted to do so will attest.Nevertheless,assume that by some means(most likely computational) a solution can be obtained for the boundary conditions,geometry,and precise form of the constitutive (11.4)at hand.Assume,furthermore,that the cracks that are predicted within the model dissipate so much energy locally that they may have further deleterious effects on the response at the next larger length scale.As an example,the so-called microcracks may in some way influence the development or extension of one or more macrocracks on the next larger length scale /2.It will be assumed that the cracks on the next larger length scale are much larger than those on the current scale and that this restriction applies to all length scales for cracks in the object of interest 141之14，l=1,n, (11.5) where n is the number of different length scales observed in the solid. Note that the above restriction is a necessary condition (but not sufficient)for the multiscale methodology proposed herein to produce reasonably accurate predictions on the larger length scale(s).If this con- dition is not satisfied,as in the case of a so-called localization problem, then there may indeed be no alternative to performing an exhaustive analysis at a single scale that takes into account all of the asperities simultaneously. 11.4.2 Homogenization Principle Connecting the Microscale to the Macroscale To perform an analysis of the solid on the next length scale up from the local scale (termed the macroscale herein for simplicity),it is necessary to find a means of linking the state variables predicted on the microscale to
the point of interest, it is sometimes called a simple (or local) model [15]. Note that a local elastic material model, such as Hooke’s law [32], is a special case of (11.4). Equations (11.2)–(11.4) must apply in the body, together with appropriate initial and boundary conditions. These are then adjoined with a fracture criterion that is capable of predicting the growth of new or existing cracks anywhere in the object. There are multiple possibilities but, for example, the Griffith criterion given by inequality (11.1) can be taken. The above then constitutes a well-posed boundary value problem, albeit nonlinear due to the crack growth criterion (perhaps as well as the constitutive model (11.4)). Obtaining solutions for this problem, even for simple geometries, is in itself a difficult challenge, as anyone who has every attempted to do so will attest. Nevertheless, assume that by some means (most likely computational) a solution can be obtained for the boundary conditions, geometry, and precise form of the constitutive (11.4) at hand. Assume, furthermore, that the cracks that are predicted within the model dissipate so much energy locally that they may have further deleterious effects on the response at the next larger length scale. As an example, the so-called microcracks may in some way influence the development or extension of one or more macrocracks on the next larger length scale l2. It will be assumed that the cracks on the next larger length scale are much larger than those on the current scale and that this restriction applies to all length scales for cracks in the object of interest 1 ll n , 1, , , µ µ + µ = … (11.5) where n is the number of different length scales observed in the solid. Note that the above restriction is a necessary condition (but not sufficient) for the multiscale methodology proposed herein to produce reasonably accurate predictions on the larger length scale(s). If this condition is not satisfied, as in the case of a so-called localization problem, then there may indeed be no alternative to performing an exhaustive analysis at a single scale that takes into account all of the asperities simultaneously. 11.4.2 Homogenization Principle Connecting the Microscale to the Macroscale To perform an analysis of the solid on the next length scale up from the local scale (termed the macroscale herein for simplicity), it is necessary to find a means of linking the state variables predicted on the microscale to Chapter 11: Multiscale Modeling of the Evolution of Damage 503
504 D.H.Allen and R.F.Soares those on the macroscale.Of course,the state variables at the microscale are predicted at an infinite collection of material points in the local domain V+V,so that there is plenty of information available to supply to the next larger length scale.However,the objective herein is to find an effi- cient means of constructing this link without sacrificing too much accuracy. In other words,it is propitious to utilize the minimum data obtained at the local scale necessary to make a sufficiently accurate prediction at the macro- scale.One way is to link the microscale to the macroscale via the use of mean fields.To see how this might work,consider the following mathe- matical expansion for the macroscale stress in terms of the microscale stress (11.6) where G.dv (11.7) is the volume averaged (or mean)stress at the microscale,and it is assumed that the local coordinate system is set at the geometric centroid of the microscale volume. Note that,since the microscale domain V+ov can be placed arbitrarily within the domain on the next larger length scale V+ the mean stress is a continuously varying function of coordinates on the next larger length scale u+1,as shown in Fig.11.1.Note also that the terms within the summation in(11.6)represent higher area moments of the stress tensor. Now,it may be said without loss of generality that microscale con- servation of momentum (11.2)also applies to the macroscale (assuming that quasistatic conditions still hold at this length scale) .61+pf=0,∈'n (11.8) By using (11.6),it can be shown that lim () (11.9) 1al→0 and (11.8)reduces to the following: .6n+pf=0,441∈V+ (11.10)
those on the macroscale. Of course, the state variables at the microscale are predicted at an infinite collection of material points in the local domain V V µ + ∂ µ , so that there is plenty of information available to supply to the next larger length scale. However, the objective herein is to find an efficient means of constructing this link without sacrificing too much accuracy. In other words, it is propitious to utilize the minimum data obtained at the local scale necessary to make a sufficiently accurate prediction at the macroscale. One way is to link the microscale to the macroscale via the use of mean fields. To see how this might work, consider the following mathematical expansion for the macroscale stress in terms of the microscale stress 1 ( ) 1 1 d , j j V j x V V x µ µ µ µ µ σ σ σσ ∞ + = = +∑ − ∫ G G (11.6) where 1 d V V V µ µ µ µ σ σ ≡ ∫ (11.7) is the volume averaged (or mean) stress at the microscale, and it is assumed that the local coordinate system is set at the geometric centroid of the microscale volume. Note that, since the microscale domain V V µ + µ ∂ can be placed arbitrarily within the domain on the next larger length scale V V µ+1 1 + ∂ µ+ , the mean stress σ µ is a continuously varying function of coordinates 1 xµ+ G on the next larger length scale µ + 1, as shown in Fig. 11.1. Note also that the terms within the summation in (11.6) represent higher area moments of the stress tensor. Now, it may be said without loss of generality that microscale conservation of momentum (11.2) also applies to the macroscale (assuming that quasistatic conditions still hold at this length scale) 1 1 1 f xV 0, . ∇⋅ + = ∀ ∈ σ ρ µ µ + + + µ G G G (11.8) By using (11.6), it can be shown that 1 1 / 0 lim ( ) l l µ µ σ µ σ + + → = (11.9) and (11.8) reduces to the following: D.H. Allen and R.F. Soares 1 1 f xV 0, . ∇⋅ + = ∀ ∈ σ ρ µ µ+ µ+ G G G (11.10) 504