IJDM_739 a Multiscale Transformation Rule of Material Defects Jie Shen Et Al. 2009

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MechanicsInternational Journal of Damage

DOI: 10.1177/10567895093466932009; 18; 739International Journal of Damage Mechanics

and Joseph M. WellsJie Shen, Jianghui Mao, German Reyes, Chi L. Chow, James Boileau, Xuming Su

A Multiresolution Transformation Rule of Material Defects

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A Multiresolution TransformationRule of Material Defects

JIE SHEN*

Department of Computer & Information Science, The University of

Michigan, Dearborn, MI 48128, USA

JIANGHUI MAO, GERMANREYES ANDCHI L. CHOWDepartment of Mechanical Engineering, The University of Michigan,

Dearborn, MI 48128, USA

JAMESBOILEAU ANDXUMING SU

Materials Research & Advanced Engineering Department, Ford

Motor Company, 2101 Village Road, Dearborn, MI 48121, USA

JOSEPH M. WELLS

JMA Associates, Mashpee, MA 02649, USA

ABSTRACT: The ability to quantify the material damage at different length scales iscritical in the multiscale analysis of material behavior from nanoscale to macroscale.In this article, on the basis of the equivalence of complementary elastic energy wepropose a multiresolution rule that transforms different levels of material defects tothe equivalent degradation of material properties. It facilitates a sequential memory-efficient processing of massive material defects in a multiresolution framework, andalso supports a functionality of partial damage conversion to serve different needs insubsequent numerical analyses. Numerical simulation was conducted with differentsettings of material defects. The analysis results indicate the efficacy of the proposed

method, offering a potential (i) to interface between multiscale material defects and(ii) as an effective method of homogenization for the determination of the damagevariable in continuum damage mechanics.

KEY WORDS: material defect, multiscale analysis, microstructure, nanomechanics.

*Author to whom correspondence should be addressed. E-mail: [email protected]

Figures 13, 58 and 10 appear in color online: http://ijd.sagepub.com

International Journal ofDAMAGEMECHANICS, Vol. 18November 2009 739

1056-7895/09/08 073920 $10.00/0 DOI: 10.1177/1056789509346693 The Author(s), 2009. Reprints and permissions:http://www.sagepub.co.uk/journalsPermissions.nav

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INTRODUCTION

MOST ENGINEERING MATERIALS contain defects of varying degrees and

sizes. These defects may range from nanoscale to macroscale. To con-duct a multiresolution analysis of material damage, it is desirable to develop

an equivalence rule that can guide the transformation of different levels of

defects into their corresponding degrees of material degradation in the form

of effective mechanical properties or damage variables. The degraded

mechanical properties or damage variables can then be brought within the

concept of continuum damage mechanics that has been proven to be valu-

able for subsequent engineering design analysis and life prediction.

In the field of plasticity, McClintock model (McClintock, 1968),

Rice

Tracey model (Rice and Tracey, 1969), Gurson model (Gurson,1977), and their modified versions (Budiansky et al., 1982; Duva and

Hutchinson, 1984; Tvergaard and Needleman, 1984; Tvergaard, 1989) pos-

tulated that all micro-defects in a material element were in the form of

spherical or cylindrical voids. This hypothesis leads to an unrealistic approx-

imation to real material defects of irregular shapes. Although the sizing

effect (Tvergaard, 1996; Fleck and Hutchinson, 1997; Liu et al., 2003;

Tvergaard and Niordson, 2004; Wen et al., 2005) and limited shape influ-

ence (Becker et al., 1989a,b; Lee and Mear, 1994; Gologanu et al., 1994;

Pardoen and Hutchinson, 2000; Pardoen and Hutchinson, 2003; Li andHuang, 2005; Gao et al., 2005) of spherical or ellipsoidal voids were

addressed, the problem associated with the above approaches or similar

ones is the recognition that the arbitrary shape of real-life material defects

inevitably plays a crucial role in affecting micro/macro-mechanical proper-

ties, as illustrated in Figure 1 for large-size defects.

Gibson and Ashby (1982) linked the mechanical properties of cellular

materials with their relative density, a ratio of the apparent density of cel-

lular materials to the density of cell wall materials. In their mathematical

models, at least one constant needs to be determined from experimental databy a statistical regression, which is thus not physics based. In addition, there

is no unified formula for different settings of cellular materials (e.g., open

cells vs. closed cells).

Scientists in nanostructured materials have made a phenomenal advance

in the reduction of grain or crystal size to 100 nm in the past three decades

(Gleiter, 1981). Many observations have been conducted on the quantum

effects, configuration entropy, thermal fluctuations, and discreteness of nano-

meter objects or systems with nanoscale components (particles, powders,

wires, rods, ribbons, or tubes). Different multiscale approaches have beenestablished to bridge between nanoscale and micro/macroscale (Tadmor

et al., 1996; Abraham et al., 1998; Rudd and Broughton, 1998; Wagner and

740 J. SHEN ET AL.



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Liu, 2003; Shilkrot et al., 2004; Xiao and Belytschko, 2004). In the aspect of

analysis, massively parallel simulation with dislocation mechanics success-

fully produced a strain hardening effect in a 10mm3 volume of a single

crystal metal (Bulatov et al., 2004). Atomistic modeling approach normally

assumes, however, no microscale, mesoscale, and macroscale defects in its

analysis domain (Iesulauro et al., 2001). Although different types of hand-shake regions were designed for a concurrent multiscale analysis with both

lattice grids and finite element (FE) meshes, an effective method of analysis

is yet to be emerged to incorporate a huge number of voids and cracks in

real-life materials into either lattice grids or FE meshes.

Traditional micromechanics provides another way to estimate the macro-

scopic mechanical properties of materials with microscale defects, including

ellipsoids and cracks (Mori and Tanaka, 1973; Budiansky and OConnell,

1976; Christensen and Lo, 1979; Hashin, 1988; Kachanov, 1992; Gelma

et al., 2009). However, multiresolution behavior of defects with differentsizes has less intensively been studied. The existing methods focus primarily

on microscale defects with little consideration on mesoscale and macro-

scopic defects, which may co-exist in real engineering structures.

The objective of this article is to establish a transformation rule that can

describe the equivalence of material damage from a multiresolution stand-

point. This rule provides a physical description of different sizes and degrees

of material defects. The rest of the article is organized as follows. In the

Nomenclature section, the notations of the problem are introduced, and in

the section following it the derivations of a new transformation rule ofmaterial damage are provided. Later, numerical experiments and discussions

are given followed by some conclusions in the final section.

(a) (b)

Threerepresentativevolume elements

with differentshapes of voids

Effectivemodulus

(GPa)

250

200

150

100

50

0

Volume fraction of material

0 0.2 0.4 0.6 0.8 1 1.2

Triangle

Rectangle

Circle

Figure 1. Influence of void shapes on the effective Youngs Modulus of representativevolume elements: (a) three representative volume elements with different shapes of voids

but the same material properties, (b) effect of volume fraction in the representative volumeelement on effective modulus.

Multiresolution Transformation Rule of Material Defects 741



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NOMENCLATURE

One way to represent material damage is the use of a symmetric second

order,D, as a damage tensor (Chow et al., 2007; Voyiadjis and Kattan, 2009).The eigenvalues ofDare written asDk(k 1,2,3), which denotes the damagevalues in three orthogonal principal directions of D. The damage effecttensor,M, is a fourth-order tensor for effective stress equation:

~r MD : r, 1where, ~r and r are effective stress and conventional stress tensors, respec-

tively. Symbol (:) is a tensor product contracted on two indices. M(D) meansthat M is dependent upon D.

At most, M(D) could have 21 independent components, and to another

extreme it may depend upon only one scalar value (DDk, k 1,3) inisotropic cases. For anisotropic damage, M(D) in its principal directionscould be simplified into the following form (Chow and Jie, 2009):

MD

11D1 0 0 0 0 0

0 11D2 0 0 0 0

0 0 11D3 0 0 0

0 0 0 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1D11D2p 0 00 0 0 0 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1D21D3p

0

0 0 0 0 0 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1D31D1p

2666666

66664

3777777

77775

:

2In this article, we focus on the simple form in Equation (2), because it

reflects the essential characteristics of damage effect just like eigenvalues

of a linear transformation matrix.

Each defect has an attribute of characteristic length, which is defined as

the maximum size among three principal directions. The principal compo-

nents analysis (Hoppe et al., 1993) is used to determine the directions atvertex p, which is defined as the centroid of each defect. The covariancematrix of the set of neighboring vertices is:

CV X

q2Nbhdpq p q p,

whereNbhd(p) is the set of all the vertices that are associated with the defectand is outer product operator of vectors. The vertices here could be eithersurface or volume points within the analysis domain of each defect. A Jacobi

transformation (Press et al., 1992) can be used to determine eigenvectors(v1, v2, v3) and eigenvalues (l1 l2 l3) of the CV. (v1, v2, v3) defines thethree principal directions of each defect.

742 J. SHEN ET AL.



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All the defects are classified into n levels (n 1) on the basis of charac-teristic length. Level 1 refers to the defects with characteristic length among

the smallest, while level n contains the defects with sizes being among the

greatest. When n equals 1, a multiresolution problem is degenerated into aproblem with only one level that includes all sizes of defects.

TRANSFORMATION RULE OF MATERIAL DAMAGE

Figure 2(a) demonstrates the concept of damage transformation of mate-

rial defects. Assume a domain that contains both a fine level and a coarse

level of defects. In one way, we calculate the material damage incrementally

by determining the effect of all the fine defects on the original Youngs

modulus, E0, first. The resulting effective modulus is E1 due to these fine

defects alone. The influence of all the coarse defects is then determined on

the degraded material with a modified effective modulus, E1. The effect of

coarse defects leads to a new effective modulus, E2, as a final material deg-

radation sequentially from these two levels of defects. Alternatively, both

levels of defects could be considered simultaneously, leading to another

effective modulus, E3, which is degraded from E0.

E2andE3in the last paragraph reflect a damage estimation determined by

two different routes.E3is calculated by a regular way, i.e., by considering all

the defects simultaneously in an analysis, while E2 is estimated in an incre-

mental way with respect to different levels of defects. Here, defects are clas-

sified into different levels on the basis of characteristic length defined in the

Nomenclature section. One effort in this article is to make the estimation of

E2 close to E3, as illustrated in Figure 2(b). The validity of the damage

transformation offers a potential method of sequential homogenization, a

critical step in conducting multiscale damage mechanics.

(a) (b)

Young

smodulus(GPa) 70

40

60

50

30

10

20E

3

E1

E2

0

E0 E1 E2

E3

Volume fraction of void

0.10 0.20 0.30 0.40

Figure 2. Transformation of material defects: (a) a simple case with two levels of defects(blue and orange colors represent two different routes in the transformation of materialdefects), (b) numerical equivalence of E2 and E3 in finite element analysis.




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For the sake of illustration, the above example is limited to two levels of

defects. In reality, there are multiple levels of defects in materials. In the

following derivations, we propose a multiresolution transformation rule of

material defects. One weakness of Lemaitres hypothesis on strain equiva-lence (Lemaitre and Chaboche, 1978) is that it leads to the asymmetry of a

stiffness matrix in the cases of anisotropic damage. By the inspiration from a

hypothesis on the equivalence of complementary elastic energy (Cordebois

and Sidoroff, 1979; Chow and Lu, 1989; Grammeneboudis et al., 2009a,b), we

assume that the complementary elastic energy at damage level itakes a form:

WerDi, Di 1

2 r

TDi

: ~C1Di

: rDi, i 0, n, 3where, We() denotes a complementary elastic energy, and D

i is a damage

tensor that represents the extent of damage accumulated from level 1 to level

i. Superscript T represents the transpose of a matrix, and rDi is a stress

tensor at damage Di. ~C1Di

refers to the inverse of an effective stiffness

tensor, and has the following special case:

~C1D0

C1, 4in whichC1 is the inverse of a stiffness tensor at damage D0 0. A generalformula for ~C

1Di

is given at a later part of this section. rD0 equals the con-

ventional stress tensor, r

.We also assume that the equivalence of complementary energy takes the

following incremental form:

WerDi1 , Di1 We ~rDi1 , Di 1

2~r

TDi1 :

~C1Di

: ~rDi1

12

~rTDi

: MdDiT : ~C1Di

:MdDi : ~rDi, i 0, n 1,5

where ~rDi

1 is an effective stress tensor at damage Di+1 and is given by:

~rDi1 MdDi : ~rDi,dDi Di1 Di, i 0, n 1:

6

Besides the above general formula, one special case for ~rDi1 is:

~rD1 MD1 : rD0 MD1 : r, 7where ris a conventional stress tensor at damage D0 0. From Equation (5),the general formula for ~C

1Di

is defined as:

~C1

Di1 MdDi

T

: ~C1

Di:M

dDi

, i

0, n

1:

8

The above equation defines a multiresolution transformation rule for mate-

rial defects. The fundamental motivation for such an incremental form is to

744 J. SHEN ET AL.



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develop an efficient means of sequential homogenization for varying sizes

and degrees of defects in a material element.

To take an example of using Equation (8), we consider a two-step incre-

mental transformation with the defects at levels i and i+ 1. Incrementalapplication of Equation (8) leads to:

~C1Di2 MdDi1T :MdDiT : ~C

1Di

:MdDi :MdDi1: 9If ~C

1Di

is represented by:

~C1

Di 1

~EDi

1 ~Di ~Di 0 0 0 ~Di 1 ~Di 0 0 0

~Di

~Di 1 0 0 00 0 0 21 ~Di 0 0

0 0 0 0 21 ~Di 00 0 0 0 0 21 ~Di

2

6666664

3

7777775,

where ~EDi and ~Di are respectively the effective elastic modulus and

Poissons ratio, we then have:

C1Di2

1

~EDi

b11 ~Dib12 ~Dib13 0 0 0 ~Dib21 b22 ~Dib23 0 0 0 ~Dib31 ~Dib32 b33 0 0 0

0 0 0 21 ~Dib12 0 00 0 0 0 21 ~Dib23 00 0 0 0 0 21 ~Dib31

266666666664

377777777775

,

10where

bst 11 dDis1 dDi1s1 dDit1 dDi1t ,wheres, t 1,2,3

in which, dDi[k] (k 1,3) and dDi+1[k] (k 1,3) are the primary componentsof damage increments at Diand Di+1, respectively.

To check the accuracy of Equation (10), a single step transformation can

be envisioned as follows:

~C1Di2 MdDi dDi1T : ~C

1Di

:MdDi dDi1, 11




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which then leads to:

~C1Di2

1

~

EDi

b011 ~Dib

012 ~Dib

013 0 0 0

~Dib021 b

022 ~Dib

023 0 0 0

~Dib031 ~Dib

032 b

033 0 0 0

0 0 0 21 ~Dib012 0 0

0 0 0 0 21 ~Dib023 0

0 0 0 0 0 21 ~Dib031

2666666664

3777777775

,

12

where

b0st 1

1 dDis dDi1s1 dDit dDi1t:

C1Di2 predicted by the incremental form in Equation (10) should be a close

approximation to ~C1Di2 calculated in Equation (12).

NUMERICAL ANALYSIS AND DISCUSSIONS

Numerical simulations are conducted by a FE package. To verify the pro-

posed method, a series of regular FE domains are constructed with different

settings of material defects embedded in each FE domain. In this article, we

consider these FE domains as representative volume elements (RVEs). When

our method is applied to the engineering analysis of real structures, RVEs will

be distributed as a local neighborhood within the domain of real structures.

One classical criticism to the RVE approach is that the RVE analysis gives

an upper bound and a lower bound of apparent stiffness with specified

displacement and traction conditions, respectively (Hollister and Kikuchi,

1992). In other words, the estimated stiffness varies with the boundary

conditions. In the context of regular FE domains, we found out a special

displacement boundary condition that almost (


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(3) Two nodes are constrained at one corner of the RVE to prevent any rigid

body movement.

The average strain and stress in the regular FE domain can be easily

computed on the basis of the FE analysis. The effective elasticity modulus

is then calculated by a ratio of average stress to average strain in one of three

coordinate directions.

To simplify the validation of Equation (10), 1D cases are considered.

Equations (10) and (12) are respectively degenerated into:

CDi2 ~EDi1 dDi121 dDi112 ~EDi1 dDi1 dDi11 B13

~CDi2

~EDi

1

dDi1

dDi

1

1

2

~ED

i1

dDi1

dDi

1

1

A,

14

where,

B dDi1dDi111 1 dDi11 dDi11 dDi1 dDi111 dDi11 dDi11,

A dDi1 dDi111 dDi1 dDi11: 15The difference between ~CDi2 and

CDi2 is then:

~CDi2

CDi2

~EDi

A

B:

16

On the basis of Equations (9) and (11), for 1D cases we have the following

three hypotheses:

I. FE prediction of CDi1 can be represented by:

CDi1 EDi1 ~EDi1 dDi12II. FE prediction ofCDi2 can be represented by:

CDi2 EDi2 ~EDi11 dDi112

III. FE prediction of ~CDi2

can be represented by

~CDi2 ~EDi2 ~EDi1 dDi1 dDi112

To ascertain the degree of approximation ofCDi2 to ~CDi2 , a numerical

scheme is designed with the procedure described in Table 1.

The shape of real defects in materials has been observed to be random and

arbitrary, and may contain c1 discontinuity at their sharp edges and corners.

Here, c1 discontinuity means the discontinuity of the first-order derivative of

surface or boundary-line functions. Accordingly, numerical simulations of

the equivalency analysis become necessary to determine the change in effec-tive elastic moduli and associated damage variables. In this article, we design

a series of numerical test examples with different levels of material defects.




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Figure 3 depicts numerical models for different defect shapes, including

sphere, ellipsoid, triangular prism, rectangular prism, polygonal prism, and

arbitrarily shaped cracks. The numerical results of two-level damage trans-

formation in one dimension are given in Table 2. In the heading of this table,

i 0 refers to original or as-received state as a starting point for a config-uration of two-level damage, and megapascal is the unit for all the moduli.

The material damage is measured in terms of effective modulus in the

vertical direction.According to Table 2, the damage transformation estimated by the incre-

mental formula ( EnumDi2 ) is quite close to the actual damage, ~E

femDi2 , for differ-

ent types of voids as well as the combination of voids and cracks. For cracks

alone, the difference between these two, however, becomes significant, as

indicated in row 6 for crack in Table 2.

One fundamental question to be addressed is the rationale behind the use of

incremental formula if the compound damage, ~EfemDi2 , can be calculated by a one-

step FE analysis. There are two main reasons for using the proposed method:

(1) The proposed method provides a sequential homogenization of material

defects. This can thus simplify what would otherwise be a time-consuming

and laborious computing in numerical modeling, meshing, and analysis of

prohibitive number of micro-defects. The simplification is achieved by

processing the defects level by level in sequence.

(2) The incremental formula facilitates partial damage transformation. This is

extremely useful in the cases where only fine defects need damage trans-

formation, while coarse defects remain in the subsequent analysis domain.

Take for example both voids and cracks as one type of inclusions in amatrix-based material. The effective modulus of the inclusions may vary at

different levels with respect to that of the matrix material. Figure 4 is a test case

Table 1. Numerical Comparison ofCDi2 and ~CDi2 .

Step 1. Conduct an FE analysis on defects at level i+ 1. On the basis of FE results, an

estimate ofEDi

1 is denoted by E

femDi

1

, from which dDi

1

jfem

1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE

femDi1=

EDi

q .

Step 2. Conduct an FE analysis on defects at level i+ 2. On the basis of FE results, an

estimate ofEDi2 is denoted byE

femDi2 , from which dDi11jfem 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE

femDi2=

EfemDi1

q .

Step 3. Conduct an FE analysis on defects at the combination of level i+ 1 and leveli+ 2.

On the basis of FE results, an estimate of ~EDi2 is denoted by ~E

femDi2 .

Step 4. Calculate EnumDi2 by using Equation (13):

EnumDi2 ~EDi1 dDi1jfem dDi11jfem A, where A is calculated by Equation (15) withdDi1jfem and dDi11jfem replacing dDi1 and dDi11, respectively. Note that for cracks,EnumDi2 should be further corrected to

EcorDi2 (see the later part of this section for details.)

Step 5. Compare EnumDi2 and ~E

femDi2 as a numerical comparison of

CDi2 and ~CDi2 .

748 J. SHEN ET AL.



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for studying the effect of the effective modulus of inclusions on the damage

transformation. Results of numerical analyses are given in Table 3, indicatingthat the smaller the ratio of inclusions modulus to matrixs modulus is, the

more remarkable the difference between EnumDi2 and ~E

femDi2 becomes. When the

(a) (b)

(c) (d)

2

13

2

13

2

13

2

13

(e) (f)

2

13

2

13

(g)

2

13

Figure 3. Numerical models of material defects with different shapes: (a) sphere, (b) ellip-soid, (c) triangular prism, (d) rectangular prism, (e) polygonal prism, (f) crack,(g) crack + void.




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ratio is zero, inclusions are solely consisted of holes, and the difference reaches

its maximum if no strong inclusion is considered. Although the differencebetween EnumDi2 and

~EfemDi2 in Table 3 is not significant for void-type inclusions

(Figure 5), this is not true when the increment of damage becomes large, as

demonstrated by the next example in Figure 6 and Table 4.

Figure 6 is a typical model with two-level circular holes: one large hole

and four small holes. It is used to investigate the effect of damage increment

on the transformation rule of material defects. Here, the damage increment

means the extent of extra damage induced at certain level of defects. Let dDi(i 0, n 1) be defined an incremental damage tensor. In 1D cases, dDi isdegenerated into one principal component of the damage tensor.

The first column of Table 4 represents the radii of these two sizes of holes

in each test case. A general tendency is that the bigger the holes are, the

2

13

Figure 4. Numerical model with weak inclusions.

Table 2. Shape effect of material defects on the transformationof material damage (i^0; unit: MPa).

Defect shape E femDi

1E femDi

2

E numDi

2

~E femDi

2

Sphere 66134.2 58412.2 58196.8 58022

Ellipsoid 69048.4 62109.6 62063.2 62000.2

Triangular prism 66211.4 51661.2 51276.9 51181.6

Rectangular prism 62877 41520.8 40471.9 39283.4

Polygonal prism 52308 38642.4 36686.7 34647.8

Crack 43609 25921 21972.8 29610.8

Crack + Void 63098.8 52734.6 52208.6 52082.8

750 J. SHEN ET AL.



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more significant the difference between EnumDi2 and ~E

femDi2 becomes. It is inter-

esting to note that even for void-type defects the difference between EnumDi2and ~E

femDi2 could increase, as illustrated in the last row of Table 4.

In the table, the larger the radii are, the bigger the damage increment is.

This is analogous to the error caused by a Taylor series expansion. When the

increment becomes large, a linear approximation becomes insufficient to

maintain a low approximation error. One solution to this is to try to use

a reasonable number of multiple levels in the multiresolution damage trans-

formation such that each damage increment is not too large at each level.

Figure 7 is a typical model with two-level cracks: one large crack and four

small cracks. The first column of Table 5 represents the lengths of these twosizes of cracks in each test case. A general tendency is that the bigger the

cracks are, the more significant the difference between EnumDi2 and ~E

femDi2

Ei: inclusion E0: matrix

50000

55000

60000

65000

70000

75000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Ei/E0

E(MPa)

E2

E3

Figure 5. Damage transformation with weak inclusions.

Table 3. Effect of weak inclusions on the transformation ofmaterial damage (i^ 0; unit: MPa).

Ratio of inclusions modulus

to matrixs modulus E femDi1 E femDi2 E numDi2 ~E femDi2

0 68681.8 52955.8 52815.5 53120.4

0.25 69279.4 61436.8 61397.4 61444.8

0.5 69618.8 65641.6 65630.9 65627.4

0.75 69841.4 68229.2 68227.4 68219

1 70000 70000 70000 70000




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becomes. As observed by other researchers (Kachanov, 1992), cracks behave

differently from void-type defects. The damage transformation of Equation

(14) may be modified specifically for the cases of cracks as:

Ccor

Di2 E

femDi2 dDi15 0:15 or dDi115 0:15

~EDi1 dDi1 dDi11 rA otherwise

8

IJDM_739 a Multiscale Transformation Rule of Material Defects Jie Shen Et Al. 2009

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