A9944792 AIAA-99-1445

Woven Fabric Composites: Part 2 - Characterization of Macro-Crack Initiation

Loads for Global Damage Analysis

Jaehyon Choi’ Kumar K. Tammat

Department of Mechanical Engineering 111 Church Street S.E.

University of Minnesota Minneapolis, MN 55455

Ph: (612) 625-1821 Fax: (612) 624-1398

Abstract

In Part 2, finite element techniques are developed to additionally investigate the global damage of woven fabric composites, focusing on plain weaves. The homogenized elastic properties of the original undamaged and the damaged woven fabric composites from the micromechanical homogenization and the micromechanical progressive damage analysis investigated earlier in Part 1 are subsequently employed in the present global damage analysis. The theory of continuum damage mechanics is utilized in the global damage analysis. The damage variables, the most important material properties of the continuum damage mechanics, are measures of average material degradation at a macro-mechanics scale. In the present study, these damage variables are calculated numerically using the results from the micromechanical damage analysis in Part 1, instead of being obtained experimentally. Subsequently, a finite element formulation for the global damage analysis of woven fabric composites is developed to predict the initiation loads of macro-cracks.

Nomenclature

x; =

& =

u; =

Ei =

(Ti =

E, =

Mij =

principal axis in Cartesian

coordinate system general axis original overall cross-sectional area

resisting area

isotropic damage variable

anisotropic damage variable

in principal axes

-damage evolution rate

actual stress actual strain

effective stress

effective strain

damage effect tensor component

c = c =

El =

w2, h3 =

El =

fin, 43 =

Fd =

P =

Bo =

m4 =

C?d =

J =

P =

*Graduate research assistant tprofessor, to receive correspondence. E-mail: ktamma@tc.umn.edu Copyright 0 1999 by the authors. Published by the American htihk of Aeronautics and A~honautic~, Inc. with permission.

complementary elastic energy undamaged stiffness tensor effective stiffness tensor

undamaged Young’s modulus in xl-direction

undamaged Poisson’s ratios damaged Young’s modulus in xl-direction

damaged Poisson’s ratios damage dissipation potential overall damage initial damage threshold increment of damage threshold effective damage equivalent stress damage characteristic tensor damage evolution constant

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1 Introduction

Whereas the micromechanical damage analysis is a unit cell level study including the observation of damage initiation and propagation in the unit cell, the global damage analysis is. concerned with pre- dicting a macro-crack initiation load in a global structure with a crack. Continuum damage me- chanics is utilized in this global damage analysis. Kachanov [l, Z] introduced the continuum damage mechanics in 1958. The concept of the continuum damage mechanics can now be adequately used to supplement the design of structures based on the theory of fracture mechanics. Fracture mechanics considers the initiation and growth of micro-cracks as a discontinuous phenomenon. In contrast to this, the continuum damage mechanics uses continuous damage variables to describe material deterioration before the initiation of macro-cracks. Although the continuum damage mechanics provides a measure of material degradation at a micro-scale level such as nucleation and growth of voids, cavities, and micro-cracks, the damage variables are introduced to reflect average material degradation at a macro- scale level associated with the theory of continuum mechanics. These damage variables are typically measured experimentally using laboratory-size spec- imens recommerrded for conventional testing stan- dards.

Based on the damage variables, constitutive equations are developed to. predict the macro- crack initiation loads [3-9]. Lemaitre [lo, 111 and Chaboche [12] used the continuum damage mechan- ics to solve fatigue problems. Leckie et al. [13], Hult [14], and Lemaitre et al. [15] used it to solve creep and creep-fatigue interaction problems. All these works are, however, restricted to isotropic damage where material damage is same in all di- rections. Murakami [16] and Krajcinovic [17] inves- tigated brittle and creep fracture using anisotropic damage models. These models lack mathemati- cal justification and mechanical consistency. Af- ter the works by Sidoroff [18] and Lee et al. [19] on anisotropic damage models which suffered cer- tain anomalies in mathematical formulations, Chow and Wang [20-221 proposed an anisotropic dam- age model of the continuum damage mechanics for isotropic materials. This anisotropic damage model of Chow and Wang was proven to show good results when compared with experimental results. Voyiad- jis et al. [23] verified the model of Chow and Wang. Their result was close to that-of Chow and Wang.

In the present study, as mentioned earlier, the damage variables are obtained numerically using the damaged and the undamaged elastic moduli and Poisson’s ratios obtained from the micromechanical

analysis instead of being measured cxperimcntally. The anisotropic damage model for isotropic materi- als proposed by Chow and Wang is modified for wo- ven fabric composites. Then a 6nite element formu- lation is developed based on the modified anisotropic model. Also, a finite element mesh for the global damage analysis, a ccntcr-cracked thin woven fabric composite plate subject to in-plane uniaxial tension, is employed. Subsequently, the macro-crack initia- tion loads are predicted by applying the in-plane uni- axial tensile force. Two yarn orientations of 0”/90” and f45” to the loading direction arc investigated for predicting the macro-crack initiation loads.

2 Application of Continuum - Damage Mechanics

2.1 Damage Variables and Effective Stresses

Kachanov [l, 2] introduced the idea of damage in the framework of continuum mechanics. Based on the theory of the continuum damage mechanics, the phenomenon of progressive material degradation is introduced by a number of damage parameters, in- cluding an effective stress tensor 6 and a damage tensor 4 which is of second-order. The concept of the damage variables may be physically illustrated by considering a damage volume element at macro- scale level as shown in Figure 1.

Let A shown in Figure 1 be the overall cross- sectional area of the element before loading with its orientation defined by the direction of a unit normal vector n. Due to material degradation caused by the presence of micro-cracks, cavities, micro-stress concentrations in the vicinity of discontinuities and the interactions between the closed defects, the area A becomes the effective resisting (remaining) area A after loading and no longer depends on the direc- tion n. For the case of isotropic damage and using the concept of effective stress, the isotropic damage variable $ is defined as

where 4 is the time rate of damage and 4 > 0 means that 4 always increases.

Figure 2 shows the isotropic damage of a bar cle- ment in uniaxial tension. In Figure 2, D is an actual stress and T = aA is the force acting on the over- all cross-sectional area A while 5 is an imaginary effective stress and T = CA is the force acting on the resisting area -4. Using the hypothesis of strain equivalence [24], the effective stress 3 can be ob- tained by equating the force acting on the area .4

2037

with the force acting on the area A as

u-4 = (T/i (4

Equation 2 can be rewritten for general cases, using tensor notation, as

aA=eii (3)

The effective stress tensor 5 based on the effective area A is related to the actual stress tensor 0, using Equation 1, by

-4 u e=“Z= l-4

For the anisotropic damage, the effective stress of Equation 4 may be expressed in a generalized form as

a = M($)u (5) where &f(4) known as the damage effect tensor is a linear symmetric operator. If the principal axes (~1, ICZ and 2s) of 5 and u coincide, the compo- nents of M(4) may be expressed in the principal coordinate system as

M;j(f$) = $~;(f#~)&j no sum over i (6)

where $(4) are the components of M(4) in the form of scalar functions, and &, us the Kronecker delta. Equation 5 becomes

(Ti = Mij(q5)Uj (7)

In terms of a general coordinate system (z;, Z; and xi), the tensor is derived using the law of coordinate transformation.

Chow and Wang [20] suggested one particular formulation for the damage effect tensor Afij in the principal coordinate system as:

(8)

The damage variables 41, 42 and 4s in Equation 8 refer to the principal damage components as before. Since 04 = (~5 = us = 0 in the principal axes, Equa- tion 7 becomes in the principal axes

(9)

2.2 Hypothesis of Elastic Energy Equivalence

Sidoroff [18] proposed the use of the elastic energy equivalence concept and postulated that the com- plementary elastic energy of a damaged material is the same in form as that of an undamaged material, except that the stress is replaced by the effective stress in the energy formulation. The complemen- tary elastic energy W(a, 0) of an undamaged mate- rial (4 = 0) is

1 W(u, 0) = jUTC% (10)

where m is the actual stress tensor and C is the un- damaged stiffness tensor. The complementary elas- tic energy W(a, 4) of a damaged material is

where 6 is the effective stiffness tensor of the dam- aged material.

By the hypothesis of the elastic energy equiva- lence concept, IV(u, 4) can also be expressed as

W(u, 0) = TV((T, 0) = ;a716 (12)

Substituting Equation 5 into Equation 12,~we have

TV(u, f)) = ;(Mu)TC-l(Mu) = ;uTMTC-‘Mu (13)

Equating Equation II with Equation 13 gives

&l=~Tc-l~ (14)

which yields

The elastic compliance matrix of woven fabric composites with the ZQ-ZZ in-plane is

-KU -42 I& 1 El -Ku -Ku El 0 0 0 El El El 0 0 0

- KlA -aa c-1 = El

0 0”’ l& 0 0 0 d & 0 0

0 0 0 o&o 0 0 0 oo&

( Substituting Equations 8 and 16 into Equation we obtain the effective elastic compliance matrix

14, for

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damaged woven fabric composites in coordinate system as:

the principal

07)

2.3 Verification of D-age Variables

From Equation 11, the constitutive damage equation of elasticity is written as

E= aw(u,+) =C-lu

au (18)

An important ingredient in the derivation of the damage tensor equations is that the damage vari- ables should be readily reduced to material param- eters measurable from a uniaxial test. For instance, by substituting Equation 17Jnto Equation 18, the constitutive equations under uniaxial tension in the xi-direction axe

El = El(ly ~1)2 = 2 -

--v12n1 E2 = El(l -Tl)(l _ 42) =-k2El = -fi12g

(19)

--y13g1 63 = El (1 -q(l _ 43) *h3c1 = --y13 2

from which El = Ei(1 - f$i)2

v12 = h2Pr $1)

1 - 42

are the damaged effective Young’s modulus and Poisson’s ratios, respectively. Accordingly, the dam- age variables $1, & and 45s can be evaluated as

43 = 1- v13(1 - 41) v13

El 3 242 and ~13 are given elastic properties. From a uniaxial tension test, El, 232 and fiis arc measured and thus 41, $2 and $3 are obtained. In the present study, El, 42 and ~13, the homoge- nized elastic properties of woven fabric composite unit cell are obtained from the micromechanical ho- mogenization study investigated earlier by authors. El, Vi2 and r33, the homogenized elastic properGs of the damaged woven fabric composite unit cell arc obtained during each load increment from the mi- cromechanical progressive damage analysis instead of being measured experimentally.

2.4 Damage Evolution

A crack in a part of a material will propagate to- wards contiguous parts when an applied load reaches the macro-crack initiation load of the material. Therefore constitutive equations for damage evolu- tion are required in the theory of continuum dam- age mechanics. To investigate the damage cvolu- tion of a damaged material, a new damage parame- ter, namely, the damage evolution constant is neces- sary in addition to the damage variables mentioned above. Chow and Wang [21] proposed the damage dissipation potential, a damage evolution criterion as

Fd(Z,B) = Sd - [Be + B(P)] = 0 (221 where p = overall damage, Bs = initial damage threshold and B(P) = increment of damage thrcsh- old depending on ,B, and the effective damage equiv- alent stress ad is defined as

ifd =

(23)

where j = MTJM (24)

The damage characteristic tensor J characterizes the damage response of the material to the effective

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stresses. It is assumed that J is independent of ma- terial positions and coordinate systems chosen. This implies that J is same everywhere and J’ = J for which J’ is the damage characteristic tensor in any coordinate system zr:. But it is evident from the above that the effective damage characteristic ten- sor J = J(4) is element position dependent due to damage induced inhomogeneity. In fact, it can be readily verified that in an arbitrary coordinate sys- tem of s:,

cf’ = [M’lTJM’ # j (25)

due to the damage induced material anisotropy as M’ is different from M except for isotropic dam- age.

Chow and Wang [21] proposed the J-formulation as:

1 P P 0 0 0 PlP cl 0 0

J=2 PP1 0 0 0 0 0 0 2(1-p) 0 0 000 0 V-P) 0

-000 0 0 20 - PI - (26)

where ~1 is a material constant. J is positive semidef- inite since from Equation 23

1 5; = ZCTJE > 0 -

from which the value of p is confined within the range of

The value of /.L = 1 implies isotropic damages while that of p = -i does the highest damage anisotropy.

Taking the damage dissipation potential as a damage evolution criterion, the damage evolution equations can be deduced as

(28)

(29)

where the Lagrange multiplier Xd is evaluated as

( >o if Fd = 0

Ad =

I or if Fd i 0

2.5 Verification of Damage Evolution

Analogous to Equation 21 which was derived to ob- tain the damage variables 41, 42 and 43, we derive a equation to obtain the damage evolution constant p. Under uniaxial tension, the damage evolution of Equation 28 becomes

{ I!}=${ $}* (30)

where 41, $2 and $3 are the damage evolution rates in the principal axes. From Equation 30, it is deduced that

db 1 - 42 -=

dh I41 - $1)

Integrating Equation 31 yields

$20 - $2/2)

p = $&Cl-41/q

(31)

(32)

Or substituting Equation 21 into Equation 32, we have

~ = l- (~I’($ l-2

(33)

It is noticed that p = 1 implies the isotropic dam- ages of 41 = 42 = 43. With the knowledge of 41 and 42 calculated from Equation 21, p can be read- ily determined using Equation 32. In fact, p is the slope of the large portion of linearity displayed by the relationship of the two fuctions, &(l - r&/2) and h(l -h/2).

The increment of damage threshold B(P) can also be readily determined. From Equation 23, we deduce under simple tension

Substituting Equations 29 and 34 into Equation 30, we have

& =L 1 - ($1

from which p is finally derived after integration as

P = $hu -b/2) (35)

On the other hand, B in Equation 22 may be de- duced under simple tension

Bz~~-&,=-

1 - 41 - B. = Cl - B,, (38)

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3 Finite Element Analysis

3.1 Finite Element Formulation

The general procedure to formulat,e the finite ele- ment equations including material damage is similar to the conventional one. The only differences are the constitutive equations of elasticity which con- tain the new variables for damage, thus necessitating the modification of the conventional stiffness matrix. For the relationship between the effective stress in- crement and the effective strain increment, we may rewrite the elastic constitutivc equation of Equa- tion 18 with the concept of elastic energy equiva- lence. To investigate a macro-crack initiation load, in-plane tensile loading is applied incrementally un- til the overall damage p reaches its critical value. Therefore the equation is cast in the incremental form given as

{da} = [C]{d?} (37)

The procedure now derives the effective instan- taneous stress-strain matrix [Cl corresponding to

{du} = [C]{de} (38)

The effective strain tensor is defined as

(5) = [M]-T(E} (39)

From the definitions of Equations 5 and 39, we have

ida) = Wfl{~~) = 4MltnT) + Wltda) (40)

{dc} = d([M]-T{c}) = d[M~-T{c} + [M]-T{de} (41)

=

and

and for the state of plane stress we have

@4 = 8{4}T ={dr$}

d[M]-T = G (dd)

I (42)

= -[A~]-T~(dc,b}[M]-’ (43)

From Equations 28 a,nd 29,

(44)

Premultiplying Equation 44 by {aF~/a$}T gives

{%}T{d+ {%}T{z}dP (45)

Rewriting the damage criterion of Equation 22 in the incremental form, we have

dad - dB(/3) = 0

or

(46)

From ed = &(f$, a), we have

Taking the partial derivatives of Equation 22 with respect to 4 and u gives respectively

(48)

(49)

Substituting Equations 47, 48 and 49 into Equa- tion 46 leads to

{$jT{dq!++{~)i{d+~d/9=0 (50)

Substituting Equation 45 into Equation 50, we have the equation of the incremental form of the overall damage as

df, = {%h,, Idu) (51)

Then from Equations 44 and 51, the incremental rc- lation between damage variables and strcssss is ob- tained as

By combining Equations 18, 37, 40-43 and 52, the effective stress-strain matrix of Equation 38 is finally derived as

[q = bfl-‘~XIKmfl-T (53)

2041

where

[xl = [II +

Also we can calculate the overall damage ,0 and the increment of damage threshold B(P) from p = $i (1 - &/2) (Equation 35) and B = Cr - BO (Equa- tion 36), from which the B-P curve can be readily

$f$ (9) [Ml-’ + [Cl ($$‘$ { 2> [nl-‘)T [Cl:’ obtained. TmThen using the B-B curve, -we can calculate.

where [I] is the identity matrix.

3.2 Geometric Global Damage Mode1

The geometric finite element models for the global damage analysis are center-cracked thin woven fab- ric composite plates subjected to in-plane uniaxial tension [25]. Figures 3 (a) and (b) show the com- posite plates with 0’/90” and f45” yarn orientation to the loading direction, respectively. These woven fabric composite plates are anisotropic (square sym- metric) but now homogenized. Because of the sym- metrical nature of the composite plates in geometry and loading, only a quarter of t,he composite plates need to be analyzed. The grid for this finite element mesh for the composite plate consists of 380 a-node quadrilateral elements and 1225 nodes. The state of plane stress is assumed since the thickness is small compared to the other dimensions. As mentioned before, in this part of the damage analysis, we pro- pose to predict the macro-crack initiation loads for the thin woven fabric composite plates with a center crack.

3.3 Determination of Damage Pa- rameters

To be used in Equation 21 for calculating the dam- age variables, the homogenized elastic properties of the original undamaged woven fabric composites, EI and ~12, are obtained from the micromechanical homogenization procedures. And the homogenized elastic properties of the damaged woven fabric com- posites, fii and fii2, are obtained during each load increment from the micromechanical damage analy- sis.

After calculating the damage variables, 41 and $2 using Equation 21, we can calculate the effective stress ai = cl/(1 - $1) from Equation 9 and the effective strain Ei = (1 - $i)ei from Equation 39, from which the Cl-El curve can be obtained. In this 31-21 curve, we obtain the initial damage threshold Bo, the stress point of the initial stiffness change. The Bo is also the stress point at which the curve of & intersects the stress-axis in the curve of $i versus ffl

6'B/i3p to include in the overall damage increment, dp from Equation 51 and the effective stress-strain matrix, c from Equation 53 which includes damage evolution. The overall damage parameter p is mon- itored in the element around the crack tip at each load increment since it is this factor that is used to determine the crack initiation.

The damage evolution constant /J is calculated from the large portion of linearity displayed by the relationship of the two functions, &( 1 - &/2) and &(l - $i/2) in Equation 32, /.L = {&(l - 42/2)}/{&(1 - &/2)}. The large portion of lin- earity is chosen from the continuous data points of close slopes.

4 Results

4.1 Damage Parameters

Before performing the finite element analysis of the global loading, all damage parameters such as the damage variables, the overall damage, the increment of damage threshold, the initial damage threshold and the damage evolution constant were calculated based on the damaged and the undamaged elastic properties obtained from the micromechanical dam- age analysis investigated earlier by authors. The two woven fabric composite materials [26,27] inves- tigated in the micromechanical damage analysis are employed to predict macro-crack initiation loads in this global damage analysis. The tensile stress-strain curves for Material [26] obtained from the microme- chanical damage analysis are shown again in Fig- ure 4(a).

The values of El and Cl2 for Material [26] showed gradual material degradation with the increase of strain. For the damage variables for Material [26] the difference of the damage variables 41 and ~$2 (the degree of the anisotropic damage) was distinct. The value of p determined from the large portion of lin- earity was calculated as ~1 = 0.365 for Material [26] which belongs to the range of -5 _< p 2 1 discussed before.

The effective stress-strain ((Tr-El) curve and the increment of damage threshold-the overall damage (B-p) curve for Material [26] are shown respectively in Figures 4(b) and 4(c). The initial damage thresh- old was obtained as B,-, = 314 MPa for Mate- rial [26]. The critical overall damage was calculated as Per = 0.0688 for hiaterial [26].

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The tensile stress-strain curves for Material [27] obtained from the micromechanical damage analy- sis are shown again in Figure 4(d). The values of Er and VIZ for Material [27] showed gradual material degradation with the increase of strain like those for Material [26]. The difference of the damage variables #pi and 42 was small like an isotropic damage. The value of p determined from the large portion of lin- earity was calculated as p = 0.167 for Material [27] which belongs tothe range of -f 5 p 5 1. _-

The effect,ive stress-strain @i-Ti) curve and the increment of damage threshold-the overall damage (B-P) curve for Material [27] are shown respectively in Figures 4(e) and 4(f). The initial damage thresh- old was obtained as Ba = 288 MPa for Material [27]. And the critical overall damage was calculated as ,& = 0.06033 for Material [27].

4.2 Macrwcrack Initiation Loads

The macro-crack initiation loads were investigated for the two yarn orientations of 0”/90” and ~t45” for the two woven fabric composite materials [26,27]. Table 1 shows the macro-crackinitiation loads calcu- lated for each case. Material [26] has higher macro- crack initiation loads than does Material [27]. For each material, the composites with f45O yarn orien- tation show higher macro-crack initiation loads than those with 0°/900 yarn orientation. At the same crit- ical overall damage per = 0.0688, which implies the crack initiation, the composite with &45” yarn orien- tation sustains higher stress and thus higher macro- crack initiation load. Similar to the case of the com- posite [26], the composite [27] with f45” yarn orien- tation sustains higher stress and thus higher macro- crack initiation l-oad at the same critical overall dam- age & = 0.06033.

5 Conclusions 1 In the present global damage-analysis of woven fab- ric composites, both the undamaged and the dam- aged homogenized elastic properties obtained from the micromechanical analysis investigated earlier by authors in Part 1 were utilized with the continuum damage mechanics. The finite element model and formulation for the global damage analysis were de- veloped to predict the macro-crack initiation loads for woven fabric composite plates with a center- crack. No experimental results are available for the global damage analysis of woven fabric composites. However, the damage variables, the most important material properties of the continuum damage me- chanics, are obtained based on the damaged and the undamaged elastic properties. These elastic prop- erties obtained from the micromechanical analysis

were shown to be in good agreement with cxpcri- mental results. Also a macro-crack initiation load for an isotropic material was shown to be in good agrcc- ment with available experimental results, which orig- inally validated the continuum damage mechanics model utilized in the present study [22]. The WO- ven fabric composite plates with ~t45’ yarn oricnta- tion showed higher macro-crack initiation loads than those of the composite plates with 0”/90’.

6 Acknowledgment Acknowledgment is due to service support in part,, by the MENET services and the Minnesota Super- computer Institute, University of Minnesota. Spe- cialized acknowledgment is due to Dr. Andrew Mark and C. Nietubicz of the U.S. Army Research Labo- ratory, the IMT efforts of the ARL/MSRC technical activities and Battelle/ARO under DAAH05-96-C- 0086.

il

PI

[p!

PI

151

161

171

PI

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0”/90° (MPU) f4iP (MPa) per B,, (MPa) Material [26] 247 324 0.0688 314 Material [27] 226 296 0.06033 288

Table 1: Macro-crack initiation loads, PO and Bo

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-

-

Figure 1: Damage volume element.

Figure 2: Isotropic damage in uniaxial tension (concept of effective stress).

I i Figure 3: Thin woven fabric composite plates with a center crack.

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(a) Tensile stress-strain curves obtained for Mate- rial [26].

(c) Increment of damage threshold B-P curve for Material [26].

(e) Effective stress-strain (El-Zl) curve for Mate- rial [27].

EffectWe Stress-Stmln Cura

(b) Effective stress-strain (S1-Sl) curve for Mate- rial [26].

(d) Tensile stress-strain curves obtained for Mate- rial [27].

Increment 01 Demage Threshold ( q - p Cura) mo-

(f) Increment of damage threshold B-P curve for Material [27].

Figure 4: Continuum damage mechanics results.

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