Failure stability of a cracked layer between dissimilar materials

Theoretical and Applied Fracture Mechanics 13 (1990) 59-68 Elsevier

59

FAILURE STABILITY OF A CRACKED LAYER BETWEEN DISSIMILAR MATERIALS C.K. C H A O and S.Y. L I N Department of Mechanical Engineering, National Taiwan Institute of Technology, Taipei, Taiwan, Republic of China

A composite is designed to fail locally such that energy can be dissipated more uniformly. Cracking of the matrix in a fiber-reinforced composite is not necessarily undesirable provided that the failure can be localized. Such a condition is investigated by depicting a cracked matrix layer between two orthotropic media whose elastic properties are those equivalent to the fiber and matrix depending on the volume fraction. An index of failure stability "1" is determined from the local and global relative minima of the strain energy density function, denoted by [(dW/dV)min]L and [ ( d W / d V ) m i n ] G , respectively. The former refers to the local coordinate systems at each point while the latter refers to the global coordinate system for the entire system. Depending on the load, geometry and material in_homogeneity, l being the distance between [(dW/dV)m~]L and [(dW/dV)m~']~ serves as a measure of the degree of failure stability. The failure of the cracked layer system is found to become more stable as the crack matrix layer thickness is reduced. Influence of load type is also analyzed in combination with changes in material properties and fiber volume fraction.

1. Introduction

Fiber reinforced composites have been used widely in engineering structures because they have a high strength-to-weight ratio. In principle, fibers are needed to provide the strength and stiffness while the matrix tends to distribute the load more uniformly. Broken fibers which are e m b e d d e d in the matrix can still carry a portion of the load. One of the difficulties in ranking composites is to include the c o m b i n e d effects of material inhomogeneity, geometry and loading conditions. The majority of past works have individually considered fiber breakage, matrix cracking, f i b e r / m a t r i x debonding and delamination [1-3]. While it is recognized that composite failure is localized, the relation between local and global failure is seldom quantified. Only in recent times has more attention been given to the failure stability condition [4-6]. One of the prerequisites for making a successful prediction of composite failure stability is a suitable criterion that is not always straightforward, particularly when the problem involves so m a n y variables. There is the additional uncertainty of inaccuracy arising from numerical analysis. Special attention should be given to the finite element gridding near regions of high energy concentration. To this end, the 1 / r decay of the strain energy density function would be preserved near the crack tip from which r is measured. This is 0167-8442/90/$3.50 © 1990, Elsevier Science Publishers B.V.

accomplished by shifting the nodes to ~ and ~ of the element side distance adjacent to the crack tip 1 [7]. It is, therefore, natural to incorporate the strain energy density failure criterion [8-10] into the finite element method.

2. Strain energy density criterion

Ambiguities arising from stress and failure analysis are not always obvious and have been discussed extensively in the literature. Conventional criteria such as stress or strain alone are n o w k n o w n to be inadequate in general as they tend to favor specific applications and contradict others. Increased applications of the strain energy density criterion [8-10] have shown its versatility in addition to consistency.

2.1. Basic assumptions C o n t i n u u m mechanics assumes that a system can be divided into finite elements that are connected in a continuous fashion. Because of variations in load, geometry and material properties, the energy stored in each unit volume of the

1 Although in linear elasticity, this also guarantees the 1/v~ crack tip stress singularity, the 1/r decay of dW/dV applies to any nonlinear problem in general.

C.K. Chao, S. Y. Lin / Failure stability of a cracked layer

60

material is not the same. For a given element, the energy density can be expressed as

dW dV

=)Co~"oig d~ij 1 =

~OijEij

,

(1)

in which o~j and c~j are, respectively, the stress and strain components. If r is the distance referenced from the point of observation such as a crack tip, then

dW/dV = S / r

(2)

and S is known as the strain energy density factor. In general, dW/dV can be computed everywhere in the system. The basic hypotheses of the strain energy density criterion as applied to yield and fracture initiation can be stated as follows: Hypotheses (1). The relative minima of the strain energy density function, (dW/dV)r~n, and maxima, (dW/dV)m~x, are assumed to coincide with the locations of fracture and yielding initiation, respectively. Hypotheses (2). Yielding and fracture are assumed to initiate when the maximum of (dW/dV)max or (dW/dV)m~Xx and maximum of (dW/dV)~,in or (dW/dV)~i~nx reach their respective critical values. More specifically, fracture initiates when (dW/dV)mi~x coincides with (dW/dV)¢. Fracture may or may not be unstable depending upon whether S is critical or not. In other words, (dW/dV)¢ guarantees only subcritical crack growth. The onset of rapid crack propagation corresponds to (1 + 1,)(1 - 2~)K2~ Sc = 2"~E '

nates (rj, 0j). If rj is taken to be fixed, say r0, then dW/dV will have a maximum [(dW/dV)r,a×]L and a minimum [(dW/dV)min]L where L denotes the local reference. Among them, there will be a set with the largest amplitude denoted by max max [(dW/dV)ma×]L and [(dW/dV)min ]L" On the other hand, every point in a system can be referred to a single coordinate system (X, Y). This is taken as the global view such that [(dW/dV)mm~x]G and [(dW/dV)mi~]G will be the unique pair. The subscript G denotes the global reference. In what follows, only fracture initiation will be considered and hence reference will be made only to the pair [(dW/dV)mi~x]L and [(dW/dV)~mi~]o where L and G are always at a finite distance away unless local and global instability coincides. The distance between L and G is denoted by " l " and known as the index of failure instability. In general, a system tends to be unstable as l increases and vice versa.

3. Problem and method of approach A schematic of the composite system to be analyzed is shown in Fig. 1. The fiber reinforced composite is modelled as an orthotropic material

J FIBER/MATRIX 2

=1

(3)

in which ~ and E are, respectively, the Poisson's ratio and Young's modulus. The ASTM valid fracture toughness value is K~¢.

MATRIX

. . . .

x

"

/

2.2. Local and global stationary values

tFIBER/MATRIX 2

The application of the local and global stationary values of dW/dV to characterize system instability has been described in ref. [11-13]. The idea was further extended to composites [4-6]. Very briefly, it is not difficult to visualize that at each point (xj, yj) ( j = 1, 2 . . . . . n) in a two-dimensional system, dW/dV possesses a unique maximum and minimum. This can be most easily found by referring to a system of polar coordi-

3

Fig. 1. Schematic of a cracked matrix layer in an orthotropic composite.

C.K. Chao, S. Y. Lin / Failure stability of a cracked layer

61

ttttttt

Table 1 Material properties of orthotropic composite Material

E1 × 106 (MPa)

E2 × 106 (MPa)

PI2

1)23

A B C

25 25 1

1 25 25

0.28 0.28 0.0111

0.37 0.28 0.37

that possesses the equivalent combined material properties of the fiber and matrix. With b and c fixed at 10 mm and 10 mm, respectively, the matrix layer thickness h can be varied. The fibers can be directed either in the 1 a n d / o r 2 direction as shown while 3 coincides with the thickness of the plate which is taken to be under the condition of plane strain. The subscripts 1, 2 and 3 will thus be used to denote quantities referred to these axes of material symmetry. Three different combinations of Young's moduli E 1 and E 2 and Poisson's ratios v12 and/-P23 are considered as given in Table 1 with case B being for an isotropic system of dissimilar materials. The matrix contains a crack of length 2a being equal to 2 mm centered at the origin of the (x, y ) coordinate system and is isotropic with a Young's modulus of E = 10.5 × 10 6 MPa and a Poisson's ratio of i, = 0.33.

3.1. Loading type Three different loading types are considered and referred to I, II and III as displayed in Figs. 2(a), 2(b) and 2(c), respectively. The first in Fig. 2(a) is a uniform constant load of P = 100 N T / m m while the second in Fig. 2(b) is a linearly varying load that depends on x as P0 (1 - I x / c I) with P0 = 2P. Type III loading is shown in Fig. 2(c) where a force F = 100 N T is concentrated at x = 0 and y = + [ b + ( h / 2 ) ] . The load path for each of the three cases will be different. This affects local crack tip stress intensification and hence failure instability.

tttt~t!

(a) Type I

(b)Type]'T

t 1

(c) Type 111" Fig. 2. Three different applied loadings. (a) Type I: Uniform distribution; (b) Type II: Triangular distribution; (c) Type III: Concentrated forces.

and 4 as the b/h ratio is altered. For b/h = 2.0, 92 elements and 490 nodes were employed as given in Fig. 3. Note that more elements were used

,,Q

\

t-

3.2. Grid pattern The twelve (12) node isoparametric finite elements will be used to discretize the composite system. Since this procedure is now well known, there is no need for a detailed elaboration. Reference can b e made to ref. [7]. Two basic grid patterns are used and they are displayed in Figs. 3

dl

\ /

\

Fig. 3. Grid pattern with 92 elements and 490 nodes for b / h = 2.0.

C.K. Chao, S. Y. Lin / Failure stability of a cracked layer

62

\

24 ~-

\

d

/

\

W

/

/

/

t.,

\

i

20 t

\

16

12 ,

\

Fig. 4. Grid pattern w i t h 80 elements and 430 nodes for b / h = 5.0 and 10.0.

near the crack tips where the stress and energy density are elevated. Only 80 elements and 430 nodes are used for the cases when b/h = 5 and 10. This is illustrated in Fig. 4.

L 0

I

I

d~ ......

4

J__

I

8

~

d__ . . . . . . .

12

L ......

16

L_

20

Fig. 5. Constant dW/dVcontours for E 2 / E 1 = 0.04, b / h = 2.0 and uniform load.

3.3. Failure instability index The condition of plane strain is employed everywhere in the system such that eq. (1) can thus be simplified by setting the transverse normal strain cz in the z-direction equal to zero. As mentioned earlier, the accurate determination of the strain energy density function d W / d V is pertinent for locating the stationary values. To this end, constant contours of d W / d V are computed to obtain the position for [(dW/dV)m~.x]G. With the presence of a crack, the location of failure initiation becomes obvious; it coincides with max [(dW/dV)mi. ]L which occurs at x = _+a, y = 0 in each of the example problems. Therefore, the failure instability index l can be found without difficulty.

4. Discussion of results Results for the three different material type A, B and C and loading type I, II and III are obtained in addition to different ratios of b/h. This will provide information on the ways with which material, loading and geometry as a combination

would affect the instability of failure initiating from a crack in the matrix layer.

4.1. Load type I - uniform distribution Referring to Fig. 2(a), the load applied on the composite system is uniform and has a magnitude of P = 100 N T / m m . With b/h = 2, 5 and 10 and material type A, B and C, the locations of L and G are found for the nine cases. Knowing that L is next to the crack tip, G can be found from a constant d W / d V contour plot as shown in Fig. 5 for E2/E 1 = 0.04 and b/h = 2.0. The location of G is distinct and easily identified. Table 2 summarizes the results for the failure instability index

Table 2 Failure instability index /(mm) for uniform loading

b/h

2 5 10

E2/E , 0.04

1.0

25

3.00 1.40 0.70

9.00 9.00 7.00

3.90 1.80 0.83

CK. Chao, S.Y. Lin / Failure stability of a cracked layer

2.5[

2.5

O a.

E

u~

63

2.0

~Z"°2

1.5

"-" 1.~

.

0

~

'

'

'

I0 ,p-

×

x

G

1,0

"O

>

1.0

"U

"0 0.5

0.5 I

0

'

'

'

'

4o

o

x-a (mm)

Fig. 6. Variations of strain energy density function with distance from crack tip for E 2 / E 1 = 25, b / h = 5 and uniform load.

l. A value of l = 3.0 m m is obtained. Failure becomes even more localized as b/h is increased to 5 and 10. Material A represents the case when the fibers are parallel to the crack which yields the smallest / for a given b / h ratio. While failure is more localized, it would occur sooner because the matrix is carrying most of the load. If the fiber is placed in the load direction, then E2/E 1 = 25. Even though I is increased slightly, this would be a more desirable orientation as the fibers would be carrying the load. Orthotropy of the material adjacent to the cracked matrix layer does enhance failure stability. Failure at large is predicted if the property of orthotropy is removed for b / h = 2 and 5 in which case G coincides with the specimen at x = c and hence l = 9.0 mm. Only for a very thin matrix layer b / h = 10 does G move inside the system to ! = 7.0 mm. Displayed in Figs. 6 and 7 are plots of d W / d V with x - a for E2/E 1 = 25 and b/h = 5 and 10, respectively. In both situations, the point G is clearly the m a x i m u m of minimum d W / d V . The local and global stationary value of the strain energy density functions demax noted by [(dW/dV)min]L and [(dW/dV)mi. m a x ]~, respectively, are given in Table 3. In general, [(dW/dV)m~n] L is approximately four orders of magnitude greater than [ ( d W / d V ) m imax n ]G" T h i s would indicate that should failure be initiated at L, tendency for the path to reach G is very weak. In this case, no failure is predicted because all of the strain energy density functions in Table 3 are

x-a

'

(mm)

/3

'

;o

Fig. 7. Variations of strain energy density function with distance from crack tip for E 2 / E 1 = 2 5 , b / h = 1 0 and uniform load.

below the critical value ( d W / d V ) ¢ = 8.40 x 10 -3 M P a for the matrix. Summarized in Fig. 8 are variations of l with the matrix layer thickness h. The parameter 1 for Materials A and C increased with increasing h. A wide layer of matrix material fails in a more unstable fashion. This is not an unexpected result. The curve for E2/E 1 = 1.0 has a weak m a x i m u m for l at h = 3.5 mm. In view of the lack of a physical explanation, the results may be suspect.

4.2. Load type H - triangular distribution Similarly, constant d W / d V contour plots of d W / d V can be found for the triangular load dis-

Table 3 Local and global stationary values of strain energy density for uniform loading

b/h

[(dW/dV)ma~]L

[(dW/dV)~'~.n] G

X 10 -3 (MPa)

× 10 -7 (MPa)

E 2 / E l = 0.04 2 5 10

6.31 x 1 0 - 3 10.70 20.40

9,46 7,37 6,29

E 2 / E 1 = 1.0 2 5 10

3.19 2.27 1.84

6.02 6.72 8.78

E a / E 1 = 25 2 5 10

3.87 4.92 5.96

10.4 10.1 11.8

64

C.K. Chao, S. E Lin / Failurestabili(v of a cracked layer

Table 4 Failure instability index /(ram) for triangular loading

12

E21E1 = 1.0

10

b/h

E2/E 1

2 5 10

8 E

i

I

r

I

I

o h (ram) f i g l 8 " In~u~nce of E m / E I arid m a t ~

laye~ thickness o~

instability index for uniform load.

t r i b u t i o n in Fig. 2(b). Again, the p o s i t i o n of G along the x-axis of s y m m e t r y can b e l o c a t e d as illustrated in Fig. 9 for E 2 / E t = 0.04 a n d b / h =

2O

0.04

1.0

25

5.70 1.47 0.83

9.00 9.00 9.00

9.00 9.00 9.00

2.0. This c o r r e s p o n d s to l = 5.7 m m which can be f o u n d in T a b l e 4. R e d u c t i o n in b / h again leads to larger 1. F i g u r e 10 d i s p l a y s the fluctuation of d W / d V for M a t e r i a l A a n d b / h = 2.0 as discussed earlier. T h e case of M a t e r i a l C a n d b / h = 5 is shown in Fig. 11 w h e r e G lies on x = c with l = 9.0 mm. A l t e r a t i o n s on b / h h a d no effect on / for E z / E t = I a n d 25. This is s u m m a r i z e d in Fig. 12 with l as a c o n s t a n t while it increased with h o n l y when E 2 / E 1 is r e d u c e d to 0.04. Referring to T a b l e 5, the local a n d g l o b a l strain energy d e n s i t y functions for the case of t r i a n g u l a r loading are tabulated. The magnitude of [ ( d W / d V ) m ~ ] L is again greater t h a n that of [ ( d W / d V ) m,x ] G b y several orders. It increases with decreasing thickness h of the m a t r i x for E z / E 1 = 0.04 a n d 25 t h a t c o r r e s p o n d to the o r t h o t r o p i c material. T h e o p p o s i t e prevails for the isotropic m a t e r i a l E z / E 1 = 1.0. F r a c t u r e initiation is pred i c t e d f o r E 2 / E l = 25 w h e r e all of the [ ( d W / d V ) m i ~ ] L values c o r r e s p o n d i n g to b / h = 2, 5 a n d 10 are greater t h a n ( d W / d V ) c = 8.40 × 10 -3 MPa. 2.5

2,0

12 o

1.5

)<

>

8

1.0 "0

0.5

o I

I

~1__J

I

I

I

~ hkkkk'xM/~

4 8 12 16 20 0 Fig. 9. Constant dW/dV contours for Ez/E I = 0.04, b/h = 2.0 and triangular load distribution.

'

'

' x-a

'

$

'

¢o

(mm)

Fig. 10. Variations of strain energy density with distance from crack tip for E2/E 1=0.04, b/h =2.0 and triangular load distribution.

C.K. Chao, S. Y. Lin / Failure stability of a cracked layer

4.0

Table 5 Local and global stationary values of strain energy density for triangular loading

"~ 3.2

b/h

a.

to

2.4

to

65

. . [( d W/dV)min]L × 10 -3 (MPa)

.

.

. [(dW/dV)min]G × 10 -7 (MPa)

.

E2/E 1 = 0.04

v.x

> 1.6

0.8 G _

0

2

4

6

~-

8

I

10

x - a (mm)

Fig. 11. Variations of strain energy density function with distance from crack tip for E 2 / E 1 = 25, b / h = 5 and triangular load distribution.

2 5 10 E 2 / E 1 = 1.0 2 5 10 E 2 / E 1 = 25 2 5 10

8.94 x 1 0 - 3 15.50 30.10

8.62 9.59 8.42

4.45 3.71 3.21

2.82 2.11 2.09

11.2 15.2 19.1

1.76×10 -2 0.42 2.29

24

4.3. Load type 111 - concentrated load Consider the situation in Fig. 2(c) where the load is now concentrated at the points x = 0 and y = +__[b+ ( h / 2 ) ] . A typical plot of d W / d V constant contours is given in Fig. 13 for E2//E1 = 0.04 and b /h = 2.0. T h e point G falls inside the composite at x = 5.7 mm. Additional plots can be obtained to find G as b / h and E 2/ E 1 are varied. Table 6 summarizes the results of calculation. The numerical values of d W / d V in Fig. 13 can be plotted as a function of the distance from the

20

16

12

12

10

v

E2/E 1 = 1 & 95

E E

0.04

4

j

0

4

8

12

16

Table 6 Failure instabifity index l(mrn) for concentrated load

b/h o

I

20

Fig. 13. Strain energy density contour for the case of a concentrated load.

I

I

I

h (mm)

Fig. 12. Influence of E 2 / E 1 and matrix layer thickness on instabifity index for a triangular load distribution.

2 5 10

E2/E 1 0.04

1.0

25

5.70 1.47 0.83

9.00 9.00 9.00

7.00 3.17 3.03

C.K. Chao, S. Y. Lin / Failure stability of a cracked layer

66 4.0

~ 0 Q.

8.0

6.4

3.2 0 ¢1.

:E

=...-

'o

v

2.4

i~O

4.8

v.x

x

>

>

1.6

"lO

-o

3.2

"0

0.8

0

•"o

I

I

I

I

2

4 x -

1

I

6

I

I

I

8

1

1.6

b

I

I

0

2

4

6---~

10

x - a (mm)

a (ram)

Fig. 14. Variations of strain energy density function with distance from crack tip for E2/E ] = 0.04, b/h = 2 and concentrated load.

Fig, 16. Variations of strain energy density function with distance from crack tip for E z / E 1 = 25, b/h = 5 and concentrated load.

crack tip which yields G as shown in Fig. 14. W h e n E 2 equals to E 1 even for b / h - 10, failure is predicted to occur at large, i.e., a crack once started would extend across the entire width of plate x = c. This is implied b y the variations of d W / d V with x - a in Fig. 15. Stability is enhanced when the m o d u l u s E 2 differed from E1 a n d shown in Fig. 16 for E z / E 1 = 25. The p o i n t G moved toward the crack tip a n d a value of l = 3.17 m m is o b t a i n e d for b / h = 5. A c c o r d i n g to the data in T a b l e 6, l did not change with h for E2/E ] = 1.0 while it increased with h for E z / E 1 = 0.04 and 25. F o r h < 2.0 mm, l tends to a c o n s t a n t

for E z / E ] = 25. This is shown by the graphical display in Fig. 17. I m p r o v e m e n t on failure stability increases as h decreases if the fibers are aligned i n the direction of the load. This corresponds to the curve for E 2 / E 1 = 25. The influence of E 2 / E 1 a n d b / h o n the strain energy density for a c o n c e n t r a t e d load is similar to that for a t r i a n g u l a r load except that the magnitude of [ ( d W / d V ) m ~ ] L is higher. The results are shown in T a b l e 7. N o t e that for E 2 / E ] = 25, [ ( d W / d V ) ~ mn a]xL has exceeded the ( d W / d V ) c =

12 5

10

E 2 / E 1 = 1.0

4

8 E E

b

6

4

"o

1 2

G 0

2

4

6

8-

-

. 4

1'0

x - = (ram)

Fig. 15. Variations of strain energy density function with distance from crack tip for E2/El=1, b/h=lO and concentrated load.

o

'

'

h (ram) Fig. 17. Influence of E2/E ] and matrix layer thickness on instability index for a concentrated load.

67

CK. Chao, S.Y. Lin / Failure stability of a cracked layer 10

Table 7 Local and global stationary values of strain energy density for concentrated loading b // h

[(dW// dV )nfin . ]L.

[( . dW// . dV )min ]G

.

× 10- 3 (MPa)

× 10- 8 (MPa)

E 2/E t = 0.04

E

2 5 10

1.44 × 10 - 2 2.60 4.98

E 2//E 1 =

63.0 135.0 125.0

6 4

1.0

2 5 10

6.39 6.16 5.64

5.42 0.091 0.023

=

62.9 144.0 188.0

F,b.r i! F'b.'l _

.k.

Crack O

E 2/E 1 = 25

2 5 10

J

2

I

i iiiiiii

10-2

1.06 4.41 9.97

i

10 "1

-,,--I--~

i iiiiill

II

"

Crack i

10 0

i iiiIill

i

101

i iiiiill

10 2

El/E=

Fig. 19. Effect of orthotropy on index of instability for b / h = 5.

8.40 X 10-3 M P a b y a large margin. Fracture initiation is, therefore, predicted. 4. 4. Effect o f orthotropy

Plots of l as a function of E 1 / E 2 are given in Figs. 18, 19 and 20 for b / h -- 2, 5 and 10, respectively. There is a general trend in all the curves; they all reached a peak and then decreased except for the load type II loading where the raised portion of the curve is not as pronounced. All the curves also intersected at E 2 / E t = 1.0 except for load type I and b / h = 10. N o t accounting for load type II, they are approximately symmetrical about the line E 2 = E~ in Figs. 18 to 20. This means that failure stability can be increased by either increasing or decreasing E l / E 2 , i.e., increasing the degree

of orthotropy. This does not apply to load type II where localized failure would be enhanced only by adding more fibers parallel to the crack. The ways with which stability is reflected b y / for E 1 / E 2 = 0.08 and 12 are qualitatively the same as those for E 1 / E 2 = 0.04 and 25, respectively, provided that the geometry and loading are the same.

5. C o n c l u s i o n s

Failure instability initiating f r o m an existing crack in a matrix layer is analyzed for a fiber reinforced composite. The composite next to this cracked layer is modelled as orthotropic materials where the fiber directions can be aligned parallel or normal to the initial crack. Application of the 10'

]].

10 8

+!

6 v

E 4 Type I 2

Fiber j.

Crack 0

10 -2

I

i iiiiill

i

10 -+

Fiber ~l-..~ i ill|ill

II

Crack i

10 0 E 1 / E=

i iiiiill

i

10 +

10-2

i iiiiiii

10 2

Fig. 18. Effect of orthotropy on index of instability for b/h = 2.

1 0 -+

10 0

101

10 2

El/E2

Fig. 20. Effect of orthotropy on index of instability for b/h = 10.

68

C.K. Chao, S. Y. Lin / Failure stability of a cracked laver

local a n d g l o b a l s t a t i o n a r y values of the strain energy density function, an i n d e x of failure instability, is defined that accounts for the c o m b i n e d effect of the system, geometry, l o a d type a n d m a t e r i a l a n i s o t r o p y . I n d e e d , c o n f i n e m e n t of failure to regions near the initial defect can be m a d e b y increasing the degree of o r t h o t r o p y . It is f o u n d desirable to have the fibers along the a p p l i e d u n i f o r m a n d c o n c e n t r a t e d l o a d direction that is n o r m a l to the crack plane. This d i d not h o l d in the case of linearly varying l o a d where failure c a n be m o r e localized b y placing fibers p a r a l l e l to the crack. In general, b o t h the variations of 1, max max [(dW//dV)min]L a n d [ ( d W / d V ) m i n ] G have to be c o n s i d e r e d for o p t i m i z i n g the design of a given system. H i g h l is n o t always u n d e s i r a b l e if the c o r r e s p o n d i n g energy density functions [ ( d W / dV)mi. m a x ]L a n d [ ( d W / d V ) m imax n ]~ are k e p t b e l o w the critical value. Situations where b o t h l a n d ( d W / dV)m"~ are high should be a v o i d e d for they i m p l y n o t only the likelihood of failure initiation, b u t once it occurs, the system collapses as a whole.

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