Generalized inverse of the compliance tensor, and behaviour of incompressible anisotropic materials — Application to damage

MechanicsResearchCommunications,Vol.24, No. 4, pp. 371-376, 1997 Copyright© 1997ElsevierScienceLid Printedin the USA. All fightsr~cn,cd 0093-tM13/97$17.00+ .00

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Pll S0093-6413(97)00038-4

GENERALIZED INVERSE OF THE C O M P L I A N C E TENSOR, AND BEHAVIOUR OF INCOMPRESSIBLE ANISOTROPIC MATERIALS - APPLICATION TO DAMAGE.

A. Loredo* and H. Klocker** + LRMT, ISAT, Universite de Bourgogne, France +÷ SMS, Ecole des Mines de Saint-Etienne, France

(Received 26 June 1996; accepted for print 24 Februa~ 1997)

Introduction

Before the final rupture, most structural materials exhibit an import damage in the form of microvoids. The overall behaviour of a damaged elastic material depends on the void volume fractionf Undamaged polymers are generally considered as incompressible elastic. Metals at high temperature may be considered as linearly viscoplastic. Thus the undamaged material may be described by an incompressible behaviour, while the overall behaviour of the damaged material is compressible depending on the void volume fraction. The transition from a compressible to an incompressible behaviour leads to a singular compliance matrix and an undefined rigidity matrix. The generalized inverse of the compliance matrix of the damaged and undamaged material shows to be of primary interest to study both incompressible and compressible materials with the same formulation. We show that incompressible behaviour of anisotropic elastic materials can be formulated in terms of generalized inverse of the compliance matrix. When incompressibility conditions are required, the compliance matrix of a material with known moduli becomes singular. The Moore-Penrose generalized inverse can help us to formulate Hooke's law for such materials. Examples of such relations are given tbr isotropic, transversally isotropic and orthotropic materials.

Incompressible behaviour

371

372

A. LOREDO and H. KLOCKER

We shall assume that the detbrmations are measured from the natural stress-free state and are small, the influence of the temperature and other fields is insignificant, and the stress tensor o is symmetric and is a linear function of the deformation tensor e. Using the classical notation with six indices, Hooke's law is written in matrix form: ¢~=Ce

or

cr/=('ljc J with ( l . J ) ~ (1.6) 2

(1)

The matrix C is symmetric and the inverse law is: ~=Sa

or

~t =SIJaJ

with

( I . J ) c (I.6) 2

(2)

With the small strains assumption, incompressibility leads to equation (3): Nr~=0 with N 7= [1

1 1 0 0

i.e.

~t+c2+e'3=0

(3)

0], This gives rise to the six following conditions on the compliance matrix: NI's=0

i.e.

SIj+S2j+.~3j:O d ~ ( 1 . 6 )

(4)

The use of equation (4) leads to a new smgTdar compliance matrix S,,c: = Si,,ca

(5)

Now, the inverse relation is not unique, and not easily computable by means of classical matrix calculus. Notice that an hydrostatic pressure does not deform the continuum, even if general anisotropy is considered (o = - p N forces = 0).

Generalized Inverse of S.j,_..c , and general solution

The general solution of equation (5), existing only if N?e = 0, can be written in function of the Moore-Penrose •

-t

generalized inverse Si,,c of Si,c: t tr = S i , ~ e - p N

with

Nle=0

(6)

where p is an undetermined scalar corresponding to the pressure. This relation may be obtained easily for isotropic materials without the use of the generalized inverse theory. On the contrary, for anisotropic materials, equation (6) is quite difficult to obtain. With the help of generalized inverses, it may be done very simply and on a perfectly natural way. The generalized inverse is defined for any kind of matrix and is unique. It possesses important properties [4]: multiplying it with a vector, gives the minimal norm least square solution (or exact if existence) of the corresponding linear system. The rigidity matrix C,, c will now be defined as the generalized inverse S,tc of Sinc, and can be computed with the following rule [1], [2]:

DAMAGE IN INCOMPRESSIBLE ANISOTROPIC MATERIALS

C,, = S , ,?c =

(SipK. +

kNNT) -1_ kNN r

k:l/llnll2:l/3

with

373

(7a)

C~,,~ is a singular matrix with the same nullspace than Sin: and from the theoretical point o f view it plays a central role For numerical applications it is better to use a regularized inverse [3] C l a = S~tI = (S,,.. + a k N N r ) -1 = S.,c ¢ +lkNNr

with

~e0andreal

(7b)

The needed relation (3) could be enforced by the help of Lagrange multipliers. lsotropic materials

We choose to write the compliance matrix S with the Young's modulus E and the Poisson's ratio v:

0",-L I -v

1

-v

0

0"_, i

0

l+v

0

cr4 I °sJ

0

0

1+ v

a6j

(8) '

,

Le6J

L

0

The six conditions (4) reduce here to the classical relation v = 1/2. Substitution o f this value in S gives S~,,c. Applying formula (7) gives: O"1 ,

4

2

2

0-2

2

4

2

0

RE Is2

2 4 E 2 0-4 = ~ .......................... 3.......6 6 84 0 0 3 0 0"5 0 0 3 L86 0"6.

1

0-3

e 1+ e2 + e3 = 0

(9)

ii -p

Transversally isotropic materials

We choose to write the matrix S in thnction o f the Young's moduli E I = E2, E3, the Poisson's ratios vt2 and vt3 = v23 and the shear moduli Gi3 = G23 ((O,x3) is the axe of symmetry):

374

A. LOREDO and H. KLOCKER

6.2

I/Ej -v)2/E l

-VI2/EI lie 1

-VI3 / E1 -vI~IE I

6.3

-Vl3/E1

-vl3/E~

I/E3

¢4

........................................................

6.1

,

] [o-l] |to- 2 I

0

/ o-3 i-~~;~ ........ ~ .................. ~- ..........

0

0 -

1/GI3

o

£6

0

o

l o-~/

(lo)

[l o-~ I

20 + v,2)/Elj [o-~]

The six conditions (4) reduce in this case to:

VI3 --

El 2E3

(11)

El 2E3

t'12 = 1 - - -

Applying the procedure we obtain: 4L O"2

-2M

-2M ] -2E~.

-2E 3

] [6' I

4L -2E 3 - 2 E s_ 4E~.

0

6.2 6"3

...................................... 9ai~ ....... ~.......... ~ 0

0

9Gi3

0

and

~4 r l `%

£1 + 6"2 + 6.3 = 0

-pi ! 01

[o6J

(~2) with

L _

E3(E3+2E1) 4 E 3 - E1

M _

E~(-2E~+SE,) . 4E 3 - E)

. E3E , G~ 2 - - 4E 3 - E)

Orthotropic media

We shall write the compliance matrix with the help of the three Young's moduli E I, E 2, E3, the three Poisson's ratios v12, v13. v23 and the shear moduli G23, G13, and G12:

[ej ]

6.4

1/E I -v12/E I -vi2/E 1 I/E 2

-vi3/E I -v2~/E 2

-v13/E 1 -v23/E 2

I/E 3

.........................................................

16., I [6.~J

The conditions (4) reduce here to:

0

0 i/G~; 0

0

........ ~ ............ o .... I/G)3

0

0

1/GI2

03)

DAMAGE IN INCOMPRESSIBLE ANISOTROPIC MATERIALS

375

1 -E2E 3 + E1E3 + EIE2 EIE~ v _ 1E2E 3 - E R E ~+E~E 2 13 - ~ E2E~ v 1E2E 3 + EIE~ - E1E 2

V2~ = - -

2

(14)

and the inverse relation is: ~37

[4L 1

-2M

-2P

0"2t

-2M

4L 2

-2Q

0

7V E,]

[-17

IIc2~

1

°iiJ:¼6:2~--~2°4~'~L '0 0"3........ 9G,.......... 3 06'11l~'oj., 51~-' -P 0 _ L 0 9G,2jL~6j Lo

and

~1 +

~'2 + 6"3 =

0

o

with LL= EtE2E3(E2E3-2E1E'-2EIE2) A L 2 - EIE2E3(-2E2E3 + EIE 3 - 2E,E2) A L 3 - EtE2E3(-2E2E3 - 2EIE 3 + EIE2) A and A = E2E322 + E2E 2 + E I2E2 2 _

(15) M

EIE2E, t E2E3 + E I E 3 - 5 E 1 E 2 , I A p = E1E2E3(E2E3 - 5E,E 3 + E1E2) A Q - E,E2E3(-5E2E3 + E1E 3 + E,E2) A 2E2E2E3. _ 2E1E2E3. _ 2E1E2E2_

Monoclinic materials, general anisotropy

The same procedure can be applied to any kind of behaviour, but for anisotropy more general than the orthotroplc case, formulas are very complicated. In the monoclinic case, the incompressibility conditions (4) reduce to 4 independent conditions on the coefficients of S, and for general anisotropy, the six conditions are independent.

Applications and future developments

In a following paper we shall show how to split Hooke's law into two parts, one describing compressible and the other incompressible material behaviour. Then it is possible to make a continuous transition from one to the other (and the contrary). This formulation is of primary interest to study microvoid damaged solids by homogenisation procedures. These procedures need to start with zero void volume fraction. If the undamaged material is incompressible, and even if it becomes immediately compressible when voids appear, it seems reasonable to apply strictly incoml~ressible behaviour to start the calculus. The continuous transition included in the formulation makes

376

A. LOREDO aM H. KLOCKER

the resolution easier. Differences between this approach and the approximate one (using nearly incompressible laws) will be evaluated.

Conclusion

The incompressibility condition is interpreted by means of a matrix relation (4), independently of the material properties. The Moore-Penrose generalized inverse gives a natural way for writing Hooke's law for incompressible materials. There is an infinite number of such relations, but the generalized inverse plays a central role, due to its interesting properties. Nevertheless, we can compute a regular matrix giving also the solution (To), which is preferred in numerical applications, The work associated with hydrostatic pressure equals zero for any incompressible solid.

References

[1] G Verchery, Regularisation du systeme de l'equilibre des structures discretes / How to solve the singular equilibrium system of discrete elastic structures, C. R. Acad. Sci. Paris-, t. 311, serie 1L p. 585-589, 1990 [2] A Loredo, Calcul numerique de la quasi-inverse d'une matrice reelle symetrique semi-definie positive / Computing the generalized inverse of a real symmetrical non negative definite matrix, ('. R. Acad. ScJ. Paris, t 321, serie I, p. 247-252, 1995. [3] A. Loredo, Reanalyse des structures - applications au contact elastique et aux materiaux incompressibles / Structural Reanalysis - Applications to Elastic Contact and Incompressibility, Dax'toral Dissertation, Universite de Lyon-I, France, 1993. [4] Campbell S. L & Meyer C. D : Generalized inverses of linear transtbrmations, Dover Publications, NewYork ( 1979).