Introduction to computational models of damage dynamics under stochastic actionst

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Probabilistic Engineering Mechanics 11 (1996) 107-112 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All fights reserved 0266-8920/96 $15.00

Introduction to computational models of damage dynamics under stochastic actions t G. Augusti & P. M. Mariano Department of Structural and Geotechnical Engineering, Universith di Roma "La Sapienza", Via Eudossiana 18, 00184 Rome, Italy

This paper reviews and discusses some basic ingredients necessary for the study of damaged continua with diffuseddefects like microcracks, pores, dislocations, etc., under stochastic loading histories and, in particular, under sequences of impulses described by Poisson arrival processes. The mechanical model of a continuum with microstructure is adopted: in other words, the state of the continuum is described by the usual displacement field and by an additional field of a secondorder non-symmetrictensor which describes the microstructural rearrangement of the material due to the presence of defects. It is shown that the time evolution of this tensor, usually assumed empirically on the basis of experimental results, is governed by a balance equation. The discretization of the problem and integral measures of damage, useful for the numerical solutions, are also discussed.

body, have also been defined with the help of the Clausius-Duhem inequality. Usually damage evolution is placed into the general framework of internal variables, and its law is assumed on the basis of experimental results. On the contrary, in this paper, the damage variable is considered as a structural variable; then, it is shown that the time evolution of the damage satisfies a balance equation like the macromotion. This model can be useful to explain the physical meaning of some stochastic models of damage evolution, in particular, models that relate the dynamics of the damage to Itr's stochastic differential equation, such as those proposed by Woo and Li s and Zhu and Lei 4. In these models the external actions are stochastic while the properties of the body are usually assumed deterministic. Note also that both quoted models consider damage as a scalar variable and therefore can describe only isotropic damage: the anisotropy induced by damage in the mechanical behaviour of the material can be taken into account only if a tensorial variable is used, as in this paper.

1 INTRODUCTION The mechanical behaviour of continua with diffused defects under loads described by random processes (and, in particular, Poisson processes) is studied in this paper with the aim of establishing a general framework useful for developing computational analyses of damage development in solids. The dynamics of the body is described adopting the microstructural point of view: the state of the continuum is described by the superposition of two continuous and differentiable fields, namely the displacement field u(x,t) and a second-order non-symmetric tensor field ~(x,t). ~ is a local measure of damage and can be either a tensorial internal variable (e.g. in the case of chemical damage of the body) or a tensor describing the irreversible rearrangement of the microstructure induced by nucleation and growth of defects (i.e. microcracks, pores, etc.) because of mechanical actions. ~'2 The regions of the body in which there is nucleation of macrocracks, can be identified with the regions where exceeds a critical condition. Integral measures of damage (usually averages of area or volume integrals: cf. Section 5 below) have often been used for computational simplicity; in fact, such a choice allows a rougher spatial discretization. Alternative global damage measures, related to the entropy flux induced by nucleation and growth of defects throughout the

2 EQUATIONS OF MOTION Consider a body ~ in the usual three-dimensional Euclidean space. On the body, the second-order non-symmetric tensor field ~(x,t) (cf. Section 1) is defined, i.e. there exists an application

tRevised text of a paper presented at the Second International Conference on Computational Stochastic Mechanics, Athens, Greece, 12-15 June 1994.

(x(X, t), t) -~ ~(x(X, t), ~) c ~r 107

(1)

108

G. Augusti, P. M. Mariano

where ~ is a finite-dimensional connected differentiable manifold, and {X} are referential coordinates. The macroscopic mechanical behaviour of the body is assumed non-linear-elastic up to the occurrence of macrocracks. Different choices can be made for the tensor field 9(x,t), e.g.: (a) For every point, 9 can be considered as the second-order term of the microcracks density function (dipole approximation: quadrupole effects are thus neglected). 2 (b) In the case of crystalline bodies, weakened by systems of fiat microcracks, every microcrack can be approximated with a continuous distribution of dislocations. Therefore, 9 can be identified with the dislocation density tensor. 9 is then equal to d~ ® curl d ~, d ~ being the vector field which describes the crystalline lattice. In particular, the quantities 9 ~ / n and m, where n = det{d c} and m = pdet{dc}, 9 ~ being the component of 9 with respect to the lattice vectors, are invariant under elastic deformations. 5 In Section 4 below, it will be shown how the proposed damage model is applied to the two definitions of 9 just quoted. Only brittle solids are considered and a non-classical Hamiltonian model is proposed. This model takes into consideration, in an approximate way, the dissipative nature of the brittle evolution of the damage by means of a weak damage potential, i.e. lower potential for the action related only to the damage evolution. Under these assumptions, the motion of the body is regulated by a Lagrangian density L = L(u, VRU , !1, 9 , V R 9 , ~ )

(2)

(u being the displacement vector field and 9 the local damage field) from which the action

If T may be put in the form 7 T = Tl(ti ) + T2(9,~),

(4)

TI (9) is a quadratic form in the tangent bundle of ~, while T2(9, ~ ) depends on the metric in ~ and can be assumed quadratic in ~. The Lagrangian density must be such that 9 is a nonnegative, non-decreasing time function. The Lagrange equation for the macromotion is s 00L Ot Oil

OL OL ~-Div = Q Ou 0VRU

(5)

The time variation of 9 is regulated by an equation of balance between the variation of the microstructural kinetics and the forces acting on the defects state: 00L Ot O~

OL

.

OL

= Q,

(6)

~ + D w 0~TR~@

OL where ~ represents the forces between the defects (e.g. OL microcracks), induced by themselves, while - OVR9 represents the microstresses acting along the defect. In obtaining the system of eqns (5) and (6), the variations of the action 3 have been taken with respect to 6u and 69, which are elements of the tangent bundle of the configuration body manifold. Q and Q' are the nonconservative external generalized forces: more precisely, Q' represents forces acting directly on the defects. Let Ij and ~ be vectors defined by [~T = (Ul, U2' U3,_@11, 922, -@33,912, "~21, 913, 931, 923, -~32)

(7)

and ~T = (Q, Q,).

(7')

The system of eqns (5) and (6) can then be written 3 =

LdVdt to

(3)

can be defined. The density L is assumed to be given by L= T-U-A~

(2I)

00L OtOb

OL OL Ob ~ - D i v ~ = ~ .

(6')

By using the classical method of Legendre transform, let

p

OL

(8)

=

where • T(u,9,~) is the kinetic energy density; • U(U,VRU,@) is the macroscopic potential density related with the elastic non-linear macroscopic mechanical behaviour; • A~ =/\~(U,~7RU,9,VR 9 ) is a weak damage potential density. The existence of such a potential, usually assumed, has been recently demonstrated in the context of a general, physically consistent, axiomatic framework of damage models. 6

The Hamiltonian density is H = p.~ - L

(8')

and the system eqn (6') can be written in the form OH OH

OH

(9)

b = op

where ~' is the external non-conservative force vector, equivalent to ~ in eqn (6').

Computational models of damage dynamics Define now a vector q by

• V6 > O, 3L6 > O, L6 E fit, such that if x, y E 9t m and

qT =

Ix[ < 6 and lyl < ~5, t E[0,T], then [zeg(x,t) ~¢i(y,t)[ < L6lx-y[ and [fg/(x,t)-f~/(y,t)[ <_Zalx-y[.

(10)

The system eqn (9) can then be written q

= d

(9')

+

where

OH

.

0~- - Olv ~

OH

109

)

~¢ =

(9")

and (fq.~)T = (~,, 0).

The initial value, q(to=qo), is a random variable Ft0--measurable with Elq012< + is the a-algebra constituted by all events which can be recognized whether they occurred or not at time t (i.e. F,0 is an element of the filtration '~t on the event space). Consequently, q(t) is a progressively measurable function. 9 In the probabilistic context, taking into account elastic deformations of the microstructure and using the definitions given in Section 2, ~ must be such that:

(9'") (a) in the case in which ~ is the dipole approximation of the microcracks density function

3 S P A C E - D I S C R E T I Z A T I O N AND S T O C H A S T I C

TIME BEHAVIOUR

P{~o.(t2)ninj < ~[~ij(tl)ninj : ~ } Vtl, t2 E [0, T], t 2 > tl, Vx E

Let ~(~k) be a lattice on the deformable body ~ , defined by the diagram ~

(13)

(b) when the microstructural variable is the dislocation density tensor of crystalline bodies

kt B(0)

= 0

~3(t) C E 3

P

(t2) < kl-n-(tl) = k

=0

Vtl, t2 E [0, T], t 2 > tl, VX E

(13')

P{m(t2) < l~lm(tl) = Ij} = 0 (ll) where 3 is a finite index set, ~(0) the reference configuration, ~(t) the present configuration determined by the placement k t, E 3 the three-dimensional Euclidean space and i an index application on ~(0). With reference to lattice ~(ijk), the space-discretization of q, ~¢, ~ are indicated by q(/jk, t), ~¢/jk(q(t),t), fgqk(q(t),t), respectively. Consequently, system (9') becomes:

il(ijk, t) : -~qk(q(t), t) + f~ijk(q(t), t).~

(12)

dq(/jk, t) = ~¢ijk(q(t), t)dt + f#iyk(q(t), t).~dt.

(12')

Vtl, t 2 E [0, T], t 2 > tl, Vx E ~.

(13")

Here, and in the following, P{.]*} indicates the probability of event {.} conditional on the event {*}. Eqn (12') points out the Markovian structure of the above-described picture and the diffusive nature of the time evolution of the damage in solids.

4 UNCOUPLED PROBLEM

or

If d p = ~ d t is a stochastic process, eqn (12') is an It6 vectorial non-linear differential equation. ~' and c~ must then be assumed to have the following properties in order to satisfy the theorem of existence and uniqueness:

• ~¢(.,.) and if(.,.) are measurable functions of their arguments; •

m (where m is the dimension of the problem), there exists a constant M E 9~, M > 0, such that I~'i(x,t)[
VXE~

The mechanical nucleation of damage is induced by destabilization of the microstructure, i.e. crystalline grains or macromolecule aggregates. This destabilization is induced by the elastic part of the macroscopic behaviour: the elastic energy is stored by the microstructure up to the time when its configuration becomes unstable. In general, in the treatment of the previous sections, the macromotion and damage time are coupled. If the damage state is small but such that the field approximation is plausible, the problem can be uncoupled: this approximation makes the search for analytical solutions much more feasible.

110

G. Augusti, P. M . Mariano

The time evolution of the field ~ can be described in either of two ways: (i) by a Lagrangian system, as in Section 2, in which the Lagrange density function depends only on ~ and its derivatives: Lup = Lup(~ , V R ~ , ~ )

(14)

(to solve this problem the method in Section 3 can be utilized); (ii) by a phenomenological model. One such model has been proposed, with reference to an abstract dynamical system, by Woo and Li, 3 who assumed that the damage depends on a potential postulated as a convex non-negative scalar function of state variables and of their conjugate fluxes: Itr's equation is obtained equating the rate of damage to the sum of (i) the derivative of the damage potential with respect to the release rate of the damage energy, and (ii) the product between a quantity proportional to the said derivative and the Wiener process. A more general stochastic model for damage evolution has been independently formulated by Zhu and Lei, 4 who related, using the Stratonovic-Kasmiskii limit theorem, an axiomatically introduced first-order stochastic differential equation for damage time evolution with the Kolmogorov-Fokker-Planck equation: in this way, the authors emphasize the diffusive character of the damage. Note that if the damage can be considered isotropic, the tensor ~ reduces to a scalar variable ~ , sufficient to describe the damage. Note that in a phenomenological model, the external stochastic forces that act on the microstructure must be related with the elastic waves induced by macroscopical motion.

~[A~dA, respectively; and dq(t) = ~'"(q(t), t)dt + fg"(q(t), t ) . d p

where s l " ( q ( t ) , t) = J'~ ~¢dV and ff"(q(t), t) = J'a~ffdV. Solutions in closed form can be obtained for some given forms of ~¢" and ~". Some solutions are given by Koch. l° For example, in the uncoupled case, when the effects of the dynamics of the microstructural change are negligible and damage is isotropic, eqn (18) is a scalar equation. Therefore, if ag"(q(t), t) = ,.wfl1 + .~/2q

(19)

~"(q(t), t) =cff 1 q- C~2q

(20)

with a/i, a¢2, ffl, f¢2 constant, then the solution can be obtained in the following way. Let rl, r2, ... rk ... (0 <_ r~< r2< ... < re< ...) be the random times of the shock arrivals and k the number of shocks arriving in [0, t] (r~ <_ t, rk+l > t). Under the hypotheses eqns (19) and (20), and ~ 2 > 0, the solution of eqn (18) is q(t) = (1 + ~2)ge ~¢2t q(O) + × (e

JA

~dA

e

~1

k

+

.e -~2~]

(1 +

(21)

J

and, for a/2 = 0 k ~1 q(t) = (1 = S2) k q(0) + E (1 + S2) i-1 (r, - TiM ) i=1 Cgl ] d~¢l (t - rg) + ~ ------------i (1 + ~d2) " + (1 + ~2) k i=1

(15)

and ~v = f ~dV. (16) Jv The equations of time stochastic evolution related with the above defined measures become, respectively: dq(i, t) = ~¢'i(q(t), t)dt + fq'i(q(t), t).dp

a/2(1 q- ,(~2)i_1

,5~'1 (e-~g2rk_ e - d 2 t) + ~¢2( 1 + ~2) k

5 MEAN DAMAGE AND SOLUTIONS

As hinted in Section 1, integral measures may be useful in applications. With direct reference to ~ , an area measure and a volume measure can be defined as follows

(17)

where ~¢/'(q(t),t) and (~'(q(t),t) are the discretizations on a one-dimensional lattice of the terms J'A ~¢dA,

(18)

(22)

In this context Pk(t) = P{q(t) <

q¢lk shocks

in [0, t]}

f JD P~l'~>'"~k(th' a2, "'" °'k)dtrld°'2""d°'k

(23)

where P = { ( r l , r 2 ..... rk)lq(t)_< q¢} and O*l,*2,...,*k is the joint probability density of rbr2 ..... Tk conditional upon the event {% < t,rk+1 > t}, which is equivalent to the event {Tk+l = t} enclosed in { r k < t}. Consequently

P~,,~2,...,~ (~1, ~2, =

..., ~ )

k! 7~'

0~0"1 ~0"2 ~ ' " - - < O ' k ~ t

0

elsewhere.

(24)

Computational models of damage dynamics If the Poisson process is generated by many mutually independent causes of shocks, let k; be the number of shocks of/-type in [0,t] then, if eqn (18) can be written in the scalar form

111

free-energy, eqn (28) can be written -

-

-

-

S'VR

1

-- ~qR'~7R -- 0Divh~ _< 0

n

dq(t) = d q ( t ) d t + Z f~iq(t)dpi i-1

(31)

(25) If ~ = ~(F, O, VR 0, ~, VR~), the following equations can be obtained:

its solution can be found in the form n

q(t) = q(0)e~t 1--I(1 + ~i) ki i=1

(26)

and

p~-~=T,

o¢

p~-~= Z,

Pk(t) = P{q(t) < qclk total shocks in [0, t]}

rl=p~,

o¢

POVR~

p

_

(Vr0)=0,

s

(32)

In this case, inequality eqn (31) becomes

=

k

Z

f l \)x J kV

Pk'k:'"k.(t) X± k.W -"

k I ,k2,'",kn =0 k I +k2+..+kn=k

i~ 1

(27)

(33)

1'

where At is the parameter of the ith Poisson process and A = ~7=1 Ai. The solution for the multidimensional case can be obtained by using the Feynmann path integrals method. 1 A different type of integral measure of damage can be defined with reference to the thermodynamical aspects of damage evolution, in particular with reference to the Clausius-Duhem inequality. In this inequality a flux of entropy, related only to the nucleation and growth of microcracks, can be introduced. (In full rigour an entropy source, related to the microcrack nucleation, should be introduced: but, if the microcrack state is such that the field approximation is physically correct, it is plausible to neglect this supplementary source.) In the microstructural framework, the ClausiusDuhem inequality can be written: pz] - Div h R -- pK >_0

p~ = T.F + Z . ~ + S . V ~ - DivqR + pr

The entropy flux hi, related to the damage evolution, is given by a constitutive equation whose functional dependence can be given by h~ = fi~ (o~,9, VR~, ~ )

(34)

where a is the surface tension of the material. Consequently, the energetic global damage measure can be defined as = J~ Div h i d V

(35)

It is obvious that, taking into consideration the entropy flux h, eqns (13), (13'), (13") change in the following way:

P{(~ijninj)'< P{(~)'<

0lfil > O} = 0

0lfi' > 0} = 0

(36)

(36')

(28)

where ~ is the entropy density, h is the entropy flux and K the entropy source. For microstructured continua, the equation of internal energy balance is 12 (29)

where e is the internal energy density, T the macroscopic stress tensor, F the displacement gradient, Z the density of volume forces acting on the microstructure, S the stress on the microstructure and q and r the heat flux and the heat source, respectively. In the following it is further assumed that qR T, "R, ki K = ~ r hR =-"~"

~ q R ' ~ 7 a 0 + 0Divh I > 0

(30a,b)

where 0 is the temperature and h i the entropy flux related to the damage evolution. By substituing eqn (29) into eqn (28) and taking into account eqns (30a,b), putting ~b= e - fly, where ~b is the

P{th < 0161 > 0} = 0

(36")

6 SOME CONCLUDING REMARKS AND

COMPARISONS WITH PREVIOUS WORK In this paper, a weak Hamiltonian model has been formulated in order to describe the dynamics of damaged brittle materials and, more specifically, of a body with diffused microcracking. Specific features of the model are in the choice of a weak damage potential which takes into account the dissipative aspects of the damage evolution, the field approximation for the microcracking state and the introduction of a supplementary entropy flux h. It is simple to introduce mean measures useful to computational analyses. In the case of stochastic actions on the body, this model leads immediately to It6's equation, and thus emphasizes the diffusive nature of damage evolution,

112

G. Augusti, P. M. Mariano

In recent literature, several approaches have been followed to tackle similar problems. For istance, in the case of bodies with continuous distributions of dislocations, the influence of the inhomogeneities on the motion have been described by means of alterations of the geometrical properties of the body manifold. 13'z4 Also, models based on an extension of the Yang-Mills minimal replacement construct for semi-simple gauge groups, have been formulated. 15 Along these lines, Lagoudas 16 proposed a model in which the damage is taken into account by the gauge potentials related to the gauge invariance with respect to the local translation: however, this model cannot take into account the dissipative nature of damage due to the evolution of the defect states (in particular the microcracks state). In the approach of the present paper, the damage state is described by the microstructural field 9, the potential A~ and the entropy flux; possible gauge invariances give additional information on the damage state, in particular on the structure of the Lagrange density function. Other authors have proposed to take into account the presence of inhomogeneities through the explicit dependence of the Lagrange density on the referential coordinates. 17-19 In these papers, with reference to dissipative solids with internal variables, a pseudopotential of dissipation, which describes memory effects, is axiomatically introduced: however, this assumption is .incompatible with the Markovian viewpoint and hence with the diffusive nature of the evolution of the microcracks. The description of damage evolution outlined in the present paper, allows the alteration of the internal structure induced by damage to be considered more explicitly, and computational problems to be solved more easily. Also, it is more consistent with the nature of the damage evolution than the description provided by empirical models. Undoubtedly, in order to apply the procedure to specific materials, a physically clear identification of the field ~ and of the potential Ae will be necessary in order to write a consistent Lagrangian density function. This can only be given by systematic experimental data.

REFERENCES 1. Onat, E. T. & Leckie, F. A., Representation of mechanical behaviour in the presence of changing internal structure. A S M E J. Appl. Mech., 55 (1988) 1-10. 2. Lubarda, V. A. & Krajcinovic, D., Damage tensors and the crack density distribution. Int. J. Solids Struct., 30 (1993) 2859-2877. 3. Woo, C. D. & Li, D. L., A general stochastic dynamic model of continuum damage mechanics. Int. J. Solids Struct., 29 (1992) 2921-2932. 4. Zhu, W. Q. & Lei, Y., A stochastic theory of cumulative fatigue damage. Probabilistic Engng Mech., 29 (1991) 222-227. 5. Davini, C., A proposal for a continuum theory of defective crystals. Arch. Rational Mech. Anal., 96 (1986)295-317. 6. Kijowski, J., Elasticith Finita e Relativistica: Introduzione ai Metodi Geometrici Della Teoria dei Campi. Pitagora Editrice, Bologna, 1991. 7. Capriz, G. & Podio-Guidugli, P., Structured continua from a Lagrangian point of view. Ann. Matemat. Pura AppL, 135 (1983) 1-25. 8. Mariano, P. M. & Augusti, G., Some axioms and theorems in damage mechanicsand fatigue of materials. (Submitted). 9. Baldi, P., Equazioni Differenziali Stocastiche ed Applicazioni. Pitagora Editrice, Bologna, 1984. 10. Koch, G., Modelli di guasto per shock ed invecchiamento. Unpublished (1987). 11. Feng, G. M., Wang, B. & Lu, Y. F., Path integral, functional method, and stochastic dynamical systems. Probabilistic Engng Mech., 7 (1992) 149-157. 12. Capriz, G., Continui Con Microstruttura. ETS Edizioni, Pisa, 1987. 13. Noll, W., Materially uniform simple bodies with inhomogeneities. Arch. Rational Mech. Anal., 27 (1967) 1-32. 14. Wang, C.-C., On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations. Arch. Rational Mech. Anal., 27 (1967) 33-94. 15. Edelen, D. G. B. & Lagoudas, D. C., Gauge Theory and Defects in Solids. North-Holland, Amsterdam, 1988. 16. Lagoudas, D. C., A gauge theory of damage. Int. J. Engng Sci., 5 (1991) 597-606. 17. Eshelby, J. D., The elastic energy-momentum tensor. J. Elastic., 5 (1975) 3-4. 18. Maugin, G. A. & Trimarco, C., Pseudomomentum and material forces in nonlinear elasticity:variational formulations and application to brittle fracture. Acta Mech., 94 (1992) 1-28. 19. Maugin, G. A., Material forces in inhomogeneous dissipative solids. Z A M M Z. angew. Math. Mech., 73(4-5) (1993) 268-270.