Damage theory: microscopic effects of vanishing macroscopic motions

Damage theory: microscopic effects of vanishing macroscopic motions

Elena Bonetti^I; Michel Frémond^II

^ILaboratoire Lagrange, Dipartimento di Matematica ''F. Casorati'', Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy, E-mail: [e-mail]

^IILaboratoire Lagrange, Laboratoire Central des Ponts et Chaussées, 58 boulevard Lefebvre, 75732 Paris Cedex 15, France, E-mail: [e-mail]

ABSTRACT

This paper deals with a mechanical model describing the evolution of damage in elastic and viscoelastic materials. The state variables are macroscopic deformations and a microscopic phase parameter, which is related to the quantity of damaged material. The equilibrium equations are recovered by refining the principle of virtual powers including also microscopic forces. After proving an existence and uniqueness result for a regularized problem, we investigate the behavior of solutions, in the case when a vanishing sequence of external forces is applied. By use of a rigorous asymptotics analysis, we show that macroscopic deformations can disappear at the limit, but their damaging effect remains in the equation describing the evolution of damage at a microscopic level. Moreover, it is proved that the balance of the energy is satisfied at the limit.

Mathematical subject classification: 74A45, 73C50, 35K60.

Key words: damage, principle of virtual power, non linear elasticity, nonlinear parabolic PDE, asymptotic analysis.

1 Introduction

These notes are concerning with a mechanical model describing the evolution of damage in elastic and viscoelastic bodies (cf. [4, 8]). It is known that the mechanical process of damage results from microscopic motions in the structure, as it is caused by microfractures and microcavities resulting in the decreasing of the material stiffness. Consequently, to provide a good macroscopic predictive theory one has to account for macroscopic effects of microscopic motions. In particular, the model is written in terms of a damage quantity within the framework of continuum mechanics and we derive the constitutive equations by two functionals, the free energy and the pseudo-potential of dissipation (cf. [11]), which are responsible for the equilibrium and the evolution of the system, respectively. The constitutive equations are coupled with equations of motion resulting from the principle of virtual power. The main idea in this modeling procedure consists in refining the power of interior forces by including also the effects of microscopic movements, which are clearly related to the damage rate. We recall that, when compared to experimental results, the model can predict the behavior of concrete structure correctly and, accounting for the gradient of damage, the structure size effect, too, which is particularly important in civil engineering (cf. [12]).

Concerning the analytical investigation of the above model, as far as we know, only some partial results have been obtained. The difficulty is mainly due to the degeneracy of the elastic coefficient in the macroscopic equilibrium equation. This is combined in the equations of microscopic motions, with the presence of subdifferential operators and quadratic nonlinearity for deformations. In [6, 7] a simplified version of the problem is investigated in the one-dimensional case, regularizing the subdifferentials. In particular, local in time existence and uniqueness results are proved both for the quasi-static situation and in the case when macroscopic accelerations are retained in the principle of virtual powers. Finally, we recall a recent paper [3] dealing with the complete 3D problem, but describing the evolution of damage only during the time interval in which the material is not completely damaged. A local in time existence result is stated.

However, these notes do not focus on the problem of existence of solutions, but mainly investigate relations between macroscopic and microscopic motions during the damaging process. Indeed, as the modeling method is based on the separation between the description of macroscopic and microscopic levels, it is natural to wonder which are the microscopic effects of vanishing macroscopic motions. Concerning this subject, in a recent paper [5], it is presented an example of vanishing macroscopic motions which retain at the limit a damaging effect in the microscopic equation. Now, we aim to show that the above behavior can be proved in a more general situation. The main idea consists in approximating the system to get a global existence and uniqueness result, and then applying a vanishing sequence of exterior forces in such a way that the resulting macroscopic motions become smaller and smaller. By use of a rigorous asymptotics analysis we show that at the limit the macroscopic motions become microscopic, in the sense that the macroscopic motions vanish but their effects remain as a source of damage, which is clearly related to microscopic motions. Moreover, we show that the balance of the energy is satisfied at the limit.

2 The model

In this section, we briefly recall the model and write the corresponding PDE's system. Concerning the mechanical aspects, we do not enter the details and refer, e.g. to [4]. Nonetheless, for the sake of completeness, let us point out the main ingredients of the modeling procedure. Assuming that the temperature is constant, the state variables are the strain tensor e(u) (u is the vector of small displacements), a phase parameter c denoting the quantity of damaged material, and the gradient Ñc accounting for local microscopic interactions. Concerning the phase parameter c, we prescribe the constraint

letting c = 1 correspond to the undamaged structure and c = 0 to the completely damaged material. Next, the dissipation due to damage is reasonably described by the dissipative variables c_t, Ñc_t related to microscopic velocities, where the subscript _t denotes the time derivative. After introducing the free energy functional and a pseudo-potential of dissipation, e.g. as in [4, 8], we can explicitly write in terms of the unknowns (u, c) the constitutive laws for microscopic and macroscopic stresses. Finally, we state the balance law for momentum, both for microscopic and macroscopic motions. More precisely, we consider a concrete structure located in a bounded smooth domain W Ì ³ with boundary ¶W =: G. We look at the damage evolution during a finite time interval [0, T] and denote by Q := W × [0, T]. Hence, the principle of virtual power, in which we include also microscopic forces, leads to the following inclusion governing the evolution of the phase parameter

whose ingredients will be specified in a moment, and to the equilibrium equation for macroscopic deformations, written in the quasi-static case

The positive constants c, k, and w in (2.2) denote the viscosity of damage, a local interaction coefficient, and the cohesion energy, respectively. The term ¶I[0,1] is the subdifferential of the indicator function I_[0,1] and it accounts for the required constraint on c prescribed by (2.1). Indeed, the domain ¶I_[0,1] is the interval [0,1], and we have ¶I[0,1](y) = 0 if y Î (0,1), ¶I[0,1](0) = (–¥,0], and ¶I[0,1](1) = [0,+¥). In (2.3), f denotes a volumic force applied to the structure. Nonetheless, in the following we consider a regularized version of the above equation, which is obtained by adding a viscosity contribution for macroscopic displacements and deformations, and an higher order nonlinear elastic term in the stress. More precisely, we substitute (2.3) by

Remark 2.1. Let us not that from a mechanical point of view (2.4) is associated with a viscous phenomenon, by the presence of the velocity terms u_t and e(u_t).

Remark 2.2. Let us stress that equations (2.2) and (2.3) or (2.4) are coupled: the macroscopic motion, i.e., displacement u, influences the microscopic motion schematised by c and governed by equation (2.2). Conversely, microscopic motion when resulting from a damage source, i.e., a right hand side in (2.2) influences macroscopic motion u governed by equation (2.3) or (2.4).

Remark 2.3. We point out that the regularizing (2.3) by (2.4) turns out to be useful to control the right hand side of (2.2). Indeed, as the elasticity coefficient c in (2.3) degenerates once the material is completely damaged, i.e. c = 0, the right hand side of (2.2) could not be controlled. In terms of mechanics, the regularization (2.4) controls the strain localisation which may occur in damage phenomena and preserves the uniqueness of solutions.

Now, we have to prescribe suitable initial and boundary conditions. Assuming, for the sake of simplicity, that at the initial time t = 0 the material is completely undamaged, we prescribe

Next, we assume that no energy is provided from the outside (cf. [4])

Finally, we fix an initial condition on the vector of displacements u

and, letting n be the outward normal unit vector to the boundary, we give a Neumann condition on the stress

from which it follows that the applied traction on the boundary is 0.

In the following, we are interested to consider the behavior of the solutions (c, u) to an initial and boundary values problem associated with (2.2), (2.4), in the case when a vanishing sequence of external forces f_t is applied, i.e. f_t0 as t tends to 0. Hence, letting (u_t, c_t) be the corresponding solutions, we can exploit an a priori estimates - passage to the limit procedure to perform the required asymptotics analysis as t 0. Thus, at a first step, we aim to find existence and uniqueness of solutions to the system for any fixed t > 0 and, on a second step, prove boundedness properties on these solutions which allow us to pass to the limit as t 0.

Here is the outline of the paper. In Section 3, we introduce the abstract formulation of the regularized system and state the main existence and uniqueness theorem (cf. Theorem 3.1). Hence, in Section 4, under suitable assumptions on the convergence of a sequence of applied forces f_t, we perform an asymptotics analysis of the solutions as t 0 (cf. Theorem 4.1). Finally, in Section 5, we investigate the balance of the energy at the limit t = 0.

3 The abstract problem

In this section, we first introduce the abstract version of the regularized damage system (2.2), (2.4) combined with initial and boundary conditions (2.5)-(2.6), (2.7)-(2.8). We prove that the resulting problem admits a unique solution and then, in the next sections, we investigate the asymptotics behavior of the solution as the external efforts applied to the structure vanish. Let us first introduce a suitable abstract setting. In regards of simplicity, but without loss of generality, in the following we take u as a scalar quantity u. In addition, fix c = k = 1 in (2.2). Hence, we start by considering the Hilbert triplet (cf. [10])

where

V := H¹(W) and H := L²(W),

H being identified as usual with its dual space. Notice that we use the same notation \norma·_X both for the norm in a functional space X and in X³. Hence, let

W := W^1,4(W)

Finally, we introduce the following abstract operators

where D_i stands for the usual partial derivative operator with respect to the variable x_i, i = 1,2,3. In particular, let us observe that (3.1) corresponds to the laplacian operator with associated Neumann homogeneous boundary condition (cf. (2.6)) in the duality pairing between V¢ and V. Analogously, (3.2) corresponds to the abstract formulation of the 4-laplacian operator with associated homogeneous boundary assumption. Now, we can make precise the abstract problem we are dealing with and state a corresponding existence and uniqueness result.

Problem P_a : Find (u, c) such that

and fulfilling for a.a. t in (0, T)

where x Î L²(0, T; H) is such that x Î ¶I[0,1](c) a.e. in Q.

The following proposition holds.

Theorem 3.1. Let

Then, Problem P_a admits a unique solution (u, c) with regularity

Proof. To prove the existence of a solution to Problem P_a stated by Theorem 3.1, we mainly exploit a fixed point argument and refer to well known results on evolution nonlinear equations. In regard of simplicity, we do not enter the details of the demonstration, but just sketch the main idea and give some useful references. We aim to apply the Schauder theorem to a suitable operator we are going to construct as it follows. We start by letting

where R will be fixed later, and substitute Ñu in (3.6) by Ñ. Thus, after observing that by the Hölder inequality the right hand side of (3.6) is in L²(Q), we can apply well-known results on evolution equations associated with maximal monotone operators (cf. e.g. [2]). We infer that there exists a unique corresponding solution

Moreover, one can prove that the following estimate

holds for a suitable constant c depending on W, T, w, and R, but not on the choice of in (3.10). Let us briefly outline the procedure and prove some formal a priori estimates on the solution. We test (3.6) by c_t and integrate over (0, t). After an application of the Young inequality

and integrating by parts in time, we have (cf. (3.4))

In particular, let us point out that in deducing (3.14), one can formally proceed as follows

Then, we test (3.6) by Ac and recall that, by monotonicity of the subdifferential operator and the chain rule, it follows

Thus, by similarly proceeding as in (3.14) we get

Moreover, owing to (3.10), (3.14), and (3.18), a comparison in (3.6) leads to

On a second step, we fix c = E () in (3.5) and look for a corresponding solution fulfilling the required Cauchy condition stated by (3.4). Thanks to (3.7) and (3.11), we owe, e.g., to [9, Theorem 1.2bis, p. 163-164] to infer existence and uniqueness of a unique solution

Hence, let us show that for a suitable choice of R in (3.10), u = G(E()) turns out to belong to X. Indeed, we can test (3.5) by u and integrate in time over (0, t). Recalling (3.1)-(3.4), a suitable application of the Young inequality yields

and the Gronwall lemma yields

where depends only W, T, and f. Thus, if we choose R > in (3.10), it follows that

maps X into itself. Then, towards the aim of applying the Schauder theorem to the operator X, and consequently prove that P admits a fixed point, we show that P is a continuous and compact operator in X with respect to the natural topology induced by L⁴(0, T; W). We perform some a priori estimates, which turn out to be useful to prove both the compactness and the continuity of P. Let us point out that some of the following estimates are only formal. Nonetheless, they can be rigorously justified prooceeding, e.g., as in [9].

We first test (3.5) by u_t and then integrate over (0, t). Hence, integrating by parts in time and applying the Young inequality, we infer that

In particular, let us point out that we have performed the following integration by parts in time

Thus, after recalling that (3.12) holds independently of , the Gronwall lemma yields

Finally, we formally test (3.5) by Au. By monotonicity arguments and applying the chain rule, we have

Then, by definition of the operators A and B, owing to (3.12), (3.7), and (3.27), we can write

Thus, owing to the regularity of c (cf. in particular (3.18)) and (3.26), the generalized version of the Gronwall introduced in [1] ensures

and finally

Thus, we can infer that

from which it easily follows that P is a compact operator in X. Moreover, the previous estimates (3.12) and (3.31), which are independent of the choice of , allow us to deduce that P is a continuous operator in X. Towards the aim of proving continuity of P, we take

and set

Our purpose is to show that

strongly in L⁴(0, T; W) (cf. (3.32)). Proceeding as above, we show that (u_n, c_n, x_n), with x_nÎ ¶I_[0,1](c_n), fulfil the following estimate (cf. (3.12) and (3.31))

independently of n. By weak and weak star compactness results, at least for some suitable subsequence, the following convergence holds

and by compactness (cf. [13])

Hence, by the boundedness of x_n stated by (3.35), we have

so that (3.37) and (3.38) ensure x Î ¶I_[0,1](c) (cf. [2]). Now, we can pass to the limit in (3.6), written for n, as n ® +¥, and by the uniqueness of the solution for the limit equation, once is fixed, we eventually identify c = E(). Moreover, let us observe that uniqueness of the solution for the limit equation extends convergences (3.36)-(3.38) to the whole sequences. On a second step, we discuss the asymptotics behavior of u_n and |Ñu_n|², in order to pass to the limit in (3.5) written for n. By (3.35), we deduce, at least up to the extraction of some subsequence,

By compactness, (3.39) leads to the strong convergence

which allows us to eventually identify d = |Ñu|². Hence, to show that P is continuous, we aim to pass to the limit as n ® +¥ in (3.5), which is possible thanks to (3.39), (3.40), and (3.41). Then, uniqueness of the solution for the limit equation allow us to identify u = G(c) in (3.41), concludes the proof. Moreover, it follows that (3.39)-(3.41) actually hold for the whole sequence. In particular, P turns out to satisfy the assumptions of the Schauder theorem and consequently it admits a fixed point u. It is now a standard matter to verify that the couple of functions (u, c = E(u)) is a solution to Problem P_a.

Hence, to complete the proof of Theorem 3.1, it remains to show that this solution is unique. First let us make some few remarks about useful notation we will use in the sequel. We assume that Problem P_a admits two solutions

whose regularity is specified by Theorem 3.1. We denote by [f] the difference of two functions f₁, f₂, i.e.

and by =, the mean value

so that we can make use of the identity

To simplify notation, but without loss of generality, in the sequel let us rewrite (3.45) and subsequent computations omitting the symbol of the mean value, namely

Now, we take the difference of (3.5) written for S₁ and S₂

test by [u], and integrate over (0, t). By monotonicity of (cf. [9]), we have

Thus, recalling the definition of the operator B, and integrating by parts in time, we write

Analogously, we take the difference of (3.6) written for S₁ and S₂, test by \diffc, and integrate over (0, t). By monotonicity of the subdifferential operator, there holds

Thus, we finally write

Now, we can add (3.49) and (3.51), apply the Young inequality to get (cf. (3.8))

Thus, owing to the Gronwall lemma and taking, e.g., d = 1/2 we finally have

which concludes the proof of uniqueness.

Remark 3.2. It is clear that Theorem 3.1. holds also in the case when the higher order nonlinear elastic term is neglected in (3.5), namely the equation governing macroscopic displacements is substituted by

Nonetheless, it turns out to be useful to keep it in order to perform the following asymptotics analysis.

Remark 3.3. Let us point out that the presence of the viscosity term involving deformations is strongly exploited in the proof of the uniqueness of the solution to Problem P_a. Analogously, we could introduce dissipation for the gradient of the phase parameter by adding Ac_t in (3.6).

Remark 3.4. An analogous result as in Theorem 3.1 could be obtained neglecting u_t in (3.5), but to substitute the boundary condition (2.8), e.g., by a Dirichlet prescription on the velocity u_t on G, to get A be coercive via Poincaré's inequality.

4 Passage to the limit for vanishing external forces

Theorem 3.1 ensures that, for f fulfilling (3.7) there exists a unique solution to Problem P_a. In this section, we consider a sequence f_t of exterior forces fulfilling (3.7) and denote by (u_t, c_t) the corresponding solution to Problem P_a, i.e.

where x_t Î ¶I_[0,1](c_t). Then, we prescribe

as t 0. We aim to investigate the asymptotics behavior of the solutions (u_t, c_t), as the parameter t 0. The following theorem can be proved.

Theorem 4.1. Let (4.3) hold. Then, as t 0 the following convergences are verified

Moreover, d and c fulfill, a.e. in (0, T),

for some x belonging to ¶I_[0,1](c) a.e. in Q.

Proof. The proof of Theorem 4.1 is performed by exploiting an a priori estimates-passage to the limit procedure. In particular, we prove some estimates on the solutions which are uniform with respect to t, and then pass to the limit by compactness arguments.

First a priori estimate. We test (4.2) by and (4.1) by , integrate in time over (0, t), and add the resulting equations. It is now a standard matter to integrate by parts and write

Notice that on the left hand side we can equivalently rewrite (cf. (3.4))

so that two terms cancel and applying the Young inequality, we finally obtain

Remark 4.2. In particular, notice that the presence of the higher order elasticity term allows us to bound |Ñu_t|² in L²(Q) in (4.2) after prescribing only that f_t is bounded in L²(0, T; V¢) (cf. Remark 3.2).

Second a priori estimate. We proceed formally and test (4.2) by Ac_t. After integrating over (0, t), by monotonicity of the subdifferential operator, standard arguments show that (cf. (4.10))

Finally, a comparison in (4.2) yields

For the sake of simplicity, we summarize the main estimates we have obtained and we exploit in the following passage to the limit argumentation,

Thus, well known weak and weak star compactness results allow us to deduce that the following convergences hold, at least up to the extraction of a suitable subsequence

In particular, let us observe that the weak convergence (4.16) is not sufficient to identify d with |Ñu|² in (4.17). Hence, by (4.12) we also infer that

Moreover, by compactness, from (4.16) and (4.18), we obtain the following strong convergences (actually something more holds)

Owing to the results presented in [2], (4.19) and (4.21) enable us to identify at the limit x Î ¶I_[0,1](c). Thus, (4.16)-(4.18), (4.19), and (4.20)-(4.21), allow us to pass to the limit as t 0 in (4.1) and (4.2) and get, a.e. in (0, T),

where

Then, as one can prove that (4.22)-(4.23) admits a unique solution (cf. the proof of Theorem 3.1), we can identify u = 0, so that (4.22) vanishes, namely there are not macroscopic displacements or deformations at the limit. Hence, let us discuss the mechanical meaning of d in (4.23). By (4.17) it follows that d is the weak limit of deformation energies associated to the vanishing sequence of macroscopic motions (cf. (4.16)) and, in general, d ¹ 0. Thus, this function, representing a source of damage in (4.23), can be interpreted as the remaining damaging effect of macroscopic motions, acting at a microscopic level. It follows that a sequence of vanishing macroscopic motions can retain its damaging effect, at the limit, as a source of damage in the equation of microscopic motions. In order to support this fact, in the next section we briefly discuss the balance of the energy of our problem.

5 Balance of the energy

We first consider the work provided by the exterior forces during a time interval (0, t), which is given by

By (4.1), we can rewrite (5.1) as it follows

We let

and then, by (4.2), we write

In particular, in (5.4) we have used the fact that

as c_t Î [0, 1] a.e. and c_t (0) = 1. Hence, we set

Thus, we can rewrite T_t as follows

Then, we investigate the limit as t 0 of T_t. By (4.18), the weak lower semicontinuity of norms, and (4.23) we can infer that

where Q_B > 0. Analogously, by (4.16)-(4.17) (where u = 0) we have that

and Q_A > 0. Finally, by (4.17) and (4.21) we have

Hence, we can take a suitable subsequence of t_j such that

Thus, denoting by

owing to (5.8)-(5.9) we can eventually infer that

where (cf. (5.8) and (5.9))

is introduced in (5.8), and

Thus, let us conclude that the work which is provided to the structure T(t) is divideed between damaging external work (t), external source of heat Q(t), and stored energy S(t). In particular, when no instantaneous damage work is applied at the final time t (d(t) = 0), from (5.15) it results S(t) = 0 and, consequently, the work which has been provided is exactly the sum of the damaging work and of the heat sources Q resulting from the dissipative phenomena.

Received: 03/XII/02.

Accepted: 01/IV/03.

#559/03.

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Publication Dates

• Publication in this collection
  20 July 2004
• Date of issue
  2003

History

• Received
  03 Dec 2002
• Accepted
  01 Apr 2003