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.
damage; principle of virtual power; non linear elasticity; nonlinear parabolic PDE; asymptotic analysis
Damage theory: microscopic effects of vanishing macroscopic motions
Elena BonettiI; Michel FrémondII
ILaboratoire Lagrange, Dipartimento di Matematica ''F. Casorati'', Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy, E-mail: bonetti@dimat.unipv.it
IILaboratoire Lagrange, Laboratoire Central des Ponts et Chaussées, 58 boulevard Lefebvre, 75732 Paris Cedex 15, France, E-mail: michel.fremond@lcpc.fr
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 ct, Ñct 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 ut and e(ut).
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 ft is applied, i.e. ft0 as t tends to 0. Hence, letting (ut, ct) 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 ft, 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 := W1,4(W)
Finally, we introduce the following abstract operators
where Di stands for the usual partial derivative operator with respect to the variable xi, 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 Pa : 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 Pa admits a unique solution (u, c) with regularity
Proof. To prove the existence of a solution to Problem Pa 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 ct 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 ut 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 (un, cn, xn), with xnÎ ¶I[0,1](cn), 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 xn 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 un and |Ñun|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 Pa.
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 Pa 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 Pa. Analogously, we could introduce dissipation for the gradient of the phase parameter by adding Act in (3.6).
Remark 3.4. An analogous result as in Theorem 3.1 could be obtained neglecting ut in (3.5), but to substitute the boundary condition (2.8), e.g., by a Dirichlet prescription on the velocity ut 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 Pa. In this section, we consider a sequence ft of exterior forces fulfilling (3.7) and denote by (ut, ct) the corresponding solution to Problem Pa, i.e.
where xt Î ¶I[0,1](ct). Then, we prescribe
as t 0. We aim to investigate the asymptotics behavior of the solutions (ut, ct), 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 |Ñut|2 in L2(Q) in (4.2) after prescribing only that ft is bounded in L2(0, T; V¢) (cf. Remark 3.2).
Second a priori estimate. We proceed formally and test (4.2) by Act. 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 ct Î [0, 1] a.e. and ct (0) = 1. Hence, we set
Thus, we can rewrite Tt as follows
Then, we investigate the limit as t 0 of Tt. By (4.18), the weak lower semicontinuity of norms, and (4.23) we can infer that
where QB > 0. Analogously, by (4.16)-(4.17) (where u = 0) we have that
and QA > 0. Finally, by (4.17) and (4.21) we have
Hence, we can take a suitable subsequence of tj 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