Fatigue accumulation in an oscillating plate

A thermodynamic model for fatigue accumulation in an oscillating elastoplastic Kirchhoffplate based on the hypothesis that the fatigue accumulation rate is proportional tothe dissipation rate, is derived for the case that both the elastic and the plasticmaterial characteristics change with increasing fatigue. We prove the existence ofa unique solution in the whole time interval before a singularity (material failure) occursunder the simplifying hypothesis that the temperature history is a priori given.

Introduction. In this paper, we pursue the study of cyclic fatigue accumulation in oscillating elastoplastic systems started in [6,7]. The main goal of our project is to model the most important experimental features of fatigue, such as material softening, heat release, and material failure in finite time. The analysis of the socalled rainflow method of cyclic damage evaluation carried out in [2] has shown a qualitative and quantitative correspondence between the damage accumulation rule and the energy dissipation. On the other hand, experimental measurements at the point of material failure confirm strong temperature increase, which manifests an energy dissipation peak. In fact, temperature tests are regularly used in engineering practice for damage analysis in high frequency regimes (e.g. in aircraft industry). Our substantial modeling hypothesis thus consists in introducing a scalar fatigue parameter m, assuming that its time derivative (the fatigue rate) is proportional to the dissipation rate, and that the material parameters depend on m. We believe that this assumption is realistic. Plastic deformations are driven by moving dislocations and ruptures of interatomic connections, which at the same time dissipate energy, and reduce the cohesion of the solid. Note that in the Gurson model for void nucleation and growth in elastoplastic materials, see [10], the elasticity 910 MICHELA ELEUTERI, JANA KOPFOVÁ AND PAVEL KREJČÍ domain shrinks, being parameterized by the plastic dissipation rate. For a more detailed discussion about the modeling issues, see [6]. In the previous papers, we have considered the case that only the hardening modulus depends on m, and that the plastic characteristics are not altered by fatigue (but may possibly depend on temperature). Here, we include the fatigue dependence into the plastic constitutive law as well.
The PDE system of momentum and energy balance equations for transversal oscillations of an elastoplastic plate under fatigue is derived in Section 1. The unknowns of the full problem are w (the transversal displacement), θ (absolute temperature), and m (fatigue). In this paper, however, we do not prove the wellposedness of the complete system resulting from a thermodynamic analysis. We only make a first step in this direction and solve the momentum balance equation coupled with the fatigue accumulation equation, assuming that the temperature history is known. An existence and uniqueness theory for the full system will be the subject of a subsequent paper.
It cannot be expected that solutions of the system with fatigue exist globally in time. The material failure in finite time is an integral part of the model. We give an efficient lower bound for the existence time.
The paper is organized as follows. In Section 1, we derive the model equations from thermodynamic principles. The mathematical problem is stated in Section 2. Some new properties of the vectorial Prandtl-Ishlinskii operator are proved in Section 3, and the proof of the main existence and uniqueness theorems is carried out in Sections 4-6.
1. The model. An elastoplastic plate subject to bending exhibits plasticized zones occurring first on the boundary and propagating towards the interior. Assuming a single yield von Mises plasticity criterion in the original 3D setting (cf. [16]), the Kirchhoff dimensional reduction to 2D carried out in [9] was shown to give rise to a Prandtl-Ishlinskii (also called "generalized St. Venant", see [16]) constitutive relation with a continuum of yield surfaces that are successively activated in agreement with natural expectations: The midsurface of the plate is subject to smaller deformations than eccentric layers, therefore plastic yielding occurs earlier far from the midsurface.
This emerging multiyield character of the elastoplastic plate bending problem does not seem to have been taken into consideration before. For instance in [1] there is no direct reference to plates; in [3,13,18] the yield condition is still described by one sharp surface of plasticity. The methods of the papers [8,19,14] based on Γ-convergence of energy minimizers have been only recently refined in [15] to obtain the Prandtl-Ishlinskii model in the Γ-limit as well. The drawback of the Γ-limit technique is that it cannot be extended to the study of oscillating systems, and of nonequilibrium problems in general.
After dimensional reduction and integration over the thickness, the strain and stress tensors are transformed into functions of the space variable x from a domain Ω ⊂ R 2 and the time variable t ∈ (0, T ), and have only three independent components. The elastoplastic plate problem is derived in [9] in the form where w is the transversal displacement, ε and σ are weighted averages of strain and stress tensors over the thickness of the plate (i.e. bulk moments), > 0 is the constant mass density, h > 0 is the thickness of the plate, B, K are positive definite symmetric matrices, g is the external load, γ ∈ L 1 (0, ∞) is a given Prandtl-Ishlinskii density function, and χ r for r > 0 are solutions of the family of variational inequalities constrained to the system of convex closed sets rZ , where Z ⊂ R 3 is a referential set, and rZ = {rz ∈ X : z ∈ Z} for r > 0. We denote by Q rZ : R 3 → rZ the projection onto rZ , orthogonal with respect to the scalar product ξ, η = Kξ · η . The mapping P which associates to each ε (in some suitable space) the integral part of (1.3) is called the vectorial Prandtl-Ishlinskii operator , see (3.7). In [6], we have included the fatigue and temperature dependence into the model by introducing a scalar fatigue parameter m(x, t) ≥ 0, assuming that the matrix B depends on m and the Prandtl-Ishlinskii density γ depends on temperature. Here, we let both the matrix B and the function γ depend on m, and complement the constitutive law (1.3) with viscosity and thermal expansion terms to obtain where θ > 0 is the absolute temperature, θ 0 > 0 is a given referential temperature, 1 is the vector (1, 1, 0), β is the thermal expansion coefficient, C is the viscosity matrix, and B(m), γ(m, r) are functions specified below in Hypothesis 2.1. By analogy to [6,9], we associate with (1.5) the free energy F defined by the formula (1.6) where c 0 is the constant specific heat capacity. The internal energy U and the entropy S thus have the form The equations for the state variables θ and m are derived from the first and the second principles of thermodynamics in the form where q is the heat flux vector that we assume in the form with a constant heat conductivity coefficient κ > 0. Then (1.9) reads (1.12) The notation is slightly ambiguous, and we hope that the reader will not get confused. For simplicity, we denote by t and m partial derivatives with respect to the corresponding variables. The index r is not a partial derivative. There is no differentiation with respect to r in the paper.
In view of (1.11), we see that the Clausius-Duhem inequality (1.10) is certainly (1.14) The last integral term in (1.14) is nonnegative by virtue of (1.4). The assumption that the fatigue accumulation rate m t is nonnegative (that is, fatigue can only increase in time) is therefore compatible with the second principle provided In other words, material softening takes place under increasing fatigue in agreement with experimental evidence similarly as in [10]. We close the system by assuming that the fatigue accumulation rate m t at a point x ∈ Ω is proportional to the dissipation rate averaged over a neighborhood of the point x, that is, , we assume that a function θ : Ω × (0, T ) → R 3 describing a combined action of thermal expansion and external load is given, set the physical constants to 1 for simplicity, and consider the problem where we denote ξ r = ε − χ r . The following hypotheses are assumed to hold.
Hypothesis 2.1. We fix a Lipschitzian domain Ω ⊂ R 2 , and denote by | · | p the L p (Ω) norm for p ≥ 1. We assume that there exists a constant ν > 0 such that for

6)
and for T > 0 we denote Ω T = Ω × (0, T ). Furthermore, (i) C, K are symmetric positive definite 3×3 matrices, and there exist constants is a given function with compact support, and Λ > 0 is a constant such that 0 ≤ λ ≤ Λ a.e.; a.e., γ m (0, r) = 0 a.e., and there exists a constant Γ > 0 such that We prove in the next sections the following existence and uniqueness results.

FATIGUE ACCUMULATION IN AN OSCILLATING PLATE 915
As an immediate consequence of (3.5), we have for all t ∈ [0, T ] that Given a nonnegative function γ ∈ L 1 (0, ∞), we define the vectorial Prandtl-Ishlinskii operator P with characteristic Z and density γ by the formula for ε ∈ W 1,1 (0, T ; X). The definition is meaningful due to the fact that, setting We need here a special Lipschitz continuity result (Theorem 3.3 below) which is not explicitly stated in the literature. Let us introduce first some necessary concepts.
With the convex closed set Z , we associate the Minkowski functional M Z : X → R + defined by the formula hold for every χ, η ∈ X . The Minkowski functional of a convex closed set containing 0 is proper, convex, and lower semicontinuous. The smoothness assumption in Hypothesis 3.1 implies that its subdifferential ∂M Z (χ) for all χ = 0 contains a single vector parallel to the unit outward normal vector n Z (χ/M Z (χ)) taken at the point χ/M Z (χ) on the boundary of Z . We define the duality mapping J Z : X → X by the formula The Minkowski functionals M Z and M rZ for r > 0 are related through a simple scaling formula. Indeed, (3.13) and hence ∂M rZ = 1 r ∂M Z . We thus conclude that Formula [5, (3.35)] can be written here in the form hence, by (3.10), In terms of the Minkowski functional, putting ξ(t) = ε(t) − χ(t), we can represent the variational inequality (3.1) by the differential inclusion χ(t) ∈ ∂M Z * (ξ t (t)), or, equivalently, by the identity a.e. (3.18) This is the so-called energetic formulation, see [17]. The energetic interpretation of (3.18) is that ε t (t), χ(t) is the power supplied to the system, part of which is used for the potential increase d dt 1 2 |χ(t)| 2 , and the other part ξ t (t), We now prove the main result of this section, namely the Lipschitz continuity of the dissipation functional. Theorem 3.3. Let ε , ε ∈ W 1,1 (0, T ; X) and r > 0 be given, and let χ i r , i = , , be solutions of the variational inequalities Set ξ i r = ε i − χ i r . Then we have
We now apply this result to our plate problem.

FATIGUE ACCUMULATION IN AN OSCILLATING PLATE 919
The definition is meaningful, as by Hypothesis 2.1, p ∈ L 1 (0, T ). We fix the number 4) where the supremum is taken over all ε ∈ E T and m ∈ L ∞ (Ω T ) (it is indeed finite), and define µ(t) as the solution of the ODĖ that is, We see thatμ(t) blows up to +∞ as t 1/(2AR). We choose a small δ ∈ (0, 1) that we keep fixed throughout the paper, and set Eq. (2.7) cannot be expected to have global solutions for the same reason as in [7]. We state the intermediate result in the following form. γ(m, r) Kχ r · (ξ r ) t dr dy (4.9) whereC > 0 is a constant which comes out from the following computation: and test (4.11) by e −2α(t) . This yields that is, We see that the mapping that with m associatesm is a contraction on M R , hence Eq. (2.7) has a unique solution m for every ε ∈ E T R . Eq. (2.1) with a given ε ∈ E T R is linear in w and the existence and uniqueness of a solution can be easily proved e.g. by Galerkin approximations. The required regularity follows by testing (2.1) successively by ϕ = w t and ϕ = (I − ∆) −1 w tt , using the assumption (2.6). To complete the proof, it remains to check that D 2 w ∈ E T R . Choosing again ϕ = w t in (2.1) and using Hypothesis 2.1, we obtain which we wanted to prove.
5. The coupled system. This section is devoted to the proof of Theorem 2.2. We start with an auxiliary result on the solution mapping of (1.4), (2.3), (2.7) which with a given ε ∈ E T R associates m ∈ M R .
Lemma 5.1. There exists a constant C 2 depending only on R and on the data of the problem such that for all ε , ε ∈ E T R , the corresponding solutions m , m ∈ M R to (2.3), (2.7) satisfy the inequality Proof. With the notation of Section 4 we have where S > 0 is a constant, and where we have used the fact that γ(m 2 , r) ≤ γ(0, r) by Hypothesis 2.1 (iv). Testing (5.2) by e −α(t) , withα from (4.12), yields that Integrating from 0 to t and using (3.27) we obtain the assertion.
Lemma 5.2. The mapping defined in Proposition 4.1 that with ε ∈ E T R associates D 2 w ∈ E T R is a contraction with respect to a suitable norm.
Proof. We test the difference of Eqs. (2.1) written for ε , ε and the corresponding solutions w , w byw t = w t − w t , and obtain d dt We have with a constant C 4 > 0, and with the notationε = ε − ε . Testing (5.6) by e −2C4t and integrating from 0 to T R , we get the inequality which completes the proof.
We are now ready to finish the proof of Theorem 2.2.
Proof. In order to prove Theorem 2.2, it suffices to combine Proposition 4.1 with Lemma 5.2 and apply the contraction principle.
6. Proof of Theorem 2.3. The main goal of this section is to remove the cut-off function Q R in (2.7). This will be done by establishing additional estimates, where the dependence on R is explicitly taken into account. The constants C 5 , . . . , C 10 which appear in the formulas below are independent of R.