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Damage processes coupled with phase separation in elastically stressed alloys
Abstract
In micro-electronic materials such as solder alloys, phase-separation and coarsening as well as damage phenomena occur at the same time and influences each other. In this note, a unifying model which couples multi-component Cahn-Hilliard systems with elasticity and uni-directional damage processes is presented. We outline the equations and their initial-boundary conditions in a classical setting and cite some existence results for weak solutions recently proved in [8, 9]. The damage is assumed to be incomplete, i.e. the maximal damaged material parts still feature elastic properties. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
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A theory of continuum damage mechanics is developed within the framework of the principle of virtual power. Because the damage in a solid results from microscopic movements, we decided to include the power of these microscopic movements in the power of the internal forces. The microscopic velocities are related to the damage rate. The power of the internal forces we choose depends on the damage velocity and its gradient to take into account the interactions. Models issued from this theory are presented. The are coherent from the mechanical and mathematical points of view. The numerical computations show no mesh sensitivity. The describe with good agreement the main experimental properties. Concrete is chosen as an example to illustrate the theory. A model using two damage quantities is also presented. It yields the correct description of the unilateral phenomenon observed in concrete. Finally, an extension of existing models is proposed to descride the fatigue damage behaviour.
A typical phase field approach for describing phase separation and coarsening phenomena in alloys is the Cahn-Hilliard model. This model has been generalized to the so-called Cahn-Larché system by combining it with elasticity to capture non-neglecting deformation phenomena, which occur during phase separation and coarsening processes in the material. In order to account for damage effects, we extend the existing framework of Cahn-Hilliard and Cahn-Larché systems by incorporating an internal damage variable of local character. This damage variable allows to model the effect that damage of a material point is influenced by its local surrounding. The damage process is described by a unidirectional rate-dependent evolution inclusion for the internal variable. For the introduced Cahn-Larché systems coupled with rate-dependent damage processes, we establish a suitable notion of weak solutions and prove existence of weak solutions.
In this paper, we analytically investigate multi-component Cahn–Hilliard and Allen–Cahn systems which are coupled with elasticity and uni-directional damage processes. The free energy of the system is of the form ∫Ω½Γ∇c : ∇c + ½|∇z|2+W
ch(c)+W
el(e,c,z)dx with a polynomial or logarithmic chemical energy density W
ch, an inhomogeneous elastic energy density W
el and a quadratic structure of the gradient of damage variable z. For the corresponding elastic Cahn–Hilliard and Allen–Cahn systems coupled with uni-directional damage processes, we present an appropriate notion of weak solutions and prove existence results based on certain regularization methods and a higher integrability result for strain e.
We analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps.
Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arclength reparametrization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.
The stability of a solid solution to all infinitesimal composition fluctuations is considered, taking surface tension and elastic energy into account. It is found that for infinite isotropic solids, free from imperfections the spinodal marks the limit of metastability to such fluctuations only if there is no change in molar volume with composition. Otherwise the elastic energy due to a fluctuation stabilizes the solution and alters the criterion for the limit of metastability. For an unstable solution the kinetics of decomposition are discussed and the expected mean particle size or wavelength of the most rapidly growing fluctuation is derived.
We consider a model describing the evolution of damage in visco-elastic materials, where both the stiffness and the viscosity properties are assumed to degenerate as the damaging is complete. The equation of motion ruling the evolution of macroscopic displacement is hyperbolic. The evolution of the damage parameter is described by a doubly nonlinear parabolic variational inclusion, due to the presence of two maximal monotone graphs involving the phase parameter and its time derivative. Existence of a solution is proved in some subinterval of time in which the damage process is not complete. Uniqueness is established in the case when one of the two monotone graphs is assumed to be Lipschitz continuous.
In this note, we study a Cahn-Hilliard equation in a deformable elastic isotropic continuum. We define boundary conditions associated with the problem and obtain the existence and uniqueness of solutions. We also study the long time behavior of the system.
The paper concerns the existence of weak solutions to the Cahn-Hilliard-Gurtin system coupled with nonstationary elas-ticity. The system describes phase separation process in elastically stressed material. It generalizes the Cahn-Hilliard equation by ad-mitting a more general structure and by coupling diffusive and elastic effects. The system is studied with the help of a singularly perturbed problem which has the form of a well-known phase field model cou-pled with elasticity. The established existence results are restricted to the homogeneous problem with gradient energy tensor and elas-ticity tensor independent of the order parameter.
This paper discusses an existence result for energetic solutions of rate-independent damage processes and the temporal regularity of the solution. We consider a body consisting of a physically nonlinearly elastic material undergoing small deformations and partial damage. The present work is a generalization of [16] concerning the properties of the stored elastic energy density as well as the suitable Sobolev space for the damage variable: While previous work assumes that the damage variable z satisfies z in W1, r(Ω) with r > d for Ω ⊂ ℝd, we can handle the case r > 1 by a new technique for the construction of joint recovery sequences. Moreover, this work generalizes the temporal regularity results to physically nonlinearly elastic materials by analyzing Lipschitz- and Hölder-continuity of solutions with respect to time.
The stability of a solid solution to all infinitesimal composition fluctuations is considered, taking surface tension and elastic energy into account. It is found that for infinite isotropic solids, free from imperfections the spinodal marks the limit of metastability to such fluctuations only if there is no change in molar volume with composition. Otherwise the elastic energy due to a fluctuation stabilizes the solution and alters the criterion for the limit of metastability. For an unstable solution the kinetics of decomposition are discussed and the expected mean particle size or wavelength of the most rapidly growing fluctuation is derived.
Tiller et al. have recently attempted(1) to reformulate the theory of spinodal decomposition. Unfortunately, the equations they propose violate the second law of thermodynamics as well as displaying other incongruities. It is shown that, even if their intentions are carried out correctly, the results are at variance with existing diffusion data. It is also demonstrated that their criticisms of the mathematics of the original treatment are invalid.RésuméTiller, Pound et Hirth ont récemment essayé(1) de donner une nouvelle formulation de la théorie de la décomposition spinodale. Malheureusement, les équations qu'ils proposent n'obéissent pas à la deuxième loi de la thermodynamique et présentent également d'autres incorrections. Les auteurs montrent que, même si leurs développements sont exécutés correctement, les résultats sont en désaccord avec les résultats déjà existant pour la diffusion: ils montrent également que les critiques de la théorie mathématique qu'ils ont proposé ne sont pas valables.ZusammenfassungTiller, Pound und Hirth haben kürzlich den Versuch unternommen, die Theorie der spinodalen Entmischung neu zu formulieren.(1) Die von ihnen vorgeschlagenen Gleichungen verletzen leider den zweiten Hauptsatz der Thermodynamik und zeigen auβerdem andere Widersprüche. Es wird gezeigt, daβ selbst bei richtiger Durchführung ihrer Intentionen die Resultate mit vorliegenden Diffusionsdaten im Widerspruch sind. Es wird auβerdem dargelegt, daβ ihre Kritik an der Mathematik der ursprünglichen Behandlung der spinodalen Entmischung nicht zutrifft.
A microscopic diffusional theory for the motion of a curved antiphase boundary is presented. The interfacial velocity is found to be linearly proportional to the mean curvature of the boundary, but unlike earlier theories the constant of proportionality does not include the specific surface free energy, yet the diffusional dissipation of free energy is shown to be equal to the reduction in total boundary free energy. The theory is incorporated into a model for antiphase domain coarsening. Experimental measurements of domain coarsening kinetics in Fe-Al alloys were made over a temperature range where the specific surface free energy was varied by more than two orders of magnitude. The results are consistent with the theory; in particular, the domain coarsening kinetics do not have the temperature dependence of the specific surface free energy.