Stress Assisted Diffusion

A detailed numerical study (Bond 2005) has shown that for the same overall fiber volume fraction, the diffusion coefficient D is rather insensitive to the spacing or distribution of circular fibers, although, in the case of composite laminates, the presence of a resin-rich interlaminar region would affect the overall value of the transverse component of D. 

On the other hand, it was demonstrated by means of a thorough computational scheme (Aditya and Sinha 1996) that D is highly sensitive (up to 30%) to fiber shape and that this sensitivity increases with volume fraction. 

Besides circular and elliptical fiber cross sections, that work investigated the effects of more complex, realistic shapes depicted in Fig. 5.3 below 

Upon the incorporation of stress effects on the free volume (Knauss and Emri 1981), namely 

While fiber spacings and distributions have but a small effect on diffusivity they play a major role in the residual hygral stresses within a fiber reinforce composite. 

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To save the space, further development deals solely with stress assisted diffusion since its generalization for the case of stress-assisted diffusion is straightforward. According to previous studies [2], it is assumed that hydrogen diffusion through material proceeds toward the sites where the lowest concentration or the higher hydrostatic stresses occur. The combination of these factors results in an equation for the stress-assisted diffusion flux of hydrogen which is ... 

Obviously, quantitative modelling of stress-assisted hydrogen diffusion requires the stress field in a testpiece of interest to be known. Even for rather simple cases, such as a notched bar being considered here, neither the exact solutions nor the closed form ones are usually available. Thus, one must count on some sort of the numerical solution of the mechanical portion of the coupled problem of the stress-assisted diffusion. The finite element method (FEM) approach, well-developed for both linear and nonlinear analyses of deformable solid mechanics, is a right choice to perform the stress analysis as a prerequisite for diffusion calculations. 

Some advanced general purpose finite-element codes, well adapted for stress analysis in particular, e.g. ABAQUS or MSC.MARC, have certain capabilities to simulate the stress-assisted diffusion, too. Unfortunately, they still are limited in some rather important aspects. As regards ABAQUS, this allows to perform simulations of the stress-assisted diffusion governed by equation (5) "over" the data of an accomplished solution of a geometrically and physically nonlinear stress-strain analysis, i.e., for the stationary stress field at the end of some preliminary loading trajectory, but not for the case of simultaneous transient loading and hydrogenation. 

With this in mind, it seems to be a reasonable compromise to consider a FEM implementation of the modelling of stress-assisted diffusion over the previously (or simultaneously) performed stress analysis taking the nodal values of stresses, obtained with a post-processing technique, as the entry data for diffusion, i.e., constructing a finite-element approximation of the stress field with the aid of the same finite-element shape functions used in the mechanical analysis to approximate the displacement fields. 

To proceed with simulations of stress-assisted diffusion with rather modest computational facilities available, it turned out to be indeed necessary to reduce the FEM-problem size. Among two possible approaches, i.e., coarsening of the mesh of the modelled "full-scale" specimen or shrinking the domain of diffusion simulation focusing on the locations of prospective hydrogen assisted fracture initiation near the notch, the second one seems to be preferable. The relevant data about stress fields may be transferred to this domain from the full scale mechanical analyses, performing their interpolation for the finite element mesh for diffusion, if convenient. 

This model is considered to be useful to improve the knowledge of the role played by the factor of hydrogen accumulation in prospective rupture sites by stress-assisted diffusion, one of the key items in hydrogen embrittlement, a very dangerous phenomenon that frequently accompanies structural metals and alloys in service. 

Figure 7.4 Stress-induced lattice diffusion in a single crystal a) Crystal under the action of tensile and compressive stresses (the arrows in the crystal show directions of vacancy motion) b) Strains produced by the stress-assisted diffusion process.

The particular evolution phenomena in material systems considered in this chapter include the transition from a nominally flat surface to a wavy surface in a stressed solid, the spontaneous growth of epitaxial islands due to deposition of a material on a substrate with lattice mismatch, stress relaxation by grain boundary diffusion, the role of stress in altering compositional variations in solid solutions, and stress-assisted diffusion in the presence of an electric field or electromigration. 

A side effect of the anodic dissolution is often, particularly at crack tips (see Sec. 5.2.5), the production of hydrogen by the reduction of water. Some studies have shown that hydrogen absorption can favor local plasticity, due to enhanced dislocation velocities with hydrogen. In the presence of a crack, accelerated hydrogen penetration can occur very near to the crack tip region by stress-assisted diffusion and dislocation transport. These effects will be discussed later. 

A comprehensive formulation of stress-assisted diffusion (Weitsman 1987a) was established upon employing fundamental concepts of irreversible thermodynamics and continuum mechanics. 

Vieth W, Howell J, Hsieh J (1976) Dual sorption theory. J Memb Sci 1 177-220 Weitsman YJ (1987a) Stress assisted diffusion in elastic and viscoelastic materials. J Mech Phys Solids 35(l) 73-94... 

Weitsman YJ, Guo Y (2002) A correlation between fluid-induced damage and anomalous fluid sorption in polymeric composites. Compos Sci Technol 62(6) 889-908 Williams ML, Landel RF, Ferry JD (1955) The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. J Am Chem Soc 77(14) 3701-3707 Wu CH (2001) The role of Eshelby stress in composition-generated and stress-assisted diffusion. J Mech Phys Solids 49(8) 1771-1794... 

Consider now stress-assisted diffusion in aging viscoelastic media. Employing (6.7), (6.21), (6.23), and (6.24) we obtain... 

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