M.W.D. van der Burg

On the linear elastic properties of regular and random open-cell foam models

Foams can be created from coagulation of gas bubbles in liquid. After removal of cell faces, an open-cell foam remains consisting of a strut framework. In the past, mechanical properties were estimated by a small unit cell consisting of only a few struts. However, the random geometry of the foam can be of importance for the linear elastic properties. Here, large foam unit cells are created using Voronoi techniques. A smooth transition from regular to random geometries is made, showing the strong sensitivity of the mechanical properties from the geometry of the microstructure. Uniaxial global loads are transmitted through chains of highly loaded struts. The deformation of the struts in the foam is a mixture of bending and normal deformation, the ratio of which shown here to be dependent on the magnitude of the density.

A numerical study of large deformations of low-density elastomeric open-cell foams

Abstract A numerical study is presented of the mechanical properties of low-density open-cell polymer foams subjected to large deformations. The foams are modelled as three-dimensional frameworks of slender struts. Regular as well as random foams are analyzed, where the latter are generated using the Voronoi technique. The macroscopic mechanical properties are determined for various types of struts properties through unit-cell analyses containing many foam cells per unit-cell. The computations make use of standard Finite Element (FE) techniques. Bending of the struts dominates the mechanical foam response at low strains. Axial deformation of the struts becomes the dominant mechanism at larger tensile strains. Strut buckling becomes the main mechanism at larger compressive strains, and causes a significant decrease in load carrying capacity of the foam. The large strain mechanical behavior of foams is found to be dependent on the weakest cross-section of the foam appearing in the random foam structure, the so-called “minimum effective cross-section”. The minimum effective cross-section determines the tangential foam modulus at large tensile strains. Regular foam structures have a uniform unit-cell cross-section and, as a result, a higher minimum effective cross-section than regular foam structures and, therefore, a higher tangent modulus in the large strain range.

VOID GROWTH DUE TO CREEP AND GRAIN BOUNDARY DIFFUSION AT HIGH TRIAXIALITIES

The growth of grain boundary voids at elevated temperatures by coupled creep and grain boundary diffusion is studied numerically using a cylindrical unit cell model. Emphasis is on the influence of the remote stress triaxiality, which is taken to cover the full range of axisymmetric stress states, from purely effective to purely hydrostatic states of stress. The motivation for extending previous results stems from the need for an accurate cavity growth model to analyse damage due to hydrogen attack, where the grain boundary voids are internally pressurized. Because of the wide range of stress states considered, numerical stability requires the use of two normalizations of the variational principle for the coupled void growth problem; one when the effective stress is dominant and the other when the mean stress is dominant. In the regime where deformation is primarily by creep, two distinct modes of deformation appear for each level of porosity; one for low triaxialities and one that takes over for sufficiently high triaxialities. Approximate models found in the literature for a dilute concentration of voids, or for finite concentrations, are explored to check their ability to represent the stress state dependence of the volumetric void growth rate. A novel approximate formula is derived for creep dominated growth and is shown to give good agreement with numerically computed void growth rates in the high triaxiality regime and for finite concentrations. A fairly abrupt transition between creep dominated void growth and diffusion dominated void growth is found when the stress triaxiality is very high, so that the interaction between creep and diffusion is then relatively unimportant. Finally, formulae are presented which give an approximate, yet fairly accurate, expression for the void volume growth rate due to coupled diffusional and creep growth over the full range of axisymmetric stress states.

INVESTIGATION OF HYDROGEN ATTACK IN 2.25Cr-1Mo STEELS WITH A HIGH-TRIAXIALITY VOID GROWTH MODEL

A model is presented to estimate the lifetime under hydrogen attack (HA) conditions. The first ingredient is the Odette-Vagarali model to calculate the equilibrium methane pressure as a function of hydrogen pressure, temperature, and type and composition of the carbides and the alloy. The second ingredient is a model for the growth to coalescence of methane-filled grain boundary cavities, possibly under the presence of (applied or residual) macroscopic stresses. This model is based on recent detailed numerical studies of the growth of voids under simultaneous grain boundary diffusion and creep of the grain material. A new, accurate analytical approximate void growth relation valid for high stress triaxialities is adapted for application to HA. The model is used to perform a study of HA, including a computation of Nelson curves, in 2.25Cr-1Mo steels with different types of carbides and for various applied stress states. Finally, the results of the model are presented in a concise, non-dimensional form that reveals the key parameters that determine HA life times.

A continuum damage relation for hydrogen attack cavitation

Abstract A continuum damage relation (CDR) is proposed to describe the failure process of hydrogen attack, i.e. grain boundary cavitation of steels under conditions of high temperature and high hydrogen pressure. The cavitation is caused by the chemical reaction of hydrogen with grain boundary carbides forming cavities filled with high pressure methane. The micromechanisms described are the grain boundary cavitation and the dislocation creep of the grains. The CDR is based on two extreme cavitation rate distribution modes. In the first mode, the cavitation rate along the facets is uniform, resulting in a hydrostatic dilatation while the creep deformations remain relatively small. In the second mode, cavitation proceeds predominantly on grain boundary facets transverse to the principal macroscopic stress. This part of the CDR builds on Tvergaards constituitive relation for intergranular creep rupture [Tvergaard, V., Acta metallurgica , 1984, 32 , 1977] where the facet cavitation is constrained by creep of the surrounding grains. The mode corresponding to the highest cavitation rate is the active mode. The two-dimensional version of the CDR is verified against detailed finite element analyses of hydrogen attack in planar polycrystalline aggregates. Finally, the generalization to a three-dimensional CDR is discussed.

A continuum damage analysis of hydrogen attack in a 2.25Cr–1Mo pressure vessel

A micromechanically based continuum damage model is presented to analyze the stress, temperature and hydrogen pressure dependent material degradation process termed hydrogen attack, inside a pressure vessel. Hydrogen attack (HA) is the damage process of grain boundary facets due to a chemical reaction of carbides with hydrogen, thus forming cavities with high pressure methane gas. Driven by the methane gas pressure, the cavities grow, while remote tensile stresses can significantly enhance the cavitation rate. The damage model gives the strain-rate and damage rate as a function of the temperature, hydrogen pressure and applied stresses. The model is applied to study HA in a vessel wall, where nonuniform distributions of hydrogen pressure, temperature and stresses result in a nonuniform damage distribution over the vessel wall. Stresses inside the vessel wall first tend to accelerate and later decelerate the cavitation rate significantly. Numerical studies for different material parameters and different stress conditions demonstrate the HA process inside a vessel in time. Also, the lifetime of the pressure vessel is determined. The analyses underline that the general applicability of the Nelson curve is questionable.

Some effects of random microstructural variations on creep rupture

Abstract High-temperature creep rupture of polycrystalline materials involves a number of physical mechanisms, such as the nucleation and diffusive growth of grain boundary cavities and grain boundary sliding, which act at different length scales in the material. This paper uses a micromechanical model to explore how random variations in the microstructure of the material affect its lifetime. The model involves a chain of size scale transitions and two size scales are considered in particular: the size scale of individual cavities and the scale of aggregates of grains. Emphasis is put on geometrical variations in the microstructure, i.e. random variations in size and shape of grains in an aggregate. The role of grain boundary sliding, and the competition between creep flow and grain boundary diffusion are highlighted. Wherever possible, regimes are indicated where such microstructural variations can be safely neglected or where they are critical in determining the lifetime.

Non-uniform hydrogen attack cavitation and the role of interaction with creep

Hydrogen attack (HA) is the development of grain-boundary porosity by cavities filled with high-pressure methane that originates from the reaction of carbides with hydrogen at high temperatures. The cavities grow by grain-boundary diffusion and by creep of the adjacent grain material till they coalesce with neighbouring cavities to form a microcrack. Earlier work on HA has focussed on unit cells containing a single cavity, using average cavitation properties. Here, non-uniform cavitation properties on the grain-size scale are assumed in a polycrystalline aggregate, and unit cell analyses are performed to investigate the influence of the adjacent grains on the development of the grain-boundary HA. The numerical results are explained in terms of two simplified models which highlight the key parameters governing the grain deformation-grain boundary cavitation interaction process.