A.M. Gokhale

Modeling stress state dependent damage evolution in a cast Al–Si–Mg aluminum alloy

Abstract Internal state variable rate equations are cast in a continuum framework to model void nucleation, growth, and coalescence in a cast Al–Si–Mg aluminum alloy. The kinematics and constitutive relations for damage resulting from void nucleation, growth, and coalescence are discussed. Because damage evolution is intimately coupled with the stress state, internal state variable hardening rate equations are developed to distinguish between compression, tension, and torsion straining conditions. The scalar isotropic hardening equation and second rank tensorial kinematic hardening equation from the Bammann–Chiesa–Johnson (BCJ) Plasticity model are modified to account for hardening rate differences under tension, compression, and torsion. A method for determining the material constants for the plasticity and damage equations is presented. Parameter determination for the proposed phenomenological nucleation rate equation, motivated from fracture mechanics and microscale physical observations, involves counting nucleation sites as a function of strain from optical micrographs. Although different void growth models can be included, the McClintock void growth model is used in this study. A coalescence model is also introduced. The damage framework is then evaluated with respect to experimental tensile data of notched Al–Si–Mg cast aluminum alloy specimens. Finite element results employing the damage framework are shown to illustrate its usefulness.

Stereological length estimation using spherical probes.

Lineal structures in biological tissue support a wide variety of physiological functions, including membrane stabilization, vascular perfusion, and cell‐to‐cell communication. In 1953, Smith and Guttman demonstrated a stereological method to estimate the total length density (Lv) of linear objects based on random intersections with a two‐dimensional sampling probe. Several methods have been developed to ensure the required isotropy of object–probe intersections, including isotropic‐uniform‐random (IUR) sections, vertical‐uniform‐random (VUR) slices, and isotropic virtual planes. The disadvantages of these methods are the requirements for inconvenient section orientations (IUR, VUR) or complex counting rules at multiple focal planes (isotropic virtual planes). To overcome these limitations we report a convenient and straightforward approach to estimate Lv and total length, L, for linear objects on tissue sections cut at any arbitrary orientation. The approach presented here uses spherical probes that are inherently isotropic, combined with unbiased fractionator sampling, to demonstrate total L estimation for thin nerve fibres in dorsal hippocampus of the mouse brain.

A void–crack nucleation model for ductile metals

Abstract A phenomenological void–crack nucleation model for ductile metals with secondphases is described which is motivated from fracture mechanics and microscale physicalobservations. The void–crack nucleation model is a function of the fracture toughness of theaggregate material, length scale parameter (taken to be the average size of the second phaseparticles in the examples shown in this writing) , the volume fraction of the second phase, strainlevel, and stress state. These parameters are varied to explore their effects upon the nucleationand damage rates. Examples of correlating the void–crack nucleation model to tension data in theliterature illustrate the utility of the model for several ductile metals. Furthermore, compression,tension, and torsion experiments on a cast Al–Si–Mg alloy were conducted to determinevoid–crack nucleation rates under different loading conditions. The nucleation model was thencorrelated to the cast Al–Si–Mg data as well.

Representative volume element for non-uniform micro-structure

Abstract A representative volume element (RVE) of micro-structure is an important input for computational-mechanics-based simulations of micro-mechanical response of heterogeneous materials such as composites. In this contribution, a methodology has been developed to arrive at a sufficiently small micro-structural window that can be regarded as a RVE of a non-uniform micro-structure of a ceramic matrix composite (CMC) containing a range of fiber sizes, and fiber-rich and -poor regions at the length scale of about 100 μm. The RVE contains about 250 fibers of 14 μm diameter average size. The absolute size of the RVE is 0.1 mm 2 . The methodology involves a unique combination of quantitative characterization of geometry and spatial arrangement of micro-structural features using stereological and image analysis techniques, development of computer simulated micro-structure model that is statistically similar to the real micro-structure, finite element (FE)-based simulations of micro-mechanical response on computer-simulated micro-structural widows of different sizes containing 60–2000 fibers, and FE-based simulations on large-area high-resolution digital image of the composite micro-structure containing about 2000 fibers. The RVE has the micro-structure that is statistically similar to that of the CMC having fiber-rich and -poor regions, its Youngs modulus is very close to that of the composite, and has local stress distribution that is comparable to that in the real composite under similar loading conditions. Therefore, such an RVE is useful for realistic simulations of damage initiation.

Unbiased estimation of curve length in 3-D using vertical slices

An efficient sampling procedure is presented for estimation of total line length per unit volume Lv. It involves the following steps: (1) choose a vertical axis in the specimen, and cut the specimen to obtain VUR vertical slices of constant thickness Δ such that parallel planes of the slices contain the vertical direction; (2) observe the projected image of a vertical slice using transmission microscopy such that beam direction is perpendicular to the slice; (3) count the number of intersections of the projected images of the lineal features of interest with cycloid‐shaped test lines whose minor axis is perpendicular to the vertical axis. The expected value of the number of intersections per unit length prj is related to Lv as follows:

Computer simulation of spatial arrangement and connectivity of particles in three-dimensional microstructure: Application to model electrical conductivity of polymer matrix composite

Computer simulation is a powerful tool for analyzing the geometry of three-dimensional microstructure. A computer simulation model is developed to represent the three-dimensional microstructure of a two-phase particulate composite where particles may be in contact with one another but do not overlap significantly. The model is used to quantify the connectedness of the particulate phase of a polymer matrix composite containing hollow carbon particles in a dielectric polymer resin matrix. The simulations are utilized to estimate the morphological percolation volume fraction for electrical conduction, and the effective volume fraction of the particles that actually take part in the electrical conduction. The calculated values of the effective volume fraction are used as an input for a self-consistent physical model for electrical conductivity. The predicted values of electrical conductivity are in very good agreement with the corresponding experimental data on a series of specimens having different particulate volume fraction.

Void growth in 6061-aluminum alloy under triaxial stress state

Abstract In numerous metals and alloys, ductile fracture involves void nucleation, growth, and coalescence. In this contribution, void growth has been quantitatively characterized in an extruded 6061-wrought Al-alloy as a function of stress state in notch tensile test specimens. Digital image analysis and Stereology have been used to estimate the volume fraction and three-dimensional number density of voids in a series of interrupted notch tensile test specimens where the local stress state is predominantly triaxial. Finite elements (FE) simulations have been used to compute the stress states at different locations in the specimens. The computed stress states and experimentally estimated average void volume are utilized to verify analytical void growth models. Lack of agreement between the predictions of the models and the experimental data is due to interactions between neighboring voids, which are ignored in the theoretical models, and continuous void nucleation. The following empirical damage evolution equation is obtained from the experimental data on void volume fraction expressed as % ( f ), and the corresponding local equivalent plastic strain ( e p ) and stress triaxiality ( I ) computed from FE simulations: f = a + b ln[ e p ]+ cI . In this equation, a , b and c are empirical constants whose values depend on the alloy chemistry, heat treatment, and microstructure. The equation is useful only for 6061(T6) Al-alloy.

Effect of gravity on three-dimensional coordination number distribution in liquid phase sintered microstructures

Abstract Gravity affects microstructural evolution when a liquid phase is present during sintering. The effect of gravity on the three-dimensional coordination number distribution of tungsten grains in liquid phase sintered heavy alloy specimens is quantitatively characterized. A combination of montage serial sectioning, digital image processing, and unbiased stereological sampling procedures is used to estimate the coordination number distribution in three-dimensional microstructures. The microgravity environment decreases the mean coordination number. However, hardly any isolated grains are observed in the specimens, liquid phase sintered in a microgravity environment. The effect of microgravity on the coordination numbers mainly resides in its effect on the mean coordination number. In all specimens, there is a strong correlation between grain size and coordination number, which can be expressed as [D c / D ] 2 =C/C 0 where C0 is the mean coordination number, D the global average size of the tungsten grains, and Dc the average size of only those grains which have coordination number C.

Relationship between microstructural extremum and fracture path in a cast Al-Si-Mg alloy

Fracture related mechanical properties of materials (ductility, toughness, etc.) depend on microstructure. Materials science literature contains large number of contributions where attempts have been made to relate the average properties of microstructure (for example, average particle size, average grain size, etc.) to the mechanical properties such as ductility, toughness, etc. However, the fracture related material properties are often sensitive to the fracture path, and the relevant fracture path does not always sample the average microstructure of the material. Rather, the fracture path often senses and goes through the microstructural extrema. In such cases, the mechanical properties are governed by the extrema in the microstructural attributes, rather than the averages. It is the purpose of this contribution to illustrate an extreme case of such behavior exhibited by cast Al-Si-Mg based A356 alloy, where the fracture path of tensile test specimens primarily goes through largest 1% Si particles. The present observations provide some insights into the reasons for wide variations in the ductility of cast A356 alloy specimens having the same chemistry and the same average microstructure.

MODELING OF NON-UNIFORM SPATIAL ARRANGEMENT OF FIBERS IN A CERAMIC MATRIX COMPOSITE

Abstract In the unidirectional fiber reinforced composites, the spatial arrangement of fibers is often non-uniform. These non-uniformities are linked to the processing conditions, and they affect the properties of the composite. In this contribution, a recently developed digital image analysis technique is used to quantify the non-uniform spatial arrangement of Nicalon fibers in a ceramic matrix composite (CMC). These quantitative data are utilized to develop a six parameter computer simulated microstructure model that is statistically equivalent to the non-uniform microstructure of the CMC. The simulated microstructure can be utilized as a RVE for the micro-mechanical modeling studies.