WHEN metals are submitted to stress, the stress/swain relation is generally regarded as consisting of two parts, the elastic region and the region bfsplastiwin which a permanent set remains upon the remova of the stress. In the elastic region the absence of a permanent set does not necessarily imply that the relation between stress and strain is linear of even single-valued. Prof. Zener uses the term ‘anelasticity’ to describe the properties of solids, as a result of which stress and strain are not uniquely related. Examples are the elastic after-effect, the dependence of elastic constants on the method of measurement, and the dissipation of energy during vibration, which is often referred to as the damping capacity or internal friction of a solid. These effects have aroused much interest in recent years, largely owing to Prof. Zener's own work, and a good book on the subject is much to be desired. Elasticity and Anelasticity of Metals By Clarence Zener. Pp. x + 170. (Chicago: University of Chicago Press, London: Cambridge University Press, 1948.) 4 dollars.

Elasticity and Anelasticity of Metals

In the last twenty years, numerical modeling has become an indispensable part of magnetism research. It has become a standard tool for both the exploration of new systems and for the interpretation of experimental data. In the last five years, the capabilities of micromagnetic modeling have dramatically increased due to the deployment of graphical processing units (GPU), which have sped up calculations to a factor of 200. This has enabled many studies which were previously unfeasible. In this topical review, we give an overview of this modeling approach and show how it has contributed to the forefront of current magnetism research.

https://iopscience.iop.org/article/10.1088/1361-6463/aaab1c/pdf

Fast micromagnetic simulations on GPU—recent advances made with $\mathsf{mumax}^3$

Elastic wave scattering at grain boundaries in polycrystalline media can be quantified to determine microstructural properties. The amplitude drop observed for coherent wave propagation (attenuation) as well as diffuse-field scattering events have been extensively studied. In all cases, the scattering shows a clear dependence on grain size, grain shape, and microstructural texture. Models used to quantify scattering experiments are often developed assuming dependence on a single spatial length scale, usually, mean grain diameter. However, several microscopy studies suggest that most metals have a log normal distribution of grain sizes. In this study, grain size distribution is discussed within the context of previous attenuation models valid for arbitrary crystallite symmetries. Results are presented for titanium using a range of distribution means and widths assuming equiaxed grains and no preferred crystallographic orientation. The longitudinal and shear attenuations are shown to vary with respect to the frequency dependence for varying distribution widths even when the volumetric mean grain size is held constant. Furthermore, the results suggest that grain size estimates based on attenuation can have large errors if the distribution is neglected. This work is anticipated to play an important role in microstructural characterization research associated with ultrasonic scattering.

Ultrasonic attenuation of polycrystalline materials with a distribution of grain sizes.

There is an increasing interest in high frequency short range guided waves to screen or monitor for corrosion. This contrasts with long range guided waves (LRGWs) which screen pipes for large patches of corrosion and have been successfully used in corrosion management for the past twenty years. The fundamental setup described in this paper uses circumferential guided waves, which are excited at a single location on a pipe and travel around the pipe wall and are detected at the same location. The study uses a finite element model assisted method to evaluate the detection capability of two short range circumferential guided wave setups which use both the reflected and transmitted signals. The setups themselves consist of either an axial array of transducers, for monitoring, or a single transducer which axially scans a pipe. Both setups have an array or scan pitch between either adjacent transducers or measurements. The detection capability of the fundamental Lamb wave modes (A0 and S0) in both reflection and transmission have been compared, as well as a hybrid shear horizontal wave setup, which uses the SH0 mode in reflection and the SH1 mode in transmission. A sensitivity analysis was conducted using two separate methods to determine the probability of detection (POD) for either the reflection or transmission signals. Both methods determine a POD for a specific defect, noise level, and array or scan pitch. Probability images are produced which map the POD for a range of defect sizes. For the parameters investigated in this study, it was found that in transmission large diameter defects have a higher detectability, whereas deep, narrow diameter defects are more detectable in reflection. A generalised overview of the sensitivity of short range guided waves is presented by combining both the reflection and transmission PODs. The data fused sensitivity of the S0 and SH hybrid modes are given as 0.6% and 0.75% cross sectional area (CSA) respectively, allowing for the comparison with LRGWs. The A0 mode was excluded from the POD analysis because it was much less sensitive than the other two modes.

Detectability of corrosion damage with circumferential guided waves in reflection and transmission

The influence of a polycrystals' grain structure on elastic wave scattering is studied with analytical and numerical methods in a broad frequency range. A semi-analytical attenuation model, based on an established scattering theory, is presented. This technique accurately accounts for the grain morphology without prior assumptions on grain statistics. This is achieved by incorporating a samples' exact spatial two-point correlation function into the theory. The approach is verified by using a finite element method (FEM) to simulate P-wave propagation in 3D Voronoi crystals with equal mean grain diameter, but different grain shape uniformity. Aluminum and Inconel serve as representatives for weak and strong scattering cubic class materials for simulations and analytical calculations. It was found that the shape of the grains has a strong influence on the attenuation curve progression in the Rayleigh-stochastic transition region, which was attributed to mode conversion scattering. Comparisons between simulations and theory show excellent agreement for both materials. This demonstrates the need for accurately taking the microstructure of heterogeneous materials into account, to get precise analytical predictions for their scattering behaviour. It also demonstrates the impressive accuracy and flexibility of the scattering theory which was used.

Influence of grain morphology on ultrasonic wave attenuation in polycrystalline media with statistically equiaxed grains

The scattering treated here arises when elastic waves propagate within a heterogeneous medium defined by random spatial fluctuation of its elastic properties. Whereas classical analytical studies are based on lower-order scattering assumptions, numerical methods conversely present no such limitations by inherently incorporating multiple scattering. Until now, studies have typically been limited to two or one dimension, however, owing to computational constraints. This article seizes recent advances to realize a finite-element formulation that solves the three-dimensional elastodynamic scattering problem. The developed methodology enables the fundamental behaviour of scattering in terms of attenuation and dispersion to be studied. In particular, the example of elastic waves propagating within polycrystalline materials is adopted, using Voronoi tessellations to randomly generate representative models. The numerically observed scattering is compared against entirely independent but well-established analytical scattering theory. The quantitative agreement is found to be excellent across previously unvisited scattering regimes; it is believed that this is the first quantitative validation of its kind which provides significant support towards the existence of the transitional scattering regime and facilitates future deployment of numerical methods for these problems.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2016.0738

Finite-element modelling of elastic wave propagation and scattering within heterogeneous media

The elastodynamic behavior of polycrystalline cubic materials is studied through the fundamental propagation properties, the attenuation and wave speed, of a longitudinal wave. Predictions made by different analytical models are compared to both numerical and experimental results. The numerical model is based on a three-dimensional Finite Element (FE) simulation which provides a full-physics solution to the scattering problem. The three main analytical models include the Far-Field Approximation (FFA), the Self-Consistent Approximation (SCA) to the reference medium, and the herein derived Second Order Approximation (SOA). The classic Stanke and Kino model is also included, which by comparison to the SOA, reveals the importance of the distribution of length-scales described in terms of the two-point correlation function in determining scattering behavior. Further comparison with the FE model demonstrates that the FFA provides a simple but satisfactory approximation, whereas the SOA shows all-around excellent agreement. The experimental wave velocity data evaluated against the SOA and SC reveal a better agreement when the Voigt reference is used in second order models. The use of full-physics numerical simulations has enabled the study of wave behavior in these random media which will be important to inform the ongoing development of analytical models and the understanding of observations.

/pdf/numerical-and-analytic-modelling-of-elastodynamic-scattering-n45bulfw6g.pdf

Numerical and analytic modelling of elastodynamic scattering within polycrystalline materials.

Microstructural noise has long hindered ultrasonic NDE of polycrystalline materials. In recent years however, arrays have enabled new possibilities to advance ultrasonic inspection of these materials. A Finite Element (FE) model is used to explore the different phenomena caused by grain scattering which may hinder detection of defects by an array. These include multiple scattering and beam deviation due to anisotropy; two aspects of the physics which are often required to be ignored in analytical models due to theoretical assumptions or computational limitations. We rely on a GPU based FE solver, Pogo, to provide fast computation and thereby enable parametric studies. The impact on array detection performance when varying center-frequency and aperture size is investigated. Preliminary results show that there exists an optimum for both the array aperture and frequency for inspection of these materials.

A finite element model investigation of ultrasonic array performance for inspecting polycrystalline materials

Ultrasonic inspection of highly scattering materials presents challenges for industry. This article describes a Finite Element modelling methodology to simulate wave propagation within polycrystalline materials. Concerns are answered regarding its required mesh sampling and ability to capture the complex scattering physics. It is shown that grain scattering phenomena are closely reproduced across a range of scattering regimes. The procedure is subsequently applied to investigate the optimal configuration of an array inspecting such a material. It is found that in certain situations, separating emitter and receiver can be advantageous as this reduces the received backscatter.

/pdf/finite-element-modelling-of-wave-propagation-in-highly-1pae87huy4.pdf

Finite element modelling of wave propagation in highly scattering materials

The finite element method solved with explicit time increments is a general approach which can be applied to many ultrasound problems. It is widely used as a powerful tool within NDE for developing and testing inspection techniques, and can also be used in inversion processes. However, the solution technique is computationally intensive, requiring many calculations to be performed for each simulation, so traditionally speed has been an issue. For maximum speed, an implementation of the method, called Pogo [Huthwaite, J. Comp. Phys. 2014, doi: 10.1016/j.jcp.2013.10.017], has been developed to run on graphics cards, exploiting the highly parallelisable nature of the algorithm. Pogo typically demonstrates speed improvements of 60-90x over commercial CPU alternatives. Pogo is applied to three NDE examples, where the speed improvements are important: guided wave tomography, where a full 3D simulation must be run for each source transducer and every different defect size; scattering from rough cracks, where many simulations need to be run to build up a statistical model of the behaviour; and ultrasound propagation within coarse-grained materials where the mesh must be highly refined and many different cases run.

A. Van Pamel

Papers

Finite-element modelling of elastic wave propagation and scattering within heterogeneous media

Numerical and analytic modelling of elastodynamic scattering within polycrystalline materials.

A finite element model investigation of ultrasonic array performance for inspecting polycrystalline materials

Finite element modelling of wave propagation in highly scattering materials

High-speed GPU-based finite element simulations for NDT