Elastic Material Parameters Research Articles

Enhanced springback prediction for bending of high-strength spring steel using material data from an inverse modelling approach

Abstract Springs for electrical components made from strip materials and flat wires are produced using stamping and forming technologies. The process design of multi-stage stamping and bending processes is still based on expert knowledge and experimental tests. The complexity of the forming processes and the high-strength materials used for springs make the FE simulation of those processes a challenging task. For the springback prediction in FEM, however, an accurate description of the elasto-plastic material behaviour is mandatory. Under process conditions such as bending of spring steel, the locally occurring strains are many times higher than the ultimate elongation of approx. 2 % that can be determined in tensile test. For those materials, standard tests fail to receive enhanced material data such as the influence of plastic deformation on the Young’s modulus. The goal of this work is to determine the elasto-plastic material parameters of high-strength spring steel (X10CrNi18-8) using an inverse modelling approach by means of a 3-point bending test. The inverse approach is used for determining the Young’s modulus and hardening parameters of Ludwik-Hollomon’s law for bending of high-strength spring steel. Furthermore, this work focuses on the unloading at the end of the bending experiment and its effect on the elastic material parameters. The results demonstrate that the inversely determined material data show lower scattering compared to material data from tensile test. Furthermore, material data of high-strength spring steel from the inverse modelling approach significantly enhances the springback prediction under bending conditions compared to FE simulations using material data from tensile test.

Computational studies of shape control of charged deformable nanocontainers.

Biological matter is often compartmentalized by soft membranes that dynamically change their shape in response to chemical and mechanical cues. Deformable soft-matter-based nanoscale membranes or nanocontainers that mimic this behavior can be used as drug-delivery carriers that can adapt to evolving physiological conditions, or as dynamic building blocks for the design of novel hierarchical materials via assembly engineering. Here, we connect the intrinsic features of charged deformable nanocontainers such as their size, charge, surface tension, and elasticity with their equilibrium shapes for a wide range of solution conditions using molecular dynamics simulations. These links identify the fundamental mechanisms that establish the chemical and materials design control strategies for modulating the equilibrium shape of these nanocontainers. We show that flexible nanocontainers of radii ranging from 10-20 nm exhibit sphere-to-rod-to-disc shape transitions yielding rods and discs over a wide range of aspect ratio λ (0.3 < λ < 5). The shape transitions can be controlled by tuning salt and/or surfactant concentration as well as material elastic parameters. The shape changes are driven by reduction in the global electrostatic energy and are associated with dramatic changes in local surface elastic energy distributions. To illustrate the shape transition mechanisms, exact analytical calculations for idealized spheroidal nanocontainers in salt-free conditions are performed. Explicit counterion simulations near nanocontainers and associated Manning model calculations provide an assessment of the stability of observed shape deformations in the event of ion condensation.

Assessment of Elastic–Plastic and Electrical Properties of Printed Silver-Based Interconnects for Flexible Electronics

In this work, the elastic–plastic properties of the printed interconnects on a glass substrate with Ag-filled polymer-conductor ink are evaluated through a theoretical framework based on finite element (FE) modeling of instrumented sharp indentation, experimental indentation, the concept of the representative strain, and dimensional analysis. Besides, the influences of the ink-solvent content and temperature on the elastic–plastic and electrical properties of the printed Ag-based interconnects are also addressed. First of all, parametric FE indentation analyses are carried out over a wide range of elastic–plastic material parameters. These parametric results together with the concept of the representative strain are used via dimensional analysis to constitute a number of dimensionless functions, and further the forward/reverse algorithms. The forward algorithm is used for describing the indentation load–depth relationship and the reverse for predicting the elastic–plastic parameters of the printed Ag-based interconnects. The proposed algorithms are validated through the correct predictions of the plastic properties of three known metals. At last, their surface morphology, microstructure, and elemental composition are experimentally characterized. Results show that the elastic–plastic properties and electrical sheet resistance of the printed Ag-based interconnects increase with the ink-solvent content, mainly due to the increase of carbon element as a result of the increased ink-solvent residue, whereas their elastic–plastic properties and electrical performance decreases with the temperature.

Identification of mesoscale model parameters for brick-masonry

Realistic assessment of existing masonry structures requires the use of detailed nonlinear numerical descriptions with accurate model material parameters. In this work, a novel numerical-experimental strategy for the identification of the main material parameters of a detailed nonlinear brick-masonry mesoscale model is presented. According to the proposed strategy, elastic material parameters are obtained from the results of diagonal compression tests, while a flat-jack test, purposely designed for in-situ investigations, is used to determine the material parameters governing the nonlinear behaviour. The identification procedure involves: a) the definition of a detailed finite element (FE) description for the tests; b) the development and validation of an efficient metamodel; c) the global sensitivity analysis for parameter reduction; and d) the minimisation of a functional representing the discrepancy between experimental and numerical data. The results obtained by applying the proposed strategy in laboratory tests are discussed in the paper. These results confirm the accuracy of the developed approach for material parameter identification, which can be used also in combination with in-situ tests for assessing existing structures. Practical and theoretical aspects related to the proposed flat-jack test, the experimental data to be considered in the process and the post-processing methodology are critically discussed.

Rheological behavior of wood in stress relaxation under compression

Rheological behavior of wood under uniaxial compression along and perpendicular to the grain in constant environment was examined. Tests with constant deformation rate until failure and stress relaxation tests with constant deformation applied stepwise were carried out. The experimental results of stress relaxation showed nonlinear material behavior over time that got more prominent under high deformation levels. Considerable amount of stress relaxed during applying the deformation. Wood experienced greater stress relaxation along the grain than perpendicular to it. Three rheological models for orthotropic material were calibrated to the experimentally determined stress–time curves in longitudinal and transverse directions simultaneously. Small deformation levels assuming linear strains were accounted for in the models. Required elastic material parameters were determined from the tests with constant deformation rate. A model including the highest number of viscoelastic material parameters was the most successful in predicting stress relaxation of wood under stepwise deformation. Modeling indicated that wood behavior was very close to linear viscoelastic in relaxation under small deformation. The obtained material parameters made the model suitable for predicting rheological behavior of wood comprehensively, under sustained deformation or load in constant conditions.

Identifying Elastic and Viscoelastic Material Parameters by Means of a Tikhonov Regularization

For studying the interaction of displacements, stresses, and acting forces for elastic and viscoelastic materials, it is of utmost importance to have a decent mathematical model available. Usually such a model consists of a coupled set of nonlinear differential equations together with appropriate boundary conditions. However, since the different material classes vary significantly with respect to their physical and mechanical behavior, the parameters which appear in these equations are unknown and therefore have to be determined before the equations can be used for further investigations or simulations. It is this very step which is addressed in this article where we consider elastic as well as viscoelastic material behavior. The idea is to compute the parameters as solutions of a minimization problem for Tikhonov functionals. Tikhonov regularization is a well-established solution technique for tackling inverse problems. On the one hand, it assures a computation that is stable with respect to noisy input data, and on the other hand, it involves desired a priori information on the solution. In this article we develop problem adapted Tikhonov functionals and prove that a Tikhonov regularization improves the accuracy especially when the underlying system is ill-conditioned.

Influence of grain direction on the time-dependent behavior of wood analyzed by a 3D rheological model. A mathematical consideration

AbstractA three-dimensional (3D) rheological model for an orthotropic material subjected to sustained load or deformation under constant climate has been mathematically formulated. The elastic and viscoelastic compliance matrices are symmetric, where the mathematical derivation of the latter is shown. The model is linear and requires constant numerical values for the elastic and viscoelastic material parameters. The model’s ability to predict the natural time-dependent response in three material directions simultaneously is demonstrated on a Douglas fir (Pseudotsuga menziesii) specimen subjected to a constant uniaxial tensile load. The material extends in a longitudinal direction and contracts in the transverse directions with time. The required material parameters are taken from the literature when possible, otherwise they are assumed. Furthermore, the influence of misalignment between the directions of observation and wood material directions on induced time-dependent strains is analyzed. The analyses show that the misalignment has a large effect on the material behavior. In some cases, the specimen under constant uniaxial tension even extends in the perpendicular transverse direction with time. The obtained results clearly demonstrate the high importance of considering the alignment of material directions precisely in order to be able to interpret the time-dependent behavior of wood correctly.

An Elasticity Solution for Vibration Analysis of Laminated Plates with Functionally Graded Core Reinforced by Multi-walled Carbon Nanotubes

In the present work, vibration characteristics of functionally graded (FG) sandwich rectangular plates reinforced by multiwalled carbon nanotubes (MWCNTs) resting on Pasternak foundation are presented. The response of the elastic medium is formulated by the Winkler/Pasternak model. Modified Halpin-Tsai equation is used to evaluate the Young’s modulus of the MWCNT/epoxy composite samples by the incorporation of an orientation as well as an exponential shape factor in the equation. The mass density and Poisson’s ratio of the MWCNT/phenolic composite are considered based on the rule of mixtures. The proposed sandwich rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The effects of two-parameter elastic foundation modulus, geometrical and material parameters together with the boundary conditions on the frequency parameters of the sandwich plates are investigated.

Elastic properties of superconductors and materials with weakly correlated spins

It is shown that in the ergodic regime, the temperature dependence of Young’s modulus is solely determined by the magnetic properties of a material. For the large class of materials with paramagnetic or diamagnetic response, simple functional forms of the temperature derivative of Young’s modulus are derived and compared with experimental data and empirical results. Superconducting materials in the Meissner phase are ideal diamagnets. As such, they display remarkable elastic properties. Constant diamagnetic susceptibility gives rise to a temperature independent elastic modulus for ceramic and single crystalline superconductors alike. The thermodynamic approach established in this report, paves the way to tailor elastic material parameters through the design of magnetic properties.

Analysing the large amplitude oscillatory shear response of a bingham-hooke model in pipkin space and by fourier Transform rheology to predict weak strain overshoots

The work presented in this paper contributes to theory behind the modelling of strain-controlled large amplitude oscillatory shear (LAOS) with weak strain overshoot. The latter term refers to a decreasing strain amplitude dependent storage modulus and a peak in loss modulus. These physical nonlinearities are restricted in this work in order to remain within the geometrically linear theory measured for an inertial system. It is also premised that yielding causes weak strain overshoot. Yielding is modelled with the aid of the Bingham–Hooke model consisting of a Hookean spring connected in series to a Bingham model. In addition, the Bingham–Hooke model is connected in parallel to an rCross model, the LAOS response of which was investigated with the aid of FT rheology in [Boisly et al., Journal of Non-Newtonian Fluid Mechanics 225 (2015) 10–27]. The LAOS behaviour of the Bingham–Hooke model is analysed in this work using Fourier Transform (FT) rheology and Lissajous plots in Pipkin space. This approach facilitates the identification of so-called characteristic points of FT harmonics. These are distinct points – for example maximum loss modulus – easy to determine experimentally and can therefore be used for the purposes of parameter identification. Analytical methods are applied to relate material parameters to characteristic points. It is therefore possible to identify material parameters deterministically without minimising residuals with the aid of nonlinear optimisation algorithms. This insight enables the authors to analytically model nonlinear phenomena based on the characteristic points of FT harmonics. The entire set of elastic, viscous and plastic material parameters – η0, η∞, K1, G, η and yield stress τy – are identified by evaluating the characteristic points of the storage and loss modulus.

The effect of carbon nanotubes agglomeration on vibrational response of thick functionally graded sandwich plates

In the present work, by considering the agglomeration effect of single-walled carbon nanotubes, free vibration characteristics of functionally graded (FG) nanocomposite sandwich plates resting on Pasternak foundation are presented. The volume fractions of randomly oriented agglomerated single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. To determine the effect of CNT agglomeration on the elastic properties of CNT-reinforced composites, a two-parameter micromechanical model of agglomeration is employed. In this research work, an equivalent continuum model based on the Eshelby-Mori-Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented straight CNTs. The 2-D generalized differential quadrature method (GDQM) as an efficient and accurate numerical tool is used to discretize the equations of motion and to implement the various boundary conditions. The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The benefit of using the considered power-law distribution is to illustrate and present useful results arising from symmetric and asymmetric profiles. The effects of twoparameter elastic foundation modulus, geometrical and material parameters together with the boundary conditions on the frequency parameters of the laminated FG nanocomposite plates are investigated. It is shown that the natural frequencies of structure are seriously affected by the influence of CNTs agglomeration. This study serves as a benchmark for assessing the validity of numerical methods or two-dimensional theories used to analysis of laminated plates.

Free vibration of four-parameter functionally graded moderately thick doubly-curved panels and shells of revolution with general boundary conditions

This paper aims to present a unified vibration analysis approach for the four-parameter functionally graded moderately thick doubly-curved shells and panels of revolution with general boundary conditions. The first-order shear deformation theory is used in this formulation. The functionally graded panels structures consists of ceramic and metal which are set to vary continuously in the thickness direction according to the general four-parameter power-law distribution, and six types of power-law distributions are considered for the ceramic volume fraction. The admissible function of the FG panels and shells of revolution is obtained by the improved Fourier series with the help of the governing equations and the boundary conditions. The solution is obtained by using the variational operation in terms of the unknown expanded coefficients. By a great many numerical examples, the rapid convergence and good reliability and accuracy of the proposed approach are validated. A variety of new results for vibration problems of the FG doubly-curved shells and panels with different elastic restraints, geometric and material parameters are presented. The effects of the elastic restraint parameters, power-law exponent, circumference angle and power-law distributions on the free vibration characteristic of the panels are also presented, which can be served as benchmark data in the research and the actual production process.

Extraction of nonlinear elastic material parameters from single-impact nonlinear ring-down spectroscopy

Volumetric microcracking is one of the early symptoms of distress in cementitious materials caused by excessive mechanical stress, chemical attacks, and environmental influences. The microcracks widen, coalesce, and develop into larger cracks with the progress of damage. Visible macro-cracks indicate severe damage that often cannot be mitigated. As such, detection of damage at the early stages of development is essential for designing optimal preventive maintenance programs for concrete structures. Nonlinear acoustics-based nondestructive testing techniques have shown great promise in identification of microscopic cracks in diverse materials including concrete. Impact-based alternatives of conventional techniques are gaining popularity for concrete testing mainly due to their field transportability. In this study, we focus on impact-based nonlinear resonance acoustic spectroscopy (INRAS). We compare the results from multi-impact INRAS, where several impacts of increasing intensities are applied, to those from single impact ring-down spectroscopy where only one impact of large intensity is used. Hilbert Huang Transform (HHT) is used to obtain the time-dependent frequency content of the single-impact ring-down. We propose several models for fitting the entire ring-down in order to extract nonlinear elastic material parameters. We demonstrate that our proposed approach for analyzing single-impact INRAS data yields material parameters compatible with those obtained from conventional testing and analysis.

A density-dependent modified Drucker-Prager Cap model for die compaction of Ag57.6-Cu22.4-Sn10-In10 mixed metal powders

Due to the bad plastic workability, silver based brazing filler metals with high proportions of additional elements such as Zn, Sn, In, Ga, are very brittle and difficult to be made into thin sheets using conventional manufacturing processes. A novel process method that combine powder compaction and sintering, can fabricate thin sheets of the fragile silver based filler metals. In this work, the densification mechanism of Ag57.6-Cu22.4-Sn10-In10 (wt.%) mixed metal powders was investigated, and a modified density-dependent Drucker-Prager Cap (DPC) model was introduced to describe its compaction behavior based on the equivalent density method. The powder compact was considered as a continuum, and a linear elasticity law as a function of relative density was then used to express the elastic behavior of the powders. An instrumented die with force transducers has been designed to determine the material parameters of the modified DPC model. The elastic and plastic material parameters such as E, υ, β, d, pa, R and pb were then determined experimentally. The friction coefficient between the powders and the die wall with a lubricated die was determined based on the Janssen–Walker theory in the instrumented die experiments. Furthermore, the modified DPC model with the linear elasticity law was validated using finite element simulation method in ABAQUS with a user subroutine (USDFLD). A good agreement between the simulations and experimental results was obtained, indicating the feasibility and applicability of the introduced modified models to describe the die compaction behavior of the mixed metal powders.

Effectiveness of Coatings With Constant, Linearly, and Exponentially Varying Elastic Parameters in Heavily Loaded Line Elastohydrodynamically Lubricated Contacts

In contacts of functionally graded elastic solids, the conditions produced are significantly different from the ones in similar contacts of homogeneous elastic materials. Especially it is true for heavily loaded lubricated contacts. The situation is even more diverse due to different dependences of the material elastic parameters on material depth. In the previous papers, the cases of lubricated contacts with coatings made of homogeneous materials are considered using asymptotic and semi-analytical methods. The goal of the paper is to determine the behavior of the coating effectiveness criteria in heavily loaded elastohydrodynamically lubricated (EHL) contacts for coatings with elastic modulus varying linearly and exponentially across the coating thickness as well as to compare the results with the case of coatings made of homogeneous materials. The above criteria include the criteria on the lubrication film thickness and friction force. The approach used for analyzing the influence of functionally graded elastic materials on parameters of heavily loaded line EHL contacts is based on the asymptotic methods earlier developed by the authors. The analysis is based on splitting the problem into two distinct parts: the problem for dry (nonlubricated) contacts and a problem for lubricated contacts. The bridge between the two problems is the asymptotic behavior of pressure in the vicinity of the end points of the contacts. More specifically, in the central part of the contact the solution of the EHL problem for functionally graded materials is close to the one for the dry contact of these materials while in the narrow zones near the inlet and exit points of the contact the lubrication effects become comparable to the effects due to the elasticity of the solids. This approach to the EHL problem solution reveals its structure.

Identification of Elastic Orthotropic Material Parameters by the Singular Boundary Method

AbstractThis article addresses the resolution of the inverse problem for the parameter identification in orthotropic materials with a number of measurements merely on the boundaries. The inverse problem is formulated as an optimization problem of a residual functional which evaluates the differences between the experimental and predicted displacements. The singular boundary method, an integration-free, mathematically simple and boundary-only meshless method, is employed to numerically determine the predicted displacements. The residual functional is minimized by the Levenberg-Marquardt method. Three numerical examples are carried out to illustrate the robustness, efficiency, and accuracy of the proposed scheme. In addition, different levels of noise are added into the boundary conditions to verify the stability of the present methodology.

Inverse detection of constituent level elastic parameters of FRP composite panels with elastic boundaries using finite element model updating

Fiber reinforced plastics (FRP) panels are extensively used in ship structures due to their superior specific stiffness and strength as compared to metals. However, their prolonged use may result in degradation of material properties as well as the boundary conditions; thus, affecting the dynamic performance. Moreover, the changes in material properties are mostly at constituent level, i.e. fiber or matrix. A vibration based inverse identification technique is proposed using finite element model updating to estimate the constituent elastic material parameters of FRP panels having elastically restrained boundary. The objective function is formed from the difference of experimental as well as finite element prediction of dynamic responses. A gradient based optimization viz. the inverse eigensensitivity method is implemented. A set of numerically simulated examples is presented to demonstrate that the prediction of material parameters can be grossly erroneous if the boundary elasticity is overlooked. The algorithm is found to be robust even when the ‘experimental’ data is sparse and contains random noise. The method can be used for condition assessment and damage detection of FRP ship panels. The technique is novel as for the first time constituent level elastic parameters of FRP panels having elastic boundaries are estimated from dynamic responses.

Non‐linear Stochastic Finite Element

AbstractThis paper investigates the uncertainty of physically non‐linear problems by modeling the elastic random material parameters as stochastic fields. For its stochastic discretization a polynomial chaos (PC) is used to expand the coefficients into deterministic and stochastic parts. Then, from experimental data for an adhesive material the distribution of the random variables, i.e. Young's modulus E(θ), the static yield point Y0 and the nonlinear hardening parameters q and b, are known. In the numerical example the distribution of the stresses obtained by the PC based SFEM and Monte Carlo simulation is compared. (© 2015 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Free vibration and vibrational displacements analysis of thick elastically supported laminated curved panels with power-law distribution functionally graded layers and finite length via 2D GDQ method

This paper deals with free vibration and vibrational displacements analysis of thick laminated curved panels with finite length resting on two-parameter elastic foundations based on the three-dimensional elasticity theory. Because of using two-dimensional generalized differential quadrature method, the present approach makes possible vibration analysis of cylindrical panels with two opposite axial edges simply supported and arbitrary boundary conditions including free, simply supported and clamped at the curved edges. The material properties vary continuously through the layers’ thickness according to a three-parameter power-low distribution. It is assumed that the inner surfaces of the functionally graded sheets are metal rich, while the outer surfaces of the layers can be metal rich, ceramic rich or made of a mixture of two constituents. The benefit of using the considered power-law distribution is to illustrate and present useful results arising from symmetric and asymmetric profiles. The effects of two-parameter elastic foundation modulus, geometrical and material parameters together with the boundary conditions on the frequency parameters of the laminated functionally graded panels are investigated. The obtained results show that the outer functionally graded material layers have significant effect on the vibration behavior of cylindrical panels. This study serves as a benchmark for assessing the validity of numerical methods or two-dimensional theories used to analysis of laminated curved panels.

Elastic parabolic equation solutions for oceanic T-wave generation and propagation from deep seismic sources.

Oceanic T-waves are earthquake signals that originate when elastic waves interact with the fluid-elastic interface at the ocean bottom and are converted to acoustic waves in the ocean. These waves propagate long distances in the Sound Fixing and Ranging (SOFAR) channel and tend to be the largest observed arrivals from seismic events. Thus, an understanding of their generation is important for event detection, localization, and source-type discrimination. Recently benchmarked seismic self-starting fields are used to generate elastic parabolic equation solutions that demonstrate generation and propagation of oceanic T-waves in range-dependent underwater acoustic environments. Both downward sloping and abyssal ocean range-dependent environments are considered, and results demonstrate conversion of elastic waves into water-borne oceanic T-waves. Examples demonstrating long-range broadband T-wave propagation in range-dependent environments are shown. These results confirm that elastic parabolic equation solutions are valuable for characterization of the relationships between T-wave propagation and variations in range-dependent bathymetry or elastic material parameters, as well as for modeling T-wave receptions at hydrophone arrays or coastal receiving stations.