Size Effect Model Research Articles

Phase-field modeling of multivariant martensitic microstructures and size effects in nano-indentation

A finite-strain phase-field model is developed for the analysis of multivariant martensitic transformation during nano-indentation. Variational formulation of the complete evolution problem is developed within the incremental energy minimization framework. Computer implementation is performed based on the finite-element method which allows a natural treatment of the finite-strain formulation and of the contact interactions. A detailed computational study of nano-indentation reveals several interesting effects including the pop-in effect associated with nucleation of martensite and the energy-lowering breakdown of the symmetry of microstructure. The effect of the indenter radius is also examined revealing significant size effects governed by the interfacial energy.

Plastic deformation behavior of a nickel-based superalloy on the mesoscopic scale

Nickel-based superalloys have become the key materials of micro-parts depending on excellent mechanical properties at high temperatures. The plastic deformation behavior is difficult to predict due to the occurrence of size effect on the mesoscopic scale. In this paper, the effect of specimen diameter to grain size ratio (D/d) on the flow stress and inhomogeneous plastic deformation behavior in compression of nickel-based superalloy cylindrical specimens was investigated on the mesoscopic scale. The results showed that when D/d is less than 9.7, the flow stress increases with the grain size. Aiming at this phenomenon, a flow stress size effect model considering compressive strain partitioning was established. The calculated flow stress values agree well with the experimental values, thus revealing the effect of D/d on the flow stress in compression of nickel-base superalloy on the mesoscopic scale. The inhomogeneous plastic deformation during compression deformation increases with the grain size. The end surface profiles evolve from a regular circular shape to an irregular shape with the grain size. The surface folding phenomenon occurs only in partially compressed specimen with a few grains across the diameter. Crystal plasticity finite-element (CPFE) simulation of compression deformation on the mesoscopic scale real-time displayed the evolution of microstructure. The study of this paper has important guiding significance for understanding the influence of D/d on the compression deformation behavior of nickel-based superalloy on the mesoscopic scale.

Radiobiological modelling of clonogen distribution, hypoxic fraction and tumour size effects on local tumour control of non-small cell lung cancer

Published clinical data show that hypoxia in human lung tumours can impede the establishment of optimum local tumour control. However, the overall effect of hypoxia on the tumour control probability (TCP) model is not clear. The focus of this project was to assess the influence of radiobiological parameters (the number of clonogens and the hypoxic fraction), as well as some treatment parameters (i.e., the tumour size), on local tumour control of early stage non-small cell lung cancer (NSCLC). A TCP model, based on LQ cell survival concept combined with the Poisson statistic, was established to predict one, two and three years of local tumour control. This TCP model was created using data from seventeen publications of early-stage NSCLC treated using one of the three radiotherapy modalities: three-dimensional conformal radiation therapy (3D-CRT), continuous hyperfractionated accelerated radiotherapy (CHART) or stereotactic ablative body radiotherapy (SABR). The variations in the TCP with the gross tumour volume (GTV) size, clonogen number and hypoxic fraction were then investigated. This issue was approached by varying the clonogen densities values (between 10̂1 and 10̂7 cm̂3), the GTV volume (20–140 cc) and the hypoxic fraction (20–90%). The optimum values used to compute the TCP model were a clonogen density of 10̂7 cm̂3 and a hypoxic fraction of 20%, which were consistent with the clinical outcome values reported in the literature for NSCLC. This radiobiological model has demonstrated the proof of concept that poor local tumour control is strongly associated with the hypoxic fraction and large tumours.

Modeling of anisotropic hardening and grain size effects based on advanced numerical methods and crystal plasticity

Modeling of anisotropic behavior as well as hardening behavior based on micromechanical quantities in combination with a spectral solver is the focus of this study. A deep drawing steel as well as two different aluminum alloys are investigated. Prediction capabilities of the proposed modeling strategy are discussed and the benefits of the micromechanical model are highlighted. Further, a comparison of the crystal plasticity (CP) results with the well established macroscopic model YLD2000-2d underlines the importance of the CP as a complementary modeling technique to the macroscopic modeling. Both models – the microscopic as well as the macroscopic – are validated on experimental data mainly gained from uniaxial and biaxial tests. In the second part of this study, strong inhomogeneous microstructures are investigated from a modeling point of view. For this purpose, a Hall–Petch phenomenological model is implemented in the CP open-source code DAMASK to take the grain size effects into account. Appropriate combinations of the grain sizes in a bimodal microstructure are presented in order to increase the strength as well as ductility of a generic aluminium alloy. The proposed numerical strategy of coupling the CP and efficient FFT-based spectral solver supports the development of new materials in an optimal way.

A simple model for size effects in constrained shear

Simple shear of a ductile layer constrained between two rigid solids is analyzed using a conventional one dimensional strain gradient plasticity formulation. The aim of the analyses is to gain insight into the scaling relations and size effects seen in the experiments of Mu et al.(2014, 2016, 2017). Our analysis presumes that the deviation from previous predictions based on conventional strain gradient plasticity theory is not due to a defect in the theory but is due to a change in boundary condition that takes place when a sufficiently large plastic shear strain gradient develops at the layer boundaries. Predictions based on this presumption are qualitatively, and in some aspects, quantitatively consistent with a range of the observations.

Transferable screened range-separated hybrids for layered materials: The cases ofMoS2and h-BN

Screened range-separated hybrid (SRSH) functionals are of potential interest as a computationally inexpensive yet accurate alternative approach for studying (opto)electronic properties in the solid state. At present, SRSH functionals are typically tuned to reproduce with high accuracy the properties of either bulk or low-dimensional structures, rendering such functionals not only material specific, but also structure specific. The transferability of tuned SRSH functionals between bulk and low-dimensional phases has not been examined systematically and is desirable for straightforward and consistent modeling of size effects in nanostructures. We present a simple yet effective approach for simultaneous tuning of the fraction of short-range exact exchange and the range-separation parameter, which delivers accurate and transferable SRSH functionals for two-dimensional and bulk phases of two prototypical layered materials, molybdenum disulfide (${\mathrm{MoS}}_{2}$) and hexagonal boron nitride (h-BN). The ground-state SRSH band structures, resulting from minimal fitting of the SRSH to a single quasiparticle energy, are found to be in excellent agreement with GW calculations over the entire Brillouin zone. Excited-state properties are predicted using time-dependent density functional theory calculations, based on the SRSH (TD-SRSH) functional. Calculated absorption spectra are found to be in excellent agreement with GW and Bethe-Salpeter equation (BSE) calculations for ${\mathrm{MoS}}_{2}$, but less so for h-BN; the failure of the TD-SRSH for the latter material is examined and a BSE approach based on the SRSH ground state is shown to restore accuracy.

Effects of Thickness and Crystallographic Orientation on Tensile Properties of Thinned Silicon Wafers

Thinning silicon wafers for stacking in limited space is essential for the 3-D integration (3DI) technology of semiconductors. Due to the lack of research on the mechanical properties of thinned silicon wafers, it is difficult to assess and improve the mechanical reliability of 3DI semiconductor devices. This article reports the effects of thickness and crystallographic orientation on the tensile properties, such as Young’s modulus, elongation, and strength, of the thinned silicon wafer. Tensile properties of a {100} silicon wafer are measured using a direct tensile testing system, where a digital image correlation method is adopted for accurate strain measurement. Femtosecond laser patterning for accurate shape control is used to fabricate dog-bone-shaped specimens with various thicknesses and crystallographic orientations. The effect of crystallographic orientation is investigated for $\langle 110\rangle $ , $\langle 320\rangle $ , $\langle 210\rangle $ , and $\langle 100\rangle $ orientations. The Young’s modulus of each orientation closely matches the theory of anisotropic elasticity. The surface energy ratios between crystallographic planes are calculated using fracture mechanics analysis. As the thickness decreases from 100 to $10~\mu \text{m}$ , the elongation and strength increase threefold, while Young’s modulus is constant along the $\langle 110\rangle $ direction. The strength results are analyzed with a Weibull statistical size effect model, where the Weibull modulus is calculated to be 2.35, which correlates strength only with thickness variation. Using this value and the Weibull size effect model, the expected strength of specific thickness can be calculated easily without additional experiments.

Conductivity Size Effect of Square Cross-Section Polycrystalline Nanowires.

A theoretical model for the electrical conductivity size effect of square nanowires is proposed in this manuscript, which features combining the three main carrier scattering mechanisms in polycrystalline nanowires together, namely, background scattering, external surface scattering, as well as grain boundary scattering. Comparisons to traditional models and experiment data show that this model achieves a higher correlation with the experiment data.

A Surface Transition Layer Model for Size Effect in T2 Copper Micro-Compression

The effects of sample size and grain size on the surface morphology and flow stress of deformed samples were investigated by means of copper micro-cylinder compression experiments at room temperature. The results of SEM showed that when the grain size increased or the sample size decreased, the deformation non-uniformity of samples’ free surfaces increased. Meanwhile, the stress–strain curves showed that during the compression process, the flow stress of the sample also tended to decrease as the grain size increased or the sample size decreased. According to the experimental results of nanoindentation, a surface transition layer model was established on the basis of the surface layer model by considering the mutual constraint of grains and the existence of transition layer grains. The experimental results indicated that the stress–strain curve calculated by the surface transition layer model can more accurately reflect the actual deformation situation of the material compared to the surface layer model.

Probabilistic modelling of notch fatigue and size effect of components using highly stressed volume approach

Modeling of the notch and size effects on fatigue behavior of materials is vital for ensuring structural integrity and reliability of engineering components. This study presents a methodology considering both effects of notch and size to analyze the fatigue life distribution of specimens with different geometries using the highly stressed volume approach. Specifically, a dynamic model coefficient considering the influence of different maximum local stresses is developed by modeling the size effect of highly stressed volumes with Weibull distribution. Experimental data of three aluminum and titanium alloys are utilized for model validation and comparison. Fatigue lives of three materials with different geometries are evaluated respectively, and predicted P-S-N curves indicate that proposed model predictions agree well with the probabilistic scatter band of experimental results.

Probabilistic modeling of fatigue life distribution and size effect of components with random defects

Engineering components made of ductile cast irons and aluminum alloys, show fatigue lives which are normally dominated by crack initiation from defects raised by manufacturing processes. This study presents a probabilistic model to account for the influence of manufacturing defects on fatigue life, based on size and position of those defects. Specifically, a correction factor considering the influence of defect surface position is developed by modeling the damage mechanism of surface initial cracks with Weibull distribution. Experimental data of three cast irons and aluminum alloys are used for model validation and comparison. Moreover, the statistical size effect influence on fatigue life distribution under constant amplitude loading is explored. Fatigue lives of three materials with different sizes are evaluated respectively, and P–S–N diagrams show that proposed model predictions agree well with the probabilistic scatter bands.

An analytical model for the flat punch indentation size effect

An analytical approximation for the indentation size effect (ISE) due to plane strain flat punch nanoindentation is derived. The flat punch ISE differs from that observed for self-similar (pointed) and spherical indenters in a number of ways: (1) the contact area does not change; (2) the contact pressure depends on two length scales not just one (the punch width and the indentation depth); (3) the profile of the punch is not differentially continuous, resulting in singular plastic strain gradients at the sharp edges, such that (4) the shape and connectivity of the plastic zones change with indentation depth and punch width, resulting in (5) changes in the proportion of the deformation accommodated by elasticity and plasticity are important, meaning that a fully elastoplastic model is required. Complete loading-unloading curves are modelled, with the calibration of geometrical parameters from finite element strain gradient plasticity simulations. As the punch width decreases, it is observed that there are increases in the indentation pressure, the relative size of the plastic zone(s) and the elastic component of the deformation. These predictions are found to compare favourably with experimental measurements in the literature. The model is extended to incorporate the consequence of imposing natural limitations on the maximum dislocation density at the edges. It is suggested that observable changes in the plastic zone morphology with the ISE make this an experimentally interesting area for the validation of size effects in plasticity.

Evaluation of 3D Embedded Element Technique in the Finite Element Analysis for the Composite

RVE combined with finite element analysis (FEA) is a very popular method to predict the mechanical property of the composite reinforced by short fibers. In the conventional method, generally the “tie” approach is used. By this method, the FE model with high fiber aspect ratio can not be achieved and the non-convergence of the numerical calculation may appear because of the complex mesh. The embedded element techinique is considered to be a replaceable method . Using this method, the mechanical behavior of composite with high fiber aspect ratio would be simulated. Therefore, in this study, the 3D solid element was employed for the FE model with multi cylinder particles. The comparisions of the Mise stress and the displacement between the embedded and conventional method indicate that compared with the stress transfer, the simulated result of composite stiffness is more accurate. In addition, the effects of model size, fiber orientated angle, fiber volume fraction and fiber aspect ratio were investigated. The numerical results were compared with the Mori-Tanaka model and the good agreements verify the applicability of the embedded element technique we studied in this paper.

Resistivity and surface scattering of (0001) single crystal ruthenium thin films

The resistivity size effect in nanoscale metals is of both scientific and technological interest, the latter due to its importance to interconnects between transistors in integrated circuits. In this work, the authors report the variation of resistivity with film thickness and with changes in surface scattering of ex situ annealed single crystal Ru thin films grown on sapphire substrates by sputter deposition. The room temperature deposition of SiO2 on the Ru sample surface was observed to increase the resistivity of films that had previously been subjected to annealing in a reducing gas ambient. These overcoated samples were also found to increase in resistivity as a result of an oxidizing anneal and reduce in resistivity as a result of a subsequent reducing gas (Ar + H2) anneal. To a large extent, the surface structure and electron scattering characteristics were found to be reversible between oxidizing and reducing gas anneals. The chemistry and structure of the Ru upper surface was characterized by low energy electron diffraction (prior to the SiO2 overcoat deposition), x-ray reflectivity, x-ray photoelectron spectroscopy, and resistivity measurements. The changes in surface structure and chemistry were related to the changes in the specularity of the Ru surface for electron scattering in the context of the Fuchs–Sondheimer semiclassical model of the resistivity size effect, and in this context a mostly specular metal/dielectric interface is reported.

Predictive model for minimum chip thickness and size effect in single diamond grain grinding of zirconia ceramics under different lubricating conditions

Abstract To address the current bottleneck of debris formation mechanism in plastic removal for hard-brittle materials, a minimum chip thickness ( h min ) model that considers lubrication conditions (represented by frictional angle β ) is developed according to strain gradient, as well as geometry and kinematics analyses. Model results show that h min decreases with increasing β . Furthermore, grinding experiments using single diamond grain under different lubricating conditions are carried out to verify the model. With increasing β , h min values are 71.6, 57.8, 52.0, 50.7, 45.6, 39.7, and 32.4 nm, thereby verifying the trend of h min decreasing with increasing β . Furthermore, the location of size effect occurs is determined according to the variation trend of single abrasive particle specific energy and unit grinding force curves. The size effect occurs in the border area of ploughing, the cutting region, and mainly, in the ploughing region. Theoretical analysis results are consistent with experimental results with a model error of 6.06%, thereby confirming the validity of the theoretical model.

A model for the indentation size effect in polycrystalline alloys coupling intrinsic and extrinsic length scales

Abstract

Numerical modeling of size effect in shear strength of FRP-reinforced concrete beams

This study presents the numerical modeling of size effect in shear strength of FRP-reinforced concrete (RC) beams with and without stirrups by finite element analysis (FEA) model. The size effect was incorporated in the model by controlling the mesh size, softening tensile stress-strain response, and dilation angle. A new modification was suggested to represent the softening tensile strength of concrete for different size of beams. The calibrated FEA model for beams with and without stirrups showed great power and ability in simulating the experimental results in terms of ultimate shear capacity, ultimate deflection, flexural and stirrup strains, and crack patterns. Additionally, the calibrated model was then verified by thirteen experimental results from four different studies, as well as a parametric study was performed.

Determination of Fracture Properties of Concrete Using Size and Boundary Effect Models

Tensile strength and fracture toughness are two essential material parameters for the study of concrete fracture. The experimental procedures to measure these two fracture parameters might be complicated due to their dependence on the specimen size or test method. Alternatively, based on the fracture test results only, size and boundary effect models can determine both parameters simultaneously. In this study, different versions of boundary effect models developed by Hu et al. were summarized, and a modified Hu-Guan’s boundary effect model with a more appropriate equivalent crack length definition is proposed. The proposed model can correctly combine the contributions of material strength and linear elastic fracture mechanics on the failure of concrete material with any maximum aggregate size. Another size and boundary model developed based on the local energy concept is also introduced, and its capability to predict the fracture parameters from the fracture test results of wedge-splitting and compact tension specimens is first validated. In addition, the classical Bažant’s Type 2 size effect law is transformed to its boundary effect shape with the same equivalent crack length as Koval-Gao’s size and boundary effect model. This improvement could extend the applicability of the model to infer the material parameters from the test results of different types of specimens, including the geometrically similar specimens with constant crack-length-to-height ratios and specimens with different initial crack-length-to-height ratios. The test results of different types of specimens are adopted to verify the applicability of different size and boundary effect models for the determination of fracture toughness and tensile strength of concrete material. The quality of the extrapolated fracture parameters of the different models are compared and discussed in detail, and the corresponding recommendations for predicting the fracture parameters for dam concrete are proposed.

Strain gradient crystal plasticity modelling of size effects in a hierarchical martensitic steel using the Voronoi tessellation method

Inelastic deformation of a high-strength martensitic steel (P91) is investigated using a strain gradient crystal plasticity model implemented using the finite element method. Voronoi tessellation is used to model the hierarchical structure, prior austenite grain (PAG)/packet/block, of the martensitic steel and the effect of PAG/packet/block size on the macro- and micro-scale mechanical response is analysed numerically. The role of lath interaction and the influence of dislocation type (statistically stored and geometrically necessary dislocations) are investigated. It is found that block size determines the overall mechanical response, consistent with the Hall-Petch relation, while packet and block diameters influence the microplastic strain distribution. A modified Hall-Petch relation is examined which provides a relationship between material flow strength and block diameter (size) which holds for a wide range of initial dislocation densities and block diameters.

Investigation of a gradient enriched Gurson-Tvergaard model for porous strain hardening materials

Size effects in a strain hardening porous solid are investigated using the Gurson-Tvergaard (GT) model enriched by a constitutive length parameter, as proposed by Niordson and Tvergaard [C.F. Niordson, V. Tvergaard, A homogenised model for size effects in porous metals, J. Mech. Phys. Solids (2019)]. The results are compared with unit cell calculations of regularly distributed voids embedded in a strain gradient enhanced matrix material. The strain gradient plasticity theory proposed by Fleck and Willis [N.A. Fleck, J.R. Willis, A mathematical basis for strain gradient plasticity theory. Part II: tensorial plastic multiplier, J. Mech. Phys. Solids 57 (2009) 1045–1057], extended to finite strains, is adopted for the cell model, consistent with the gradient enriched Gurson model. The gradient model allows for a material length parameter to enter the constitutive framework for dimensional consistency, while the enriched GT model has the same length parameter introduced through prefactors of the usual q1 and q2 factors. The continuum model featuring size-dependent Tvergaard-constants is used to investigate a strain hardening material with the strain gradient plasticity enriched cell model as reference. The two models are compared for three triaxialities, three initial void volume fractions, and three hardening exponents. The enriched GT model captures the effect of elevated yield point and suppressed void growth with increasing length parameter for all the cases investigated. The agreement between the models is good until severe void distortion or plastic flow localisation between neighbouring voids. The response curves and void growth curves for the enriched GT model deviate from those of the cell model at high axial strains. Void shape plots, which are only available for the cell model, show that the length parameter influences the shape of the void which in turn has impact on the material response curves and the void evolution. This is not captured by the enriched GT model as the voids are accounted for solely through a volume fraction parameter.