Size Effect Model Research Articles

Nonlocal integral elasticity for third-order small-scale beams

Small-scale beams are basic structural components of miniaturized electro-mechanical systems whose design requires accurate modeling of size effects. In this research, the size-dependent behavior of nonlocal elastic beams is investigated by adopting the stress-driven elasticity theory. Kinematics of beams is modeled by the Reddy variational third-order beam theory accounting for the effective distribution of shear stresses on cross sections without needing the evaluation of shear correction factors. Stress-driven integral elasticity is thus extended to third-order small-scale beams providing an equivalent constitutive formulation with boundary conditions. The relevant nonlocal elastic equilibrium problem is formulated and an analytical strategy is proposed to obtain closed-form solutions. The present approach is elucidated by solving some structural problems of current interest in Nanotechnology.

Discrete modeling of deterministic size effect of normal-strength and ultra-high performance concrete under compression

Size effect of heterogeneous granular materials under tensile loading is a well known phenomenon and the subsequent size effect laws are widely available in the scientific literature. On the contrary, size effect in compression is scarcely studied, which, especially for concrete, is more significant for its brittle failure mode in practical applications. This paper investigates the deterministic size effect in compression of concretes including a normal strength concrete (NSC) and an ultra high performance concrete (UHPC). The Lattice Discrete Particle Model (LDPM) is selected to determine the compressive strength of a broad range of sizes utilizing experimental-based calibrated data sets. The simulated tests include both cylinders and prisms with the diameter/width range from 150 mm to 600 mm for normal strength concrete and from 50 mm to 200 mm for UHPC. The investigated aspect ratio ranges from 1 to 16. It is found that the simulated strengths show no size effect under low friction loading, and the magnitude of size effect decreases with rising aspect ratio under high friction loading. The size effect laws available in the literature are not able to fit the obtained compressive strength data and a new compressive size effect law formula named ComSEL3D is proposed in terms of diameter and aspect ratio. The correlation by ComSEL3D for the simulated strengths is above 0.99, which shows its excellent ability in describing and predicting the deterministic size effect behavior in compression for both concretes in cylindrical and prism shapes. The ComSEL3D also outperforms the available formulas in the literature in matching the experimental results of various concretes under compression.

Initial Assessment on Hydraulic Conductivity of Rock Mass and its Size Effect Using Drilling Data

Predicting hydraulic conductivity and characterizing its size effect in jointed rock masses is fundamental to various geoscience and engineering disciplines. A new empirical method, using drilling data, is established to predict the hydraulic conductivity of a rock mass in the scanline direction. The effect of the scanline unit length on the distribution characteristics of hydraulic conductivity for boreholes in jointed rock masses is studied. In addition, the scanline length of boreholes was used to study the size effect of hydraulic conductivity of jointed rock masses, and a new model was proposed to determine the size effect of hydraulic conductivity. A mathematical method for representative volume element (RVE) determination was proposed based on the proposed size effect model of hydraulic conductivity. Some packer tests were conducted to validate the developed method for predicting hydraulic conductivity. The results show that (a) the predicted hydraulic conductivities are close to those determined from the packer tests; (b) the scanline unit lengths influence the distribution characteristics of hydraulic conductivity; (c) a large but fluctuating increase in hydraulic conductivity with increasing scanline length, and the fluctuation decreases with fewer fractures and joints; (d) the mathematical method for predicting RVE and KRVE is effective to make an initial evaluation on jointed rock masses with high fracture frequency.

Decoupling indentation size and strain rate effects during nanoindentation: A case study in tungsten

Materials indented at small scales may simultaneously exhibit indentation size and strain rate effects which complicate the identification of the mechanisms that control deformation and strength. Here, we explore the possibility that indentation size and rate effects in some materials can be decoupled in a simple way. Nanoindentation tests with various load-time histories were carried out to measure the hardness of a tungsten single crystal over a wide range of indentation depths (∼500–3600 nm) and indentation strain rates (∼5 × 10–5–2 × 10–1 s–1). Under these conditions, this material exhibits significant indentation size and rate effects, but the size effect is, to a good approximation, independent of strain rate. It is shown that this behavior can be understood by the Nix-Gao model for the indentation size effect modified to include the effects of a strain rate dependent friction stress. As a consequence, the size and rate dependencies of the hardness can be expressed as the sum of two independent terms:H(ε˙i,hc)=Hf(ε˙i)+(H0−Hf)1+h*hc,where H(ε˙i,hc) is the hardness at given indentation strain rate (ε˙i) and contact depth (hc), Hf(ε˙i) is the hardness contributed by the rate dependent friction stress, and (H0−Hf) and h* are size and rate independent constants that follow from the Nix-Gao analysis. This formula, together with an expression for the rate dependence of Hf, was successfully applied to decouple the indentation size and rate effects observed in tungsten. In addition, the physics underlying the rate independence of indentation size effect is discussed, which provides guidance for application of the proposed approach to other materials.

Asymptotic is Better than Bollen-Stine Bootstrapping to Assess Model Fit: The Effect of Model Size on the Chi-Square Statistic

Previous research on bootstrapped p-values for the chi-square test of model fit has been limited to small models (around 10 variables), revealing that these p-values are accurate provided the sample size is not too small. For small sample sizes (N < 100), usual p-values, obtained using asymptotic methods, are more accurate. However, as the number of variables increases asymptotic p-values incorrectly suggest that models fit poorly. We investigate whether Bollen-Stine (1992) bootstrapped p-values can overcome this problem using normal and non-normal data. We found that as model size increases, Bollen-Stine bootstrapped p-values become too conservative and less accurate than asymptotic p-values obtained using robust methods (i.e., mean and variance corrections). Bollen-Stine p-values cannot be recommended to assess model fit.

Indentation Size Effect Model of Ti6Al4V Alloy by Combining the Macroscopic Power‐Law Constitutive Relation and Strain Gradient Theory

Indentation size effect (ISE) is a critical feature for investigating the local properties of materials using the indentation test method. Although there are numerous works on the indentation response, there is a lack of data in the literature on the association between nanoindentation and uniaxial compression. Herein this study, uniaxial compression tests combined with Digital Image Correlation Technique are conducted on Ti6Al4V alloy with a focus on investigating the elastic–plastic properties and obtaining the accurately constitutive relation. Then, an ISE model of Ti6Al4V is established based on this constitutive relation and strain gradient theory. Indentation hardness, predicted by this proposed model, agrees well with the measurement by nanoindentation experimental techniques. This model can be used to bridge the gap between the uniaxial stress–strain and the depth‐dependent hardness, and clarifies the effect of the strain hardening on indentation hardness. The significance of this model is that it provides a direct and reasonable theory foundation to investigate ISE by uniaxial compression, and promotes the potential application of nanoindentation.

Aspects of size effect on discrete element modeling of concrete

Abstract Present paper focuses on the modeling of size effect on the compressive strength of normal strength concrete with the application of discrete element method, considering specimen of different concrete mixes and shapes. An equation was derived to estimate the parallel bond strength from the compressive strength. The results showed a good agreement with the literature and the derived estimation models showed strong correlation with the measurements. The results indicated that size effect is stronger on concretes with lower strength class and that it is more significant on cube specimens than on cylinders. The relationship of model size and computational time was analyzed and a method to decrease the computational time (iterations) was proposed.

Uncovering the Mechanism of Size Effect on the Thermomechanical Properties of Highly Cross-Linked Epoxy Resins.

Epoxy resins are widely used as matrix resins, especially for carbon-fiber-reinforced plastic, due to their outstanding physical and mechanical properties. To date, most research into cross-linking processes using simulation has considered only a distance-based criterion to judge the probability of reaction. In this work, a new algorithm was developed for use with the large-scale atomic/molecular massively parallel simulator (LAMMPS) simulation package to study the cross-linking process; this new approach combines both a distance-based criterion and several kinetic criteria to identify whether the reaction has occurred. Using this simulation framework, we investigated the effect of model size on predicted thermomechanical properties of three different structural systems: diglycidyl ether of bisphenol A (DGEBA)/4,4'-diaminodiphenyl sulfone (4,4'-DDS), DGEBA/diethylenetriamine (DETA), and tetraglycidyl diaminodiphenylmethane (TGDDM)/4,4'-DDS. Derived values of gel point, volume shrinkage, and cross-linked resin density were found to be insensitive to model size in these three systems. Other thermomechanical properties, i.e., glass-transition temperature, Young's modulus, and yield stress, were found to reach stable values for systems larger than ∼40 000 atoms for both DGEBA/4,4'-DDS and DGEBA/DETA. However, these same properties modeled for TGDDM/4,4'-DDS did not stabilize until the system size reached 50 000 atoms. Our results provide general guidelines for simulation system size and procedures to more accurately predict the thermomechanical properties of epoxy resins.

Pore-scale study on Rayleigh-Bénard convection formed in the melting process of metal foam composite phase change material

Rayleigh-Bénard convection significantly affects the melting process of phase change material (PCM). It has been verified to exist in the model with bottom heating. Nevertheless, researchers rarely focus on the Rayleigh-Bénard convection in the metal foam composite PCMs (MFCPCMs). This paper took numerical investigations to study the melting behavior of the MFCPCMs when Rayleigh-Bénard convection was involved, and a visualized experiment was used to verify their accuracy. By analyzing the number of Bénard cells, dimensionless parameters of Rayleigh number and Nusselt number, we found three regimes, including the conductive regime, linear regime, and coarsening regime, forming in the models. The liquid fraction contour captured the evolution of the melting front, and the state of convective heat transfer was characterized by Bénard cells. Results show that the intensity of Rayleigh-Bénard convection in the MFCPCM is obviously weaker than that in the pure PCM. Moreover, the porosity of metal foam has a significant influence on the development of Rayleigh-Bénard convection. With the increase of the porosity, the suppression of liquid PCM flow is weakened, so the convection is more developed. The inhomogeneous evolution of the melting front is more pronounced. Besides, to study the effect of model size on the melting behavior, we built three groups of models with width-height ratios (γ) of 2, 1, and 0.5, respectively. Results show that the model with large γ has more significant inhibition on the development of Rayleigh-Bénard convection. The solid/liquid interface is flat or with a slight fluctuation. On the contrary, the model with small γ has more developed Rayleigh-Bénard convection. The interface of it has a violent fluctuation, which shows as a humped arch. This work interprets the internal heat-transfer characteristics of the MFCPCM model with Rayleigh-Bénard convection. It paves the way for the structural optimization of the metal skeleton to gain efficient heat storage.

Modeling size effects in microwire torsion: A comparison between a Lagrange multiplier-based and a [formula omitted] gradient crystal plasticity model

The size-dependent response of metallic microwires under monotonic and cyclic torsion is modeled taking a reduced-order strain gradient crystal plasticity approach involving a single scalar-valued micromorphic variable. It is compared with the response predicted by the CurlFp model proposed in Kaiser and Menzel (2019a), which is based on the complete dislocation density tensor. It is shown that, in cyclic non-uniform plastic deformation processes, the gradient of the scalar-valued internal variable in the reduced-order model predicts isotropic hardening in contrast to kinematic-type hardening produced by the CurlFp model due to a dislocation-induced back-stress component. The arising size effect in the monotonic torsion tests is described by the normalized torque T/R3 as a function of the ratio of the microwire radius R and the characteristic length scale ℓ. In the size-dependent domain, characterized by an inflection point on the corresponding curve, the scaling law T/R3∼(R/ℓ)n can be identified, and explicit relations are found for the power n. The relative evolution of Statistically Stored Dislocation (SSD) and Geometrically Necessary Dislocation (GND) densities during torsion is described in detail.

Multiscale model for the scale effect of tensile strength of hardened cement paste based on pore size distribution

Mechanical properties of cement-based materials are significantly affected by pore structure. This paper aims to determine the effect of pore size distribution on the multiscale tensile strength of hardened cement pastes, which is named “scale effect” of tensile strength in this paper. A multiscale model for the scale effect of tensile strength is proposed by combining Griffith fracture theory and Bazant’s size effect law. Splitting tensile tests were conducted and the load was applied at the speed of 0.05 MPa/s, and the obtained splitting tensile strength of the specimen was 4.2 MPa. The proposed model is verified by results of mechanical tests and simulations together with the data of pore size distribution. The model provides a good description of the variation of tensile strength with increasing scale, and it is applicable in a wider range of scale compared with the existing size effect models.

Numerical modelling of size effects in micro-cutting of f.c.c. single crystal: Influence of strain gradients

A size effect is exhibited by metals in cutting processes at small scales: a nonlinear escalation in the cutting energy per unit volume with a decrease in the thickness of the uncut chip. To understand this phenomenon, a numerical model of orthogonal micro-cutting of f.c.c. single-crystal copper was developed, and three different depths of cut (0.8 μm, 1.4 μm and 2.0 μm) were considered. Enhanced models of crystal plasticity (EMCP) and strain-gradient crystal plasticity (EMSGCP) implemented in ABAQUS/Explicit through a VUMAT Fortran subroutine were used. Obtained results were compared with the experiments reported in the literature. The influence of size effect on deformation response of a single-crystal material was investigated extensively in terms of the distribution of field variables such as strain gradients, stresses, lattice rotations and activity of different slip systems. The influence of crystallographic orientation was also studied. It was observed that the size-effect phenomenon was predicted more accurately with the EMSGCP theory, where polar dislocations evolving during deformation were found to affect it significantly.

Interfacial thermal transport of graphene/β-Ga2O3 heterojunctions: a molecular dynamics study with a self-consistent interatomic potential.

Graphene/β-Ga2O3 heterojunctions are widely used in high-power and high-frequency devices, for which thermal management is vital to the device operation and life. Here we apply molecular dynamics simulation to calculate the interfacial thermal resistance (ITR) between graphene and β-Ga2O3. Based on the rigid ion model, a self-consistent interatomic potential with a set of parameters that can well reproduce the basic physical properties of crystal β-Ga2O3 is fitted. Using this potential, the effects of model size, interface type, temperature, vacancy defects and graphene hydrogenation on the ITR of graphene/β-Ga2O3 heterojunctions are evaluated. The results show that there is no obvious dependence of ITR on the size of graphene and β-Ga2O3. It is reported that the ITR values of the (100), (010) and (001) interfaces are 7.28 ± 0.35 × 10-8 K m2 W-1, 6.69 ± 0.44 × 10-8 K m2 W-1 and 5.22 ± 0.35 × 10-8 K m2 W-1 at 300 K, respectively. Both temperature increase and vacancy defect increase can prompt the energy propagation across graphene/β-Ga2O3 interfaces due to the enhancement of phonon coupling. In addition, graphene hydrogenation provides new channels for in-plane and out-of-plane phonon coupling, and thus reduces the ITR between graphene and β-Ga2O3. This study provides basic strategies for the thermal design and management of graphene/β-Ga2O3 based photoelectric devices.

The effect of model size and boundary conditions on the representativeness of digital material representation simulations of ferritic-pearlitic sample compression

The effect of model size and boundary conditions on the representativeness of digital material representation simulations of ferritic-pearlitic sample compression

Fatigue Damage Analysis On 42CrMo4+QT Via Critical Volume Approach

The influence of critical volume on fatigue strength is assessed based on fatigue experiments of six specimen types (both unnotched and notched) manufactured from 42CrMo4+QT steel. Three fatigue size effect models are evaluated. To deal with them, additional coinciding fatigue strength influencing factors, such as surface roughness, had to be analyzed on a data set comprised of both unnotched and notched specimens.

Assessment of EFD and CFD capability for KRISO Container Ship added power in head and oblique waves

EFD and CFD capability assessment for KCS added power (AP) in head and oblique waves are conducted based on experiments from three facilities using three different model sizes and CFD from five institutes. The analysis includes the standard deviation (SD) in both CFD and EFD to identify facility biases, scale effects and CFD errors for motions, self-propulsion (SP), propulsive efficiency (η) and AP. The overall SD%D (D: EFD values) for all calm water SP variables and AP variables is 9% and 11%, respectively. SP correlates with Re via advance coefficient J(Re) and SP points lie along nondimensional propeller load curves. AP vs. λ/L correlates with large bow relative motion such that the wave effects on J(λ/L) have the same scaling as the model size effect J(Re). Logarithmic derivative analysis of EFD data shows that for head waves the added resistance (AR) and η are responsible for 70 vs. 30%AP, respectively, whereas for oblique waves the AR and η are responsible for 55 vs. 38%AP, respectively. The overall conclusion is that the experimental and CFD approaches are of sufficient accuracy to be useful for design.

Size effect and comprehensive mathematical model for gas-sensing mechanism of SnO2 thin film gas sensors

Tin oxide (SnO2) is widely used in metal-oxide-semiconductor for gas-sensing materials due to its unique physical and chemical properties. The grain size is one of the major influencing factors that determine the gas-sensing characteristics. In this work, a facile hydrolysis-oxidation-hydrothermal method is used to prepare the size-controllable SnO2 quantum dots (QDs) of 4.7–8.9 nm, and the gas-sensing characteristics of the SnO2 QDs sensors are evaluated by C4H10, H2 and C2H5OH at room temperature. The experimental results show that the resistance has a negative correlation with hydrothermal time, and the maximum response is obtained when the grain radius is comparable to the depletion layer width. A comprehensive model is proposed by considering all gas-sensing procedures of the receptor function, the transducer function and the utility factor. The sensor properties are formulated as functions of grain size, depletion layer width, film thickness, oxygen vacancy density, gas concentration, pore size as well as operating temperature. The present model provides a comprehensive mathematical interpretation of the size effects of SnO2 from partial depletion to volume depletion.

Size effect mechanism of cross-section deformation and section hollow coefficient-bending degree of the thin-walled composite bending tube

The composite tube has been widely used in various fields of piping systems, because of saving precious metals and combining physical and chemical properties of bi-material. There are important correlations between the bending forming accuracy and cross-section deformation mechanism of the composite tube, and its own size factors, that is the size effect. However, the correlations are not clear, and the effects of size factors are coupled or superimposed. Therefore, a brand-new research method of size effect is established. Two dimensionless variables of section hollow coefficient λ and bending degree w are introduced to describe the sectional dimensional characteristics. The mathematical model of size effect is established as well as the relationships between cross-section deformation Δδ and λ, w and λ-w. Finally, the distribution of Δδ in λ-w space has been developed based on the above research. The results are shown as follows. (1) All the relationships of Δδ and λ-w can be described by the product of the double Boltzmann function, which is also the mathematical model of the size effect. (2) There are “un-formable zone”, “bendable forming zone” and “best forming zone” of the composite tube in λ-w space. The “best forming zone” is Zb = {λ, w|0.2 ≤ λ ≤ 0.58, 0.23 ≤ w ≤ 0.90}.

Aspects of size effect on discrete elementmodeling of normal strength concrete

Present paper focuses on the modeling of size effect on the compressive strength of normal concrete with the application of Discrete Element Method (DEM). Test specimens with different size and shape were cast and uniaxial compressive strength test was performed on each sample. Five different concrete mixes were used, all belonging to a different normal strength concrete class (C20/25, C30/37, C35/45, C45/55, and C50/60). The numerical simulations were carried out by using the PFC 5 software, which applies rigid spheres and contacts between them to model the material. DEM modeling of size effect could be advantageous because the development of micro-cracks in the material can be observed and the failure mode can be visualized. The series of experiments were repeated with the model after calibration. The relationship of the parallel bond strength of the contacts and the laboratory compressive strength test was analyzed by aiming to determine a relation between the compressive strength and the bond strength of different sized models. An equation was derived based on Bazant's size effect law to estimate the parallel bond strength of differently sized specimens. The parameters of the equation were optimized based on measurement data using nonlinear least-squares method with SSE (sum of squared errors) objective function. The laboratory test results showed a good agreement with the literature data (compressive strength is decreasing with the increase of the size of the specimen regardless of the shape). The derived estimation models showed strong correlation with the measurement data. The results indicated that the size effect is stronger on concretes with lower strength class due to the higher level of inhomogeneity of the material. It was observed that size effect is more significant on cube specimens than on cylinder samples, which can be caused by the side ratios of the specimens and the size of the purely compressed zone. A limit value for the minimum size of DE model for cubes and cylinder was determined, above which the size effect on compressive strength can be neglected within the investigated size range. The relationship of model size (particle number) and computational time was analyzed and a method to decrease the computational time (number of iterations) of material genesis is proposed.

Phase-field modelling of size effect on strength and structural brittleness

Phase-field modelling of size effect on strength and structural brittleness