Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress

Duan, Huiling; Wang, J.; Huang, Zhuping; Karihaloo, Bhushan Lal

doi:10.1016/j.jmps.2005.02.009

Cited by 660 publications

(328 citation statements)

References 48 publications

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“…Gurtin and Murdoch [1,2] developed a generic continuum model incorporating the surface/interface effects where the surface/interface is modeled as a zero-thickness layer with its own physical properties. This surface/interface elasticity model has been extensively used to model nanostructures [3][4][5][6][7][8], and it was shown that the size-dependent behaviors obtained from the surface elasticity model agree very well with the atomistic simulations [4,9].…”

Section: Introductionmentioning

confidence: 76%

A hybrid smoothed extended finite element/level set method for modeling equilibrium shapes of nano-inhomogeneities

Zhao

Bordas

2013

Comput Mech

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Interfacial energy plays an important role in equilibrium morphologies of nanosized microstructures of solid materials due to the high interface-to-volume ratio, and can no longer be neglected as it does in conventional mechanics analysis. The present work develops an effective numerical approach by means of a hybrid smoothed extended finite element/level set method to model nanoscale inhomogeneities with interfacial energy effect, in which the finite element mesh can be completely independent of the interface geometry. The Gurtin-Murdoch surface elasticity model is used to account for the interface stress effect and the Wachspress interpolants are used for the first time to construct the shape functions in the smoothed extended finite element method. Selected numerical results are presented to study the accuracy and efficiency of the proposed method as well as the equilibrium shapes of misfit particles in elastic solids. The presented results compare very well with those obtained from theoretical solutions and experimental observations, and the computational efficiency of the method is shown to be superior to that of its most advanced competitor.Keywords: Smoothed extended finite element method; Level set method; Wachspress shape function; Interface excess energy; Equilibrium shape. IntroductionRecent advances in nanotechnology exacerbate the need for computational tools that are capable of capturing the effects of interfaces, which play an important role in nanostructured materials due to a characteristically high interface-to-volume ratio. Although atomic level computational tools such as molecular dynamic (MD) and first principle calculations are able to simulate interface effects, these methods are computationally intensive, thus their applications are usually limited to nano-scale samples and nano-second time durations. Many engineering problems, however, occur at much larger spatial and temporal scales. For example, simulating the formation and morphological evolution of precipitates in superalloys involves length scales ranging from several nanometers to tens of micrometers, and the physical processes last up to hours in time. In these cases, it is necessary to use a continuum level model that can capture the interfacial effects of particles at different length and time scales.

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Section: Introductionmentioning

confidence: 76%

A hybrid smoothed extended finite element/level set method for modeling equilibrium shapes of nano-inhomogeneities

Zhao

Bordas

2013

Comput Mech

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“…Analytical methods for modeling reinforced composite materials considering imperfect interface conditions have been developed recently [149][150][151][152][153][154][155][156][157][158][159][160][161][162][163]. Also, the importance of the interphase zone in modeling composite materials has been discussed in Refs.…”

Section: Introductionmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

190

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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“…On the other hand, Miller and Shenoy [8] developed a simple theoretical model incorporating the surface stress effect to predict the size-dependence of the effective elastic modulus for nano materials. Duan et al also took into account the surface/interface stress effect and showed the size-dependence of effective elastic constants for nano-inhomogeneity solid [9] . Sun and Zhang [10] applied a semi-continuum model to analysis of the size effect of plate-like nanostructured materials, and concluded that the predicted Young's modulus is dependent on the number of atom layers in the thickness direction.…”

Section: Introductionmentioning

confidence: 99%

“…Wang et al [14] , and Zhu and Zheng addressed the strain gradient effect and modified the elastic properties of nanostructures [15] . The popularly used approaches for investigating the size effects mainly include the atomistic simulation [11,16] , the continuum methods [8,9,17] , as well as the multi-scale analysis [18] . In the present study, we intend to adopt a simple quasi-continuum model, the Cauchy-Born rule (CBR) model, to investigate the size effects of material properties.…”

Section: Introductionmentioning

confidence: 99%

Size effects of elastic modulus of fcc metals based on the Cauchy-Born rule and nanoplate models

Liu

Song

Wei

2014

Acta Mechanica Solida Sinica

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In the present research, a simple quasi-continuum model, the Cauchy-Born rule model, is used to investigate the size effects of elastic modulus for fcc metals. By considering a nanoplate model and calculating the strain energy for the nano-sized plate under tension and bending, the relationship between the elastic modulus and the plate thickness is found. Size effects of the elastic modulus are displayed by the relative differences of the elastic modulus between the nano-sized plate sample and the bulk sample. By comparing the present results with those of others, the effectiveness of the Cauchy-Born rule model in studying the size effects of material properties are shown.

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Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress

Cited by 660 publications

References 48 publications

A hybrid smoothed extended finite element/level set method for modeling equilibrium shapes of nano-inhomogeneities

A hybrid smoothed extended finite element/level set method for modeling equilibrium shapes of nano-inhomogeneities

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Size effects of elastic modulus of fcc metals based on the Cauchy-Born rule and nanoplate models

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