Elastic Constitutive Relation Research Articles

Elastic crack propagation model for crystalline solids using a self-consistent coupled atomistic–continuum framework

Deformation and failure processes of crystalline materials are governed by complex phenomena at multiple scales. It is necessary to couple these scales for physics-based modeling of these phenomena, while overcoming limitations of modeling at individual scales. To address this issue, this paper develops self-consistent elastic constitutive and crack propagation relations of crystalline materials containing atomic scale cracks, from observations made in a concurrent multi-scale simulation system coupling atomistic and continuum domain models. The concurrent multi-scale model incorporates a finite temperature atomistic region containing the crack, a continuum region represented by a self-consistent crystal elasticity constitutive model, and a handshaking interphase region. Atomistic modeling is done by the molecular dynamics code LAMMPS, while continuum modeling is conducted by the finite element method. For single crystal nickel a nonlinear and nonlocal crystal elasticity constitutive relation is derived, consistent with the atomic potential function. An efficient, staggered solution scheme with parallel implementation is designed for the coupled problem. The atomistic–continuum coupling is achieved by enforcing geometric compatibility and force equilibrium in the interphase region. Quantitative analyses of the crack propagation process focuses on size dependence, strain energy release rate, crack propagation rate and degradation of the local stiffness. The self-consistent constitutive and crack propagation relations, derived from the concurrent model simulation results are validated by comparing results from the concurrent and full FE models. Excellent accuracy and enhanced efficiency are observed in comparison with pure MD and concurrent model results.

Elastic and elastoplastic fracture analysis of a reactor pressure vessel under pressurized thermal shock loading

Abstract Due to the limitation of the manufacturing process and improper operation, there may be defects near the inlet nozzles of reactor pressure vessel (RPV). Under pressurized thermal shock (PTS), the crack region is subjected to a high tensile stress, and the material toughness is gradually decreased in the cooling process. So it is necessary to evaluate the integrity of RPV in the thermo-mechanical coupling field. The 3-D finite element model is established for the beltline region around the inlet nozzles of a RPV. According to the calculated nil-ductility reference temperature, the elastic constitutive relation is firstly established for the fracture analysis. The stress intensity factors along the crack tips are compared with the material toughness. Using the XFEM, the crack propagation path is obtained to verify the predicted results. At relatively low reference temperatures, the ductility characteristics of the nuclear material can be exhibited. In this case, the ultimate bearing capacity of the structure is demonstrated by the elastoplastic fracture analysis. Finally, the safe range of reference temperature is summed up to prevent the surface crack from running through the whole wall thickness of the vessel during the PTS transient.

Classifications of ideal 3D elastica shapes at equilibrium

We investigate the equilibrium configurations of the ideal 3D elastica, i.e., inextensible, unshearable, isotropic, uniform, and naturally straight and prismatic rods, with linear elastic constitutive relations. Infinite solution trajectories are expressed analytically and classified in terms of only three parameters related to physical quantities. Orientation of sections and mechanical loading are also well described analytically with these parameters. Detailed analysis of solution trajectories yields two main results. First, all particular trajectories are completely characterized and located in the space of these parameters. Second, a general geometric structure is exhibited for every ideal 3D elastic rod, where the trajectory winds around a core helix in a tube-shaped envelope. This remarkable structure leads to a classification of the general case according to three properties called chirality components. In addition, the geometry of the envelope provides another characterization of the ideal 3D elastica. For both results, the domains and the frontiers of every class are plotted in the space of the parameters.

Common nonlocal elastic constitutive relation and material-behavior modeling of nanostructures

This article reviews conventional nonlocal elasticity constitutive relation which is frequently used for mechanical analyses of nanostructures. It is shown here that since this constitutive relation has been essentially derived based on infinite-body assumption, it cannot consider the nonlocal effects at all points of a nanoscale body accurately. Also, it is shown that although the nonlocal constitutive relations can potentially consider the surface effects, that constitutive relation has been obtained substantially by ignoring those effects. So, it cannot also consider the surface effects accurately. Therefore, the conventional nonlocal constitutive relation generally is not accurate for material-behavior modeling and consequently mechanical analysis of nanostructures. Furthermore, common nonlocal constitutive law is examined in buckling problem of Timoshenko beam-columns to show another limitation of that constitutive law. Finally, some special cases for which that constitutive relation can be used more accurately are proposed.

Proper ferroelastic phase transitions of hexagonal crystals

ABSTRACTThree dimensional slowness surfaces are used to study ferroelastic phase transitions of hexagonal crystals. The most prominent softening direction and the soft mode have been determined by the maximum slowness. For hexagonal crystals, the components of the elastic stiffness coefficients c44 or c66 has direct relationship with ferroelastic phase transitions. The spontaneous strains are calculated by using the elastic constitutive relation. The symmetry of ferroelastic phases are determined by using the Curie principle.

Ferroelastic phase transition of LiCsSO4 crystal

ABSTRACTThe ferroelastic phase transition in LiCsSO4 crystal is vividly shown by the change of slowness surfaces between 293 K and 202 K. The soft mode theory is used to investigate the phase transition. The pure shear mode which is corresponding to the elastic constant c66 starts softening at 206 K and softens completely at 202 K. According to the elastic constitutive relations and the Curie principle, LiCsSO4 crystal is in the same symmetry between 202 K and 206 K. There isn't an intermediate phase during the phase transition.

Passive manipulation of free-surface instability by deformable solid bilayers.

This study deals with the elastohydrodynamic coupling that occurs in the flow of a liquid layer down an inclined plane lined with a deformable solid bilayer and its consequences on the stability of the free surface of the liquid layer. The fluid is Newtonian and incompressible, while the linear elastic constitutive relation has been considered for the deformable solid bilayer, and the densities of the fluid and the two solids are kept equal. A temporal linear stability analysis is carried out for this coupled solid-fluid system. A long-wave asymptotic analysis is employed to obtain an analytical expression for the complex wavespeed in the low wave-number regime, and a numerical shooting method is used to solve the coupled set of governing differential equations in order to obtain the stability criterion for arbitrary values of the wave number. In a previous work on plane Couette flow past an elastic bilayer, Neelmegam et al. [Phys. Rev. E 90, 043004 (2014)PLEEE81539-375510.1103/PhysRevE.90.043004] showed that the instability of the flow can be significantly influenced by the nature of the solid layer, which is adjacent to the liquid layer. In stark contrast, for free-surface flow past a bilayer, our long-wave asymptotic analysis demonstrates that the stability of the free-surface mode is insensitive to the nature of the solid adjacent to the liquid layer. Instead, it is the effective shear modulus of the bilayer G_{eff} (given by H/G_{eff}=H_{1}/G_{1}+H_{2}/G_{2}, where H=H_{1}+H_{2} is the total thickness of the solid bilayer, H_{1} and H_{2} are the thicknesses of the two solid layers, and G_{1} and G_{2} are the shear moduli of the two solid layers) that determines the stability of the free surface in the long-wave limit. We show that for a given Reynolds number, the free-surface instability is stabilized when G_{eff} decreases below a critical value. At finite wave numbers, our numerical solution indicates that additional instabilities at the free surface and the liquid-solid interface can be induced by wall deformability and inertia in the fluid and solid. Interestingly, the onset of these additional instabilities is sensitive to the relative placements of the two solid layers comprising the bilayer. We show that it is possible to delay the onset of these additional instabilities, while still suppressing the free-surface instability, by manipulating the ratio of the shear moduli and the thicknesses of the two solid layers in the bilayer. At moderate Reynolds number and finite wave number, we demonstrate that an exchange of modes occurs between the gas-liquid and liquid-solid interfacial modes as the solid bilayer becomes more deformable. We demonstrate further that dissipative effects in the individual solid layers have an important bearing on the stability of the system, and they could also be exploited in suppressing the instability. This study thus shows that the ability to passively manipulate and control interfacial instabilities increases substantially with the use of solid bilayers.

ICCM2015: Study on Elastic Matrix Model of the Bi-modulus Finite Element Numerical Method

Bi-modulus materials exhibit the different modulus in tension and compression. The value of elastic modulus and Poisson ratio of every point in the bi-modulus elastic body not only depend on the material itself, but also the stress state and the strain state of the point. The uncertainty and nonlinearity of the elastic constitutive relation result in that the bi-modulus elastic problem is the complicated nonlinear problem This paper aims at studying the bi-modulus elastic constitutive equation employed in the bi-modulus finite element numerical method (FEM). The new elastic matrix model is proposed based on Ye’s principal strain criterion with the assumption that the Poisson ratio maintain constant whenever in tension or in compression, and the elastic matrix is symmetric by equivalent transmitting. The shear modulus expression of this elastic matrix model is derived to enable the elastic matrix completely and improve the convergence of the FEM calculation. The statically indeterminate bi-modulus beam is analyzed by means of FEM employing the proposed elastic matrix model. The effects of the tensile modulus to compressive modulus ratio and the boundary condition on the stress and deflection of the bi-modulus beam is studied.

Modeling discontinuous rock mass based on smoothed finite element method

This paper aims at developing a method for modeling rock mass with preexisting multiple discontinuities within the framework of the smoothed finite element method (SFEM). The discontinuity is simulated by an interface element with zero thickness, the stiffness matrix of which are derived explicitly based on the SFEM. An elastic damage constitutive relation with residual strength is introduced in order to describe the nonlinear mechanical behavior of the discontinuities. The computation codes of the present method were developed. The present method has been verified to be a sound approach for modeling discontinuous rock mass, inheriting the advantages of the SFEM.

Numerical investigation of parameters influencing debonding of FRP sheets in shear-strengthened beams

An effective technique for strengthening of concrete members is to provide extra reinforcements in the form of externally bonded fibre-reinforced polymer (FRP) sheets. Nonetheless, debonding often prevents strengthened beams from gaining their full load capacity and experimental studies have confirmed this. The current study proposes a non-linear three-dimensional (3D) finite element model to look into debonding failure of FRP sheets and its consequential effects in various strengthening schemes such as side-bonded and U-wrap (i.e. continuous FRP or discontinuous strips). 3D elements with linear elastic orthotropic constitutive relation was assumed for the FRP sheet and appropriate interface elements are used to simulate bond–slip behaviour between concrete, internal steel reinforcements and external FRP reinforcement. Once the model was validated against published experimental results, a parametric study was conducted to research the influence of some parameters on the shear capacity and failure mode, with a particular emphasis on the interfacial behaviour and debonding phenomena. The results show the significance of each parameter as well as FRP configuration on failure of retrofitted beams.

Cylindrical Borehole Failure in a Poroelastic Medium

Drilling a cylindrical borehole is the first and important step in oil mining. Borehole design and strength check are big problems of utmost importance. Biot introduced a poroelastic constitutive theory for porous rock with freely moving fluid inside. In this paper, by using Biot poroelastic model, we analyze a borehole with drilling fluid in an infinite porous rock with three-dimensional in situ stresses and obtain whole domain solutions for instantaneous, short-time, and long-time stress distributions. Maximum and minimum allowable drilling pressures are given for tensile failure and shear failure criterions, and allowable drilling pressure regions are drawn in the space of in situ hydrostatic stress P0, deviatoric stress S0, and pore pressure p0. By comparing with classical elastic constitutive relations, or Hooke's model, the necessity of Biot poroelastic constitutive relations is shown.

Nonlinear FEA of Pile Group Subjected to Lateral Load

A finite element method based program has been developed to perform the static nonlinear analysis of pile group with six different configurations subjected to lateral loads. The pile has been assumed to remain elastic all the time whereas the soil has been assumed to undergo plastic yielding following von Mises yield criterion. The formulation of elasto-plastic analysis following von Mises yield criterion has been explained. The effect of Drucker-Prager and Mohr Coulomb yield criteria on the response of pile group is also investigated. The whole analysis is based on incremental load application. The external load is applied in small increments and the stresses are initially computed assuming elastic constitutive relation. Significant effect of soil nonlinearity is observed at smaller pile spacing which reduces with increase in spacing.

Proper Ferroelastic Phase Transitions of Tetragonal Crystal Classes 4 mm, 422,4¯2 m, and 4/mmm

The proper ferroelastic phase transitions of tetragonal crystal classes , 422,and are investigated by the method of three dimensional slowness surfaces. There are three types of slowness surfaces for the four classes stated above. The elastic constants, and have relations with the classification of slowness surfaces and the ferroelastic phase transitions. The spontaneous strains are determined by using elastic constitutive relations. The symmetry of ferroelastic phases are obtained by using the Curie principle.

Linearly elastic constitutive relations and consistency for GBT-based thin-walled beams

The present paper focuses on the constitutive assumptions, both for the isotropic and orthotropic cases, and consistency in the framework of the Generalized Beam Theory. In particular, a novel approach based on energetic arguments, able to automatically select appropriate constitutive relations in accordance with the GBT kinematics, is presented. Furthermore, the concept of consistency of a GBT-based model is established and a consistency analysis is presented. This yields a formal rational basis to investigate the effects of the various families of cross-section deformation modes in terms of predictive capabilities of the GBT model. Some numerical examples illustrate the arguments exposed in the paper.

The performance of buildings adjacent to excavation supported by inclined struts

Limitations in the design method used for the support systems of urban buildings make them vulnerable to damage by adjacent excavations. This paper examines a traditional system used to support excavation sites and adjacent buildings in which inclined struts are connected to the wall or foundation of the adjacent building. This method can be considered to be a type of shoring or underpinning. The performance of buildings and the criteria for deformation control during excavation are introduced. Next, a 2D finite element analysis is presented in which an excavation is modeled considering the parameters from the adjacent building and the inclined struts. The numerical model is capable of simulating the overall excavation and installation of the support system. The soil is modeled using an elastic perfectly-plastic constitutive relation based on the Mohr-Coulomb criterion. The finite element model is validated using Rankine earth pressure and in situ data was measured during an excavation. The effect of different variables on performance and acceptable limits for the inclined strut are discussed. The model used for the parametric study shows the influence of the characteristics of the adjacent building, soil parameters, geometry of excavation, type of excavation and effect of strut installation. It was found that one type of strut arrangement produced the best possible result. The results can be used as a primary approximation of small-to-medium depth excavations in which struts are used to reduce the deflections.

재료 비선형과 연속체 손상역학을 고려한 복합 적층판의 강도 예측

이 논문에서는 복합 적층판의 점진적 파손해석 기법을 개발하고 검증하였다. 강도 및 강성 예측의 정확성을 높이기 위해 재료 비선형 효과와 연속체 손상역학을 동시에 고려하였다. 파손 시작점과 성장을 예측하기 위한 식으로 Hashin의 판별식이 사용되었으며, 파손 모드는 수지인장/전단, 섬유 인장의 2가지 파손모드를 고려하였다. 비선형 탄성 및 점탄성의 구성방정식을 고려한 평형을 계산하기 위해 Newton-Raphson 방법이 사용되었다. 실험을 통해 얻어진 복합재료 단층의 물성을 이용하여 노치가 없는 시편에 인장력을 가했을 때 예상되는 적층복합재의 강도 및 변형률을 예측하였다. 이 경우 선형 물성과 저하계수만을 고려하여 예측된 강성/강도보다 실험결과에 근사하게 나타남을 확인하였다. This paper presents development and verification of the progressive failure analysis upon the composite laminates. Strength and stiffness of the fiber-reinforced composite are analyzed by property degradation approach with emphasis on the material nonlinearity and continuum damage mechanics (CDM). Longitudinal and transverse tensile modes derived from Hashin's failure criterion are used to predict the thresholds for damage initiation and growth. The modified Newton-Raphson iterative procedure is implemented for determining nonlinear elastic and viscoelastic constitutive relations. Laminar properties of the composite are obtained by experiments. Prediction on the un-notched tensile (UNT) specimen is performed under the laminate level. Stress-strain curves and strength results are compared with the experimental measurement. It is concluded that the present nonlinear CDM approach is capable of predicting the strength and stiffness more accurately than the corresponding linear CDM one does.

Elastic Theory of Nanomaterials Based on Surface-Energy Density

Recent investigations into surface-energy density of nanomaterials lead to a ripe chance to propose, within the framework of continuum mechanics, a new theory for nanomaterials based on surface-energy density. In contrast to the previous theories, the linearly elastic constitutive relationship that is usually adopted to describe the surface layer of nanomaterials is not invoked and the surface elastic constants are no longer needed in the new theory. Instead, a surface-induced traction to characterize the surface effect in nanomaterials is derived, which depends only on the Eulerian surface-energy density. By considering sample-size effects, residual surface strain, and external loading, an explicit expression for the Lagrangian surface-energy density is achieved and the relationship between the Eulerian surface-energy density and the Lagrangian surface-energy density yields a conclusion that only two material constants—the bulk surface-energy density and the surface-relaxation parameter—are needed in the new elastic theory. The new theory is further used to characterize the elastic properties of several fcc metallic nanofilms under biaxial tension, and the theoretical results agree very well with existing numerical results. Due to the nonlinear surface effect, nanomaterials may exhibit a nonlinearly elastic property though the inside of nanomaterials or the corresponding bulk one is linearly elastic. Moreover, it is found that externally applied loading should be responsible for the softening of the elastic modulus of a nanofilm. In contrast to the surface elastic constants required by existing theories, the bulk surface-energy density and the surface-relaxation parameter are much easy to obtain, which makes the new theory more convenient for practical applications.

General lattice model of gradient elasticity

In this paper, new lattice model for the gradient elasticity is suggested. This lattice model gives a microstructural basis for second-order strain-gradient elasticity of continuum that is described by the linear elastic constitutive relation with the negative sign in front of the gradient. Moreover, the suggested lattice model allows us to have a unified description of gradient models with positive and negative signs of the strain gradient terms. Possible generalizations of this model for the high-order gradient elasticity and three-dimensional case are also suggested.

Numerical Simulation of Hysteretic Live Load Effect in a Soil-Steel Bridge

Abstract The paper presents numerical simulation of hysteretic live load effect in a soil-steel bridge. The effect was originally identified experimentally by Machelski [1], [2]. The truck was crossing the bridge one way and the other in the full-scale test performed. At the same time, displacements and stress in the shell were measured. The major conclusion from the research was that the measured quantities formed hysteretic loops. A numerical simulation of that effect is addressed in the present work. The analysis was performed using Flac finite difference code. The methodology of solving the mechanical problems implemented in Flac enables us to solve the problem concerning a sequence of load and non-linear mechanical behaviour of the structure. The numerical model incorporates linear elastic constitutive relations for the soil backfill, for the steel shell and the sheet piles, being a flexible substructure for the shell. Contact zone between the shell and the soil backfill is assumed to reflect elastic-plastic constitutive model. Maximum shear stress in contact zone is limited by the Coulomb condition. The plastic flow rule is described by dilation angle ψ = 0. The obtained results of numerical analysis are in fair agreement with the experimental evidence. The primary finding from the performed simulation is that the slip in the interface can be considered an explanation of the hysteresis occurrence in the charts of displacement and stress in the shell.

Large-strain elastic viscoplastic consolidation analysis of very soft clay layers with vertical drains under preloading

A very soft clay layer is highly compressible and exhibits significant creep under loading. The classical linearly elastic constitutive relationship and small-strain assumption are not suitable for the consolidation analysis of very soft clays. This paper presents a new large-strain consolidation model that incorporates the Yin–Graham elastic viscoplastic (EVP) constitutive equation for use in studying the consolidation of very soft clay layers with vertical drains under preloading. First, the large-strain fluid continuity equation and the EVP constitutive equation are incorporated into a quadratic differential equation of pore-water pressure and its integral terms. Second, the alternating-direction implicit (ADI) method and virtual node method are adopted to obtain the finite difference solution. A computer program named “BSSDS” is developed for large-strain EVP consolidation analysis of clay layers with vertical drains, taking into account the complicated in situ conditions, such as resistance of vertical drains, smear effects, variation of permeability with void ratio, and multilayered soils. Third, the new large-strain numerical method is applied to the consolidation modeling of very soft clay layers with vertical drains under preloading at a site that is part of the Hong Kong – Shenzhen Western Corridor Link Project. It is found that the foundation settlements of the new large-strain EVP consolidation model have good agreement with the measured data. Finally, three different consolidation models are used to calculate the average degree of consolidation and settlements of the clay layers. The analysis shows that it is essential to consider both large-strain compression and creep effects in the analysis of very soft clay layers with vertical drains under loading.