Macrohomogeneity condition for strain gradient homogenization of periodic heterogeneous media with interfacial strong discontinuities

2020 ◽  
pp. 108128652095875
Author(s):  
JF Ganghoffer ◽  
XN Do ◽  
G Maurice
Keyword(s):  
Strain Gradient ◽  
Heterogeneous Media ◽  
Weak Form ◽  
Displacement Discontinuity ◽  
Order Effects ◽  
Interfacial Behavior ◽  
Damage Models ◽  
Strong Discontinuities ◽  
Polynomial Expression ◽  
The Hill

The Hill macrohomogeneity condition is revisited in the context of strain gradient homogenization for heterogeneous materials prone to interfacial displacement jumps. The consideration of strain gradient effects is motivated by their use as a regularization method for strain-softening constitutive damage models leading to strain localization and displacement discontinuity. Starting from the weak form of the boundary value problem formulated at the microscopic level, a polynomial expression of the virtual velocity is adopted as the minimum microscopic kinematics consistent with the selected macroscopic kinematics of the strain gradient effective continuum. The effective volumetric and interfacial mesoscopic strains and stress measures for the effective substitution are obtained versus the microscopic strains and stresses. The Hill macrohomogeneity condition is successively formulated for continuous interfaces and discontinuous interfaces witnessing strong discontinuities. It highlights the expressions of the effective stress measures associated to the volumetric and interfacial behavior for both classical and higher-order effects.

scholarly journals Strain-gradient vs damage-gradient regularizations of softening damage models

2018 ◽  
Vol 340 ◽  
pp. 424-450 ◽  
Author(s):  
Duc Trung Le ◽  
Jean-Jacques Marigo ◽  
Corrado Maurini ◽  
Stefano Vidoli
Keyword(s):  
Strain Gradient ◽  
Damage Models

scholarly journals SOME EFFECTS OF 2537 Å ON GREEN ALGAE AND CHLOROPLAST PREPARATIONS

1951 ◽  
Vol 34 (5) ◽  
pp. 627-645 ◽  
Author(s):  
A. S. Holt ◽  
I. A. Brooks ◽  
W. A. Arnold
Keyword(s):  
Photosynthetic Apparatus ◽  
Total Duration ◽  
Hill Reaction ◽  
Endogenous Respiration ◽  
Order Effects ◽  
Hydrogen Metabolism ◽  
Bicarbonate Buffer ◽  
First Order ◽  
Solid Dose ◽  
The Hill

1. The kinetics of the inactivation of photosynthesis by 2537 Å in Chlorella pyrenoidosa and Scenedesmus D1 indicate that, while the destruction process is largely a first order effect, higher order effects also occur, which become evident at low exposures. In agreement with previous observations, endogenous respiration is insensitive to exposures which inactivate photosynthesis. 2. In Scenedesmus D1 a solid dose of ultraviolet has no more effect on the photosynthetic apparatus than a dose of equal total duration interrupted by periods of photosynthesis. Nor is any difference noted if the cells are in a different buffer, e.g. 0.05 M KH2PO4, or carbonate-bicarbonate buffer 9. 3. In C. pyrenoidosa, a solid dose and an interrupted dose cause equal effects on photosynthesis when neutral phosphate buffer is used. If the ultraviolet exposure schedules are identical, equal effects are also noted in cells suspended in buffer 9, and in 0.05 M phosphate (pH 6.2). Solid exposures are, however, much more effective than interrupted exposures, when buffer 9 is used. 4. Oxygen evolution (Hill reaction), photosynthesis, and photoreduction in Scenedesmus D1 are equally sensitive to a given dose of ultraviolet. The mechanism responsible for adaptation to hydrogen metabolism is not more sensitive to ultraviolet than is the photosynthetic mechanism. The O2/H2/CO2 reaction in darkness is less sensitive to ultraviolet than any of the above reactions. 5. Glucose oxidation by C. pyrenoidosa, and colony formation in Scenedesmus D1 are far more sensitive to a given dose of ultraviolet than photosynthesis in these organisms. 6. The photosynthetic apparatus of C. pyrenoidosa is more sensitive to ultraviolet than that of Scenedesmus D1. 7. The Hill reaction in chloroplast fragments is also inactivated by 2537 Å by a first order process. Exposures which inactivate this reaction completely have no effect on polyphenol oxidase, cytochrome oxidase, or catalase in the same chloroplast preparation. 8. After irradiation, the survival of photosynthesis in Scenedesmus D1 and of the Hill reaction in chloroplast fragments are independent of the light intensity used to measure these processes. 9. No significant changes occur in the ultraviolet absorption of chloroplasts after an exposure to 2537 Å, which completely inactivates the Hill reaction.

scholarly journals Simulation of elastic wave propagation across fractures using a nodal discontinuous Galerkin method—theory, implementation and validation

2019 ◽  
Vol 219 (3) ◽  
pp. 1900-1914 ◽  
Author(s):  
T Möller ◽  
W Friederich
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Elastic Wave ◽  
Discontinuous Galerkin ◽  
Heterogeneous Media ◽  
Rheological Behaviour ◽  
Displacement Discontinuity ◽  
Signal Frequency ◽  
Elastic Wave Equation ◽  
Numerical Flux

SUMMARY An existing nodal discontinuous Galerkin (NDG) method for the simulation of seismic waves in heterogeneous media is extended to media containing fractures with various rheological behaviour. Fractures are treated as 2-D surfaces where Schoenberg’s linear slip or displacement discontinuity condition is applied as an additional boundary condition to the elastic wave equation which is in turn implemented as an additional numerical flux within the NDG formulation. Explicit expressions for the new numerical flux are derived by considering the Riemann problem for the elastic wave equation at fractures with varying rheologies. In all cases, we obtain further first order differential equations that fully describe the temporal evolution of the particle velocity jump at the fracture. Our flux formulation allows to separate the effect of a fracture from flux contributions due to simple welded interfaces enabling us to easily declare element faces as parts of a fracture. We make use of this fact by first generating the numerical mesh and then building up fractures by selecting appropriate element faces instead of adjusting the mesh to pre-defined fracture surfaces. The implementation of the new numerical fluxes into NDG is verified in 1-D by comparison to an analytical solution and in 2-D by comparing the results of a simulation valid in 1-D and 2-D. Further numerical examples address the effect of fracture systems on seismic wave propagation in 1-D and 2-D featuring effective anisotropy and coda generation. Finally, a study of the reflective and transmissive behaviour of fractures indicates that reflection and transmission coefficients are controlled by the ratio of signal frequency and relaxation frequency of the fracture.

scholarly journals Perfectly matched absorbing layer for modelling transient wave propagation in heterogeneous poroelastic media

2019 ◽  
Author(s):  
Yanbin He ◽  
Tianning Chen ◽  
Jinghuai Gao
Keyword(s):  
Wave Equations ◽  
Heterogeneous Media ◽  
Near Field ◽  
Weak Form ◽  
Transient Wave ◽  
Absorbing Boundary ◽  
Poroelastic Media ◽  
Elastic Wave Equations ◽  
Absorbing Layer ◽  
Biot’S Equations

Abstract The perfectly matched layer (PML) has been demonstrated to be an efficient absorbing boundary for near-field wave simulation. For heterogeneous media, the property of the PML needs to be carefully specified to avoid numerical instability and artificial reflection because part of it lies at the discontinuous interface. Coupled acoustic-poroelastic (A-P) media or coupled elastic-poroelastic (E-P) media often arise in the field of geophysics. However, PMLs that appropriately terminate these heterogeneous poroelastic media are still lacking. The main purpose of this paper is to explore the application of unsplit PMLs for transient wave modeling in infinite, heterogeneous, coupled A-P media or coupled E-P media. To this end, a consistent derivation of memory-efficient PML formulations for the second-order Biot's equations, elastic wave equations and acoustic wave equations is performed based on complex coordinate transformation using auxiliary differential equations. Furthermore, the interface boundary conditions inside the absorbing layer are rigorously derived for the considered A-P and E-P cases. Finally, the weak form of PML formulations for coupled poroelastic problems is presented. The finite element method is used to validate the proposed PML based on several two-dimensional benchmarks. The accuracy and stability of weak PML formulations are investigated. In particular, for coupled acoustic-poroelastic PML, two extreme (open-pore and sealed-pore) interface conditions are considered and PML results are compared with known analytical solutions. This study demonstrates the ability of the PML to effectively eliminate outgoing bulk waves and surface waves in coupled poroelastic media.

scholarly journals A Finite Element Formulation for Kirchhoff Plates in Strain-gradient Elasticity

2019 ◽  
Author(s):  
Alireza Beheshti
Keyword(s):  
Finite Element ◽  
Strain Gradient ◽  
Virtual Work ◽  
Hermite Polynomials ◽  
Gradient Elasticity ◽  
Weak Form ◽  
Element Formulation ◽  
Strain Gradient Elasticity ◽  
Rectangular Plates ◽  
Kirchhoff Plates

The current contribution is centered on bending of rectangular plates using the finite element method in the strain-gradient elasticity. To this aim, following introducing stresses and strains for a plate based on the Kirchhoff hypothesis, the principle of the virtual work is adopted to derive the weak form. Building upon Hermite polynomials and by deeming convergence requirements, four rectangular elements for the static analysis of strain-gradient plates are presented. To explore the performance of the proposed elements, particularly in small scales, some problems are solved and the results are compared with analytical solutions.

scholarly journals Finite Element Analyses of the Modified Strain Gradient Theory Based Kirchhoff Microplates

Surfaces ◽  
2021 ◽  
Vol 4 (2) ◽  
pp. 115-157
Author(s):  
Murat Kandaz ◽  
Hüsnü Dal
Keyword(s):  
Finite Element ◽  
Variational Problem ◽  
Strain Gradient ◽  
Degrees Of Freedom ◽  
Weak Form ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Shape Functions ◽  
Lagrange Equations ◽  
Modified Strain Gradient Theory

In this contribution, the variational problem for the Kirchhoff plate based on the modified strain gradient theory (MSGT) is derived, and the Euler-Lagrange equations governing the equation of motion are obtained. The Galerkin-type weak form, upon which the finite element method is constructed, is derived from the variational problem. The shape functions which satisfy the governing homogeneous partial differential equation are derived as extensions of Adini-Clough-Melosh (ACM) and Bogner-Fox-Schmit (BFS) plate element formulations by introducing additional curvature degrees of freedom (DOF) on each node. Based on the proposed set of shape functions, 20-, 24-, 28- and 32- DOF modified strain gradient theory-based higher-order Kirchhoff microplate element are proposed. The performance of the elements are demonstrated in terms of various tests and representative boundary value problems. Length scale parameters for gold are also proposed based on experiments reported in literature.

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