scholarly journals A degenerating PDE system for phase transitions and damage

2014 ◽  
Vol 24 (07) ◽  
pp. 1265-1341 ◽  
Author(s):  
Elisabetta Rocca ◽  
Riccarda Rossi
Keyword(s):  
Evolution Equations ◽  
Momentum Equation ◽  
Damage Parameter ◽  
Absolute Temperature ◽  
Degenerate System ◽  
Weak Formulation ◽  
Damage Models ◽  
Pure Phases ◽  
Damage Processes ◽  
Weak Solvability

In this paper, we analyze a PDE system arising in the modeling of phase transition and damage phenomena in thermoviscoelastic materials. The resulting evolution equations in the unknowns ϑ (absolute temperature), u (displacement), and χ (phase/damage parameter) are strongly nonlinearly coupled. Moreover, the momentum equation for u contains χ-dependent elliptic operators, which degenerate at the pure phases (corresponding to the values χ = 0 and χ = 1), making the whole system degenerate. That is why, we have to resort to a suitable weak solvability notion for the analysis of the problem: it consists of the weak formulations of the heat and momentum equation, and, for the phase/damage parameter χ, of a generalization of the principle of virtual powers, partially mutuated from the theory of rate-independent damage processes. To prove an existence result for this weak formulation, an approximating problem is introduced, where the elliptic degeneracy of the displacement equation is ruled out: in the framework of damage models, this corresponds to allowing for partial damage only. For such an approximate system, global-in-time existence and well-posedness results are established in various cases. Then, the passage to the limit to the degenerate system is performed via suitable variational techniques.

scholarly journals CONTACT PROBLEMS FOR NONLINEARLY ELASTIC MATERIALS: WEAK SOLVABILITY INVOLVING DUAL LAGRANGE MULTIPLIERS

2010 ◽  
Vol 52 (2) ◽  
pp. 160-178 ◽  
Author(s):  
A. MATEI ◽  
R. CIURCEA
Keyword(s):  
Lagrange Multipliers ◽  
Variational Equation ◽  
Small Deformation ◽  
Contact Problems ◽  
Point Theory ◽  
The Body ◽  
Elastic Materials ◽  
Weak Formulation ◽  
Nonlinearly Elastic ◽  
Weak Solvability

AbstractA class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.

scholarly journals Stability of Leapfrog Constant-Coefficients Semi-Implicit Schemes for the Fully Elastic System of Euler Equations: Case with Orography

2005 ◽  
Vol 133 (5) ◽  
pp. 1065-1075 ◽  
Author(s):  
P. Bénard ◽  
J. Mašek ◽  
P. Smolíková
Keyword(s):  
Euler Equations ◽  
Evolution Equations ◽  
Momentum Equation ◽  
Primitive Equations ◽  
Prognostic Variables ◽  
Flat Terrain ◽  
Implicit Schemes ◽  
Constant Coefficients ◽  
The Stability ◽  
The One

Abstract The stability of constant-coefficients semi-implicit schemes for the hydrostatic primitive equations and the fully elastic Euler equations in the presence of explicitly treated thermal residuals has been theoretically examined in the earlier literature, but only for the case of a flat terrain. This paper extends these analyses to a case in which an orography is present, in the shape of a uniform slope. It is shown, with mass-based coordinates, that for the Euler equations, the presence of a slope reduces furthermore the set of the prognostic variables that can be used in the vertical momentum equation to maintain the robustness of the scheme, compared to the case of a flat terrain. The situation appears to be similar for systems cast in mass-based and height-based vertical coordinates. Still for mass-based vertical coordinates, an optimal prognostic variable is proposed and is shown to result in a robustness similar to the one observed for the hydrostatic primitive equations system. The prognostic variables that lead to robust semi-implicit schemes share the property of having cumbersome evolution equations, and an alternative time treatment of some terms is then required to keep the evolution equation reasonably simple. This treatment is shown not to modify substantially the stability of the time scheme. This study finally indicates that with a pertinent choice for the prognostic variables, mass-based vertical coordinates are equally suitable as height-based coordinates for efficiently solving the compressible Euler equations system.

scholarly journals Single-molecule imaging with longer X-ray laser pulses

IUCrJ ◽  
2015 ◽  
Vol 2 (6) ◽  
pp. 661-674 ◽  
Author(s):  
Andrew V. Martin ◽  
Justine K. Corso ◽  
Carl Caleman ◽  
Nicusor Timneanu ◽  
Harry M. Quiney
Keyword(s):  
Single Molecule ◽  
Laser Pulses ◽  
Bragg Diffraction ◽  
Single Molecule Imaging ◽  
X Ray ◽  
Damage Models ◽  
Statistical Fluctuations ◽  
Damage Processes ◽  
The Impact ◽  
Serial Femtosecond Crystallography

During the last five years, serial femtosecond crystallography using X-ray laser pulses has been developed into a powerful technique for determining the atomic structures of protein molecules from micrometre- and sub-micrometre-sized crystals. One of the key reasons for this success is the `self-gating' pulse effect, whereby the X-ray laser pulses do not need to outrun all radiation damage processes. Instead, X-ray-induced damage terminates the Bragg diffraction prior to the pulse completing its passage through the sample, as if the Bragg diffraction were generated by a shorter pulse of equal intensity. As a result, serial femtosecond crystallography does not need to be performed with pulses as short as 5–10 fs, but can succeed for pulses 50–100 fs in duration. It is shown here that a similar gating effect applies to single-molecule diffraction with respect to spatially uncorrelated damage processes like ionization and ion diffusion. The effect is clearly seen in calculations of the diffraction contrast, by calculating the diffraction of the average structure separately to the diffraction from statistical fluctuations of the structure due to damage (`damage noise'). The results suggest that sub-nanometre single-molecule imaging with 30–50 fs pulses, like those produced at currently operating facilities, should not yet be ruled out. The theory presented opens up new experimental avenues to measure the impact of damage on single-particle diffraction, which is needed to test damage models and to identify optimal imaging conditions.

scholarly journals On a nonlocal Cahn–Hilliard model permitting sharp interfaces

2021 ◽  
pp. 1-38
Author(s):  
Olena Burkovska ◽  
Max Gunzburger
Keyword(s):  
Inequality Constraints ◽  
Coupled System ◽  
Energy Functional ◽  
Sharp Interface ◽  
Nonlocal Model ◽  
Weak Formulation ◽  
Diffuse Interfaces ◽  
Spatial Dimensions ◽  
Pure Phases ◽  
Sharp Interfaces

A nonlocal Cahn–Hilliard model with a non-smooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg–Landau energy functional is defined which recovers the classical (local) model as the extent of nonlocal interactions vanish. In contrast to the local Cahn–Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. Here, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak formulation, lead to a coupled system of variational inequalities which at each time instance can be restated as a constrained optimization problem. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions, and derive the conditions under which pure phases are admitted. Moreover, we develop discretizations of the problem based on finite element methods and implicit–explicit time-stepping methods that can be realized efficiently. Finally, we illustrate our theoretical findings through several numerical experiments in one and two spatial dimensions that highlight the differences in features of local and nonlocal solutions and also the sharp interface properties of the nonlocal model.

Simulations of Cyclic Plasticity and Fatigue Behavior of Structural Steels under Multiaxial Loading

2006 ◽  
Vol 514-516 ◽  
pp. 1414-1418
Author(s):  
Bin Li ◽  
Luís G. Reis ◽  
Manuel de Freitas
Keyword(s):  
Fatigue Behavior ◽  
Cyclic Stress ◽  
Damage Parameter ◽  
Cyclic Plasticity ◽  
Strong Interactions ◽  
Stress Strain ◽  
Damage Models ◽  
Asme Code ◽  
Strain Responses ◽  
Plastic Stress

The local cyclic elastic-plastic stress-strain responses were simulated using the incremental plasticity procedures of ABAQUS finite element code. It is shown a better understanding on the evolutions of the local cyclic stress-strain and the strong interactions between the most stressed material element and its neighboring material elements in the plastic deformations and stress redistributions. Based on the stress/strain states of the stabilized cycle, a new damage parameter, proposed on modification of the ASME code parameter, is applied and improved correlations between the predicted and the experimental fatigue lives are shown. It is concluded that the improvement of fatigue life prediction depends not only on the fatigue damage models, but also on the accurate evaluations of the cyclic elasto-plastic stress/strain responses.

scholarly journals Weak formulation of free-surface wave equations

2001 ◽  
Vol 5 (2) ◽  
pp. 75-85
Author(s):  
A. D. Sneyd
Keyword(s):  
Free Surface ◽  
Evolution Equation ◽  
Wave Equations ◽  
Evolution Equations ◽  
Surface Displacement ◽  
Wave Dispersion ◽  
The Body ◽  
Weak Formulation ◽  
Free Surface Displacement ◽  
Free Surface Wave

An alternative method for deriving water wave dispersion relations and evolution equations is to use a weak formulation. The free-surface displacement η is written as an eigenfunction expansion, η=∑n=1∞αn(t)En where the αn(t) are time-dependent coefficients. For a tank with vertical sides the En are eigenfunctions of the eigenvalue problem, ∇2+λ2E=0,  ∇E⋅n^=0 on the tank side walls. Evolution equations for the αn(t) can be obtained by taking inner products of the linearised equation of motion, ρ∂v∂t=−1ρ∇P+F with the normal irrotational wave modes. For unforced waves each evolution equation is a simple harmonic oscillator, but the method is most useful when the body force F is something more exotic than gravity. It can always be represented by a forcing term in the SHM evolution equation, and it is not necessary to assume F irrotational. Several applications are considered: the Faraday experiment, generation of surface waves by an unsteady magnetic field, and the metal-pad instability in aluminium reduction cells.

scholarly journals A Review on Multiscale Bone Damage: From the Clinical to the Research Perspective

Materials ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1240
Author(s):  
Federica Buccino ◽  
Chiara Colombo ◽  
Laura Maria Vergani
Keyword(s):  
Clinical Practice ◽  
Bone Fractures ◽  
Imaging Techniques ◽  
Actual State ◽  
Micro Computed Tomography ◽  
Research Perspective ◽  
Damage Models ◽  
Age Related ◽  
Bone Damage ◽  
Damage Processes

The investigation of bone damage processes is a crucial point to understand the mechanisms of age-related bone fractures. In order to reduce their impact, early diagnosis is key. The intricate architecture of bone and the complexity of multiscale damage processes make fracture prediction an ambitious goal. This review, supported by a detailed analysis of bone damage physical principles, aims at presenting a critical overview of how multiscale imaging techniques could be used to implement reliable and validated numerical tools for the study and prediction of bone fractures. While macro- and meso-scale imaging find applications in clinical practice, micro- and nano-scale imaging are commonly used only for research purposes, with the objective to extract fragility indexes. Those images are used as a source for multiscale computational damage models. As an example, micro-computed tomography (micro-CT) images in combination with micro-finite element models could shed some light on the comprehension of the interaction between micro-cracks and micro-scale bone features. As future insights, the actual state of technology suggests that these models could be a potential substitute for invasive clinical practice for the prediction of age-related bone fractures. However, the translation to clinical practice requires experimental validation, which is still in progress.

A Mesh Independent GTN Damage Model and Its Application in Simulation of Ductile Fracture Behaviour

2008 ◽  
Author(s):  
M. K. Samal ◽  
M. Seidenfuss ◽  
E. Roos ◽  
B. K. Dutta ◽  
H. S. Kushwaha
Keyword(s):  
Ductile Fracture ◽  
Fracture Resistance ◽  
Damage Mechanics ◽  
Wide Spectrum ◽  
Weighted Average ◽  
Damage Parameter ◽  
Nonlocal Model ◽  
Fe Method ◽  
Diffusion Type ◽  
Damage Models

Ductile fracture process involves the typical stages of nucleation, growth and coalescence of voids in the micro-scale. In order to take the effects of these voids on the stress carrying capability of a mechanical continuum during simulation, damage mechanics models, such as those of Rousselier and Gurson-Tvergaard-Needleman (GTN) are widely used. These have been highly successful in simulating the fracture resistance behaviour of different specimens and components made of a wide spectrum of engineering steels. However, apart from the material parameters, a characteristic length parameter has to be used as a measure of the size of the discretisation in the zone of crack propagation. This inherent limitation of these local damage models prevents them from simulating the stress distribution near the sharp stress gradients satisfactorily, especially at transition temperature regime. There have been efforts in literature to overcome the effect of mesh-dependency by development of nonlocal and gradient damage models. A nonlocal measure (weighted average of a quantity over a characteristics volume) of damage is usually used in the material constitutive equation. In this paper, the authors have extended the GTN model to its nonlocal form using damage parameter ‘d’ as a degree of freedom in the finite element (FE) formulation. The evolution of the nonlocal damage is related to the actual void volume faction ‘f’ in ductile fracture using a diffusion type equation. The coupled mechanical equilibrium and damage diffusion equations have been discretised using FE method. In order to demonstrate the mesh independent nature of the new formulation, a standard fracture mechanics specimen (i.e., 1T compact tension) has been analysed using different mesh sizes near the crack tip and the results have been compared with those of experiment. The results of the nonlocal model have also been compared with those of the local model. The effect of different GTN parameters on the fracture resistance behaviour of this specimen has been studied for the nonlocal model and these results have been compared with those of experiment.

scholarly journals Momentum Balance and Eliassen–Palm Flux on Moist Isentropic Surfaces

2016 ◽  
Vol 73 (3) ◽  
pp. 1293-1314 ◽  
Author(s):  
Ray Yamada ◽  
Olivier Pauluis
Keyword(s):  
Momentum Equation ◽  
Momentum Balance ◽  
Similar Increase ◽  
Moist Air ◽  
Momentum Exchange ◽  
Weak Formulation ◽  
Horizontal Structure ◽  
Vertical Coordinate ◽  
Momentum Budget ◽  
Conditional Averaging

Abstract Previous formulations for the zonally averaged momentum budget and Eliassen–Palm (EP) flux diagnostics do not adequately account for moist dynamics, since air parcels are not differentiated by their moisture content when averages are taken. The difficulty in formulating the momentum budget in moist coordinates lies in the fact that they are generally not invertible with height. Here, a conditional-averaging approach is used to derive a weak formulation of the momentum budget and EP flux in terms of a general vertical coordinate that is not assumed to be invertible. The generalized equation reduces to the typical mass-weighted zonal-mean momentum equation for invertible vertical coordinates. The weak formulation is applied here to study the momentum budget on moist isentropes. Recent studies have shown that the meridional mass transport in the midlatitudes is twice as strong on moist isentropes as on dry isentropes. It is shown here that this implies a similar increase in the EP flux between the dry and moist frameworks. Physically, the increase in momentum exchange is tied to an enhancement of the form drag associated with the horizontal structure of midlatitude eddies, where the poleward flow of moist air is located in regions of strong eastward pressure gradient.

Application of Damage Models to Reliability Investigations for Input/Output Electronic Connectors

2006 ◽  
Author(s):  
C.-C. Chang ◽  
K.-C. Liao
Keyword(s):  
Contact Force ◽  
Signal Transmission ◽  
Damage Parameter ◽  
Contact Forces ◽  
Input Output ◽  
Minimum Requirement ◽  
Damage Models ◽  
Terminal Design ◽  
Force Reduction ◽  
Number Of Cycles

Contact forces between terminals of an electronic connector and the corresponding counterparts play an important role on signal transmission. The mated terminal with insufficient contact force might severely raise electrical resistance and induce intermittence or disconnection of current eventually. For input/output connectors, however, the contact force of the terminal usually decays dramatically after several thousand mating/unmating cycles. Critical plane approaches are adopted to estimate the "application duration" which indicates the number of cycles as the contact force of the terminal degrades beneath the corresponding minimum requirement in the present study. Damage parameters based on various criteria are evaluated for the terminal under cyclic loading conditions. Relationships among the damage parameter, the contact force reduction ratio, and the number of cycles are then constructed by linking numerical results to experimental measurements. Proposed models could be further applied to reliability investigations for the similar terminal design.

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