Energy Principles and Deflection of Beam

2021 ◽  
pp. 157-180
Author(s):  
Debabrata Podder ◽  
Santanu Chatterjee
Keyword(s):  
Energy Principles

scholarly journals RESIDUAL DISPLACEMENTS‘ PROGRESIVE ANALYSIS OF THE MULTISUPPORTED BEAM / DAUGIAATRAMĖS SIJOS LIEKAMŲJŲ POSLINKIŲ PROGRESINĖ ANALIZĖ

2014 ◽  
Vol 6 (5) ◽  
pp. 461-467 ◽  
Author(s):  
Liudas Liepa ◽  
Agnė Gervytė ◽  
Ela Jarmolajeva ◽  
Juozas Atkočiūnas
Keyword(s):  
Numerical Methods ◽  
Mathematical Models ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Analysis Problem ◽  
Residual Displacements ◽  
Repeated Load ◽  
Energy Principles

This paper focuses on a shakedown behaviour of the ideally elasto-plastic beams system under variable repeated load. The mathematical models of the analysis problems are created using numerical methods, extremum energy principles and mathematic programming. It is shown that during the shakedown process the residual displacements vary non-monotonically. By solving analysis problem, where the load locus is being progressively expanded, it is possible to determine the upper and lower bounds of residual displacements. Suggested methods are ilustrated by solving multisupported beam example problem. The results are obtained considering principle of the small displacements. Nagrinėjamas idealiai tampriai plastinės lenkiamos strypinės sistemos prisitaikomumo būvis, veikiant kartotinei kintamajai apkrovai. Analizės uždavinių matematiniai modeliai sudaromi, pasitelkus skaitinius metodus, ekstreminius energinius principus ir matematinį programavimą. Parodoma, kad prisitaikant konstrukcijai jos liekamieji poslinkiai gali kisti nemonotoniškai. Išsprendus analizės uždavinį, kuriame progresyviai plečiama apkrovos veikimo sritis, galima nustatyti viršutines ir apatines liekamųjų poslinkių kitimo ribas. Siūloma metodika iliustruota daugiaatramės sijos liekamųjų poslinkių skaičiavimo pavyzdžiu. Rezultatai gauti, esant mažų poslinkių prielaidai.

scholarly journals Minimum energy bounds on longitudinal elastic constants of transversely isotropic unidirectional composites

2020 ◽  
Vol 476 (2239) ◽  
pp. 20190752
Author(s):  
Duc-Chinh pham
Keyword(s):  
Elastic Constants ◽  
Minimum Energy ◽  
Young Modulus ◽  
Transversely Isotropic ◽  
Longitudinal Axis ◽  
Arithmetic Average ◽  
Energy Bounds ◽  
Cylindrical Phase ◽  
Elastic Composites ◽  
Energy Principles

We consider the n -component transversely isotropic unidirectional elastic composites, the longitudinal axis of which is parallel to those of the transversely isotropic components as well as the generators of the cylindrical phase boundaries between them. From the minimum energy and complementary energy principles, with appropriate constant strain and piece-wise constant stress trial fields, optimization and iteration techniques, a set of bounds for the macroscopic (effective) longitudinal elastic constants of the composites (including the simple lower arithmetic average estimate for longitudinal Young modulus E eff  ≥  E V ) are constructed. Numerical examples are provided to illustrate the obtained results.

scholarly journals The compton scattering and the new statistics

1929 ◽  
Vol 125 (796) ◽  
pp. 231-237 ◽  
Keyword(s):  
Compton Scattering ◽  
Electron Gas ◽  
Compton Effect ◽  
Maxwellian Distribution ◽  
Great Success ◽  
Free Electrons ◽  
Electron Theory ◽  
Conservation Of Momentum ◽  
New Statistics ◽  
Energy Principles

Great success has been achieved by Sommerfeld in the electron theory of metals by assuming that there are free electrons in them which obey the Fermi-Dirac statistics. It has been assumed in the case of univalent metals that on the average one electron per atom is free. In general, however, the valency electrons can be considered as free. These free electrons will take part in the Compton scattering. The analysis of such a Compton effect reduces to the analysis of the collisions between radiation quanta and an electron gas. The general features of such a scattering was first considered by Dirac. But he has assumed a Maxwellian distribution for the electrons which will not be applicable to the case under consideration, because the electrons in a conductor being degenerate do not obey the Maxwell's law, but the Fermian distribution. In considering such a process we take it that the conservation of momentum and energy principles are satisfied for each particular collision just as in Compton's theory—only we are here dealing with moving electrons instead of stationary electrons which Compton considers. Thus electrons of different momenta components will produce different Compton shifts, and the intensity of any particular shift will depend on the number of electrons in that state. Thus we have to average for the radiation falling on an assembly of electrons whose momenta are distributed according to the Fermi-Dirac law.

scholarly journals Bounding of Flow and Transport Analysis in Heterogeneous Saturated Porous Media: A Minimum Energy Dissipation Principle for the Bounding and Scale-Up

Hydrology ◽  
2019 ◽  
Vol 6 (2) ◽  
pp. 33 ◽  
Author(s):  
Nelson ◽  
Williams
Keyword(s):  
Porous Media ◽  
Energy Dissipation ◽  
Heterogeneous System ◽  
Minimum Energy ◽  
Scale Up ◽  
Heterogeneous Porous Media ◽  
Porous Media Flow ◽  
Rotational Flow ◽  
Energy Principles

We apply minimum kinetic energy principles from classic mechanics to heterogeneous porous media flow equations to derive and evaluate rotational flow components to determine bounding homogenous representations. Kelvin characterized irrotational motions in terms of energy dissipation and showed that minimum dynamic energy dissipation occurs if the motion is irrotational; i.e., a homogeneous flow system. For porous media flow, reductions in rotational flow represent heterogeneity reductions. At the limit, a homogeneous system, flow is irrotational. Using these principles, we can find a homogenous system that bounds a more complex heterogeneous system. We present mathematics for using the minimum energy principle to describe flow in heterogeneous porous media along with reduced special cases with the necessary bounding and associated scale-up equations. The first, simple derivation involves no boundary differences and gives results based on direct Kelvin-type minimum energy principles. It provides bounding criteria, but yields only a single ultimate scale-up. We present an extended derivation that considers differing boundaries, which may occur between scale-up elements. This approach enables a piecewise less heterogeneous representation to bound the more heterogeneous system. It provides scale-up flexibility for individual model elements with differing sizes, and shapes and supports a more accurate representation of material properties. We include a case study to illustrate bounding with a single direct scale-up. The case study demonstrates rigorous bounding and provides insight on using bounding flow to help understand heterogeneous systems. This work provides a theoretical basis for developing bounding models of flow systems. This provides a means to justify bounding conditions and results.

An energy-based approach for modelling the behaviour of packaging material during processing

2004 ◽  
Vol 218 (1) ◽  
pp. 105-118 ◽  
Author(s):  
B J Hicks ◽  
C Berry ◽  
G Mullineux ◽  
C J McPherson ◽  
A J Medl
Keyword(s):  
Theoretical Model ◽  
Minimum Energy ◽  
Packaging Material ◽  
Parametric Models ◽  
Machine Material ◽  
Software Model ◽  
Energy Principles ◽  
Improved Methods ◽  
Material Interaction ◽  
Modelling Approach

This paper deals with the investigation of improved methods for considering machine-material interaction during the design and production of packaging machinery. Minimum energy principles are used to create a theoretical model of the response of the packaging material during processing. The complex non-linear properties of the packaging material are encapsulated in parametric models generated through analysis of the physical measurement of the changing properties during processing. These two techniques are incorporated into a software model that represents the behaviour of a skillet during the erection process. This software model considers the material, the pack design and the machine system. The overall modelling approach is validated by comparison with a physical system, which shows a good correlation with the theoretical model.

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