scholarly journals Thermodynamic Foundation of Generalized Variational Principle

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Variational Principle ◽  
Multiplier Method ◽  
Generalized Variational Principle ◽  
Lagrangian Multiplier ◽  
Stress Strain ◽  
Strain Relation ◽  
Stress Strain Relation ◽  
Lagrangian Multiplier Method ◽  
Open Question

One open question remains regarding the theory of the generalized variational principle, that is, why the stress-strain relation still be derived from the generalized variational principle while the Lagrangian multiplier method is applied in vain? This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics. As long as the functional has one of the combination $A(\epsilon_{ij})-\sigma_{ij}\epsilon_{ij}$ or $B(\sigma_{ij})-\sigma_{ij}\epsilon_{ij}$, its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. It is proved herein that the Hu-Washizu functional $\Pi_{HW}[u_i,\epsilon_{ij},\sigma_{ij}]$ and Hu-Washizu variational principle comprise a real three-field functional.

scholarly journals Thermodynamic Foundation of Generalized Variational Principle

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Variational Principle ◽  
Multiplier Method ◽  
Generalized Variational Principle ◽  
Lagrangian Multiplier ◽  
Stress Strain ◽  
Strain Relation ◽  
Stress Strain Relation ◽  
Lagrangian Multiplier Method ◽  
Open Question ◽  
Special Case

One long-standing open question remains regarding the theory of the generalized variational principle, that is, why can the stress-strain relation still be derived from the generalized variational principle while the method of Lagrangian multiplier method is applied in vain? This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics. As long as the functional has one of the combination $A(\epsilon_{ij})-\sigma_{ij}\epsilon_{ij}$ or $B(\sigma_{ij})-\sigma_{ij}\epsilon_{ij}$, its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. It is proved herein that the Hu-Washizu functional $\Pi_{HW}[u_i,\epsilon_{ij},\sigma_{ij}]$ and Hu-Washizu variational principle comprise a real three-field functional. In addition, that Chien's functional $\Pi_{Q}[u_i,\epsilon_{ij},\sigma_{ij},\lambda]$ is a much more general four-field functional and that the Hu-Washizu functional is its special case as $\lambda=0$ are confirmed.

Parametric Variational Principle for Bi-Modulus Materials and Its Application to Nacreous Bio-Composites

2016 ◽  
Vol 08 (06) ◽  
pp. 1650082 ◽  
Author(s):  
Liang Zhang ◽  
Huiting Zhang ◽  
Jian Wu ◽  
Bo Yan ◽  
Mengkai Lu
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Variational Principle ◽  
Principal Stress ◽  
Stress Strain ◽  
Element Analysis ◽  
Parametric Variational Principle ◽  
Strain Relation ◽  
Stress Strain Relation ◽  
Nonlinear Stress

Bi-modulus materials have different moduli in tension and compression and the stress–strain relation depends on principal stress that is unknown before displacement is determined. Establishment of variational principle is important for mechanical analysis of materials. First, parametric variational principle (PVP) is proposed for static analysis of bi-modulus materials and structures. A parametric variable indicating state of principal stress is included in the potential energy formulation and the nonlinear stress–strain relation is evolved into a linear complementarity constraint. Convergence of finite element analysis is thus improved. Then the proposed variational principle is extended to a dynamic problem and the dynamic equation can be derived based on Hamilton’s principle. Finite element analysis of nacreous bio-composites is performed, in which a unilateral contact behavior between two hard mineral bricks is modeled using the bi-modulus stress–strain relation. Effective modulus of composites can be determined numerically and stress mechanism of “tension–shear chain” in nacre is revealed. A delayed effect on stress propagation is found around the “gaps” between mineral bricks, when a tension force is loaded to nacreous bio-composites dynamically.

scholarly journals Limit analysis of narrow support elements in W7-X considering the serration effect of the stress–strain relation at 4K

2011 ◽  
Vol 86 (6-8) ◽  
pp. 1462-1465 ◽  
Author(s):  
E. Briani ◽  
C. Gianini ◽  
F. Lucca ◽  
A. Marin ◽  
J. Fellinger ◽  
...  
Keyword(s):  
Limit Analysis ◽  
Stress Strain ◽  
Strain Relation ◽  
Stress Strain Relation

Analytical stress-strain relation for soils

2021 ◽  
Author(s):  
Kristian Krabbenhoft ◽  
J. Wang
Keyword(s):  
Test Data ◽  
Narrow Range ◽  
Strain Range ◽  
Data Sets ◽  
Soil Tests ◽  
Stress Strain ◽  
Strain Relation ◽  
Stress Strain Relation ◽  
Typical Test ◽  
Typical Soil

A new stress-strain relation capable of reproducing the entire stress-strain range of typical soil tests is presented. The new relation involves a total of five parameters, four of which can be inferred directly from typical test data. The fifth parameter is a fitting parameter with a relatively narrow range. The capabilities of the new relation is demonstrated by the application to various clay and sand data sets.

Stress-Strain Relations and Vibrations of a Granular Medium

1957 ◽  
Vol 24 (4) ◽  
pp. 585-593
Author(s):  
J. Duffy ◽  
R. D. Mindlin
Keyword(s):  
Energy Dissipation ◽  
Contact Forces ◽  
Face Centered Cubic ◽  
Stress Strain ◽  
Differential Stress ◽  
Elastic Bodies ◽  
Strain Relation ◽  
Stress Strain Relation ◽  
Face Centered ◽  
Experimental Values

Abstract A differential stress-strain relation is derived for a medium composed of a face-centered cubic array of elastic spheres in contact. The stress-strain relation is based on the theory of elastic bodies in contact, and includes the effects of both normal and tangential components of contact forces. A description is given of an experiment performed as a test of the contact theories and the differential stress-strain relation derived from them. The experiment consists of a determination of wave velocities and the accompanying rates of energy dissipation in granular bars composed of face-centered cubic arrays of spheres. Experimental results indicate a close agreement between the theoretical and experimental values of wave velocity. However, as in previous experiments with single contacts, the rate of energy dissipation is found to be proportional to the square of the maximum tangential contact force rather than to the cube, as predicted by the theory for small amplitudes.

scholarly journals CONCENTRIC LOADING TESTS OF CONCRETE FILLED SPIRAL STEEL TUBES AND STRESS-STRAIN RELATION OF CONCRETE CONFINED BY STEEL TUBES

2009 ◽  
Vol 65 (4) ◽  
pp. 548-563 ◽  
Author(s):  
Mitsuyoshi AKIYAMA ◽  
Hideki NAITO ◽  
Kiyoshi ONO ◽  
Nobutaka SHIRAHAMA ◽  
Daisuke MATSUMOTO ◽  
...  
Keyword(s):  
Stress Strain ◽  
Steel Tubes ◽  
Strain Relation ◽  
Stress Strain Relation ◽  
Loading Tests ◽  
Concentric Loading
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