Back to Search
Start Over
New insights on free energies and Saint-Venant’s principle in viscoelasticity
- Source :
- Articles
- Publication Year :
- 2014
- Publisher :
- Elsevier BV, 2014.
-
Abstract
- Explicit expressions for the minimum free energy of a linear viscoelastic material and Noll’s definition of state are used here to explore spatial energy decay estimates for viscoelastic bodies, in the full dynamical case and in the quasi-static approximation. In the inertial case, Chirita et al. obtained a certain spatial decay inequality for a space–time integral over a portion of the body and over a finite time interval of the total mechanical energy. This involves the work done on histories, which is not a function of state in general. Here it is shown that for free energies which are functions of state and obey a certain reasonable property, the spatial decay of the corresponding space–time integral is stronger than the one involving the work done on the past history. It turns out that the bound obtained is optimal for the minimal free energy. Two cases are discussed for the quasi-static approximation. The first case deals with general states, so that general histories belonging to the equivalence class of any given state can be considered. The continuity of the stress functional with respect to the norm based on the minimal free energy is proved, and the energy measure based on the minimal free energy turns out to obey the decay inequality derived Chirita et al. for the quasi-static case. The second case explores a crucial point for viscoelastic materials, namely that the response is influenced by the rate of application of loads. Quite surprisingly, the analysis of this phenomenon in the context of Saint-Venant principles has never been carried out explicitly before, even in the linear case. This effect is explored by considering states, the related histories of which are sinusoidal. The spatial decay parameter is shown to be frequency-dependent, i.e. it depends on the rate of load application, and it is proved that of those considered, the most conservative estimate of the frequency-dependent decay is associated with the minimal free energy. A comparison is made of the results for sinusoidal histories at low frequencies and general histories.
- Subjects :
- Inertial frame of reference
Context (language use)
02 engineering and technology
01 natural sciences
Dissipation rate
State function
Materials Science(all)
0203 mechanical engineering
Modelling and Simulation
Saint Venant principle
Integral element
General Materials Science
Computer Engineering
Free energy
0101 mathematics
Equivalence class
Mechanical energy
Mathematics
State in viscoelasticity
Spatial decay
Saint-Venant's Principle
Applied Mathematics
Mechanical Engineering
Mathematical analysis
Viscoelasticity
State (functional analysis)
Condensed Matter Physics
010101 applied mathematics
020303 mechanical engineering & transports
Mechanics of Materials
Modeling and Simulation
Residual stress decay
Subjects
Details
- ISSN :
- 00207683
- Volume :
- 51
- Database :
- OpenAIRE
- Journal :
- International Journal of Solids and Structures
- Accession number :
- edsair.doi.dedup.....1586c97d78cb837edb29ed768a94ff87