ὅδε οἶκος, ὦ ἑταῖρε, μνημεῖον ἐστιν ζῴων τῶν σοφῶν ἀνδρῶν, καὶ τῶν ἔργων αὐτῶν

**Mechanics Colloquim **

**PROGRAM**

MATEMATIČKI INSTITUT SANU

ODELJENJE ZA MEHANIKU

__PROGRAM ZA MAJ 2010.__

Pozivamo Vas da učestvujete u radu sednica Odeljenja i to:

__SREDA, 05. maj 2010. u 18 sati:__

* Ljubica Oparnica, Matematicki institut SANU *

** TALASI U VISKOELASTICNOM MEDIJU **

[1] T. M. Atanackovic and A. Guran. Theory of Elasticity for Scientists and Engineers. Birkhauser, Boston, 2000. [2] S. Konjik, Lj. Oparnica, and D.Zorica. Waves in viscoelastic media described by linear fractional model. Int. Transf. Spec. Func., 2011. [3] S. Konjik, Lj. Oparnica, and D. Zorica. Waves in fractional zener type viscoelastic media. J. Math. Anal. Appl., 365:259{286; doi: 10.1016/j.jma.2009.10.043, 2010.

__SREDA, 12. maj 2010. u 18 sati:__

* Prof. Milan Micunovic, Masinski fakultet Kragujevac *

** TWO DIMENSIONAL PLASTIC WAVES IN QRI VISCOPLASTIC MATERIALS **

It is known fact that initial yield stress under dynamic loading depends on strain rate or stress rate: at higher stress rates the initial stress yield is larger. On the other hand, the phenomenon of delayed yielding inherent to some metals and alloys is observed [4]: stress under dynamic loading exceeds its static value and plasticity starts after a certain time called delay time. Let plastic deformation commence at time t*. Denote by Y0 the initial equivalent static yield stress, and by the initial equivalent dynamic yield stress. Then, the accumulated plastic strain is governed by corresponding constitutive equation having the following form [2]: , where the kernel and . Here is an universal constant (which means the same for uniaxial tension, biaxial tension and shear) introduced and determined for AISI 316H steel in [3] as well as for ASME 537 steel in [1] within a very wide range of strain rates from to . In this lecture a simplest version of endochronic evolution is used permitting scaling of plastic strain rate, being very useful for calibration from low to almost impact strain rates. Let us introduce the stress tensor invariants: s1 = tr S, s2=tr SD2. Then, the model is described by the following tensor representation: with (1) A scalar coefficient ? is responsible for rate dependence whereas the tensor generators are: scalars: and ?(x) ? the Heaviside's function, ? ? a material constant. The stresses Y and Y0 are explained above. At first sight (1) seems to be rate independent. However, the rate dependence appears in stress rate dependent value of the initial yield stress Y, which has a triggering role for inelasticity onset. The model is named as QRI i.e quasi rate independent. According to experimental evidence [1] we assume that shears are neglected i.e. that in the Kroners decomposition rule elastic and plastic distortions as well as the total deformation gradient tensor are diagonal. Now, making use of evolution equations (1), geometric relations and momentum balance equations we arrive at wave equations (with velocity in-plate components ) (2) Here is transpose of dynamic state vector which completely describes viscoplastic acceleration wave. Like in [2], a solution of (2) is assumed to have the form where c is a wave speed and wave propagation takes place in the direction This gives rise to the Christoffels equation leading to the characteristic equation (3) Since the ( ) acoustic tensor has the form with non-constant static state vector we obtain like in [2] that plastic speeds are not constant unless small strains of few percents take place. This is contrary to the traditional viscoplastic wave theory (cf. [4]) where only elastic waves with constant wave speeds exist due to the fact that plastic stretching in traditional setting depends on stress and strain but not on stress rate as in quasi rate independent theory. The governing equations of the problem are solved numerically by means of a Finite Volume based procedure [5], which was implemented by using the software package UG (Unstructured Grids). The analysis concers a quadratic thin plate and has been performed by diverse boundary and initial conditions. Then the corresponding FEM analysis for quadratic thin plate has been performed for diverse boundary and initial conditions. Some numerical characteristic solutions are investigated at the boundary of initial elastic range as well as subsequent elastic range with advanced previous static plastic strain. References [1] M. Micunovic, C. Albertini, G. Pino, G. Maresca, Some results concerning damage acquired by cruciform Hopkinson bar technique, Proc. 10th EMMC10 (eds. Nowacki, W.K, Zhao, H.), Kazimerz Dolny, Poland, pp. 185-194, 2007. [2] M.Micunovic, A.Baltov, Plastic wave propagation in Hopkinson bar revisited, Archives of Mechanics, 54/5-6, 577-602, 2002. [3] M. Micunovic, C. Albertini, M. Montagnani, M. High strain rate viscoplasticity of AISI 316 stainless steel from tension and shear experiments, In: Solid Mechanics, ed., P. Miljanic, Serbian Acad. Sci.-Sci. Meetings 87, Dept. Techn. Sci. 3, 97-106, 1997. [4] W. K. Nowacki, Stress Waves in Non-Elastic Solids, Pergamon, Oxford, 1978. [5] M.Micunovic, A.Grillo, I.Muha, G.Wittum, Two Dimensional Plastic Waves in Quasi Rate Independent Viscoplastic Materials, ESMC2009, 7th Euromech Solid Mechanics Conference, Lisbon, 2009 [6] M.Micunovic, Thermomechanics of Viscoplasticity Fundamentals and Applications, Springer, New York, 2009.

__SREDA, 19. maj 2010. u 18 sati: __

* A. S. Franck, Potsdam Institute for Climate impact Research and Institute for Physics and Astronomy of Potsdam University, Germany *

Is there life beyond planet Earth? This is one of the grand enigmas which humankind tries to solve through scientific research. Recent progress in astrophysical measurement techniques has confirmed the existence of a multitude of extra-solar planets. On the other hand, up to now, we know only life forms existing on our home planet Earth. Therefore, we start our investigations with a model for the evolution of the Earth system under the external forcing of an increasing solar luminosity. The model for the global carbon cycle of the Earth contains the reservoirs mantle, ocean floor, continental crust, biosphere, and the kerogen, as well as the combined ocean and atmosphere reservoir. The model is specified by introducing three different types of biosphere: procaryotes, eucaryotes, and complex multicellular life. During the entire existence of the biosphere procaryotes are always present. 2 Gyr ago eucaryotic life first appears. The emergence of complex multicellular life is connected with an explosive increase in biomass and a strong ecrease in Cambrian global surface temperature at about 0.54 Gyr ago. In the long-term future the three types of biosphere will die out in reverse sequence of their appearance. A simplified Earth system model is applied for the investigation of the habitable zone (HZ). The HZ around a given central star is defined as the region within which an Earthlike planet might enjoy the moderate surface temperatures required for advanced life forms. Super-Earths are rocky planets from one to ten Earth masses with the same chemical land mineral composition as the Earth. We use scaling laws to obtain the total radius, mantle thickness and average density as a function of planetary mass. The HZ around Gl 581 for super-Earths with five and eight Earth masses has been calculated. Our results can be used to determine the average number of planets per planetary system that are within the HZ. With the help of a segment of the Drake equation, the number of habitable planets in the Milky Way is estimated. This leads to the thoroughly educated guess that there should exist about 50 millions habitable planets in our galaxy.

__PONEDELJAK, 24. maj 2010. OD 09-12 sati:__

* *

** PRISTUPNA PREDAVANjA NOVOIZABRANIH REDOVNIH CLANOVA SANU IZ OBLASITI MATEMATIKE FIZIKE, GEO NAUKA I TEHNICKIH NAUKA svecana sala SANU, drugi sprat **

Sednice se održavaju u zgradi SANU, Knez Mihailova 35, u sali 2 na prvom spratu.

Sekretar Odeljenja

Bojan Međo

Bojan Međo

Upravnik Odeljenja

Akademik Teodor Atanacković, s.r.

Akademik Teodor Atanacković, s.r.