String theory and cosmic connection during superplastic flow

J.D. Munoz-Andrade

The objective of this work is to describe the string theory andcosmic connection during super plastic flow in spatially extendedpolycrystalline systems (SEPCS), where the phenomenology andmechanics of the crystalline structure of the universe in a relativisticframework at Max Planck scale plays an important role. Accordingto string theory as the unified theory of every thing, dislocationsdynamics is related in nature form with the string theory. Conse-quently, dislocations at Max Plank scale are the universal wayof working of every thing. In this physical framework, the theore-

tical model of Munoz-Andrade allows obtained the activation en-ergy for super plastic flow in SEPCS. Finally, the main results ofthis work are analyzed in the context of the unified interpretation ofstring theory as cosmic connection during Hubble flow, plastic flowand super plastic flow.

Keywords: String Theory, Quantum Mechanics, Special Relativ-ity Theory, Cosmic Micro Mechanics Connection, Super PlasticFlow, Plastic Flow, Hubble Flow, Dislocation Dynamics.

1 Introduction

In physics the universal validity of a mathematical modelallows to define a theory, for example in the especial relativitytheory development by Albert Einstein, the relationship E =mc2 defined their legality. In this framework, the cosmic mi-cromechanics connection during super plastic flow has beenstudied and worked out in order to become a part of the unifiedtheory of every thing in the universe. At this point, it is im-portant to mentioned that the main results obtained until now,tell us that there is a correspondence law between the expan-sion process of the universe at Max Planck scale (MPS), Edw-ing Hubble scale (EHS) and plastic flow (PF) in spatially ex-tended single crystals and polycrystalline systems at atomicscale and super plastic flow (SPF) in spatially extended poly-crystalline systems (SEPCS) at mesoscopic scale. Inside thecrystalline state of substances their linear defects named dis-locations perform as important role in atomic scale and in thepolycrystalline substances the cellular dislocations dynamicsplays an essential responsibility during creep, plastic flow andsuper plastic flow (SPF) at mesoscopic scale. Therefore, it isinteresting to examine that the dislocation line cannot end in-side a crystal. It must whichever leave the crystal with edgeend at the crystal surface or form a closed dislocation loop.Consequently, to evaluate the dislocations dynamics as vibrat-ing strings, it is necessary to know the energy and effectivemass of a moving dislocation in the elastic field of a crystalduring creep, plastic flow and SPF. For that reason, disloca-tions mechanics as a basis of the theory of creep, plasticity andsuper plasticity could be associated with string theory as theunified theory of every thing, where electrons and quarks,which are the elementary particles of atoms, are actuallytiny loops of vibrating string. In view of that, the string theoryhas been resolved the incompatibility between quantum me-chanics and general relativity [1]. In this context a mathema-tical model to obtain the activation energy for SPF is pro-posed.

2 Quantum mechanics and specialrelativity theory during super plastic flow

An essential consideration of the phenomenology and me-chanics of SPF in SEPCS associated with quantum mechanicsand the especial relativity theory is that by similarity with theradiant energy of a photon of a given frequency (m) only beemitted and absorbed in “quanta” of energy given by E = hm,where h is the Planck constant (h = 6.6260755 x 10-34 J sec =6.626 x 10-27 erg sec), in this manner the relationship betweendeformation or disturbance of the elastic field in a crystal lat-tice of a given frequency n? could only be emitted and ab-sorbed in “quanta” of given by E? = hn?, and the typical ther-mal energy for an oscillator is given by kT, where , k is theBoltzmann constant (k = 8.617x10-5eV/K = 1.38x10-23J/K), Tis the absolute temperature. Also, n? is the strain rate for theirreversible deformation processes associated with the Oro-wan equation for plastic flow (n?= q? t? k?.), where k? isthe Burgers vector for dislocations, q? is the density of dis-locations, t? is the average glide velocity of dislocations. Ac-cording to this assumption, the phenomenology of a pair pro-duction process from a photon into a positron and an electronor a par annihilation where a positron combines with an elec-tron to form a bound system called a positronium, which has ashort lived, decaying into photons as a pair annihilation [2], itis an equivalent process that could be possible under the cor-respondence law in the crystalline state related to the move-ment of the unitary dislocations on the elastic field associatedto the dislocation dissociation reaction in two partial disloca-tions, the displacement vector associated to dislocation iscalled the Burgers vector of a dislocation b or k? in thiswork, as it shown in Fig. 1. Consequently, with the similaritiesof both phenomenons, dislocations of opposite sign merge inthe crystal lattice annihilation takes place, resulting in the eli-mination of two linear defects inside the crystal, with the re-unification of the regular atomic place [3].

Quantum mechanics theory is of fundamental significancefor physics and can be used to predict the behaviour of atomicand subatomic systems and also in the macroscopic domainwhere classical physics is applicable. According to de Brogliecontributions for matter and for radiation in a similar way thetotal energy E? of an entity, in this case a dislocation, is related

Mat.-wiss. u. Werkstofftech. 2008, 39, No. 4-5 DOI: 10.1002/mawe.200800307

F 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 363

to the frequency n? of the wave or strain rate associated withits motion by the equation:

E? ¼ hn?: ð1Þ

And the moment p? of the entity (dislocation) is related tothe wavelength k? of the associated wave by the relationship:

p? ¼ h

k?: ð2Þ

Consequently, the wavelength of a dislocation k? wave as-sociated with its motion having a moment p?, mass m? andvelocity t?, it predicts by the Broglie wavelength:

k? ¼ h

p?¼ h

m?t?: ð3Þ

Therefore, the phase velocity de Broglie t?ph for a dislo-cation is then:

t?ph ¼ k?n? ¼ h

p?

� �E?h

� �¼ E?

p?: ð4Þ

Recently, under these physical theories, it was proposed byMunoz-Andrade the follow theoretical model to obtain the ac-tivation energy for SPF in SEPCS related with the cosmic mi-cromechanics connection (CMC) [4 – 7]:

Q? ¼ �mkT lnp?t?k

2?

c

� �¼ �mkT ln

n?k?c

� �: ð5Þ

Where: Q? is the activation energy for glide cellular dislo-cations in SEPS, m is the strain rate sensitiveðm ¼ lnðr2=r1Þ=lnðn?2=n?1Þ:, where r1 and r2 are the ap-plied stresses, k is the Boltzmann constant (k = 8.617x10-5

eV/K = 1.38x10-23J/K), T is the absolute temperature, n? isthe strain rate for the irreversible deformation processes asso-ciated with the Orowan equation for plastic flow (n?= q? t?k?.), k? is the Burgers vector for dislocations, q? is the den-sity of dislocations, t? is the average glide velocity of dislo-cations and c is the speed of light (c = 299792458 m/s).

Consequently, by introducing the de Broglie phase velocityt?ph of the equation (4) into the equation (5), it is possible torewrite such equation as follow

Q? ¼ �mkT lnt?ph

c

� �¼ �mkT ln

E?p?c

� �: ð6Þ

A central part of the phenomenology and mechanics of SPFin SEPCS associated with quantum mechanics and the espe-cial relativity theory is that by analogy with the energy of aphoton E = hm, the relationship between deformation or dis-turbance of the elastic field in a crystal lattice of a given fre-quency n? could only be emitted and absorbed in “quanta” ofgiven by E? = hn?, and the typical thermal energy for an os-cillator given by kT, in order to obtain in a equivalent way ofthe Eq. (5) and Eq. (6), the activation energy associated withdislocation dynamics for plastic flow or cellular dislocationdynamics for super plastic flow, as follow:

Q? ¼ �mkT lnhn?kT

� �: ð7Þ

At the present, by applying the essential equations de-scribed above from the analyses of quantum mechanics andrelativity theory during plastic flow and SPF of SEPCS, inthe next section some experimental researches related withthe activation energy for irreversible deformations processesof advanced structural materials are considered in order toevaluate the validation of this contribution.

3 Activation energy for super plasticflow of Al-Cu eutectic alloy

The evaluation of the activation energy for SPF of SEPCSby experimental techniques and theoretical methods are verysignificant for engineering proposed. In this field the values ofstrain rate sensitive (m) of the flow stress are exceptionallyimportant to establish the super plastic behavior in advancedmaterials, under certain thermo mechanical conditions m va-lues of 0.5 or higher could be development in SEPCS duringirreversible deformation process. Therefore, in order to findout the activation energy for SPF dependence on strain ratesensitive m, let us to analyze the experimental data reportedin the past for the Al–Cu eutectic alloy at 793 K [8]. In Fig. 2the dependence of strain rate sensitivity on strain rate at 793 K,it is shown. From that information and by the application ofthe Eq. (7) the activation energy at different values of strainrate sensitive has been obtained in order to established theirdependence, as it is shown in Fig. 3. Additionally, the depend-ence of the activation energy for SPF on the phase velocity deBroglie, also a cellular dislocation k? wave were obtained,these results are shown in Fig. 4.

4 Discussion

The size of the strings of string theory are related with thedimension of Max Planck length kP = Planck length =1.62x10-35m. Recently in this framework the cosmic micro-mechanics connection associated with the unified interpreta-tion of Hubble flow, plastic flow and super plastic flow was

Fig. 1. Schematic representation of the pair production of partialdislocations in a similar pair production of an electron (e-) and po-sitron (e+), the pair, from a high energy photon, which lost all of itsenergy E = hm in encounter with a nucleus.

364 J.D. Munoz-Andrade Mat.-wiss. u. Werkstofftech. 2008, 39, No. 4-5

proposed by obtained the density of dislocations or distur-bances in the cosmic structure during Hubble flow at severalrecession velocities of galaxies, by the analogous applicationof the Orowan equation used widely in plastic deformation,where the magnitude of Burgers vector for the cosmic struc-

ture of the universe as crystalline system formed by the darkmatter and dark energy, it is the Planck length. Also, in a si-milar method to the SPF the expansion rate of the universewas obtained by using the rate reaction theory with the strongtemperature dependence of strain rate as follow:

Fig. 2. Dependence of strain rate sensitivity (m) of theflow stress during thermo mechanical worked of Al-Cueutectic alloy on strain rate n? at 793 K, with grain sized = 1.7lm. From [8].

Fig. 3. Dependence of activation energy for super plas-tic flow of thermo mechanical worked Al-Cu eutecticalloy on strain rate sensitivity at 793 K, with grain sized = 1.7lm.

Fig. 4. Dependence of activation energy for super plas-tic flow of thermo mechanical worked Al-Cu eutecticalloy on velocity of phase de Broglie at 793 K, withgrain size d = 1.7lm.

Mat.-wiss. u. Werkstofftech. 2008, 39, No. 4-5 String theory and cosmic connection during super plastic flow 365

H0 ¼ nu ¼ c

kP

� �exp

QP

kTP

� �ð8Þ

Where, QP = the Planck activation energy of the system atthe Planck scale (QP = 1.221x1028eV), where the quantum ef-fects and gravitational effects are of equal importance, (c/kP)= the overall frequency factor, k = the Boltzmann constant (k =8.617x10-5eV/K), TP = the Planck temperature (TP =1.010285625x1030K). This Eq. (8) was used to obtain the va-lue of the expansion rate of the universe related with the Hub-ble parameter, nu = H0 =70 (km/sec)/Mpc = 2.26854593 x10-

18s-1[7].Inclusively the first experimental evidence for the pair pro-

duction process, and the existence of positrons, was obtainedduring an investigation of the cosmic radiation.

In this construction, the application of the quantum me-chanics relativistic mathematical model presented in thiswork, allows to determine the activation energy for super plas-tic flow of Al – Cu eutectic alloy and the results are in an ex-cellent closed agreement with the experimental values re-ported in the past [9]. In this approach it is feasible to geta better understanding of the phenomenology and mechanicsof SPF in advanced materials in a universal point of view.

5 Summary

The string theory and cosmic micromechanics connectiontheory during SPF in SEPCS proposed in this work provides aprospect to enrich the understanding of the spatially extendednature related to the unified interpretation of every thing.

The quantum mechanics and relativistic mathematicalmodel presented in this work allows obtain in a practical tech-nique the activation energy for SPF dependence on strain rateand phase velocity de Broglie. As well, the nature determina-tion of the wavelength of the cellular dislocations k? waveassociated with SPF. As a result, SPF associated with coop-erative grain boundary sliding and self accommodation pro-cess is assisted in nature manner by cellular dislocations dy-namics.

6 References

1. B. Greene, “The Elegant Universe”, Vintage Books, N. Y. 2003,136.

2. R. Eisberg, R. Resnick, “Quantum Physics”, John Wiley & Sons,N. Y. 1974, 3 – 53.

3. A. M. Kossevich, “The Crystal Lattice”, Wiley-VCH, Berlin1999, 218 – 220.

4. J. D. Munoz-Andrade, Doctorate Thesis, “Physical Theory ofSuper Plastic Flow in Spatially Extended Polycrystalline Sys-tems” Facultad de Ingenierıa de la Universidad Central de Ve-nezuela, Caracas, Venezuela, 2007, In preparation for presenta-tion.

5. J. D. Munoz-Andrade, “A Mathematical Model for Plasticityand Cosmology”, CP908, NUMIFORM’07, Materials Proces-sing and Design: Modeling, Simulation and Applications Ed.By J. M. A. Cesar de Sa and A. D. Santos, American Instituteof Physics 2007, 1337 – 1342.

6. J. D. Munoz-Andrade, Key Engineering Materials 2007, 345–346, 577.

7. J. D. Munoz-Andrade, “Unified Interpretation of Hubble Flow,Plastic Flow and Super Plastic Flow”, Proceedings of the 8th

ESAFORM Conference on Material Forming, Editor: Prof. D.BANABIC, Cluj Napoca, Romania. The Publishing House ofthe Romanian Academy 2005, 603 – 606.

8. W. F. Hosford, R. M. Caddell, “Metal Forming: Mechanics andMetallurgy” Prentice Hall, Inc., Englewood Cliffs, N. J. 1983,80 – 91.

9. S.S. Bhattacharya, K. A. Padmanabhan, Mater. Sci. Forum 1997,243–245, 67 – 50.

J. D. Munoz-Andrade, Departamento de Materiales, Division deCiencias Basicas e Ingenierıa, Universidad Autonoma Metropoli-tana Unidad Azcapotzalco, Av. San Pablo 180, Col. Reynosa Ta-maulipas, 02200 Mexico Distrito Federal, Mexico, e-mail: jdma@-correo.azc.uam.mx

Received in final form: January 31, 2008 [T 307]

366 J.D. Munoz-Andrade Mat.-wiss. u. Werkstofftech. 2008, 39, No. 4-5