On mechanics of deformation and crushing processes
The mechanics of crushing and breaking of particles is one of the most intractable problems in materials science. The stressed states of processed materials are significantly inhomogeneous, and thus the deformation and disintegration mechanisms vary greatly. Two techniques have been developed for re...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2008
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| Цитувати: | On mechanics of deformation and crushing processes / L. Berka // Проблемы прочности. — 2008. — № 1. — С. 64-68. — Бібліогр.: 21 назв. — англ. |
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irk-123456789-484582013-08-19T22:27:24Z On mechanics of deformation and crushing processes Berka, L. Научно-технический раздел The mechanics of crushing and breaking of particles is one of the most intractable problems in materials science. The stressed states of processed materials are significantly inhomogeneous, and thus the deformation and disintegration mechanisms vary greatly. Two techniques have been developed for realizing these processes as a quasi-homogeneous transition. The device and method developed by Enikolopov transform a solid polymer spontaneously into powder. The same loading system is now used for obtaining fine-grained metals, similarly as when using the ECAP device developed by Valiev. Both techniques are now used for obtaining nanostructured materials. The common feature of both types of methods is the formation of new physical surfaces. These are particle-free oversurfaces or grain boundaries. The method requires a supply of energy in the form of mechanical work, and this is mostly done by simultaneous action of pressure and shear stress. The formation offree oversurfaces in stressed solid bodies is the subject offracture mechanics. The Griffith equation is employed to describe the problem. Механика дробления и разрушения частиц является одной из трудноразрешимых проблем материаловедения. Напряженное состояние обработанных материалов значительно неоднородно, и поэтому механизмы деформирования и разрушения существенно отличаются. Разработаны два метода для реализации этих процессов как квазиоднородного перехода. С помощью устройства и метода, разработанных Ениколоповым, твердый полимер самопроизвольно преобразуется в порошок. Аналогичная система нагружения используется для получения мелкозернистых металлов, подобно использованию устройства для равноканального углового прессования, разработанного Валиевым. Оба метода используются в настоящее время для получения наноструктурных материалов. Образование новых физических поверхностей является общей чертой обоих методов. Они представляют собой свободные от частиц верхние поверхности, над- поверхности или границы зерен. В соответствии с методом Валиева, требуется подача энергии в виде механической работы, что обычно осуществляется одновременным воздействием давления и касательного напряжения. Поскольку образование свободных надповерхностей в нагруженных твердых телах является предметом механики разрушения, для решения этой задачи используется уравнение Гриффитса. 2008 Article On mechanics of deformation and crushing processes / L. Berka // Проблемы прочности. — 2008. — № 1. — С. 64-68. — Бібліогр.: 21 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/48458 539. 4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Berka, L. On mechanics of deformation and crushing processes Проблемы прочности |
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The mechanics of crushing and breaking of particles is one of the most intractable problems in materials science. The stressed states of processed materials are significantly inhomogeneous, and thus the deformation and disintegration mechanisms vary greatly. Two techniques have been developed for realizing these processes as a quasi-homogeneous transition. The device and method developed by Enikolopov transform a solid polymer spontaneously into powder. The same loading system is now used for obtaining fine-grained metals, similarly as when using the ECAP device developed by Valiev. Both techniques are now used for obtaining nanostructured materials. The common feature of both types of methods is the formation of new physical surfaces. These are particle-free oversurfaces or grain boundaries. The method requires a supply of energy in the form of mechanical work, and this is mostly done by simultaneous action of pressure and shear stress. The formation offree oversurfaces in stressed solid bodies is the subject offracture mechanics. The Griffith equation is employed to describe the problem. |
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Article |
| author |
Berka, L. |
| author_facet |
Berka, L. |
| author_sort |
Berka, L. |
| title |
On mechanics of deformation and crushing processes |
| title_short |
On mechanics of deformation and crushing processes |
| title_full |
On mechanics of deformation and crushing processes |
| title_fullStr |
On mechanics of deformation and crushing processes |
| title_full_unstemmed |
On mechanics of deformation and crushing processes |
| title_sort |
on mechanics of deformation and crushing processes |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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2008 |
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Научно-технический раздел |
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http://dspace.nbuv.gov.ua/handle/123456789/48458 |
| citation_txt |
On mechanics of deformation and crushing processes / L. Berka // Проблемы прочности. — 2008. — № 1. — С. 64-68. — Бібліогр.: 21 назв. — англ. |
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Проблемы прочности |
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2025-07-04T08:58:29Z |
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1836706179650158592 |
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UDC 539. 4
O n M e c h a n ic s o f D e fo r m a t io n a n d C r u s h in g P r o c e s s e s
L. B erk a 1,a
1 Czech Technical University, Faculty of Civil Engineering, Department o f Building Structures,
Prague, Czech Republic
a berka@fsv.cvut.cz
The mechanics o f crushing and breaking o f particles is one o f the most intractable problems in
materials science. The stressed states o f processed materials are significantly inhomogeneous, and
thus the deformation and disintegration mechanisms vary greatly. Two techniques have been
developed fo r realizing these processes as a quasi-homogeneous transition. The device and method
developed by Enikolopov transform a solid polymer spontaneously into powder. The same loading
system is now used fo r obtaining fine-grained metals, similarly as when using the ECAP device
developed by Valiev. Both techniques are now used fo r obtaining nanostructured materials. The
common feature o f both types o f methods is the formation o f new physical surfaces. These are
particle-free oversurfaces or grain boundaries. The method requires a supply o f energy in the form
o f mechanical work, and this is mostly done by simultaneous action o f pressure and shear stress.
The formation offree oversurfaces in stressed solid bodies is the subject offracture mechanics. The
Griffith equation is employed to describe the problem.
K eyw o rd s : technology, processes, solids, crushing, m echanics, polar continuum, particle,
oversurface, grain, grain boundary.
In trod uction . Granulation forms the basis o f m any m echanical technologies that
aim to change the material substructure into a bulk. Two techniques have been developed
for realizing these processes as a quasi-hom ogeneous transition [1, 2]. The d evice and
m ethod developed by Enikolopov [3] transforms a solid polym er spontaneously into
powder. The sam e loading system is n ow used for obtaining fine-grained polycrystals,
sim ilarly as w hen using the ECAP device [4] developed by Valiev. Both techniques are
now used for obtaining nanostructured materials [5]. It is necessary to clarify the
m icrom echanism s o f these processes for the purposes o f materials science as w ell as
practical applications.
The initial phase o f the processes in question is the state w ith large deform ations
w here loca l rotation as a part o f the deform ation gradient cannot be neglected. This effect
has been studied experim entally using X -ray techniques [6 , 7] and also m icroscopically on
the polished surface o f specim ens [8 -1 0 ]. The theoretical analysis o f deform ations
assum ing loca l rotations is know n as the Cosserat continuum theory. This theory was
developed in the second h a lf o f the 20th century [11, 12]. A number o f problem s
involving form ation o f the substructure and grain size were solved using a coupled stress
theory [13, 14].
The second phase o f the spontaneous fragmentation process o f quasi-hom ogeneous
solids, w hich takes place under pressure and shear stress, results in the formation o f new
physical surfaces, i.e ., particle-free oversurfaces [3] and a n ew grain structure w ith new
grain boundaries [4]. The lim it states o f individual shear cracks and shear bands are now
under very intensive theoretical and experim ental study. A n early paper [15] provided an
elastic solution for the in-plane crack problem as w ell as the out-of-plane crack problem.
3D shear cracks were studied in [1 6 -1 8 ] using the extended finite elem ent m ethod and
continuum -discontinuum m odeling.
A description o f the final state o f the process n ow arises from the superposition o f
the tw o phases. This show s the formation o f the field o f deform ations and rotations and
shear spherical cracks that they induce. The deform ation field around a spherical
© L. BER K A , 2008
64 ISSN 0556-171X. npo6n.eubi npounocmu, 2008, N 1
mailto:berka@fsv.cvut.cz
On Mechanics o f Deformation and Crushing Processes
macrocrack w as studied in [19], using an interaction energy integral method. Attem pts to
solve the above m entioned problem s are based on the assum ption o f a hom ogeneous
continuum w ith individual defects [20, 21]. The aim o f this paper is to put forward a
m odel o f the granulation process, in w hich a quasi-hom ogeneous solid changes its grain
size and structure in the w hole body volum e.
A M od el o f th e G ran u la tion P rocess in a Solid B ody. The general principle o f an
idealized granulation process is a transition o f a hom ogeneous solid body into a bulk o f
hom ogeneous particles having surface tension and the sam e m ass as in the original body.
The process is comparable w ith brittle fracture, but the stress states and crack m odes are
different. Brittle fracture takes place m ainly under tensile stress and in the form o f the first
crack opening m ode. A ccording to the experim ental results introduced above, the
analyzed process continues under the com bination o f both shear and pressure stresses.
This stress state then results in cracks o f the 2nd and 3rd m odes. The energy balance
principle used in the study o f crack problem s is expressed by the Griffith equation:
where E is the total potential energy o f a cracked body, S is the surface energy o f the
crack, U is the strain energy o f a deform ed body w ith a crack, and W is the potential
energy o f the applied loads. The presented relation betw een W and U is valid on
condition that the strains are elastic.
T h e P o lar C on tin u u m M ech an ics E q u ation s. D eform ations in technological
processes are alw ays large and the experim ental results introduced above show that in
theoretical description kinem atic rotations o f a deform ed material cannot be neglected.
The effect o f local rotation is described in the m echanics o f deform able bodies by
m icropolar continuum theories. There, the w hole system o f deform ations (Fig. 1) and
stresses is supplem ented by the introduction o f the m om ent stress tensor n ÿ and
distortion tensor k ÿ into the equations that describe its m echanical behavior. The
deform ation o f the differential representative volum e elem ent o f a material is expressed
by the fo llow in g relations [ 11]:
The stresses o jj and my that occur in the centers o f the volum e elem ent planes are
described by the fo llow ing equations:
dE = dS + d U - d W = 0, d W = 2 d U , ( 1)
Fig. 1. system o f deformations [11].
(2 )
ISSN 0556-171X. npoôëeMbi npounocmu, 2008, N 1 65
L. Berka
^ ij, j 0, ^ nm ^ nm + £ imn № ij, j 0 (3)
The elastic behavior o f isotropic materials is then defined by the fo llow ing relations:
where G and v are elastic constants and a and t are parameters o f the material
substructure.
The strain energy d U in the differential volum e elem ent d V is determ ined by the
fo llow ing formula:
Sp h erica l S h ear C racks in a S p atia lly C om p ressed M ateria l. A crucial point in
the granulation process is connected w ith the origin o f the spherical particle form. It is
necessary to take into account the continuous field o f local rotations resulting from large
shear strain. Experimental results show that at a certain level o f the shear strain, rotations
stay discontinuous [6 ]. W hen the material is assum ed to be isotropic, the originating
discontinuities acquire the form o f a sphere as these rotations are alw ays spatial.
Furthermore, this transition is possib le ow ing to the energy flux com ing from the
com pressed volum e elem ents into the surface layers and originating along the spherical
discontinuities, w hich can be seen as frozen eddies w ith boundary layers. To reach the
conditions for the transition lim it state, the Griffith equation is used. The quantities o f the
surface energy S and strain potential energy U, therefore, have to be determined.
Let us n ow assum e spherical shear cracks on the surface o f a sphere p laced inside a
representative volum e elem ent cube (Fig. 2). The cracks are bounded by circles, w hich
originate from sections parallel to the sides o f the cube. Their surface area is the same
as that o f the six spherical segm ents. The differentials o f their surface area and volum e are
determ ined by the fo llow ing formulas:
where R is the radius o f the sphere and <p is the angle betw een its normal and meridian
plane.
The crack surface energy dS is n ow obtained by m ultiplying the crack surface area
2dA by the specific surface energy y, w hich also contains the energy o f the plastically
deform ed surface layer.
(4)
d U = u d V , u = s ij £ ij + mij k j j . (5)
dA = 12tzR 2 sin <pd<p, d V = 6mR3 s in 3 <pd<p, (6 )
N 0 + N 2
dA
ds = Rd<p
dh = ds sin p = R sin p
c = R sin p
Fig. 2. Volume element cube.
66 ISSN 0556-171X. npoöxeMU npouHocmu, 2008, № 1
On Mechanics o f Deformation and Crushing Processes
The strain energy density u in the material w ith polar stresses is now calculated by
the substitution o f inverse relation (4) for K tj in Eq. (5), and further arranged and
expressed in the form containing the invariants S ( and M ( o f the stress and m om ent
tensors s H and mH:v lJ
Ц
u =
l j l j 3 (1 + f t ) 11
2G
+ -
■Vmijmji
4 a 2G ( 1 - ] 2 )
1
2G
S (1)
3(1+ ,и )
+ S (2) +
M (2) - ] M (S2)
2a 2G ( 1 - ] 2 )
(7)
The strain energy d U is n ow expressed using Eqs. (5 )-(7 )
d U = \ —
12G
S (1)
3( 1+ ,и )
+ S (2)
M
+ (2)
2a 2G ( 1 - ] 2 )
\R 3 s in 3 pdp. (8)
Substituting both quantities dS and d U in the Griffith equation, Eq. (1) results in
the fo llow ing formula:
dE = ’ (1)
3( 1+ ,и )
+ S (2) +
M (2) ] M (S2)
2a 2G (1 - ] 2 )
R sin ip 6nR sin p = 0. (9 )
A n approxim ation is accepted for the purpose o f sim plifying the equation. The angle
<p is assum ed for the octahedral plane, i.e ., the value o f s in 2 <p then equals 3/4. The
character o f the parameters a and R should be pointed out. Both express the length and
stand for the characteristic dim ension o f structural elem ents, i.e ., the particle size.
Therefore, an equation taking into account the condition R = a leads, after som e
rearrangement, to the fo llow ing quadratic form regarding the structural elem ent size a :
a 2 ( 1- ] 2 ) (1) + 3S D)
1+ ft
■ 1 6 y a G (1 -] 2 ) + 3( M (2)- ' ] M (2) ) = ( 10)
The solution o f this form w ill be the subject o f the next theoretical and experimental
analyses.
C onclu sions. The solution o f the problem under study is based on the account o f the
mutual coupling o f shear deform ations w ith loca l rotations. The rotations predetermine the
origin o f shear spherical cracks in all internal points o f the m acrovolum e o f the material.
The energy supplied to the system by the applied spherical pressure then leads, together
w ith other physicochem ical auxiliary effects, to new conditions for thermo- dynamic
equilibrium o f the process and to the form ation o f n ew physical surfaces.
Acknowledgment. The author appreciates the support o f GA ASCR for the project No.
IAA200710604.
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L. Berka
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Received 28. 06. 2007
68 ISSN 0556-171X. npoöxeMU npounocmu, 2008, № 1
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