Some Considerations on Failure of Solids and Liquids
Процесс разрушения как твердых, так и жидких тел в сплошной среде можно рассматривать как единое физическое явление. Математически описать его можно при выполнении некоторого предельного условия, в котором обычно учитывается напряженное состояние материала. Анализ классических теорий разрушения тв...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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irk-123456789-1117192017-01-14T03:03:31Z Some Considerations on Failure of Solids and Liquids Brighenti, R. Carpinteri, A. Научно-технический раздел Процесс разрушения как твердых, так и жидких тел в сплошной среде можно рассматривать как единое физическое явление. Математически описать его можно при выполнении некоторого предельного условия, в котором обычно учитывается напряженное состояние материала. Анализ классических теорий разрушения твердых тел (с точки зрения пластичности и механики разрушения) и современной теории разрушения жидких тел (явление кавитации с учетом максимального значения главного напряжения) позволил обнаружить общие закономерности в условиях их разрушения. В рамках общего подхода к разрушению тел предложено использовать для описания предельного условия разрушения такой параметр поля напряжений как трехосность напряженного состояния. Показано, что функции коллапса для твердых и жидких тел в сплошной среде имеют подобный вид. Процес руйнування твердих і рідких тіл у суцільному середовищі можна розглядати як єдине фізичне явище. Математично описати його можна при виконанні деякої граничної умови, в якій зазвичай враховується напружений стан матеріалу. Аналіз класичних теорій руйнування твердих тіл (із точки зору пластичності і механіки руйнування) і сучасної теорії руйнування рідких тіл (явище кавітації з урахуванням максимального значення головного напруження) дозволив виявити загальні закономірності в умовах їх руйнування. У рамках загального підходу до руйнування тіл запропоновано використовувати для описування граничної умови руйнування такий параметр поля напружень, як тривісний напружений стан. Показано, що функції коллапса для твердих і рідких тіл у суцільному середовищі подібні. 2010 Article Some Considerations on Failure of Solids and Liquids/ R. Brighenti, A. Carpinteri// Проблемы прочности. — 2010. — № 2. — С. 39-54. — Бібліогр.: 28 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/111719 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Brighenti, R. Carpinteri, A. Some Considerations on Failure of Solids and Liquids Проблемы прочности |
| description |
Процесс разрушения как твердых, так и жидких тел в сплошной среде можно рассматривать как единое физическое явление. Математически описать его можно при выполнении
некоторого предельного условия, в котором обычно учитывается напряженное состояние
материала. Анализ классических теорий разрушения твердых тел (с точки зрения пластичности и механики разрушения) и современной теории разрушения жидких тел (явление
кавитации с учетом максимального значения главного напряжения) позволил обнаружить
общие закономерности в условиях их разрушения. В рамках общего подхода к разрушению тел
предложено использовать для описания предельного условия разрушения такой параметр
поля напряжений как трехосность напряженного состояния. Показано, что функции коллапса для твердых и жидких тел в сплошной среде имеют подобный вид. |
| format |
Article |
| author |
Brighenti, R. Carpinteri, A. |
| author_facet |
Brighenti, R. Carpinteri, A. |
| author_sort |
Brighenti, R. |
| title |
Some Considerations on Failure of Solids and Liquids |
| title_short |
Some Considerations on Failure of Solids and Liquids |
| title_full |
Some Considerations on Failure of Solids and Liquids |
| title_fullStr |
Some Considerations on Failure of Solids and Liquids |
| title_full_unstemmed |
Some Considerations on Failure of Solids and Liquids |
| title_sort |
some considerations on failure of solids and liquids |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2010 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/111719 |
| citation_txt |
Some Considerations on Failure of Solids and Liquids/ R. Brighenti, A. Carpinteri// Проблемы прочности. — 2010. — № 2. — С. 39-54. — Бібліогр.: 28 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT brighentir someconsiderationsonfailureofsolidsandliquids AT carpinteria someconsiderationsonfailureofsolidsandliquids |
| first_indexed |
2025-07-08T02:35:05Z |
| last_indexed |
2025-07-08T02:35:05Z |
| _version_ |
1837044446405853184 |
| fulltext |
UDC 539.4
Some Considerations on Failure of Solids and Liquids
R. Brighenti and A. Carpinteri
Department of Civil and Enviromental Engineering & Architecture, University of Parma,
Parma, Italy
ÓÄÊ 539.4
O íåêîòîðûõ îáùèõ çàêîíîìåðíîñòÿõ ðàçðóøåíèÿ òâåðäûõ è æèäêèõ
òåë
Ð. Áðèãåíòè, À. Êàðïèíòåðè
Îòäåëåíèå ãðàæäàíñêîãî ñòðîèòåëüñòâà, ýêîëîãèè è àðõèòåêòóðû, Óíèâåðñèòåò ã. Ïàðìû,
Ïàðìà, Èòàëèÿ
Ïðîöåññ ðàçðóøåíèÿ êàê òâåðäûõ, òàê è æèäêèõ òåë â ñïëîøíîé ñðåäå ìîæíî ðàññìàò-
ðèâàòü êàê åäèíîå ôèçè÷åñêîå ÿâëåíèå. Ìàòåìàòè÷åñêè îïèñàòü åãî ìîæíî ïðè âûïîëíåíèè
íåêîòîðîãî ïðåäåëüíîãî óñëîâèÿ, â êîòîðîì îáû÷íî ó÷èòûâàåòñÿ íàïðÿæåííîå ñîñòîÿíèå
ìàòåðèàëà. Àíàëèç êëàññè÷åñêèõ òåîðèé ðàçðóøåíèÿ òâåðäûõ òåë (ñ òî÷êè çðåíèÿ ïëàñòè÷-
íîñòè è ìåõàíèêè ðàçðóøåíèÿ) è ñîâðåìåííîé òåîðèè ðàçðóøåíèÿ æèäêèõ òåë (ÿâëåíèå
êàâèòàöèè ñ ó÷åòîì ìàêñèìàëüíîãî çíà÷åíèÿ ãëàâíîãî íàïðÿæåíèÿ) ïîçâîëèë îáíàðóæèòü
îáùèå çàêîíîìåðíîñòè â óñëîâèÿõ èõ ðàçðóøåíèÿ. Â ðàìêàõ îáùåãî ïîäõîäà ê ðàçðóøåíèþ òåë
ïðåäëîæåíî èñïîëüçîâàòü äëÿ îïèñàíèÿ ïðåäåëüíîãî óñëîâèÿ ðàçðóøåíèÿ òàêîé ïàðàìåòð
ïîëÿ íàïðÿæåíèé êàê òðåõîñíîñòü íàïðÿæåííîãî ñîñòîÿíèÿ. Ïîêàçàíî, ÷òî ôóíêöèè êîë-
ëàïñà äëÿ òâåðäûõ è æèäêèõ òåë â ñïëîøíîé ñðåäå èìåþò ïîäîáíûé âèä.
Êëþ÷åâûå ñëîâà: ðàçðóøåíèå, èçëîì, êàâèòàöèÿ, òðåõîñíîñòü íàïðÿæåííîãî
ñîñòîÿíèÿ, òâåðäûå òåëà, æèäêîñòè.
N o m e n c l a t u r e
a – characteristic dimension of a defect
B – material’s mechanical parameter
D u[ ] – symmetric part of the gradient of the velocity field u
E G, � – Young’s and shear moduli, respectively
g – acceleration of gravity
G G c, I – fracture energy and critical fracture energy under Mode I, respectively
I1 – first invariant of the stress tensor
J 2 – second invariant of the deviatoric stress tensor
K I – stress intensity factor (SIF)
K cI – fracture toughness (plane strain condition)
p – hydrostatic pressure in a liquid under static condition
p Tc ( ) – liquid’s cavitation threshold pressure
© R. BRIGHENTI, A. CARPINTERI, 2010
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2 39
S, sij – deviatoric stress tensor
S ij – principal stresses of the deviatoric stress tensor S (i j, , ,�1 2 3)
t T, – stress triaxiality and temperature, respectively
u�{ }u v w – velocity field in the x x x1 2 3, , coordinate system
Y – dimensionless SIF or geometric correction factor
� – liquid dynamic viscosity
� – hydrostatic stress component of the stress tensor
� �, – parameter’s of the failure criterion adopted for solids
� – mass density
s s, – normal and shear stress, respectively
– stress tensor
i – principal stresses of the stress tensor (i�1 2 3, , )
0 – remote applied stress
[ ]u – shear stress tensor in a moving liquid
Introduction. Failure in continuum media, solids or liquids, can be regarded
as a single physical phenomenon which is determined by the fulfilment of a limit
condition which generally involves the stress field in the medium under study.
Failure conditions can be easily and conveniently written in terms of the stress
invariants and the deviatoric stress invariants, and this approach is used in the
following.
By considering the classical theory of plasticity or fracture mechanics concepts
in the context of failure of solids and a recent theory on the cavitation phenomenon
in the context of liquids, a unified discussion is herein presented, showing as the
collapse functions present a similar form for both classes of continuum materials.
Furthermore, by introducing a stress field parameter (the stress triaxiality), the
limit conditions of collapse for both classes of materials can be simply represented
through the same equation by only setting the involved material’s parameters
appropriately. The values assumed by such parameters can give us useful information
about the kind of expected failure.
Local collapse of continuum media, for both solids (mainly brittle solids) and
liquids, frequently appears as a loss of continuity of the material. The above
phenomenon occurs through the formation of cracks or high strain deformation due
to plastic flow in solids, or as the appearance of voids or bubbles (due to the
so-called cavitation phenomenon) in liquids.
Several studies have been carried out on collapse of solids. The classical
plasticity has been described by a well established theory [1, 2] (up to recent
studies on crystal plasticity [3, 4]), as well as fracture mechanics that dates back to
the pioneering researches carried out by Griffith [5, 6] and to the rigorous
mathematical approach performed by Westergaard [7], up to subsequent works by
Liebowitz [8] and Sih [9] and the recent studies of fracture at microscopic and
nanoscale level [10, 11]. Liquid collapse, also known as liquid cavitation, has also
been studied by several authors [12–18]. By considering a stress-based approach to
the mentioned failure phenomena in solids and liquids [4, 12], similar governing
equations can be found.
R. Brighenti and A. Carpinteri
40 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2
1. Failure in Solids: Plastic Collapse and Fracture Collapse. Failure in
solids can be regarded as an irreversible phenomenon which involves a great
rearrangement of the solid structure accompanied by energy dissipation. Such an
irreversible modification of the material structure can appear as a plastic flow due
to the sliding – at the microscopic scale – of the lattice structure of the material,
characterized by high plastic strain level (typical of the so-called ductile materials)
or as a loss of continuity, due to a large strain localization (as usually occurs in
brittle materials).
Fracture in solids is a common phenomenon which usually takes place due to
the growth and coalescence of microcracks or voids, produced by the stressed state
in the material, up to the formation of a macrodefect which can be mathematically
represented as a strain localization in a narrow band which identifies the fracture
position.
Typical approaches to the fracture phenomenon are the well-known SIF
criterion {descending from the classical Westergaard solution in linear elastic
fracture mechanics (LEFM) [7]} and the energy criterion introduced by Griffith [5,
6].
The critical SIF criterion for catastrophic fracture under pure Mode I can be
written as follows:
K Y a K cI I� � �0 , (1)
where Y is the so-called geometric correction factor (or dimensionless SIF), 0 is
the applied remote stress, a is a characteristic size of the existing crack, and K cI
is the fracture toughness of the material under given environmental conditions.
As mentioned above, the fracture phenomenon can also be analyzed through
an energy-based approach. For example, by considering a tensile loading of infinite
plate containing a straight crack with length 2a [in such a case: Y �1 in Eq. (1)],
the energy criterion for collapse states that [5, 6]:
G a E G c� �� 0
2
I , (2)
where G is the fracture energy, i.e., the energy per unit crack surface, available (in
the mechanical system) for crack extension.
The above two approaches to fracture can be linked by the classical
relationships:
G K Ec cI I� 2 (plane stress), G K Ec cI I�
( )1 2 2� (plane strain). (3)
Such fracture mechanics approaches assume that fracture process occurs
starting from a pre-existing crack or defect having a characteristic size a, which
instantaneously grows when the critical condition [either (1) or (2)] is fulfilled. In
other words, in the (theoretical) case of absence of any defect, fracture mechanics
approach leads to the erroneous conclusion that no failure occurs, irrespective of
the stress level in the body.
Some Considerations on Failure of Solids and Liquids
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2 41
In macroscopically non-damaged solids, the failure process can be regarded as
a breaking or a decohesion phenomenon produced by a sufficiently high stress
level, which must be greater than the tensile strength, f t , of the material.
The collapse condition in a material under uniform tension, 0 , can be
generally stated as follows:
�1�
�
�
�
�
�
�
�
�min ,f
EG
at
cI
or
�1�
�
�
�
�
�
�min , ,f
K
Y a
t
cI
(4)
where the maximum principal stress 1 0� is used.
From such an equation, a characteristic flaw dimension a0 , below which
collapse is strength-like and above which collapse is fracture-like, can be identified:
a
EG
f
c
t
0 2
� I
�
(fracture-like collapse), (5a)
a
EG
f
c
t
0 2
� I
�
(plastic-like collapse). (5b)
In the context of plasticity, a more general and common failure criterion,
which has a wide applicability for several materials, is the so-called Drucker
criterion [2], usually expressed by means of the stress invariants. In the present
paper, the criterion is written as follows:
I J k1 2 0�
�� � | with k p�
3 , ��
�
��
�
�
�
�
�3 3
3k f
f
c
c
t
t
, (6)
where the material parameters � and k can be determined by considering two
different stressed states at failure: a hydrostatic failure compressive stress field p ,
and a uniform tensile failure stress field f t . The dimensionless ratio c in Eq. (6) is
defined as c f t p� . For compressive hydrostatic-insensitive materials, the
parameter � is approximately equal to
3 3 and k is equal to
3 p , and the
criterion (6) becomes I J p1 23 3 3
�
| | or equivalently:
� �� 3 2| |J p , (7)
where � is the mean stress or hydrostatic pressure (� � �tr ii3 3).
Among the various parameters to identify the stressed state ‘quality’ in a point
of a continuum, the stress triaxiality factor [19–23] can be used:
t tr J J
I
J
� � �
1
3 3
2 2
1
2
�| | | |
| |
, (8)
42 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2
R. Brighenti and A. Carpinteri
i.e., the ratio between the mean hydrostatic stress and a quantity that measures the
root of the distortional energy (equivalent to the von Mises stress). From the above
definition, it can be deduced that t�� for purely hydrostatic stressed states,
t�0 for purely shear stressed states, and t�1 3 for an uniaxial 3D stressed
state. By using such a definition, the condition (4) can be written as
| |( ) min ,J t f
EG
at
c
2 1� �
�
�
�
�
�
�
�
�
I
�
or | |( ) min , ,J t f
K
Y a
t
c
2 1� �
�
�
�
�
�
�I
�
(9)
where the maximum principal stress is written as 1 2 1�
| |( ),J t as is discussed
below.
On the other hand, by considering the Druker plastic collapse criterion, we
get:
| |J t
k
2 3 3
�
�
�
�
�
�
�
�� (10)
and, in the particular case expressed by Eq. (7), the criterion becomes:
| |( ) .J t p2 3
� � (11)
The above failure expressions (9) and (11) can be summarized in the
following general equation:
| |( ) ,J t B2 � � �� (12)
where � and B are material constants as is discussed below.
It can be observed [25–28] that the stress triaxiality can provide a simple
parameter to identify the type of collapse: stress triaxiality lower than zero
( t�0 corresponds to a pure shear situation) produces a shear-type plastic flow
collapse, which can be identified as a special case of fracture process, while
positive triaxiality produces a void formation-type fracture process (fracture along
a direction normal to the principal stress) [27, 28]. Experimental observations [21]
have shown that fracture never occurs for t�
1 3.
2. “Fracture” in Liquids: The Cavitation Phenomenon. Generally speaking,
liquid failure can be identified by the appearance of voids or bubbles in the liquid
with loss of continuity and, consequently, the well-known corrosion in solid parts
which are in contact which such broken zones produced by the so-called cavitation
phenomenon.
The knowledge of the stressed state in a liquid is fundamental to predict the
possibility of liquid breaking. The stressed state � ij can be generally written
as
�
� �p 1 u[ ], (13)
where p is a scalar that represents the hydrostatic pressure (a positive value of p
indicates hydrostatic compression), 1 is the identity tensor, and
is the shearing
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2 43
Some Considerations on Failure of Solids and Liquids
stress tensor which can be determined by an appropriate constitutive law depending
on the liquid under consideration.
Generally, the shearing stress tensor
can be written as an appropriate
function of the velocity. As an example, a simple linear constitutive equation can
be assumed for a Newtonian liquid:
�[ ] [ ],u D u� �2 (14)
where � is the dynamic viscosity, and D u[ ] is the symmetric part of the gradient
of the velocity vector ui , D u u u[ ] ( ) ( ),, ,� � � � �D u uij
T
i j j i1 2 1 2� � i�1, 2,
3. According to the above hypothesis, the stress tensor can be rewritten as
� �ij ij i j j ip u u�
� �( ),, , (15)
where the symbol � ij denotes the Kroneker delta function.
As is well-known, the following relationship holds for an incompressible liquid:
div u tr ui i� � �D u[ ] , 0 (mass conservation equation). From the previous equation,
it can be stated that tr tr
�� �( [ ]) .2 0D u In this case, the mean hydrostatic
pressure becomes:
p tr ii�
�
1
3
1
3
. (16)
The above equation is true for a liquid in a static condition for which the
hydrostatic pressure p represents the average of the principal stresses.
In a flowing liquid, the average of the principal stresses has not significance
[16, 18], and the hydrostatic pressure p also depends on the stresses given by
(14). In such a general case, p tr�
1 3 ( )
since tr
0, as occurs for a
constitutive relationship different from the classical constitutive relationship (which
holds for the so-called non-Newtonian liquids) expressed by Eq. (14). In this last
case, Eq. (13) can be rewritten as
��
� � �
� �p S1 1 , (17)
where p tr�
1 3 ( ),
�
�
�
1 3 1 3tr p tr , and S is the deviatoric stress
tensor for which tr sijS� �0 holds (the scalars and are assumed positive for
hydrostatic compression states).
The breaking phenomenon in a liquid, commonly known as cavitation, takes
place by the formation of bubbles inside the liquid and is classically identified by
the following condition [18]:
p p Tc� ( ) (cavitation), (18a)
p p Tc! ( ) (no cavitation), (18b)
where p Tc( ) is the cavitation threshold (negative) pressure of the considered
liquid at a given temperature T. The above condition can be rewritten for liquids in
44 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2
R. Brighenti and A. Carpinteri
motion, by considering the principal stresses instead of the average pressure p
which can loose significance in such a case [18]. The previous hypothesis requires
the determination of the maximum (principal) stresses and of the corresponding
directions in a point of the liquid. Principal stresses in the coordinate system
coincident with the principal stress directions can be determined by considering
that the stress tensor has a diagonal form in such a coordinate system:
diag diag diag � ��
� � �
� �( )1 1 SS (19a)
or
�
1
2
3
0 0
0 0
0 0
1 0 0
0 1 0
0 0 1
"
#
$
$
$
%
&
'
'
'
�
"
#
$
$
$
%
&
'
'
'
�diagS. (19b)
In two dimensions, the above relationship becomes:
� �
1
2
11
22
0
0
1 0
0 1
0
0
1 0
0 1
"
#
$
%
&
'�
"
#
$
%
&
'�
"
#
$
%
&
'�
"
#
$
S
S
%
&
'�
"
#
$
%
&
'
S
S
11
11
0
0
since S S11 22 0� � ,
(20)
where the principal stresses of the deviatoric tensor S, S 11 , S 22 , are equal to
S s s11 11
2
12
2 1 2� �( ) ,/ S s s22 11
2
12
2 1 2�
�( ) / (since s s11 22�
), and the term sij �
� � � �ij ij is the generic component of S. Since the principal stresses of the
deviatoric tensor S are:
S 11 1 1 2
1
2
�
� ( ), S 22 2 1 2
1
2
�
� ( ), (21)
the maximum difference between the maximum ( 1) and the minimum ( 2)
principal stress is equal to 1 2 11 22 112
�
�S S S .
The condition of rupture (18), formulated in terms of the maximum principal
stress, becomes:
1� pc (22a)
or
� � � �1 11 11
2
12
2
1 1
2
1 2 2 1
2 1 2�
� �
� � �
�S s s u u u( ) [ ( ) ], , ,
/ pc . (22b)
When the above condition is fulfilled, a slit vacuum cavity initially opens in a
direction perpendicular to the maximum tension axis; subsequently the flow
vorticity can rotate the major axis of the ellipse, and vapour fills the created cavity
[18]. If also �2 22�
�S pc then the cavity opens also along the other
principal direction.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2 45
Some Considerations on Failure of Solids and Liquids
The above criterion must be corrected in presence of impurities (such as
bubbles, etc.) that facilitate the cavitation phenomenon. If a defect having a
characteristic dimension a, is supposed to exist in a point of the liquid, Eq. (22)
can be written as follows [16]:
1� � �p a T K T a p Tc c c( , ) ( ) ( ) , (23)
where K Tc( ) is a material parameter for a given temperature T.
Similar to the case of collapse in solids, the cavitation phenomenon, in
general, takes place when the following condition is fulfilled [16]:
1�
�
�
�
�
�
�
�
��min ( ) ,
( ) ( )
.p T
K T
a
K T
a
c
c c
(24)
Defining the stress triaxiality factor similar to the case of solids (see Section 1):
t J�� 2 with J s s s2 11 22 12
2�
, (25)
the maximum principal stress can be rewritten as follows:
� � �1 11
2
12
2 1 2
11 22 12
2 1 2
2� �
�
�
�
( ) ( ) ./ /s s s s s J (26)
Then, the cavitation phenomenon takes place when
1 2 1�
�J t pc( ) . (27)
As can be also observed under pure shear (t�0), liquid rupture can occur
when J pc2 � . If a high level of hydrostatic compressive pressure exists
( ,t!!0 since �!0 for hydrostatic compression), 11 2 1 0�
�J t( ) and no
cavitation occurs.
Some examples of fracture collapse in solids and fluids are given in Fig. 1.
46 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2
R. Brighenti and A. Carpinteri
Fig. 1. Fracture collapse in a ductule (a) or brittle (b) solid material and rupture (cavitation) in a
liquid (c).
a cb
3. Discussion. From the previous Sections, we can deduce that the failure
condition in flawed solids [see Eqs. (9) and (11)] and that in liquids [see Eq. (27)]
present a similar form which can be synthesized as follows:
1 2 1 21� � �J t C C( ) min( , ), (28a)
where
C C
f
EG
a
K
Y a
p T
K T
a
t
c c
c
c
1 2,
, ,
( ) ,
( )
�
�I I
for solids
for
� �
liquids,
(
)
*
*
+
*
*
(28b)
where the same definition of � is used for both solids and liquids, i.e., � �1 3 tr
(instead of � �
1 3 tr as usually made for liquids). Equation (29) is valid for
both solids and liquids once the constants C1 and C 2 are appropriately defined.
In the above general relationship, a difference in the meaning of the stress 1
must be considered: the failure in liquids takes place in the point where the
principal stress attains the maximum value (the imperfections are assumed to be
equally distributed in the liquid domain), while the stress in solids must be
interpreted as the reference stress (applied to the structure) that produces the
critical condition of the SIF.
By considering the previous Sections, it can also be noted that the failure
conditions in unflawed solids [see Eqs. (9) and (10)] and liquids [see Eq. (27)]
have a similar form which can be synthesized in this way:
J t B2 ( ) ,� �� � (29)
where the material constants �, �, and B depend on the mechanical material
behavior and the presence of defects. The values assumed by the three parameters
involved in Eq. (29) are summarized in Table 1.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2 47
Some Considerations on Failure of Solids and Liquids
T a b l e 1
Parameters Involved in the General Failure Function, J t B2 ( )� �� � ,
for Both Unflawed and Flawed Solids and Liquids
Material Mechanical behavior
and defects
Failure parameters
� � B
Solids Unflawed brittle 1 1 ft
Unflawed ductile 1 � 3 k 3
Flawed 1 1 EG acI ( )�
Liquids Unflawed
1 1 p Tc ( )
Flawed
1 1 p a T K T ac c( , ) ( )�
In order to compare the failure phenomenon occurrence in unflawed solids
and liquids, two 2D examples related to collapse in simple situations are discussed
below.
3.1. Collapse of a Solid Undefined 2D Strip under Shear Load. As an
example of application of the collapse functions determined above, an undefined
2D strip in plane stress condition, under self weight and an applied shear load, is
considered in the present Section (Fig. 2). The stressed state in the material can be
approximately defined by the following equations:
��
"
#
$
%
&
'�
� � �
11 12
21 22
1 S
�
�
�
�
�
"
#
$
%
&
'�
�g h e x e
h e x e2
1 2 1 0
0 1 2 1
2
2
( ) ( )
( ) ( )
�
� � �
�
� �
�
�
g
h e x e G
d
h
G
d
h
g
h e x e
2
1 2 1
2
1 2 1
2
2
[ ( ) ( )]
[ ( ) ( )]
"
#
$
$
$
%
&
'
'
'
, (30)
where �, g , and �G are solid density, acceleration of gravity and shear modulus,
respectively. Furthermore, d is the displacement applied to the top edge of the
strip, and e x y� ( ) ( )0 0 is the ratio between the horizontal stress and the
vertical stress at the bottom of the strip.
In the principal stress coordinate system, the stress tensor becomes:
diag
�
�
�
�
�
�
"
#
$
%
&
'�
g h e x e
h e x e
J
2
1 2 1 0
0 1 2 1
2
2
2( ) ( )
( ) ( )
0
0 2
"
#
$
$
%
&
'
'J
,
(31)
where J
g
h e x e
G d
h
2
2 2
2
2
2 2
24
1 2 1�
�
�
��
[ ( ) ( )] .
48 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2
R. Brighenti and A. Carpinteri
Fig. 2. 2D infinite strip under self weight and shear load.
It can be observed that, at the top free surface, we have:
�
�
��
"
#
$
%
&
'�
�
�
"
#
$
$
$
%
&
'
'
'
ghe
ghe
G
d
h
G
d
h
ghe2
1 0
0 1
2
2
, (32)
diag
� �
( )x h
ghe g h e G d
h
2
2 2 2 2 2 2
22
1 0
0 1 4
1 0
0 1
� �
"
#
$
%
&
'� �
�
"
#
$
%
&
',
and
J
g h e G d
h
2
2 2 2 2 2
24
� �
��,
.
By introducing the stress triaxiality t (which is a function of x 2 only):
t x
J
g
h e x e
g
h e x e
( )
( ) ( )
[ ( ) ( )
2
2
2
2 2
2
2
1 2 1
4
1 2 1
� �
�
�
�
�
�
�
]
,
2
2 2
2
�
�G d
h
(33)
the maximum principal stress can be expressed as follows:
�
�
1 2 2 2 2
2
1
2
( ) ( ( ))
( )
x J J t x
g h x
� � � � �
�
�
�
��2 2
2
2 2 2
24
g h x G d
h
( )
, (34)
and the failure conditions can be written as
1 2 2 21( ) ( ( )) ,x J t x f t� � �
J t
k
2 3 3
�
�
�
�
�
�
�
�� ,
(35)
for an undamaged brittle or ductile material, respectively, or in dimensionless
form:
F
x
f
J
t x
ft t
1
1 2
2
2
1
1
1 0�
�
�
�
( ) ( ( ))
, F
J
k
t2
2
3 3
1 0� �
�
�
�
�
�
�
�
�
. (36)
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2 49
Some Considerations on Failure of Solids and Liquids
In Fig. 3a, the qualitative collapse function patterns are displayed. As can be
observed, the collapse for a ductile solid occurs first at the base of the strip, while
the collapse for a brittle solid occurs first at the top of the strip. In Fig. 3b, the
stress triaxiality factor along a vertical line of the solid strip is represented: such a
parameter increases from the base to the top of the strip.
3.2. Collapse of a Liquid under Shear Flow. Fracture in liquids can occur
even under shear flow if the maximum tension stress criterion described above is
adopted. By assuming a one-dimensional flow with a velocity field expressed as
follows (Fig. 4):
{ } { },u v w Ux h� 2 0 0 (37)
the constitutive equation (14) becomes:
�[ ] .u
U h
U h�
"
#
$
$
$
%
&
'
'
'
0 0
0 0
0 0 0
(38)
The liquid is assumed to be in the gravity field (the direction of gravity is
x 2). Under the previous hypothesis, the stressed state in the liquid (only
quantities on plane x y
are considered) can be expressed as follows:
�
� ��
� � �
"
#
$
%
&
'�
"
#
$
%
&
'�1 S
11 12
21 22
2
1 0
0 1
0
g h x( )
U h
U h 0
"
#
$
%
&
', (39)
where � and g are the liquid density and the acceleration of gravity, respectively.
In the principal stress coordinate system, the stress tensor becomes:
50 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2
R. Brighenti and A. Carpinteri
Fig. 3. Brittle or ductile solid strip under shear: (a) collapse functions; (b) stress triaxiality factor.
a b
diag � ��
"
#
$
%
&
'�
"
#
$
%
&
'g h x
U h
U h
( ) .2
1 0
0 1
0
0
(40)
Therefore, at the free surface we have:
diag �( ) .x h
U h
U h2
0
0
� �
"
#
$
%
&
'
By introducing the stress triaxiality t (which is a function of x 2 only):
t x
J
g h x h
U
( )
( )
,2
2
2� �
� �
� (41)
the maximum principal stress can be expressed in terms of such a parameter as
�
�1 2 2 2 21( ) ( ( )) ( ).x J t x
U
h
g h x�
�
(42)
The condition of liquid fracture (or cavitation) can be finally written as
follows:
� �1 2 2 2 21( ) ( ( )) ( ) ,x J t x U g h x pc�
�
� (43)
or in dimensionless form:
F
x
p p
U
gh
p
h x
h
c c c
1
1 2 2
1 1 0�
�
�
� �( ) ( )
. (44)
In Fig. 5, the qualitative collapse function pattern is displayed. As can be
observed, since ( ) ,h x
!2 0 the cavitation phenomenon takes place first at the top
free surface of the liquid located at x h2� . In Fig. 5b, the stress triaxiality factor
along a vertical line of the liquid is represented: it increases from the base to the
top of the liquid domain.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2 51
Some Considerations on Failure of Solids and Liquids
Fig. 4. Shear flow of a liquid in the gravitational field.
Conclusions. In the present paper, the failure phenomena in continuum media,
solids or liquids, have been regarded as a single physical phenomenon which is
determined by the fulfilment of a limiting condition.
By considering the classical plasticity or fracture mechanics theory to describe
the failure of solids and the theory of the cavitation phenomenon to describe the
failure of liquids, a unified description has been proposed, showing that collapse
functions present a similar form for both classes of continuum materials.
The stress triaxiality, as a stress field parameter, allows us to express the limit
conditions of collapse for both classes of materials through a single equation, i.e.,
the structure of the governing failure equation in both solids or liquids failure is the
same. By appropriately setting the involved material parameters, the failure
function can be explicitly written. The values of such parameters provide useful
information about the type of expected failure.
Acknowledgements. The authors gratefully acknowledge the research support
for this work provided by the Italian Ministry for University and Technological and
Scientific Research (MIUR).
Ð å ç þ ì å
Ïðîöåñ ðóéíóâàííÿ òâåðäèõ ³ ð³äêèõ ò³ë ó ñóö³ëüíîìó ñåðåäîâèù³ ìîæíà
ðîçãëÿäàòè ÿê ºäèíå ô³çè÷íå ÿâèùå. Ìàòåìàòè÷íî îïèñàòè éîãî ìîæíà ïðè
âèêîíàíí³ äåÿêî¿ ãðàíè÷íî¿ óìîâè, â ÿê³é çàçâè÷àé âðàõîâóºòüñÿ íàïðóæåíèé
ñòàí ìàòåð³àëó. Àíàë³ç êëàñè÷íèõ òåîð³é ðóéíóâàííÿ òâåðäèõ ò³ë (³ç òî÷êè
çîðó ïëàñòè÷íîñò³ ³ ìåõàí³êè ðóéíóâàííÿ) ³ ñó÷àñíî¿ òåî𳿠ðóéíóâàííÿ ð³äêèõ
ò³ë (ÿâèùå êàâ³òàö³¿ ç óðàõóâàííÿì ìàêñèìàëüíîãî çíà÷åííÿ ãîëîâíîãî íàïðó-
æåííÿ) äîçâîëèâ âèÿâèòè çàãàëüí³ çàêîíîì³ðíîñò³ â óìîâàõ ¿õ ðóéíóâàííÿ. Ó
ðàìêàõ çàãàëüíîãî ï³äõîäó äî ðóéíóâàííÿ ò³ë çàïðîïîíîâàíî âèêîðèñòîâó-
52 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2
R. Brighenti and A. Carpinteri
a b
Fig. 5. Liquid under shear flow: (a) collapse function along a vertical line of the liquid; (b) stress
triaxiality factor.
âàòè äëÿ îïèñóâàííÿ ãðàíè÷íî¿ óìîâè ðóéíóâàííÿ òàêèé ïàðàìåòð ïîëÿ
íàïðóæåíü, ÿê òðèâ³ñíèé íàïðóæåíèé ñòàí. Ïîêàçàíî, ùî ôóíêö³¿ êîëëàïñà
äëÿ òâåðäèõ ³ ð³äêèõ ò³ë ó ñóö³ëüíîìó ñåðåäîâèù³ ïîä³áí³.
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Received 10. 09. 2009
54 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2010, ¹ 2
R. Brighenti and A. Carpinteri
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken voor kwaliteitsafdrukken op desktopprinters en proofers. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/ENU (Use these settings to create Adobe PDF documents for quality printing on desktop printers and proofers. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /NoConversion
/DestinationProfileName ()
/DestinationProfileSelector /NA
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure true
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /NA
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|