DEVIATORIC
AND HYDROSTATIC MODE INTERACTION HARD AND SOFT TISSUE STEPHEN
IN
C. COWIN
Department of Mechanical Engineering. City College of CUNY,
New York. NY 10031, U.S.A.
Abstract-It has been established that many hard and soft tissues have anisotropic material symmetries. These materials are contrasted with traditional structural metals which have isotropic material symmetry. It is noted here that the deviatoric and hydrostatic modes interact with each other in a general anisotropic elastic material. in the special case of isotropic, linear elastic, materials these modes are non-interactive. As a consequenceof the interaction of these modes encountered in anisotropic materials. the decomposition into hydrostaticanddeviatoricmodes,and deviatoricmodeconceptssuchas the von Mises effective stress are not appropriate for anisotropic materials in general. The implications of this observation for the presentation of computationally generated stresscontours for hard and soft tissuesare discussed.It is also pointed out that the mode coupling and mode interaction raise the question of whether anisotropic living tissuesrespond directly to stress or to some other physical quantity such as strain or strain energy, in view of the recent hypothesis concerning the proliferation and ossification of cartilage.
INTRODUCIION
plications of the coupling of the hydrostatic and deviatoric modes in the presentation of the results of computational stress analysis are discussed.It is also noted that, in view of the hypothesis of Carter (1987) concerning the processesof proliferation, maturation and ossification of cartilage, the mode coupling and mode interaction raise the question of whether living tissues respond to stress or some other physical quantity. In the following section, the non-interaction of the hydrostatic and deviatoric modes in isotropic materials is developed. In the section after that, the same results are calculated for anisotropic materials and the interaction of modes is demonstrated. The formulas for the decomposition in anisotropic materials arc then illustrated using data on the orthotropic stressstrain relations for bone tissue.The case for the lack of usefulness of the deviatoric stress in hard and soft tissue mechanics is summarized in the Discussion.
In the analysis of the Row and fracture of isotropic structural metals the unique and complete decomposition of stressesand strains into hydrostatic (or dilatational in the case of strain) and deviatoric (or distortional) parts has been an extremely useful feature from which considerable physical insight has been obtained. In the case of the plastic deformation of structural metals it was found that, to a first approximation, the yielding was little influenced by a moderate hydrostatic pressureor tension. If the components T,, of the stresstensor are written as a matrix T, the deviatoric portion of T is denoted by T and delined by T szT - (trT/3)1
(1)
where I is the unit tensor or matrix and (trT/3) is the hydrostatic stress, trT = T,, = T, , + T,, + T,,. The developments in the plastic theory of structural metals caused considerable emphasis to be placed on the deviatoric stressT and the scalar that is the second invariant of the deviatoric stress(t&*/2). This scalar. whose importance was first clarified by von Mises (1928) has many different names, including the von Mises stress.octahedral shear stress,etc. (Paul, 1968). The purpose of this note is to show that for an anisotropic elastic material, while it is still possible to decompose stress and strain into hydrostatic and deviatoric components, the advantage of the decomposition is lost because the hydrostatic and deviatoric modes are coupled. Also, the further advantage of the use of decomposition in isotropic metals does not exist because, for anisotropic materials, the yield or fracture condition for these materials is not, in general, independent of hydrostatic pressure. The im-
TllE NON-INTC:RACTION OF tIVDROSTATIC AND DEVIATORIC MODFS IN ISOTROFIC ELASTIC h1ATERtAl.S
The decompositions of the total stress T and the total strain E into deviatoric and hydrostatic (or dilatational) components is possible in any situation. As was the case with the formula (I). the following formulas are simply definitions of the deviatoric stress and strain tensors:
T,, = f,, + (trT/W,,.
(2)
E,, = I?,) + (tr E/3)6,.
(3)
It is possible to accomplish these decompositions independent of the constitutive equation of the material and, in particular, independent of the material symmetry characterizing the material response.
Rcceiued in jnoi form 3 October 1988. I1
s. c. COWIN
12 In isotropic
elastic materials
plete decomposition
the unique and com-
into hydrostatic
modes can be extended
and deviatoric
to the stress-strain
relations
and to the strain energy. For an isotropic
material.
Hooke’s law can be written
THE ISTER.4CTIOS OF THE HYDROST\TIC DEI’IATORIC MODES IZ .\\;ISOTROPIC .M.\TERI.\LS
The
in the form
unique and complete
stress-strain cj = i.S,,(trE)
+ 2pEij.
(4)
deviatoric
law
and
decompositions
the total
and hydrostatic
strain
where i and ,u are the Lame elastic moduli. When the
vious section can be accomplished materials
and materials
of the
energy
into
parts presented in the pre-
decompositions
(2) and (3) of the stress and strain,
\VD
EL.\STIC
only for isotropic
with cubic symmetry.
respectively, are substituted in (4). (4) can be rewritten as two equations.
for isotropic materials and materials with cubic symmetry.
trT = (3L + Z&rE.
iii = 7~l&~.
(5)
between trT2 and trcr
Also.
the proportionality
In this section, the algebra
The first of these equations shows that the hydrostatic
material.
extensions of the decomposition
dilatational
strain.
trE.
and
factor of proportionality. deviatoric strain
to the hydrostatic that
(i.+Zrr)/3
or
is the
The second shows that the
stress is proportional
to illustrate the dihiculties
hydrostatic
that prohibit
into deviatoric
the and
parts for these materials.
The generalized
Hooke’s
law is written
to the deviatoric
and that Zyl is the factor
of proportionality.
Thus. for isotropic materials the hydrostatic ents of stress and strain
of the previous
section is repeated, this time for a general anisotropic
stress (trT/3)
is proportional
holds only
compon-
arc proportional.
and the
dcviatoric components ofstrcss and strain are proportional.
where the Ci+,, are the components tensor. There arc three important
of the elasticity
symmetry
rcstric-
(ions on the tensor Cijr,.
Cijlrm= Cjikm* Cijrm = Ci,“~,
C,j~m= Chmij. (I ‘)
The strain energy per unit volume Z is given by These restrictions ZC = tr(TE) When the decompositions strain. rcspcctivcly,
= TjE,j.
(6)
(2) and (3) of the stress and
arc substituted
into the formula
(6) for the strain energy, it can be written z = &* where
strain
strain energy of the deviatoric
the requirement
energy.
i.e.. the
stress and dcviatoric
of the
of the strain tensor, and be produced
by the
in a closed loading cycle. rospcctivoly.
The dihrriod
m~dulrt.~ wrwr
C is dcfincd by
Cij f Cijkl 9 Cklij,
(13)
and represents the stress response to
a
strain. The totally dcviatoric
tensor c+.,,
elasticity
dilatational is
defined in terms of Cok, by
strains: 2x,,,
= tr(i.k)
= fijiij
= l/lZi,Eij
= 2/ltr2’;
(8)
eijkm
s
of
the
strain energy, i.e., the strain
hydrostatic
stress and
dilatational
-
C*$w
+ and X3,,, is the hydrostatic cncrgy
from the symmetry
that no work
elastic material
(7)
+ &I
is the dcviatoric
Id_
in the form
follow
stress tensor, the symmetry
(
l/3)c,jiilm
- (l/3)Si,c’,,
( I/9hiijS,,trC.
(14)
Note that cijx_, is traceless with respect to two pairs of indices,
strain:
dijkk = Ckkij = 0. 2 &I
= ((trT)/3)trE
= ((32. -t- 2jO/3)trE)‘.
The result (7) shows that. in an isotropic material, total strain energy can be uniquely decomposed hydrostatic
into
the
and completely
the sum of the distortional
and
Introducing
bcon particularly
the devintoric
part
of the dilatational
modulus tensor denoted by C, (14) may be written
C,jkm = Cij&_ + (1/3)Ci,ls,, the quantity
useful in the analysis
tr?’
in
the form
strain energy.
As noted in the introduction,
(15)
(9)
has
+ (l/3)cSijC,,
+ (l/P)tS,S,,trC.
of experi-
(16)
mental data from the plastic deformation
of metals.
Substitution
The quantity
arc associ-
formula (16) into the gcncralized
Hooke’s law permits
the stress to
into
trf’
or a constant times trT*
ated with a number of ditfercnt names, including Mises stress and octahedral
shear stress. From
second of equations (5) it is easy to see that trT’ tr i?’ are proportional
for an isotropic c&tic
von the
hydrostatic
be decomposed
components,
and trT Comparison
= trCE = t&
strain,
terials the deviatoric
shear strain,
stress and
etc. arc proportional
elastic material.
and
(17)
+ (1/3)(trC)trE.
(18)
of these results with those for isotropic
materials
shear
drviatoric
thus
(10)
It follows that the von Mises stress and von Mises the octahedral
(1) and (3) and the
fsj = CijLn,ZLm + (I/3)Ccj(trE).
material,
thus tr?-’ = 4$tri12.
of the decompositions
octahedral
in an isotropic
dilatational Also.
given by (5) shows that for anisotropic strain
stress is now proportional as well as the deviatoric
the hydrostatic
stress is proportional
mato the
strain. to the
Deviatoric
deviatoric
and hydrostatic
strain as well as the dilatational
anisotropic
materials.
From
strain in
(18) it can be
(17) and
seen that, in order to have the proportionality by isotropic
materials.
the deviatoric
e_njoyed
tensor C must
vanish. This tensor only vanishes for isotropic materConsider
again the formula
(6) for the total strain
energy. Substitute the decomposition
is given by trT = (42S)trE
(2) of stress into
(6), then use (17) and (18) to express T and trT in terms
- 4.4i,,
where, in this equation are GPa.
-
l.6ilL
+ 5.9i,,,
(27)
and the one above, the units
The expressions (26) and (27) demonstrate
the numerical
ials and materials with cubic symmetry.
13
mode interaction
deviatoric
values of the coupling
stress and dilatational
between
the
strain on one hand,
and the hydrostatic stress and deviatoric strain, on the other hand, for human femoral cortical bone tissue.
of strain, thus z = Lc* + &I is the deviatoric
where &
+
DISCUSSIOY
(19)
xi”4
While
strain energy
the concepts of the deviatoric
and hydro-
static modes of stress and strain have been important 2X,,,
= Cij&j&,.
ZhYd is the hydrostatic
(20)
in the development
of the plasticity
general anisotropic 2Zhrd = (trC/9)trE2,
(21)
and Zin, is the term representing interaction
between
the energy of the
the deviatoric
and hydrostatic
E.
(22)
This result shows that the interaction
energy vanishes
only if C vanishes. The
quantity
quantity
trc*
trTz
is simply
not related
for an anisotropic
elastic
to the
material.
From (17) it follows that trT*
material or, in particular,
and soft tissue. There and hydrostatic
(23)
mentally
of the stress-strain
proved that the hydrostatic
influence on the plastic deformation tals. the dcviatoric
cortical
bone tissue equation
(I I) can be written as
Neither
of these two
hydrostatic
elastic domain
is any less, or any more,
+ l0.7E,,. + 27.6E,,,
T,,=5.6lE,,.
material.
Comparing stress T in
elastic materials,
respcct-
material,
Similarly,
but to only f? in an the hydrostatic
stress
is coupled to both c and trE in an anisotropic
material, equation (18). but to only trE in an isotropic
T,,=4.52E,,.
where the data are taken from Ashman
et al. (1984)
tissue, equation (18) representing the deviatoric
equation (5). The total strain energy in the
anisotropic
case contains a term associated with the
interaction
between
components,
stress
has the form
isotropic
- 4.4(trE/3),
the hydrostatic
equation
case, equation
are proportional
(7). Finally,
in an isotropic
(IO), they are only indirectly
viatoric (26)
into hydrostatic
material
of these modes.
T,, and T,, are the same as in equation (25). but with hats on all the Ts and
not just biological
materials.
Es. The scalar formula
strain fields obtained
stress
matcr-
because it contains
the interaction
apply to all anisotropic
(I 8) for the hydrostatic
mode
and de-
for anisotropic
These observations
where the formulas for T,,,
equation
of the corresponding
modes inappropriate
a term representing
+ 5.9(trE/3),
material.
related in an anisotropic
ials. This basic difficulty is retlcctcd in the total strain energy for an anisotropic
- 4.87&s
the von
bctwecn the stress mode component
renders the decomposition
- 1.6(trE/3),
while
material, equation (23). In general, the lack of a direct and the strain component
= 2.lSE^,, +7.13&s
and deviatoric
(19); this term vanishes in the
Mises etfective stress and von Mises effective strain
proportionality - 4.53i,,
Second, the two modes
in an anisotropic
material.
material,
and all numbers have the units of GPa. For this bone
+ 9.53&,
portion.
in the
important
ively. it is easy to see that it is coupled to both k and
+ 20.2E,,
(24)
evidence that the
and anisotropic
TJJ = IO.1 E,, + 10.7E,,
?,,, = - 4.53&I
for anisotropic
isotropic
(trT/3)
-4.87i,,
portion of the total energy.
facts is true
(5) and (17) for the deviatoric
isotropic
(29
i,,
mc-
the significant
formulas
+ IO.1 E,,,
= 6.8E^,, - 2.152,s
mode has little
mode of bone tissue deformation
trE in an anisotropic
, + 9.98E,,
and
i,,
law and the
of structural
mode represents
stress and the significant
are coupled
EXAMP1.B HUMAN CORTICAL HONE TWUE
T*,=6.23E,,.
the deviatoric
total energy and, second. since it has been experi-
than the deviatoric
T12 = 9.98E,,
reasons for this
modes represent a unique and com-
lar. First, there is no experimental
+ (trE/3)*trC2
+ (2trE/3)~i,Lm?,,&m.
T,, = 18.01:‘,
for a
for hard
materials in general and hard or soft tissue in particu-
= 6,,,,6,,,,i,,&,
For human femoral
are several
difference. First, in isotropic materials plete decomposition
component Xi,,, = (t&/3)tr
theory for mate-
rially isotropic metals. they are not appropriate
strain energy
materials.
In view of these results, the nature of the stress and from the finite element analysis
I4
s. c.
of anisotropic lar,
objects in general.
is not illuminated
and bones in particu-
by providing
contour
plots of
the von Mises stress. Such contour plots are useful for isotropic
objects in generrtl, and structural
particular,
metals in
because they provide a convenient means
of illustrating
regions of potential yield or failure. It is
for this reason that many commercial finite element programs output.
provide
von Mises stress as an optional
However, for anisotropic
stress, not just structural Cowin.
materials.
the total
the von Mises stress, contributes
yielding or failure (Hayes and Wright,
to
1977;
1979; Stone et al., 1983). thus a measure of the
COWIN
due to mode interaction
the response cannot be op-
posite for the corresponding
strain energy modes be-
cause the strain energy associated with a single stress mode will be spread across two strain energy modes. Thus,
the Carter hypothesis
that the adaptive mech-
anisms of cartilage respond in opposite fashions deviatoric and hydrostatic
to
stress states suggests that
the adaptive mechanisms of cartilage respond specifically to stress and not to strain nor to strain energy. This
brings to light a very interesting
ant question
in physiology:
and import-
do living tissues respond
to stress or to some other mechanical stimulus such as
total stress field should be used. A display of contour
strain
values of the total strain energy, suggested by others,
answered here. The issue deserves further
‘or total strain
energy? This
question
is not
discussion
is endorsed here. The total strain energy is a sum of
and investigation.
squares of all the stress components (or all the strain
authors that living tissue responds to strain or to total
components) weighted by the anisotropic
strain
stants.
It is a measure that works
isotropic
and anisotropic
elastic con-
equally well for of
intermittent
deviatoric stress will accelerate and that
intermittent
compressive hydrostatic stress will retard
the processes of proliferation,
and ossilication
maturation,
of cartilage in the appendicular skel-
eton. For linear isotropic elastic materials the Carter hypothesis
is equivalent
to making
ment in terms of equivalent namely
the deviatoric
volumetric
energy. Some aspects of a rationale for strain
response have been put forward
by Cowin (1984).
materials.
Carter (1987) has proposed that the application
or arrest
It has been suggested by several
strain
the same state-
mode strain
Arlinow/~~d~rmunls-This investigation was supported by USPHS. Research Grant DE 06859 from the National Institute of Dental Research, National Institutes of Health, Bcthcsda, MD 20205, U.S.A. This research was also supported (in part) by grant number 669301 from the PSC-CUNY Rcscarch Award Program of the City University of New York. I would like to thank Richard T. Hart and William C. Van Buskirk for helpful comments on an earlier draft this manuscript.
of
measures,
and the compressive
strain, or in terms of the equivalent mode
REFERENCES
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elastic
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and dcviatoric
the
and strain cn-
modes no longer
holds due to interaction and coupling of the modes. A purely deviatoric stress state (or a purely hydrostatic stress state) will produce a strain combination product
state that is some
of dcviatoric and hydrostatic
a strain
deviatoric (or hydrostatic)
Carter’s
one
and one that is an intcr-
action of the deviatoric and hydrostatic for anisotropic
and it will
energy with two components,
modes. Thus,
linear and non-linear cartilage tissues,
hypothesis
suggests that the adaptive mech-
anisms of the tissue will respond in opposite fashions for deviatoric stress and hydrostatic
stress, but due to
mode coupling the responses will nut be opposite for the dcviatoric
strain
and the hydrostatic
strain
and
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