Deviatoric and hydrostatic mode interaction in hard and soft tissue

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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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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