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Errors in the orientation of the principal stress axes if bone tissue is modeled as isotropic
TECHNICALNOTE ERRORS IN THE ORIENTATION OF THE PRINCIPAL STRESS AXES IF BONE TISSUE IS MODELED AS ISOTROPIC STEPHEN C. COWIN*
and
RICHARD T. HART?
‘Department of Mechanical Engineering. City College of the City University of New York. New York. NY 10031. U.S.A. and +Department of Biomedical Engineering. Tulane University. New Orleans. Louisiana 70118. U.S.A.
Abstract-The error in the prediction of the orientation of the principal axes of stress in bone (issue is determined in the case ashen the tissue is modeled as elastically isotropic rather than as orthotropic. the probable symmetry of bone tissue. Results are two-dimensional and assume 1he same underlying strain slate for both the orthotropic and isotropic cases. The maximum error is 45’. and the typical error is generally signiticant.
STATEXlEST OF TttE PRORLEM In this note we consider the error in the prediction of the orientation of the principal axes of stress if 1he bone tissue is modeled as a linear isotropic elastic solid. rather than as an ortho1ropIc enc. Both cancellous and cortical bone tissue can probably be mod&d as linear orrhorropically claslic materials (Cowin. 19X9). The problem is formulated as follows: two plane situations are considered; in one the ma1crial is orthotropic. and in the other 1he material is isotropic. For the orthotropic material the plane is a plane of symmetry. The oul-of-plane shearing stresses and shearing strains are zero for both materials. Thus. in the direction perpendicular IO the plane, the principal direc1ions of stress and strain are coincident for both the isolropic and anisotropic maIeri~ls. Le1 x, and .x2 be IWO in-plant normals [or the planes of symmetry of the orthotropic material. The components of the in-plane strain referred IO these axes are E,i. E,, and E,a. If the material is assumed IO be isotropic, then the in-plane principal strain direction makes an angle 0, with the x, direction as illusIrated in Fig. I. In an isotropic material. the principal axes of stress are coincident with the prmcipal axes of strain, thus 0, is the principal s1ress direction closest IO the x, direction if the material is isorropic. The angle 0, is used IO denote the principal stress direction closest IO the X, direction if the material is orIho1ropic. The error in principal s1res.s axis prediction. $=0,-U,. caused by assuming isotropy when the material is actually orthotropic, is determined assuming the same strain slate E,,. E,, and E,, exists in both the isotropic and orthotropic materials. The angles 0, and 0, and Ihc x,. x2 axes arc illus1rdtcd in Fig. 1. The problem addressed in this note Is the following: determine the error $ in the prediction or the orientation ol the principal axes or stress if the material is assumed IO be isotropic, but is actually orthotropic with known technical elaslic constants E,. E:. Y,~. Y,, and G,2. The error $ is a function of the strain SI~IC.
x, Fig. I. The reference coordinate system and 1he principal directions of interest. The x,, x2 coordinate system is selected so tha1 the x, and x1 directions are normals to the planes of symmelry of the orthotropic material. The angle U, specifics the orientation of 1he principal stress axis I if the malerial is isotropic. The angle U, specilies the orientation of the principlll stress axis A il the material is orthotropic. The angle J, is the dtlfcrcnce in orienlalion between these two axes.
F0RML’t.A FOR THE ERROR In the planar situation considered. the stress-strain tions for an ortho1ropic material are given by
El 7.1,= -(E,,+ I -“lz”zI
T,z=‘G,,E,,.
rela-
“21 E,,).
(1)
where T, , . T,, and T, 2 are the components of in-plane stress. E, and E, are Young’s moduli, G,, is the in-plane shear modulus, and V, 2 and vz, arc Poisson’s ratios. The 1an 20, is 339
350
Technical
given by 7;2 ( T,, - Tz21.If the material were isotropic rather than orthotropic. the principal axes of stress would correspond with the principal axes of strain. The tan:@, is then given by E,,,YE,, -E,,). It is well known that the strains E,, . E,, and E, l can be represented in terms of the angle 0, and the quantities C and R by
E,,=C+RcostP,. E,,=C-Rcos~O,,
Etz=Rsin2U,.
(2)
where
ZC=E,,+E,,.
(E,,-I&Y
R=
+&f,
4
1 .
(31
Expressing e,, in terms of 0, and then using equation (3) to express the strains in terms of C, R and 0,. one obtains the following expression for tan Y,. sin 20, tan 20, = Qs + iLf cos ZU,
(4)
’
where C ‘=
Q= M=
i
E,,+E,, = &E,,
-Ez2)’
+4E;,]
'
E,(I+V,,)-E,(I +y,*1 4G,z(l
4G,,(l
0, and 0”. thus sin2e,(Qs+(.V
(6) 1+c0sze,1~~+~.~f-i)c0s2e,~
This is the desired formula for the error $. It has a number of significant properties. First note that because the sine is an odd function and the cosine an even function, tan24 is an odd function of 0,. Thus we need to consider only positive values of 0, because negative values of 8, simply reverse the sign of tan 24. Second. note that tan 2$ given by (6) becomes -tan 2JI when s and cos20, are replaced by -s and - cos 20,. respectively. Since cos(n - 20,) = -cos 20, and sin(n - 20,) = sin 20, this means that tan 2$ evaluated at s and 0, is equal to -tan 2@ evaluated at -sand n/2 -0,. Thus we need only to investigate the formula (6) in the range of 0, from 0 to 45” for all s because a reversal of the sign of s and tan 2$ covers the range 4S-90’. These conclusions can also be obtained from geometric as well as algebraic arguments. Third, note that the ends of the range O-45” are represented by very special cases of (6). The curves associated with 0, =O’ and U,=45” are tan 2$ =0 and tan 211 =Qs, respectively. In between the extremes of O,=r and 0,=45’, the function (6) behaves quite differently. It vanishes for s equal to so. so=[(I -izf)/Q]cos_‘C),. and it has a singularity a1 s equal to s,. s,=soI/Qcos?U,. When the definitions of s, and so are substituted into (6). it can be rewritten tan t$ =
-v,z)
(3
-~,~vz,)
If the orthotropic material is allowed IO become isotropic (i.e..ifwesctE,=E~,~,2=v1,=v.~~nd2(~,,(l+v)=E)thcn Q =O. hf = I and equation (4) shows that 11,= 0” as one would expect. The angle $ will now be calculated using the trigonometric formula for the tangent of twice the difference between
-I)cos?~,)
tan 2$ =
’
-v,zv~,)
E,(I -v,,)+E,(I
Note
s - so tan 20,. ( s-s, >
From this result it is easy to .scc that the limit as s tends to plus or minus inlinity is 0,. A typical error curve is a curve of $ vs s given by (6) or (7) for some lixcd value of0, between 0 and 45” but not equal to 0 or 45”. (Recall that there arc spcu-ial error curves for 0, equal to 0 or 45”.) It is casicr to illustrate a typical curve if a specific example of a typical error curve is selected. As a specilic
_____---___---_--___-45*
-60’
1’ -60
-40
-20
s
of
(7)
Fig. 2. A plot a typical error curve. the one for 0, = 30”. The symbols A, B. C. and D are explained in the text. Note that the point B is not at the origin. although it appears to be in this illustration. It is slightly IO the left of the origin at so= -0.13.
Technical
example. we consider the curve 6, = 30” for human cortical bone tissue with the technical elastic constants E, = 19.8GPa. E,=5.61 GPa. G,r=5.61 GPa, v,,=0.?27and v,,=0.385.ThusQ=0.4&M=1.10,s,=-0.13.s,=-5.19 and the error curve for 6, = 30” is then given by Z+ = arctan [(s+O.13) (s +519)].Thiscurveis plotted in Fig. 2. There are lour points of interest in the plot of this error curve. These four points are illustrated in Fig. 3 as well as Fig 2. First. consider the two points labeled A in Fig. 2 and Fig. 3A. Equation (7) shows that as s tends to plus or minus infinity, $
*2 t
351
Note
tends to 6, = 30”. as shown in the illustration. Thus the two extremes of the error curve plotted in Fig. ? show (L tending to 0, = 30’. Figure 3A illustrates this value of II, in the real plane. Second,at s=so= -0.13 in the illustrated case. JI =O’. This is point B of Fig. 2. and Fig. 38 ihustrates the situation in the real plane. Third. as s tends to s, = - 5. I9 from above. $ tends to -45’ which is illustrated at point C on Fig. 2 and in Fig. 3C. Finally. as s tends to s, = - 5. I9 from below, 4 tends to 45” which is illustrated at point Don Fig. 2 and in Fig. 3D. Using the data on the technical constants for human cortical bone tissue given above, plots of these error curves given by equation (6) are shown in Fig. 4. The percentage error, lOO[(0,-e,),,U,]. can be calculated for each error curve. Plots of these percentage error curves are given in Fig. 5 for human cortical bone tissue.
EXAMPLE:
te1
(01
(Cl
-
Fig. 3. Thr rcfcrcnce coordinate system and the principal directions of interest for the error curve corresponding to 0, = 30 The Figures 3A. 3U. 3C and 3D corrcspond to the situation illustrated at points A, B, C and D, respectively, in Fig. 2. See the text for a more dctailcd explanation.
CALCL’LATIOfi
OF ERROR
The purpose of this section is to illustrate the use of the formula and curves generated above to estimate the error in the principal stress axis orientation induced by the assumption of isotropy in the bone tissue. Suppose a stress analysis ol a human femur was accomplished assuming the bone tissue was isotropic. At one point of interest in the femur the isotropic analysis predicts that a principal stress axis ma&es an angle of 30’ with the long axis or the bone. The x, axis is taken to be coincident with the long axis of the bone and the x1 axis is in a direction perpendicular to the bone’s external surface: thus 0, is 30” in this case. Assume that the strain state predicted at that point is E , , = 2000II”, E,, = 660pz and EL2 = Il6O/rr. What is the error in the orientation of the principal stress axis due to the assumption of isotropy? The answer can bc obtained tither from the direct application of equation (6) or from inspection of Fig. 4. We illustrate the use ol equation (6) first. From the values of E,, . E,, and k:,, given ahovc. s =0.993. Using this value of s. 0, = 3@‘. Q =0.40 and M = 1.10; the value of tan ZJI predicted by (6) is 0.313 and the associated 3/ value is X.69’. representing the difTerence in the principal stress axis orientation. This is an error ol 29%
60’
Y
-60
Fig. 4. Plots of error curves for human cortical bone tissue. The ordinate is the error, JI. the abscissa is the parameter s. and the curves are for various values 0,.
352
Technical Note
-60
-40
-20
0
20
60
t
s
Fig. 5. Plots ol pcrcentagc error for human cortical bone tissue. Note that the curves for various values 0, arc superposed upon one another in these plots.
due to the assumption of isotropy. To obtain the same result from Fig. 4, the value s is estimated to bc approximately one and this vaiuc is used to locate a point on the curve 0, = 30”. The value ol J, associated with this point is seen to be approiimatcly IO”. IMSCL’SSION
The results displayed in Fig. 4 show that the error involved in predicting the orientation of the principal stressaxes of the cortical bone tissue if it is assumed to isotropic rather than orthotropiccan frequently be as much as 45”. Figure 5 shows the percentage error as a function of the paramrtrr s for various values of 0, between 0 and 45”. Very similar results arc obtained for canccllous bone tissue. Observe that for absolute values of s greater than twenty the error is constant at 100%. independent of the values of 0,. The observation is independent of the value ol0, because all of the curves of various values for 0, lie almost on top of one another in these domains. The curves also lie on top of one another in the domain of s between zero and twenty, although the error in that domain is not equal to 100%. One should note, of
course, that the percentage error can be large when the error it.sclfis small. For example, an error of I” in a measurement of just I” is an error of 100%. although it is not an error of serious concern. Acknowfudyemmrs-This investigation has been supported by JJSMHS, Research Grant DE06859 from the National Institute of Dental Research, National Institutes of Health, Bethesda, MD. to Tulane University and by NSF Grant No. BSC-8822401 to the City College of the City University of New York. This research was also supported (in part) by grant number 669301 from the PSC-CUNY Research Award Program of the City University of New York. All calculations and plots were produced at Tulane University on the Department of Biomedical Engineering Local Area VAXclustcr. and we appreciate the partial support of the computer system that has been provided by the Digital Equipment Corporation.
REFERENCE
Cowin.S. C. (1989) Bone Mechanics. CRC Press, Boca Raton.
and
RICHARD T. HART?
‘Department of Mechanical Engineering. City College of the City University of New York. New York. NY 10031. U.S.A. and +Department of Biomedical Engineering. Tulane University. New Orleans. Louisiana 70118. U.S.A.
Abstract-The error in the prediction of the orientation of the principal axes of stress in bone (issue is determined in the case ashen the tissue is modeled as elastically isotropic rather than as orthotropic. the probable symmetry of bone tissue. Results are two-dimensional and assume 1he same underlying strain slate for both the orthotropic and isotropic cases. The maximum error is 45’. and the typical error is generally signiticant.
STATEXlEST OF TttE PRORLEM In this note we consider the error in the prediction of the orientation of the principal axes of stress if 1he bone tissue is modeled as a linear isotropic elastic solid. rather than as an ortho1ropIc enc. Both cancellous and cortical bone tissue can probably be mod&d as linear orrhorropically claslic materials (Cowin. 19X9). The problem is formulated as follows: two plane situations are considered; in one the ma1crial is orthotropic. and in the other 1he material is isotropic. For the orthotropic material the plane is a plane of symmetry. The oul-of-plane shearing stresses and shearing strains are zero for both materials. Thus. in the direction perpendicular IO the plane, the principal direc1ions of stress and strain are coincident for both the isolropic and anisotropic maIeri~ls. Le1 x, and .x2 be IWO in-plant normals [or the planes of symmetry of the orthotropic material. The components of the in-plane strain referred IO these axes are E,i. E,, and E,a. If the material is assumed IO be isotropic, then the in-plane principal strain direction makes an angle 0, with the x, direction as illusIrated in Fig. I. In an isotropic material. the principal axes of stress are coincident with the prmcipal axes of strain, thus 0, is the principal s1ress direction closest IO the x, direction if the material is isorropic. The angle 0, is used IO denote the principal stress direction closest IO the X, direction if the material is orIho1ropic. The error in principal s1res.s axis prediction. $=0,-U,. caused by assuming isotropy when the material is actually orthotropic, is determined assuming the same strain slate E,,. E,, and E,, exists in both the isotropic and orthotropic materials. The angles 0, and 0, and Ihc x,. x2 axes arc illus1rdtcd in Fig. 1. The problem addressed in this note Is the following: determine the error $ in the prediction or the orientation ol the principal axes or stress if the material is assumed IO be isotropic, but is actually orthotropic with known technical elaslic constants E,. E:. Y,~. Y,, and G,2. The error $ is a function of the strain SI~IC.
x, Fig. I. The reference coordinate system and 1he principal directions of interest. The x,, x2 coordinate system is selected so tha1 the x, and x1 directions are normals to the planes of symmelry of the orthotropic material. The angle U, specifics the orientation of 1he principal stress axis I if the malerial is isotropic. The angle U, specilies the orientation of the principlll stress axis A il the material is orthotropic. The angle J, is the dtlfcrcnce in orienlalion between these two axes.
F0RML’t.A FOR THE ERROR In the planar situation considered. the stress-strain tions for an ortho1ropic material are given by
El 7.1,= -(E,,+ I -“lz”zI
T,z=‘G,,E,,.
rela-
“21 E,,).
(1)
where T, , . T,, and T, 2 are the components of in-plane stress. E, and E, are Young’s moduli, G,, is the in-plane shear modulus, and V, 2 and vz, arc Poisson’s ratios. The 1an 20, is 339
350
Technical
given by 7;2 ( T,, - Tz21.If the material were isotropic rather than orthotropic. the principal axes of stress would correspond with the principal axes of strain. The tan:@, is then given by E,,,YE,, -E,,). It is well known that the strains E,, . E,, and E, l can be represented in terms of the angle 0, and the quantities C and R by
E,,=C+RcostP,. E,,=C-Rcos~O,,
Etz=Rsin2U,.
(2)
where
ZC=E,,+E,,.
(E,,-I&Y
R=
+&f,
4
1 .
(31
Expressing e,, in terms of 0, and then using equation (3) to express the strains in terms of C, R and 0,. one obtains the following expression for tan Y,. sin 20, tan 20, = Qs + iLf cos ZU,
(4)
’
where C ‘=
Q= M=
i
E,,+E,, = &E,,
-Ez2)’
+4E;,]
'
E,(I+V,,)-E,(I +y,*1 4G,z(l
4G,,(l
0, and 0”. thus sin2e,(Qs+(.V
(6) 1+c0sze,1~~+~.~f-i)c0s2e,~
This is the desired formula for the error $. It has a number of significant properties. First note that because the sine is an odd function and the cosine an even function, tan24 is an odd function of 0,. Thus we need to consider only positive values of 0, because negative values of 8, simply reverse the sign of tan 24. Second. note that tan 2$ given by (6) becomes -tan 2JI when s and cos20, are replaced by -s and - cos 20,. respectively. Since cos(n - 20,) = -cos 20, and sin(n - 20,) = sin 20, this means that tan 2$ evaluated at s and 0, is equal to -tan 2@ evaluated at -sand n/2 -0,. Thus we need only to investigate the formula (6) in the range of 0, from 0 to 45” for all s because a reversal of the sign of s and tan 2$ covers the range 4S-90’. These conclusions can also be obtained from geometric as well as algebraic arguments. Third, note that the ends of the range O-45” are represented by very special cases of (6). The curves associated with 0, =O’ and U,=45” are tan 2$ =0 and tan 211 =Qs, respectively. In between the extremes of O,=r and 0,=45’, the function (6) behaves quite differently. It vanishes for s equal to so. so=[(I -izf)/Q]cos_‘C),. and it has a singularity a1 s equal to s,. s,=soI/Qcos?U,. When the definitions of s, and so are substituted into (6). it can be rewritten tan t$ =
-v,z)
(3
-~,~vz,)
If the orthotropic material is allowed IO become isotropic (i.e..ifwesctE,=E~,~,2=v1,=v.~~nd2(~,,(l+v)=E)thcn Q =O. hf = I and equation (4) shows that 11,= 0” as one would expect. The angle $ will now be calculated using the trigonometric formula for the tangent of twice the difference between
-I)cos?~,)
tan 2$ =
’
-v,zv~,)
E,(I -v,,)+E,(I
Note
s - so tan 20,. ( s-s, >
From this result it is easy to .scc that the limit as s tends to plus or minus inlinity is 0,. A typical error curve is a curve of $ vs s given by (6) or (7) for some lixcd value of0, between 0 and 45” but not equal to 0 or 45”. (Recall that there arc spcu-ial error curves for 0, equal to 0 or 45”.) It is casicr to illustrate a typical curve if a specific example of a typical error curve is selected. As a specilic
_____---___---_--___-45*
-60’
1’ -60
-40
-20
s
of
(7)
Fig. 2. A plot a typical error curve. the one for 0, = 30”. The symbols A, B. C. and D are explained in the text. Note that the point B is not at the origin. although it appears to be in this illustration. It is slightly IO the left of the origin at so= -0.13.
Technical
example. we consider the curve 6, = 30” for human cortical bone tissue with the technical elastic constants E, = 19.8GPa. E,=5.61 GPa. G,r=5.61 GPa, v,,=0.?27and v,,=0.385.ThusQ=0.4&M=1.10,s,=-0.13.s,=-5.19 and the error curve for 6, = 30” is then given by Z+ = arctan [(s+O.13) (s +519)].Thiscurveis plotted in Fig. 2. There are lour points of interest in the plot of this error curve. These four points are illustrated in Fig. 3 as well as Fig 2. First. consider the two points labeled A in Fig. 2 and Fig. 3A. Equation (7) shows that as s tends to plus or minus infinity, $
*2 t
351
Note
tends to 6, = 30”. as shown in the illustration. Thus the two extremes of the error curve plotted in Fig. ? show (L tending to 0, = 30’. Figure 3A illustrates this value of II, in the real plane. Second,at s=so= -0.13 in the illustrated case. JI =O’. This is point B of Fig. 2. and Fig. 38 ihustrates the situation in the real plane. Third. as s tends to s, = - 5. I9 from above. $ tends to -45’ which is illustrated at point C on Fig. 2 and in Fig. 3C. Finally. as s tends to s, = - 5. I9 from below, 4 tends to 45” which is illustrated at point Don Fig. 2 and in Fig. 3D. Using the data on the technical constants for human cortical bone tissue given above, plots of these error curves given by equation (6) are shown in Fig. 4. The percentage error, lOO[(0,-e,),,U,]. can be calculated for each error curve. Plots of these percentage error curves are given in Fig. 5 for human cortical bone tissue.
EXAMPLE:
te1
(01
(Cl
-
Fig. 3. Thr rcfcrcnce coordinate system and the principal directions of interest for the error curve corresponding to 0, = 30 The Figures 3A. 3U. 3C and 3D corrcspond to the situation illustrated at points A, B, C and D, respectively, in Fig. 2. See the text for a more dctailcd explanation.
CALCL’LATIOfi
OF ERROR
The purpose of this section is to illustrate the use of the formula and curves generated above to estimate the error in the principal stress axis orientation induced by the assumption of isotropy in the bone tissue. Suppose a stress analysis ol a human femur was accomplished assuming the bone tissue was isotropic. At one point of interest in the femur the isotropic analysis predicts that a principal stress axis ma&es an angle of 30’ with the long axis or the bone. The x, axis is taken to be coincident with the long axis of the bone and the x1 axis is in a direction perpendicular to the bone’s external surface: thus 0, is 30” in this case. Assume that the strain state predicted at that point is E , , = 2000II”, E,, = 660pz and EL2 = Il6O/rr. What is the error in the orientation of the principal stress axis due to the assumption of isotropy? The answer can bc obtained tither from the direct application of equation (6) or from inspection of Fig. 4. We illustrate the use ol equation (6) first. From the values of E,, . E,, and k:,, given ahovc. s =0.993. Using this value of s. 0, = 3@‘. Q =0.40 and M = 1.10; the value of tan ZJI predicted by (6) is 0.313 and the associated 3/ value is X.69’. representing the difTerence in the principal stress axis orientation. This is an error ol 29%
60’
Y
-60
Fig. 4. Plots of error curves for human cortical bone tissue. The ordinate is the error, JI. the abscissa is the parameter s. and the curves are for various values 0,.
352
Technical Note
-60
-40
-20
0
20
60
t
s
Fig. 5. Plots ol pcrcentagc error for human cortical bone tissue. Note that the curves for various values 0, arc superposed upon one another in these plots.
due to the assumption of isotropy. To obtain the same result from Fig. 4, the value s is estimated to bc approximately one and this vaiuc is used to locate a point on the curve 0, = 30”. The value ol J, associated with this point is seen to be approiimatcly IO”. IMSCL’SSION
The results displayed in Fig. 4 show that the error involved in predicting the orientation of the principal stressaxes of the cortical bone tissue if it is assumed to isotropic rather than orthotropiccan frequently be as much as 45”. Figure 5 shows the percentage error as a function of the paramrtrr s for various values of 0, between 0 and 45”. Very similar results arc obtained for canccllous bone tissue. Observe that for absolute values of s greater than twenty the error is constant at 100%. independent of the values of 0,. The observation is independent of the value ol0, because all of the curves of various values for 0, lie almost on top of one another in these domains. The curves also lie on top of one another in the domain of s between zero and twenty, although the error in that domain is not equal to 100%. One should note, of
course, that the percentage error can be large when the error it.sclfis small. For example, an error of I” in a measurement of just I” is an error of 100%. although it is not an error of serious concern. Acknowfudyemmrs-This investigation has been supported by JJSMHS, Research Grant DE06859 from the National Institute of Dental Research, National Institutes of Health, Bethesda, MD. to Tulane University and by NSF Grant No. BSC-8822401 to the City College of the City University of New York. This research was also supported (in part) by grant number 669301 from the PSC-CUNY Research Award Program of the City University of New York. All calculations and plots were produced at Tulane University on the Department of Biomedical Engineering Local Area VAXclustcr. and we appreciate the partial support of the computer system that has been provided by the Digital Equipment Corporation.
REFERENCE
Cowin.S. C. (1989) Bone Mechanics. CRC Press, Boca Raton.