international journal for numerical methods in biomedical engineering volume aop issue aop 2012 [doi

8/12/2019 International Journal for Numerical Methods in Biomedical Engineering Volume Aop Issue Aop 2012 [Doi

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING Int. J. Numer. Meth. Biomed. Engng. (2012)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.2491

Algorithms for a strain-based plasticity criterion for bone

Pankaj Pankaj * , and Finn E. Donaldson

School of Engineering, The University of Edinburgh, Kings Buildings, Edinburgh, EH9 3JL, UK

SUMMARY

A range of stress-based plasticity criteria have been employed in the nite element analysis of the post-elastic behaviour of bone. There is some recognition now that strain-based criteria are more suitable for thismaterial because they better represent its behaviour. Moreover, because bone yields at relatively isotropicstrains, a strain-based criterion requires fewer material parameters unlike those required for a stress-basedcriterion. Based on a minimum and maximum principal strain criterion, a robust strain-based plasticityalgorithm is developed. As the criterion comprises six piecewise linear surfaces in principal strain space,it has a number of singular regions. Singularity indicators are developed to direct the algorithm to makeappropriate plastic corrector returns when singularity regions are encountered. The developed algorithmspermit a plastic corrector to be achieved in a single iterative step in all cases. A range of benchmark testsare developed and conducted after implementing the algorithm in a nite element package. These tests showthat the constitutive behaviour is as expected. Copyright 2012 John Wiley & Sons, Ltd.

Received 21 February 2012; Accepted 26 March 2012

KEY WORDS : stress-based; anisotropy; singularity indicators; benchmark tests

1. INTRODUCTION

Several stress-based criteria have been considered for bone. These range from the extensively usedisotropic von Mises criterion to the anisotropic Tsai-Wu [1] criterion [211]. Although some of thesimpler isotropic criteria, perhaps employed because they were readily available, have been shownto be unsuitable for bone [3, 8], other anisotropic ones require too many parameters, and their accu-racy is not sufcient to justify the additional experimental effort required for the determination of the parameters [3, 12].

There has been some debate as to whether stress-based or strain-based criteria are best for rep-resenting bone yielding [13]. Some studies have indicated that strain-based criteria offer greateraccuracy [4, 14]. For bone, a strain-based criterion is more suitable than stress-based criteria fortwo compelling reasons. Firstly, there is now some evidence to suggest that yielding and failure of bone is based on strain rather than stress [15] and strain-based criteria offer greater accuracy [4,14].Secondly, bone is an anisotropic material, which can be approximated by orthotropy [16, 17]. Thisis reected both in its elastic properties and in its strength. As a consequence, the yield criterion for

bone has to be anisotropic in the stress space. However, in the strain space, bone yields at relativelyisotropic strains, and the yield strain is not dependent upon apparent elastic stiffness or density[4, 14, 1820]. An isotropic, strain-based criterion requires relatively few material properties to beevaluated and no a priori identication of material orientation. Contrastingly, an anisotropic stress-based criterion requires numerous material parameters to be determined and must be oriented withthe physical structure of the bone. These reasons suggest that for bone, a strain-based criterion is

*Correspondence to: Pankaj Pankaj, School of Engineering, The University of Edinburgh, Kings Buildings, Edinburgh,EH9 3JL, UK.

E-mail: [email protected]

Copyright 2012 John Wiley & Sons, Ltd.


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P. PANKAJ AND F. E. DONALDSON

not only biodelic but also numerically more convenient to employ. Interestingly, although severalstudies have found strain-based yield criteria to be more suitable for bone, development of plastic-ity algorithms has been extremely limited [21]. In fact, some studies have simply used an elasticanalysis to predict regions where strains violate a strain-based criterion [8,22].

Strain-based plasticity was rst discussed about four decades ago by Naghdi and Trapp [23] andsubsequently developed by others [2428]. Over the decades, it has received very little attention in

comparison with stress-based theories, and generally, nite element codes do not incorporate strain-based plasticity. It should be noted that for any static stressstrain state, a yield criterion formulatedin stress space can be converted into its equivalent in strain space. However, key differences becomeapparent as yielding progresses [23, 29, 30]. First identied by Yoder and Iwan [27], it was illus-trated by Lan et al . [25] that the sign of the derivative of yield function uniquely identies loading,neutral and unloading conditions in strain space, but is ambiguous in stress space.

Gupta et al . [21] developed a yield criterion for trabecular bone based upon an isotropic modi-ed super ellipsoid dened in principal strain space by Bayraktar et al . [18]. The criterion has fourparameters, two of which are essentially the yield strains in compression and tension. This yieldsurface is smooth and is therefore free from the issues associated with singularities at sharp cornersin piecewise linear surfaces such as those in MohrCoulomb, Tresca and Rankine criteria; althoughalgorithms have been developed to effectively handle issues associated with singular yield surfaces[3133]. It has been demonstrated by previous researchers [3, 8] that reasonable accuracy can beachieved through the use of a maximum (Saint Venant) and minimum principal strain criterion.This yield criterion can be visualised as a cube in principal strain space. Indeed, it consists of sixplanar yield surfaces and requires only two parameters (yield strains in tension and compression)to be dened. Furthermore, it has been shown that accurate returns to such piecewise linear sur-faces (in a predictorcorrector algorithm) can be achieved in a single iterative step [31, 33]. Thisgreatly reduces the numerical cost of the overall algorithm and also improves its stability and accu-racy. Such algorithms can be effectively used in biomedical engineering problems [34]. For thesereasons, the minimum and maximum elastic principal strain criterion was selected for this work.Although it has previously been used as a limit in studies of bone failure [35], it has not previouslybeen developed as a full plasticity algorithm. The aim of this study was to develop algorithms forthis simple strain-based constitutive model for inclusion in nite element codes.

2. FORMULATION AND DEVELOPMENT OF ALGORITHMS

2.1. Denition of the criterion

The yield criterion is dened by limiting values of the elastic principal strains. The criterion is fullydened by specication of tensile and compressive yield strains. The yield criterion g , is a functionof the principal elastic strains, e1 ,

e2 ,

e3 i.e.

g D g e1 ,e2 ,

e3 (1)

As there is little available data on post-yield behaviour of bone, the algorithms developed in thisstudy are limited to perfect plasticity. The yield criterion is represented in the principal strain space

by the six yield planes

g1 t D e1 Y t I g2 t De2 Y t I g3t D

e3 Y t

g1 c D e1 C Y c I g2 c De2 C Y c I g3 c D

e3 C Y c (2)

where Y t and Y c are the tensile and compressive yield strains. As usual, tension is taken as positiveand compression as negative. In principal strain space, this yield criterion can be visualised as acube with sides Y t C j Y c j as illustrated in Figure 1. It is important to note that even under conditionsof elastic isotropy, a yield surface in principal strain and principal stress space do not have thesame shape.

Copyright 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. (2012)

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ALGORITHMS FOR A STRAIN-BASED PLASTICITY CRITERION FOR BONE

(a)

(b)

Figure 1. The 3D maximum principal strain criterion (a) and its shape in stress space (b) under conditionsof elastic isotropy. Note Y t and Y c indicate tensile and compressive yield strains, respectively. The line of coloured locations indicate equivalent points on the two yield surfaces for 1 D Y t , 2 D 0, 3 D Y c ! Y t .

2.2. Predictorcorrector theory

Clausen et al . [36] and Huang and Grifths [37] considered piecewise linear yield functions instress-based plasticity. Here, we follow their general arguments. For piecewise linear yield func-tions, the ow vectors are constant during the corrector phase if it is performed in the principalcoordinate system. This fact can be used to greatly simplify the return mapping process. To realisethese simplications, it is necessary to rotate the predictor state into its principal orientation beforeperforming the plastic return. The consistent tangent matrix can then be dened in principal coordi-nates before it and the returned state are rotated back into general coordinates. In the following, thesymbols

eand O

edenote the same term in general and principal coordinate systems. The rotation of

strain and stress vectors in Voigt notation can be expressed as

O

eD A

eeor

eD A

e1 O

e(3)

O

eD A

eT

eor

eD A

eT O

e(4)

and elasticity and compliance tensors as

OD

e

eD A

e

T D

e

e A

e

1 or D

e

e D A

e

T OD

e

eA

e

(5)


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OC

ee

D A

eC

ee A

eT or C

ee D A

e1 OC

eeA

eT (6)

where A

eis the 6 6 rotation matrix, which can be derived by following the arguments of

Clausen et al . [36] as

A

eD 26666666664

cxOx cxOx c

yOx c

yOx c

Ox c

Ox c

xOx c

yOx c

Ox c

xOx c

yOx c

Ox

cxOy

cxOy

cyOy

cyOy

cOy

c Oy

cxOy

cyOy

cOx

cxOy

cyOy

c OycxO c

xO c

yO c

yO c

Oc

O c

xO c

yO c

Oc

xO c

yO c

O

2c xOx cxOy 2c

yOx c

yOy 2c

Ox c

Oy c

xOx c

yOy C c

xOy c

yOx c

Ox c

xOy C c

Oy c

xOx c

yOx c

Oy C c

yOy c

Ox

2c xO cxOx 2cyO c

yOx 2c

Oc

Ox c

xO c

yOx C c

xOx c

yO c

OcxOx C c

Ox c

xO c

yO c

Ox C c

yOx c

O

2c xOy cxO 2c

yOy c

yO 2c

Oy c

O c

xOy c

yO C c

xO c

yOy c

Oy c

xO C c

Oc

xOy c

yOy c

O C c

yO c

Oy

37777777775(7)

in which c iOj D cosiOj

, or the direction cosine of the angle iOj from axis Oj to axis i .

Hookes law can be written in terms of the elastic and plastic parts of the total strain as

O

eD OD

e

eO

eO

e

p D OD

e

eO

e

e (8)

where OD

ee

is the elasticity matrix in principal strain orientations.The associative plastic ow rule for stress-based criterion f , and equivalent strain-based criterion

g , can be expressed in rate form as

PO

ep D d

@f @O

eD d OC

ee @g

@O

ee (9)

where OC

ee

D OD

ee 1

, is the compliance matrix in principal strain orientations and d is the plastic

multiplier. The presence of OC

ee

in Equation (9) is required to satisfy Iliushins postulate [38]. Thepredictorcorrector phase can be written in terms of stress and strain as

O

eD O

eT d OD

ee @f

@O

e(10)

and OD

ee

O

ee D OD

ee

O

eeT d

@g@O

ee (11)

respectively, where O

eT and O

eeT are the trial or predictor states of stress and elastic strain. Rearranging

Equation (11) and premultiplying by OC

ee, we have

d OC

ee @g

@O

ee D O

eeT O

ee (12)

Premultiplying Equation (12) by @g=@O

e

e T and alternatively @g=@O

e

eT

T gives

d @g

@O

ee

T O

C ee @g

@O

ee D

@g

@O

ee

T

Oee

T Oee

(13)

and d @g@O

eeT

T OC

ee @g

@O

ee D

@g@O

eeT

T

O

eeT O

ee (14)

respectively. Following the analysis of convex surfaces by Hiriart-Urruty and Lemarechal [39] andyield planes by Huang and Grifths [37], we can write

@f @O

e

T

O

eT O

e6 f O

eT 6

@f @O

e

T

T

O

eT O

e (15)


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and @g@O

ee

T

O

eeT O

ee 6 g O

eeT 6

@g@O

eeT

T

O

eeT O

ee (16)

Substituting the left hand sides of Equations (13) and (14) into Equation (16) we get

@g

@Oee

T OC

ee @g

@Oee 6

g O

eeT

d

6 @g

@OeeT

T OC

ee @g

@Oee (17)

If the yield surface is piecewise linear and the return is to a single surface then @g=@O

ee D @g=@O

eeT

will remain constant throughout the return in principal coordinates. Therefore, Equation (17)becomes

@g@O

ee

T OC

ee @g

@O

ee D

g O

eeT

d D

@g@O

eeT

T OC

ee @g

@O

ee

giving d Dg O

eeT

@g@O

ee

T OC

e

e @g@O

ee

Dg O

eeT

@g@O

eeT

T OC

e

e @g@O

eeT

(18)

Substituting Equation (18) into Equation (11) we have

OD

ee

O

ee D OD

ee

O

eeT 26664

g O

eeT

@g@O

eeT

T OC

ee @g

@O

eeT

37775@g

@O

eeT

O

e

e D O

e

eT 26664

g O

eeT

@g@O

eeT

T OC

ee @g

@O

eeT

37775OC

e

e @g@O

e

eT

(19)

Equation (19) represents the core return mapping process for returns to a plane. However, in thepresent yield criterion, it is also necessary for states of strain that lie in singular regions to bereturned to the following: lines (intersection of two yield planes) and points (intersections of threeyield planes). Singularity indicators are developed later in this section to determine if a return tothese locations is required.

We rst consider the situation in which the singularity indicators reveal that a return to a line isrequired. The equation of a line on the yield surface can be expressed as the parametric equation

O

ee D t Or

el C O

el (20)

where t is a scalar parameter, Or

e

l is a vector parallel to the line and O

e

l is an arbitrary point on theline. An innite number of predictor states may require a return to a given point on a line, therefore,the direction of d O

ep is unknown. However, the fact that the rst derivatives of the yield surfaces

are orthogonal to the line can be used to derive a closed form expression for the corrected state.Substitution of Equation (20) into Equation (12) gives

d O

ep D d OC

ee @g

@O

ee D hOe

eT t Or

el C O

el i (21)

which when premultiplied by OD

ee

gives

d @g@O

e

e D OD

ee hOe

eT O

el t Or

eli (22)


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Rearranging Equation (22) and premultiplying by Or

el T produces

d Or

e

l T @g@O

e

e C t Or

e

l T OD

e

eOr

e

l D Or

e

l T OD

e

eO

e

eT O

e

l (23)

Noting that

Or

el T @g

@O

ee D 0 (24)

because of the orthogonality condition, the parameter t can be evaluated as

t DOr

el T OD

ee

O

eeT O

el

Or

el T OD

ee

Or

el

(25)

Substitution of Equation (25) into Equation (20) gives the corrected state as

O

ee D 264

Or

el T OD

ee

O

ee

T O

el

Or

el T OD

ee

Or

el 375Ore

l C O

el (26)

In the case of a return to a point, the corrected state can simply be written down as the coordinatesof the point in the principal space, that is

O

ee D Y t =c Y t =c Y t =c 0 0 0 T (27)

where Y t =c represents the yield strain in tension or compression as appropriate.

2.3. Derivation of the singularity indicators

The equations derived in Section 2.2 can be used to return any predictor state of strain to the yield

surface (Figure 2). It remains to identify which states of strain are required to be returned to a plane,a line or a point. This problem is confounded by the fact that the return vectors in principal strainspace are not orthogonal to the yield surface (e.g. the predictorcorrector line plotted in Figure 2).Note that although @g=@O

ee is orthogonal to g , the term OC

ee.@g=@O

e/ e is not. Predictor states of strain

therefore exist, which are in violation of only one yield surface, yet when corrected are in viola-tion of another. Such strain states are a subset of the singular states. To overcome this obstacle, werequire the derivation of singularity indicators similar to those derived by de Borst et al . [40] andPankaj and Bicanic [33] for the MohrCoulomb yield criterion.

Following an argument similar to that of Pankaj and Bicanic [33] for stress space, we continuein principal strain space, which simplies the algebra required. The process used here effectivelychecks whether the elastic principal strains have changed order following a return(s) to the yieldsurface(s). By denition, the predictor principal strains, are ranked eT 1 >

eT 2 >

eT 3 . A singular

predictor state must exist if, following a return(s), the nal elastic principal strains are rankedother than e1 > e2 >

e3 . In principal strain space, the rst derivatives of the yield functions

(Equation (2)) are

@g1t@O

ee D 8


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(a)

(b)

Figure 2. The strain-based yield surface (a) in principal strain space and (b) plane representation.A predictorcorrector step is shown between states of strain

ei and

ei C 1 .

Writing Equation (19) in vector form, we therefore get one of the following

8


8/22


If this return results in e2 > e1 , then the predictor strain state is in a singular regime, and we can

dene a singularity indicator for the tensiontension region as

tt D e2e1 > 0

or tt D

(eT 2 g1 t

OC e21OC e

11

)

eT 1 g1t (33)

A value of t t > 0 will indicate the need for a return to the line at the intersection of the yieldplanes g1 t and g2 t .

2.3.2. Example 2 - Singular in tensiontensiontension region. Consider a situation with tt > 0 ,indicating that the return needs to be made to the line as discussed earlier. A return to the linemarking the intersection of yield planes g1 t and g2 t leads to

O

ee D t Or

el C O

el

8


9/22


substituting from Equation (38) for d O

ee in Equation (39) gives

d g D @g@O

e

e

T d Oed OC

ee @g

@O

e

e D 0 (40)

Rearranging Equation (40) gives

@g@O

ee

T d O

eD d

@g@O

ee

T OC

ee @g

@O

ee (41)

and

d D

@g@O

ee

T d O

e@g@Oe

e

T OC

ee @g

@O

ee

(42)

which can be substituted into Equation (37) to produce

d O

eD 26664

OD

ee

@g@O

ee

@g@O

ee

T

@g@O

ee

T OC

ee @g

@O

ee

37775d O

ed O

eD OD

eep

d O

e(43)

2.4.2. Tangential modulus matrixstrain state on a line. In the case of a strain state on a line at theintersection of multiple yield surfaces, OD

e

epcan be evaluated as the summation of two 6 6 matrices

in the form adopted by Clausen et al . [36] for isotropic stress-based yield criteria in principal space

OD

eep

D h OD

ei

ep

C OG

e(44)

where h OD

ei ep contains elements relating only to normal strain components and is thus non-zero

only in the rst three rows and columns, and OG

econtains only the shear stiffness terms rotated into

principal directions (following Equation (5)), that is

OG

eD A

eT 24

0

e3 30

e3 30e3 3D

eeij

3 3

35Ae1 , for i , j D 4, 5, 6 (45)

For a return to a line, the direction of the elastic strain increment, dened in principal strain space,must be parallel to the line. Therefore,

d O

ee D d Or

el (46)

where d is a scalar multiple. The corresponding increment of stress must be dened as

d O

eD d OD

ee

Or

el (47)

and OD

eep

must be singular with respect to any vector orthogonal to OD

ee

Or

el . Hence, from Equation (37)

and its denition in Equation (43), OD

eep

can be written as

OD

e

epd O

e

e D OD

e

ed O

e

e (48)


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and therefore

h OD

ei ep Or

el D OD

ee

Or

el

h OD

ei ep Or

e

l Or

e

l T OD

e

eD OD

e

eOr

e

l Or

e

l T OD

e

e

h OD

ei ep D ODe

e

Or

el

Or

el T

OD

ee

Or

el T OD

ee

Or

el

(49)

2.4.3. Tangential modulus matrixstrain state at a point. In the present yield criterion, onlysingularity points dened by the intersection of three yield surfaces exist. Following the argumentsof [36] at such a point, the tangent modulus matrix is singular with respect to any direction inprincipal space, but not with respect to the shear directions. Hence,

OD

eep

D OG

e(50)

where OG

eis dened in Equation (45).

2.5. Consistent tangent matrix

Use of the tangent modulus matrix OD

eep

derived in Section 2.4 has been shown to impair thequadratic rate of convergence of the global iteration scheme [41]. Simo and Taylor [42] developeda consistent tangent matrix to restore the quadratic convergence of the global Newton scheme. Thiscan be expressed as

OD

eepc

D T

eOD

eep

(51)

where T

eis a modication matrix generally dened as

T

eD I

eC D

ee

d d f = d

ed

e!1 (52)

in which the derivative is taken at the corrected state, and I

eis the identity matrix. In the case of

linear yield criteria (such as the present one), it has been established that T

ecan be evaluated at the

predictor state as

T

eD I

eD

ee

d d f = d

ed e(53)

thus avoiding the inversion required in Equation (52) [43]. Conversion of Equation (53) for a yieldsurface dened in strain space gives

T

eD I

eC

ee

d d g=d

e

e

d

ee D I

e

d C

e

e d g=d

e

e

d

ee D I

e d

e

p

d

ee (54)

where the plastic corrector in general strain space,

ep can be expressed in terms of its

representation in principal strain space, using Equations (3) and (7), as

T

eD I

e d

ep

d

ee D I

ed A

e1 O

ep

d

ee (55)

which can be expanded, noting the matrix identity d A

e1 =d x D A

e1 d A

e=d x A

e1 [44,45], with

the substitution

ei D A

e1 d A

ed

e

ei

A

e1 O

ep


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such that

T

eD I

e

ex

ey

e

exy

ex

ey (56)

The term d A

e=d

e

ei represents the rate of change of the principal strain axes, which can be expressed

using the following geometrical arguments, similar to Clausen et al . [36], for principal stress axes.Expanding d A

e=d

eei with the chain rule gives

d A

edeei

D @A

e@ xd x

d

eei

C @A

e@ yd y

d

eei

C @A

e@ d

d

eei

(57)

where d x , d y , and d are innitesimal rotation angles about the x -axis, y -axis and -axis. Inthe case that the principal ( Ox , Oy , O) and general ( x , y , ) axes are aligned, the angles between themcan be represented as the tensor

e

0 D 264xOx

xOy

xO

yOx

yOy

yO

Ox

Oy

O

375D 24

0 2 22 0

2

2 2 035 (58)

from which the corresponding transformation tensor can be derived as

e0 D cos

e0 D 24

1 0 00 1 00 0 1

35 (59)Innitesimal rotations of the coordinate system about each of the axes are illustrated in Figure 3 andproduce the transformation tensors

ex D cos24

0 2 22 d

x 2 d

x

2 2 C dx d x

35D 241 0 00 1 d x

0 d x 135 (60)

ey D cos24

dy

2 2 C dy

2 0

2

2 dy

2 dy 35D 24

1 0 dy

0 1 0d y 0 1 35 (61)

and

e D cos24

d 2 d

2

2 C d d 2

2 2 035D 24

1 d 0d 1 00 0 1

35 (62)

Figure 3. Innitesimal rotation angles about the three coordinate axes; (a) angle d x about the x -axis, (b)angle d y about the y -axis, and (c) angle d about the -axis.


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respectively. It may be noted that these equations are slightly different from those obtained byClausen et al . [36]. From Equations (59)(62) the change in transformation matrix A

e(Equation (7))

resulting from a rotation around each axis can be expressed as

@Ae@ x dx D A

eex A

ee0 D

26666664

0 0 00

e3 3

0 0 d x

0 0 d x0 0 0 0 d x 00 0 0 d x 0 00 2d x 2d x 0 0 0

37777775

(63)

@A

e@ y dy D A

eey A

ee0 D

26666664

0 d y 00

e3 30 0 0

0 d y 00 0 0 0 0 d y

2d y 0 2d y 0 0 00 0 0 d y 0 0

37777775

(64)

and

@A

e@ d D A

ee A

ee0 D

26666664

d 0 00

e3 3d 0 0

0 0 02d 2d 0 0 0 0

0 0 0 0 0 d

0 0 0 0 d 0

37777775(65)

where quadratic terms are neglected. To evaluate the terms of Equation (57), it is also necessaryto calculate d x =d

eei , d

y =d

eei and d

=d

eei . Innitesimal changes in the normal strains do not

affect the orientation of the principal strain directions, hence,

d i

d ej D 0

e)

d A

ed ej D 0e)

ej D 0

e, for i , j D x , y , (66)

Through consideration of the strain Mohrs circles (Figure 4) the rate of change of the rotationangles with the shear strains exy ,

ex and

ey can be found as

Figure 4. Mohrs circles of strain for innitesimal rotation angles d x , d y and d .


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sin .2d x / D12 d

ey

12

e2

e3

, d x

d eyD

12 e2

e3

(67)

sin .2d y / D12 d

ex

1

2

e

1

e

3

, d y

d ex

D 1

2 e1

e

3

(68)

sin .2d / D12 d

exy

12

e1

e2

, d

d exyD

12 e1

e2

(69)

noting that for small angles, sin . Substitution of Equation (67) into Equation (63) gives

@A

e@ ey

D 1

2 e2e3

26666664

0 0 00

e3 30 0 1

0 0 10 0 0 0 1 00 0 0 1 0 00 2 2 0 0 0

37777775

(70)

and similarly from Equation (68) and Equation (64),

@A

e@ ex D 1

2 e1e3

26666664

0 1 00

e3 30 0 0

0 1 00 0 0 0 0 12 0 2 0 0 00 0 0 1 0 0

37777775(71)

and Equation (69) into Equation (65)

@A

e@ exy D 1

2 e1e2

26666664

1 0 00

e3 31 0 0

0 0 02 2 0 0 0 00 0 0 0 0 10 0 0 0 1 0

37777775(72)

The consistent modier T

e

can be calculated by inserting Equations (66) and (70)(72) intoEquation (56) such that

T

eD I

e 0

e 0

e 0

e

exy

ex

ey (73)

2.6. Overview of the algorithm

1.Initialisation:Pass in, d

e,

ee ,

ep

Set, Y t , Y c , D

ee , C

ee D D

ee 1 ,

eeT D

ee C d

e2.Transform eeT , D

ee and C

ee into principal coordinates O

eeT , OD

ee

andOC

e

eusing Equations (3), (5) and (6)


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3.Check yield conditionsIF all g. O

ee / 6 0: no return required,

O

ee D O

eeT , OD

eepc

D OD

ee

GO TO 5ELSE: strain return required,GO TO 4

4.Check singularity indicators to identify type of returnIF return to a plane:Calculate d O

ep from Equation (19) and OD

eep

from Equation (43)O

ee D O

eeT d O

ep

ELSE IF return to a line:Calculate O

ee from Equation (26) and OD

eep

from Equations (44) and (49)ELSE IF return to a point:Calculate O

ee from Equation (27) and OD

eep

from Equation (50)END IFCalculate T

e

from Equation (73)OD

eepc

D T

eOD

eep

5.Rotate O

ee and OD

eepc

back into general coordinates

ee and D

eepc

using Equations 3 and 5

6.Update stress and state variables

eD D

ee

ee ,

ep D

eee

7.Return

e, D

eepc ,

ee and

ep

3. BENCHMARK TESTS

The algorithm was coded as a Fortran user subroutine appended to Abaqus Standard (ABAQUS,Inc., Providence, RI, USA). The applied conditions were uniaxial tension and compression; hydro-

static tension and compression; triaxial combinations of tension and compression; and planar andtriaxial pure shear. Uniaxial tests (Figure 5(a) and (b)) were applied by xing all nodes on one facein the direction and one of these also in the x and y directions; equal displacement was applied tothe remaining nodes in the direction. In hydrostatic tests (Figure 5(c) and (d)), one node was xedin all directions, and all nodes with a common face were restrained against translation orthogonallyto that face; equal displacements were applied orthogonally to nodes on the unrestrained faces. Intriaxial tension and compression tests (Figure 5(e) and (f)), nodes were restrained identically to thehydrostatic case, but the displacements were applied such that the ratio of tensile to compressivestrain was Y t =Y c . In the planar shear test (Figure 5(g)), one node was fully xed, and other nodeswere displaced to produce a symmetrical diamond shape within the testing plane, without restraintto displacement out of that plane. In the triaxial shear test (Figure 5(h)), one node was fully xed,and all others were displaced to produce a symmetrical diamond in 3D. The tests were performedusing tensile and compressive yield strains of 0.5% and 0.7%, respectively. These are typical yieldstrains reported for bone [46]. In the benchmark tests, the material was assumed to be isotropic,to maintain transparency of the solution, with Youngs modulus of 10 GPa and Poissons ration of 0.3. In the following, the results shown are the three principal strain components. Their elastic andplastic components are plotted against the largest principal component of applied total strain in eachcase. The results of the benchmark tests are considered in the following paragraphs. A discussion of the results is provided in the following section.

Results of uniaxial tests are illustrated in Figure 6(a) (compression) and (b) (tension). Underuniaxial compression, the elastic strain increased linearly until the yield strain in compression wasreached. Initial plastic returns were made to a single yield surface. Following rst yield, there was nofurther increment in the elastic strain component. The minimum elastic principal strain component


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(a) (b) (c)

(d) (e)

(g)

(h)

(f)

Figure 5. Single element benchmark test boundary conditions for (a) uniaxial compression, (b) uniaxialtension, (c) hydrostatic compression, (d) hydrostatic tension, (e) compressivecompressivetensile,(f) tensiletensilecompressive, (g) plane pure shear, and (h) triaxial pure shear. Note D t , D c and D s are

the applied tensile, compressive and shear displacements, respectively.

remained at a constant value of 0.7%. All further applied strain became plastic strain. A similarpattern of yielding occurred in tension; rst yield occurred when the maximum elastic principalstrain reached the tensile yield strain and remained at a constant value of 0.5% thereafter. In bothcases, it was noted that once rst yield occurred in one direction, the elastic strain in other unyieldeddirections did not increase.

The results of hydrostatic tests are illustrated in Figure 6(c) (compression) and (d) (tension).In Figure 6(c), the plastic strain remained zero until the compressive yield surface was reached.Plastic returns were made to three yield surfaces. This occurred simultaneously in all principal


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(a) (b)

(c) (d)

Figure 6. Single element uniaxial and hydrostatic tests of the strain-based yield criterion under (a) uniaxialcompressive, (b) uniaxial tensile, (c) hydrostatic compressive and (d) hydrostatic tensile displacements.

directions. Under further displacement, the elastic strain remained constant at a value of 0.7%,and all additional strain became plastic. Equivalent behaviour was observed under conditions of hydrostatic tension (Figure 6(d)). The elastic strain reached the tensile yield strain simultaneouslyin all principal directions; further strain became plastic. The limiting strain was 0.5%.

Under compressioncompressiontension conditions, the response was as shown in Figure 7(a).The ratio of tensile to compressive applied displacements was Y t =Y c . The principal elastic strainsincreased until rst yield occurred; in this case, at two compressive and one tensile yield planessimultaneously. Further strain induced plastic strain in all principal directions. Once yielded, theelastic strain components remained constant at their respective yield values. Further applied strainbecame plastic. Similar behaviour was observed under tensiontensioncompression conditions,illustrated in Figure 7(b). In this case, tensile yield occurred for both the maximum and intermedi-ate principal strain component and compressive yield for the minimum. At post-yield, the principalelastic strains remained at the yield strain values.

Figure 7(c) shows the response under pure shear displacements in a single plane. Under these con-ditions, the applied shear strain, app can be resolved into principal strains of 1 D app =2, 2 D 0,and 3 D app =2. Therefore, because the yield strain was lower in tension, it was the mode of rst yield. Following rst yield, a component of plastic strain was induced in all principal directionscaused by the Poisson effect. This reduced the rate of increase of the elastic minimum principal


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(a) (b)

(c) (d)

Figure 7. Single element triaxial and shear tests of the strain-based yield criterion. Applied boundary condi-tions were (a) compressivecompressivetensile, (b) tensiletensilecompressive, (c) plane pure shear and

(d) triaxial pure shear displacements.

strain component (see kinks at points A and B in Figure 7(c)), and induced an equal and oppositecomponent of elastic strain in the direction of the middle principal strain to satisfy the boundarycondition 2 D 0. The minimum principal elastic strain then increased until the compressive yieldstrain was reached. At this point, no further elastic strain accumulated; further displacement inducedonly plastic strain. The principal elastic strains in yielded directions remained at the yield values.

The results of a triaxial state of pure shear are presented in Figure 7(d). The applied shear strains of 12 D 23 D 31 D app resulted in applied principal strains of 1 D app , and 2 D 3 D app =2.

The elastic strains increased until rst yield occurred in tension. Further straining induced plas-tic strain in all principal directions. This reduced the rate of accumulation of elastic strain in theunyielded middle and minimum principal directions (see kinks at points A and B in Figure 7(c)).Elastic strain accumulated in the unyielded directions until the compressive yield strain was reached.Following yield, all elastic principal strain components remained at their respective yield strains.

Cyclic loading tests were conducted to examine the behaviour of the developed algorithm underunloading and reloading. The elastic and plastic strain components, which developed in the singleelement subjected to several cycles of uniaxial tensile and compressive displacements are plottedin Figure 8(a) and (b). Under the initial tensile conditions, the behaviour was linear until it reached


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(a) (b)

Figure 8. Single element test under cycles of uniaxial tensile and compressive applied strain in the direction; (a) elastic and (c) plastic strain components.

the tensile yield strain Y t . Prescribed displacement was then progressed until 0.1% plastic strain atwhich point the applied displacements were reduced. The elastic strain returned along the pre-yieldgradient with a non-zero total strain component, where the elastic strain was reduced to zero. Addi-tional compressive displacement was applied until yield occurred on the compressive yield surface.This was encountered at a total strain value of 0.6%, 0.1% higher than the assigned yield strain asa result of the preceding tensile plastic strain in the direction. Compressive displacement was con-tinued up to 0.3% plastic strain. Two further cycles of tensile followed by compressive loading wereapplied and produced similar behaviour. The principal elastic strain components remained withinthe assigned yield strains throughout all cycles.

4. DISCUSSION OF BENCHMARK TESTS

Under uniaxial tension and compression, it was demonstrated that the elastic predictorplasticcorrector algorithm worked correctly for non-singular states of strain. The various tests under hydro-static and combinations of tension and compression involved singular strain states in violation of both two and three yield surfaces. In every case, the algorithm was seen to achieve a valid plas-tic return to the yield surface. The planar and triaxial pure shear conditions involved the greatestpossible rotation of the principal strain state from the model coordinate system. This representedan additional test for the developed algorithm. Under these conditions, the singular strain stateswere resolved successfully. Under repeated cyclical uniaxial tensile and compressive strains, thealgorithm correctly resolved the elastic and plastic strain components. These results indicate thatthe developed algorithm is able to accurately model the maximum/minimum principal strain yieldcriterion in general FE analyses, including loading, unloading and reverse loading conditions.

4.1. Yielding and the Poisson effect In the results of uniaxial single element tests, it was observed that following yielding in one direc-tion, the elastic Poisson effect in other directions ceased. More specically, the component of elasticstrain induced by Poissons effect was replaced by an equivalent plastic strain. For example, thisbehaviour can be seen in Figure 6(a) with uniaxial compression. Preceding rst yield, the elas-tic compressive strain increased and was accompanied by associated tensile Poisson strains in theorthogonal directions. Following rst yield, both the compressive and tensile elastic strain compo-nents remained constant, although only the former had exceeded the yield surface. This behaviourwas accompanied by the accumulation of plastic strains at an equivalent rate to the pre-yield state.It is important to note that the Poisson effect is related to elastic strain, and once the elastic strain


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becomes constant in the principal loading direction, there will be no additional elastic strain becauseof the Poisson effect in the orthogonal directions.

In the case of multiaxial and shear tests, similar behaviour was observed. However, following rstyield in these cases, the rate of increase of elastic strain reduced, but did not cease in the unyieldedorthogonal directions. Although the applied loading at the element boundary continued to increaseat the same rate, both pre-yield and post-yield, the fact that the rate of increase of elastic strain

reduced in the unyielded directions requires explanation.To explain this, we rst consider the uniaxial case in more detail. Consider an isotropic solid withelastic compliance tensor OC

ee. If we assume an initial state of strain, which lies on the tensile yield

surface and apply an additional tensile unit stress O

e, the elastic predictor, or trial strain increment

in principal space is

O

eeT D OC

ee

O

eD 1E 241

1 1

358


20/22


directions must be reduced by the amount of plastic strain induced by the Poisson effect. To illus-trate this, consider the case of a planar pure shear boundary condition on an isotropic material(Figure 7(c)). As previously noted, an applied shear strain of app resolves into principal strains of

1 D app =2, 2 D 0, and 3 D app =2. The initial state lies upon the tensile yield surface, and anadditional stress O

eis applied

OeeT D OC e

e

O eD 1E 241

1 1

358


21/22


CONFLICT OF INTEREST STATEMENT

The authors have no conict of interest with respect to the presented work.

REFERENCES

1. Tsai SW, Wu EM. A general theory of strength for anisotropic materials. Journal of Composite Materials 1971;5 :5880.

2. Donaldson F, Pankaj P, Law AH, Simpson AH. Virtual trabecular bone models and their mechanical response.Proceedings of the Institution of Mechanical Engineers, H: The Journal of Engineering in Medicine 2008;222 :11851195.

3. Fenech CM, Keaveny TM. A cellular solid criterion for predicting the axialshear failure properties of bovinetrabecular bone. Journal of Biomechanical Engineering 1999; 121 :414422.

4. Ford CM, Keaveny TM, Hayes WC. The effect of impact direction on the structural capacity of the proximal femurduring falls. Journal of Bone and Mineral Research 1996; 11 :377383.

5. Keyak JH. Improved prediction of proximal femoral fracture load using nonlinear nite element models. Medical Engineering & Physics 2001; 23 :165173.

6. Keyak JH, Rossi SA, Jones KA, Skinner HB. Prediction of femoral fracture load using automated nite elementmodeling. Journal of Biomechanics 1998; 31 :125133.

7. Keyak JH, Rossi SA, Jones KA, Les CM, Skinner HB. Prediction of fracture location in the proximal femur usingnite element models. Medical Engineering & Physics 2001; 23 :657664.

8. Keyak JH, Rossi SA. Prediction of femoral fracture load using nite element models: an examination of stress- andstrain-based failure theories. Journal of Biomechanics 2000; 33 (2):209214.

9. Lotz JC, Cheal EJ, Hayes WC. Fracture prediction for the proximal femur using nite element models: Part 1 linearanalysis. Journal of Biomechanical Engineering 1991; 113 :353360.

10. Lotz JC, Cheal EJ, Hayes WC. Fracture prediction for the proximal femur using nite element models: Part2nonlinear analysis. Journal of Biomechanical Engineering 1991; 113 :361365.

11. Voor MJ, Anderson RC, Hart RT. Stress analysis of halo pin insertion by non-linear nite element modeling. Journalof Biomechanics 1997; 30 :903909.

12. Niebur GL. A computational investigation of multiaxial failure in trabecular bone. PhD Thesis , MechanicalEngineering, Graduate Division, The University of California, Berkley, USA, 2000.

13. Doblare M, Garcia JM, Gomez MJ. Modelling bone tissue fracture and healing: a review. Engineering Fracture Mechanics 2004; 71 :18091840.

14. Keaveny TM, Wachtel EF, Ford CM, Hayes WC. Differences between the tensile and compressive strengths of bovine tibial trabecular bone depends on modulus. Journal of Biomechanics 1994; 27 (9):11371146.

15. Nalla RK, Kinney JH, Ritchie RO. Mechanistic fracture criteria for the failure of human cortical bone. Nature Materials 2003; 2 :164168.

16. Cowin SC, Mehrabadi MM. Identication of the elastic symmetry of bone and other materials. Journal of Biomechanics 1989; 22 (6/7):503515.

17. Donaldson F, Pankaj P, Cooper DML, Thomas CDL, Clement JG, Simpson AHRW. Relating age and micro-architecture with apparent-level elastic constants: a FE study of female cortical bone from the anterior femoralmidshaft. Proceedings of the Institution of Mechanical Engineers, H: The Journal of Engineering in Medicine 2011;225 :585596.

18. Bayraktar HH, Morgan EF, Niebur GL, Morris GE, Wong EK, Keaveny TM. Comparison of the elastic and yieldproperties of human femoral trabecular and cortical bone tissue. Journal of Biomechanics 2004; 37 (1):2735.

19. Mercer C, He MY, Wang R, Evans AG. Mechanisms governing the inelastic deformation of cortical bone andapplication to trabecular bone. Acta Biomaterialia 2006; 2 :5968.

20. Vahey JW, Lewis JL, Vanderby R. Elastic moduli, yield stress, and ultimate stress of cancellous bone in the canineproximal femur. Journal of Biomechanics 1987; 20 :2933.

21. Gupta A, Bayraktar HH, Fox JC, Keaveny TM, Papadopoulos P. Constitutive modeling and algorithmic implemen-tation of a plasticity-like model for trabecular bone structures. Computational Mechanics 2007; 40 :6172.

22. Schileo E, Taddei F, Cristofolini L, Viceconti M. Subject-specic nite element models implementing a maximumprincipal straincriterion areable to estimate failure risk and fracture location on humanfemurs tested in vitro . Journalof Biomechanics 2008; 41 :356367.

23. Naghdi PM, Trapp JA. The signicance of formulating plasticity theory with reference to loading surfaces instrainspace. International Journal of Engineering Science 1975; 13 :785797.

24. Farahat AM, Kawakami M, Ohtsu M. Strainspace plasticity model for the compressive hardeningsofteningbehaviour of concrete. Construction and Building Materials 1995; 9 :4559.

25. Lan YM, Sotelino ED, Chen WF. The strainspace consistent tangent operator and return mapping algorithmfor constitutive modelling of conned concrete. International Journal of Applied Science and Engineering 2003;1 :1729.

26. Mizuno E, Hatanaka S. Compressive softening model for concrete. Journal of Engineering Mechanics 1992;118 :15461563.


DOI: 10.1002/cnm


22/22


27. Yoder PJ, Iwan WD. On the formulation of strainspace plasticity with multiple loading surfaces. Journal of Applied Mechanics 1981; 48 :773778.

28. Youquan Y. Stress space and strain space formulation of the elasto-plastic constitutive relations for singular yieldsurface. Acta Mechanica Sinica 1986; 2 :169177.

29. Casey J, Naghdi PMa. Discussion: on the formulation of strainspace plasticity with multiple loading surfaces. ASME Journal of Applied Mechanics 1982; 49 :460462.

30. Casey J, Naghdi PM. On the nonequivalence of the stress space and strain space formulations of plasticity theory.

ASME Journal of Applied Mechanics 1983; 50

:350354.31. de Borst R. Integration of plasticity equations for singular yield functions. Computers and Structures 1987;26 :823829.

32. Knockaert R, Chastel Y, Massoni E. Rate-independent crystalline and polycrystalline plasticity, application to FCCmaterials. International Journal of P 2000; 16 :17198.

33. Pankaj P, Bicanic N. Detection of multiple active yield conditions for mohr coulomb elasto-plasticity. Computersand Structures 1997; 62 :5161.

34. Donaldson F, Pankaj P, Simpson A. Investigation of factors affecting loosening of Ilizarov ring-wire external xatorsystems at the bone-wire interface. Journal of Orthopaedic Research 2012; 30 :726732.

35. Oden ZM, Selvitelli DM, Bouxsein ML. Effect of local density changes on the failure load of the proximal femur. Journal of Orthopaedic Research 1999; 17 :661667.

36. Clausen J, Damkilde L, Andersen L. Efcient return algorithms for associated plasticity with multiple yield planes. International Journal for Numerical Methods in Engineering 2006; 66 :10361059.

37. Huang J, Grifths DV. Observations on return mapping algorithms for piecewise linear yield criteria. International Journal of Geomechanics, ASCE 2008; 8 :253265.

38. Iliushin AA. On the postulate of plasticity. Journal of Applied Mathematics and Mechanics 1961; 25

:746752.39. Hiriart-Urruty JB, Lemarechal C. Convex Analysis and Minimization Algorithms , Vol. 1. Springer-Verlag: Berlin,1993.

40. de Borst R, Pankaj P, Bicanic N. A note on singularity indicators for MohrCoulomb type yield criteria. Computersand Structures 1991; 39 :219220.

41. Nagtegaal JC. On the implementation of inelastic constitutive equations with special reference to large deformationproblems. Computer Methods in Applied Mechanics and Engineering 1982; 33 :469484.

42. SimoJC, Taylor RL. Consistent tangent operators for rate-independent elastoplasticity. Computer Methods in Applied Mechanics and Engineering 1985; 48 :101118.

43. Criseld MA. Nonlinear Finite Element Analysis of Solids and Structures: Advanced Topics , Vol. 2. Wiley:New York, 1997.

44. Petersen KB, Pedersen MS. The matrix cookbook, Oct 2008. http://www2.imm.dtu.dk/pubdb/p.php?3274,accessed15/11/2010, version 20081110.

45. Selby S. Standard Mathematical Tables . CRC Press: Cleveland, Ohio, 1975.46. Ebacher V, Tang C, McKay H, Oxland TR, Guy P, Wang R. Strain redistribution and cracking behaviour of human

bone during bending. Bone 2007; 40 :12651275.

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