Experimental study of micropolar and couple stress elasticity in compact bone in bending

Yang, Jianming; Lakes, Roderic S.

doi:10.1016/0021-9290(82)90040-9

Cited by 239 publications

(125 citation statements)

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“…Contradictory size effects have been reported previously in human cortical bone [Yang & Lakes, 1982;Choi et al, 1990] where samples were loaded in bending experiments in each case. While the earlier work reported a positive size effect the later reported an opposite, negative effect.…”

Section: Discussionsupporting

confidence: 56%

“…Behaviour like this has been observed in both polymeric foams [Lakes, 1983[Lakes, , 1986Anderson & Lakes, 1994] and materials comprised of a two dimensional homogeneous matrix perforated by a regular array of circular voids Waseem et al, 2013;McGregor & Wheel, 2014]. However, there are materials such as cortical bone, for which both increasing and decreasing stiffness with reducing size have been reported [Yang & Lakes, 1982;Choi et al, 1990]. This contradictory behaviour has been attributed to surface effects induced in sample preparation that increase compliance and result in a distorted size effect.…”

Section: Introductionmentioning

confidence: 79%

“…However, there are some materials that do exhibit size dependent behaviour when loaded. Such behaviour has been observed in fabricated materials like foams [Lakes, 1983[Lakes, , 1986Anderson & Lakes, 1994] as well as biological tissues such as bone [Yang & Lakes, 1982;Choi et al, 1990]. The size dependency apparently results from the size scale of their microstructure which may be sufficient to influence their macroscopic behaviour.…”

Section: Introductionmentioning

confidence: 82%

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Paradoxical Size Effects in Composite Laminates and Other Heterogeneous Materials

Wheel

Frame

Riches

2016

Advanced Structured Materials

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Size effects in which there is an apparent increase in stiffness with reducing size scale are forecast in those heterogeneous materials that have constitutive behaviour described by more generalized continuum theories such as couple stress, micropolar or micromorphic elasticity. This short paper considers possibly the simplest heterogeneous material exhibiting such size effects, a two phase composite laminate consisting of alternating layers of stiff and compliant material, and shows that when loaded in bending the nature of the size effect actually depends on the composition of the sample surfaces. The laminate material is apparently capable of exhibiting a diversity of size effects some of which are compatible with the predictions of generalized continuum theories while others are contradictory. Another heterogeneous material consisting of a periodic or regular array of voids within a classically elastic matrix is then considered. Detailed finite element analysis shows that the diversity of size effects encountered in the laminate material may also be observed in this more representative material thereby providing some insight into the contradictory size effects that have sometimes been reported elsewhere in the literature.

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Section: Discussionsupporting

confidence: 56%

Section: Introductionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 82%

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Paradoxical Size Effects in Composite Laminates and Other Heterogeneous Materials

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Riches

2016

Advanced Structured Materials

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“…No attempt was made to correlate the modulus and specimen size in their study. On the other hand, contradicting data have been reported by Yang and Lakes (1982). where modulus increased as specimens became smaller @=X2-1.4 mm).…”

Section: E =A-b(s)+c(s)"mentioning

confidence: 71%

The elastic moduli of human subchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus

Choi

Kuhn

Ciarelli

et al. 1990

Journal of Biomechanics

452

214

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Abstract-The elastic moduli of human subchondral. trabecular. and cortical bone tissue from a proximal tibia were experimentally determined using three-point bending tests on a microstructural level. The mean modulus of subchondral specimens was I.ISGPa, and those of trabecular and cortical specimens was 4.59 OPaland 5.44 CPa respectively. Significant differences were found in the modulus values between bone tissues, which may have mainly resulted from the difference in the microstructures ofeach bone tissue rather than in the mineral density. Furthermore, the size-dependency of the modulus was examined using eight different sjza ofcortical specimens(heights h= 100-1000 am). While the modulus values for relatively large specimens (h>5OO~m) remained fairly constant (approximately I5 GPa). the values decreased as the specimens became smaller. A significant correlation was round between the modulus and specimen size. The surface area to volume ratio proved to be a key variable to explain the size-dependency.

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“…Other accounts of micropolar elasticity and Cosserat modeling of cellular solids can be found in [25][26][27]. We remark that in [18][19][20][21][22][23][24][25][26][27], Lagrangian description is used for solid matter, however the mathematical descriptions are purely derived using strain energy density functional and principle of virtual work. This approach works well for elastic solids in which mechanical deformation is reversible.…”

mentioning

confidence: 99%

A Non-Classical Internal Polar Continuum Theory for Finite Deformation of Solids Using First Piola-Kirchhoff Stress Tensor

2016

Jour. Pure Appl. Math. Adv. Appl.

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This paper presents a non-classical internal polar continuum theory for finite deformation of isotropic, homogeneous compressible and incompressible solid continua. The classical continuum theories only incorporate partial physics of deformation in the thermodynamic framework. Since the Jacobian of deformation J J J J J J J J J is fundamental measure of deformation in solid continua, J J J J J J J J J in its entirety must be incorporated in the thermodynamic framework. Polar decomposition of J J J J J J J J J into right stretch tensor S S S r r r S S S r r r S S S r r r and pure rotation tensor R R R R R R R R R shows that entirety of J J J J J J J J J implies entirety of S S S r r r S S S r r r S S S r r r and R R R R R R R R R. The classical continuum theories for isotropic and homogeneous solid continua are derived purely using S S S r r r S S S r r r S S S r r r , thus ignoring the influence of R R R R R R R R R altogether. The purpose of this research is to present a new and more complete thermodynamic framework for 2010 Mathematics Subject Classification: 74A35, 74A20.Keywords and phrases: non-classical continuum theory, internal polar continuum theory, solid continua, Jacobian of deformation, polar decomposition, stretch tensor, rotation tensor, finite deformation. This can be accomplished by incorporating the additional physics due to R R R R R R R R R in the current theories as these theories already contain the physics due to S S S r r r S S S r r r S S S r r r . We note that the rotation tensor R R R R R R R R R results due to deformation of solid continua, hence arises in all deforming solid continua.Thus, this theory can be referred to as internal polar non-classical theory for solid continua. The use of internal polar non-classical is appropriate as the theory considers internal rotations .If the varying internal rotations and the rotation rates are resisted by the solid continua, then there must exist internal moments that are conjugate to the rotations which together with rotations and rotation rates can result in additional energy storage, dissipation, and memory.Derivations of conservation and balance laws are presented for internal polar non-classical continuum theory for solid continua for finite deformation. Necessity of additional conservation and balance laws is discussed and their derivations are presented. The resulting mathematical model is compared with the mathematical models resulting from the current continuum theories for finite deformation to illustrate the differences in them due to incorporating the additional physics associated with R R R R R R R R R and thereby incorporating J J J J J J J J J in its entirety. The non-classical continuum theory for solid continua presented here is not to be confused with the micropolar theories, stress-couple theories, or strain gradient theories as demonstrated in this paper. The objective of the theory presented here is to present new thermodynamic framework for solid continua with finite deformation that is consistent with the deform...

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Experimental study of micropolar and couple stress elasticity in compact bone in bending

Cited by 239 publications

References 19 publications

Paradoxical Size Effects in Composite Laminates and Other Heterogeneous Materials

Paradoxical Size Effects in Composite Laminates and Other Heterogeneous Materials

The elastic moduli of human subchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus

A Non-Classical Internal Polar Continuum Theory for Finite Deformation of Solids Using First Piola-Kirchhoff Stress Tensor

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