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Research Papers

Two-Way Coupled Multiscale Model for Predicting Mechanical Behavior of Bone Subjected to Viscoelastic Deformation and Fracture Damage OPEN ACCESS

[+] Author and Article Information
Taesun You

Department of Civil Engineering,
362H WHIT,
2200 Vine Street,
University of Nebraska,
Lincoln, NE 68583
e-mail: tae-sun.you@unl.edu

Yong-Rak Kim

Professor
Department of Civil Engineering,
362N WHIT,
2200 Vine Street,
University of Nebraska,
Lincoln, NE 68583
e-mail: ykim3@unl.edu

Taehyo Park

Professor
Department of Civil and
Environmental Engineering,
Hanyang University,
Seoul 133-791, South Korea
e-mail: cepark@hanyang.ac.kr

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received June 2, 2016; final manuscript received December 5, 2016; published online February 9, 2017. Assoc. Editor: Xi Chen.

J. Eng. Mater. Technol 139(2), 021016 (Feb 09, 2017) (8 pages) Paper No: MATS-16-1164; doi: 10.1115/1.4035618 History: Received June 02, 2016; Revised December 05, 2016

This paper presents a two-way linked computational multiscale model and its application to predict the mechanical behavior of bone subjected to viscoelastic deformation and fracture damage. The model is based on continuum thermos-mechanics and is implemented through the finite element method (FEM). Two physical length scales (the global scale of bone and local scale of compact bone) were two-way coupled in the framework by linking a homogenized global object to heterogeneous local-scale representative volume elements (RVEs). Multiscaling accounts for microstructure heterogeneity, viscoelastic deformation, and rate-dependent fracture damage at the local scale in order to predict the overall behavior of bone by using a viscoelastic cohesive zone model incorporated with a rate-dependent damage evolution law. In particular, age-related changes in material properties and geometries in bone were considered to investigate the effect of aging, loading rate, and damage evolution characteristics on the mechanical behavior of bone. The model successfully demonstrated its capability to predict the viscoelastic response and fracture damage due to different levels of aging, loading conditions (such as rates), and microscale damage evolution characteristics with only material properties of each constituent in the RVEs.

FIGURES IN THIS ARTICLE
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Bone is a mineralized biological material that has a hierarchical structure at multiple length scales [1]. Bone at a macroscale is composed of bone tissues. At a mesoscale, the bone tissue has three main components: yellow marrow and two types of bones—compact (cortical) bone and spongy (cancellous) bone. The compact bone at a microscale consists of osteons and interstitial lamellae that occupy the space between osteons.

Failure and fracture in bone can lead to further disability and complications, and the loss of bone mass and quality with aging increases the risk of bone failure and fracture [2]. Consequently, it is necessary to diagnose accurately the bone quality and its mechanical properties, including stiffness, strength, and fracture toughness, in order to prevent any failure and fracture in bone with aging. Although measuring bone mass or bone mineral density is one of the traditional methods widely used to assess bone fracture risk, it was found that they are not good indicators of bone fracture and strength [3,4]. Fracture mechanics have been applied widely for experimentally evaluating bone fractures [58]. Although the experimental studies provided fundamental insight into bone fracture behavior, it is rarely possible to investigate the effect of individual factors on bone fractures due to the variability in donor bones and testing.

In order to overcome those drawbacks, the use of computational approaches, including the FEM, has been attractive due to their ability to evaluate the effect of a single factor on bone fracture response. In many FEM studies, various length scales, ranging from micro- to macro-scale, were considered. At a macroscale level, bone failure load [9], bone strength [10], and bone stiffness [11] were estimated by the FEM analysis. Microscale FEM models of bone mostly focused on the effect of the microstructures of osteons and interstitial lamellae and their material properties on the elasticity [12,13] and fracture behavior [1417] of compact bone. Although these microscale computational studies could account for the geometric heterogeneity of bone and its fracture behavior, such single-scale modeling requires a high amount of computational costs to model every microstructural detail. This limitation has led researchers to seek efficient alternative approaches that can consider the hierarchical structure with heterogeneous materials.

Multiscale modeling is one such approach and is used widely for various composite materials [1829]. In this approach, a macroscopic body is divided with multiple length scales, for which a separate scale analysis is performed. Multiscale modeling is based on a homogenization principle that assumes solutions at any smaller length scales are statistically homogenized to produce field equations for the next length scale. In order to model the damage explicitly, fracture and/or damage mechanics can be incorporated with the analysis at each length scale. Thus, multiscale modeling is more accurately able to predict the behavior of the hierarchical structure with less computational efforts because it requires only constituents' fundamental properties.

To date, only limited studies have applied the multiscale modeling approach to bone modeling. Ghanbari and Naghdabadi [30] used a hierarchical multiscale scheme that accounted for macroscale and microscale in compact bone. The mechanical properties of compact bone, including elastic moduli and Poisson's ratio in two major directions, and shear modulus were obtained by this method. Hamed and Jasiuk [31] presented a bottom–up multiscale model that took into account various deformation and failure mechanisms occurring at different length scales. The strength of bone obtained at a lower level was used for computations at a higher level. Consequently, bone strength and elastic moduli at three length scales were predicted by this method with cohesive finite elements. However, these studies did not consider the viscoelastic response of bone, whereas numerous studies [3241] have reported the rate-dependent nature of bone behavior. Materials such as bone tissues in which significant inelastic deformation and various damage, including cracks in different length or time scales, that progress with time, require concurrent scale-linking techniques, consideration of microscale damage evolution, and multiple energy dissipation mechanisms including material viscoelasticity and fracture process. This would lead to a more accurate modeling and simulation.

The primary objective of this study is to present a multiscale computational model for predicting the mechanical behavior of viscoelastic bone subjected to fracture damage. Based on the continuum thermomechanics, the multiscale model is implemented through finite-element formulation. Two length scales (global scale of bone and local scale of compact bone) are two-way coupled in the quasi-static model framework by linking a homogenized object at a global scale to heterogeneous RVEs at a local scale. The global object has three phases—compact bone, spongy bone, and yellow marrow—while the local-scale RVEs are composed of elastic osteons and viscoelastic interstitial lamellae with fracture. Age-related quality and geometric changes in bone are considered when generating the global objects and local-scale RVEs. Then, with this model framework, the effect of aging, loading rate, and damage evolution on fracture response of bone is investigated by using a viscoelastic cohesive zone model [4244] and a rate-dependent damage law proposed by Allen and Searcy [42].

Multiscale modeling has been an attractive tool for problems with evolving an object that is statistically homogeneous at the global scale but microscopically heterogeneous at the local scale. Figure 1 presents a human bone where each component is considered statistically homogeneous at the global scale (bone level) but highly heterogeneous at the local scale (compact bone level). By multiscale modeling, the structural constitutive relations in human bone (global scale) are determined based on the behavior of the individual constituents and their interactions in the compact bone (local scale). It also can account for the effect of microstructures, crack growth, and inelasticity and nonlinearity of materials at the local scale.

Global-Scale Modeling.

A global-scale initial boundary value problem (IBVP) can be posed by an appropriate set of initial-boundary conditions and a set of governing equations, such as the conservation of linear momentum (Eq. (1)), the conservation of angular momentum (Eq. (2)), the small strain–displacement relation (Eq. (3)), and the constitutive equation (Eq. (4)) Display Formula

(1)σji,jG=0inVG
Display Formula
(2)σjiG=σijGinVG
Display Formula
(3)εijG=12(ui,jG+uj,iG)inVG
Display Formula
(4)σijG(t)=Ω¯τ=τ=t{εklG(τ)}inVG

where σijG is the global-scale stress tensor; εijG is the global-scale strain tensor; uiG is the global-scale displacement vector; and VG is the volume of the global-scale body. Ω¯τ=τ=t is the global-scale functional that describes the constitutive behavior at each position in the global-scale object, which is determined during the multiscale analysis by homogenization principles. This functional is obtained by averaging the response of the local-scale RVE in which cracks (or damage) and/or time-dependent effects, such as viscoelasticity, can be considered. It is assumed that there are no body forces, inertial effects, and large deformations.

Local-Scale Modeling.

Considering that the local-scale RVE is large enough that continuum mechanics theories are still valid at the local-scale, a similar local-scale IBVP to the global one exists with the following assumptions:

  • (a)The global-length scale is much larger than the local-length scale.
  • (b)The local-length scale is much larger than the crack-associated length scale at the local-scale.

Now, a well-posed local IBVP can be written by Display Formula

(5)σji,jL=0inVL
Display Formula
(6)σjiL=σijLinVL
Display Formula
(7)εijL=12(ui,jL+uj,iL)inVL
Display Formula
(8)σijL(t)=Ωτ=τ=t{εklL(τ)}inVL

where σijL is the local-scale stress tensor; εijL is the local-scale strain tensor; uiL is the local-scale displacement vector; VL is the volume of local-scale body; and Ωτ=τ=t is the local-scale functional mapping that describes the constitutive behavior in the local-scale object. It is noted that cracks in the local object can only grow (i.e., crack healing is not considered), and crack growth can only occur if the fracture energy release rate (GiL) at a particular position in the local scale overcomes the critical energy release rate of the material (GiCL) as follows: Display Formula

(9)GiLGiCLt(VIL)>0inVL

where VIL is internal boundary, such as cracks in the local-scale object.

In Eq. (8), the local-scale functional (Ωτ=τ=t) can account for a variety of constitutive responses, and the functional representing linear elastic (Eq. (10)) or linear viscoelastic (Eq. (11)) behavior is considered for the constituent in the local-scale RVE in this study Display Formula

(10)σijL=CijklLεklLinVL
Display Formula
(11)σijL(t)=tCijklL(tτ)εklLτdτinVL

where CijklL is the elastic tensor; t is time of interest; and τ is the integration variable. CijklL(t) is the viscoelastic stress relaxation modulus tensor, which is time-dependent and determined by a laboratory relaxation test whose results can be represented by a Prony series on the generalized Maxwell model as follows: Display Formula

(12)CijklL(t)=Cijkl,L+p=1qCijkl,pLexp(tρpL)

where Cijkl,L and Cijkl,pL are spring constants; ρpL is the relaxation time; and q is the number of Prony series terms in the generalized Maxwell model.

Crack propagation in Eq. (9) at the local-scale was modeled by a cohesive zone, as illustrated in Fig. 1. Cohesive zone models are well-established tools that can remove stress singularities ahead of crack tips and consider fracture as a gradual phenomenon in which separation takes place across an extended crack tip or cohesive zone (fracture process zone). In order to simulate damage growth due to cracks in the viscoelastic media such as bones, a nonlinear viscoelastic cohesive zone formulated by Allen and Searcy [42], Yoon and Allen [43], and Allen and Searcy [44] was used in this study. This model is appropriate for predicting the damage evolution and fracture failure of highly inelastic and heterogeneous particulate composite materials, including bones, by accounting for nonlinear viscoelastic damage growth [43]. The traction–displacement relationship for the nonlinear viscoelastic cohesive zone can be written as follows [42]: Display Formula

(13)TiL(t)=uiL(t)λL(t)δiL[1αL(t)][σiL+t0tECZL(tτ)λL(τ)τdτ]onVCZL

where TiL(t) is the local-scale cohesive zone traction; uiL(t) is the local-scale cohesive zone opening displacement; δiL is the cohesive zone length parameter; αL(t) is the internal damage parameter; σiL is the required stress level to initiate cohesive zone damage; VCZL is the internal boundary occupied by the cohesive zone; and subscript i is n (normal) or r (tangential) for two-dimensional (2D) objects. λL(t) is the Euclidean norm of the cohesive zone displacement for 2D objects, as follows: Display Formula

(14)λL(t)=([unL(t)]δnL)2+([urL(t)]δrL)2

ECZL(t) is the uniaxial relaxation modulus of a single fibril in the cohesive zone and can be presented in terms of the Prony series similar to Eq. (12) for the bulk viscoelastic material as follows: Display Formula

(15)ECZL(t)=ECZ,L+p=1qECZ,pLexp(tρCZ,pL)

where ECZ,L and ECZ,pL are spring constants and ρCZ,pL is the relaxation time.

The internal damage parameter αL(t) reflects the area of the fraction of voids with respect to the cross-sectional area of the idealized cohesive zone and can be determined by performing fracture tests [47]. Alternatively, a phenomenological form of the damage evolution also can be employed to represent rate-dependent fracture. In this study, the following phenomenological form (Eq. (16)) proposed by Allen and Searcy [42] has been selected since it is sufficient to demonstrate rate-dependent damage growth of bones: Display Formula

(16){α˙L(t)=A[λL(t)]mwhenλ˙L(t)>0andαL(t)<1α˙L(t)=0whenλ˙L(t)0orαL(t)=1

where A and m are the microscale phenomenological material constants that govern damage evolution behavior.

Homogenization for Linking Two Scales.

The global and local length scales can be connected by the homogenization principles [4851]. Structural degradation due to accumulated damage (microcracks) in the local-scale RVE affects the constitutive behavior of the global-scale object. The results from the local-scale analysis are homogenized and linked to the global-scale problem. The concept of homogenization is applicable when the heterogeneous medium satisfies statistical homogeneity. Homogenization is central to the idea of multiscale modeling and can be conducted through the average process of local-scale fields within the heterogeneous medium as follows: Display Formula

(17)fG(xiG,t)=f¯L1VLVLfL(xiL,t)dV

where fG and fL are the functions at the global and local scales, respectively. f¯ is the volume average of a generic function f, and xiG and xiL are the spatial coordinates at the global and local scales, respectively.

The use of the divergence theorem enables to transform the volumetric integral in Eq. (17) to a surface integral equation, and hence, in the case of the homogeneous boundary conditions at the local-scale, the homogenized stresses at the global-scale in terms of the local-scale stresses can be expressed as follows: Display Formula

(18)σijG=σ¯ijL1VLVELσkiLnkLxjLdS

where VEL is the external boundary of the local-scale RVE, and nkL is the unit outer normal vector to the volume of local-scale RVE. Similarly, the homogenized strains at the global-scale with regard to local-scale strains can be written as follows: Display Formula

(19)εijG=ε¯ijL=EijG+eijG
Display Formula
(20)EijG=E¯ijL1VLVEL12(uiLnjL+ujLniL)dS
Display Formula
(21)eijG=e¯ijL1VLVIL12(uiLnjL+ujLniL)dS

where E¯ijL is the external boundary average strain tensor at the local-scale; e¯ijL is the internal boundary average strain tensor that represents an averaged measure of damage due to cracks in the local-scale RVE; and VEL is the internal boundary of the local-scale RVE.

The use of Eqs. (18)(21) is called a mean field theory of homogenization because only the mean stress and strain tensors evaluated at the boundary of the local-scale RVE are used when the global-scale behavior is determined. In the case of quasi-static problems such as the one discussed in this paper, the computation of the homogenized constitutive tensor is obtained in order to calculate the global-scale tangent stiffness matrix, Ω¯τ=τ=t in Eq. (4).

Two-Way Coupled Multiscale Algorithm.

The multiscale model here is considered a two-way linked model because the displacements computed from the global-scale strain tensors apply on the boundary of the local-scale (global-to-local linking), and the solution of the local-scale IBVP is homogenized and used to calculate the global-scale tangent tensor (local-to-global linking). Figure 2 presents the two-way linked multiscale algorithm in a flowchart form, where there are seven steps: (1) read the global and local inputs; (2) calculate the initial homogenized global tangent tensor from the local-scale inputs; (3) solve the global-scale problem at a given time step; (4) apply the global-scale solution to the local-scale boundary value problem; (5) solve the local-scale problem at a given time step; (6) homogenize the local-scale solution; and (7) update the homogenized local-scale solution to the global-scale object at each integration point for the next time step.

As shown in Fig. 3, a global-scale object is represented with three primary phases: compact bone, spongy bone, and yellow marrow. The left and right sides of the global-scale object were fixed, and different loading rates were applied to the center at the bottom of the global-scale object. Compact bone is a homogeneous and viscoelastic material in the global point of view, but it is modeled through two-way coupled multiscaling by linking global-scale elements to respective heterogeneous local-scale RVEs. Thus, the nonlinear and inelastic behavior of the compact bone due to viscoelasticity, microstructural characteristics, and the directional-progressive evolution of cohesive zones and microcracks is addressed automatically by the scale linking. Only 40 global-scale elements in a black color in the same figure were selected for this partial multiscale analysis, although all elements in the three phases can be modeled through the multiscale method with their corresponding RVEs. The rest of the compact bone phase in a white color in Fig. 3 was modeled as a homogenous viscoelastic material [52]. The yellow marrow was considered as an isotropic elastic material [53], while the spongy bone was regarded as an orthotropic elastic material [54]. Table 1 summarizes the material properties of each phase in the global-scale object.

In general, bone becomes slender and less dense with aging. Parfitt et al. [55] and Russo et al. [56] found that there was about a 17% reduction of compact bone area and a 7% thickness decrease in spongy bone for aged bone (80 years) when compared to young bone (30 years). Based on that, two global-scale objects were generated, as shown in Fig. 4. The average thickness of the spongy phase in the model for a young group was 950 μm, while that for an aged group was 886 μm. The areas of the compact bone phase for the young and aged groups were 221 mm2 and 184 mm2, respectively [45,55,56].

The local-scale RVE for the compact bone was composed of two phases—osteon and interstitial lamellae—and different RVEs for the young and aged groups were modeled, as shown in Fig. 5. Based on the previous experimental measurements [1,57], the local-scale RVE for the aged group contained a greater number of osteons with smaller diameters, compared to those of the young group. The osteon at the local-scale was considered an orthotropic elastic material, whereas the interstitial lamellae phase was viscoelastic with fracture [31,58]. Adaptive insertion of the viscoelastic cohesive zone in Eq. (13) is allowed in the interstitial lamellae to simulate rate-dependent damage in the form of discrete cracks at the local scale. Table 2 summarizes the material properties of the osteon and interstitial lamellae for the young and aged groups [17,52,57,59], respectively. It is important to note that the interstitial lamellae are stiffer but more brittle for the aged group.

As shown in Fig. 6(a), three local-scale RVEs (i.e., RVE-1, RVE-2, and RVE-3) for the young group and one local-scale RVE (RVE-1) for the aged group were selected strategically for demonstration purposes. Local-scale RVE-1 and RVE-2 represent the bone behavior at the top of the compact bone under the loading, while RVE-3 was selected to investigate the bone behavior on the surface of the compact bone beneath the loading.

As can be seen clearly in Fig. 7, bone presented higher stresses as the applied loading increased. Considerable compressive stress occurred at the outside of the compact bone due to the applied loading, while significant tensile stress was observed at the inside of the compact bone, which caused potential regions for crack propagation in the opening mode. Moreover, Fig. 8 shows the deformation contours of each RVE. A number of cohesive zone elements (represented by solid lines) were embedded, and some macrocracks were developed in RVE-1, due to the highest tensile stress. Some cohesive zone elements were inserted in RVE-2, while RVE-3 did not show any significant damage. It is noted that because the adaptive cohesive zone element insertion technique was used in this study, the development of the cohesive elements implies high potential for microcracking followed by crack coalescence (i.e., damage localization).

Effect of Aging.

As shown in Figs. 4 and 5, two global objects and two local-scale RVEs were modeled to investigate the effect of aging on the fracture behavior of bone. Different material properties for the young and aged groups were also used for the simulations (see Tables 1 and 2). Figure 9 shows stress–strain plots of RVE-1 for the young and aged groups when various loading rates were applied. Figure 9 clearly shows that the strength of the aged bone was greater than that of young bone, while aged bone failed earlier. These findings agree with the previous experimental tests [8,60]. Additionally, the figure presents rate-dependence of bone, which was found in other studies [36,6163]. It is shown for both the young and aged groups that strength and strain at failure of bone were getting greater, as the loading rates increased.

Effect of Microscale Damage Evolution Characteristics.

In this study, the viscoelastic cohesive zone model incorporated with rate-dependent damage evolution was used to account for nonlinear viscoelastic damage growth. As noted in Eq. (13), the cohesive zone traction induced by nodal displacements at the interface is affected by the damage evolution law in Eq. (16). The rate of viscoelastic damage evolution (dα/dt) due to crack growth is governed by the damage parameters, A and m. Consequently, it appears that the two damage parameters significantly influence the overall behavior of bone fracture and damage. Several different values (1.5, 3.0, and 5.0 for value A, and 0.1, 0.3, and 0.5 for value m) were applied in order to investigate the sensitivity of the damage parameters for microscale crack growth and eventual macroscale failure of the bone.

The stress–strain plots of RVE-1 for the young group are presented in Fig. 10, indicating that the values chosen can have a significant impact on the bone behavior. The microscale damage evolution rate (dα/dt) is directly proportional to higher values of the microscale damage evolution coefficient, A. On the other hand, the microscale damage evolution rate increases as the microscale damage evolution exponent, m decreases, since λL in Eq. (16) is typically less than 1.0 before complete failure of the cohesive zone. More rapid growth of damage contributes to a reduction of resultant traction in the cohesive zones and a corresponding decrease in average stress of the overall composite.

This paper described the multiscale modeling framework and its application to predict the mechanical behavior of viscoelastic bone subjected to fracture damage. Two physical length scales (global-scale of bone and local-scale of compact bone) were two-way coupled in the framework by linking a homogenized object at a global-scale to heterogeneous local-scale RVEs. Heterogeneity, viscoelastic deformation, and rate-dependent fracture damage at the local-scale (i.e., compact bone) were considered in predicting the overall behavior of bone based on the two-way linked multiscale framework. The global object with three phases, including compact bone, spongy bone, and yellow marrow, and local-scale RVEs were modeled in consideration of age-related changes in material properties and geometries in bone. Then, the effect of aging, loading rates, and damage evolution characteristics on the fracture behavior of bone were investigated by using the viscoelastic cohesive zone model, which is incorporated with the rate-dependent damage evolution function.

The model successfully demonstrated its capability to distinguish fracture damage of young and aged groups, estimate the rate-dependent response of bone, and investigate the effect of damage evolution on the overall response of bone. It also presented clear potential and significant benefits compared to the previous approaches, such as single-scale modeling and one-way multiscale modeling (e.g., hierarchical bottom–up modeling). This is because only material properties and microstructure characteristics in the small-scale RVE are necessary to model the damage-associated behavior of larger-scale objects. Additionally, high computational efficiency can be achieved by using the homogenization process and parallel computing capabilities. It is expected that a successfully developed multiscale computational model such as the model presented here can be a purely mechanistic assessment tool for various biological materials such as bone.

The authors are grateful for the financial support received from the National Science Foundation (Grant No. CMMI-0644618).

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Oden, J. T. , Vemaganti, K. , and Moës, N. , 1999, “ Hierarchical Modeling of Heterogeneous Solids,” Comput. Methods Appl. Mech. Eng., 172(1), pp. 3–25. [CrossRef]
Oden, J. T. , and Zohdi, T. I. , 1997, “ Analysis and Adaptive Modeling of Highly Heterogeneous Elastic Structures,” Comput. Methods Appl. Mech. Eng., 148(3), pp. 367–391. [CrossRef]
Raghavan, P. , Moorthy, S. , Ghosh, S. , and Pagano, N. , 2001, “ Revisiting the Composite Laminate Problem With an Adaptive Multi-Level Computational Model,” Compos. Sci. Technol., 61(8), pp. 1017–1040. [CrossRef]
Ghanbari, J. , and Naghdabadi, R. , 2009, “ Nonlinear Hierarchical Multiscale Modeling of Cortical Bone Considering Its Nanoscale Microstructure,” J. Biomech., 42(10), pp. 1560–1565. [CrossRef] [PubMed]
Hamed, E. , and Jasiuk, I. , 2013, “ Multiscale Damage and Strength of Lamellar Bone Modeled by Cohesive Finite Elements,” J. Mech. Behav. Biomed. Mater., 28, pp. 94–110. [CrossRef] [PubMed]
Adharapurapu, R. R. , Jiang, F. , and Vecchio, K. S. , 2006, “ Dynamic Fracture of Bovine Bone,” Mater. Sci. Eng. C, 26(8), pp. 1325–1332. [CrossRef]
Crowninshield, R. , and Pope, M. , 1974, “ The Response of Compact Bone in Tension at Various Strain Rates,” Ann. Biomed. Eng., 2(2), pp. 217–225. [CrossRef]
Katsamanis, F. , and Raftopoulos, D. D. , 1990, “ Determination of Mechanical Properties of Human Femoral Cortical Bone by the Hopkinson Bar Stress Technique,” J. Biomech., 23(11), pp. 1173–1184. [CrossRef] [PubMed]
Lewis, J. , and Goldsmith, W. , 1975, “ The Dynamic Fracture and Prefracture Response of Compact Bone by Split Hopkinson Bar Methods,” J. Biomech., 8(1), pp. 27–40. [CrossRef] [PubMed]
McElhaney, J. H. , 1966, “ Dynamic Response of Bone and Muscle Tissue,” J. Appl. Physiol., 21(4), pp. 1231–1236. [PubMed]
Melnis, A. , and Knets, I. , 1982, “ Effect of the Rate of Deformation on the Mechanical Properties of Compact Bone Tissue,” Mech. Compos. Mater., 18(3), pp. 358–363. [CrossRef]
Roberts, V. , and Melvin, J. , “ The Measurement of the Dynamic Mechanical Properties of Human Skull Bone,” Applied Polymer Symposia, pp. 235–247.
Tennyson, R. , Ewert, R. , and Niranjan, V. , 1972, “ Dynamic Viscoelastic Response of Bone,” Exp. Mech., 12(11), pp. 502–507. [CrossRef]
Wood, J. L. , 1971, “ Dynamic Response of Human Cranial Bone,” J. Biomech., 4(1), pp. 1–12. [CrossRef] [PubMed]
Wright, T. , and Hayes, W. , 1976, “ Tensile Testing of Bone Over a Wide Range of Strain Rates: Effects of Strain Rate, Microstructure and Density,” Med. Biol. Eng., 14(6), pp. 671–680. [CrossRef] [PubMed]
Allen, D. H. , and Searcy, C. R. , 2001, “ A Micromechanical Model for a Viscoelastic Cohesive Zone,” Int. J. Fract., 107(2), pp. 159–176. [CrossRef]
Yoon, C. , and Allen, D. H. , 1999, “ Damage Dependent Constitutive Behavior and Energy Release Rate for a Cohesive Zone in a Thermoviscoelastic Solid,” Int. J. Fract., 96(1), pp. 55–74. [CrossRef]
Allen, D. H. , and Searcy, C. R. , 2000, “ Numerical Aspects of a Micromechanical Model of a Cohesive Zone,” J. Reinforced Plast. Compos., 19(3), pp. 240–248. [CrossRef]
Savalli, U. , 2013, “ Cross Section of a Bone,” Arizona State University, Phoenix, AZ, accessed Mar. 3, 2016, http://www.Savalli.us/BIO201/Labs/05-Bones/BoneHistologyLabel.Html
Pearson Education, 2004, “ Microscopic Structure of Bone,” Pearson PLC, London, accessed Mar. 3, 2016, http://www.Worcesterthinktank.com/Sites/wtk.Drupalgardens.com/Files/Uploads/APcompANDspongyBone.pdf
Williams, J. J. , 2002, “ Two Experiments for Measuring Specific Viscoelastic Cohesive Zone Parameters,” Master's thesis, Texas A & M University, College Station, TX.
Allen, D. H. , 2001, “ Homogenization Principles and Their Application to Continuum Damage Mechanics,” Compos. Sci. Technol., 61(15), pp. 2223–2230. [CrossRef]
Nemat-Nasser, S. , and Hori, M. , 2013, Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier, Amsterdam, The Netherlands.
Allen, D. , and Yoon, C. , 1998, “ Homogenization Techniques for Thermoviscoelastic Solids Containing Cracks,” Int. J. Solids Struct., 35(31), pp. 4035–4053. [CrossRef]
Roters, F. , Eisenlohr, P. , Hantcherli, L. , Tjahjanto, D. D. , Bieler, T. R. , and Raabe, D. , 2010, “ Overview of Constitutive Laws, Kinematics, Homogenization and Multiscale Methods in Crystal Plasticity Finite-Element Modeling: Theory, Experiments, Applications,” Acta Mater., 58(4), pp. 1152–1211. [CrossRef]
Lakes, R. S. , Katz, J. L. , and Sternstein, S. S. , 1979, “ Viscoelastic Properties of Wet Cortical Bone—I. Torsional and Biaxial Studies,” J. Biomech., 12(9), pp. 657–678. [CrossRef] [PubMed]
Jansen, L. E. , Birch, N. P. , Schiffman, J. D. , Crosby, A. J. , and Peyton, S. R. , 2015, “ Mechanics of Intact Bone Marrow,” J. Mech. Behav. Biomed. Mater., 50, pp. 299–307. [CrossRef] [PubMed]
Van Rietbergen, B. , Majumdar, S. , Pistoia, W. , Newitt, D. , Kothari, M. , Laib, A. , and Ruegsegger, P. , 1998, “ Assessment of Cancellous Bone Mechanical Properties From Micro-FE Models Based on Micro-CT, pQCT and MR Images,” Technol. Health Care, 6(5–6), pp. 413–420. [PubMed]
Parfitt, A. , Mathews, C. , Villanueva, A. , Kleerekoper, M. , Frame, B. , and Rao, D. , 1983, “ Relationships Between Surface, Volume, and Thickness of Iliac Trabecular Bone in Aging and in Osteoporosis. Implications for the Microanatomic and Cellular Mechanisms of Bone Loss,” J. Clin. Invest., 72(4), pp. 1396–1409. [CrossRef] [PubMed]
Russo, C. , Lauretani, F. , Bandinelli, S. , Bartali, B. , Di Iorio, A. , Volpato, S. , Guralnik, J. , Harris, T. , and Ferrucci, L. , 2003, “ Aging Bone in Men and Women: Beyond Changes in Bone Mineral Density,” Osteoporosis Int., 14(7), pp. 531–538. [CrossRef]
Bernhard, A. , Milovanovic, P. , Zimmermann, E. , Hahn, M. , Djonic, D. , Krause, M. , Breer, S. , Püschel, K. , Djuric, M. , and Amling, M. , 2013, “ Micro-Morphological Properties of Osteons Reveal Changes in Cortical Bone Stability During Aging, Osteoporosis, and Bisphosphonate Treatment in Women,” Osteoporosis Int., 24(10), pp. 2671–2680. [CrossRef]
Giner, E. , Arango, C. , Vercher, A. , and Fuenmayor, F. J. , 2014, “ Numerical Modelling of the Mechanical Behaviour of an Osteon With Microcracks,” J. Mech. Behav. Biomed. Mater., 37, pp. 109–124. [CrossRef] [PubMed]
Wang, X. , Osborn, R. , Wolf, J. , and Puram, S. , “ Age-Related Changes in the Mechanical Properties of Interstitial Bone Tissues,” 50th Annual Meeting of the Orthopaedic Research Society, p. 0037.
Green, J. O. , Wang, J. , Diab, T. , Vidakovic, B. , and Guldberg, R. E. , 2011, “ Age-Related Differences in the Morphology of Microdamage Propagation in Trabecular Bone,” J. Biomech., 44(15), pp. 2659–2666. [CrossRef] [PubMed]
Launey, M. E. , Buehler, M. J. , and Ritchie, R. O. , 2010, “ On the Mechanistic Origins of Toughness in Bone,” Annu. Rev. Mater. Res., 40, pp. 25–53. [CrossRef]
Sanborn, B. , Gunnarsson, C. , Foster, M. , and Weerasooriya, T. , 2014, “ Quantitative Visualization of Human Cortical Bone Mechanical Response: Studies on the Anisotropic Compressive Response and Fracture Behavior as a Function of Loading Rate,” Exp. Mech., 56(1), pp. 81–95. [CrossRef]
Weerasooriya, T. , Sanborn, B. , Gunnarsson, C. A. , and Foster, M. , 2016, “ Orientation Dependent Compressive Response of Human Femoral Cortical Bone as a Function of Strain Rate,” J. Dyn. Behav. Mater., 2(1), pp. 74–90. [CrossRef]
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References

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Fish, J. , and Belsky, V. , 1995, “ Multi-Grid Method for Periodic Heterogeneous Media Part 2: Multiscale Modeling and Quality Control in Multidimensional Case,” Comput. Methods Appl. Mech. Eng., 126(1), pp. 17–38. [CrossRef]
Fish, J. , and Shek, K. , 2000, “ Multiscale Analysis of Composite Materials and Structures,” Compos. Sci. Technol., 60(12), pp. 2547–2556. [CrossRef]
Fish, J. , and Wagiman, A. , 1993, “ Multiscale Finite Element Method for a Locally Nonperiodic Heterogeneous Medium,” Comput. Mech., 12(3), pp. 164–180. [CrossRef]
Ghosh, S. , Lee, K. , and Raghavan, P. , 2001, “ A Multi-Level Computational Model for Multi-Scale Damage Analysis in Composite and Porous Materials,” Int. J. Solids Struct., 38(14), pp. 2335–2385. [CrossRef]
Haj-Ali, R. M. , and Muliana, A. H. , 2004, “ A Multi-Scale Constitutive Formulation for the Nonlinear Viscoelastic Analysis of Laminated Composite Materials and Structures,” Int. J. Solids Struct., 41(13), pp. 3461–3490. [CrossRef]
Kim, Y.-R. , Souza, F. V. , and Teixeira, J. E. S. L. , 2013, “ A Two-Way Coupled Multiscale Model for Predicting Damage-Associated Performance of Asphaltic Roadways,” Comput. Mech., 51(2), pp. 187–201. [CrossRef]
Lee, K. , Moorthy, S. , and Ghosh, S. , 1999, “ Multiple Scale Computational Model for Damage in Composite Materials,” Comput. Methods Appl. Mech. Eng., 172(1), pp. 175–201. [CrossRef]
Oden, J. T. , Vemaganti, K. , and Moës, N. , 1999, “ Hierarchical Modeling of Heterogeneous Solids,” Comput. Methods Appl. Mech. Eng., 172(1), pp. 3–25. [CrossRef]
Oden, J. T. , and Zohdi, T. I. , 1997, “ Analysis and Adaptive Modeling of Highly Heterogeneous Elastic Structures,” Comput. Methods Appl. Mech. Eng., 148(3), pp. 367–391. [CrossRef]
Raghavan, P. , Moorthy, S. , Ghosh, S. , and Pagano, N. , 2001, “ Revisiting the Composite Laminate Problem With an Adaptive Multi-Level Computational Model,” Compos. Sci. Technol., 61(8), pp. 1017–1040. [CrossRef]
Ghanbari, J. , and Naghdabadi, R. , 2009, “ Nonlinear Hierarchical Multiscale Modeling of Cortical Bone Considering Its Nanoscale Microstructure,” J. Biomech., 42(10), pp. 1560–1565. [CrossRef] [PubMed]
Hamed, E. , and Jasiuk, I. , 2013, “ Multiscale Damage and Strength of Lamellar Bone Modeled by Cohesive Finite Elements,” J. Mech. Behav. Biomed. Mater., 28, pp. 94–110. [CrossRef] [PubMed]
Adharapurapu, R. R. , Jiang, F. , and Vecchio, K. S. , 2006, “ Dynamic Fracture of Bovine Bone,” Mater. Sci. Eng. C, 26(8), pp. 1325–1332. [CrossRef]
Crowninshield, R. , and Pope, M. , 1974, “ The Response of Compact Bone in Tension at Various Strain Rates,” Ann. Biomed. Eng., 2(2), pp. 217–225. [CrossRef]
Katsamanis, F. , and Raftopoulos, D. D. , 1990, “ Determination of Mechanical Properties of Human Femoral Cortical Bone by the Hopkinson Bar Stress Technique,” J. Biomech., 23(11), pp. 1173–1184. [CrossRef] [PubMed]
Lewis, J. , and Goldsmith, W. , 1975, “ The Dynamic Fracture and Prefracture Response of Compact Bone by Split Hopkinson Bar Methods,” J. Biomech., 8(1), pp. 27–40. [CrossRef] [PubMed]
McElhaney, J. H. , 1966, “ Dynamic Response of Bone and Muscle Tissue,” J. Appl. Physiol., 21(4), pp. 1231–1236. [PubMed]
Melnis, A. , and Knets, I. , 1982, “ Effect of the Rate of Deformation on the Mechanical Properties of Compact Bone Tissue,” Mech. Compos. Mater., 18(3), pp. 358–363. [CrossRef]
Roberts, V. , and Melvin, J. , “ The Measurement of the Dynamic Mechanical Properties of Human Skull Bone,” Applied Polymer Symposia, pp. 235–247.
Tennyson, R. , Ewert, R. , and Niranjan, V. , 1972, “ Dynamic Viscoelastic Response of Bone,” Exp. Mech., 12(11), pp. 502–507. [CrossRef]
Wood, J. L. , 1971, “ Dynamic Response of Human Cranial Bone,” J. Biomech., 4(1), pp. 1–12. [CrossRef] [PubMed]
Wright, T. , and Hayes, W. , 1976, “ Tensile Testing of Bone Over a Wide Range of Strain Rates: Effects of Strain Rate, Microstructure and Density,” Med. Biol. Eng., 14(6), pp. 671–680. [CrossRef] [PubMed]
Allen, D. H. , and Searcy, C. R. , 2001, “ A Micromechanical Model for a Viscoelastic Cohesive Zone,” Int. J. Fract., 107(2), pp. 159–176. [CrossRef]
Yoon, C. , and Allen, D. H. , 1999, “ Damage Dependent Constitutive Behavior and Energy Release Rate for a Cohesive Zone in a Thermoviscoelastic Solid,” Int. J. Fract., 96(1), pp. 55–74. [CrossRef]
Allen, D. H. , and Searcy, C. R. , 2000, “ Numerical Aspects of a Micromechanical Model of a Cohesive Zone,” J. Reinforced Plast. Compos., 19(3), pp. 240–248. [CrossRef]
Savalli, U. , 2013, “ Cross Section of a Bone,” Arizona State University, Phoenix, AZ, accessed Mar. 3, 2016, http://www.Savalli.us/BIO201/Labs/05-Bones/BoneHistologyLabel.Html
Pearson Education, 2004, “ Microscopic Structure of Bone,” Pearson PLC, London, accessed Mar. 3, 2016, http://www.Worcesterthinktank.com/Sites/wtk.Drupalgardens.com/Files/Uploads/APcompANDspongyBone.pdf
Williams, J. J. , 2002, “ Two Experiments for Measuring Specific Viscoelastic Cohesive Zone Parameters,” Master's thesis, Texas A & M University, College Station, TX.
Allen, D. H. , 2001, “ Homogenization Principles and Their Application to Continuum Damage Mechanics,” Compos. Sci. Technol., 61(15), pp. 2223–2230. [CrossRef]
Nemat-Nasser, S. , and Hori, M. , 2013, Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier, Amsterdam, The Netherlands.
Allen, D. , and Yoon, C. , 1998, “ Homogenization Techniques for Thermoviscoelastic Solids Containing Cracks,” Int. J. Solids Struct., 35(31), pp. 4035–4053. [CrossRef]
Roters, F. , Eisenlohr, P. , Hantcherli, L. , Tjahjanto, D. D. , Bieler, T. R. , and Raabe, D. , 2010, “ Overview of Constitutive Laws, Kinematics, Homogenization and Multiscale Methods in Crystal Plasticity Finite-Element Modeling: Theory, Experiments, Applications,” Acta Mater., 58(4), pp. 1152–1211. [CrossRef]
Lakes, R. S. , Katz, J. L. , and Sternstein, S. S. , 1979, “ Viscoelastic Properties of Wet Cortical Bone—I. Torsional and Biaxial Studies,” J. Biomech., 12(9), pp. 657–678. [CrossRef] [PubMed]
Jansen, L. E. , Birch, N. P. , Schiffman, J. D. , Crosby, A. J. , and Peyton, S. R. , 2015, “ Mechanics of Intact Bone Marrow,” J. Mech. Behav. Biomed. Mater., 50, pp. 299–307. [CrossRef] [PubMed]
Van Rietbergen, B. , Majumdar, S. , Pistoia, W. , Newitt, D. , Kothari, M. , Laib, A. , and Ruegsegger, P. , 1998, “ Assessment of Cancellous Bone Mechanical Properties From Micro-FE Models Based on Micro-CT, pQCT and MR Images,” Technol. Health Care, 6(5–6), pp. 413–420. [PubMed]
Parfitt, A. , Mathews, C. , Villanueva, A. , Kleerekoper, M. , Frame, B. , and Rao, D. , 1983, “ Relationships Between Surface, Volume, and Thickness of Iliac Trabecular Bone in Aging and in Osteoporosis. Implications for the Microanatomic and Cellular Mechanisms of Bone Loss,” J. Clin. Invest., 72(4), pp. 1396–1409. [CrossRef] [PubMed]
Russo, C. , Lauretani, F. , Bandinelli, S. , Bartali, B. , Di Iorio, A. , Volpato, S. , Guralnik, J. , Harris, T. , and Ferrucci, L. , 2003, “ Aging Bone in Men and Women: Beyond Changes in Bone Mineral Density,” Osteoporosis Int., 14(7), pp. 531–538. [CrossRef]
Bernhard, A. , Milovanovic, P. , Zimmermann, E. , Hahn, M. , Djonic, D. , Krause, M. , Breer, S. , Püschel, K. , Djuric, M. , and Amling, M. , 2013, “ Micro-Morphological Properties of Osteons Reveal Changes in Cortical Bone Stability During Aging, Osteoporosis, and Bisphosphonate Treatment in Women,” Osteoporosis Int., 24(10), pp. 2671–2680. [CrossRef]
Giner, E. , Arango, C. , Vercher, A. , and Fuenmayor, F. J. , 2014, “ Numerical Modelling of the Mechanical Behaviour of an Osteon With Microcracks,” J. Mech. Behav. Biomed. Mater., 37, pp. 109–124. [CrossRef] [PubMed]
Wang, X. , Osborn, R. , Wolf, J. , and Puram, S. , “ Age-Related Changes in the Mechanical Properties of Interstitial Bone Tissues,” 50th Annual Meeting of the Orthopaedic Research Society, p. 0037.
Green, J. O. , Wang, J. , Diab, T. , Vidakovic, B. , and Guldberg, R. E. , 2011, “ Age-Related Differences in the Morphology of Microdamage Propagation in Trabecular Bone,” J. Biomech., 44(15), pp. 2659–2666. [CrossRef] [PubMed]
Launey, M. E. , Buehler, M. J. , and Ritchie, R. O. , 2010, “ On the Mechanistic Origins of Toughness in Bone,” Annu. Rev. Mater. Res., 40, pp. 25–53. [CrossRef]
Sanborn, B. , Gunnarsson, C. , Foster, M. , and Weerasooriya, T. , 2014, “ Quantitative Visualization of Human Cortical Bone Mechanical Response: Studies on the Anisotropic Compressive Response and Fracture Behavior as a Function of Loading Rate,” Exp. Mech., 56(1), pp. 81–95. [CrossRef]
Weerasooriya, T. , Sanborn, B. , Gunnarsson, C. A. , and Foster, M. , 2016, “ Orientation Dependent Compressive Response of Human Femoral Cortical Bone as a Function of Strain Rate,” J. Dyn. Behav. Mater., 2(1), pp. 74–90. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Human bone (cross section) with two length scales (homogenous global-scale and heterogeneous local-scale RVE with fracture) [45,46]

Grahic Jump Location
Fig. 2

Flowchart describing the two-way linked multiscale algorithm

Grahic Jump Location
Fig. 3

Multiscale modeling of typical human bone with two length scales

Grahic Jump Location
Fig. 4

Global-scale objects for: (a) young group and (b) aged group

Grahic Jump Location
Fig. 5

Local scale RVEs for: (a) young group and (b) aged group

Grahic Jump Location
Fig. 6

Selected local-scale RVEs for: (a) young group and (b) aged group

Grahic Jump Location
Fig. 7

Stress contour of the global object for the young group at: (a) 0.2 s, (b) 0.5 s, and (c) 1.2 s

Grahic Jump Location
Fig. 8

Embedded cohesive zone elements and crack development at 1.2 s for the young group in: (a) RVE-1, (b) RVE-2, and (c) RVE-3

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Fig. 9

Stress–strain plots for the young and aged groups with different loading rates

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Fig. 10

Stress–strain plots for various parameters in damage evolution law (Eq. (16)): (a) A and (b) m

Tables

Table Grahic Jump Location
Table 1 Material properties for each phase in a global scale object [5254]
Table Grahic Jump Location
Table 2 Material properties of each phase in local-scale RVEs for young and aged groups [17,52,57,59]

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