* Corresponding author. Fax: #1-905-524-2121. Journal of Biomechanics 32 (1999) 1071}1079 A fabric-dependent fracture criterion for bone S. Pietruszczak*, D. Inglis, G.N. Pande Department of Civil Engineering, McMaster University, 1280 Main St. West Hamilton, Ont., Canada L8S 4L7 Department of Civil Engineering, University of Wales Swansea, Swansea SA2 8PP, UK Received 3 August 1998; accepted 3 May 1999 Abstract A fracture criterion for bone tissue is proposed. Bone material is considered to be anisotropic and its properties are described by invoking the concept of directional variation of porosity. The fracture criterion is expressed as a scalar-valued function of the stress tensor and it incorporates an orientation-dependent distribution of compressive/tensile strength. The proposed mathematical framework is applied to a numerical analysis of fracture in the proximal femur due to a fall from standing height. The risk of fracture is assessed in the context of two di!erent porosity distributions, simulating a healthy and an osteoporotic bone.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Fabric tensor; Constitutive law; Fracture criterion; Proximal femur 1. Introduction Over the last decade, a number of three-dimensional "nite-element analyses simulating fracture in the human proximal femur have been carried out (Lotz et al., 1991a,b; McNamara et al., 1996; Merz et al., 1996; Taylor et al., 1996). Most of these analyses make use of simple failure criteria which have been developed for a class of isotropic, non-porous materials. In particular, a number of authors have used von Mises criterion (e.g. Lotz et al., 1991a,b;Merz et al., 1996) which was originally developed for metals. In general, assessment of the risk of fracture based on these criteria may not be reliable as bone tissue is porous and strongly anisotropic. A substantial amount of work has provided evidence of anisotropy in bone, which may be quanti"ed using an orientation distribu- tion function characterized by &fabric' tensors (Odgaard, 1997). In addition, previous studies (Reilly and Burstein, 1975; Burstein et al., 1976; Carter and Spengler, 1978; Torzilli et al., 1981; Stone et al., 1983; Keaveny et al., 1994) have clearly demonstrated that the mechanical behaviour is, in general, inelastic and the strength in tension is considerably lower than in compression. Given this evidence, the speci"cation of the conditions at failure should include both these e!ects, i.e. the directional dependence of strength characteristics as well as their sensitivity to all three basic stress invariants. The primary objective of this work is to develop such a general frac- ture criterion which incorporates a measure of internal bone architecture. The related objective is to outline a methodology which involves the use of this criterion for the assessment of risk of hip and other bone fractures. The general approach to the formulation of failure criteria for anisotropic materials, which employs proper tensor generators and invariants, is mathematically com- plex and requires extensive experimental data for identi- "cation of material parameters. In view of this, a simpler approach is pursued here. In particular, the proposed fracture criterion incorporates a scalar-valued function de"ning the &directional porosity' which is perceived as a measure of material fabric. This measure describes the spatial distribution of void fraction in the neighbourhood of the material point and is identi"able from 3D micro- computed tomography. It is noted that other authors have used di!erent measures of fabric such as mean intercept length (e.g., Zysset and Curnier, 1996; Cowin, 1989), volume orientation (Odgaard et al., 1990), etc. The former de"nes the distribution of the mean path length between bone/marrow interfaces (Whitehouse, 1974; Harrigan and Mann, 1984), whereas the latter one speci- "es for each point within the structure the local orienta- tion of the solid volume. It is important to note that the 0021-9290/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 9 9 ) 0 0 0 9 6 - 2