Eur. Phys. J. AP 8, 257–263 (1999) T HE EUROPEAN P HYSICAL JOURNAL APPLIED PHYSICS c  EDP Sciences 1999 Remodeling of the bone material containing microcracks: A theoretical analysis S. Ramtani a and M. Zidi Universit´ e Paris Val de Marne, Facult´ e des Sciences et Technologie, Laboratoire de M´ ecanique Physique b , 61 avenue du g´ en´ eral de Gaulle, 94010 Cr´ eteil Cedex, France Received: 22 January 1999 / Revised and Accepted: 16 September 1999 Abstract. The question is, what happens when the bone loses its ability for load-driven adaptation, when damage is no longer repaired as it seems to be the case for bone loss associated with age, medication or disease? In this study, we tempt to show how damage can influence the remodeling process. A thermody- namic theoretical framework is therefore provided as a basis for a consistent formulation of bone remodeling involving a chemical reaction and mass transfer between two constituents in presence of microcracks. PACS. 46.05.+b General theory of continuum mechanics of solids – 46.50.+a Fracture mechanics, fatigue and cracks 1 Introduction The bone remodeling process is generally viewed as bone material response to functional demands and muscle at- tachments by continual process of growth, reinforcement and resorption which occur in living situations [1–3]. Ev- eryday activities damage bone, and this damage is nor- mally repaired in a continuous remodeling process. Rapid generation of damage exceeds the repair rate and leads to fractures in response to minor traumas [4,5] or acts as a stimulus for bone remodeling [6–10] and adaptation [11–13]. It is now generally accepted that bone adapts to altered loading and that a such functional adaptation is of major clinical importance [14]. As an example, Pren- dergast and Taylor [12] developed a remodeling law which accounts for the stress history to which the bone tissue has been exposed since its formation, in order to predict the time-course of bone adaptation. It is based on two hy- potheses : (a) there is an amount of damage, in the form of microcracks, present in the bone at remodeling equi- librium and, the stimulus for remodeling is the change in damage from this equilibrium amount and (b) the rate of damage repair is determined by the homeostatic stress. Based on these previous hypotheses, an integral math- ematical law has been derived and used in a finite ele- ment analysis, for the prediction of bone adaptation in the Lanyon sheep forelimb model [15]. For such a theory to work, bone tissue would need to be able to sense dam- age as it happens and presume that there is a sensor in the bone which collects the mechanical stimulus and uses it to signal a biological response [16–19]. These approaches in a e-mail: ramtani@univ-paris12.fr b ESA 7052 du CNRS which bone remodeling is mathematically described to be a self organizational process, suffer from their non ther- modynamical basis. The objective of this paper is the formulation of a con- tinuum model in order to predict both local strain and damage controlled remodeling properties of bone. An out- line of the paper is as follows: in Section 2, we assume that the proposed remodeling theory that attempts to ac- count for the mechanical objectives of the remodeling pro- cess should be cast in terms of both local bone strain and damage. In Section 3, internal state variables describing the damaged porous bone and the remodeling process are chosen. Fundamental assumptions are made in Section 4, and continuum thermodynamics is used for the description of the damaged bone remodeling process. In particular, representations for the rate of work equation and the en- tropy production inequality involving mass transfer to and from the porous solid are postulated and, an original re- sult showing how damage could affect the remodeling pro- cess is derived. In Section 5, constitutive assumptions are made and, a condition concerning the damage-dissipation is stated as a new thermodynamical restriction. Section 6, is devoted to a summary of the proposed model in a par- ticular case of an isothermal process, and to the analysis of some specific remodeling rate equations. 2 Damaged bone remodeling process Schematically, bone contains primarily three types of cells: (1) osteoblasts, which create (deposit) bone; (2) osteoclasts, which destroy (resorb) bone;