Modeling Progressive Damage Accumulation in Bone Remodeling Explains the Thermodynamic Basis of Bone Resorption by Overloading

Abstract

Computational modeling of skeletal tissue seeks to predict the structural adaptation of bone in response to mechanical loading. The theory of continuum damage–repair, a mathematical description of structural adaptation based on principles of damage mechanics, continues to be developed and utilized for the prediction of long-term peri-implant outcomes. Despite its technical soundness, CDR does not account for the accumulation of mechanical damage and irreversible deformation. In this work, a nonlinear mathematical model of independent damage accumulation and plastic deformation is developed in terms of the CDR formulation. The proposed model incorporates empirical correlations from uniaxial experiments. Supporting elements of the model are derived, including damage and yielding criteria, corresponding consistency conditions, and the basic, necessary forms for integration during loading. Positivity of mechanical dissipation due to damage is proved, while strain-based, associative plastic flow and linear hardening describe post-yield behavior. Calibration of model parameters to the empirical correlations from which the model was derived is then accomplished. Results of numerical experiments on a point-wise specimen show that damage and plasticity inhibit bone formation by dissipation of energy available to biological processes, leading to material failure that would otherwise be predicted to experience a net gain of bone.

Appendices

Proof of the Necessity to Use a Differential Form of Hooke’s Law

The formulation in Sect. 3 requires mathematical support for path-dependent changes in the material state during loading for any combination of elastic and plastic strains. As such, the elastoplastic stiffness $C_{ijkl}^{ep}$ must account for changes in the material state with or without the occurrence of plastic strains, and vice versa, over some increment of stresses $d\sigma _{ij}$.

Assume that (21) is true and for total strains $\varepsilon _{ij}$ and plastic strains $\varepsilon _{ij}^{pl}$, $\varepsilon _{ij}^{pl} = \varepsilon _{ij}^{pl} \left( \varepsilon _{kl} \right)$. Then

Proof of Positive Definite Damage Differential

Consider the eigenvalue problem,

$$\begin{aligned} \text {d}D_{ij} n_j^k = \text {d}D_{ji} n_j^k = \lambda _k n_i^k , \end{aligned}$$

(59)

where $\lambda _k$ is the kth eigenvalue of the damage tensor differential $\text {d}D_{ij}$ with corresponding eigenvector $n_i^k$. By substitution of (59) into the differential form of (2),

$$\begin{aligned} \text {d}\left( h_{il} h_{lj} \right) n_j^k&= -\lambda _k n_i^k , \end{aligned}$$

(60a)

$$\begin{aligned}&= \text {d}\left( h_{jl} h_{li} \right) n_j^k. \end{aligned}$$

(60b)

Rewriting (34) in the frame of the spectral norm of the total strain and applying the chain rule,

$$\begin{aligned} \text {d}\left( h_{ij} h_{jk} h_{kl} h_{lm} \right)&= -c \text {d}\varepsilon h^o_{ij} h^o_{jk} h^o_{kl} h^o_{lm}, \end{aligned}$$

(61a)

$$\begin{aligned}&= h_{kl} h_{lm} \text {d}\left( h_{ij} h_{jk} \right) + h_{ij} h_{jk} \text {d}\left( h_{kl} h_{lm} \right) , \end{aligned}$$

(61b)

where $c \text {d}\varepsilon > 0$ and $h^o_{ij}$ is $h_{ij}$ at the onset of damage. By substitution of (60) into (61),