Figure 1.

Schema of salient configurations for arterial G&R. New constituents k = c and m having natural (stress-free) configurations ${\kappa }_n^k(·)$ (typically unmeasurable and denoted by dashed bodies), are incorporated into evolving in vivo mixture configurations κ(·) (potentially measurable and denoted by solid bodies); deposition tensors G^k(τ) describe motions from individual natural configurations to the in vivo mixture configuration κ(τ) at intermediate time τ ∈ [0, s]. Deformation gradients F(τ) and F(s) describe motions from the reference configuration κ(0) to κ(τ) and κ(s), respectively. Thus, ${\mathbf{F}}_{n(\tau )}^k(s)=\mathbf{F}(s){\mathbf{F}}^{-1}(\tau ){\mathbf{G}}^k(\tau )$ describes motions from ${\kappa }_n^k(\tau )$ to κ(s). Note that the concept of constant target states and deposition stretches requires G^k(τ) ≡ G^k. ${\mathbf{F}}_{n(\tau )}^k(s)$ (not shown) is identically G^k. The special case F(τ) = I, representing tissue maintenance, recovers ${\mathbf{F}}_{n(\tau )}^k(s)\equiv {\mathbf{F}}_{n(0)}^k(s)$, as expected. Letting F(τ) = G^k = I, where the reference configuration is identically the natural configuration, recovers the classical result ${\mathbf{F}}_{n(\tau )}^k(s)\equiv \mathbf{F}(s)\forall \tau \in [0,s]$, as desired.