Micro-displacement tensor

TL;DR

This work introduces an extended kinematics for elastic solids by incorporating a micro-displacement tensor $U$ to capture micro-scale incompatibilities and their relaxation. The framework writes the elastic strain as $\varepsilon_e = \text{sym}(\nabla u + \text{Curl} U)$ and couples boundary micro-activities via a layering ansatz to produce nonlocal boundary controls and residual stresses. It demonstrates the approach in three case studies: surface deposition with active micro-forces generating pre-stress, crystallization from a melt under controlled pre-stress, and winding with prescribed pre-stretch, together with models of bulk relaxation of inelastic strain. The results show that microscopic activity can create and erase incompatibility in a thermodynamically consistent way, offering a route to design smart, growth-enabled solids and to inform 3D printing and bio-inspired materials engineering.

Abstract

We propose an extended kinematics of nominally elastic continuum solids allowing one to describe their mechanical interaction with micro-scale loading devices. The main new ingredient is the concept of a micro-displacement tensor which extends the conventional description of the deforming elastic solids in terms of macroscopic displacement vectors. We show that micro-displacement tensors are particularly useful in dealing with active incompatibility acquisition and its subsequent passive relaxation. We use the proposed approach to describe the energetics of surface deposition while accounting for the presence of micro-mechanical controls.To illustrate the effectiveness of the new conceptual scheme we present two case studies: crystallization from a melt resulting in pre-stress, and winding of a coil with controlled pre-stretch.

Figures (11)

• Figure 1: Residual stresses in an incompatible arrangement of $m$ layers, oriented perpendicularly to the direction $\boldsymbol{n}$. The layers, having initially uneven stress-free lengths, see $(a)$, are stretched or compressed to be of the same size and then glued together, see $(b)$.
• Figure 2: The growing body which retains a memory of its internal layering emerging during the deposition process. The local orientation of the layer $\boldsymbol{n}(\boldsymbol{x})$ is determined at the moment when the growth surface passes through the point $\boldsymbol{x}$ and remains unchanged thereafter.
• Figure 3: Schematic representation of a solid under non-hydrostatic stress, in contact with its melt. Vertical sliding walls enforce homogeneous lateral deformation, with tractions $\boldsymbol{s}_w=-P_w\boldsymbol{e}_r$. The liquid experiences a hydrostatic pressure $\boldsymbol{s}_n = -P\boldsymbol{n}$ with ($P>0$). The growth front, positioned at $\psi(t)$, carries controllable "active" surface stress $\sigma_a=-P_a$.
• Figure 4: (a) Regime diagram in the space of parameters $({p_a}, p_w)$ for the evolution problem $\dot\psi = \mathcal{G}(\psi)$ with $\psi(0) = \psi_0$, see \ref{['evolutionpb']}. In regions 1 and 2, $\mathcal{G}'(\psi_e) < 0$, producing asymptotically stable regimes. In regions 3 and 4, $\mathcal{G}'(\psi_e) > 0$, indicating unstable regimes. (b)–(c) The trajectories of interest on the phase plane $(\dot\psi, \psi / \psi_0)$. Here $\psi_0$ is the initial location of the growth surface (blue dot). (b) Stable trajectories (regions 1 and 2) converge asymptotically to equilibria (red dots), which are either above ($Q_1$) or below ($Q_2$) the initial state. (c) Unstable trajectories (regions 3 and 4) either extending to infinity ($Q_4$) or collapsing to zero ($Q_3$). All plots are obtained with Poisson ratio $\nu = 0.3$. The choices of the parameters ${P},E,{p_a},p_w$ only contribute to rescaling of the trajectories shown in (b) and (c).
• Figure 5: (a) Evolution of growing surfaces $\psi(t)$ represented by the solutions of the problem $\dot\psi = \mathcal{G}(\psi)$ with initial condition $\psi(0) = 1$, and parameters corresponding to the points $Q_1 = (0.9, 1.1)$ and $Q_2 = (0.9, 1.4)$ in Fig. \ref{['papw']}. In both cases, $\psi(t)$ converges to a finite equilibrium, which is higher than the initial value in the first case (solidification) and lower in the second (melting). (b) Behavior of the function $\psi(t)$ corresponding to points $Q_3 = (1.7, 1.4)$ and $Q_4 = (1.7, 1.1)$ in Fig. \ref{['papw']}. In these cases, no stabilization is possible: $\psi(t)$ collapses to zero in the former case, while it blows up in the latter case. Parameters are fixed at $\nu = 0.3$ and ${P} = E = 1$.