Topic A — A 3D-Printed Hydrogel Scaffold

periodic homogenization of a printed lattice · large-strain viscoelastic compression · tier: standard

Motivation

Bioprinted hydrogel scaffolds (cartilage, bone, vascular tissue) must match the stiffness of the tissue they replace and must not collapse under the first loading. Their stiffness comes from two sources: the hydrogel itself (soft, rate-dependent) and the printed architecture (strand spacing, layer stacking, porosity). You will separate the two: first the effective stiffness of a printed lattice by periodic homogenization, then the time-dependent large-strain response of the scaffold in confined and unconfined compression.

Related fields: biomaterials and tissue engineering (injectable and printable hydrogels, 3D bioprinting); rheology of soft materials and porous polymers. Course weeks: 1 (hyperelasticity, load steps), 5 (Prony series, internal variables), 9 (periodic homogenization with dolfinx_mpc).

Model

Part 1 — homogenization. A unit cell of the printed lattice: two layers of strands of diameter d at spacing s, rotated by 90° (log-pile), in 2D first (a square cell with one circular or rectangular strand section, plane strain) and then in 3D. Linear elastic hydrogel (E_g, \nu_g), pores empty (or filled with a much softer phase, E_p=10^{-3}E_g, to keep the problem well-posed). Periodic fluctuation \tilde{\mathbf u} with a macroscopic strain \mathbf E: \mathbf u=\mathbf E\mathbf x+\tilde{\mathbf u}; average stress \langle\boldsymbol\sigma\rangle=\mathbb C^{\rm hom}:\mathbf E from six (3D) or three (2D) unit load cases.

Part 2 — compression. The homogenized scaffold, or a few unit cells resolved, compressed between rigid plates to 20–30 % nominal strain at a constant rate, held, released. Compressible neo-Hookean equilibrium response plus one or two Maxwell branches (finite-strain extension of Week 5: a multiplicative or a simple additive-in-stress form — justify the choice); friction-free or bonded plates.

Design outputs: E^{\rm hom}/E_g against porosity; the peak and relaxed compressive stress at 20 % strain; the relaxation time of the scaffold.

Suggested parameters

E_g=50 kPa, \nu_g=0.45 (nearly incompressible — mind locking), strand diameter 200 µm, spacing 400–800 µm (porosity 0.4–0.75); one Prony branch with E_1=E_g, \tau=10 s; compression rate 1 %/s.

Required tasks

  1. 2D unit cell, three load cases, \mathbb C^{\rm hom}, convergence in the mesh size; symmetry of \mathbb C^{\rm hom}.
  2. E^{\rm hom}/E_g against porosity (at least five spacings); compare 2D and 3D for one spacing.
  3. Large-strain compression of a homogenized block (equilibrium only), then with viscoelasticity: load–time and stress–strain curves.
  4. Mesh and time-step independence of the peak stress.

Verification specific to this topic

  • Laminate: a unit cell made of two layers (gel / soft phase) with equal Poisson ratios has exact effective Young’s moduli — the Voigt average for loading parallel to the layers, the Reuss average across them. The exact fluctuation is piecewise linear, so with the interface on mesh lines your code must reproduce them to round-off. (Unequal Poisson ratios: closed-form laminate formulas in the homogenization literature, e.g. Yvonnet 2019.)
  • Bounds: every E^{\rm hom} lies between the Reuss and the Voigt bound of the two phases; check the symmetry of \mathbb C^{\rm hom} and positivity.
  • Homogeneous compression: a frictionless block in uniaxial compression has a homogeneous solution; the lateral stretch follows from one scalar equation for the compressible neo-Hookean law (solve it with NumPy) — the FE solution must match to solver tolerance.
  • Relaxation: a step strain held constant must relax exactly as the Prony series in the small-strain limit (Week 5).
  • MMS for the periodic cell problem (a smooth periodic fluctuation with its body force).

Stretch goals

  1. Layer offset in the log-pile (aligned vs staggered strands) — which is stiffer?
  2. Poroelasticity (fluid flow through the gel network) instead of viscoelasticity: biphasic (\mathbf u,p) mixed formulation.
  3. Inverse problem: identify E_g, E_1, \tau from a synthetic relaxation curve.

Milestones

M1 2D periodic cell with the laminate check · M2 porosity study · M3 hyperelastic compression with the homogeneous check · M4 viscoelastic compression and relaxation · M5 3D cell or resolved scaffold.

Pitfalls

  • \nu_g=0.45: P1 displacements lock in bending of thin strands — use P2 (Week 1) and check against \nu_g=0.3.
  • Empty pores make the cell problem singular where strands do not connect (rigid strands floating): keep a soft filler or check connectivity.
  • Periodic constraints with dolfinx_mpc: the corner nodes need consistent master–slave pairs (Week 9); a wrong pairing shows up as an unsymmetric \mathbb C^{\rm hom}.

References

  • J. Yvonnet, Computational Homogenization of Heterogeneous Materials with Finite Elements, Springer 2019.
  • G. A. Holzapfel, Nonlinear Solid Mechanics, Wiley 2000 — compressible neo-Hookean, finite viscoelasticity.
  • J. Malda et al., “25th anniversary article: Engineering hydrogels for biofabrication”, Adv. Mater. 25 (2013) 5011–5028.