Buildability Assessment of 3D-Printed Concrete Structures Using Smoothed Mohr-Coulomb Plasticity with Isotropic Hardening

1. Introduction

• Buildability ends at the maximum layer count before either plastic collapse, when base stress reaches yield strength, or elastic buckling, when stability is lost below material strength.Which mode governs depends on geometry, print speed, layer dimensions and material properties.Source: §1 · p. 2
• Analytical structural models combine buckling and plastic-collapse limits but are confined to idealised straight walls, while fluid simulations of structural build-up are computationally demanding.Full-scale printing also limits experimental design exploration, and geometry and boundary-condition choices can materially shift predicted buildability.Source: §1 · pp. 2–3
• Earlier Mohr–Coulomb simulations captured straight-wall buckling but overpredicted hollow-cylinder buildability by about 60%.Liu et al. (2022) added nonlinear hardening to Drucker–Prager, reducing but not eliminating the cylinder discrepancy.Source: §1 · pp. 3–4
• The proposed framework combines a smoothed Mohr–Coulomb surface with nonlinear cohesion hardening, addressing prior models that represented pressure dependence and hardening separately.Its hardening uses one additional parameter calibrated from standard uniaxial compression data, and damage is omitted because simulated failures occur at or before peak strength.Source: §1 · p. 5
• The first zero crossing of the smallest tangent-stiffness eigenvalue identifies instability, reproduces analytical self-weight buckling load, and is insensitive to small geometric imperfections.The introduction contrasts this with convergence, first-yield and prescribed-deformation criteria, which may not coincide with structural instability.Source: §1 · p. 5
• The study validates the framework against printed wall and hollow-cylinder experiments and examines how geometry and hardening rate relate to buildability and strength mobilisation.The introduction frames the model as a numerical approach to assessing buildability and its constitutive basis.Source: §1 · p. 5

2. Theoretical Framework

• Small-strain elastoplasticity on a smoothed Mohr–Coulomb surface with nonlinear cohesion hardening is embedded in an updated Lagrangian formulation with a co-rotational stress update.The formulation models the quasi-static response of a growing structure under self-weight.Source: §2 · p. 6

2.1. Thermodynamic Constitutive Setting

• A scalar hardening variable raises cohesion from its initial value toward a finite saturated value as plastic deformation accumulates.The hardening force evolves at a rate set by ω; its derivative saturates to represent strength approaching a finite limit.Source: §2.1 · pp. 6–7

2.2. Smoothed Mohr–Coulomb Yield Surface

• The Abbo–Sloan construction smooths Mohr–Coulomb corners, while the cohesion term expands the yield surface isotropically as strength builds.The surface represents pressure and deviatoric-stress dependence; the transition angle sets the corner-smoothing region.Source: §2.2 · p. 7, Figure 2 · p. 8

2.3. Return Mapping and Algorithmic Tangent

• An implicit backward-Euler return mapping updates stress and hardening state from a strain increment, and its Jacobian supplies the algorithmic tangent.The local Newton solve enforces the flow rule, consistency condition and hardening evolution.Source: §2.3 · pp. 9–10

2.4. Updated Lagrangian Weak Form

• Resetting the reference to the current configuration at every step accounts for accumulated deflection while keeping each incremental displacement small.The weight of each new layer is applied over several time steps.Source: §2.4 · p. 10
• Finite-element discretisation expresses equilibrium as equality of the internal and external force vectors, with the material tangent supplying the equilibrium tangent.The stress in the internal force is returned by the material update at integration points.Source: §2.4 · p. 11
• The small-strain incremental formulation omits stiffness from changes in geometry, which is reinstated separately for stability analysis.Source: §2.4 · p. 11
• Co-rotation transports stress objectively through a midpoint frame, where the material update is evaluated before stress is transported back.The stress transport linearisation is omitted from the iteration matrix, yielding linear rather than quadratic asymptotic convergence.Source: §2.4 · pp. 11–12
• The solver accepts a step when the residual norm falls below tolerance; omitting stress-transport linearisation slows convergence but does not reduce robustness.The omitted term is of the order of the step increment, which the layer-wise schedule keeps small.Source: §2.4 · p. 12

2.5. Linearised Tangent and Stability

• The material stiffness omits the destabilising work of existing stress against geometric change, so initial-stress stiffness is added to the tangent for stability analysis.The approximation is justified while displacement gradients remain small up to the critical point.Source: §2.5 · pp. 12–13
• The stability tangent omits a stress-transport linearisation term of the order of the step increment, as does the equilibrium iteration.Source: §2.5 · p. 13
• Instability is detected when the restricted tangent on free printed-layer degrees of freedom loses positive definiteness; its smallest eigenvalue reaching zero marks bifurcation.A positive smallest eigenvalue indicates stable equilibrium, and its sign change is geometric because the hardening law has no softening branch.Source: §2.5 · p. 13

3.1. Printing Simulation Setup

• The shared quasi-static setup activates self-weight and stiffness layer by layer, gives each layer a local age clock, and uses clamped bases with traction-free surfaces.The whole mesh is present initially, with elements activated as deposition reaches them; the two cases differ in material law and dimensions.Source: §3.1 · p. 14

3.2. Buildability Criterion

• The detector tracks the smallest total-tangent eigenvalue’s sign change; the predictor’s frozen-state buckling factor approaches unity at the same event as step resolution increases.The criterion is converted to a fractional layer count because failure may occur part-way through deposition.Source: §3.2 · pp. 14–15

3.3. Stopping-Criterion Comparison

• The stopping-criterion comparison uses an age-independent linear-elastic material and sweeps initial lateral imperfections to distinguish stability loss from continuation failure.The wall provides a closed-form buckling reference; the cylinder tests whether numerical agreement carries over without such a reference.Source: §3.3 · p. 16

3.3.1. Straight wall

• 22.65 layers: the wall detector and predictor lie within 0.7% of the 22.51-layer analytical reference for ξ≤10^-3, while Newton divergence overpredicts buildability.Newton divergence falls from 33.17 layers for the perfect mesh to 25.50 at ξ=10^-2.Source: §3.3.1 · p. 17, Table 3 · p. 18

3.3.2. Cylinder

• 30.59 layers: the cylinder detector and predictor agree within 0.1 layer for ξ≤10^-3, while Newton divergence occurs later and shifts with imperfection.The cylinder’s governing local wall mode has no closed-form reference; the eigenvalue criteria are mutually coincident and nearly insensitive to small imperfections.Source: Figure 4 · p. 18, Table 3 · p. 18, §3.3.2 · p. 19

3.3.3. Adopted criterion

• The study adopts the first sign change of the smallest total-tangent eigenvalue as its buildability criterion, stopping at instability rather than solver divergence.The predictor is only a frozen-state multiplier, whereas printing changes the active domain, self-weight, material age, and tangent at every layer.Source: §3.3.3 · p. 19

4.1.1. Reference cases and model parameters

• Both material laws reproduce the reference stress–strain response across tested ages; Mohr–Coulomb cohesion is mapped from von Mises strengths and its curves sit slightly lower because of smoothing.The offset is small and conservative and does not affect the conclusions.Source: §4.1.1 · pp. 21–22, Figure 6 · p. 22

4.1.2. Straight wall

• 20.1 layers: both material laws reproduce the wall’s 21-layer experimental collapse within 4.3%, and the earlier finite-element prediction is 20 layers.The two laws reach different peak von Mises stresses at failure, 5.9 kPa for Mohr–Coulomb versus 4.7 kPa for von Mises, without changing buildability because the wall buckles globally.Source: §4.1.2 · pp. 22–23, Table 5 · p. 23

4.1.3. Hollow cylinder

• 24% versus 59%: the framework’s cylinder prediction over-predicts the experimental mean less than the earlier simulation, while matching the wall benchmark simulation.The cylinder prediction range reaches the measured spread, whereas the earlier simulation’s range lies above all five prints.Source: Table 6 · p. 23, §4.1.3 · p. 24
• 35.9 versus 38.4 layers: the cylinder reaches its critical state earlier with Mohr–Coulomb than with von Mises, and its prediction is closer to experiment than the earlier simulation.Scaling stiffness and cohesion by ±17.5% gives a Mohr–Coulomb range of 30.8–41.8 layers; pressure dependence affects this curved, confined and bending structure more than the slender wall.Source: Table 6 · p. 23, §4.1.3 · pp. 23–24

4.2.1. Parametric design

• The 6 × 6 cylinder sweep varies diameter and cohesion hardening rate while holding other material parameters constant, so it shows relative trends rather than the buildable height of a specific mix.Layer time is fixed at 6 s across diameters; the implied deposition speed may not be feasible on every printer at the largest diameters.Source: §4.2.1 · p. 24, Table 7 · p. 25

4.2.2. Buildability map

• Buildability rises monotonically with hardening rate and peaks near D = 300 mm before falling toward the widest cylinders.Across the lowest-to-highest hardening rates, the increase is about 46% at D = 100 mm and 23% at D = 300 mm; slender columns and widest shells are buckling-limited.Source: §4.2.2 · pp. 25–26, Figure 8 · p. 26

4.2.3. Observed failure modes

• At detected failure, slender cylinders lean about their base, intermediate cylinders bulge in their lowest layers, and the widest deform into a barrel-like profile.The displacements are magnified five times; stresses are highest at intermediate diameter and concentrate in a narrow base ring in the widest cylinders.Source: §4.2.3 · p. 26, Figure 9 · p. 27

4.2.4. Strength mobilisation at failure

• At the weakest hardening, strength mobilisation is 0.14 for D = 75 mm but 0.58 for D = 300 mm; low mobilisation marks instability before substantial hardening, while near-full mobilisation marks plastic-collapse control.Mobilisation peaks at D = 300 mm, matching the buildability peak; at the perfectly plastic limit, every cylinder with D ≥ 200 mm fails fully mobilised.Source: §4.2.4 · p. 28

5. Conclusions

• The constitutive framework uses a smoothed Mohr–Coulomb surface with nonlinear isotropic cohesion hardening, updated Lagrangian formulation, and layer activation.Its total tangent stiffness combines material stiffness with geometric stiffness from existing stress.Source: §5 · pp. 28–29
• The smallest tangent eigenvalue reaches zero at 22.67 wall layers, within 0.7% of analytical self-weight buckling height, and is unaffected by imposed geometric imperfection.Newton divergence occurs up to fifteen layers later and decreases as imperfection grows.Source: §5 · p. 29
• The Mohr–Coulomb model predicts hollow-cylinder collapse at 35.9 layers versus an experimental mean of 29, and its 30.8–41.8-layer material-property range reaches the observed 25–31-layer spread.For the straight wall, the framework reproduces the observed 21-layer collapse within 4.3%.Source: §5 · pp. 29–30
• Buildability rises with hardening rate and peaks near D = 300 mm, where intermediate cylinders are more stable than slender and widest cylinders.The study samples six diameters and six hardening rates; Figure 10 shows strength mobilisation peaking at D = 300 mm and deepening as hardening rate decreases.Source: Figure 10 · p. 29, §5 · p. 30
• At weakest hardening, failure mobilisation is 0.14 for D = 75 mm and 0.58 for D = 300 mm; at the perfectly plastic limit, cylinders of D ≥200 mm fail fully mobilised.Low mobilisation indicates instability before the material spends its available strength, while full mobilisation indicates plastic collapse; all failures occur while the most heavily loaded point is yielding.Source: §5 · p. 30
• Faster hardening adds layers when slender cylinders become unstable with cohesion gain unspent, but has almost no layer-count effect when intermediate cylinders fail fully mobilised.The hardening rate controls buildability in instability-driven cases, whereas saturated cohesion sets the limit in collapse-driven cases.Source: §5 · p. 30
• The formulation lacks probabilistic parameter treatment and softening, which would represent buildability as a distribution and capture post-instability cracking and degradation.Scaling cylinder stiffness and cohesion shifts predicted buildability by eleven layers.Source: §5 · p. 30

Appendix A. Consistent Tangent of the Smoothed Mohr–Coulomb Model

• The appendix records the gradients, Jacobian, and consistent tangent of the return mapping.These quantities are required to reproduce the constitutive update but not to follow its mechanics.Source: Appendix A · p. 30
• The listed derivatives support reproduction of the constitutive update.The passage identifies the return mapping of Section 2.3 as the update being documented.Source: Appendix A · p. 30
• The appendix distinguishes implementation detail from the mechanics needed to follow the model.It states that the recorded quantities are not necessary for understanding those mechanics.Source: Appendix A · p. 30

Appendix A.1. Gradient of the plastic potential

• The derivatives of G with respect to the invariants are specified before applying the chain rule through those invariants.The passage introduces the invariant derivatives and then states that the chain rule gives the gradient.Source: Appendix A.1 · p. 31
• The chain rule connects the invariant-level derivatives to the gradient expression.Source: Appendix A.1 · p. 31
• The yield-surface gradient has the same structure as the plastic-potential gradient, with φ, K = K(θ, φ), and rF = q.Source: Appendix A.1 · p. 31

Appendix A.2. Hessian of the plastic potential

• The appendix gives the second derivative of J3 as part of constructing the plastic-potential Hessian.Source: Appendix A.2 · p. 31
• It identifies the derivative of (s2)ik with respect to smn and maps it to Mandel form as Tmandel(s).Source: Appendix A.2 · p. 31
• The scalar second derivatives of G are specified as components of the Hessian construction.Source: Appendix A.2 · p. 32
• The full Hessian is assembled by combining the contributions through the chain rule.Source: Appendix A.2 · p. 32

• Newton iteration solves the return-mapping residual using the state vector y = [σn+1, ∆λ, αn+1]⊤.Source: Appendix A.3 · p. 32
• The 8 × 8 Jacobian uses the flow direction, its Hessian, the yield normal, and the hardening slope.Source: Appendix A.3 · p. 32
• At convergence, the consistent algorithmic tangent is the top-left block of the inverse Jacobian scaled by elastic stiffness.It gives dσ = Cep dε, with Cep = Ce in the elastic case.Source: Appendix A.3 · p. 32

• For |θ| > θT, the Mohr–Coulomb shape function is replaced by a quadratic blend in the vertex region.Source: Appendix B · p. 33
• The blend coefficients use φ for the yield surface or ψ for the plastic potential and enforce C2 continuity.Source: Appendix B · p. 33
• The first two derivatives of K with respect to the Lode angle are required for the gradients and Hessian and are listed in Table B.8.Source: Appendix B · p. 33

• Table C.9 collects the parameters obtained by the calibration in Section 4.1.1.Source: Appendix C · p. 33