Lightweight phase-field surrogate for modelling ductile-to-brittle transition through phenomenological elastoplastic coupling

Abstract

The ductile-to-brittle transition (DBT) in body-centred cubic systems is a central design constraint for cryogenic structures. Performing parametric studies to enhance the understanding on DBT using fully coupled thermomechanical continuum DBT models is computationally expensive. Therefore, in this work, a lightweight phase-field surrogate is proposed. This surrogate approach captures key \emph{DBT-like} trends within a standard isothermal two-field (displacement--damage) setting by prescribing temperature dependence through three phenomenological mechanisms: (i) a temperature-dependent degradation exponent $n(T)$ that sharpens stiffness loss from gradual (ductile-like, $n=2.0$ at 293\,K) to abrupt (brittle-like, $n=3.5$ at 77\,K), (ii) temperature-dependent yield stress and elastic modulus to modulate the balance between plastic dissipation and elastic energy storage, and (iii) an effective fracture toughness and driving-force scaling to represent reduced crack-tip shielding at cryogenic temperatures. The model is implemented in FEniCSx using small-strain $J_2$ return mapping and a staggered solution scheme. Simulations of a single-edge-notched specimen over 77--293\,K demonstrate a systematic progression from brittle-like to ductile-like response, characterised by reduced displacement to unstable fracture, a transition from abrupt post-peak load drop to extended softening, and a shift from narrow, localised damage bands with confined plasticity to broader process zones. A sensitivity study comparing four interpolation schemes (linear, smoothstep, exponential, hybrid) shows that the qualitative transition trends are robust, with interpolation primarily affecting intermediate-temperature responses while endpoint behaviours remain unchanged.

Figures (9)

• Figure 1: Mesh convergence study at $T = 77$ K. (a) Force--displacement curves for three refinement levels. (b) Peak load convergence. (c) Summary table with wall-clock times.
• Figure 2: Plasticity coupling verification at $T = 293$ K. Left: force--displacement for pure elastic vs. elastoplastic models. Right: evolution of maximum equivalent plastic strain.
• Figure 3: Comparison of 293 K and 77 K simulations. (a) Force--displacement curves. (b) Evolution of maximum damage $d_{\max}$. (c) Evolution of maximum equivalent plastic strain $\bar{\varepsilon}^p_{\max}$; dashed vertical lines mark the peak load for each temperature.
• Figure 4: Damage field $d(\bm{x})$ at four displacement levels ($\bar{u} = 1.3$, 2.1, 2.9, 3.8 $\mu$m). Top row: 293 K. Bottom row: 77 K. Colour scale: $d = 0$ (dark, intact) to $d = 1$ (bright, fully fractured).
• Figure 5: Von Mises stress distribution at a representative pre-peak load step (cell-averaged). (a) 293 K ($\sigma_y=350$ MPa). (b) 77 K ($\sigma_y=700$ MPa). Note the different colour bar scales.