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

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.