Quasi-Brittle Fracture: The Blended Approach

TL;DR

This work develops a mathematically well-posed, nonlocal blended model that couples peridynamics-inspired interactions with a two-point history-dependent phase field to predict quasi-brittle fracture. Displacement and damage evolve from a momentum balance IBVP, and the framework yields an explicit Griffith-type energy release and positive damage dissipation without ad hoc diffusion equations. Calibrated with a minimal set of material parameters, the model captures elastic response, strength degradation, fracture energy, and size effects, and demonstrates robust performance across mode-I, mixed-mode, and dynamic fracture, including corner singularities and fast crack propagation. A dimension-free formulation and Gamma-convergence-based calibration underpin the robustness and applicability to complex quasi-brittle materials, with numerical results showing strong agreement with experiments in concrete beams, L-shaped panels, and glass sheets. The approach offers a compact, physically grounded alternative to conventional phase-field methods for quasi-brittle fracture with seamless cross-regime capabilities and scalable predictions.

Abstract

A field theory is presented for predicting damage and fracture in quasi-brittle materials. The approach taken here is new and blends a non-local constitutive law with a two-point phase field. In this formulation, the material displacement field is uniquely determined by the initial boundary value problem. The theory naturally satisfies energy balance, with positive energy dissipation rate in accord with thermodynamics. Notably, these properties are not imposed but follow directly from the constitutive law and evolution equation when multiplying the equation of motion by the velocity and integrating by parts. In addition to elastic constants, the model requires at most three key material parameters: the strain at the onset of nonlinearity, the ultimate tensile strength, and the fracture toughness. The approach simplifies parameter identification while ensuring representation of material behavior. The approach seamlessly handles fracture evolution across loading regimes, from quasi-static to dynamic, accommodating both fast crack propagation and quasi-brittle failure under monotonic and cyclic loading. Numerical simulations show quantitative and qualitative agreement with experiments, including three-point bending tests on concrete. The model successfully captures the cyclic load-deflection response of crack mouth opening displacement, the structural size-effect related to ultimate load and specimen size, fracture originating from corner singularities in L- shaped domains, and bifurcating fast cracks.

Figures (25)

• Figure 1: (a) The profile $g(r)$ used in characterizing the failure envelope. (b) Bond stiffness on the failure envelope of unloading laws $g'(r)$ given by dashed curve. The bond stiffness becomes nonlinear at $r^L$, exhibits strain hardening between $r^L$ and $r^C$ and goes smoothly to zero at $r^F$. Unloading laws are linear elastic with a softer stiffness controlled by the phase field and shown in blue. The bond offers zero tensile stiffness for bonds broken in tension (red), see Definition \ref{['def: Second']}. However the bond continues to elastically resist negative strains. The bond stiffness given in \ref{['constpos']}, \ref{['constneg']}.
• Figure 2: $\gamma(\mathbf{u})(\mathbf{y},\mathbf{x},t))$. It is one for $r^\ast\leq r^L$ then decays to zero for $r^L<r^\ast(t,\mathbf{y},\mathbf{x},\mathbf{u})<r^F$.
• Figure 3: The constitutive law for a bond. The failure envelope is the black dashed curve. The linear unloading is blue. The slope of the linear unloading curve decreases when the bond strain at the present time is equal to its maximum over all earlier times. This is indicated by the arrows on the dashed curve. The force - strain constitutive law for a bond unable resist tension is red. All bonds are linear elastic under compressive strain.
• Figure 4: Domain $\Omega$ with prescribed Dirichlet data on $\Omega_D^\epsilon$. The union is denoted by $\Omega^*$
• Figure 5: Bonds in damage zone: Bond damage energy/Volume for all points on the blue unloading line connecting $(0,0)$ to $(r^U,g'(r^U))$ is the area of the dark gray region.

Theorems & Definitions (15)

• Definition 1: Bond
• Definition 2: Broken Bond
• Definition 3: Jump set
• Lemma 1: Relationship between jump discontinuities and bonds broken in tension
• Theorem 1: Existence and Uniqueness of Solution of Force-Controlled Fracture Evolution
• Lemma 2
• Lemma 3: Lipschitz continuity of $\mathcal{L}^\epsilon [\cdot]$
• Theorem 2: Existence and Uniqueness of Solution of the Displacement and Force Controlled Fracture Evolution
• Lemma 4: Growth of the Damage Energy and Process Zone
• Remark 1: Condition for Damage