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The peridynamic correspondence model (PDCM) provides the stress–strain relation that can introduce many classical constitutive models, however, the high computational consumption and zero-energy mode of PDCM certainly limit its further application to practical engineering crack problems. To solve these limitations and exploit the advantage of PDCM, we propose a simple and effective method that adaptively couples dual-horizon peridynamic element (DH-PDE) with finite element (FE) to simulate the quasi-static fracture problems. To this end, a stabilized dual-horizon peridynamic element for DH-PDCM is firstly developed that the peridynamic strain matrices for the bond and material point are constructed respectively. The nonlocal ordinary and correctional peridynamic element stiffness matrices are derived in detail and calculated by the proposed dual-assembly algorithm. Subsequently, a unified variational weak form of this adaptive coupling of DH-PDE and FE is proposed based on the convergence of peridynamics to the classical model in the limit of vanishing horizon. Therefore, the integrals of the peridynamic element and finite element in this coupling method are completely decoupled in the viewpoint of numerical implementation, which makes it easier to realize the proposed adaptive coupling by switching integral element. Moreover, the proposed adaptive coupling is implemented in Abaqus/UEL to optimize the calculational efficiency and real-time visualization of calculated results, which has potential for dealing with the engineering crack problems. Two-dimensional numerical examples involving mode-I and mixed-mode crack problems are used to demonstrate the effectiveness of this adaptive coupling in addressing the quasi-static fracture of cohesive materials. [ABSTRACT FROM AUTHOR]

The size and viscosity effects are noticeable at the micro-/nano scale. In the present work, the strain gradient viscoelastic solution of the mode-III crack in an infinite quasi-brittle advanced material is proposed based on the strain gradient viscoelasticity theory using the Wiener–Hopf method. The solutions to the gradient-dependent viscoelastic crack problem are obtained directly by using the correspondence principle between the strain gradient viscoelasticity and strain gradient elasticity in Maxwell's standard linear solid model. In this model, the stress near the crack tip is time-dependent and size-dependent. Besides, the stress near the crack tip is more significant than that based on gradient elasticity theory. Compared with the elastic strain gradient effect, the viscous gradient effect makes the stress field at the crack tip harden. The location and the value of maximum stress change with time, which differs from the case in strain gradient elasticity theory. The time that the normalized stress takes to stabilize also changes with the distance from the crack tip. When the viscosity effect is neglected or time tends to infinity, the strain gradient viscoelasticity theory can be reduced to the classical strain gradient elasticity theory. [ABSTRACT FROM AUTHOR]

• The strain gradient viscoelastic solutions of the Mode-I and Mode-II crack in an infinite quasi-brittle advanced material are proposed based on the strain gradient viscoelasticity theory using the Wiener-Hopf method and the correspondence principle in Maxwell's standard linear solid model. The strain gradient viscoelasticity theory are successfully used to explain the fracture of microscale and nanoscale materials with viscosity and size effect. • The stress near the crack tip is time-dependent and size-dependent. The stress near the crack tip is more significant than that based on gradient elasticity theory. Compared with the elastic strain gradient effect, the viscous gradient effect makes the stress field at the crack tip harden. • The location and the value of maximum stress change with time, which differs from the case in strain gradient elasticity theory. The time that the normalized stress takes to stabilize also changes with the distance from the crack tip. When the viscosity effect is neglected or time tends to infinity, the strain gradient viscoelasticity theory can be reduced to the classical strain gradient elasticity theory. The impacts of size and viscosity become distinctly evident when considering micro- and nano-scale phenomena for advanced materials with micro- and nanostructures. In this study, the mechanical behavior of the advanced material is characterized by using the strain gradient viscoelasticity theory, and a novel solution is presented for the mode-I and mode-II cracks, which is formulated based on the strain gradient viscoelasticity theory, employing the Wiener-Hopf method. Besides, the gradient-dependent viscoelastic crack solutions are directly derived by applying the correspondence principle that aligns strain gradient viscoelasticity with strain gradient elasticity within the Maxwell's standard linear solid model. Relative to the influence of elastic strain gradient effects, the involvement of viscous gradient effects instigates a reinforcement to the stress field around the crack tip, thereby offering a more reasonable representation of advanced materials crack behavior. When the viscosity effect is omitted or as time tends to infinity, the solutions based on strain gradient viscoelasticity theory converges to those on the classical strain gradient elasticity theory. [ABSTRACT FROM AUTHOR]