PAMM · Proc. Appl. Math. Mech. 17, 255 – 256 (2017) / DOI 10.1002/pamm.201710097 Experimental and numerical study of glass fracture using J-integral and phase-field modelling approaches Ziyuan Li 1, * , Yousef Heider 1 , and Bernd Markert 1 1 Institute of General Mechanics, RWTH Aachen University, Templergraben 64, 52062, Aachen, Germany In this contribution, an experimental study and FE simulations based on J-integral theory and the phase-field modelling ap- proaches are presented in order to systematically study the temperature and strain-rate dependency of glass fracture behaviour. First, a series of three-point bending tests are successfully carried out under different stain-rates and temperatures. Secondly, numerical modelling of the bending tests with the introduction of a micro crack yields the stress-strain response, which serves to the calculation of J-integral values, in order to describe the glass fracture resistance in terms of energy. At the end, the critical energy release rate serves as a bridge connecting the J-integral theory with the phase-field modelling, where a dynamic fracture model with crack propagation is realised as a new direction for further researches. c  2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction The problem of fracture of glass materials finds applications in different fields of glass manufacturing processes, such as in hot forming processes like precision glass molding process (PGM). In this, the working temperature is near the glass transition temperature (T g ) and the glass material represents either linear elastic or viscoelastic behaviour, which leads to the temperature and strain-rate dependency of glass fracture behaviour. The fracture of glass is initiated by micro cracks and propagates under tensile stresses. One of the most commonly used experimental methods for the analysis of glass fracture is the 3-point bending test, in which a glass specimen with a rectangular or circular cross section is bent until fracture. To describe the behaviour of glass fracture on a macroscopic scale, the approach based on the J-integral theory [2] is applied considering both linear elastic and visoelastic model assumptions. At the mean time, the crack of the glass specimen at ambient temperature is described as a brittle crack and described using a phase-field modelling (PFM) approach [3, 4, 6]. A combination of the two approaches is realised by the cirtical Griffith energy release rate (G c ) [1], which will be further explained. 2 Mathematical modelling The starting point of the PFM is the Griffith’s energy-based criteria for brittle fracture [1], where the global potential energy function Ψ of a cracked linear elastic, isotropic solid material can be defined as the sum of the elastic strain energy Ψ el and the crack energy Ψ cr integrated over the whole spatial domain. This integration can be achieved by inclusion of the phenomenological phase-field variable φ to distinguish between the cracked (φ =0) and the undamaged (φ =1) states of the material: Ψ = Ψ el + Ψ cr = G 4ǫ (1 - φ) 2 + Gǫ| grad φ| 2 +  (1 - η)(φ) 2 + η  Ψ + el + Ψ - el . (1) Herein, ǫ is the internal length that is related to the width of the transition area between the cracked and the unbroken states and η is the residual stiffness. The fracture energy part is obtained from the non-local quadratic approximation; while the elastic energy part is divided into positive and negative parts, which represent tensile and compressive modes, respectively. We assume a dynamic state for the present problem and neglect the body forces. By applying the Ginzburg-Landau evolution equation [6], the momentum balance equation and the phase-field evolution equation can be expressed as div σ = ρ ˙ v and ˙ φ = ∂φ ∂t = -M ∂ψ ∂φ = -M  2(1 - η)φψ + elast - G c 2ǫ (1 - φ) - 2ǫG c div(grad φ)  , (2) where σ is the stress tensor, ρ(x,t) is the mass density and v(x,t) is the velocity. M ≥ 0 represents a scalar-valued kinetic parameter related to the interface mobility (time dependency). G c is the critical Griffith energy release rate, which represents the critical fracture energy. The further application of the J-integral approach extends the material model to the viscoelastic range. The J-integral was first given by Rice [2] as a path-independent line integral around the crack tip, which takes the form J =  L  Wdx 2 - t · ∂ u ∂x 1 ds  , (3) ∗ Corresponding author: e-mail ziyuan@iam.rwth-aachen.de, phone +49 241 80 90035, fax +49 241 80 92231 c  2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim