PSI - Issue 13

Josef Květoň et al. / Procedia Structural Integrity 13 (2018) 1367 – 1372 Josef Kveˇtonˇ / Structural Integrity Procedia 00 (2018) 000–000

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e ff ect of inertia at the lower scale as well as other phenomena playing role at a high loading rate, such as viscosity of free water, must be captured phenomenologically using rate dependent constitutive law. Heterogeneous composites like concrete exhibit high spatial fluctuation of material properties. Various attempts to capture this e ff ect are proposed in literature. Modeling material properties as independent random realizations at every location has been used e.g. in van Mier (2013). More realistic approaches take into account also dependency of random properties at certain point on surrounding material, which results in imposition of mutual correlation between them. The later approach is used within this contribution by applying random field on fracture material properties Elia´sˇ et al. (2015). The contribution presents simulations of experimental series published in Ozˇbolt et al. (2015). E ff ect of fracture material properties on model response and necessity of strain rate dependent constitutive law as well as influence of spatial fluctuations of material properties are addressed.

2. Computational framework

2.1. Discrete model - geometry, kinematics and statics

Discrete particle model is used within this contribution. It is a simplified version of model from Cusatis and Cedolin (2007). The material is represented by a system of interconnected particles with three translational and three rotational degrees of freedom. Geometry of the particles comes from Voronoi tessellation generated on a set of points randomly placed in a volume domain within a prescribed minimum distance, which dictates the length scale of the model. In case of 3D model, Voronoi cells are convex polyhedrons that represent larger concrete aggregates with surrounding cement matrix, smaller aggregates are omitted. The particles are assumed to be ideally rigid and their interaction is prescribed at their contacting facets. The kinematics of the model is provided by rigid-body motion of the particles that results in displacement jump between them. This displacement jump, divided by contact length and projected into local coordinate system, repre sents contact strain vector e . Damage-based constitutive equation provides relation between strain and stress vectors at the meso-scale

s eq E 0 e eq

s = (1 − D ) E 0 α e

(1)

D = 1 −

where s is the stress vector, E 0 is the meso-scale elastic modulus, α stands for diagonal matrix with first diagonal element equal to 1 (normal direction) and the second and third diagonal element is equal to tangential / normal sti ff ness ratio α (two shear directions). D is damage parameter (between 0 and 1) that describes the material integrity. It is calculated in an equivalent space from values of equivalent stress s eq and strain e eq . Evolution of s eq , described in Eq. (3), is dependent on the straining direction which reflects the complexity of combination of normal and tangential loading of each contact and several material parameters. Two governing parameters for material in nonlinear regime are tensile strength f t and fracture energy in tension G t , the remaining parameters are derived from them according to the recommendations from Cusatis and Cedolin (2007).

2.2. Balance equation - time integration

The calculations are performed in dynamic regime, the time dependent response is obtained from the solution of equations of motion M ¨ u + C ˙ u + Ku = F , Where M , C and K stay for mass, damping and sti ff ness matrix respectively, F is a loading vector and u is vector of unknown displacements and rotations, dotted symbols represent first and second time derivative – accelerations and velocities respectively. Implicit Newmark method is utilized Bathe (1996), therefore the stability of solution is not dependent on time step length. The system is in nonlinear regime damped by dissipation of energy, damping matrix C is therefore neglected. Assuming numerical time-derivatives of displacement,