Although many aspects of the fracturing process of concrete are now well understood and successfully simulated with various models, it is still very difficult to properly simulate the different failure mechanisms observed in a concrete structure induced by ballistic impact. In this paper, an enhanced version of the effective-rate-dependent nonlocal damage model [Eng. Fracture Mechanics, 176 (2017)] is proposed to simulate the response of concrete in such events. Hydrostatic damage has been added to the formulation in order to take the damage of the material matrix observed while porosity reduces during compaction into account. Besides controlling the evolution of the nonlinear volumetric response of the material, this new damage variable contributes to the deterioration of the material stiffness upon confinement. It is demonstrated that the description of the nonlinear volumetric response of concrete by an equation of state (EOS) as a plasticity phenomenon, as it is commonly done in hydrodynamic constitutive modeling, is unrealistic for concrete. Such formulations fail to represent the effect of the loss of cohesion observed during compaction on the deviatoric response of the material. By taking this phenomenon into consideration, the proposed model systematically predicts the relevant failure modes (cratering, tunneling, radial cracking and spalling) observed during ballistic impact on a concrete plate as a function of the projectile velocity and plate thickness.

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This contribution presents a numerical study towards the propagation and branching of cracks in quasi-brittle materials, using a new effective rate-dependent damage model, enhanced by a stress-based nonlocal (SBNL) regularization scheme. This phenomenological model is mesh objective and reproduces the major phenomena associated with crack propagation and branching in quasi-brittle materials. It is discussed and demonstrated that the branching phenomenon is not controlled by a specific, material dependent, crack speed. Instead, it is governed by the evolution of the principal stresses at the crack tip, which are controlled by the evolution of damage. It is demonstrated that, with increasing crack speeds, the principal stresses at the crack tip tend to evolve from a mode-I to a mixed-mode state. Beyond a certain (critical) crack speed, the stress distribution around the crack tip reaches a critical state at which a single crack is no longer stable. When this condition is met, crack branching occurs whenever the stress field at the crack tip is destabilized by either a physical discontinuity or an interfering stress wave reflected at the specimen boundaries.

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From a macroscopic point of view, the dynamic tensile response of concrete is mainly due to the viscous behavior of the bulk material and inertia effects at multi-scale levels. It has been suggested that almost all of these mechanisms have to be considered as intrinsic material properties and explicitly included in the constitutive relations of the continuum model. It is discussed and demonstrated that the use of semi-empirical dynamic strength increase factor (DIF) functions to numerically describe rate effects do not characterize true constitutive relations of the material. A strain-rate dependent formulation is used to describe the strength and fracture energy increase of concrete under dynamic tensile loading conditions. However, instead of the commonly used ε̇ (instantaneous strain-rate) to update the constitutive law, an effective rate (R) is considered. With this new concept a time scale is introduced in the constitutive law which restrains the ‘evolution of rate’, to represent the inherent dynamic properties of concrete. This has a weak regularization effect and acts as a localization limiter. Mesh objectivity is recovered with the addition of a material length scale to the constitutive relations, here accomplished by an explicit stress-based nonlocal regularization scheme. Two sets of modified split Hopkinson bar tests are simulated for validation, using respectively notched and un-notched specimens. The results are objective and in good agreement with the experiments.

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Extraordinary actions such as blast loadings and high velocity impact are rare, but usually have devastating effects. Thus, making critical infrastructures, such as military and governmental facilities, power-plants, dams, bridges, hospitals, etc., more resilient against these hazards is one of the best ways to protect ourselves and our societies. Since concrete is a very common construction material, the development of realistic numerical tools to efficiently simulate its failure behavior under extreme dynamic loading conditions is of paramount importance, but still a major challenge. This thesis presents a new stress-based nonlocal effective rate-dependent damage model, developed to simulate the dynamic response and failure of concrete during ballistic impact. The proposed isotropic damage formulation combines the effect of three damage modes: (i ) tension (mode I), (i i ) compressive-shear (mode II and mixed-mode) and (i i i ) hydrostatic damage to describe crushing of the cement matrix under pressure. The strain-rate dependent update of the constitutive relations to express the dynamic increase of strength and fracture energy in tension and compression is made a function of an effective rate, instead of the commonly used instantaneous strain rate. An enhanced version of the stress-based nonlocal regularization scheme is used to correct spurious mesh sensitivity. The proposedmodel was developed solely in the effective strain-space, following an entirely explicit computation scheme.@en

The development of realistic numerical tools to efficiently model the response of concrete structures subjected to close-in detonations and high velocity impact has been one of the major quests in defense research. Under these loading conditions, quasi-brittle materials undergo a multitude of failure (damage) mechanisms. Dynamic tensile failure (e.g. spalling), characterized by a significant strength increase associated with loading rate, has revealed to be particularly challenging to represent. In this contribution, a rate-dependent stress-based nonlocal damage model has been introduced for the simulation of dynamic tensile failure of quasi-brittle materials. The recently proposed stress-based nonlocal criterion has been updated in order to be consistently combined with a rate-dependent version of the well-known Mazars damage model. The model was implemented in LS-DYNA using a fully explicit computational scheme. Two sets of numerical examples have been presented. First, one-dimensional numerical analyses were conducted to evaluate the model capabilities, applicability and limitations. Second, the model has been validated against experimental results. It has been shown that the proposed model, in addition to correcting spurious mesh sensitivity, also provides a more realistic representation of damage initiation and growth, in particular around discontinuities (notches and free boundaries) and damaged areas.

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Stress-based nonlocal model, Damage, Rate dependency, Dynamic crack-branching Abstract. In concrete often complex fracture and fragmentation patterns develop when subjected to high straining loads. The proper simulation of the dynamic cracking process in concrete is crucial for good predictions of the residual bearing capacity of structures in the risk of being exposed to extraordinary events like explosions, high velocity impacts or earthquakes. As it is well known, concrete is a highly rate dependent material. Experimental and numerical studies indicate that the evolution of damage is governed by complex phenomena taking place simultaneously at different material scales, i.e. micro, meso and macro-scales. Therefore, the constitutive law, and its rate dependency, must be adjusted to the level of representation. For a proper phenomenological (macroscopic) representation of the reality, the constitutive law has to explicitly describe all phenomena taking place at the lower material scales. Macro-scale inertia effects are implicitly simulated by the equation of motion. In the current paper, dynamic crack propagation and branching is studied with a new rate-dependent stress-based nonlocal damage model. The definition of rate in the constitutive law is changed to account for the inherent meso-scale structural inertia effects. This is accomplished by a new concept of effective rate which governs the dynamic delayed response of the material to variations of the deformation (strain) rate, usually described as micro-inertia effects. The proposed model realistically simulates dynamic crack propagation and crack branching phenomena in concrete. @en