PSI - Issue 28

N.V. Mikhailova et al. / Procedia Structural Integrity 28 (2020) 2026–2031 / Structural Integrity Procedia 00 (2020) 000–000

2027

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the spall experiment on the rod. Therefore, this type of experimental scheme was used in the modelling. Also, the calculation of stress was carried out using the solution of an elastic wave problem, therefore and concrete was used as a material. The modelling results were verified by experimental data from work (Schuler et al., 2006) where tests were carried out on concrete rods and the places of spallation were recorded. Nomenclature

w displacement stress c speed of sound density impulse amplitude �� � impulse form length of the sample incubation time � static tensile strength ∗ time of fracture criterion fulfilment ∗ coordinate of spall section 2. Model of spall fracture

In modeling the stress state in the sample, the linear elastic behavior of the material is assumed. The experimental scheme of spall fracture allows using a one-dimensional formulation of the problem. Then the displacements of the particles in the material � � � �� � � satisfy the wave equation with the initial conditions in displacements (2) and boundary conditions in stress (3), where � � � � � � � . � � � � � �� � � � � � �� � � � � , (1) ��� � � , ��� � � �� � , (2) ��� � � , � � � � ��� � � . (3) The solution of this wave problem allows to determine the loading history at any section � : � � � �� � � � � � � � � � � � � � �� (4) Spall fracture occurs under the action of tensile stress. Fracture conditions are determined by the incubation time criterion. For spalling the fracture condition are determined by static strength and incubation time: � � ∗ � ∗ �� � �� � � (5) By substituting (4) into (5), the time of fulfillment of the criterion t* for each section can be calculated. Fracture occurs in the section where the time t* is minimal. After the spall formation two parts of the sample are formed where elastic waves continue to propagate. Further action of waves on the material can lead to spallation again. To calculate the fracture region, it is necessary to determine stress-time dependence in each section considering the wave movement in parts of sample.