Analysis of Concrete Failure on the Descending Branch of the Load-Displacement Curve

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## 2. Materials and Methods

#### 2.1. Mechanical Model: A Brief Description

• Concrete is viewed as a structure composed of interacting mesoscale elements.
• The material of each element obeys Hooke’s law.
• The modulus of elasticity, strength and other physical and mechanical properties of the material of each element do not depend on its size and do not change over time.
• With an increase in the external load, and hence displacement, individual mesoscale elements are destroyed, as a result of which the effective area decreases, and the load is redistributed to the elements that remain intact. As a result, the average statistical value of the effective stresses in the material of the remaining intact mesoscale elements increases.
• The destruction of mesoscale elements and their conglomerates leads to a decrease in the effective area and a decrease in the resistance of the macrostructure to external force, which corresponds to the descending branch of the “load – displacement” diagram. However, effective stresses (i.e., stresses in the material of mesoscale elements) increase. The growth of effective stresses is limited by the ultimate strength of the material of mesoscale elements.
• Stresses determined without taking damage into account can be called apparent stresses [12].
• The Poisson effect can cause some growth in the transverse dimensions and a corresponding change in the cross-sectional area of the sample under uniaxial compression. Thus, two trends should be analyzed: first, a decrease in cross-sectional area due to destruction of mesoscale elements and, second, an increase in area due to the Poisson effect.
• The primary source of information for the mathematical description of the model and obtaining numerical results is the load-displacement diagram (Figure 1).

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 Number of Samples: 1 2 3 4 5 6 7 8 9 Fiber volume, % 0.00 0.50 0.75 1.00 0.00 0.50 0.75 1.00 1.25 , MPa [22] 28.19 29.34 29.94 30.87 54.65 54.86 57.94 59.82 56.91 , mε; [22] (1 mε = 0.001) 1.950 2.657 2.931 2.954 2.050 3.08 3.000 3.080 3.080 , MPa, (13) (if = ) 76.63 79.75 81.39 83.91 148.55 149.12 157.50 162.61 154.70 , MPa; (8) 39,297 30,017 27,767 28,407 72,465 48,417 52,499 52,795 50,226 , MPa; [22] 25,260 25,090 25,900 25,990 45,210 46,570 47,160 47,400 46,540 1, MPa; [18] 31,515 24,073 22,269 22,782 58,116 38,829 42,103 42,340 40,280