Analytical Model with Independent Control of Load–Displacement Curve Branches for Brittle Material Strength Prediction Using Pre-Peak Test Loads

Abstract

The relevance of problems related to the fracturing of engineering materials and structures will not decrease over time. Fracture mechanics methods continue to be developed, which, combined with numerical methods of computer modeling, are implemented in software packages. However, this is only one facet of the complex of actual problems related to modeling and analyzing the behavior of brittle materials. No less important are the problems of developing not only numerical, but also new analytical models. In this paper, analytical models of only one class are considered, the distinguishing feature of which is that they describe the full load–strain curve using only one equation. However, the determination of model parameters requires tests for which the destruction of the test object is necessary, which may be unacceptable if controlled destruction is technically impossible or economically unreasonable. At the same time, in practice, it is possible to obtain values of stresses and strains caused by loads smaller than the peak load. Pre-peak loads can be used to predict strength using numerical methods, but it is desirable to have a suitable analytical model to extend the capabilities and to reduce the cost of applied research. Such a model was not found in the known literature, which motivated this work, which aims to modify the analytical model to predict strength and the full load–displacement (or stress–strain) curve using only pre-peak loading. This study is based on the analysis of known data and synthesis using mathematical modeling and fracture mechanics. The input data for the model do not include the particle size distribution and other physical and mechanical properties of the components of the material under study. These properties may remain unknown, but their influence is taken into account indirectly according to the “black box” methodology. Restrictions of the scope of the model are defined. The simulation results are consistent with experiments known from the literature.

Differential Energy Criterion for Brittle Fracture: Conceptualization and Application to the Analysis of Axial and Lateral Deformation in Uniaxial Compression of Rocks

“…Thus, model (14) and a modification of this model (17) represent two new steps in the progress of models of this class. The universality of models (14)–(17) is explained by the possibility of using them to independently control the pre-peak and post-peak branches of the full load–displacement (stress–strain) curve [ 6 , 32 ]. The parameters and specified in (17) can be integers and fractions.…”

“…Note that according to [ 33 ], the values of the post-peak modulus significantly depend on the stiffness of the test machine and, for example, during uniaxial compression of marble, these values can vary in the range from −550 to −0.49 GPa. To improve the accuracy of modeling the post-peak stage of axial deformation, it is reasonable to use independent control of the parameters of the pre-peak and post-peak branches of the full stress–strain curve [ 32 ]; details of this issue are beyond the scope of this paper.…”

“…The modulus of elasticity ( ) can be determined as shown above in Section 2.8 ; the determination of the parameters and is discussed by example in Section 2.9 . The peak strain and stress ( and ) are determined experimentally, but in some cases these values can be predicted using experimental data only for the pre-peak stage [ 32 , 39 ]. Additional graphical explanations of Equation (34) shown in Figure 7 .…”

Differential Energy Criterion of Brittle Fracture as a Criterion for Wood’s Transition to the Plastic Deformation Stage

“…This paper uses the Blagojevich analytical model [34], which belongs to the class of simple but sufficiently versatile load-displacement models, a review of which can be found in [35]. The application of modifications of this model to the analysis of frozen soils and some rocks is considered in [36,37]. However, when applying this model to the analysis of wood, it is necessary to take into account features that have not been previously considered in sufficient detail in the known literature.…”

“…In this paper, we use the Blagojevich analytical model [34], which belongs to the class of simple but rather universal load-displacement models, a review of which can be found in [35]. Using [36,37], we write the basic equation of the model in terms of load-displacement (𝐹-𝑢):…”