PSI - Issue 13

Stefan Reich et al. / Procedia Structural Integrity 13 (2018) 28–33 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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2. Equilibrium of brittle fracture 2.1. Energetic approach of brittle materials

Different theories were developed to describe the interaction, the development and the propagation of microscopic and macroscopic flaws of cracks, the mechanism of dislocations and the geometry of the material. One of the most known theory is the energy concept of Griffith (1921), later on developed by Irwin (Kerkhoff (1970)). According to his approach the potential energy U pot of a system is the sum of the mechanical energy U M and the surface energy U  formed by the propagation of a crack. U pot = U M - U γ (1) where U pot U M U  the total potential energy of the system mechanical energy the increase of the elastic-surface energy caused by the formation of the crack surfaces The mechanical energy consists of the inner elastic strain energy U 0 (including the outer strain applied by a force displacement) stored in the elastic material glass and the potential energy of the external load (external work), that is applied on a glass. Eq. 1 can be expanded using these energies. where U 0 the elastic strain energy of the uncracked pane, U E the decrease of the elastic energy caused by introducing the crack in the plate, W E the external work.With the assumption of a fracture near equilibrium, that means the full transfer of elastic energy into crack creation and no stored strain energy in an annealed glass eq. (2) simplifies to U 0 +W E = U γ (3) which means that all internal elastic strain energy U 0 and all external load W E is transferred into surface energy. The elastic strain energy U 0 and the deformation work WE contain all energy that is stored in or applied on the pane and usable for crack creation and crack growth. Elastic strain energy is typically in the glass stored as mechanically or thermally (thermal tempering) applied load. A load applied on a linear elastic body proportionally deforms this body to the load. The total elastic strain energy of a body is obtained by integrating the strain energy density χ(x,y,z) over the body volume. This is stored in the body as an equal amount of energy, the elastic stra in energy (1 st law of thermodynamics). A load applied on a linear elastic body proportionally deforms this body to the load. The equations (4) and (5) are valid for materials under Hooke’s law. χ x,y,z = 1 2 εσ (4) U E = ∫ χ ( x,y,z ) dxdydz = 1 2 ∑ σ ij ε ij Δ V (5) where  x,y,z strain energy density. Breaking a body into two or more parts requires the degrading of (ionic, atomic or molecular) bonds between the material components. Therefore, the force which is creating the bond has to perform a negative material specific work. This work is the surface energy necessary to create new surfaces. A crack is forming generally two surfaces, so that the surface energy U  for one crack is the product of the double specific surface energy  s , the thickness of the glass t and the crack length x: U γ = 2  s xt (6) where  s specific surface energy, t glass thickness, x crack length The material property  s is the surface energy required per unit of crack area. Several authors let the material property be constant, but suggest different values. Table 1 summarizes experimental results (  s = 1.7 to 11 Nm/m²) that significantly differ to the theoretical values (  s = 0.3 Nm/m²). U pot = U 0 - U E +W E - U γ (2)