Аннотация:Summary. The results of experimental studies of irreversible deformation of heterogeneous materials that contain micro-cracks, pores, inclusions, and other structural heterogeneities are analysed. The dependencies of deformation properties of the materials on loading conditions are examined and common mechanisms and some features of their behaviour are specified. The constitutive relations for elastoplastic materials, which properties are susceptible to the stress state types are proposed. The features of irreversible bulk deformation are studied and the conditions for unique solution of boundary value problem are formulated. The applicability of proposed constitutive relations is demonstrated on the solution of plane stress crack problem for materials under consideration.
MECHANICAL PROPERTIES OF HETEROGENEOUS METERIALS
For heterogeneous materials there is no single curve of the dependence between von Mises equivalent stress and equivalent strain. The equivalent stress-strain curves depend on loading conditions [1]. These effects are revealed in structural graphite materials, rocks, concrete, refractory ceramics, cast iron, some composite materials and others. The stress state type can be defined by the parameter ξ=σ/σ_0, where σ=1/3 σ_ii is the hydrostatic component of the stresses and σ_0 is von Mises equivalent stress, σ_0=√(3/2 S_ij S_ij ), S_ij=σ_ij-σδ_ij. This parameter determines on the average the ratio of normal stresses to shear stresses in solids and it is named stress triaxiality. The mechanical properties of media under consideration can be illustrated by experimental data for rocks. Diagrams of the dependence between effective stress σ_0 [MPa] and effective strain ε_0=√(2/3 e_ij e_ij ), where e_ij=ε_ij-1/3 εδ_ij, ε=ε_ii is bulk strain, obtained under proportional loading of cylindrical specimens of white marble are shown in Fig. 1. The experiments were carried out under axial compression and lateral pressure with different ratios of axial stress to lateral pressure. Curve 1 corresponds to uniaxial compression, ξ=-0.333, and curves 2–7 correspond to the following values of parameter ξ: −0.407 (2), −0.464 (3), −0.55 (4), −0.63 (5), −0.79 (6), −1.39 (7). The samples of white marble had the density 2.71 g/cm3 and the porosity 0.92%. The relation between the hydrostatic component of stresses and bulk strain for the same types of proportional loading are shown in Fig. 2. Instead of a single curve there is a set of curves σ_0∼ε_0. It means that the shear and bulk properties of this material are interrelated. From Fig. 2 one can see that under conditions of compressive stresses, the dilatation of a material is possible, so that the hydrostatic stress and the bulk strain may have different signs, although under the uniform triaxial compression, the material exhibits a linear dependence of the bulk strain on the hydrostatic stress.
Fig. 1: Equivalent stress-strain curves for white marble under different conditions of proportional loading. Fig. 2: Relations between hydrostatic component of stresses and bulk strain for white marble. Fig. 3: Equivalent stress-strain curves for cast iron
Similar equivalent stress – strain curves can be obtained for the cast iron SCh 15-32. Curves 1 and 6 correspond to the uniaxial tension (ξ=0.333) and the uniaxial compression (ξ=-0.333), curve 3 corresponds to the simple shear (ξ=0), and curves 2, 4, and 5 correspond to the proportional loading with the values of parameter ξ equal to 0.232, 0.064, and 0.126, respectively. These diagrams are essentially nonlinear. The volume residual change can exceed 0.6%, which is commensurable with the shear strains.
CONSTITUTIVE RELATIONS
The constitutive equations for the media under consideration can be formulated on the base of the deformation theory of plasticity using the complementary potential energy function, that can be represented in the form Φ=1/2 (A+Bξ^2 ) σ_0^2+[1+κ(ξ)]g(σ_0 ). For the power law function g_0 (σ_0 )=kσ_0^n/n the generalized stress-strain relations are
ε_ij=3/2 [A+λ(ξ)kσ_0^(n-2) ] S_ij+1/3 [B+Λ(ξ)kσ_0^(n-2) ]σδ_ij,
λ(ξ)=1+κ(ξ)-κ^' (ξ)ξ/n, Λ(ξ)=κ'(ξ)/(ξn ).
The equivalent bulk strain and bulk strain are determined by the equations
ε_0=[A+λ(ξ)kσ_0^(n-2) ] σ_0, ε=[B+Λ(ξ)kσ_0^(n-2) ]σ.
The bulk strain and the equivalent strain are interrelated, and this relationship depends on the type of stress state in a medium. A quite satisfactory correspondence between theoretical stress - strain curves shown in Fig. 1 and Fig. 2 by solid lines and experimental values of stresses and strains marked by points corresponding to different types of loading is observed. The approximations for material functions are suggested. The conditions for the uniqueness of solution of boundary value problem are the following:
A+[(n-1)λ(ξ)-ξλ^' (ξ)]kσ_0^(n-2)>0,
B+[Λ(ξ)+ξΛ^' (ξ)]kσ_0^(n-2)-[λ^' (ξ)kσ_0^(n-2) ]^2/(A+[(n-1)λ(ξ)-ξλ^' (ξ)]kσ_0^(n-2) )>0,
1/2 (A+Bξ^2 ) σ_0^2+[λ(ξ)+ξ^2 Λ(ξ)]kσ_0^n/n>0.
The crack problem under remote tensile stress normal to the crack plane is considered to study the influence of features of material’s behavior on stress and strain fields near a crack tip and conditions for crack growth.
CONCLUSIONS
On the base of analysis of results of experimental studies of different heterogeneous materials, the constitutive relations are proposed to describe the stress state susceptibility of materials properties, materials dilatation and other features of materials behaviour. Quite satisfactory correspondence between experimental data and theoretical dependencies is demonstrated. For specific material functions, the distributions of stresses, strains, and displacements near the crack tip are studied. The conditions of the crack growth initiation are determined with the use of the invariant integral. It is shown that the amplitude of singularity for the material with stress-state dependent properties is substantially lower than the corresponding value obtained for the incompressible material.
References
[1] Lomakin E.V. Constitutive models of mechanical behavior of media with stress state dependent properties. In: Altenbach H., Maugin G.A., Erofeev V. (ed) Mechanics of generalized continua, Advanced Structured Materials, 7:339-350. Heidelberg: Springer, 2011.