Constitutive Relation of Yunjialing Anthracite Under Medium Strain Rate

By means of the split Hopkinson pressure bar （SHPB） testing system, this paper presents a dynamic constitu- tive relation of anthracite at a strain rate of ε =5-85s^-1. Generally, the dynamic stress-strain curve for this kind of anthracite under uni-axial compression has the following four stages： a non-linear loading stage, a plastic yielding stage, a strain-strengthening stage and an unloading breakage stage. Correspondingly, the initial elastic modulus Eb, the yielding strength σs and the ultimate strength σb increase along with an increasing strain rate. The time-dependent elasticity was identified when we analyzed the mechanical properties of anthracite. Based on characteristics of measured dynamic stress-strain curves and an analysis of existing rock dynamic constitutive models, as well as a preparatory simulation, a new visco-elastic damage model has been introduced in this paper. A linear spring is put parallel to two Maxwell units with different relaxation times to express two distinct plastic flows. The damage D is equal to Eb- E（εi）]/Eb, where Eb is the beginning modulus and the E（εi） is the slope of a connected line between the origin point and any other point on a tested stress-strain curve. In the new constitutive model, one Maxwell unit with low relaxation time φ is used to describe the response of anthracite to a low strain rate, while the other, with a high relaxation time φ describes the response of anthracite to a high strain rate. Simulated stress-strain curves from the new model are consistent with the measured curves.

Constitutive Relation of Yunjialing Anthracite Under Medium Strain Rate

By means of the split Hopkinson pressure bar (SHPB) testing system, this paper presents a dynamic constitutive relation of anthracite at a strain rate of ε =5-85s-1. Generally, the dynamic stress-strain curve for this kind of anthracite under uni-axial compression has the following four stages: a non-linear loading stage, a plastic yielding stage, a strain-strengthening stage and an unloading breakage stage. Correspondingly, the initial elastic modulus Eb, the yielding strength σs and the ultimate strength σb increase along with an increasing strain rate. The time-dependent elasticity was identified when we analyzed the mechanical properties of anthracite. Based on characteristics of measured dynamic stress-strain curves and an analysis of existing rock dynamic constitutive models, as well as a preparatory simulation, a new visco-elastic damage model has been introduced in this paper. A linear spring is put parallel to two Maxwell units with different relaxation times to express two distinct plastic flows. The damage D is equal to Eb- E(εi)]/Eb, where Eb is the beginning modulus and the E(εi) is the slope of a connected line between the origin point and any other point on a tested stress-strain curve. In the new constitutive model, one Maxwell unit with low relaxation time (ψ)1 is used to describe the response of anthracite to a low strain rate, while the other, with a high relaxation time (ψ)2 describes the response of anthracite to a high strain rate. Simulated stress-strain curves from the new model are consistent with the measured curves.