Reducing the linear system of two first order equilibrium equations involving normal stress σ(ρ,θ) and shearing stress v(ρ,θ), by elimination, to two decoupled second order equations in σ and v, ...Reducing the linear system of two first order equilibrium equations involving normal stress σ(ρ,θ) and shearing stress v(ρ,θ), by elimination, to two decoupled second order equations in σ and v, we find that, for pressure only case, v(ρ,θ) vanishes in the half space. Consequently, the second order equation in σ can be simplified. In the language of linear system analysis, the medium(system) function, characterizing the mechanical behavior of a particulate medium in pressure only case, is obtained from the simplified second order equation ( 2 ρ+ 2 θ)σ(ρ,θ)=0 and can be inverted to give impulse reponse explicitly. Thus, response σ α(ρ,θ) may be computed directly from input, i.e., the surface pressure φ α(ρ) , by integration. Some explicit formulas for transmission problems, including response to input of strip linearly increasing pressure, are given in the paper.展开更多
文摘Reducing the linear system of two first order equilibrium equations involving normal stress σ(ρ,θ) and shearing stress v(ρ,θ), by elimination, to two decoupled second order equations in σ and v, we find that, for pressure only case, v(ρ,θ) vanishes in the half space. Consequently, the second order equation in σ can be simplified. In the language of linear system analysis, the medium(system) function, characterizing the mechanical behavior of a particulate medium in pressure only case, is obtained from the simplified second order equation ( 2 ρ+ 2 θ)σ(ρ,θ)=0 and can be inverted to give impulse reponse explicitly. Thus, response σ α(ρ,θ) may be computed directly from input, i.e., the surface pressure φ α(ρ) , by integration. Some explicit formulas for transmission problems, including response to input of strip linearly increasing pressure, are given in the paper.
