拉梅常数的力学意义与剪切模量出现于纵波速度公式的原因

Note on the Lamé constant and the reason for the presence of the shear modulus in the compressional wave speed formula

弹性纵波俗称为膨胀波,为什么其速度公式中出现剪切模量?本文证明了平面纵波和球面纵波均在引起体应变的同时引起偏应变.由于体应变和偏应变同时发生,纵波速度公式就既含反映体积恢复能力的体积模量K、又含反映形状恢复能力的剪切模量μ.纵波速度也常用剪切模量和拉梅常数λ表示,后者表示一维应变状态下横向应力与纵向应变的比值,而出现于纵波速度公式中的λ+2μ是发生一维应变时应变方向上应力与应变的比值.这个值用体积模量与剪切模量表达,就是K+(4/3)μ.

Both plane and spherical compressional waves are shown to cause deviatoric strain as well as volumetric strain. As a result,the compressional wave speed depends on both the bulk modulus associated to the volumetric strain and the shear modulus associated to the deviatoric strain. The compressional wave speed can be alternatively expressed in terms of the shear modulus and the Laméconstant λ,the later being ratio of the transverse normal stress to the only nonzero linear strain in the longitudinal direction under one-dimensional strain state. And the combined modulus λ ＋ 2μ is the ratio of the longitudinal normal stress to the nonzero linear strain in the longitudinal direction. That modulus is equal to K ＋（ 4/3） μwhere when it is expressed in terms of the bulk modulus K and the shear modulus.