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首页> 外文会议>NATO Advanced Research Workshop on Surface Waves in Anisotropic and Laminated Bodies and Defects Detection; 20011030-1102; Moscow(RU) >EXPLICIT SECULAR EQUATIONS FOR SURFACE WAVES IN AN ANISOTROPIC ELASTIC HALF-SPACE FROM RAYLEIGH TO TODAY
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EXPLICIT SECULAR EQUATIONS FOR SURFACE WAVES IN AN ANISOTROPIC ELASTIC HALF-SPACE FROM RAYLEIGH TO TODAY

机译：从Rayleigh到今天，各向异性弹性半空间中的表面波的显式方程组

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An explicit secular equation for surface waves propagating in an elastic half-space x_2 ≥ 0, in the direction of the x_1 -axis, was first obtained by Rayleigh (1885) for isotropic materials. Explicit secular equations were derived by Stoneley (1949) and Alshits and Lothe (1978) for transversely isotropic (hexagonal) materials, by Stoneley (1955) for cubic materials, by Sveklo (1948) and Stoneley (1963) for orthotropic materials, and by Currie (1979), Destrade (2001) and Ting (2002a,b,c) for monoclinic materials with the symmetry plane at x_3 = 0. For monoclinic materials with the symmetry plane at x_1 = 0 or x_2 = 0, explicit secular equations were presented by Ting (2002a,b, 2003). Explicit secular equations for general anisotropic materials were obtained by Taziev (1989) and Ting (2002b, 2003). The secular equations mentioned above employed different derivations. In most cases, the same derivation for a general anisotropic material and for a special anisotropic material has to be carried out separately. We show here that all derivations can be presented using the Stroh (1962) formalism or its modified version (Ting, 2002a,b,c). We also show how the derivations can be improved or made more general. While numerical schemes are available for computing the surface wave speed, an explicit secular equation allows us to analyze the dependence of the surface wave speed on the elastic constants. For instance, for the special case of monoclinic materials with the symmetry plane at x_3 =0, the secular equation is independent of the reduced elastic compliances s′_(16) and s′_(26) when (a) s′_(16) =0, (b) s′_(12) =0 and s′_(16) = 2s′_(26), or (c) s′_(16) = s′_(26) and s′_(12)s′_(66) + s′(1,2) = s′_(16)~2. It is shown that, when s′_(12) =0 (of which (b) is a special case), the secular equation reduces to a quadratic equation so that an exact expression of the surface wave speed is obtained. Other special materials such as (c) are presented for which an exact expression of the surface wave speed can be obtained.
机译：Rayleigh（1885）首次针对各向同性材料，获得了沿x_1轴方向在弹性半空间x_2≥0中传播的表面波的显式长期方程。对于横观各向同性（六边形）材料，Stoneley（1949）和Alshits and Lothe（1978），对于三次方材料，由Stoneley（1955），对于正交异性材料，由Sveklo（1948）和Stoneley（1963），以及Currie（1979），Destrade（2001）和Ting（2002a，b，c）对于x_3 = 0对称平面的单斜材料。对于x_1 = 0或x_2 = 0对称平面的单斜材料，显式世俗方程为由Ting（2002a，b，2003）提出。 Taziev（1989）和Ting（2002b，2003）获得了一般各向异性材料的显式长期方程。上述世俗方程采用不同的推导。在大多数情况下，必须分别对通用各向异性材料和特殊各向异性材料进行相同的推导。我们在这里表明，可以使用Stroh（1962）形式主义或其修改版本（Ting，2002a，b，c）来表示所有推导。我们还将展示如何改进派生方法或使其更通用。虽然可以使用数值方案来计算表面波速度，但显式的世俗方程使我们能够分析表面波速度对弹性常数的依赖性。例如，对于在x_3 = 0处具有对称平面的单斜晶材料的特殊情况，世俗方程与（a）s'_（）时降低的弹性柔量s'_（16）和s'_（26）无关。 16）= 0，（b）s'_（12）= 0且s'_（16）= 2s'_（26），或（c）s'_（16）= s'_（26）和s '_（12）s'_（66）+ s'（1,2）= s'_（16）〜2。结果表明，当s′_（12）＝ 0（其中（b）是特例）时，世俗方程简化为二次方程，从而获得了表面波速的精确表示。提出了诸如（c）的其他特殊材料，针对这些材料可以获得表面波速的精确表示。

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