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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 >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：从瑞利到今天的各向异性弹性半空空间中表面波的明确等方程

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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：面波在弹性半空间X_2≥0传播，在X_1的方向明确世俗方程轴，首先由瑞利（1885），用于各向同性材料获得。明确的世俗方程由斯通利波（1949）和Alshits和Lothe（1978），用于横向各向同性（六方晶）材料，由斯通利波（1955）为立方材料，通过Sveklo（1948）和斯通利波（1963），用于正交各向异性材料来源的，并通过居里（1979），Destrade（2001）和汀（2002年a，b，C），用于与在X_3 = 0为了与对称平面的单斜材料对称平面的单斜材料在X_1 = 0或X_2 = 0时，显式世俗方程分别为通过汀呈现（2002年a，b，2003）。一般各向异性材料明确世俗方程由Taziev（1989）和汀（2002年b，2003）获得。上述采用的不同的推导中提到世俗方程。在大多数情况下，对于一般的各向异性材料和一个特殊的各向异性材料相同的推导，必须分别进行。我们在这里展示的所有推导可以使用Stroh说（1962年），形式主义或它的修改版本（亭，2002年，B，C）提交。我们还展示了如何推导可以改善或变得更加普遍。虽然数值方案可以用于计算表面波速度，显式特征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，对于与该对称平面的单斜材料中的X_3 = 0的特殊情况下，该特征方程是独立的减小弹性柔S'_（16）和S'_（26）当（a）S'_（ 16）= 0，（b）中S'_（12）= 0和S'_（16）= 2S'_（26），或（c）S'_（16）= S'_（26）和s '_（12）S'_（66）+ S'（1,2）= S'_（16）〜2。它表明，当s'_（12）= 0（其中（B）的是一种特殊的情况下），使得获得的表面波速度的精确表达世俗方程简化为一个二次方程式。其他特殊的材料，如（c）中提出了能够获得的表面波速度的精确表达。

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