Abstract Currently, the sheet-intrusion paths and geometries, including the sheet opening/thickness as well as the depth to sheet tip, are commonly determined from geodetic surface data using elastic dislocation models. These models assume the volcanic zone/central volcano to be an elastic half space of uniform mechanical properties. Field observations, however, show that volcanic zones/volcanoes are composed of numerous layers whose mechanical properties (primarily Young's modulus) vary widely. Here we provide new numerical models on the effects of a typical variation in Young's modulus in an active volcanic zone/central volcano on the internal and surface stresses and displacements induced by a sheet-intrusion whose tip is arrested at a depth below the surface of 100 m. The sheet has a dip dimension (height) of 2 km. Its opening (thickness) depends on the magmatic overpressure, sheet dimension and host-rock Young's modulus. For the values used here, sheet thickness would be in the range of 0.5–1.4 m, similar to commonly measured sheet thicknesses in the field. The only loading is internal magmatic overpressure in the sheet of 5 MPa. The modelled crustal segment/volcano consists of 5 layers, all with the same Poisson's ratio (0.25). Each of the 4 uppermost layers is 10 m thick. Layer 1 (the top or surface layer) has a Young's modulus of 3 GPa, layer 2 a modulus of 20 GPa, layer 3 a modulus of 30 GPa, and layer or unit 5 a modulus of 40 GPa. We vary the Young's modulus or stiffness of the fourth layer from 10 GPa to 0.01 GPa, while the dip of the sheet takes the following values: 30°, 45°, 60° (for an inclined sheet) and 90° (for a dike). The resulting displacement and stresses are highly asymmetric across the sheet tip (except for the dike), with the main surface stresses and displacements being above the dipping sheet and highest for the 30°-dipping sheet. For comparison, three elastic half-space models of the same sheet configuration and loading but uniform Young's modulus in each model (40GPa, 20GPa, and 10 GPa) all yield much higher surface stresses and displacements than any of the layered models. As the stiffness of layer 4 decreases the surface stresses gradually decrease while changes in vertical displacements are comparatively small but greater in horizontal displacements. In particular, as the stiffness of layer 4 decreases from 10 GPa to 0.01 GPa, for the 30°-dipping sheet the maximum surface shear stress decreases from about 6.6 MPa to 2.2 MPa and the maximum tensile stress from about 6.9 MPa to about 2.3 MPa. Thus, even a single comparatively thin (10 m) soft layer close to the surface of a central volcano/volcanic zone (where such layers are almost universal) may cause a great change in the maximum sheet-induced stresses at the surface and, thereby, in any sheet-induced fracture pattern. Furthermore, the stress peaks in the layered models do not coincide with the displacement peaks; fracture formation is most likely at the location of the stress peaks. The results have important implications for the correct interpretation of geodetic data and fracturing during unrest periods with magma-chamber rupture and sheet injection.

中文翻译：

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倾斜（锥）片引起的层状岩石中的应力和位移

摘要 目前，片材侵入路径和几何形状，包括片材开口/厚度以及到片材尖端的深度，通常是使用弹性位错模型从大地表面数据确定的。这些模型假设火山带/中央火山是具有均匀机械特性的弹性半空间。然而，实地观察表明火山带/火山由许多层组成，其机械特性（主要是杨氏模量）变化很大。在这里，我们提供了新的数值模型，研究了活跃火山区/中央火山中杨氏模量的典型变化对内部和表面应力以及由片状侵入引起的位移的影响，该片状侵入的尖端在 100米。该片材的倾角尺寸（高度）为 2 公里。它的开口（厚度）取决于岩浆超压、薄片尺寸和主岩杨氏模量。对于此处使用的值，板材厚度将在 0.5-1.4 m 的范围内，类似于通常在现场测量的板材厚度。唯一的载荷是片内 5 MPa 的内部岩浆超压。模拟的地壳段/火山由 5 层组成，所有层都具有相同的泊松比 (0.25)。最上层 4 层中的每层厚 10 m。层1（顶层或表面层）的杨氏模量为3GPa，层2的模量为20GPa，层3的模量为30GPa，层或单元5的模量为40GPa。我们将第四层的杨氏模量或刚度从 10 GPa 改变为 0.01 GPa，而板的倾角采用以下值：30°、45°、60°（对于倾斜的板）和 90°（对于堤坝） ）。由此产生的位移和应力在整个片材尖端（堤坝除外）高度不对称，主要表面应力和位移在浸渍片上方，并且在 30° 浸渍片中最高。为了进行比较，每个模型（40GPa、20GPa 和 10GPa）中具有相同片材配置和载荷但杨氏模量均匀的三个弹性半空间模型都产生比任何分层模型都高得多的表面应力和位移。随着第 4 层刚度的降低，表面应力逐渐降低，而垂直位移的变化相对较小，但水平位移的变化更大。特别是，随着层 4 的刚度从 10 GPa 降低到 0.01 GPa，对于 30° 浸渍板，最大表面剪切应力从大约 6.6 MPa 降低到 2。2 MPa 和最大拉伸应力从约 6.9 MPa 到约 2.3 MPa。因此，即使靠近中央火山/火山带（这些层几乎是普遍的）表面的单个相对较薄（10 m）的软层也可能导致表面的最大板材应力发生很大变化，从而, 在任何板材引起的断裂模式中。此外，分层模型中的应力峰值与位移峰值不一致；裂缝形成最有可能出现在应力峰值位置。这些结果对于正确解释大地测量数据和岩浆房破裂和片状注入动荡时期的压裂具有重要意义。即使是靠近中央火山/火山带（这些层几乎是普遍的）表面的单个相对较薄（10 m）的软层也可能导致表面的最大板材应力发生很大变化，因此，任何板材引起的断裂模式。此外，分层模型中的应力峰值与位移峰值不一致；裂缝形成最有可能出现在应力峰值位置。这些结果对于正确解释大地测量数据和岩浆房破裂和片状注入动荡时期的压裂具有重要意义。即使是靠近中央火山/火山带（这些层几乎是普遍的）表面的单个相对较薄（10 m）的软层也可能导致表面的最大板材应力发生很大变化，因此，任何板材引起的断裂模式。此外，分层模型中的应力峰值与位移峰值不一致；裂缝形成最有可能出现在应力峰值位置。这些结果对于正确解释大地测量数据和岩浆房破裂和片状注入动荡时期的压裂具有重要意义。分层模型中的应力峰值与位移峰值不一致；裂缝形成最有可能出现在应力峰值位置。这些结果对于正确解释大地测量数据和岩浆房破裂和片状注入动荡时期的压裂具有重要意义。分层模型中的应力峰值与位移峰值不一致；裂缝形成最有可能出现在应力峰值位置。这些结果对于正确解释大地测量数据和岩浆房破裂和片状注入动荡时期的压裂具有重要意义。