Stability of the Solidified Magma Region for Direct Utilization of Magma Energy

从固体力学的观点出发，研究了直接从熔融岩浆中提取能量的凝固区的稳定性。首先，针对单轴应力-蠕变关系，采用诺顿定律描述蠕变应变对应力的依赖关系，研究了凝固岩浆的本构关系。关于蠕变应变对时间的依赖性，如通常在第二蠕变阶段中使用的那样，将蠕变应变设置为相对于时间是线性的，并且引入阿耳茨纽斯定律来表示蠕变应变的温度依赖性。基于单轴应力-蠕变应变关系，采用J_2流动理论，利用偏应力第二不变量蠕变势建立了本构方程。介绍了花岗岩、斜长岩和橄榄石三种岩石。至于温度条件，考虑以下两种情况。第一种情况是熔融岩浆流的对流非常弱，因此凝固区域的大小主要由固相线温度决定。第二种情况是，岩浆的对流足够强，使得凝固的岩浆的大小几乎由从凝固区域外部提取的热量与对流岩浆提供的热量之间的热平衡决定。基于上述本构方程和温度场，分析了两种情况下的温度分布，阐明了凝固区的应力分布和变形，揭示了凝固区内裂纹网络的形成。结果表明，从裂隙网络形成的角度来看，橄榄石是两种情况下最有潜力的候选者，其次是斜长岩。花岗岩不适合直接提取岩浆能量。

Stability of the solidified region for the direct extraction of energy from molten magma was examined from the view point of solid mechanics. Firstly, the constitutive law of the solidified magma was investigated, by using the Norton law to express the dependency of creep strain on stress, for the uni-axial stress-creep strain relation. With regard to the dependency of creep strain on time, the creep strain was set to be linear with respect to time as usually used in the secondary creep stage and the Arrhenius law is introduced to express the temperature dependency of creep strain. Based on the uni-axial stress-creep strain relation, the constitutive equation was constructed by employing the J_2 flow theory which utilized the second invariant of deviatric stress the creep potential. Three kinds of rock, i.e. granite, anorthosite and olivine, were introduced. As for the temperature condition, following two cases was considered. The first case is that convection of molten magma flow is very weak and thus the size of the solidified region is primarily determined by the solidus temperature. The second case is that the convection of magma is sufficiently strong so that the size of the solidified magma is almost determined by the heat balance between the heat extracted and the heat supplied by the convective magma from the outside of the solidified region. The temperature distribution was analyzed for the both cases and the stress distribution and the deformation were clarified, shedding light on the formation of fracture network within the solidified region, based on the constitutive equations and the temperature fields mentioned above. It was revealed that, for both of the two cases, olivine was the most potential candidate from the view point of formation of fracture network and anorthosite came next. Granite would not be appropriate for the direct extraction of magma energy.