Analytic solutions for effective elastic moduli of isotropic solids containing oblate spheroid pores with critical porosity

IF 1.8 3区 地球科学 Q3 GEOCHEMISTRY & GEOPHYSICS
Zhaoyun Zong, Fubin Chen, Xingyao Yin, Reza Rezaee, Théo Le Gallais
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Abstract

Accurate characterization for effective elastic moduli of porous solids is crucial for better understanding their mechanical behaviour and wave propagation, which has found many applications in the fields of engineering, rock physics and exploration geophysics. We choose the spheroids with different aspect ratios to describe the various pore geometries in porous solids. The approximate equations for compressibility and shear compliance of spheroid pores and differential effective medium theory constrained by critical porosity are used to derive the asymptotic solutions for effective elastic moduli of the solids containing randomly oriented spheroids. The critical porosity in the new asymptotic solutions can be flexibly adjusted according to the elastic moduli – porosity relation of a real solid, thus extending the application of classic David-Zimmerman model because it simply assumes the critical porosity is one. The asymptotic solutions are valid for the solids containing crack-like oblate spheroids with aspect ratio α $\alpha $ < 0.3, nearly spherical pores (0.7 <  α $\alpha $ < 1.3) and needle-like prolate pores with α $\alpha $  > 3, instead of just valid in the limiting cases, for example perfectly spherical pores ( α $\alpha $ = 1) and infinite thin cracks ( α $\alpha \ \to $ 0). The modelling results also show that the accuracies of asymptotic solutions are weakly affected by the critical porosity ϕ c ${{\phi }_{\mathrm{c}}}$ and grain Poisson's ratio v 0 ${{v}_0}$ , although the elastic moduli have appreciable dependency of ϕ c ${{\phi }_{\mathrm{c}}}$ and v 0 ${{v}_0}$ . We then use the approximate equations for pore compressibility and shear compliance as inputs into the Mori–Tanaka and Kuster–Toksoz theories and compare their calculations to our results from differential effective medium theory. By comparing the published laboratory measurements with modelled results, we validate our asymptotic solutions for effective elastic moduli.

包含具有临界孔隙率的扁球形孔隙的各向同性固体有效弹性模量的解析解
准确表征多孔固体的有效弹性模量对于更好地理解其力学行为和波的传播至关重要,这在工程、岩石物理和勘探地球物理领域有很多应用。我们选择不同长宽比的球面来描述多孔固体中的各种孔隙几何形状。利用球形孔隙的可压缩性和剪切顺应性近似方程以及临界孔隙度约束的微分有效介质理论,推导出含有随机取向球形体的固体的有效弹性模量渐近解。新渐近解中的临界孔隙率可根据实际固体的弹性模量-孔隙率关系灵活调整,从而扩展了经典戴维-齐默尔曼模型的应用范围,因为该模型只是假设临界孔隙率为一。渐近解适用于包含长宽比为 α $\alpha $ < 0.3 的裂缝状扁球体、近似球形孔隙(0.7 < α $\alpha $ < 1.3)和针状突起孔隙(α $\alpha $ >3),而不只是在极限情况下有效,例如完全球形孔隙(α $\alpha $ = 1)和无限细裂缝(α → $\alpha \ \to $ 0)。建模结果还表明,尽管弹性模量与临界孔隙率 ϕ c ${{\phi }_{mathrm{c}}$ 和晶粒泊松比 v 0 ${{v}_0}$ 有明显的相关性,但渐近解的精确度受临界孔隙率 ϕ c ${{\phi }_{mathrm{c}}$ 和晶粒泊松比 v 0 ${{v}_0}$ 的影响较小。然后,我们将孔隙可压缩性和剪切顺应性的近似方程作为 Mori-Tanaka 和 Kuster-Toksoz 理论的输入,并将它们的计算结果与我们从微分有效介质理论得出的结果进行比较。通过比较已公布的实验室测量结果与建模结果,我们验证了有效弹性模量的渐近解。
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来源期刊
Geophysical Prospecting
Geophysical Prospecting 地学-地球化学与地球物理
CiteScore
4.90
自引率
11.50%
发文量
118
审稿时长
4.5 months
期刊介绍: Geophysical Prospecting publishes the best in primary research on the science of geophysics as it applies to the exploration, evaluation and extraction of earth resources. Drawing heavily on contributions from researchers in the oil and mineral exploration industries, the journal has a very practical slant. Although the journal provides a valuable forum for communication among workers in these fields, it is also ideally suited to researchers in academic geophysics.
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