基于弹塑性本构理论的岩石相变折曲研究
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摘要
岩石折曲带在自然界广泛存在,是一种重要的地质构造,许多学者做过岩石折曲的实验研究,但没有从理论上进行深入的分析,因此以前一直没有合适的理论模型来揭示折曲带的形成机理和对折曲现象进行定量预测。本文将岩石折曲视为应力导致的相变,建立了沉积岩折曲带形成的相变折曲分析模型。对一些岩石矿物相变的详细过程研究发现某些相变过程确系一种塑性变形过程,从而推测岩石中的折曲带是应力导致的相变大变形留下的地质构造形迹。
通过对超弹性材料相变发生条件进行的研究表明,具有非凸应变能函数的超弹性材料可以发生相变,进一步分析表明:相变可以在能应变软化的弹塑性材料中发生;相变的Maxwell应力、弹性相和弹塑性相的应变都是确定的。文中证明,对任一条假设的应变软化曲线,Maxwell应力直线和应变软化曲线所围面积的代数和总是等于零,这和Ericksen对非线性弹性杆相变研究得到的结论相一致。
本文考虑了跨越折曲带界面的变形梯度的不连续和应力的不连续,强加了跨越界面的力的连续性和相变Maxwell条件,给出了平面应变条件下的相变控制方程。利用相变控制方程和岩石弹塑性本构方程,建立了沉积岩折曲带形成的相变折曲分析模型。折曲分析转化为在给定的简单加载条件下寻求应力张量最大主值的最小值,在该值处,由相变控制方程和弹塑性本构方程导出的方程组具有唯一的物理意义上可接受的实数解,所得的方程组实数解可以由同伦算法求解。数值结果依赖于所选用的本构方程,分别应用基于Drucker-Prager屈服准则的大变形岩石的弹塑性本构方程和引入微结构张量导出的横观各向同性弹塑性本构方程进行了求解。通过平面应变数值算例表明,折曲应力、折曲带内外区域中的应力与应变、折曲带的倾角以及折曲角等都可以由此得出,且与实验结果相吻合。计算表明当塑性切线模量与弹性模量之比较小时,跨越相变界面的应变跳越较大,验证了折曲带的产生是岩石在高围压下发生的韧性剪切的结果,得到相变发生取决于材料性质、静水压力和应力偏张量的结论。
Rock kink bands have been widely observed in nature. Rock kinking involves important information of the earth’s crust movement. Many researchers have devoted their work to experimental research on rock kinking, but no theoretical model has been given so far to reveal its formation mechanism and to predict it exactly. Regarded as a result of stress-induced phase transition for the formation of kink bands in sedimentary rocks, an elastoplastic phase transition model is suggested in the present paper. The precise studies to the rock mineral phase transition discovered some of the deformation process is truly plastic thus concluded that kink bands are the marks of geologic structure resulted from stress-induced phase transition
The theoretical studies discovered that phase transition may occur to superelastic materials having non-convex strain energy function. Further analysis revealed phase transition may occur in such elastoplastic materials with strain softening behavior and the Maxwell stress and the strains inside both elastic phase and elastoplastic phase are all determined after phase transformation has occured. It is proved that for any assumed strain-softening curve the algebraic sum of areas enclosed by the Maxwell stress straight line and strain-softening cure is always equal to zero that agrees with the result given by Ericksen for the analysis of phase transformations in nonlinearly elastic bars.
Considering the discontinuity of deformation gradient and stress across interfaces between kink band and un-kinked areas, imposing the continuity of traction across interfaces and the Maxwell relation the phase transformation control function under planar strain is established in this paper. With the control function and the elastoplastic constitutive equation for sedimentary rocks, the phase transition analysis model for rocks kinking formation is provided. With the aid of phase transition model, kinking analysis is transferred into seeking the minimum first principal value of the stress tensor on which the equations reduced by phase transition conditions and the constitutive laws have a unique, physically acceptable real solution which can be found by homotopy continuation methods. The results solved numerically depend on the constitutive laws selected so the elasoplastic constitutive laws both based on Drucker-Prager yield criteria for rock definite deformation and embodied rock transverse isotropy by incorporating a microstructure tensor are applied respectively. The numerical example of planar strain state illustrated that critical kinking stress, stress and strain in and out the kink bands, both the inclination angle of the kink band and the kink orientation can all be predicted which are accordance with the experimental measurements. As is shown by the calculations that the smaller the rate of plasticity tangent module to elasticity module the larger the strain jump quantity across phase transformation interface is, which verified the kink band formation was due to rock subjected ductile shear in high confine pressure. The conclusions that phase transformation occurrence was dependent with material behavior, mean pressure and stress deviator was drawn.
通过对超弹性材料相变发生条件进行的研究表明,具有非凸应变能函数的超弹性材料可以发生相变,进一步分析表明:相变可以在能应变软化的弹塑性材料中发生;相变的Maxwell应力、弹性相和弹塑性相的应变都是确定的。文中证明,对任一条假设的应变软化曲线,Maxwell应力直线和应变软化曲线所围面积的代数和总是等于零,这和Ericksen对非线性弹性杆相变研究得到的结论相一致。
本文考虑了跨越折曲带界面的变形梯度的不连续和应力的不连续,强加了跨越界面的力的连续性和相变Maxwell条件,给出了平面应变条件下的相变控制方程。利用相变控制方程和岩石弹塑性本构方程,建立了沉积岩折曲带形成的相变折曲分析模型。折曲分析转化为在给定的简单加载条件下寻求应力张量最大主值的最小值,在该值处,由相变控制方程和弹塑性本构方程导出的方程组具有唯一的物理意义上可接受的实数解,所得的方程组实数解可以由同伦算法求解。数值结果依赖于所选用的本构方程,分别应用基于Drucker-Prager屈服准则的大变形岩石的弹塑性本构方程和引入微结构张量导出的横观各向同性弹塑性本构方程进行了求解。通过平面应变数值算例表明,折曲应力、折曲带内外区域中的应力与应变、折曲带的倾角以及折曲角等都可以由此得出,且与实验结果相吻合。计算表明当塑性切线模量与弹性模量之比较小时,跨越相变界面的应变跳越较大,验证了折曲带的产生是岩石在高围压下发生的韧性剪切的结果,得到相变发生取决于材料性质、静水压力和应力偏张量的结论。
Rock kink bands have been widely observed in nature. Rock kinking involves important information of the earth’s crust movement. Many researchers have devoted their work to experimental research on rock kinking, but no theoretical model has been given so far to reveal its formation mechanism and to predict it exactly. Regarded as a result of stress-induced phase transition for the formation of kink bands in sedimentary rocks, an elastoplastic phase transition model is suggested in the present paper. The precise studies to the rock mineral phase transition discovered some of the deformation process is truly plastic thus concluded that kink bands are the marks of geologic structure resulted from stress-induced phase transition
The theoretical studies discovered that phase transition may occur to superelastic materials having non-convex strain energy function. Further analysis revealed phase transition may occur in such elastoplastic materials with strain softening behavior and the Maxwell stress and the strains inside both elastic phase and elastoplastic phase are all determined after phase transformation has occured. It is proved that for any assumed strain-softening curve the algebraic sum of areas enclosed by the Maxwell stress straight line and strain-softening cure is always equal to zero that agrees with the result given by Ericksen for the analysis of phase transformations in nonlinearly elastic bars.
Considering the discontinuity of deformation gradient and stress across interfaces between kink band and un-kinked areas, imposing the continuity of traction across interfaces and the Maxwell relation the phase transformation control function under planar strain is established in this paper. With the control function and the elastoplastic constitutive equation for sedimentary rocks, the phase transition analysis model for rocks kinking formation is provided. With the aid of phase transition model, kinking analysis is transferred into seeking the minimum first principal value of the stress tensor on which the equations reduced by phase transition conditions and the constitutive laws have a unique, physically acceptable real solution which can be found by homotopy continuation methods. The results solved numerically depend on the constitutive laws selected so the elasoplastic constitutive laws both based on Drucker-Prager yield criteria for rock definite deformation and embodied rock transverse isotropy by incorporating a microstructure tensor are applied respectively. The numerical example of planar strain state illustrated that critical kinking stress, stress and strain in and out the kink bands, both the inclination angle of the kink band and the kink orientation can all be predicted which are accordance with the experimental measurements. As is shown by the calculations that the smaller the rate of plasticity tangent module to elasticity module the larger the strain jump quantity across phase transformation interface is, which verified the kink band formation was due to rock subjected ductile shear in high confine pressure. The conclusions that phase transformation occurrence was dependent with material behavior, mean pressure and stress deviator was drawn.
引文
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