, 屈展STUDY ON STRENGTH CHARACTERISTICS OF MICROPOROUS CLAY IN SHALE BASED ON HOMOGENIZATION THEORY1)
HanQiang, QuZhan中图分类号:TE135
文献标识码:A
通讯作者:
收稿日期:2018-06-28
网络出版日期:2019-05-18
版权声明:2019力学学报期刊社 所有
基金资助:
展开
摘要
关键词：
Abstract
Keywords：
-->0
PDF (301KB)元数据多维度评价相关文章收藏文章
本文引用格式导出EndNoteRisBibtex收藏本文-->
引 言
作为页岩油气开发的重要基础研究之一,页岩岩石力学的表征贯穿整个地质勘探、钻完井、油气开采的全过程[1-3].如何在勘探、钻井、开采早期,超前或及时获取页岩力学性质并准确做出异常情况预警,是目前岩石力学急需解决的问题.现有的室内常规实验存在耗时长、精度低、完整岩芯不易获取等问题,受测井技术和井下工况的制约,获取的测井资料品质难以保证,导致页岩力学测量值与真实值差距较大[4].同时,常规力学实验与测井解释等方法较难反映页岩的微观力学特征,无法从根源上反映页岩力学性质形成的机理.近年来随着仪器化压入技术的长足发展,通过材料微尺度结构和显微力学特性研究,评估宏观等效力学性能的均匀化理论逐渐建立起来.基于材料在微尺度下的组成与力学关系分析,采用等效均匀化方法探寻局部对全局行为和性能的影响规律,最终完成宏观性能预测或通过微结构设计改进材料性能的目标[5-7].Cheng等[8]通过材料微观力学实验和量纲函数分析,构建了微观力学量纲函数模型.Cariou等[9]通过对多孔黏弹性介质压入硬度与组成的量纲分析,提取出多孔黏弹性材料的细观强度参数.Ortega等[10]基于前述研究,进一步建立了黏性、摩擦性材料的强度模型,分析了微观尺度下的硬度--组成特征参数.在微尺度力学实验方面,Constantinides等[11]通过开展水泥胶结材料的微观力学分析,评价了该技术对非均匀材料的适用性.
由于页岩组成的多样性和非均质,考虑多尺度均匀化理论开展其力学分析的起步较晚.Zeszotarski等[12]将微观力学测量引入到页岩多尺度力学的研究中,开展了微观黏土矿物的力学分析.Ulm等[13-14]通过对泥页岩的微结构和组成矿物评价,将泥页岩按组成尺度大小的不同划分为：单晶黏土矿物、多孔黏土矿物(微观)、具有非黏土夹的复合介质(细观)和宏观页岩.根据页岩多尺度组成模型分类,黏土单晶的尺度不大于10$^{ - 9}$m;微观尺度的研究对象为多孔黏土介质,量级为10$^{ - 7}\sim $10$^{ -6}$ m;细观尺度研究对象为夹杂非黏土相的复合介质,量级10$^{ -5}\sim $10$^{ - 4}$ m;宏观尺度不小于10$^{ - 3}$m,为常规实验测量范围[15].在此基础上,文献[16,17,18,19]开展了页岩微观弹性力学性质评价,明确了排水弹性多孔介质的有效弹性张量.Chen等[20-21]通过微米力学测试技术开展了页岩细观力学参数评价,对预测页岩宏观力学性质给出了指导性意见.本文基于均匀化理论,建立适用页岩微观多孔黏土强度准则的均匀化函数模型,开展页岩微观多孔黏土力学测试,分析纯黏土矿物的力学特征,通过量纲分析和有限元模拟,建立页岩微观多孔黏土强度参数预测模型.该研究不仅有利于认知页岩微观黏土的力学属性,同时为页岩细观强度均匀化分析和宏观强度的预测提供新的思路和方法.
1 强度均匀化理论
作为研究材料局部强度和多尺度预测的基础,强度均匀化理论是基于材料塑性均匀化本构方程,材料组成特征、耗散能原理和变分准则,建立强度均匀化模型.1.1 耗散能原理
当材料受到的有效应力大于临界强度时,会由弹性变形转为塑性变形,其应力、应变不再满足线性关系[22-23].塑性变形中,以塑性变形(热能形式)耗散掉的外力做功和材料内部存储应变能的变化,被称为塑性耗散能[24-25].局部与全局的塑性应变率满足关系
\begin{equation}\label{eq1} D = \overline { d: B} = \overline { B^{\rm T}: d}\tag{1}\end{equation}
式中, $ D$为全局塑性应变张量,$ d$为局部塑性应变张量,$ B$为弹性应力局部化张量. 全局和局部的塑性耗散能满足[26]
\begin{equation} \label{eq2} D^{\rm e} = \overline { d^{\rm e}} = \overline { \sigma : d} = \overline {( \varSigma : B^{\rm T} + \sigma ^{\rm r}): d}\tag{2}\end{equation}
式中,$ D^{\rm e}$为全局塑性耗散能,$ d^{\rm e}$为局部塑性耗散能,$ \varSigma $为全局应力张量,$ \sigma $为局部应力张量,$ \sigma^{\rm r}$为残余应力张量.
根据残余应变率的协调性和希尔引理关于可能应力场和位移场的讨论,有
\begin{equation}\label{eq3} D^{\rm e} = \varSigma : D + \overline{ \sigma ^{\rm r}: d} = \varSigma : D - \overline{ \sigma ^{\rm r}: L: \sigma ^{\rm r}}\tag{3} \end{equation}
式中,$ L$为局部柔度张量.该结果表明,局部塑性耗散能只是全局塑性能的一部分.当单元体达到某一塑性状态时,定义真实应力场与静力可能应力场塑性功的差值为
\begin{equation}\label{eq4} W_{\rm p} - W_{\rm p}^ * = \left( \sigma - \sigma ^* \right): d\tag{4}\end{equation}
式中,$W_{\rm p}$为真实塑性功,$W_{\rm p}^ *$为静力可能的塑性功,$ \sigma^ * $为静力可能的局部应力张量.
根据屈服理论,此时屈服面对原点外凸且两矢量夹角介于$0^\circ\sim 90^\circ$,此时真实应力场与可能应力场塑性功满足
\begin{equation}\label{eq5} W_{\rm p} \ge W_{\rm p}^ *\tag{5}\end{equation}
对于某一塑性状态下,真实应力与任意一个静力可能满足条件
\begin{equation}\label{eq6} \left. {{\begin{array}{l} { \sigma ^* (x) = B(x): \varSigma ^* + \sigma ^{\rm r}(x)} \\ { \sigma (x) = B(x): \varSigma + \sigma ^{\rm r}(x)} \\\end{array} }} \right\}\tag{6}\end{equation}
式中,$ \varSigma ^* $为静力可能的应力张量,$x$为位移.由式(6)可以得到
\begin{equation}\label{eq7} \sigma (x) - \sigma ^ * (x) = B(x):\left({ \varSigma - \varSigma ^ * } \right)\tag{7}\end{equation}
联立式(1)、式(5)和(7),得到
\begin{equation}\label{eq8} \left( \varSigma - \varSigma^ * \right):\left({ D(x)} \right)^{\rm T} \ge 0\tag{8}\end{equation}
通过支撑函数[27],将局部耗散能简化为
\begin{equation}\label{eq9} \pi ( d) = \mathop {\sup \sigma^{*}}\limits_{\sigma ^ * \in G} : d\tag{9}\end{equation}
式中,$ G$为强度域.
对于任意局部应力场和与其对应的全局应力场,根据耗散能原理有
\begin{equation}\label{eq10} \varSigma : D = \overline { \sigma : d} =\overline {\pi ( d)}\tag{10}\end{equation}
由屈服理论[28],对应的全局均匀化$\varPi $函数可类似地定义为
\begin{equation}\label{eq11} \varPi ^{\hom }( D) = \mathop {\sup}\limits_{\varSigma \in G} \varSigma : D\tag{11}\end{equation}
根据局部应力关系建立全局应力的屈服条件和流动法则,由于加载会产生塑性应变,需通过弹塑性变形方程求解塑性应变矢量.这里根据屈服条件和流动法则的描述,基于塑性应变凸函数性质的外功率为
\begin{equation}\label{eq12} \delta W = \overline { \sigma : d} = \varSigma : D \le \overline {\pi \left( d\right)}\tag{12}\end{equation}
由机动可能速度场得到的应变速率,其$\varPi$函数满足关系
\begin{equation}\label{eq13} \varPi ^{\hom }\left( D \right) = \overline {\pi \left( d \right)} \le \overline {\pi \left( { d^ * }\right)}\tag{13}\end{equation}
根据数学变换,可以给出单元体真实耗散能的另一简洁表达
\begin{equation}\label{eq14} \varPi ^{\hom }\left( D \right) = \inf \overline{\pi \left( { d^ * } \right)}\tag{14}\end{equation}
式中,$ d^ * $为机动可能的应变张量.
1.2 强度均匀化模型
对于多相复合材料,在单元体上的塑性均匀化积分,可以得到式(14)的变分问题\begin{equation}\label{eq15} \varPi ^{\hom }( D) = \inf \frac{1}{\left|\varOmega \right|}\int_\varOmega {\pi {\rm d}V} = \inf \overline\pi\tag{15}\end{equation}
通过上述推导,将全局均匀化耗散能问题转换为与应力和应变有关的局部耗散能求解,即局部等效刚度张量[29].
定义材料局部的应变能为
\begin{equation}\label{eq16} \omega _0 ( d) = \frac{1}{2} d: C: d\tag{16}\end{equation}
式中,$ C$为单元体刚度张量. 定义局部的耗散能与应变能差值函数为
\begin{equation}\label{eq17} v\left( C \right) = \sup \left\{ {\pi ( d) -\omega _0 ( d)} \right\}\tag{17}\end{equation}
当刚度张量为极小值时,有
\begin{equation}\label{eq18} \pi ( d) \le \inf \left\{ {\omega _0 ( d) +v\left( C \right)} \right\}\tag{18}\end{equation}
则$\varPi $函数的变分问题可表示为
\begin{equation}\label{eq19} \varPi ^{\hom }({ D}) = \inf \overline {\inf\left\{ {\omega _0 ( d) + v\left( C \right)}\right\}}\tag{19}\end{equation}
联立式(18)和式(19)得到
\begin{equation}\label{eq20} \varPi ^{\hom }({ D}) = \inf \left\{ {W_0 ( D)+ V\left( C \right)} \right\}\tag{20}\end{equation}
通过上式的变换,将全局均匀化耗散能分解为应变能和差值函数的求和问题,通过对材料中各相介质属性分析,可得到$\varPi$函数的近似解.
2 微观多孔黏土强度均匀化模型
根据组成矿物的尺寸大小,可以将页岩组成划分为不同的量级,如表1所示[14].由于多孔黏土与非黏土夹杂相的尺度相差较大,不适宜将两个尺度进行统一分析.因此,开展页岩的微观多孔黏土的强度分析,评价页岩微观黏土矿物的组成--力学关系,为进一步深入研究页岩细观强度参数和宏观强度预测奠定基础.Table 1
表1
表1页岩多尺度组成分类
Table 1The multi-scale composition of shale
 |
新窗口打开
2.1 多孔黏土应变能
由土力学提出的Drucker-Prager(D-P)准则,与主应力存在直接关系,充分考虑了颗粒材料的内聚性和摩擦性[30].微观力学研究表明,连续D-P准则能够较好地表征多晶体的刚性夹杂问题,可以将黏土颗粒简化为满足D-P准则的均匀模型,分析黏性摩擦材料的界面特征[31].因此,本文通过满足D-P准则的页岩微观黏土强度分析,构建页岩微观多孔黏土应变能.由塑性能与应变速率的关系,给出黏土晶体颗粒强度的定义
\begin{equation} \label{eq21} \left. {{\begin{array}{l} {f\left( \sigma \right) = \sqrt {J_2 } + \alpha _{\rm s} \sigma _{\rm m} - c_{\rm s} \le 0} \\[1mm] {\pi \left( d \right) = \sup \left( { \sigma : d} \right) = \dfrac{c_{\rm s} }{\alpha _{\rm s} }d_{\rm v} } \\ \end{array} }} \right\}\tag{21} \end{equation}
式中,$J_{2}$为应力偏量,$\alpha _{\rm s}$为摩擦系数,$c_{\rm s}$为内聚系数,$\sigma _{\rm m}$为主应力.
D-P准则的屈服条件满足
\begin{equation} \label{eq22} f\left( \sigma \right) = 1 - \left( {\frac{\sigma _{\rm m} - S_0 }{A_{\rm D} }} \right)^2 + \left( {\frac{\sqrt {J_2 } }{B_{\rm D} }} \right)^2 \le 0\tag{22} \end{equation}
其中
\begin{equation} \label{eq23} \left. {{\begin{array}{l} {B_{\rm D} = \alpha _{\rm s} A_{\rm D} } \\[1mm] {S_0 = \dfrac{c_{\rm s} }{\alpha _{\rm s} }} \\[1mm] {A_{\rm D} \to 0} \\ \end{array} }} \right\}\tag{23} \end{equation}
根据塑性流动法则和屈服条件,对$ \sigma$求偏导
\begin{equation} \label{eq24} \frac{\partial f}{\partial \sigma }\left( \sigma \right) = \frac{1}{3}\frac{\partial f}{\partial \sigma _{\rm m}}\left( {\sigma _{\rm m},\sqrt {J_2 } } \right) I + \frac{\partial f}{\partial \sigma _{\rm d}}\left( {\sigma _{\rm m} ,\sqrt {J_2 } } \right)\frac{\delta B_{\rm D}^2 }{\lambda \sigma _{\rm d}}\tag{24} \end{equation}
则局部耗散能可表示为
\begin{equation} \label{eq25} \pi \left( {d_{\rm v}, d_{\rm d}} \right) = S_0 d_{\rm v} - \sqrt {\left( {A_{\rm D} d_{\rm v}} \right)^2 - \left( {\sqrt 2 B_{\rm D} d_{\rm d}} \right)^2}\tag{25} \end{equation}
式中,$d_{\rm v}$为局部体应变,$d_{\rm d}$为局部偏应变.
通过变换将满足D-P准则的强度关系简化为与应变有关的$\pi $函数,强度均匀化转换为对函数的均匀化求解问题.
多孔黏土介质的应力--应变方程可表示为
\begin{equation}\label{eq26} \sigma = C: d + \tau\tag{26}\end{equation}
其中
\begin{equation} \label{eq27} \left. {{\begin{array}{l} { C = 3k J + 2\mu K} \\ { \tau = \tau _{\rm i} I, V_{\rm s}} \\ { \tau = 0, V_{\rm p} } \\ { I = J + K} \\ \end{array} }} \right\}\tag{27} \end{equation}
式中,$k$为体积模量,$\mu $为剪切模量,$ \tau$为预应力张量,$\tau _{\rm i}$为预应力张量分量,$V_{\rm s}$为固相体积,$V_{\rm p}$为孔隙体积.
考虑多孔黏土复合介质的黏土固相(黏土堆积密度$\eta$),相对应的微观均匀化的应力--应变方程为
\begin{equation}\label{eq28} \varSigma = C^{\hom }: D + T\tag{28}\end{equation}
其中
\begin{equation} \label{eq29} \left. {{\begin{array}{l} { C^{\hom } = 3k^{\hom } J + 2\mu ^{\hom } K} \\ { T = \tau :\eta \overline { A_{\rm s} } } \\ {k^{\hom } = \eta k J:\overline { A_{\rm s}} } \\ {\mu ^{\hom } = \eta \mu K:\overline { A_{\rm s}} } \\ \end{array} }} \right\}\tag{29} \end{equation}
式中,$ C^{\hom}$为微观均匀化刚度张量,$k^{\hom}$为有效体积模量,$\mu ^{\hom}$为有效剪切模量,$ A_{\rm s}$固相平均应变张量.
基于Levin理论对于复合材料的基本假设,均匀化应变能可表示为[32]
\begin{equation}\label{eq30} W_0 = \overline{\frac{1}{2} d: C: d} +\overline { \tau : d}\tag{30}\end{equation}
通过上式的变换,将含预应力的多孔黏土问题分解成线弹性和存在预应力问题展开讨论.黏土颗粒与粒间孔隙的力学特性满足条件
\begin{equation}\label{eq31} \left. {{\begin{array}{l} { \tau _{\rm s} = \tau _{\rm sm} I} \\[2mm] { C_{\rm p} = 0} \\[2mm] { \tau _{\rm P} = 0} \\\end{array} }} \right\}\tag{31}\end{equation}
微观均匀化应变能满足条件
\begin{equation}\label{eq32} \left. {{\begin{array}{l} W_0 = \dfrac{1}{2} D: C: D + \tau _{\rm s} : C_{\rm s}^{ - 1} : C^{\hom }: D+ \\[5mm] {\qquad \dfrac{1}{2} \tau _{\rm s} :( C_{\rm s}^{ - 1} : C^{\hom } - \eta I): C^{ -1}: \tau _{\rm s}} \\[2mm] { T = \tau _{\rm s} :C_{\rm s}^{ - 1} : C^{\hom }} \\[2mm] { \varSigma = C^{\hom }: D + T} \\\end{array} }} \right\}\tag{32}\end{equation}
式中,$ \tau _{\rm s}$为黏土颗粒的预应力张量,$ \tau _{\rm sm}$为预应力张量分量,$ \tau _{\rm p}$为孔隙预应力张量,$ C_{\rm p}$为孔隙刚度张量,$C_{\rm s}$为黏土颗粒刚度张量.
引入页岩黏土堆积密度参数,根据线性微观力学理论和强度均匀化方法[33],得到多孔黏土的应变能的应变表达式为
$$W_0 \left( {D_{\rm v}, D_{\rm d} } \right) =\frac{1}{2}k^{\hom}D_{\rm v}^2 +\\ \mu ^{\hom }D_{\rm d}^2 +\frac{k^{\hom }}{k}\tau D_{\rm v} + \frac{1}{2k}\left({\frac{k^{\hom }}{k} - \eta } \right)\tau ^2\tag{33}$$
式中,$D_{\rm v}$为全局体应变,$D_{\rm d}$为全局偏应变.
2.2 非线性函数
通过1.2节的讨论,可以给出固相介质的非线性函数\begin{equation} \label{eq34} \left. {{\begin{array}{l} {v_{\rm s} = \mbox{stat}\left\{ {\pi ( d) - w_{\rm s} (\rm d)} \right\}}\\[2mm] {w_{\rm s} = \dfrac{1}{2}kd_{\rm v}^2 + \mu d_{\rm d}^2 + \tau d_{\rm v}} \\ \end{array} }} \right\}\tag{34} \end{equation}
根据驻值定理,对式(34)求应变速率的偏导
\begin{equation} \label{eq35} \left. {{\begin{array}{l} \dfrac{\partial v_{\rm s}}{\partial d_{\rm v}} = \dfrac{\partial \left( {\pi - w_{\rm s}} \right)}{\partial d_{\rm v}} =\\[3mm] \qquad S_0 - \dfrac{A_D^2 d_{\rm v}}{\sqrt {\left( {A_{\rm D} d_{\rm v} } \right)^2 - \left( {\sqrt 2 B_{\rm D} d_{\rm d}} \right)^2} } - kd_{\rm v} - \tau = 0 \\[3mm] \dfrac{\partial v_{\rm s} }{\partial d_{\rm d} } = \dfrac{\partial \left( {\pi - w_{\rm s}} \right)}{\partial d_{\rm d}} =\\[3mm] \qquad \dfrac{2B_{\rm D}^2 d_{\rm d} }{\sqrt {\left( {A_{\rm D} d_{\rm v} } \right)^2 - \left( {\sqrt 2 B_{\rm D} d_{\rm d} } \right)^2} } - 2\mu d_{\rm d} = 0 \\ \end{array} }} \right\}\tag{35} \end{equation}
其中
\begin{equation} \label{eq36} \left. {{\begin{array}{l} {k = \dfrac{A_D^2 }{\sqrt {\left( {A_{\rm D} d_{\rm v} } \right)^2 - \left( {\sqrt 2 B_{\rm D} d_{\rm d} } \right)^2} } > 0} \\[3mm] {\mu = \dfrac{B_{\rm D}^2 }{\sqrt {\left( {A_{\rm D} d_{\rm v}} \right)^2 - \left( {\sqrt 2 B_{\rm D} d_{\rm d} } \right)^2} } > 0} \\ \end{array} }} \right\}\tag{36} \end{equation}
根据式(35)和式(36),可以得到
\begin{equation} \label{eq37} \frac{k}{\mu } = \frac{A_{\rm D}^2 }{B_{\rm D}^2 } = \frac{1}{\alpha _{\rm s}^2 }\tag{37} \end{equation}
将上式代入式(34),有
\begin{equation} \label{eq38} v_{\rm s} = \left( {\frac{B_{\rm D} \left( {S_0 - \tau } \right)}{2A_{\rm D} }} \right)^2\frac{1}{\mu } - \frac{1}{2}\frac{B_{\rm D}^2 }{\mu }\tag{38} \end{equation}
2.3 均匀化函数
将推导出的$W_{0}$和$v_{\rm s}$代入$\varPi $函数模型中,求解满足驻值条件的方程\begin{equation} \label{eq39} \varPi ^{\hom } = \mbox{stat}\left[ {W_0 \left( {D_{\rm v}, D_{\rm d}} \right) + \eta v_{\rm s}} \right]\tag{39} \end{equation}
将式(39)分别对剪切模量和预应力求偏导,可以得到
\begin{equation} \label{eq40} \left.{{\begin{array}{l} {\dfrac{\partial \varPi ^{\hom }}{\partial \mu } = \dfrac{\partial k}{\partial \mu }\dfrac{\partial W_0 }{\partial k} + \dfrac{\partial W_0 }{\partial \mu } + \eta \dfrac{\partial v_{\rm s}}{\partial \mu } = 0} \\[3mm] {\dfrac{\partial \varPi ^{\hom }}{\partial \tau } = \dfrac{\partial W_0 }{\partial \tau } + \eta \dfrac{\partial v_{\rm s}}{\partial \tau } = 0} \\ \end{array} }} \right\}\tag{40} \end{equation}
将式(31)、式(33)和式(38)代入式(40)中,得到
\begin{equation} \label{eq41} \left. {{\begin{array}{l} {\dfrac{\partial \varPi ^{\hom }}{\partial \tau }\dfrac{\mu }{k}K_{\rm m} D_{\rm v} + \dfrac{\tau }{k^2}\left( {\mu K_{\rm m} - k\eta } \right) + \eta \dfrac{B_{\rm D}^2 }{A_{\rm D}^2 }\left( {\dfrac{\tau - S_0 }{2\mu }} \right) = 0} \\[3mm] {\mu ^2 = \dfrac{B_{\rm D}^2 }{A_{\rm D}^2 }\dfrac{\eta \left[ {\eta A_{\rm D}^2 \left( {S_0^2-A_{\rm D}^2 } \right) + K_{\rm m} B_{\rm D}^2 \left( {2A_{\rm D}^2 - S_0^2 } \right)} \right]}{\left[ {\eta K_{\rm m} A_{\rm D}^2 D_{\rm v}^2 + \left( {2\eta M_{\rm m} A_{\rm D}^2 - 4K_{\rm m} M_{\rm m} B_{\rm D}^2 } \right)D_{\rm d}^2 } \right]}}\\ \end{array} }} \right\}\tag{41} \end{equation}
联立式(40)和式(41),有
$$\varPi ^{\hom } = \varSigma ^{\hom }D_{\rm v}- \\ \mbox{sign}\left( {2K_{\rm m} B_{\rm D}^2 - \eta A_{\rm D}^2 } \right)\sqrt {(A^{\hom })^2D_{\rm v}^2 + 2(B^{\hom })^2D_{\rm d}^2 }\\ \tag{42}$$
其中
\begin{equation} \label{eq43} \left. {{\begin{array}{l} {A^{\hom } = \sqrt {\dfrac{c_{\rm s} \eta ^2K_{\rm m} \left( {\eta - \alpha _{\rm s}^2 K_{\rm m}} \right)}{\left( {\eta - 2\alpha _{\rm s}^2 K_{\rm m} } \right)^2}} } \\[5mm] {B^{\hom } = \sqrt {\dfrac{c_{\rm s} \eta ^2M_{\rm m} \left( {\eta - \alpha _{\rm s}^2 M_{\rm m}} \right)}{\left( {\eta - 2\alpha _{\rm s}^2 M_{\rm m} } \right)^2}} } \\[5mm] {\varSigma ^{\hom } = \dfrac{\eta \alpha _{\rm s} K_{\rm m} }{2\alpha _{\rm s}^2 K_{\rm m} - \eta }} \\ \end{array} }} \right\}\tag{43} \end{equation}
通过上述推导,建立了表征黏土组成的堆积密度、强度参数和均匀化$\pi$函数之间的联系.下面通过微观力学实验外推出纯黏土的基本力学参数,采用量纲分析构建多孔黏土强度--硬度模型,应用有限元模拟页岩微观力学实验,开展微观均匀化的求解.
3 页岩微观力学测试与模型求解
由于目前单晶黏土矿物力学属性的获取较为困难,采用微观力学测试开展多孔黏土力学参数分析、组成--力学关系评价,为微观强度均匀化的求解提供基础参量. 通过$\varPi$函数模型的量纲分析,构建页岩微观力学--组成的量纲表达,并进行数值模拟求解.3.1 页岩微观力学测试
(1)试样制备测试岩样选自四川盆地志留系龙马溪组页岩露头,根据标准ISO14577和GB/T 22458-2008制作块体试样尺寸60 mm$\times $25 mm$\times$15mm,试样上下表面平行度小于0.3$^\circ$,侧面与工作面垂直度小于0.3$^\circ$,试样表面进行二次精细抛光处理,烘干、密封待用.
(2)实验设备与测量步骤
测量选用Triboindenter纳米测试仪,压头试件选用玻氏压头.单次力学测试分为三个阶段：加载阶段,将压头与试样表面垂直零接触,确定试样表面无预应力,然后将压头以恒定速率垂直加载于试样表面;保持阶段,当加载达到最大压入载荷时,保持载荷15s;卸载阶段,以恒定速率将载荷卸载为零,压头提升远离试样表面[34-35].
(3)测试结果分析
基于页岩微观力学分析和页岩的组成与孔隙特征评价,获取页岩试样的黏土堆积密度(页岩多尺度组成综合表征参量[20],根据页岩的微观力学测试分析其弹性模量和压入硬度分布,建立堆积密度与力学属性关联图版,外推出垂直/平行层里面的纯黏土弹性模量和压入硬度[21].由图1可以看出：黏土堆积密度在0.7$\sim$0.95的弹性模量分布介于4.89$\sim$ 20.5GPa,弹性模量与黏土堆积密度正相关分布,且在平行层理方向上的值要大于垂直层理方向.通过对黏土堆积密度和弹性模量、硬度的拟合,外推出纯黏土矿物的弹性模量分别为$E_{h}=24.2$ GPa和$E_{\rm v}=15.8$ GPa.图2为压入硬度与黏土堆积密度关系图版,由于在平行和垂直层理面方向上的硬度变化较小,将其统一处理,外推出黏土颗粒的硬度为$h_{\rm s}=0.51$ GPa (图2).
显示原图|下载原图ZIP|生成PPT图1微观弹性模量--黏土堆积密度关系
-->Fig. 1The relationship between elastic modulus and clay packing density
-->
显示原图|下载原图ZIP|生成PPT图2微观压入硬度--黏土堆积密度关系
-->Fig. 2The relationship between hardness and clay packing density
-->
3.2 量纲分析
根据微观力学测试响应关系,页岩硬度是局部区域力学响应在一定条件下的整体表现,是页岩对外界物体机械作用局部抵抗能力的表现,反映着黏土相凝聚和胶结强度的程度[31-32].压入测试的量纲函数关系满足\begin{equation}\label{eq44} \frac{H}{c_{\rm s}} = \varPi \left( {\theta,\frac{E}{c_{\rm s}},\alpha _{\rm s},\eta } \right)\tag{44}\end{equation}
式中,$H$为压入硬度,$\theta $为压头等效半锥角,$E$为黏土弹性模量.由硬度定义建立硬度与强度的关系
\begin{equation}\label{eq45} H = \frac{1}{A_{\rm c}}\inf \int_\varOmega {\varPi^{\hom }} \left( D \right){\rm d}\varOmega\tag{45}\end{equation}
通过上述变换,建立均匀化耗散能与量纲函数的关系表达,下面通过模型求解与分析,评价各因素间的相互关系.
3.3 模型求解与分析
根据单元等效化理论,通过离散微结构的周期性微观场方法,将微观多孔黏土等效为组成周期性排列的特征体积单元,包含组成和微结构等基本组成特征.根据对局部单元体的属性扰动变化分析,采用均匀化方法求解全局的属性参数(图3).
显示原图|下载原图ZIP|生成PPT图3周期性胞元体属性波动函数示意图
-->Fig. 3Schematic diagram of periodic cell body propertu fluctuation function
-->
通过对多孔黏土的微观力学测试分析和周期性局部分布的假设,确定数值模拟单元和输入参数,采用二维等效方法进行微观力学测试的模拟与分析.根据页岩多尺度组成分类,建立二维基质--孔隙模型,选取孔隙直径为0.25$\mu $m,样试样大小10 $\mu $m$\times $10 $\mu $m.基于微观力学测试确定纯黏土输入的基本参数：平行层理面弹性模量24.2GPa,垂直层理面弹性模量15.8 GPa,压入硬度0.51GPa,多组强度参数和黏土堆积密度,在压入载荷4.8mN下开展微观力学实验的有限元模拟.图4是根据微观力学测试有限元模拟结果,获取的孔黏土压入硬度、强度参数和黏土堆积密度分布图版.当摩擦系数一定时,多孔黏土硬度与内聚系数的比值与黏土堆积密度正相关.当黏土堆积密度一定时,硬度与内聚系数的比值受摩擦系数影响较大,为非线性递增.
显示原图|下载原图ZIP|生成PPT图4页岩微观强度参数的有限元模拟结果
-->Fig. 4Finite element simulation results of shale micro strength parameters
-->
根据微观黏土的四参数有限元力学测试模拟结果建立量纲函数,分析硬度、摩擦系数、内聚系数和堆积密度的分布规律,建立微观多孔黏土硬度的量纲表达函数
\begin{equation}\label{eq46} H = h_{\rm s} \left( {c_{\rm s}, \alpha _{\rm s}}\right)\times \varPi \left( {\alpha _{\rm s},\eta }\right)\tag{46}\end{equation}
通过有限元模拟,构造压入硬度表达式为
$$h_{\rm s} = c_{\rm s} \times 4.59\bigg[ 1 + 2.90\alpha_{\rm s} + \left( {1\times 10^{- 8}\alpha _{\rm s}} \right)^3 +\\ \left( {1\times 10^{-8}\alpha _{\rm s}} \right)^{10}\bigg]\tag{47}$$
建立与摩擦系数和堆积密度相关的微观量纲函数表达式为
\begin{equation}\label{eq48} \varPi = \varPi _1 \left( \eta \right) + \alpha _{\rm s} \left( {1 - \eta } \right)\varPi _2 \left( {\alpha _{\rm s},\eta } \right)\tag{48}\end{equation}
其中
\begin{equation} \label{eq49} \left. {{\begin{array}{l} {\varPi _1 = \eta \left( {1 + a\left( {1 - \eta } \right) + b\left( {1 - \eta } \right)^2 + c\left( {1 - \eta } \right)\eta ^3} \right)}\\ {\varPi _2 = \alpha _{\rm s} \eta ^2\left( {d + e\left( {1 - \eta } \right) + f\left( {1 - \eta } \right)\alpha _{\rm s} + g\alpha _{\rm s} ^3} \right)}\\ \end{array} }} \right\}\tag{49} \end{equation}
式中各系数的拟合结果为
\begin{equation} \label{eq50} \left. {{\begin{array}{l} {a = - 6.53;~b = 10.10;~c = 4.58;~d = 1.0\times 10^{-8}} \\ {e = 14.30;~f = -16.80;~g = 4.59} \\ \end{array} }} \right\}\tag{50} \end{equation}
通过上述问题的讨论与求解,可进一步揭示页岩微观多孔黏土矿物在组成、硬度和强度参数间的关系,为深入研究页岩细观和宏观力学形成机理奠定了基础.
4 结 论
本文基于均匀化理论,开展了页岩微观多孔黏土强度均匀化模型研究,进行了页岩微观力学测试,分析了多孔黏土力学参数与黏土堆积密度的关系,采用量纲分析和有限元模拟,求解与硬度--堆积密度和强度参数相关的均匀化函数.得出以下结论：(1)基于耗散能原理和强度均匀化理论,构建满足D-P准则的$\pi$函数,可以将页岩多孔黏土强度的均匀化问题转换为对$\pi$函数的均匀化求解,通过微观多孔黏土应变能和非线性函数的建立,推导出与多孔黏土组成和力学参数相关的均匀化函数模型.
(2)页岩微观多孔黏土力学测试表明,当黏土堆积密度一定时,弹性模量在平行层理面的值大于垂直层理面,且弹性模量与黏土堆积密度正相关.通过对微观力学测试结果的外推分析,得到纯黏土矿物的固有属性,为后续的研究奠定了理论基础.
(3)基于量纲分析和有限元模拟的页岩微观组成和力学参数分析表明,当摩擦系数一定时,堆积密度越大,硬度与内聚系数的比值也越大.当黏土堆积密度一定时,硬度与内聚系数的比值受摩擦系数影响较大,为非线性递增.通过求解得到多孔黏土硬度--强度参数模型,为页岩细观含非黏土夹杂的强度均匀化评价和宏观强度预测奠定了基础.
The authors have declared that no competing interests exist.
参考文献 原文顺序
文献年度倒序
文中引用次数倒序
被引期刊影响因子
| [1] | . 为更好地指导我国页岩气资源高效开发,在概述我国页岩气资源和开采现状的基础上,从地质特征预测、安全快速钻井、环保高效开采等方面系统总结了我国页岩气开采面临的工程地质难题,指出页岩非线性工程地质力学特征与预测理论、多重耦合下的页岩油气安全优质钻井理论、页岩地层动态随机裂缝控制机理与无水压裂技术、页岩油气多尺度渗流特征与开采理论等是需要重点解决的关键理论问题,钻采过程中页岩储层物理力学化学特征演化规律与数学表征,多场耦合条件下非连续页岩与钻井完井流体作用机理,页岩地层动态随机裂缝控制、长效导流机制与无水压裂技术,页岩微纳尺度吸附/解吸机制、尺度升级及多场耦合的多相渗流理论等是亟需解决的关键前沿理论问题,并针对各前沿关键力学问题综述了研究进展和发展趋势,对促进我国页岩油气的科学、有效开发具有一定的借鉴作用。 . 为更好地指导我国页岩气资源高效开发,在概述我国页岩气资源和开采现状的基础上,从地质特征预测、安全快速钻井、环保高效开采等方面系统总结了我国页岩气开采面临的工程地质难题,指出页岩非线性工程地质力学特征与预测理论、多重耦合下的页岩油气安全优质钻井理论、页岩地层动态随机裂缝控制机理与无水压裂技术、页岩油气多尺度渗流特征与开采理论等是需要重点解决的关键理论问题,钻采过程中页岩储层物理力学化学特征演化规律与数学表征,多场耦合条件下非连续页岩与钻井完井流体作用机理,页岩地层动态随机裂缝控制、长效导流机制与无水压裂技术,页岩微纳尺度吸附/解吸机制、尺度升级及多场耦合的多相渗流理论等是亟需解决的关键前沿理论问题,并针对各前沿关键力学问题综述了研究进展和发展趋势,对促进我国页岩油气的科学、有效开发具有一定的借鉴作用。 |
| [2] | . <p>页岩气蕴藏在页岩层中，页岩层的层理性构造使其水力压裂裂纹扩展与常规均质储层不同.为研究页岩储层水力压裂的裂纹扩展规律，基于复变函数保角变换，得出裂纹尖端应力集中解，考虑页岩非均质、强度各向异性特点，通过比较裂纹沿各方向扩展所需的裂缝尖端水压力，推导出水力压裂裂纹垂直于最小地应力方向稳定扩展过程中在斜交层理后的扩展判据.分别定义了水力压裂裂纹在层理处起裂和沿层理扩展的弱层和岩石基体临界强度比，根据两个临界强度比确定水力压裂裂纹遇层理时在层理处起裂和沿层理扩展的层理弱面强度范围，以此表示水力压裂裂纹转向层理扩展的难易程度.通过对裂纹扩展判据的分析得出：层理起裂弱层和岩石基体临界强度比随层理走向线与第一主应力夹角和层理倾角的减小以及第三主应力和岩石基体强度的增大而增大；层理走向角小于35.26°时，层理起裂弱层和岩石基体临界强度比随第一主应力的减小以及第二主应力的增大而增大；反之，层理起裂弱层和岩石基体临界强度比随第一主应力的减小以及第二主应力的增大而减小；层理扩展弱层和岩石基体临界强度比随层理走向线与第一主应力夹角、层理倾角和地应力差的减小以及岩石基体抗拉强度的增大而增大.层理起裂条件与层理扩展条件同时满足时，水力压裂裂纹转向层理方向扩展.</p> . <p>页岩气蕴藏在页岩层中，页岩层的层理性构造使其水力压裂裂纹扩展与常规均质储层不同.为研究页岩储层水力压裂的裂纹扩展规律，基于复变函数保角变换，得出裂纹尖端应力集中解，考虑页岩非均质、强度各向异性特点，通过比较裂纹沿各方向扩展所需的裂缝尖端水压力，推导出水力压裂裂纹垂直于最小地应力方向稳定扩展过程中在斜交层理后的扩展判据.分别定义了水力压裂裂纹在层理处起裂和沿层理扩展的弱层和岩石基体临界强度比，根据两个临界强度比确定水力压裂裂纹遇层理时在层理处起裂和沿层理扩展的层理弱面强度范围，以此表示水力压裂裂纹转向层理扩展的难易程度.通过对裂纹扩展判据的分析得出：层理起裂弱层和岩石基体临界强度比随层理走向线与第一主应力夹角和层理倾角的减小以及第三主应力和岩石基体强度的增大而增大；层理走向角小于35.26°时，层理起裂弱层和岩石基体临界强度比随第一主应力的减小以及第二主应力的增大而增大；反之，层理起裂弱层和岩石基体临界强度比随第一主应力的减小以及第二主应力的增大而减小；层理扩展弱层和岩石基体临界强度比随层理走向线与第一主应力夹角、层理倾角和地应力差的减小以及岩石基体抗拉强度的增大而增大.层理起裂条件与层理扩展条件同时满足时，水力压裂裂纹转向层理方向扩展.</p> |
| [3] | . <p>本文以实际岩体工程为背景，利用WDT-1500 仪器开展了轴向、侧向同时卸荷条件下砂岩的三轴试验. 结果表明：轴、侧向同卸荷这种卸荷路径下，砂岩试样破坏时并没有出现应力峰值，为了定义试样的破坏强度，将最大与最小主应力差随最小主应力的变化关系曲线上应力跌落的拐点处的应力值定义为破坏强度. 砂岩变形初始段发生应力跌落和轴向应变回弹，破坏前无明显的弹性和屈服阶段；试验的过程中，砂岩的侧向变形明显大于轴向变形，其体积应变一直处于膨胀状态；相对于砂岩的常规三轴试验结果，试样破坏时的强度在轴向、侧向同时卸荷条件下有所降低. 初始轴压和初始围压对试样的力学特征有十分显著的影响，但围压的卸荷速率却并不显著. 砂岩的破坏特征主要是以张-拉为主的混合张剪的破坏.</p> . <p>本文以实际岩体工程为背景，利用WDT-1500 仪器开展了轴向、侧向同时卸荷条件下砂岩的三轴试验. 结果表明：轴、侧向同卸荷这种卸荷路径下，砂岩试样破坏时并没有出现应力峰值，为了定义试样的破坏强度，将最大与最小主应力差随最小主应力的变化关系曲线上应力跌落的拐点处的应力值定义为破坏强度. 砂岩变形初始段发生应力跌落和轴向应变回弹，破坏前无明显的弹性和屈服阶段；试验的过程中，砂岩的侧向变形明显大于轴向变形，其体积应变一直处于膨胀状态；相对于砂岩的常规三轴试验结果，试样破坏时的强度在轴向、侧向同时卸荷条件下有所降低. 初始轴压和初始围压对试样的力学特征有十分显著的影响，但围压的卸荷速率却并不显著. 砂岩的破坏特征主要是以张-拉为主的混合张剪的破坏.</p> |
| [4] | . <p>页岩气勘探开发在中国刚刚起步，缺乏相关技术经验。为此，介绍了国外页岩气钻完井技术现状及最新进展。国外页岩气开发先后经历了直井、单支水平井、多分支水平井、丛式井、丛式水平井钻井（PAD水平井）的发展历程；目前水平井已成为页岩气开发的主要钻井方式，页岩气水平井钻井要考虑其成本，垂直井段的深度不超过3 000 m，水平井段的长度介于500～2 500 m；PAD水平井钻井利用一个钻井平台作为钻井点，先后钻多口水平井，可以降低成本、节约时间，是比较新的页岩气钻井技术；页岩气固井主要采用泡沫水泥固井技术，完井方式以套管固井后射孔完井为主。在综合分析国外页岩气钻完井技术及钻完井难点的基础上，指出国内页岩气钻井应着力解决在保持井壁稳定、预防事故、降低钻井成本、研发配套仪器和优选钻井液配方等方面存在的问题。</p> . <p>页岩气勘探开发在中国刚刚起步，缺乏相关技术经验。为此，介绍了国外页岩气钻完井技术现状及最新进展。国外页岩气开发先后经历了直井、单支水平井、多分支水平井、丛式井、丛式水平井钻井（PAD水平井）的发展历程；目前水平井已成为页岩气开发的主要钻井方式，页岩气水平井钻井要考虑其成本，垂直井段的深度不超过3 000 m，水平井段的长度介于500～2 500 m；PAD水平井钻井利用一个钻井平台作为钻井点，先后钻多口水平井，可以降低成本、节约时间，是比较新的页岩气钻井技术；页岩气固井主要采用泡沫水泥固井技术，完井方式以套管固井后射孔完井为主。在综合分析国外页岩气钻完井技术及钻完井难点的基础上，指出国内页岩气钻井应着力解决在保持井壁稳定、预防事故、降低钻井成本、研发配套仪器和优选钻井液配方等方面存在的问题。</p> |
| [5] | . 利用基于岩石矿物种类及其含量提出的岩石矿物细胞元随机性参数赋值方法，针对两矿物细胞元混合的非均质岩石模型进行参数赋值，通过数值试验研究了非均质岩石材料矿物细胞元含量及其力学参数对岩石宏观力学参数的影响；根据数值试验结果，探讨了岩石材料矿物细胞元弹性模量与宏观模量之间的函数关系。研究结果表明，岩石宏观弹性模量与矿物细胞元弹性模量、矿物细胞元含量都具有十分显著的线性相关关系，即基于岩石矿物种类及其含量的矿物细胞元随机性参数赋值方法建立的岩石有限元模型宏细观力学参数之间呈显著的线性关系；宏观力学参数可用细观力学参数的数学期望值即加权平均值来表示。 . 利用基于岩石矿物种类及其含量提出的岩石矿物细胞元随机性参数赋值方法，针对两矿物细胞元混合的非均质岩石模型进行参数赋值，通过数值试验研究了非均质岩石材料矿物细胞元含量及其力学参数对岩石宏观力学参数的影响；根据数值试验结果，探讨了岩石材料矿物细胞元弹性模量与宏观模量之间的函数关系。研究结果表明，岩石宏观弹性模量与矿物细胞元弹性模量、矿物细胞元含量都具有十分显著的线性相关关系，即基于岩石矿物种类及其含量的矿物细胞元随机性参数赋值方法建立的岩石有限元模型宏细观力学参数之间呈显著的线性关系；宏观力学参数可用细观力学参数的数学期望值即加权平均值来表示。 |
| [6] | . A new modal analysis method with second-order two-scale (SOTS) asymptotic expansion is presented for axisymmetric and spherical symmetric structures. The symmetric structures considered are periodically distributed with homogeneous and isotropic constituent materials. By the asymptotic expansion of the eigenfunctions, the homogenized modal equations, the effective materials coefficients, the first- and second-order correctors are obtained. The derived homogenized constitutive relationships are the same as the ones which serve to homogenize the corresponding static problems. The eigenvalues are also expanded to the second-order terms and using the so called 鈥渃orrector equation鈥, the correctors of the eigenvalues are expressed in terms of the first- and second-order correctors of the eigenfunctions. The anisotropic materials are obtained by homogenization with different properties in the circumferential direction. Especially for the two-dimensional axisymmetric layered structure, the one-dimensional plane axisymmetric and spherical symmetric structures, the homogenized eigenfunctions and eigenvalues, as well as their corresponding correctors are all solved analytically. The finite element algorithm is established, three typical numerical experiments are carried out and the necessity of the second-order correctors is discussed. Based on the numerical results, it is validated that the SOTS asymptotic expansion homogenization method is effective to identify the eigenvalues of the axisymmetric and spherical symmetric structures with periodic configurations and the original eigenfunctions with periodic oscillation can be reproduced by adding the correctors to the homogenized eigenfunctions. |
| [7] | . For the first time, a book is being edited to address how results from one scale can be shifted or related to another scale, say from macro to micro or vice versa. The new approach retains the use of the equilibrium mechanics within a scale level such that cross scale results can be connected by scale invariant criteria. Engineers in different disciplines should be able to understand and use the results. |
| [8] | |
| [9] | . Recent progress in instrumented nanoindentation makes it possible today to test in situ phase properties and structures of porous materials that cannot be recapitulated ex situ in bulk form. But it requires a rigorous indentation analysis to translate indentation data into meaningful mechanical properties. This paper reports the development and implementation of a multi-scale indentation analysis based on limit analysis, for the assessment of strength properties of cohesive-frictional porous materials from hardness measurements. Based on the separation-of-scale condition, we implement an elliptical strength criterion which results from the nonlinear homogenization of the strength properties of the constituents (cohesion and friction), the porosity and the microstructure, into a computational yield design approach to indentation analysis. We identify the resulting upper bound problem as a second-order conical optimization problem, for which advanced optimization algorithms became recently available. The upper bound yield design solutions are benchmarked against solutions from comprehensive elastoplastic contact mechanics finite element solutions and lower bound solutions. Furthermore, from a detailed parameter study based on intensive computational simulations, we identify characteristic hardness鈥損acking density scaling relations for cohesive-frictional porous materials. These scaling relations which are developed for two pore-morphologies, a matrix鈥損ore morphology and a polycrystal (perfect disordered) morphology, are most suitable for the reverse analysis of the strength parameters of cohesive-frictional solids from indentation hardness measurements. |
| [10] | . This paper introduces a novel micromechanics method for strength homogenization of cohesive-frictional porous composites. Within a yield design formulation, the inherently nonlinear homogenization problem associated with strength upscaling is treated by the linear comparison composite (LCC) theory, which resolves the strength properties of the heterogeneous medium by estimating the effective properties of a suitable linear comparison composite with similar underlying microstructure. The LCC homogenization method rationalizes the development of strength criteria for cohesive-frictional materials affected by the presence of porosity and rigidlike inclusions. Modeling results for benchmark microstructures improve existing micromechanics formulations by allowing the consideration of the complete range of frictional behaviors for the Drucker-Prager solid and by lifting the restriction on the incompressibility of the solid for the estimation of morphology factors that describe the mechanical interaction between material phases. The LCC strength homogenization is implemented in a multiscale thought model applicable to geomaterials, which serves as a generalized framework for quantitative assessment of effects of material composition, grain-scale properties, microstructure, and interface conditions on the overall strength of the porous composite. |
| [11] | . Recent progress in experimental and theoretical nanomechanics opens new venues in materials science for the nano-engineering of cement-based composites. In particular, as new experimental techniques such as nanoindentation provide unprecedented access to micromechanical properties of materials, it becomes possible to identify the mechanical effects of the elementary chemical components of cement-based materials at the scale where physical chemistry meets mechanics, including the properties of the four clinker phases, of portlandite, and of the C-S-H gel. In this paper, we review some recent results obtained by nanoindentation, which reveal that the C-S-H gel exists “mechanically” in two different forms, a lowdensity form and a high-density form, which have different mean stiffness and hardness values and different volume fractions. While the volume fractions of the two phases depend on mix proportions, the mean stiffness and hardness values do not change from one cement-based material to another; instead they are intrinsic properties of the C-S-H gel. |
| [12] | . Most analyses of kerogens rely on samples that have been isolated by dissolving the rock matrix. The properties of the kerogen before and after such isolation may be different and all sample orientation information is lost. We report a method of measuring kerogen mechanical properties in the rock matrix without isolation. An atomic force microscope (AFM) based nanoindenter is used to measure the hardness and reduced modulus of the kerogen within Woodford shale. The same instrument also provides useful images of polished rock sections on a submicrometer scale. Measurements were carried out both parallel and perpendicular to the bedding plane. |
| [13] | . Despite their ubiquitous presence as sealing formations in hydrocarbon bearing reservoirs affecting many fields of exploitation, the source of anisotropy of this earth material is still an enigma that has deceived many decoding attempts from experimental and theoretical sides. Sedimentary rocks, such as shales, are made of highly compacted clay particles of sub-micrometer size, nanometric porosity and different mineralogy. In this paper, we present, for the first time, results from a new experimental technique that allows one to rationally assess the elasticity content of the highly heterogeneous clay fabric of shales from nano- and microindentation. Based on the statistical analysis of massive nanoindentation tests, we find (1) that the in-situ elasticity content of the clayfabric at a scale of a few hundred to thousands nanometers is almost an order of magnitude smaller than reported clay stiffness values of clay minerals, and (2) that the elasticity and the anisotropy scale linearly with the clay packing density beyond a percolation threshold of roughly 50%. Furthermore, we show that the elasticity content sensed by nano- and microindentation tests is equal to the one that is sensed by (small strain) velocity measurements. From those observations, we conclude that shales are nanogranular composite materials, whose mechanical properties are governed by particle-to-particle contact and by characteristic packing densities, and that the much stiffer mineral properties play a secondary role. |
| [14] | . Concrete, bone and shale have one thing in common: their load-bearing mineral phase is a hydrated nanocomposite. Yet the link between material genesis, microstructure, and mechanical performance for these materials is still an enigma that has deceived many decoding attempts. In this article, we advance statistical indentation analysis techniques that make it possible to assess, in situ , the nanomechanical properties, packing density distributions, and morphology of hydrated nanocomposites. These techniques are applied to identify intrinsic and structural sources of anisotropy of hydrated nanoparticles: calcium–silicate–hydrate (C–S–H), apatite, and clay. It is shown that C–S–H and apatite, the binding phase in, respectively, cement-based materials and bone, are intrinsically isotropic; this is most probably due to a random precipitation and growth process of particles in calcium oversaturated pore solutions, which can also explain the nonnegligible internanoparticle friction. In contrast, the load-bearing clay phase in shale, the sealing formation of most hydrocarbon reservoirs, is found to be intrinsically anisotropic and frictionless. This is indicative of a 'smooth' deposition and compaction history, which, in contrast to mineral growth in confined spaces, minimizes nanoparticle interlocking. In all cases, the nanomechanical behavior is governed by packing density distributions of elementary particles delimitating macroscopic diversity. |
| [15] | . 页岩储层的力学行为和工程性质,是影响页岩油气安全、高效、经济开采的关键因素。目前,页岩的物理力学特征研究以宏观测试为主,存在制样、实验耗时长,对目的层精细研究效果不理想等问题。为此,从页岩多尺度组成、微/纳米力学测试、影响因素和跨尺度均匀化4个层面,综述了国内外页岩多尺度力学研究现状:基于多尺度方法建立的页岩微观-细观-宏观组分模型,为页岩多尺度组成-力学耦合提供理论基础；通过微/纳米力学测试技术,明确适合页岩的力学测量标准,完善影响页岩多尺度测量因素评价；基于开展多尺度力学耦合模型研究,评价多尺度间的组成-力学关系。该研究对深入认知页岩岩石力学特征及破坏机理、丰富岩石力学特性测试与表征方法具有重要作用。 . 页岩储层的力学行为和工程性质,是影响页岩油气安全、高效、经济开采的关键因素。目前,页岩的物理力学特征研究以宏观测试为主,存在制样、实验耗时长,对目的层精细研究效果不理想等问题。为此,从页岩多尺度组成、微/纳米力学测试、影响因素和跨尺度均匀化4个层面,综述了国内外页岩多尺度力学研究现状:基于多尺度方法建立的页岩微观-细观-宏观组分模型,为页岩多尺度组成-力学耦合提供理论基础；通过微/纳米力学测试技术,明确适合页岩的力学测量标准,完善影响页岩多尺度测量因素评价；基于开展多尺度力学耦合模型研究,评价多尺度间的组成-力学关系。该研究对深入认知页岩岩石力学特征及破坏机理、丰富岩石力学特性测试与表征方法具有重要作用。 |
| [16] | |
| [17] | |
| [18] | |
| [19] | , |
| [20] | . In view of the difficulty to get mechanical characteristics of shale reservoirs, a quantitative evaluation method based on micro-indentation test technology was proposed to research the meso-mechanical properties of shale. Through micro-indentation test of shale outcrop samples from the Cambrian Longmaxi Formation in the Changning area, Sichuan Basin, the meso-mechanical properties of shale were analyzed, and the relationship between the macro-scale and meso-scale shale mechanical properties was evaluated. The analysis results of micro-indentation tests show that the mesoscopic elastic modulus and indentation hardness are heterogeneous in distribution. The comparison of macro and meso experiments shows that the statistical mean value of meso-elastic modulus is approximate to the value of the macro-elastic modulus. The relationship between composition and mechanical properties of shale was obtained based on the packing density model theory. The result shows that mesoscopic elastic modulus and indentation hardness increase nonlinearly with the increase of the packing density. Based on the hardness-packing density model, the reverse analysis of cohesion and friction angles shows that the mesoscopic value is slightly smaller than the macroscopic value. The micro-indentation test technology can evaluate shale meso-mechanical properties and predict the macro-mechanical properties effectively. |
| [21] | . |
| [22] | . 复合材料以其优越的力学性能,在工程中得到了广泛应用。相比于传统材料,复合材料不但具有强度高、韧性强等特点,而且它还具有可设计性。微观结构对复合材料的宏观力学性能具有至关重要的影响,通过合理地设计复合材料微观结构可以得到期望的宏观性能。均质化方法是一种有效的设计方法,它从微观结构的角度出发,利用均匀化的概念,实现了对复合材料的宏观力学性能的预测和设计。但是,当考虑了非线性因素,均质化的实现就非常困难。因此,本文通过理论推导对复合材料的非线性弹性均质化问题进行了研究,并利用直接迭代法对问题进行了求解。本文通过双渐近展开的方法,将位移按照宏观位移和微观位移展开,推导了非线性弹性均质化方程。通过直接迭代法,对非线性弹性均质化方程进行了求解,给出了具体的迭代方法和实现步骤。根据迭代步骤和非线性弹性均质化方程,使用MATLAB程序语言进行编程,得到非线性弹性均质化问题的迭代程序。利用MATLAB程序三种典型本构关系的多孔材料和两相复合材料进行计算,对比和细致模型的应变能、最大位移和等效泊松比,对程序及迭代方法的准确性进行了验证。通过对一种三元橡胶基复合材料,进行了多尺度分析,将其分为芯丝尺度和层间尺度。用线弹性的均质化方法得到了芯丝尺度的等效弹性参数,并将其作为层间尺度的材料参数。在层间尺度应用非线性弹性均质化方法对结构进行计算,得到材料的宏观等效性能,并与实验结果进行对比。利用MATLAB调用ABAQUS进行联合计算,将有限元计算的理论通过商业软件进行实现,避免了每调换一次有限元格式就要对程序进行大幅度修改的麻烦,并用联合计算方法对三维支架模型进行了计算,验证了方法的准确性。本文对非线性弹性均质化问题进行了理论推导,并且通过MATLAB程序进行了实现。通过对多尺度问题和联合计算问题的研究,拓展了非线性弹性均质化方法的应用。对非线性均质化方法和复合材料微观结构设计具有一定的指导意义和参考价值。 . 复合材料以其优越的力学性能,在工程中得到了广泛应用。相比于传统材料,复合材料不但具有强度高、韧性强等特点,而且它还具有可设计性。微观结构对复合材料的宏观力学性能具有至关重要的影响,通过合理地设计复合材料微观结构可以得到期望的宏观性能。均质化方法是一种有效的设计方法,它从微观结构的角度出发,利用均匀化的概念,实现了对复合材料的宏观力学性能的预测和设计。但是,当考虑了非线性因素,均质化的实现就非常困难。因此,本文通过理论推导对复合材料的非线性弹性均质化问题进行了研究,并利用直接迭代法对问题进行了求解。本文通过双渐近展开的方法,将位移按照宏观位移和微观位移展开,推导了非线性弹性均质化方程。通过直接迭代法,对非线性弹性均质化方程进行了求解,给出了具体的迭代方法和实现步骤。根据迭代步骤和非线性弹性均质化方程,使用MATLAB程序语言进行编程,得到非线性弹性均质化问题的迭代程序。利用MATLAB程序三种典型本构关系的多孔材料和两相复合材料进行计算,对比和细致模型的应变能、最大位移和等效泊松比,对程序及迭代方法的准确性进行了验证。通过对一种三元橡胶基复合材料,进行了多尺度分析,将其分为芯丝尺度和层间尺度。用线弹性的均质化方法得到了芯丝尺度的等效弹性参数,并将其作为层间尺度的材料参数。在层间尺度应用非线性弹性均质化方法对结构进行计算,得到材料的宏观等效性能,并与实验结果进行对比。利用MATLAB调用ABAQUS进行联合计算,将有限元计算的理论通过商业软件进行实现,避免了每调换一次有限元格式就要对程序进行大幅度修改的麻烦,并用联合计算方法对三维支架模型进行了计算,验证了方法的准确性。本文对非线性弹性均质化问题进行了理论推导,并且通过MATLAB程序进行了实现。通过对多尺度问题和联合计算问题的研究,拓展了非线性弹性均质化方法的应用。对非线性均质化方法和复合材料微观结构设计具有一定的指导意义和参考价值。 |
| [23] | . 基于热力学第一定律和非局部塑性理论,提出了一种求解应变局部化问题的非局部方法.对材料的每一点定义了局部和非局部两种状态空间,局部状态空间的内变量通过非局部权函数映射到非局部空间,成为非局部内变量.在应变软化过程中,局部状态空间中的塑性变形服从正交流动法则,材料的软化律在非局部状态空间中被引入.通过两个状态空间的塑性应变能耗散率的等效,得到了应变软化过程中明确定义的局部化区域以及其中的塑性应变分布.应用本方法导出了一维应变局部化问题的解析解.解析解表明,应变局部化区域的尺寸只与材料内尺度有关;对于高斯型非局部权函数,局部化区域的尺寸大约是材料内尺度的6倍.一维算例表明,局部化区域的塑性应变分布以及载荷-位移曲线仅与材料参数和结构几何尺寸有关,变形局部化区域的尺寸随着材料内尺度的减小而减小,同时塑性应变也随着材料内尺度的减小变得更加集中.当内尺度趋近于零时,应用本文方法得到的解与采用传统的局部塑性理论得到的解相同. . 基于热力学第一定律和非局部塑性理论,提出了一种求解应变局部化问题的非局部方法.对材料的每一点定义了局部和非局部两种状态空间,局部状态空间的内变量通过非局部权函数映射到非局部空间,成为非局部内变量.在应变软化过程中,局部状态空间中的塑性变形服从正交流动法则,材料的软化律在非局部状态空间中被引入.通过两个状态空间的塑性应变能耗散率的等效,得到了应变软化过程中明确定义的局部化区域以及其中的塑性应变分布.应用本方法导出了一维应变局部化问题的解析解.解析解表明,应变局部化区域的尺寸只与材料内尺度有关;对于高斯型非局部权函数,局部化区域的尺寸大约是材料内尺度的6倍.一维算例表明,局部化区域的塑性应变分布以及载荷-位移曲线仅与材料参数和结构几何尺寸有关,变形局部化区域的尺寸随着材料内尺度的减小而减小,同时塑性应变也随着材料内尺度的减小变得更加集中.当内尺度趋近于零时,应用本文方法得到的解与采用传统的局部塑性理论得到的解相同. |
| [24] | . 本文在二阶计算均匀化框架下提出了颗粒材料损伤-愈合与塑性的多尺度表征方法.颗粒材料结构在宏观尺度模型化为梯度Cosserat连续体,在其有限元网格的每个积分点处定义具有离散颗粒介观结构的表征元.建立了表征元离散颗粒系统的非线性增量本构关系.表征元周边介质作用于表征元边界颗粒的增量力与增量力偶矩以表征元边界颗粒的增量线位移与增量转动角位移、当前变形状态下表征元离散介观结构弹性刚度、以及凝聚到表征元边界颗粒的增量耗散摩擦力表示.基于平均场理论与Hill定理,导出了基于介观力学信息的梯度Cosserat连续体增量非线性本构关系.在等温热动力学框架下定义了表征颗粒材料各向异性损伤-愈合和塑性的损伤、愈合张量因子与综合损伤、愈合效应的净损伤张量因子和塑性应变.此外,定义了损伤和塑性耗散能密度与愈合能密度,以定量比较材料损伤、愈合、塑性对材料失效的效应.应变局部化数值例题结果显示了所建议的颗粒材料损伤-愈合-塑性表征方法的有效性. . 本文在二阶计算均匀化框架下提出了颗粒材料损伤-愈合与塑性的多尺度表征方法.颗粒材料结构在宏观尺度模型化为梯度Cosserat连续体,在其有限元网格的每个积分点处定义具有离散颗粒介观结构的表征元.建立了表征元离散颗粒系统的非线性增量本构关系.表征元周边介质作用于表征元边界颗粒的增量力与增量力偶矩以表征元边界颗粒的增量线位移与增量转动角位移、当前变形状态下表征元离散介观结构弹性刚度、以及凝聚到表征元边界颗粒的增量耗散摩擦力表示.基于平均场理论与Hill定理,导出了基于介观力学信息的梯度Cosserat连续体增量非线性本构关系.在等温热动力学框架下定义了表征颗粒材料各向异性损伤-愈合和塑性的损伤、愈合张量因子与综合损伤、愈合效应的净损伤张量因子和塑性应变.此外,定义了损伤和塑性耗散能密度与愈合能密度,以定量比较材料损伤、愈合、塑性对材料失效的效应.应变局部化数值例题结果显示了所建议的颗粒材料损伤-愈合-塑性表征方法的有效性. |
| [25] | |
| [26] | The organization of the elementary components within the ultrastructure of mineralized tissues (bone and mineralized tendons) has provoked some controversy; especially with regard to its impact on the mechanical properties of the ultrastructure. Herein, we aim at shedding some light on the issue, by developing and verifying three different continuum-micromechanics representations of the collagen鈥搈ineral interaction in the elasticity of mineralized tissues: (i) mineral foam matrix with collagen inclusions, (ii) interpenetrating network of hydroxyapaptite crystals and collagen molecules, (iii) composite of fibrils (collagen鈥揾ydroxyapatite network) embedded in a collagen-free extrafibrillar mineral foam matrix. The validation of the different concepts, based on independent sets of experiments, shows remarkable predictive capabilites of the different micromechanical representations. Still, there are significant differences in the performance of these three different micromechanical concepts, related to the sophistication with which the ultrastructure of bone is modelled. Consideration of the fibrillar organization of bone ultrastructure improves over simpler concepts like an interpenetrating network of mineral crystals and collagen molecules, which in turn is superior to a crystal-foam representation with collagen inclusions. In fact, the most advanced concept treated here integrates the two others to a consistent whole: Each fibril is regarded as an interpenetrating network of collagen molecules and mineral crystals. The fibrils host the minority of the mineral crystals present in the tissues. At a higher observation scale, the fibrils function as templates or reinforcement in an extrafibrillar crystal foam-type matrix, which hosts the majority of the minerals present in the bone ultrastructure. The reinforcement function corresponds to low-mineralized tissues (such as deer antler) where the extrafibrillar mineral foam is softer than the fibrils, whereas the template function corresponds to high-mineralized tissues (such as cow tibia) where the extrafibrillar mineral foam is stiffer than the fibrils. The collagen is clearly represented as the governing element in inducing the anisotropy of the tissues, by (i) the anisotropy of molecular collagen itself, (ii) the anisotropy of the fibrils, and (iii) the oriented morphology of the cylindrical fibrils in the isotropic extrafibrillar space. |
| [27] | |
| [28] | |
| [29] | . I n this paper , a variational method for bounding the effective properties of nonlinear composites with isotropic phases, proposed recently by ponte casta帽eda ( J. Mech. Phys. Solids 39 , 45, 1991), is given full variational principle status. Two dual versions of the new variational principle are presented and their equivalence to each other, and to the classical variational principles, is demonstrated. The variational principles are used to determine bounds and estimates for the effective energy functions of nonlinear composites with prescribed volume fractions in the context of the deformation theory of plasticity. The classical bounds of Voigt and Reuss for completely anisotropic composites are recovered from the new variational principles and are given alternative, simpler forms. Also, use of a novel identity allows the determination of simpler forms for nonlinear Hashin-Shtrikman bounds, and estimates, for isotropic, particle-reinforced composites, as well as for transversely isotropic, fiber-reinforced composites. Additionally, third-order bounds of the Beran type are determined for the first time for nonlinear composites. The question of the optimality of these bounds is discussed briefly. |
| [30] | . Interfaces are often believed to play a role in the mechanical behavior of mineralized biological and biomimetic materials. This motivates the micromechanical description of the elasticity and brittle failure of interfaces between crystals in a (dense) polycrystal, which serves as the skeleton of a porous material defined one observation scale above. Equilibrium and compatibility conditions, together with a suitable matrix-inclusion problem with a compliant interface, yield the homogenized elastic properties of the polycrystal, and of the porous material with polycrystalline solid phase. Incompressibility of single crystals guarantees finite shear stiffness of the polycrystal, even for vanishing interface stiffness, while increasing the latter generally leads to an increase of polycrystal shear stiffness. Corresponding elastic energy expressions give access to effective stresses representing the stress heterogeneities in the microstructures, which induce brittle failure. Thereby, Coulomb-type brittle failure of the crystalline interfaces implies Drucker鈥揚rager-type (brittle, elastic limit-type) failure properties at the scale of the polycrystal. At the even higher scale of the porous material, high interfacial rigidities or low interfacial friction angles may result in closed elastic domains, indicating material failure even under hydrostatic pressure. This micromechanics model can satisfactorily reproduce the experimental strength data of different (brittle) hydroxyapatite biomaterials, across largely variable porosities. Thereby, the brittle failure criteria can be well approximated by micromechanically derived criteria referring to ductile solid matrices, both criteria being even identical if the solid matrix is incompressible. |
| [31] | The strength of granular media is characterized by the influence of the mean stress which is typical of frictional materials. The aim of the paper is to link the strength of the interfaces between grains and the strength of the granular assembly. The later is viewed as a porous polycrystal. A generalized self-consistent scheme in which the grain is represented by a rigid core surrounded by a deformable interface is developed. First, the linear elastic effective behavior is considered. Then, assuming that the failure of the interfaces is ductile, the effective strength is determined by means of non-linear homogenization techniques. Finally, the case of brittle interfaces is also considered. It is shown that the nature of failure is controlled by the porosity. |
| [32] | |
| [33] | . An inverse micromechanics approach allows interpretation of nanoindentation results to deliver cohesive-frictional strength behavior of the porous clay binder phase in shale. A recently developed strength homogenization model, using the Linear Comparison Composite approach, considers porous clay as a granular material with a cohesive-frictional solid phase. This strength homogenization model is employed in a Limit Analysis Solver to study indentation hardness responses and develop scaling relationships for indentation hardness with clay packing density. Using an inverse approach for nanoindentation on a variety of shale materials gives estimates of packing density distributions within each shale and demonstrates that there exists shale-independent scaling relations of the cohesion and of the friction coefficient that vary with clay packing density. It is observed that the friction coefficient, which may be interpreted as a degree of pressure-sensitivity in strength, tends to zero as clay packing density increases to one. In contrast, cohesion reaches its highest value as clay packing density increases to one. The physical origins of these phenomena are discussed, and related to fractal packing of these nanogranular materials. Copyright 漏 2010 John Wiley & Sons, Ltd. |
| [34] | . |
| [35] | . A methodology for interpreting instrumented sharp indentation with dual sharp indenters with different tip apex angles is presented by recourse to computational modeling within the context of finite element analysis. The forward problem predicts an indentation response from a given set of elasto-plastic properties, whereas the reverse analysis seeks to extract elasto-plastic properties from depth-sensing indentation response by developing algorithms derived from computational simulations. The present study also focuses on the uniqueness of the reverse algorithm and its sensitivity to variations in the measured indentation data in comparison with the single indentation analysis on Vickers/Berkovich tip (Dao et al. Acta Mater 49 (2001) 3899). Finite element computations were carried out for 76 different combinations of elasto-plastic properties representing common engineering metals for each tip geometry. Young’s modulus, , was varied from 10 to 210 GPa; yield strength, , from 30 to 3000 MPa; and strain hardening exponent, , from 0 to 0.5; while the Poisson’s ratio, , was fixed at 0.3. Using dimensional analysis, additional closed-form dimensionless functions were constructed to relate indentation response to elasto-plastic properties for different indenter tip geometries (i.e., 50°, 60° and 80° cones). The representative plastic strain , as defined in Dao et al. (Acta Mater 49 (2001) 3899), was constructed as a function of tip geometry in the range of 50° and 80°. Incorporating the results from 60° tip to the single indenter algorithms, the improved forward and reverse algorithms for dual indentation can be established. This dual indenter reverse algorithm provides a unique solution of the reduced Young’s modulus , the hardness and two representative stresses (measured at two corresponding representative strains), which establish the basis for constructing power-law plastic material response. Comprehensive sensitivity analyses showed much improvement of the dual indenter algorithms over the single indenter results. Experimental verifications of these dual indenter algorithms were carried out using a 60° half-angle cone tip (or a 60° cone equivalent 3-sided pyramid tip) and a standard Berkovich indenter tip for two materials: 6061-T6511 and 7075-T651 aluminum alloys. Possible extensions of the present results to studies involving multiple indenters are also suggested. |
