近十几年来, 以页岩油气为代表的非常规能源的大量应用彻底改变了世界能源格局. 作为页岩中产油产气的主要烃源, 干酪根的物理化学与力学性质受到越来越多的关注. 对干酪根力学响应的精确描述具有极其重大的意义, 其不仅可以完善力——能学的基本研究框架, 更能为改进和发展非常规能源的采集技术 (例如水力压裂和原位转化) 奠定坚实的理论基础. 与普通固体材料不同, 以干酪根为代表的一大类非线性弹性材料具有显著的温度变化, 相对较小的尺寸以及承受不可避免的原位应力. 热效应, 尺寸效应和初始应力三者的共存对非线性弹性本构理论提出了巨大的挑战. 目前没有理论框架能够描述这三种因素的共存对本构关系产生的影响, 也缺乏对这三者之间耦合与相互作用的基本理解. 建立恰当的初始应力本构理论不仅是理性力学的基本难题, 更是工程科学的前沿方向. 基于上述背景, 本文针对热效应, 尺寸效应和初始应力的耦合与共存对超弹性本构关系的影响开展研究. 意图解决三个关键科学问题: 带有初始应力的热弹性本构关系, 带有初始应力的应变梯度热弹性本构关系以及热初始应力与非协调性的本构描述. 采用理论分析方法, 基于变形 Riemann 几何, 依托有限变形物质场论, 沿着力学几何化的思路开展研究工作, 着力填补初始应力本构理论的空白. 首先根据物质应变加法分解提出了初始应力内禀嵌入法, 该方法可以计入初始应力热弹性变形历史的影响. 一旦在自然状态上指定了经典热弹性自由能密度和内部约束, 随即给出带有任意初始应力的热弹性本构关系. 通过对所导出的初始应力热弹性本构方程进行线性化, 给出了 Cauchy 应力和质量熵的具体表达式. 随后将初始应力嵌入经典 Saint Venant-Kirchhoff 本构模型, 详细分析了由等温或绝热初始变形导致的初始应力对弹性和热学系数的影响. 与三种经典初始应力本构理论相比, 新提出的本构方程在不同的热力耦合条件下呈现出全新的特性, 相关结果完善了初始应力弹性本构关系, 填补了在非等温情况下的理论空白. 其次将初始应力本构理论拓展到了非简单物质, 通过将物质应变加法分解进一步扩展为物质应变梯度的分解, 建立了考虑应变梯度的初始应力内禀嵌入法. 该分解公式表明总物质应变梯度不仅包含初始物质应变梯度与后继物质应变梯度, 还出现了正比于后继物质应变的牵连项, 后者揭示了由应变梯度导致的初始应力非局部性, 并诱导了应变与应变梯度的耦合, 要求使用与初始应力卸载相关的微分方程来代替原有的代数方程. 一旦求得该微分方程的解, 随即给出带有任意初始应力的应变梯度热弹性本构关系, 这是目前唯一能同时考虑热效应, 尺寸效应和初始应力三者影响的弹性本构模型. 随后利用物质应变梯度的分解给出了嵌入初始应力的五种应变梯度不变量的详细形式, 并将均匀各向同性的等温初始应力嵌入经典可压缩 neo-Hookean 超弹性应变能, 给出了弹性模量和尺寸参数与初始应力的函数关系, 证明初始压缩应力会增强弹性模量, 但也会减弱尺寸效应. 最后通过引入中间状态, 将自然状态拉回 Euclid 空间成为自然构形, 类比热应力建立了基于物质应变三相分解的协调性破缺——曲率补偿框架. 进一步根据自由热膨胀会改变连续介质基本力学响应的理论证明, 提出了带有自由热膨胀的新不变量分解法. 一旦在自然构形上指定了无初始应力的经典热弹性模型, 就可以利用协调性破缺——曲率补偿框架构造带有热初始应力的热弹性本构关系. 随后利用物质 Riemann 曲率给出了非线性协调方程, 由此导出了自然温度方程, 该微分方程与广义相对论中的 Einstein 场方程形式相似, 这种相似性深刻地暗示了有关曲率补偿导致初始应力产生的标准机制. 作为实例, 将五种球对称热初始应力嵌入经典 neo-Hookean 热弹性本构关系以说明新本构框架的使用方法和可行性. 自然温度方程边值问题的解揭示出初始应力的非局部效应, 这从根本上改变了一直以来认为的 “初始应力不变量函数理论是初始应力超弹性本构关系的最一般形式” 这一观点, 为理解热初始应力的性质和影响开辟了一个全新的视角. 本研究既能够在应用层面给出适用于干酪根等受到初始应力, 热效应和尺寸效应共同影响的材料的非线性本构模型, 又可以在理论上突破经典的初始应力嵌入法, 为建立初始应力本构关系这一理性力学的基本难题提供新思想和新方法.

As the main hydrocarbon source of oil and gas production in shale, the physical, chemical and mechanical properties of kerogen have attracted more and more attention. It is of great significance to accurately describe the mechanical response of kerogen, which can not only improve the basic research framework of mechano-energetics, but also lay a solid theoretical foundation for improving and developing unconventional energy production technologies (such as hydraulic fracturing and in-situ conversion). Different from ordinary solid materials, a large class of nonlinear elastic materials represented by kerogen has significant temperature changes, relatively small size and inevitable in-situ stress and thermal effect. The coexistence of thermal effects, size effects, and initial stresses poses a great challenge to the nonlinear elastic constitutive theory. At present, there is no theoretical framework to describe the influence of the coexistence of these three factors on the constitutive relation, and there is a lack of basic understanding of the coupling and interaction among them. Therefore, establishing an appropriate initial stress constitutive theory is not only a basic problem of rational mechanics, but also a frontier direction of engineering science. Based on the above background, this dissertation investigates the influence of the coupling and coexistence of thermal effects, size-effects, and initial stresses on the hyperelastic constitutive relations. Three key scientific problems are addressed: the thermoelastic constitutive relation with initial stresses, the strain gradient thermoelastic constitutive relation with initial stresses, and the constitutive description of the thermal initial stress and strain incompatibility. The theoretical and analytical approach is based on the Riemannian deformation geometry and the finite deformation material field theory, and the study is carried out along the lines of the geometrization of solid mechanics, which fills the gap in the constitutive theory with initial stresses. First, the intrinsic embedding method of initial stresses is proposed based on the additive decomposition of material strains, which takes the effect of the thermoelastic deformation history into account. Once the classical thermoelastic free energy density and the internal constraint are specified on the natural state, the thermoelastic constitutive relations with arbitrary initial stresses are then obtained. The linearization of the derived thermoelastic constitutive equations with initial stresses gives specific expressions for the Cauchy stress and the mass entropy. After that, the initial stresses are embedded into the classical Saint Venant-Kirchhoff constitutive model, and the effects of the isothermal or adiabatic initial stresses on the elastic and thermal coefficients are analysed in detail. Compared with the three existing initially-stressed constitutive theories, the newly proposed constitutive equations show new features under different thermo-mechanically coupled conditions. The related results improve the initially-stressed elastic constitutive theory and fill the theoretical gap in non-isothermal cases. Secondly, the initially-stressed constitutive theory is extended to non-simple materials, and the new intrinsic embedding method of initial stresses that applies for strain gradient framework is established by further extending the additive decomposition of the material strain to the decomposition of the material strain gradient. The corresponding decomposition shows that the total material strain gradient not only contains the initial and subsequent material strain gradients, but also has an entangled term that is proportional to the subsequent material strain. This term reveals the nonlocality of the initial stress and induces the coupling between strains and strain gradients. It also requires the use of differential equations related to the unloading of the initial stress instead of the original algebraic equations. Once the solution of this equation is obtained, the strain gradient thermoelastic constitutive relation with arbitrary initial stresses can then be determined. This is the only hyperelastic constitutive model that can take into account thermal effects, size-effects, and initial stresses. After that, the detailed forms of the five initially-stressed strain gradient invariants are deduced based on the decomposition. Moreover, the uniform and isotropic isothermal initial stress is embedded into the classical compressible neo-Hookean hyperelastic strain energy. The elastic moduli and the length parameter are derived as the functions of the initial stress, which proves that initially compressive stresses can enhance the elastic modulus but reduce the size-effect. Finally, by introducing an intermediate state, the natural state is pulled back to Euclid space to restore the natural configuration. A compatibility broken-curvature compensation framework based on the three-factor decomposition of material strains is established by analogy with thermal stresses. A new invariant decomposition takes free thermal expansion into account is proposed based on the fact that free thermal expansion changes the underlying mechanical responses. Once the classical thermoelastic model without initial stresses is specified on the natural configuration, the compatibility broken-curvature compensation framework can be used to construct the thermoelastic constitutive relations with thermal initial stresses. after that, a nonlinear compatibility equation is derived using the material Riemannian curvature. The corresponding natural temperature equation has a similar form of the Einstein field equations in general relativity. Such similarity may deeply imply a standard mechanism concerning the curvature compensation leading to the initial stress. As an example, five spherically symmetric thermal initial stresses are embedded into the classical neo-Hookean thermoelastic constitutive relation to illustrate the feasibility of the new constitutive framework. The solution of the boundary-value problem of the natural temperature equation reveals the non-local effects of the initial stress, which fundamentally changes the long-standing view that “the invariant functional theory of initial stresses is the most general form of the hyperelastic constitutive relation of initial stresses”, and opens up a completely new perspective to understand the nature and effects of the thermal initial stress. The results of this dissertation not only provide a nonlinear constitutive model for elastic materials affected by initial stresses, thermal effects, and size effects, but also break through the classical initial stress embedding method in theory. These results provide new insights and new methods for establishing the initial stress constitutive relation, which is a fundamental problem of rational mechanics.