PERIDYNAMIC MODELING OF WAVE PROPAGATION AND THE EFFECT OF MICROSTRUCTURE IN ANISOTROPIC MATERIALS
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摘要: 金属的微观结构显著地影响其宏观力学性能, 深入理解金属在动态条件下的力学响应具有重要意义. 近场动力学是一种新兴的非局部理论, 在研究冲击作用下材料的损伤与破坏方面具有天然理论优势. 文章以非常规态型近场动力学为理论框架, 模拟了波在各向异性材料中的传播过程以及波与微观结构之间的相互作用. 为了在非局部模型中准确捕获冲击波, 采用非迭代近似黎曼解修正非局部力态中的界面压力, 保证了冲击波界面物理量的守恒性与求解稳定性, 兼顾了计算效率. 基于晶体塑性理论和均匀化理论, 将有限变形晶体黏塑性本构引入近场动力学的非局部理论框架. 与解析解的对比表明, 所提出的模型能够有效消除波阵面后的非物理振荡、准确捕捉弹塑性双波结构. 与实验结果对比表明, 高效显式晶体黏塑性近场动力学模拟所得的宏观力学响应吻合较好. 在此基础上, 探讨了不同的冲击速度、材料的初始微观结构对冲击响应的各向异性和塑性变形的非均匀性的影响. 晶体取向决定了Hugoniot弹性极限和材料的软硬取向, 进而影响弹塑性波结构的演化, 晶界导致Hugoniot状态下应力分布和累积滑移应变呈现高度非均匀性.Abstract: The microstructure of metals significantly impacts their macroscopic mechanical properties, and a deep understanding of the mechanical response of metals under dynamic conditions is of great significance. Peridynamics, an emerging nonlocal theory, has a natural theoretical advantage in studying the damage and failure of materials under impact. In this paper, the theory of non-ordinary state-based peridynamics is used as the theoretical framework to simulate the propagation of waves in anisotropic materials and to investigate the interaction between waves and microstructures. To accurately capture shock waves in the nonlocal model of peridynamics, the interface pressure in the nonlocal force state is corrected using an approximate Riemann solution, ensuring the conservation and solution stability of the physical quantities at the shock wave interface. To improve computational efficiency, a non-iterative solver is implemented in the peridynamic framework. Based on the rate dependent crystal plasticity theory and homogenization theory, the finite deformation crystal viscoplastic constitutive model is introduced into the nonlocal theoretical framework of peridynamics. Comparisons with analytical solutions show that the proposed model can effectively eliminate non-physical oscillations behind the wavefront and accurately capture the elastic as well as elastoplastic wave structures. Comparisons with experimental results of annealed oxygen-free high conductivity (OFHC) copper subjected to compression show that the macroscopic mechanical response obtained from the crystal viscoplastic peridynamics simulation is in good agreement, and the explicit computation exhibits high efficiency. On this basis, the effects of different impact velocities and the initial microstructure of the material on the anisotropy of the impact response are discussed. Specifically, the heterogeneity of plastic deformation is captured in the simulations. The crystal orientation determines the Hugoniot elastic limit and also the soft and hard orientations of the material, thereby affecting the evolution of the elastoplastic wave structure. Grain boundaries lead to highly heterogeneous stress distributions and cumulative slip strains in the Hugoniot state.
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Keywords:
- peridynamics /
- impact /
- crystal /
- microstructure /
- macro responses
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引 言
金属材料在冲击载荷作用下的动态力学响应是力学、材料科学及物理等学科研究的交叉领域[1]. 金属材料的动态塑性变形是一种宏微观耦合的多尺度力学行为. 在冲击波的作用下, 微观结构的响应与演化不仅决定了金属材料的宏观响应特征, 还可能对材料的服役安全和使用寿命产生重要影响. 因此, 深入理解金属的动态力学响应对许多新技术及其应用的发展至关重要[2-3].
金属材料通常由大量晶粒构成, 尽管其宏观塑性变形可表现出均匀性以及一定程度的各向同性, 但在微观尺度上塑性变形既非均匀也非各向同性[4]. 经典的材料宏观本构模型不包含塑性变形的微观机制. 以晶体学为基础的晶体塑性理论考虑了晶体内微观滑移剪切机制, 认为塑性变形发生在晶体内部特定面和特定方向上, 通过构建塑性变形张量, 能有效描述金属多晶体材料的各向异性、织构演化和非均匀塑性变形[5]. 随着计算硬件、数值计算方法和微观组织建模技术的迅速发展, 基于晶体塑性理论的有限元模拟方法已发展成为研究晶体材料各向异性、非均匀变形的重要手段[6-8], 有效实现了考虑制造工艺影响的材料力学性能准确预测[9-11]. 此外, 晶体塑性有限元方法还能与其他方法进行耦合[12-13], 在更细观的尺度描述材料力学行为和组织演化. 尽管晶体塑性有限元模型在联系微观变形机制与宏观力学响应方面具有显著优势, 但基于局部思想以偏微分方程为基础的连续模型难以描述空间上的不连续性, 在处理材料性质和力学行为中的含奇性问题时面临极大挑战[14], 往往需要借助额外的数值技术, 如扩展有限元中能描述不连续现象的附加函数等[15].
近场动力学是一种描述连续介质的非局部理论, 其引入了非局部特征长度并通过积分方程描述质点运动, 在统一的连续模型中处理不连续现象和连续过程[16]. 鉴于此优势, 国内外学者就近场动力学理论与应用进行了广泛研究[17-18]. 键型(bond-based)近场动力学模型最早被提出, 大量应用于准脆性材料的变形、损伤和冲击破坏模拟[19]. 但是, 键型近场动力学的本构模型较为简单, 难以描述材料各种非线性和复杂力学行为. 随后, 态型(state-based)近场动力学模型被提出[20], 并进一步分为常规态型和非常规态型两类. 其中, 非常规态型近场动力学能方便地与经典连续介质力学本构模型中的物理量联系, 实现经典本构模型的非局部重构.
以此为基础, 在基于非局部思想的近场动力学理论框架内考虑材料微观变形机制是近年来新的探索方向[21-24]. Lakshmanan等[25]提出了用于预测多晶微结构在弹塑性变形中的细观局部化现象的三维近场动力学模型, 成功地模拟了实验中的晶粒平均应变. Dong等[26]以近场动力学为理论框架模拟了基于位错的弹塑性变形和断裂. Gu等[27]在近场动力学模型框架内引入晶体塑性本构关系, 并开发了显式动态求解器和隐式准静态非线性求解器. Hu等[28]通过开发一种基于键型近场动力学的非局部连续体模型, 模拟晶体固体中的中尺度位错运动或晶体滑移和剪切裂纹. 然而, 在近场动力学非局部模型中, 对冲击波的准确模拟常常被忽视, 非局部参数如何影响冲击波的传播尚不清楚, 冲击载荷作用下微观结构的影响及其随时空演化过程的模拟准确性更是难以保证.
针对上述问题, 本文基于非迭代式黎曼求解实现近场动力学模型中冲击波间断面捕获, 并采用了完全显示的晶体黏塑性本构模型来描述微观塑性滑移变形和微观结构演化, 整体模拟框架如图1所示. 本文探讨了材料初始微观结构包括晶体取向和晶界对冲击响应各向异性和塑性变形非均匀性的影响.
1. 近场动力学理论
1.1 近场动力学基本方程
近场动力学理论将有限距离的材料点的相互作用纳入运动方程, 同时用积分方程代替偏微分方程. 近场动力学中点${\boldsymbol{x}}$的积分微分运动方程为[29]
$$ \rho \ddot{{\boldsymbol{u}}}[{\boldsymbol{x}},t] = {\boldsymbol{L}}[{\boldsymbol{x}},t] + {\boldsymbol{b}}[{\boldsymbol{x}},t] $$ (1) 式中, $\rho $为质量密度, $\ddot {\boldsymbol{u}}$为加速度, ${\boldsymbol{b}}$为体力密度矢量, 描述非局部相互作用的${\boldsymbol{L}}[{\boldsymbol{x}},t]$定义为
$$ \boldsymbol{L}[{\boldsymbol{x}}, t] = \int_{\mathcal{H}_{x}}\left\{\underline{\boldsymbol{T}}[{\boldsymbol{x}}, t]\left\langle {\boldsymbol{x}}^{\prime}-\boldsymbol{x}\right\rangle-\underline{\boldsymbol{T}}\left[{\boldsymbol{x}}^{\prime}, t\right]\left\langle {\boldsymbol{x}}-{\boldsymbol{x}}^{\prime}\right\rangle\right\} {\mathrm{d}} V_{x^{\prime}} $$ (2) 式中, $\underline{{\boldsymbol{T}}} $是力矢量状态场. 材料点${\boldsymbol{x}}$和${\boldsymbol{x}}'$之间的初始相对位置向量定义为
$$ {\boldsymbol{\xi}} = {\boldsymbol{x}}' - {\boldsymbol{x}} $$ (3) 当前构型中位置由${\boldsymbol{y}} = {\boldsymbol{x}} + {\boldsymbol{u}}$和${\boldsymbol{y}}' = {\boldsymbol{x}}' + {\boldsymbol{u}}'$计算, 其中${\boldsymbol{u}}$和${\boldsymbol{u}}'$表示位移. $t$时刻的变形矢量态被定义为
$$ \underline{{\boldsymbol{Y}}} [{\boldsymbol{x}},t]\left\langle {\boldsymbol{\xi}} \right\rangle = {\boldsymbol{y}}' - {\boldsymbol{y}} $$ (4) 非局部作用积分域定义为
$$ {\mathcal{H}_x} = \mathcal{H}({\boldsymbol{x}},\delta ) = \{ {\boldsymbol{x}}' \in {\mathbb{R}^d}:\{ \parallel {\boldsymbol{x}}' - {\boldsymbol{x}}\parallel \leqslant \delta \} ,d = 1,2,3\} $$ (5) 式中, $\delta $称为点${\boldsymbol{x}}$的“作用域”, 为非局部特征长度参数. 材料点${\boldsymbol{x}}$和${\boldsymbol{x}}'$的力态分别定义为$\underline{{\boldsymbol{T}}} [{\boldsymbol{x}},t]$和$\underline{{\boldsymbol{T}}} [{\boldsymbol{x}}',t]$. 在非常规态型近场动力学理论中, 力矢量状态场$\underline{{\boldsymbol{T}}} $与${\boldsymbol{\xi}} $之间的关系为
$$ \underline{{\boldsymbol{T}}} [{\boldsymbol{x}},t]\left\langle {\boldsymbol{\xi}} \right\rangle = \underline{\omega } \left\langle {\boldsymbol{\xi}} \right\rangle {\boldsymbol{P}}{{\boldsymbol{K}}^{ - 1}}{\boldsymbol{\xi}} $$ (6) 式中, ${\boldsymbol{P}}$是${\boldsymbol{x}}$点的第一皮奥拉-基尔霍夫应力张量, $\underline{\omega } \left\langle {\boldsymbol{\xi}} \right\rangle $是权重函数, 反映材料点相互作用与距离之间的关系. 形状张量${\boldsymbol{K}}[{\boldsymbol{x}}]$定义为
$$ \boldsymbol{K}[\boldsymbol{x}] = \int_{\mathcal{H}_{x}} \underline{\omega}\langle{\boldsymbol{\xi}}\rangle({\boldsymbol{\xi}} \otimes {\boldsymbol{\xi}}) {\mathrm{d}} V_{x_{j}} $$ (7) 式中, 符号$ \otimes $表示两个向量的并积. 非局部变形梯度张量采用下式[29]
$$ \begin{split} &\boldsymbol{F}[{\boldsymbol{x}}, t] = \left[\int_{\mathcal{H}_{x}} \underline{\omega}\langle{\boldsymbol{\xi}}\rangle \underline{\boldsymbol{Y}} \otimes {\boldsymbol{\xi}} {\mathrm{d}} V_{x^{\prime}}\right] \boldsymbol{K}^{-1} = \\ &\qquad \boldsymbol{I} + \left[\int_{\mathcal{H}_{x}} \underline{\boldsymbol{ {\omega}}} \langle{\boldsymbol{\xi}}\rangle \boldsymbol{\eta} \otimes {\boldsymbol{\xi}} {\mathrm{d}} V_{{{x}}'}\right] \boldsymbol{K}^{-1} \end{split}$$ (8) 1.2 冲击波捕捉
为了有效地使用近场动力学描述冲击波作用下的材料动力学响应, 首先应该解决基本的冲击波物理问题包括冲击雨贡纽和冲击波面动力学过程等. Godunov方法使用黎曼问题解来捕获间断面, 其优点在于考虑了冲击波界面物理量的守恒性与求解稳定性, 避免了使用可调人工黏性项捕捉冲击波. 但缺点是精确求解非线性黎曼问题的难度大和计算成本高. 本文采用了Dukowicz[30]提出的非迭代式黎曼解算器提高求解效率.
首先, 将式(2)中两相互作用材料点力态之差表示为
$$ \underline {\hat {\boldsymbol{T}}} = \underline{{\boldsymbol{T}}} [{\boldsymbol{x}},t]\left\langle {\boldsymbol{\xi}} \right\rangle - \underline{{\boldsymbol{T}}} [{\boldsymbol{x}}',t]\left\langle { - {\boldsymbol{\xi}} } \right\rangle $$ (9) 柯西应力与第一皮奥拉-基尔霍夫应力的关系为
$$ {\boldsymbol{P}} = {\mathrm{det}}({\boldsymbol{F}}){\boldsymbol{\sigma}} {{\boldsymbol{F}}^{ - {\mathrm{T}}}} $$ (10) 考虑对称关系$\underline{\omega } \left\langle {\boldsymbol{\xi}} \right\rangle = \underline{\omega } \left\langle { - {\boldsymbol{\xi}} } \right\rangle $, 两点相互作用力态之差简化为
$$ \begin{split} &\underline {\hat {\boldsymbol{T}}} = \underline{{\boldsymbol{\omega}} } \left\langle {\boldsymbol{\xi}} \right\rangle [ {\mathrm{det}}({\boldsymbol{F}}[{\boldsymbol{x}},t]){\boldsymbol{\sigma}} [{\boldsymbol{x}},t]{\boldsymbol{F}}{[{\boldsymbol{x}},t]^{ - {\mathrm{T}}}}{\boldsymbol{K}}{[{\boldsymbol{x}}]^{ - 1}} +\\ &\qquad {\mathrm{det}}({\boldsymbol{F}}[{\boldsymbol{x}}',t]){\boldsymbol{\sigma}} [{\boldsymbol{x}}',t]{\boldsymbol{F}}{[{\boldsymbol{x}}',t]^{ - {\mathrm{T}}}}{\boldsymbol{K}}{[{\boldsymbol{x}}']^{ - 1}}] {\boldsymbol{\xi}} \end{split}$$ (11) 将应力分解为球张量${\sigma _m}{\boldsymbol{I}}$和偏张量${\boldsymbol{s}}$, 则$\underline{\hat {\boldsymbol{T}}} $可相应地分解成两部分, 分别记为
$$ \begin{split} & {\underline {\hat {\boldsymbol{T}}} ^m} = \underline{{\boldsymbol{\omega}} } \left\langle {\boldsymbol{\xi}} \right\rangle [ {\mathrm{det}}({\boldsymbol{F}}[{\boldsymbol{x}},t]){{{\sigma}} _m}[{\boldsymbol{x}},t]{\boldsymbol{F}}{[{\boldsymbol{x}},t]^{ - {\mathrm{T}}}}{\boldsymbol{K}}{[{\boldsymbol{x}}]^{ - 1}} + \\ &\quad {\mathrm{det}}({\boldsymbol{F}}[{\boldsymbol{x}}',t]){{{\sigma}} _m}[{\boldsymbol{x}}',t]{\boldsymbol{F}}{[{\boldsymbol{x}}',t]^{ - {\mathrm{T}}}}{\boldsymbol{K}}{[{\boldsymbol{x}}']^{ - 1}}] {\boldsymbol{\xi}} \end{split} $$ (12) $$ \begin{split} &{\underline {\hat {\boldsymbol{T}}} ^d} = \underline{{\boldsymbol{\omega}} } \left\langle {\boldsymbol{\xi}} \right\rangle [ {\mathrm{det}}({\boldsymbol{F}}[{\boldsymbol{x}},t]){\boldsymbol{s}}[{\boldsymbol{x}},t]{\boldsymbol{F}}{[{\boldsymbol{x}},t]^{ - {\mathrm{T}}}}{\boldsymbol{K}}{[{\boldsymbol{x}}]^{ - 1}} +\\ &\quad {\mathrm{det}}({\boldsymbol{F}}[{\boldsymbol{x}}',t]){\boldsymbol{s}}[{\boldsymbol{x}}',t]{\boldsymbol{F}}{[{\boldsymbol{x}}',t]^{ - {\mathrm{T}}}}{\boldsymbol{K}}{[{\boldsymbol{x}}']^{ - 1}}] {\boldsymbol{\xi}} \end{split}$$ (13) 其中, 材料点${\boldsymbol{x}}$与${\boldsymbol{x}}'$的平均正应力${\sigma _m}[{\boldsymbol{x}},t]$和${\sigma _m}[{\boldsymbol{x}}',t]$用于定义黎曼问题间断面两侧的压力, 界面等效法线方向为
$$ {\boldsymbol{n}} = \frac{\underline {\hat {\boldsymbol{T}}} ^m}{\underline {\hat {\boldsymbol{T}}} ^d} $$ (14) 值得注意的是, Godunov求解方法采用外法线方向, 因此设定间断面两侧的压力为${p_L} = {\sigma _m}[{\boldsymbol{x}}',t]$和${p_R} = {\sigma _m}[{\boldsymbol{x}},t]$. 最终, 界面压力${p^ * }$可通过下式实现非迭代式近似黎曼解算[30]
$$ \begin{split} &{p_R}{A_R}|{w^ * } - w_{{\mathrm{min}}}^ * |({w^ * } - w_{{\mathrm{min}}}^ * ) + {p_L}{A_L}|{w^ * } - \\ &\qquad w_{{\mathrm{max}}}^ * |({w^ * } - w_{{\mathrm{max}}}^ * ) + p_R^ * - p_L^ * = 0 \end{split}$$ (15) 界面压力${p^ * }$求解完成后用于替换${\underline {\hat {\boldsymbol{T}}} ^m}$表达式中的平均正应力, 记为${\underline {\hat {\boldsymbol{T}}} ^{m * }}$
$$\begin{split} &{\underline {\hat {\boldsymbol{T}}} ^{m * }} = \underline{{\boldsymbol{\omega}} } \left\langle {\boldsymbol{\xi}} \right\rangle {p^ * }[ {\mathrm{det}}({\boldsymbol{F}}[{\boldsymbol{x}},t]){\boldsymbol{F}}{[{\boldsymbol{x}},t]^{ - {\mathrm{T}}}}{\boldsymbol{K}}{[{\boldsymbol{x}}]^{ - 1}} +\\ &\qquad {\mathrm{det}}({\boldsymbol{F}}[{\boldsymbol{x}}',t]){\boldsymbol{F}}{[{\boldsymbol{x}}',t]^{ - {\mathrm{T}}}}{\boldsymbol{K}}{[{\boldsymbol{x}}']^{ - 1}}] {\boldsymbol{\xi}}\end{split} $$ (16) 两点相互作用力态之差表示为
$$ \underline {\hat {\boldsymbol{T}}} = {\underline {\hat {\boldsymbol{T}}} ^{m * }} + {\underline {\hat {\boldsymbol{T}}} ^d} $$ (17) 2. 显式晶体黏塑性本构模型
将变形梯度${\boldsymbol{F}}$做乘法分解, 将其分解为与晶格弹性拉伸和刚体旋转相关的弹性分量${{\boldsymbol{F}}^e}$, 以及对应于晶格滑移诱导剪切的非弹性分量${{\boldsymbol{F}}^p}$
$$ {\boldsymbol{F}} = {{\boldsymbol{F}}^e}{{\boldsymbol{F}}^p} $$ (18) 速度梯度张量对应地分解为
$$ {\boldsymbol{L}} = {\dot {\boldsymbol{F}}^e}{{\boldsymbol{F}}^{e - 1}} + {{\boldsymbol{F}}^e}{\dot {\boldsymbol{F}}^p}{{\boldsymbol{F}}^{p - 1}}{{\boldsymbol{F}}^{e - 1}} = {{\boldsymbol{L}}^e} + {{\boldsymbol{L}}^p} $$ (19) 由于塑性变形是由滑移系统上的位错滑移引起的, 在中间构型下的塑性速度梯度张量可表示为[31]
$$ {\bar {\boldsymbol{L}}^p} = {\dot {\boldsymbol{F}}^p}{{\boldsymbol{F}}^{p - 1}} = \sum\limits_{\alpha = 1}^{{n_{{\mathrm{active}}}}} {{{\dot \gamma }^\alpha }{\boldsymbol{m}}_0^\alpha \otimes {\boldsymbol{n}}_0^\alpha } $$ (20) 式中, ${\dot \gamma ^\alpha }$是第$\alpha $个滑移系统相对于未变形晶格的滑移速率, ${n_{{\mathrm{active}}}}$是晶体中活跃滑移系统的总数, ${\boldsymbol{m}}_0^\alpha $和 ${\boldsymbol{n}}_0^\alpha $分别是定义滑移系统$\alpha $的滑移方向和滑移面法线的单位向量. 在当前构型下, 速度梯度张量可写为
$$ {\boldsymbol{L}} = {{\boldsymbol{L}}^e} + {{\boldsymbol{L}}^p} = {{\boldsymbol{L}}^e} + \sum\limits_{\alpha = 1}^{{n_{{\mathrm{active}}}}} {{{\dot \gamma }^\alpha }{{\boldsymbol{m}}^\alpha } \otimes {{\boldsymbol{n}}^\alpha }} = {\boldsymbol{D}} + {\boldsymbol{W}} $$ (21) 式中
$$ {{\boldsymbol{m}}^\alpha } = {{\boldsymbol{F}}^e}{\boldsymbol{m}}_0^\alpha $$ (22) $$ {{\boldsymbol{n}}^\alpha } = {\boldsymbol{n}}_0^\alpha {{\boldsymbol{F}}^{e - 1}} $$ (23) 变形率和旋转率张量分别写为
$$ {\boldsymbol{D}} = {{\boldsymbol{D}}^e} + \sum\limits_{\alpha = 1}^{{n_{{\mathrm{active}}}}} {{{\dot \gamma }^\alpha }{{\boldsymbol{P}}^\alpha }} $$ (24) $$ {\boldsymbol{W}} = {{\boldsymbol{W}}^e} + \sum\limits_{\alpha = 1}^{{n_{{\mathrm{active}}}}} {{{\dot \gamma }^\alpha }{{\boldsymbol{\varOmega}} ^\alpha }} $$ (25) 式中
$$ {{\boldsymbol{D}}^e} + {{\boldsymbol{W}}^e} = {\dot {\boldsymbol{F}}^e}{{\boldsymbol{F}}^{e - 1}} = {{\boldsymbol{L}}^e} $$ (26) $$ {{\boldsymbol{P}}^\alpha } = \frac{1}{2}({{\boldsymbol{m}}^\alpha } \otimes {{\boldsymbol{n}}^\alpha } + {{\boldsymbol{n}}^\alpha } \otimes {{\boldsymbol{m}}^\alpha }) $$ (27) $$ {{\boldsymbol{\varOmega}} ^\alpha } = \frac{1}{2}({{\boldsymbol{m}}^\alpha } \otimes {{\boldsymbol{n}}^\alpha } - {{\boldsymbol{n}}^\alpha } \otimes {{\boldsymbol{m}}^\alpha }) $$ (28) 弹性响应由弹性存储能势${w_e}$定义为
$$ {w_e} = {w_e}\left( {{{\underline{{\boldsymbol{E}}} }^e}} \right) = \frac{1}{2}{\underline{{\boldsymbol{E}}} ^e}:{\underline{{\boldsymbol{K}}} ^e}:{\underline{{\boldsymbol{E}}} ^e} $$ (29) 式中
$$ {\underline{{\boldsymbol{E}}} ^e} = \frac{1}{2}({{\boldsymbol{F}}^{e{\mathrm{T}}}}{{\boldsymbol{F}}^e} - {\boldsymbol{I}}) $$ (30) 第二皮奥拉-基尔霍夫应力张量为
$$ {\underline{{\boldsymbol{S}}} ^e} = \frac{{\partial {w_e}({{\underline{{\boldsymbol{E}}} }^e})}}{{\partial {{\underline{{\boldsymbol{E}}} }^e}}} = {\underline{{\boldsymbol{K}}} ^e}:{\underline{{\boldsymbol{E}}} ^e} $$ (31) 弹性模量${\underline{{\boldsymbol{K}}} ^e}$的4阶张量解释了立方晶格的弹性各向异性, 并假设对塑性变形不变. 以与晶格相关的正交基表示
$$ {{{\boldsymbol{K}}_{ij}}} = \left[ {\begin{array}{*{20}{c}} {{c_{11}}}&{{c_{12}}}&{{c_{12}}}&0&0&0 \\ {{c_{12}}}&{{c_{11}}}&{{c_{12}}}&0&0&0 \\ {{c_{12}}}&{{c_{12}}}&{{c_{11}}}&0&0&0 \\ 0&0&0&{{c_{44}}}&0&0 \\ 0&0&0&0&{{c_{44}}}&0 \\ 0&0&0&0&0&{{c_{44}}} \end{array}} \right] $$ (32) 式中, ${c_{11}}$, ${c_{12}}$和${c_{44}}$是3个独立的弹性常数.
作用在未变形晶格中第$\alpha $个滑移系统上的泰勒-施密德应力表示为
$$ {\tau ^\alpha } = {\boldsymbol{\tau}} :{{\boldsymbol{P}}^\alpha } = {{\boldsymbol{m}}^\alpha }{\boldsymbol{\tau}} {{\boldsymbol{n}}^\alpha } $$ (33) Kirchhoff应力张量${\boldsymbol{\tau}} $可由第二皮奥拉-基尔霍夫应力张量${\underline{{\boldsymbol{S}}} ^e}$计算
$$ {\boldsymbol{\tau}} = {{\boldsymbol{F}}^e}{\underline{{\boldsymbol{S}}} ^e}{{\boldsymbol{F}}^{e{\mathrm{T}}}} $$ (34) 因此
$$ \begin{split} &{{{\tau}} ^\alpha } = {\boldsymbol{m}}_0^\alpha {{\boldsymbol{F}}^{e{\mathrm{T}}}}{{\boldsymbol{F}}^e}{\underline{{\boldsymbol{S}}} ^e}{{\boldsymbol{F}}^{e{\mathrm{T}}}}{{\boldsymbol{F}}^{e - {\mathrm{T}}}}{\boldsymbol{n}}_0^\alpha = \\ &\qquad {\boldsymbol{m}}_0^\alpha {{\boldsymbol{C}}^e}{\underline{{\boldsymbol{S}}} ^e}{\boldsymbol{n}}_0^\alpha = {{\boldsymbol{C}}^e}{\underline{{\boldsymbol{S}}} ^e}:({\boldsymbol{m}}_0^\alpha \otimes {\boldsymbol{n}}_0^\alpha )\end{split} $$ (35) 流动法则将每个滑移系统的滑移率与当前屈服应力${\tau ^\alpha }$和滑移阻力${g^\alpha }$联系起来
$$ {\dot \gamma ^\alpha } = \hat f({\tau ^\alpha },{g^\alpha }) $$ (36) 滑移系统$\alpha $上的剪切速率由幂律关系定义[31]
$$ {\dot \gamma ^\alpha } = {\dot \gamma _0}{\left| {\frac{{{\tau ^\alpha }}}{{{g^\alpha }}}} \right|^{\tfrac{1}{m}}}{\mathrm{sgn}}({\tau ^\alpha }) $$ (37) 式中, ${\dot \gamma _0}$是参考剪切速率, $m$是瞬时应变率敏感性, ${g^\alpha }$代表滑移阻力.
滑移阻力演化表达为
$$ {\dot g^\alpha } = \sum\limits_\beta {{h^{\alpha \beta }}|{{\dot \gamma }^\beta }|} = \sum\limits_\beta {h({\gamma _{{\mathrm{tot}}}}){q^{\alpha \beta }}|{{\dot \gamma }^\beta }|} $$ (38) 其中, 反映晶内塑性变形程度的晶内累积滑移
$$ {\gamma _{{\mathrm{tot}}}} = \sum\limits_\alpha {\int {|{{\dot \gamma }^\alpha }|{\mathrm{d}}t} } $$ (39) 自硬化函数表示为
$$ h(\gamma ) = {h_0}{{{\mathrm{sech}}} ^2}\left( {\frac{{{h_0}{\gamma _{{\mathrm{tot}}}}}}{{{g_s} - {g_0}}}} \right) $$ (40) 以及硬化系数
$$ {q^{\alpha \beta }} = \left\{\begin{aligned} & {1,\quad \alpha = \beta } \\ & {q,\quad \alpha \ne \beta } \end{aligned} \right. $$ (41) 泰勒方法假设所有晶粒都受到相同的变形梯度. 它可以被视为一种均质化技术, 其中假设每个连续体材料点都由有限数量的${N_g}$个与纹理相关的晶粒组成. 由此获得材料点处的体积平均柯西应力
$$ \left\langle {\boldsymbol{\sigma}} \right\rangle = \sum\limits_{g = 1}^{{N_g}} {{v_g}{{\boldsymbol{\sigma}} ^g}} $$ (42) 式中, ${v_g}$是晶粒$g$的体积分数, ${{\boldsymbol{\sigma}} ^g}$是晶粒$g$中的柯西应力张量. 如果假设所有颗粒具有相同的体积, 则
$$ \left\langle {\boldsymbol{\sigma}} \right\rangle = \frac{1}{{{N_g}}}\sum\limits_{g = 1}^{{N_g}} {{{\boldsymbol{\sigma}} ^g}} $$ (43) 泰勒近似不能确保晶界处的应力平衡, 也不能考虑晶粒内存在的应力和应变梯度.
本文采用显式积分算法[32], 在增量开始时计算滑移和滑移阻力率后, 得到塑性速度梯度, 并在增量结束时计算并归一化塑性变形梯度. 然后更新滑移、滑移阻力和弹性变形梯度, 最后确定柯西应力和织构, 如算法1所示.
算法 1. 晶体黏塑性本构显式积分 1. 计算各滑移系统的分切应力 $ {\tau }^{\alpha } $
2. 计算各滑移系统的滑移率 $ {\dot{\gamma }}^{\alpha } $
3. 计算滑移阻力速率 $ {\dot{g}}^{\alpha } $
4. 计算塑性速度梯度 $ {\bar{\boldsymbol{L}}}^{p} $
5. 更新第n + 1步的塑性变形梯度
$ {\boldsymbol{F}}_{n + 1}^{p} = {\left(\boldsymbol{I}-{\bar {\boldsymbol{L}}}^{p}\Delta t\right)}^{-1}{\boldsymbol{F}}_{n}^{p} $
6. 归一化塑性变形梯度
7. 更新第n + 1步的滑移以及滑移阻力
$ {\gamma }_{n + 1}^{\alpha } = {\gamma }_{n}^{\alpha } + {\dot{\gamma }}_{n}^{\alpha } $, $ {g}_{n + 1}^{\alpha } = {g}_{n}^{\alpha } + {\dot{g}}_{n}^{\alpha } $
8. 计算弹性变形梯度 $ {\boldsymbol{F}}_{n + 1}^{e} $
9. 计算柯西应力 $ {\boldsymbol{\sigma }}_{n + 1} $
10. 通过极分解得到旋转张量 $ {\boldsymbol{R}}_{n + 1}^{e} $
11. 更新晶体取向3. 数值算例
由于表征材料高速冲击响应和本构关系的实验可合理地近似为一维模拟, 因此在数值模拟中直接施加相关的变形约束将极大减少计算成本. 本文采用的一维计算模型如图2所示, 变形只允许沿轴向发生, 横向则施加滑轮约束. 此外, 为简化碰撞接触过程的模拟, 将入射杆和透射杆视为一整体, 其左半部分赋予向右的非零初始速度, 右半部分的初始速度为0.
3.1 冲击波捕捉与验证
为了验证基于Godunov方法改进的近场动力学的准确性, 本小节对各向同性均质材料中的冲击波进行模拟分析. 假设材料为理想弹塑性体, 其杨氏模量$ E = 77.11\;\mathrm{G}\mathrm{P}\mathrm{a} $, 泊松比$ \nu = 0.334 $, 屈服强度$ Y = 270\;\mathrm{M}\mathrm{P}\mathrm{a} $, 屈服准则采用经典von Mises准则.
首先, 设定入射杆的初始速度为$ 30\;\mathrm{m}/\mathrm{s} $. 由于冲击速度较低, 杆内将仅产生弹性波. 透射杆中的轴向应力如图3所示. 由图3(a)可见, 直接采用近场动力学方法模拟所得的轴向应力剧烈振荡, 紧随波阵面之后的最大振幅可高达近100 MPa. 基于Godunov方法改进的近场动力学方法则有效地消除了波阵面后的非物理振荡, 同时也无需任何参数调试. 由图3(b)可知, 理论计算可得轴向应力为265.56 MPa, 基于Godunov方法改进的近场动力学与理想波阵面吻合.
当撞击速度较大时, 杆中的应力峰值大于材料的屈服强度, 此时杆中传播的应力波为弹塑性双波结构. 为此, 设定入射杆的初始速度为300 m/s. 理论计算可得弹性波速为6549.72 m/s, 塑性波速为5351.83 m/s; Hugoniot弹性极限为541.63 MPa, 塑性波的轴向应力为2.27 GPa. 图4(a)给出了两种离散间距$ \mathrm{\Delta }X $所捕捉到的弹塑性双波结构. 结果显示, 近场动力学模拟获得的Hugoniot弹性极限、Hugoniot态应力大小、弹塑性波速均与解析解吻合较好. 波阵面厚度则受离散间距大小影响, 随着离散间距减小, 模拟所得的波阵面厚度逐渐变窄并趋于解析解. 图4(b)给出了在离散间距$ \mathrm{\Delta }X $ = 0.01 mm情况下近场动力学作用域半径大小对模拟结果的影响. 总体来说, 作用域半径越小, 模拟所得的波阵面越陡峭, 越接近解析解; 当$ \delta = \mathrm{\Delta }X $时, 近场动力学模拟结果收敛于局部解, 因此结果与解析解差异最小.
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图 4 撞击速度300 m/s产生的弹塑性双波结构Figure 4. Elastic-plastic wave generated by an impact velocity of 300 m/s3.2 多晶塑性本构模型验证
为了验证多晶塑性本构模型的准确性, 对退火无氧(oxygen-free high conductivity, OFHC)铜的压缩力学行为进行模拟, 晶体塑性模型参数见表1. 首先随机生成50组Bunge欧拉角($ {\varphi }_{1},{\varPhi },{\varphi }_{2} $)作为初始晶体取向(见图5(a)), 然后施加等容变形
表 1 退火无氧铜晶体塑性模型参数Table 1. Constitutive parameters for annealed OFHC copper$ {\dot{\gamma }}_{0} $/s−1 $ m $ $ {g}_{0} $/MPa $ {g}_{\mathrm{s}\mathrm{a}\mathrm{t}} $/MPa $ {h}_{0} $/MPa $ a $ $ q $ $ {c}_{11} $/GPa $ {c}_{12} $/GPa $ {c}_{14} $/GPa 10−3 0.012 16 148 180 2.25 $ 1.4 $ 186 93 46.5 ![]()
图 5 (a)随机初始晶体取向; (b)退火无氧铜压缩力学行为模拟(SD: 标准差)Figure 5. (a) Random initial crystal orientation. (b) Simulation of the compressive mechanical behavior of OFHC (standard deviation, SD)$$ {\boldsymbol{F}} = \left[ {\begin{array}{*{20}{c}} {{\mathrm{exp}}( - 0.5\varepsilon )}&0&0 \\ 0&{{\mathrm{exp}}( - 0.5\varepsilon )}&0 \\ 0&0&{{\mathrm{exp}}(\varepsilon )} \end{array}} \right] $$ (44) 式中$ \varepsilon = -0.01 t $. 考虑到实验中施加的径向正应力为0, 轴向应力可通过应力偏量进行近似计算
$$ {\sigma _{33}} = {s_{33}} - \frac{1}{{2({s_{11}} + {s_{22}})}} $$ (45) 图5(b)给出了压缩过程不同初始晶体取向的应力区间, 结果显示“软”“硬”晶体取向之间的轴向应力差异可接近100 MPa. 宏观轴向应力依据泰勒模型采用体积平均, 结果显示模拟所得的宏观力学响应与实验结果[33]吻合较好.
3.3 单晶中冲击波传播模拟与分析
下面对冲击波在一维单晶模型中的传播进行分析, 离散间距$ \mathrm{\Delta }X $设为10 μm. 选取[111]方向作为冲击加载方向, 冲击速度50和100 m/s作用下的弹塑性波结构演化如图6所示. 随着传播距离的增加, 宏观波结构呈现明显的弹、塑性分离双波结构, 即弹性前驱波和塑性冲击波. 图7给出了同一时刻下不同冲击速度产生的弹塑性双波结构, 由图可知, 更高的冲击速度对弹性前驱波未产生明显影响, 但会导致塑性波波阵面更为陡峭.
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图 6 沿单晶[111]方向不同冲击速度下的弹塑性波演化Figure 6. Evolution of elastoplastic waves under different impact velocities along the [111] direction of a single crystal![]()
图 7 沿单晶[111]方向不同冲击速度下的冲击响应比较Figure 7. Comparison of impact response under different impact velocities along the [111] direction of a single crystal各向异性材料动态力学性能受到材料初始晶体取向影响. 图8显示了沿[111]和[100]方向冲击加载产生的波阵面. 由结果可知, 不同的冲击加载方向对Hugoniot弹性极限产生显著差异. 对于面心立方晶体结构, [100]方向加载下可激活滑移系的取向因子为$ 1/\sqrt{6} $, 而[111]方向加载下可激活滑移系的取向因子为$ \sqrt{6}/9 $. 二者相比, 沿[100]方向加载为软取向, 而[111]为硬取向, 因此, 仿真结果符合理论预期. 此外, 沿[100]方向加载产生的Hugoniot态轴向应力也略低于沿[111]方向加载, 但其差异要小于Hugoniot弹性极限的差异.
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图 8 冲击速度100 m/s沿不同方向的单晶冲击响应比较Figure 8. Comparison of impact response of single crystals at an impact velocity of 100 m/s along different directions3.4 多晶中冲击波传播模拟与分析
模拟冲击波跨越晶界过程对于深入理解冲击波与多晶金属材料的相互作用和塑性机理有重要意义. 下面对冲击波在一维多晶模型中的传播行为进行分析. 离散间距$ \mathrm{\Delta }X $采用10 μm, 每个晶粒随机地由8 ~ 12个离散点组成, 即晶粒大小在80 ~ 120 μm之间取整随机均匀分布, 各晶粒的初始晶体取向随机生成.
图9出给了多晶铜一维模型中的弹塑性波演化过程. 随传播距离增加, 弹性波与塑性波分离, 逐渐形成稳定的双波结构. 由模拟结果可见, 多晶各个晶粒之间的晶体取向差异对Hugoniot态产生明显影响, 轴向应力分布呈现非均匀性. 图10对比了同一时刻多晶与单晶中的冲击波. 观察可知, 多晶铜的Hugoniot弹性极限低于单晶铜沿[111]方向加载的Hugoniot弹性极限; 多晶铜的非稳定塑性前驱阶段斜率更高, 因此更快形成稳定塑性波; 多晶铜与单晶铜的稳定塑性波阶段几乎完全重合.
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图 9 多晶铜一维模型中弹塑性波演化Figure 9. Evolution of elastoplastic waves in a one-dimensional polycrystalline copper model![]()
图 10 单晶与多晶弹塑性波比较Figure 10. Comparison of elastoplastic waves between single crystals and polycrystals晶内累积滑移应变能较好地反应晶内的塑性变形程度. 为此, 图11比较了单晶与多晶内累积滑移应变分布. 达到Hugoniot态后, 单晶中累积滑移应变均匀分布, 而多晶中累积滑移应变呈现高度非均匀性, 且累积滑移应变大小居于沿单晶[111]和[100]方向加载所产生的累积滑移应变量之间. 实验中单晶与多晶的流动应力比较呈现相同规律, 即多晶拉伸的流动应力居于沿单晶[111]和[100]方向拉伸的流动应力之间[34].
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图 11 多晶与单晶的累积滑移分布比较Figure 11. Comparison of accumulated slip between single crystals and polycrystals面心立方单晶体单轴加载时, [111]和[100]均为几何稳定取向[3], 晶格不发生旋转, 因而施密特因子在加载过程中保持不变. 沿单晶[111]方向加载, 共6个可动滑移系($ \mathcal{N} = 6 $), 施密特因子$ {S}_{0} = \sqrt{6}/9 $; 沿单晶[100]方向加载, 共8个可动滑移系($ \mathcal{N} = 8 $), 施密特因子$ {S}_{0} = 1/\sqrt{6} $. 单轴加载所施加的变形通过滑移协调, 应变率$ \dot{E} $与滑移率之间的关系为 $ \dot{E} = \displaystyle\sum _{\alpha }{\dot{\gamma }}^{\alpha }{S}_{0} $[14]. 因此, 当单轴加载应变一定时, 晶内累积滑移应变$ {\gamma }_{{\mathrm{tot}}} $与施密特因子呈反比. 沿单晶[111]和[100]方向加载累积滑移应变大小之比的理论值为$ (1/\sqrt{6})/(\sqrt{6}/9) $, 即$ 3/2 $. 观察可知, 模拟结果(见图11)与该理论值十分接近.
图12显示了塑性波到达后多晶中累积滑移应变的轴向分布. 图中纵向条带的宽度代表晶粒大小, 条带颜色交替代表晶界. 由结果可知, 本文所提出的模拟方法成功地捕捉了晶界对滑移的影响. 图12所观察到的累积滑移应变高度非均匀性是由晶界即两侧晶体取向差异所致. 相邻晶粒间的累积滑移应变差异与可动滑移系和施密特因子有关, 此外, 当加载方向未沿着稳定晶体取向时, 晶粒将快速扭转并影响材料强化行为[35].
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图 12 晶界对累积滑移的影响. 纵向条带的宽度代表晶粒大小, 条带颜色处交替代表晶界Figure 12. The effect of grain boundaries on accumulated slip. The width of the longitudinal stripes represents the size of the grains, and the alternating colors of the stripes represent the grain boundaries4. 结论与展望
本文以非局部近场动力学为理论框架, 引入非迭代式黎曼求解和显式晶体黏塑性本构模型, 准确捕获了近场动力学模型中的冲击波, 模拟分析了材料的初始微观结构包括晶体取向和晶界等对冲击响应的各向异性和塑性变形的非均匀性影响. 本文得出的主要结论如下.
(1)验证了Godunov改进方法的近场动力学的准确性. 波阵面厚度受离散间距大小和非局部作用域大小的影响. 随着离散网格间距减小, 模拟所得的波阵面厚度逐渐变窄并趋于解析解; 随着作用域半径越小, 模拟所得的波阵面越陡峭, 从而越接近解析解; 当作用域半径接近一倍的网格间距时, 近场动力学模拟结果收敛于局部解.
(2)验证多晶塑性本构模型的准确性, 模拟所得的宏观力学响应与实验结果吻合较好.
(3)冲击波在一维单晶模型中的传播特性显著受到冲击速度和晶体取向的影响, 其中冲击速度主要影响塑性波的波阵面陡峭度, 而晶体取向则决定了Hugoniot弹性极限和材料的软硬取向, 进而影响弹塑性波结构的演化.
(4)多晶材料中冲击波的传播受到晶粒取向和晶界的影响, 导致Hugoniot态的轴向应力分布和累积滑移应变呈现高度非均匀性, 而单晶材料中这些量则相对均匀, 这一现象与晶体学取向和施密特因子密切相关.
本文提出的面向冲击问题的细观近场动力学模拟框架, 为进一步研究金属在冲击响应下的损伤和破坏行为提供了基础, 但其仍存在一些局限性. 本文基于非常规态型近场动力学理论, 其优势在于能够在非局部框架下更有效地重构晶体黏塑性本构模型. 针对如射流等高速冲击场景, 欧拉近场动力学方法更具适用性[36-37]. 因此, 为有效模拟宽域冲击速度工况, 发展本文所提出的模拟框架与欧拉近场动力学的耦合模型及自适应转换算法是下一步研究重点.
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图 4 撞击速度300 m/s产生的弹塑性双波结构
Figure 4. Elastic-plastic wave generated by an impact velocity of 300 m/s
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图 5 (a)随机初始晶体取向; (b)退火无氧铜压缩力学行为模拟(SD: 标准差)
Figure 5. (a) Random initial crystal orientation. (b) Simulation of the compressive mechanical behavior of OFHC (standard deviation, SD)
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图 6 沿单晶[111]方向不同冲击速度下的弹塑性波演化
Figure 6. Evolution of elastoplastic waves under different impact velocities along the [111] direction of a single crystal
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图 7 沿单晶[111]方向不同冲击速度下的冲击响应比较
Figure 7. Comparison of impact response under different impact velocities along the [111] direction of a single crystal
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图 8 冲击速度100 m/s沿不同方向的单晶冲击响应比较
Figure 8. Comparison of impact response of single crystals at an impact velocity of 100 m/s along different directions
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图 9 多晶铜一维模型中弹塑性波演化
Figure 9. Evolution of elastoplastic waves in a one-dimensional polycrystalline copper model
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图 10 单晶与多晶弹塑性波比较
Figure 10. Comparison of elastoplastic waves between single crystals and polycrystals
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图 11 多晶与单晶的累积滑移分布比较
Figure 11. Comparison of accumulated slip between single crystals and polycrystals
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图 12 晶界对累积滑移的影响. 纵向条带的宽度代表晶粒大小, 条带颜色处交替代表晶界
Figure 12. The effect of grain boundaries on accumulated slip. The width of the longitudinal stripes represents the size of the grains, and the alternating colors of the stripes represent the grain boundaries
表 1 退火无氧铜晶体塑性模型参数
Table 1 Constitutive parameters for annealed OFHC copper
$ {\dot{\gamma }}_{0} $/s−1 $ m $ $ {g}_{0} $/MPa $ {g}_{\mathrm{s}\mathrm{a}\mathrm{t}} $/MPa $ {h}_{0} $/MPa $ a $ $ q $ $ {c}_{11} $/GPa $ {c}_{12} $/GPa $ {c}_{14} $/GPa 10−3 0.012 16 148 180 2.25 $ 1.4 $ 186 93 46.5 
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