图  1  固体推进剂细观脱粘机理：(a)固体推进剂微观结构SEM图像；(b)无量纲裂纹长度$ \bar l = {\raise0.7 ex\hbox{$l$} \mathord{\left/ {\vphantom {l d}}\right.} \lower0.7 ex\hbox{$d$}} $^[10]

Figure  1.  Microscopic debonding mechanism of solid propellants: (a) SEM image of solid propellants microstructure; (b) Dimensionless crack length $ \bar l = {\raise0.7 ex\hbox{$l$} \mathord{\left/ {\vphantom {l d}}\right.} \lower0.7 ex\hbox{$d$}} $^[10]

图  2  55 vol%颗粒增强粘合剂基体复合材料代表性体积元(RVE)模型及其在三种变形下的仿真应力云图，应力计量对基体剪切模量${\mu _{\text{m}}}$进行归一化

Figure  2.  Geometry model and simulation stress clouds under three different deformations for the representative volume element (RVE) of 55 vol% particle reinforced binder matrix composite with stress metrology normalized to the matrix shear modulus ${\mu _{\text{m}}}$

图  3  数值仿真获取的55 vol%颗粒增强粘合剂基体复合材料RVE在单轴拉伸(UT)、等双轴拉伸(ET)、纯剪切(PS)下的2方向应力应变曲线，RVE中有50%区域脱粘(即$\bar l = 0.50$)

Figure  3.  The 2-directional stress-strain curves obtained by numerical simulation of 55 vol% particle-reinforced binder matrix composites RVE under uniaxial tensile (UT), equibiaxial tensile (ET), and pure shear (PS), with 50% region of debonding in RVE (i.e. $\bar l = 0.50$)

图  4  数值仿真获取的不同程度损伤下55 vol%颗粒增强粘合剂基体复合材料RVE的弹性模量(2方向)(${E_2}$)、体积模量($K$)、剪切模量($G$)散点图

Figure  4.  Scatter plots of Young's modulus (2-direction) (${E_2}$), bulk modulus ($K$), and shear modulus ($G$) of 55 vol% particle-reinforced binder matrix composites RVE under different levels of damage obtained by numerical simulation

图  5  含损伤力学本构神经网络训练框架(以变形梯度${\boldsymbol{F}}$和无量纲裂纹长度$\bar l$为输入，名义应力${\boldsymbol{P}}({\boldsymbol{F}},\bar l)$为输出。其中，力学本构神经网络部分以变形梯度${\boldsymbol{F}}$的不变量${\bar I_1}$、${\bar I_2}$、$J$为输入，输出标量自由能函数$\psi ({\boldsymbol{F}})$；损伤折减神经网络部分以无量纲裂纹长度$\bar l$为输入，输出损伤变量$L(\bar l)$。各隐藏层节点设置可训练权重$ {w_{m,n}} $，$m$为单个网络中的隐藏层编号，$n$为隐藏层中的节点编号；$ {f}_{n}(·) $为激活函数，$n$不同则函数不同)

Figure  5.  Damage-containing constitutive neural network training framework with deformation gradient ${\boldsymbol{F}}$ and dimensionless crack length $\bar l$ as inputs and nominal stress ${\boldsymbol{P}}({\boldsymbol{F}},\bar l)$ as output (In this framework, the Constitutive neural network part inputs invariant deviation ${\bar I_1}$, ${\bar I_2}$, $J$ of deformation gradient ${\boldsymbol{F}}$,outputs the scalar free energy function $\psi ({\boldsymbol{F}})$. The damage neural network part inputs the dimensionless crack lengths $\bar l$, and outputs the damage variables $L(\bar l)$. Each hidden layer node is set with trainable weights $ {w_{m,n}} $, $m$ is the number of hidden layers in a single network and $n$ is the number of nodes in the hidden layer. $ {f}_{i}(·) $ is the activation function, which is different for different $i$)

图  6  基于变形梯度不变量、自由能的力学本构神经网络训练框架(采用不变量偏差$[{\bar I_1} - 3]$、$[{\bar I_2} - 3]$、$[J - 1]$作为隐藏层的输入变量，第一层隐藏层以线性$ {(·)}^{1} $、二次函数$ {(·)}^{2} $作为激活函数，第二层以恒等函数$ (·) $和指数函数$ (\mathrm{exp}(·)-1) $作为激活函数)

Figure  6.  Constitutive neural network training framework based on deformation gradient invariants, free energy theory (The invariant deviations $[{\bar I_1} - 3]$, $[{\bar I_2} - 3]$, $[J - 1]$ are used as the input variables of the hidden layer, the first hidden layer uses linear function $ {(·)}^{1} $ , quadratic function $ {(·)}^{2} $ as the activation function, and the second hidden layer uses the identity $ (·) $ and exponential function $ (\mathrm{exp}(·)-1) $ as the activation function)

图  7  激活函数选择：(a)线性函数；(b)二次函数；(c)线性指数函数；(d)二次指数函数

Figure  7.  Activation function selection: (a) Linear; (b) Quadratic; (c) Linear exponential; (d) Quadratic exponential

图  8  基于Boltzmann生长函数的损伤折减神经网络训练框架(单节点传播结构，以线性函数$ ((·)-1) $和$ \text{Sigmiod}(·) $作为激活函数)

Figure  8.  Damage neural network training framework based on Boltzmann growth function (Single node propagation structure with linear function $ ((·)-1) $ and $ \text{Sigmiod}(·) $ as activation functions)

图  9  以单一变形数据作训练集，力学本构神经网络模型对未损伤RVE在UT/ET/PS下应力-变形关系的预测结果(其中${R^2}$为预测评估数，${R^2} = 1.00$表示基本吻合)

Figure  9.  Prediction of stress-deformation relationship of undamaged RVE under UT/ET/PS by constitutive neural network model using single deformation data as training set (Where ${R^2}$ is the number of predicted evaluations, and ${R^2} = 1.00$ indicates substantial agreement)

图  10  以UT、BT、PS三种变形数据作训练集，力学本构神经网络模型对未损伤RVE在UT/ET/PS下应力-变形关系的预测结果

Figure  10.  Prediction results of stress-deformation relationship for undamaged RVE under UT/ET/PS by constitutive neural network model using three kinds of deformation data(UT, BT, and PS)as the training set

图  11  损伤折减神经网络模型对RVE在UT/ET/PS下脱粘程度$\bar l$与损伤变量$L(\bar l)$映射关系的预测结果

Figure  11.  Prediction results of damage neural network model for mapping the degree of particle debonding $\bar l$ to damage variables $L(\bar l)$ of the RVE under UT/ET/PS

图  12  损伤折减神经网络模型对RVE模量${E_2}$/$K$/$G$随$\bar l$劣化的预测结果

Figure  12.  Predictions of modulus ${E_2}$/$K$/$G$ degradation with $\bar l$ for the RVE by damage neural network model

图  13  损伤本构神经网络模型对损伤为$\bar l = 0.50$的RVE在UT/ET/PS下的应力-变形关系预测结果

Figure  13.  Predictions of the stress-deformation relationship for the RVE with damage of $\bar l = 0.50$ under UT/ET/PS by the damage-containing constitutive neural network model

图  14  损伤本构神经网络模型对损伤为$\bar l = 0.25$和$\bar l = 0.75$的RVE在UT/ET/PS下的应力-变形关系预测结果

Figure  14.  Predictions of the stress-deformation relationship for the RVE with damage of $\bar l = 0.25$ and $\bar l = 0.75$ under UT/ET/PS by the damage-containing constitutive neural network model

图  15  RVE在损伤为$\bar l = 0.25$、$\bar l = 0.50$和$\bar l = 0.75$下的UT仿真应力云图

Figure  15.  UT simulation stress clouds of the RVE at damage of $\bar l = 0.25$,$\bar l = 0.50$and$\bar l = 0.75$