A non-local (gradient) plasticity model for porous metals that accounts for deformation-induced anisotropy is presented. The model is based on the work of Ponte Castañeda and co-workers on porous materials containing randomly distributed ellipsoidal voids. It takes into account the evolution of porosity and the evolution/development of anisotropy due to changes in the shape and the orientation of the voids during plastic deformation. A “material length” $\ell$ is introduced and a “non-local” porosity is defined from the solution of a modified Helmholtz equation with appropriate boundary conditions, as proposed by Geers et al. (2001); Peerlings et al. (2001). At a material point located at $\mathbf{x}$, the non-local porosity $f\left(\mathbf{x}\right)$ can be identified with the average value of the “local” porosity $f_{\mathrm{loc}}\left(\mathbf{x}\right)$ over a sphere of radius $R\simeq 3\ell$ centered at $\mathbf{x}$. The same approach is used to formulate a non-local version of the Gurson isotropic model. The mathematical character of the resulting incremental elastoplastic partial differential equations of the non-local model is analyzed. It is shown that the hardening modulus of the non-local model is always larger than the corresponding hardening modulus of the local model; as a consequence, the non-local incremental problem retains its elliptic character and the possibility of discontinuous solutions is eliminated. A rate-dependent version of the non-local model is also developed. An algorithm for the numerical integration of the non-local constitutive equations is developed, and the numerical implementation of the boundary value problem in a finite element environment is discussed. An analytical method for the required calculation of the eigenvectors of symmetric second-order tensors is presented. The non-local model is implemented in ABAQUS via a material “user subroutine” (UMAT or VUMAT) and the coupled thermo-mechanical solution procedure, in which temperature is identified with the non-local porosity. Several example problems are solved numerically and the effects of the non-local formulation on the solution are discussed. In particular, the problems of plastic flow localization in plane strain tension, the plane strain mode-I blunt crack tip under small-scale-yielding conditions, the cup-and-cone fracture of a round bar, and the Charpy V-notch test specimen are analyzed.

提出了一种多孔金属的非局部（梯度）塑性模型，该模型解释了变形引起的各向异性。该模型基于PonteCastañeda和同事在多孔材料上的工作，这些材料包含随机分布的椭圆形空隙。它考虑到了由于塑性变形过程中空隙的形状和取向的变化而引起的孔隙度的演化和各向异性的演化/发展。“材料长度”$\ell$如Geers等人所提出的，引入了修正的Helmholtz方程，并在适当的边界条件下求解，从而定义了“非局部”孔隙度。（2001）；Peerlings等。（2001）。在位于$\mathbf{X}$，非局部孔隙度 $F（\mathbf{X}）$ 可以用“局部”孔隙率的平均值来识别 $F_{\mathrm{位置}}（\mathbf{X}）$ 在半径范围内 $\mathrm{[R}\simeq 3\ell$ 集中于 $\mathbf{X}$。使用相同的方法来表示Gurson各向同性模型的非本地版本。分析了非局部模型产生的增量弹塑性偏微分方程的数学特征。结果表明，非局部模型的硬化模量总是大于相应的局部模型的硬化模量。结果，非局部增量问题保留了其椭圆特征，并且消除了不连续解的可能性。还开发了非本地模型的依赖于速率的版本。提出了非局部本构方程数值积分的算法，并讨论了有限元环境下边值问题的数值实现。提出了一种计算对称二阶张量特征向量所需的解析方法。通过材料“用户子程序”（UM​​AT或VUMAT）和耦合的热机械求解程序在ABAQUS中实现非局部模型，其中温度通过非局部孔隙率确定。数值解决了几个示例问题，并讨论了非局部公式对解决方案的影响。尤其是在平面应变张力中塑性流动局部化，在小规模屈服条件下平面应变模式-I钝化裂纹尖端，圆棒的杯形和圆锥形断裂以及夏比V型缺口试验等问题标本进行了分析。通过材料“用户子程序”（UM​​AT或VUMAT）和耦合的热机械求解程序在ABAQUS中实现非局部模型，其中温度通过非局部孔隙率确定。数值解决了几个示例问题，并讨论了非局部公式对解决方案的影响。尤其是在平面应变张力中塑性流动局部化，在小规模屈服条件下平面应变模式-I钝化裂纹尖端，圆棒的杯形和圆锥形断裂以及夏比V型缺口试验等问题标本进行了分析。通过材料“用户子程序”（UM​​AT或VUMAT）和耦合的热机械求解程序在ABAQUS中实现非局部模型，其中温度通过非局部孔隙率确定。数值解决了几个示例问题，并讨论了非局部公式对解决方案的影响。尤其是在平面应变张力中塑性流动局部化，在小规模屈服条件下平面应变模式-I钝化裂纹尖端，圆棒的杯形和圆锥形断裂以及夏比V型缺口试验等问题标本进行了分析。