We present a general form of the gradient-enhanced damage theory and its numerical implementation using finite element method (FEM) in modeling quasi-static brittle crack growth in one- (1D), two- (2D) and three-dimensional (3D) bodies. Coupled equations of the equilibrium and a new implicit gradient damage formulation are introduced to govern the deformation of the solid and evolution of the damage. The resulting nonlocal damage evolution equation featuring the growth of diffusive crack is integrated with a characteristic length scale to eliminate the common mesh-bias issue in FEM implementation. In contrast to the traditional gradient-enhanced damage approaches, the nonlocal damage field here is defined as the primary variable of the damage evolution equation without interpolation mismatch between the displacement and nonlocal damage fields. For derivation of the material constitutive law and local damage parameter, a novel strain energy density (SED) function based on the energy limiter theory for brittle crack growth problems under small strain regime is introduced. To further improve the performance of the developed model, an initial SED threshold, which is used for determining the critical point when damage starts to initiate in the material, is integrated into the novel energy limiter theory. For preventing nonphysical failure in compression domains, the spectral decomposition technique for the strain tensor is adopted to split the reference SED. With integrating the energy limiter into the developed theory, unlike the conventional nonlocal damage theories where the interpretation of the length scale is still ambiguous, the developed nonlocal damage model defines the length scale parameter as the problem-dependent factor. The performance and ability of the proposed model are demonstrated via a set of representative numerical examples in 1D, 2D and 3D fracture problems.

我们提出了梯度增强损伤理论的一般形式及其使用有限元法 (FEM) 模拟一维 (1D)、二维 (2D) 和三维 (3D) 准静态脆性裂纹扩展的数值实现身体。引入平衡耦合方程和新的隐式梯度损伤公式来控制固体变形和损伤演化。由此产生的以扩散裂纹增长为特征的非局部损伤演化方程与特征长度尺度相结合，以消除 FEM 实施中常见的网格偏差问题。与传统的梯度增强损伤方法相比，这里的非局部损伤场被定义为损伤演化方程的主要变量，位移和非局部损伤场之间没有插值失配。为了推导材料本构定律和局部损伤参数，引入了一种基于能量限制器理论的新型应变能密度 (SED) 函数，用于解决小应变条件下的脆性裂纹扩展问题。为了进一步提高所开发模型的性能，将用于确定材料开始发生损伤时的临界点的初始 SED 阈值集成到新型能量限制器理论中。为了防止压缩域中的非物理故障，采用应变张量的谱分解技术来拆分参考 SED。将能量限制器整合到发展的理论中，与传统的非局部损伤理论不同，传统的非局部损伤理论对长度尺度的解释仍然模棱两可，开发的非局部损伤模型将长度尺度参数定义为问题相关因素。通过 1D、2D 和 3D 断裂问题中的一组代表性数值示例展示了所提出模型的性能和能力。