In this paper, we propose a finite element formulation for solving coupled mechanical/diffusion problems. In particular, we study hydrogen diffusion in metals and its impact on their mechanical behaviour (i.e. hydrogen embrittlement). The formulation can be used to model hydrogen diffusion through a material and its accumulation within different microstructural features of the material (dislocations, precipitates, interfaces, etc.). Further, the effect of hydrogen on the plastic response and cohesive strength of different interfaces can be incorporated. The formulation adopts a standard Galerkin method in the discretisation of both the diffusion and mechanical equilibrium equations. Thus, a displacement-based finite element formulation with chemical potential as an additional degree of freedom, rather than the concentration, is employed. Consequently, the diffusion equation can be expressed fundamentally in terms of the gradient in chemical potential, which reduces the continuity requirements on the shape functions to zero degree, $${\mathcal {C}}_{0}$$ C 0 , i.e. linear functions, compared to the $${\mathcal {C}}_{1}$$ C 1 continuity condition required when concentration is adopted. Additionally, a consistent interface element formulation can be achieved due to the continuity of the chemical potential across the interface—concentration can be discontinuous at an interface which can lead to numerical problems. As a result, the coding of the FE equations is more straightforward. The details of the physical problem, the finite element formulation and constitutive models are initially discussed. Numerical results for various example problems are then presented, in which the efficiency and accuracy of the proposed formulation are explored and a comparison with the concentration-based formulations is presented.

中文翻译：

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工程部件的耦合氢扩散和机械行为的建模框架

在本文中，我们提出了一种用于解决耦合机械/扩散问题的有限元公式。特别是，我们研究了金属中的氢扩散及其对它们的机械行为（即氢脆）的影响。该公式可用于模拟氢通过材料的扩散及其在材料的不同微观结构特征（位错、沉淀、界面等）内的积累。此外，可以结合氢对不同界面的塑性响应和内聚强度的影响。该公式在扩散和机械平衡方程的离散化中采用了标准 Galerkin 方法。因此，采用基于位移的有限元公式，将化学势作为额外的自由度，而不是浓度。因此，扩散方程可以从根本上用化学势梯度表示，这将形状函数的连续性要求降低到零度，$${\mathcal {C}}_{0}$$ C 0 ，即线性函数，与采用浓度时所需的 $${\mathcal {C}}_{1}$$ C 1 连续性条件相比。此外，由于界面上化学势的连续性，可以实现一致的界面元​​素公式 - 界面处的浓度可能不连续，这会导致数值问题。因此，有限元方程的编码更加直接。最初讨论了物理问题、有限元公式和本构模型的细节。然后给出各种示例问题的数值结果，