基于Kr的多晶体四阶规范不变梯度塑性模型欧纳不相容张量
A fourth order gauge-invariant gradient plasticity model for polycrystals based on Kröner's incompatibility tensor
摘要：
本文导出了一个新的具有各向同性硬化和Kr的无限小率无关梯度塑性四阶规范不变唯象模型öner不相容张量$inc(ε_p):= Curl[(Curl ε_p)^T]$，其中$ε_p=sym p$是对称的无穷小塑性应变张量，$p$是（非对称的）无穷小塑性变形。在这里，规范不变性表示在参考配置的复纯重新参数化下的不变性，适当地适应于几何线性设置。该模型的特点是缺陷能量的贡献，这是二次的张量$inc(ε_p)$，它包含各向同性硬化的基础上的对称无穷小塑性应变张量$\dot{ε_p}$的速率。我们通过引入一种新的梯度塑性旋转不变性的要求，我们称之为微观随机性，适合于描述多晶聚集体在介观尺度上，不符合经典的各向同性的要求，我们激励新的模型。这个新条件有效地减少了非对称无限小塑性变形$\dot{p}$到对称变形$\dot{ε_p}$的增量。在多晶的情况下，这个条件是关于对任意叠加的晶粒旋转不敏感的陈述。我们制定了一个适当的正规化非规范不变模型的数学存在性结果。正则化模型在包含无穷小共形映射的参考构形的重新参数化下是相当不变的。
英文摘要：
In this paper we derive a novel fourth order gauge-invariant phenomenological model of infinitesimal rate-independent gradient plasticity with isotropic hardening and Kröner's incompatibility tensor $inc(ε_p):= Curl[(Curl ε_p)^T]$, where $ε_p=sym p$ is the symmetric infinitesimal plastic strain tensor and $p$ is the (non-symmetric) infinitesimal plastic distortion. Here, gauge-invariance denotes invariance under diffeomorphic reparametrizations of the reference configuration, suitably adapted to the geometrically linear setting. The model features a defect energy contribution which is quadratic in the tensor $inc(ε_p)$ and it contains isotropic hardening based on the rate of the symmetric infinitesimal plastic strain tensor $\dot{ε_p}$. We motivate the new model by introducing a novel rotational invariance requirement in gradient plasticity, which we call micro-randomness, suitable for the description of polycrystalline aggregates on a mesoscopic scale and not coinciding with classical isotropy requirements. This new condition effectively reduces the increments of the non-symmetric infinitesimal plastic distortion $\dot{p}$ to their symmetric counterpart $\dot{ε_p}$. In the polycrystalline case, this condition is a statement about insensitivity to arbitrary superposed grain rotations. We formulate a mathematical existence result for a suitably regularized non-gauge-invariant model. The regularized model is rather invariant under reparametrizations of the reference configuration including infinitesimal conformal mappings.
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