We extend to finite elasticity the Data-Driven formulation of geometrically linear elasticity presented in Conti et al. (Arch Ration Mech Anal 229:79–123, 2018). The main focus of this paper concerns the formulation of a suitable framework in which the Data-Driven problem of finite elasticity is well-posed in the sense of existence of solutions. We confine attention to deformation gradients $$F \in L^p(\Omega ;{\mathbb {R}}^{n\times n})$$ F ∈ L p ( Ω ; R n × n ) and first Piola-Kirchhoff stresses $$P \in L^q(\Omega ;{\mathbb {R}}^{n\times n})$$ P ∈ L q ( Ω ; R n × n ) , with $$(p,q)\in (1,\infty )$$ ( p , q ) ∈ ( 1 , ∞ ) and $$1/p+1/q=1$$ 1 / p + 1 / q = 1 . We assume that the material behavior is described by means of a material data set containing all the states ( F , P ) that can be attained by the material, and develop germane notions of coercivity and closedness of the material data set. Within this framework, we put forth conditions ensuring the existence of solutions. We exhibit specific examples of two- and three-dimensional material data sets that fit the present setting and are compatible with material frame indifference.

中文翻译：

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数据驱动的有限弹性

我们将 Conti 等人提出的几何线性弹性的数据驱动公式扩展到有限弹性。(Arch Ration Mech Anal 229:79–123, 2018)。本文的主要重点是制定一个合适的框架，在该框架中，有限弹性的数据驱动问题在解决方案存在的意义上是适定的。我们将注意力限制在变形梯度 $$F \in L^p(\Omega ;{\mathbb {R}}^{n\times n})$$ F ∈ L p ( Ω ; R n × n ) 和第一个 Piola -Kirchhoff 强调 $$P \in L^q(\Omega ;{\mathbb {R}}^{n\times n})$$ P ∈ L q ( Ω ; R n × n ) ，其中 $$(p ,q)\in (1,\infty )$$ ( p , q ) ∈ ( 1 , ∞ ) 和 $$1/p+1/q=1$$ 1 / p + 1 / q = 1 。我们假设材料行为是通过包含材料可以达到的所有状态（ F ， P ）的材料数据集来描述的，并发展材料数据集的矫顽力和封闭性的密切概念。在这个框架内，我们提出了确保解决方案存在的条件。我们展示了适合当前设置并与材料框架无差异兼容的二维和三维材料数据集的具体示例。