Growth occurs in a wide range of systems ranging from biological tissue to additive manufacturing. This work considers surface growth, in which mass is added to the boundary of a continuum body from the ambient medium or from within the body. In contrast to bulk growth in the interior, the description of surface growth requires the addition of new continuum particles to the body. This is challenging for standard continuum formulations for solids that are meant for situations with a fixed amount of material. Recent approaches to handle this have used, for instance, higher-dimensional time-evolving reference configurations.

In this work, an Eulerian approach to this problem is formulated, enabling the side-stepping of the issue of constructing the reference configuration. However, this raises the complementary challenge of determining the stress response of the solid, which typically requires the deformation gradient that is not immediately available in the Eulerian formulation. To resolve this, the approach introduces additional kinematic descriptors, namely the relaxed zero-stress deformation and the elastic deformation; in contrast to the deformation gradient, these have the important advantage that they are not required to satisfy kinematic compatibility. The zero-stress deformation and the elastic deformation are used to eliminate the deformation gradient from the formulation, with the evolution of the elastic deformation shown to be governed by a transport equation. The resulting model has only the density, velocity, and elastic deformation as variables in the Eulerian setting. The proposed method is applied to simplified examples that demonstrate non-normal growth and growth with boundary tractions.

The introduction in this formulation of the relaxed deformation and the elastic deformation provides a description of surface growth whereby the added material can bring in its own kinematic information. Loosely, the added material “brings in its own reference configuration” through the specification of the relaxed deformation and the elastic deformation. This kinematic description enables, e.g., modeling of non-normal growth using a standard normal growth velocity and a simple approach to prescribing boundary conditions.

生长发生在从生物组织到增材制造的广泛系统中。这项工作考虑了表面生长，其中质量从环境介质或身体内部添加到连续体的边界。与内部的体积增长相反，表面增长的描述需要向主体添加新的连续粒子。这对于用于固定材料量情况的固体的标准连续体配方具有挑战性。例如，最近处理这个问题的方法使用了更高维的时间演化参考配置。

在这项工作中，制定了解决此问题的欧拉方法，从而避免了构建参考配置的问题。然而，这提出了确定固体应力响应的补充挑战，这通常需要在欧拉公式中无法立即获得的变形梯度。为了解决这个问题，该方法引入了额外的运动学描述符，即松弛的零应力变形和弹性变形；与变形梯度相比，这些具有重要的优势，即不需要满足运动学兼容性。零应力变形和弹性变形用于消除公式中的变形梯度，弹性变形的演变表明受输运方程控制。生成的模型只有密度、速度和弹性变形作为欧拉设置中的变量。所提出的方法应用于证明非正常增长和边界牵引增长的简化示例。

在此公式中引入松弛变形和弹性变形提供了表面生长的描述，由此添加的材料可以带来其自身的运动学信息。松散地，添加的材料通过松弛变形和弹性变形的规范“带来自己的参考配置”。这种运动学描述能够使用标准的正常生长速度和一种简单的方法来对非正常生长进行建模，以规定边界条件。