Tissues in vivo are not stress-free. As we grow, our tissues adapt to different physiological and disease conditions through growth and remodeling. This adaptation occurs at the microscopic scale, where cells control the microstructure of their immediate extracellular environment to achieve homeostasis. The local and heterogeneous nature of this process is the source of residual stresses. At the macroscopic scale, growth and remodeling can be accurately captured with the finite volume growth framework within continuum mechanics, which is akin to plasticity. The multiplicative split of the deformation gradient into growth and elastic contributions brings about the notion of incompatibility as a plausible description for the origin of residual stress. Here we define the geometric features that characterize incompatibility in biological materials. We introduce the geometric incompatibility tensor for different growth types, showing that the constraints associated with growth lead to specific patterns of the incompatibility metrics. To numerically investigate the distribution of incompatibility measures, we implement the analysis within a finite element framework. Simple, illustrative examples are shown first to explain the main concepts. Then, numerical characterization of incompatibility and residual stress is performed on three biomedical applications: brain atrophy, skin expansion, and cortical folding. Our analysis provides new insights into the role of growth in the development of tissue defects and residual stresses. Thus, we anticipate that our work will further motivate additional research to characterize residual stresses in living tissue and their role in development, disease, and clinical intervention.

体内组织不是没有压力。随着我们的成长，我们的组织会通过生长和重塑来适应不同的生理和疾病状况。这种适应发生在微观尺度上，其中细胞控制其直接细胞外环境的微观结构以实现体内平衡。该过程的局部和异质性质是残余应力的来源。在宏观尺度上，可以使用类似于塑性的连续介质力学中的有限体积增长框架准确捕获增长和重塑。变形梯度乘法分裂为增长和弹性贡献带来了不相容的概念，作为对残余应力起源的合理描述。在这里，我们定义了表征生物材料不相容性的几何特征。我们为不同的增长类型引入了几何不相容性张量，表明与增长相关的约束会导致不相容性指标的特定模式。为了数值研究不兼容措施的分布，我们在有限元框架内进行分析。首先显示简单的说明性示例以解释主要概念。然后，对三种生物医学应用进行了不相容性和残余应力的数值表征：脑萎缩、皮肤扩张和皮质折叠。我们的分析提供了关于生长在组织缺陷和残余应力发展中的作用的新见解。因此，我们预计我们的工作将进一步激发更多研究，以表征活组织中的残余应力及其在发育、疾病、