We develop a general class of thermodynamically consistent, continuum models based on mixture theory with phase effects that describe the behavior of a mass of multiple interacting constituents. The constituents consist of solid species undergoing large elastic deformations and incompressible viscous fluids. The fundamental building blocks framing the mixture theories consist of the mass balance law of diffusing species and microscopic (cellular scale) and macroscopic (tissue scale) force balances, as well as energy balance and the entropy production inequality derived from the first and second laws of thermodynamics. A general phase-field framework is developed by closing the system through postulating constitutive equations (i.e., specific forms of free energy and rate of dissipation potentials) to depict the growth of tumors in a microenvironment. A notable feature of this theory is that it contains a unified continuum mechanics framework for addressing the interactions of multiple species evolving in both space and time and involved in biological growth of soft tissues (e.g., tumor cells and nutrients). The formulation also accounts for the regulating roles of the mechanical deformation on the growth of tumors, through a physically and mathematically consistent coupled diffusion and deformation framework. A new algorithm for numerical approximation of the proposed model using mixed finite elements is presented. The results of numerical experiments indicate that the proposed theory captures critical features of avascular tumor growth in the various microenvironment of living tissue, in agreement with the experimental studies in the literature.

我们基于混合理论开发了一类热力学一致的连续介质模型，其相效应描述了大量相互作用的成分的行为。其成分由经历大弹性变形的固体物质和不可压缩的粘性流体组成。构成混合理论的基本构件包括扩散物质的质量平衡定律和微观（细胞尺度）和宏观（组织尺度）力平衡，以及从第一和第二定律导出的能量平衡和熵产生不等式。热力学。通过假设本构方程（即特定形式的自由能和耗散势能速率）来闭合系统来描述肿瘤在微环境中的生长，从而开发出通用相场框架。该理论的一个显着特点是，它包含一个统一的连续介质力学框架，用于解决在空间和时间上进化并参与软组织（例如肿瘤细胞和营养物质）生物生长的多个物种的相互作用。该公式还通过物理和数学上一致的耦合扩散和变形框架解释了机械变形对肿瘤生长的调节作用。提出了一种使用混合有限元对所提出的模型进行数值近似的新算法。数值实验结果表明，所提出的理论捕捉了活体组织各种微环境中无血管肿瘤生长的关键特征，与文献中的实验研究一致。