This article discusses the thermodynamics of elastic materials that exhibit heat conduction and viscosity. It begins by outlining the fundamental concepts of classical continuum mechanics, such as body, configuration, and force system, and emphasizes the importance of precise mathematical representations of these concepts. It then introduces the additional thermodynamic concepts required for the analysis, including temperature, specific internal energy, specific entropy, heat flux, and heat supply. The article proposes that thermodynamics should retain the general principles of mechanics while introducing two new principles: the first law of thermodynamics (the law of balance of energy) and the second law, expressed as the Clausius-Duhem inequality. The constitutive assumptions presented include a caloric equation of state, a temperature equation, a stress tensor composed of elastic and viscous terms, and a heat flux dependent on strain, specific entropy, and temperature gradient. These assumptions are not the most general but are sufficiently broad to cover many applications, including thermoelasticity and viscous fluids with heat conduction. The article also emphasizes the need to define thermodynamic fields within the framework of general laws and constitutive assumptions, avoiding non-mathematical concepts. It defines a thermodynamic process as a time-dependent set of configurations, force systems, and thermodynamic fields, and states that an admissible process must satisfy the constitutive assumptions and the Clausius-Duhem inequality. The article aims to provide a rigorous mathematical foundation for the thermodynamics of elastic materials without relying on vague or non-mathematical concepts.This article discusses the thermodynamics of elastic materials that exhibit heat conduction and viscosity. It begins by outlining the fundamental concepts of classical continuum mechanics, such as body, configuration, and force system, and emphasizes the importance of precise mathematical representations of these concepts. It then introduces the additional thermodynamic concepts required for the analysis, including temperature, specific internal energy, specific entropy, heat flux, and heat supply. The article proposes that thermodynamics should retain the general principles of mechanics while introducing two new principles: the first law of thermodynamics (the law of balance of energy) and the second law, expressed as the Clausius-Duhem inequality. The constitutive assumptions presented include a caloric equation of state, a temperature equation, a stress tensor composed of elastic and viscous terms, and a heat flux dependent on strain, specific entropy, and temperature gradient. These assumptions are not the most general but are sufficiently broad to cover many applications, including thermoelasticity and viscous fluids with heat conduction. The article also emphasizes the need to define thermodynamic fields within the framework of general laws and constitutive assumptions, avoiding non-mathematical concepts. It defines a thermodynamic process as a time-dependent set of configurations, force systems, and thermodynamic fields, and states that an admissible process must satisfy the constitutive assumptions and the Clausius-Duhem inequality. The article aims to provide a rigorous mathematical foundation for the thermodynamics of elastic materials without relying on vague or non-mathematical concepts.