The thermodynamics of elastic materials with heat conduction and viscosity is discussed by Coleman and Noll. They begin by defining the basic concepts of classical continuum mechanics: body, configuration, and force system. These concepts are formalized as mathematical entities, allowing for precise statements of general principles and constitutive assumptions. The principles of mechanics are the same for all continuum mechanics, but the constitutive assumptions vary depending on the application. To discuss thermodynamics, five additional concepts are introduced: temperature, specific internal energy, specific entropy, heat flux, and heat supply. These are represented as mathematical fields over the body. The authors argue that thermodynamics should retain all general principles of mechanics and add two new principles: the first law of thermodynamics (energy balance) and the second law (Clausius-Duhem inequality). The constitutive assumptions 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, entropy, and temperature gradient. The constitutive assumptions are not the most general but are sufficiently general to cover many applications, including thermoelasticity and viscous fluids with heat conduction. The authors emphasize the importance of thermodynamic fields in the general laws of thermodynamics and constitutive assumptions to fix the mathematics of thermodynamics and allow a rational development without non-mathematical concepts. A thermodynamic process is defined as a time-dependent set of configurations, force systems, and thermodynamic fields compatible with mechanics and energy conservation. An admissible process must satisfy the Clausius-Duhem inequality. The paper aims to provide a rigorous mathematical framework for the thermodynamics of elastic materials with heat conduction and viscosity.The thermodynamics of elastic materials with heat conduction and viscosity is discussed by Coleman and Noll. They begin by defining the basic concepts of classical continuum mechanics: body, configuration, and force system. These concepts are formalized as mathematical entities, allowing for precise statements of general principles and constitutive assumptions. The principles of mechanics are the same for all continuum mechanics, but the constitutive assumptions vary depending on the application. To discuss thermodynamics, five additional concepts are introduced: temperature, specific internal energy, specific entropy, heat flux, and heat supply. These are represented as mathematical fields over the body. The authors argue that thermodynamics should retain all general principles of mechanics and add two new principles: the first law of thermodynamics (energy balance) and the second law (Clausius-Duhem inequality). The constitutive assumptions 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, entropy, and temperature gradient. The constitutive assumptions are not the most general but are sufficiently general to cover many applications, including thermoelasticity and viscous fluids with heat conduction. The authors emphasize the importance of thermodynamic fields in the general laws of thermodynamics and constitutive assumptions to fix the mathematics of thermodynamics and allow a rational development without non-mathematical concepts. A thermodynamic process is defined as a time-dependent set of configurations, force systems, and thermodynamic fields compatible with mechanics and energy conservation. An admissible process must satisfy the Clausius-Duhem inequality. The paper aims to provide a rigorous mathematical framework for the thermodynamics of elastic materials with heat conduction and viscosity.