To develop constitutive models in continuum mechanics, one must consider kinematics, balance of momentum, balance of energy, and free-energy imbalance. Next, one must propose constitutive response functions that express dependent variables (e.g., free energy, stress, entropy, flux, and chemical potential) as functions of independent variables (e.g., deformation gradient, temperature, temperature gradient, and concentration of species). So, what are the general principles to determine which variables are dependent variables and which are independent variables?
Interesting question!
Following it for my own knowledge.
As a naive answer I would think the problem would dictate which variables are independent, and which variables are dependent.
Naveed Salman Thanks for your response. I completely agree with you—the underlying physics determines the relevant variables; experimental observations help a lot. In addition, some variables (e.g., entropy, chemical potential, species concentration) may be taken as either independent variables or dependent variables, depending on specific problems and conditions of interest. I was curious if there are certain guiding principles that we must bear in mind when making decisions.
Hi Dawei Song, this is a great question. I agree with you and Naveed that this question could be problem dependent. Here I have some thoughts: if your problem can be written into some variational form (for example Hu-Washizu Variational Principle for elasticity problem), all the conjugate variables are dependent. Or if you have some governing PDEs, maybe you could figure out how many independent equations to determine how many variables are independent.
Hello, Jin Yang, thank you for your helpful comments. I agree with you on both ends. Below are my thoughts.
It is a great idea to treat conjugate variables as dependent variables. This strategy appears to work well even if a problem has no convenient variational forms, in which case the free-energy imbalance can still provide sets of conjugate variables. For instance, guided by the free-energy imbalance, entropy is conjugate to temperature, stress is conjugate to strain, heat flux is conjugate to temperature gradient, chemical potential is conjugate to species concentration, etc.
It is also very helpful to use the number of governing PDEs to identify the number of unknown, or independent, variables. Then, constitutive laws can be treated as additional equations that serve to close the system—any remaining quantities that are NOT independent variables should be expressed as functions of those independent ones.
After applying these principles, one may be already a long way toward identifying independent and dependent variables. For many problems, however, this kind of classification is non-unique. This is because some variables can be written in terms of each other by simply inverting the related functions, thus interchanging the role of independent and dependent variables. Which choice is most convenient really depends on conditions of interest; for those cases, it might be hard to articulate a general guiding principle—one has to bring in intuitions in physics.
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