Materials with memory

It has been shown that the thermodynamic foundations of plasticity may be considered within the framework of the continuum mechanics of materials with memory. A nonlinear material with memory is defined by a system of constitutive equations in which some state functions such as the stress tension or the internal energy, the heat flux, etc., are determined as functionals of a function which represents the time history of the local configuration of a material particle. 

The second approach, called the thermodynamic theory of materials with memory. 

The theories of elastic and viscoelastic materials can be obtained as particular cases of the theory of materials with memory. This theory enables the description of many important mechanical phenomena, such as elastic instability and phenomena accompanying wave propagation. The applicability of the methods of the third approach is, on the other hand, limited to linear problems. It does not seem likely that further generalization to nonlinear problems is possible within the framework of the assumptions of this approach. The results obtained concern problems of linear viscoelasticity. 

Coleman, B. D., Thermodynamics of Materials with Memory Treatise, Arch. Rat. Mech. Anal., 1964. 

Higher order terms can be obtained by writing the inner and outer solutions as expansions in powers of e and solving the sets of equations obtained by comparing coefficients. This enzymatic example is treated extensively in [73] and a connection with the theory of materials with memory is made in [82]. The essence of the singular perturbation analysis, as this method is called, is that there are two (or more in some extensions) time (or spatial) scales involved. If the initial point lies in the domain of attraction of steady states of the fast variables and these are unique and stable, the state of the system will rapidly pass to the stable manifold of the slow variables and, one might... 

State is given by independent variables of constitutive equations modeling the properties of such system, e.g., density, temperature, their gradients and time derivatives, deformation rate, etc. Constitutive equations need not be only lunctions, but, e.g., ffinctionals where state variables may also be lunctions of time (histories in materials with memory, cf. example in Rem. 3 in Chap. 2) or space (nonuniform or nonlocal systems). Sometimes (e.g., for eneigy) the state is determined also by velocity and other external influences, e.g., gravitation or radiation, cf. also Chap. 2. 

Coleman, B.D. Thermodynamics of materials with memory. Arch. Ration. Mech. Anal. 17, 1-46(1964)... 

Bowen, R.M., Chen, P.J. Acceleration waves in a mixture of chemically reacting materials with memory. Acta Mech. 19(3 ), 201-214 (1974)... 

Kestin, J., BataiUe, J. Materials with memory An essay in thermodynamics. J. Non-EquUib. Thermodyn. 5, 19-33 (1980)... 

The first synthetic materials with memory for a template were obtained by Dickey in 1949 using a silica gel matrix. Imprinted silica materials were produced by acid precipitation of aqueous solution of sodium silicate in the presence of dyes as templates (e.g., methyl orange). In the following years research on... 

The theory of thermomechanics adopts the concept of materials with memory . According to this concept the constitutive quantities depend on the history of the independent variables, and not only on their actual value. In other words, it is not definite that the instantaneous value of state variables (i.e. the state) completely determines the state. 

S. R. Lustig, R. M. Shay, and J. M. Caruthers, Thermodynamic Constitutive Equations for Materials with Memory on a Material Time Scale , J. Rheol. 40, 69-106 (1996). 

Coleman BD (1964) Thermodyntunics of materials with memory. Arch Ration Mech Anal 17 1 7... 

Green AE, Rivlin RS (1957) The mechanics of non-linear materials with memory. Arch Rational Mech Anal 1 1-21... 

Golden JM (2005) A proposal concerning the physical rate of dissipation in materials with memory. Q Appl Math 63 117-155... 

Golden JM (2001) Consequences of non-uniqueness in the free energy of materials with memory. Int J Eng Sci 39 53-70... 

The basic physical picture may be described as follows. During the motion, various points within a body experience different loading histories. For inelastic materials with memory, the deformation response of the material depends upon the loading history. This leads to asymmetric distributions of normal pressure and deformation even if the initial geometry of the static problem is symmetric with respect to an axis normal to the direction of motion. The asymmetry gives rise to a resistance to the motion. The resistance is known as the deformation friction. 

Colemann D. Bernard (1930-) US. math., researched hydrodynamics of non-classical fluids, theoiy of wave propagation in materials with memory, co-founder of rational thermodynamics Comenius (Komensky) Jan Amos (1592-1670) Czech educator and expatriate, known as the teacher of nations who necessitated spontaneity ( Janna Linguarum reserata , Orbis Sensualium Pictus or Didactica ragna )... 

This chapter deals with fundamental definitions, constitutive equations of a viscoelastic medium subject to infinitesimal strain, and the nature and properties of the associated viscoelastic functions. General dynamical equations are written down. Also, the boundary value problems that will be discussed in later chapters are stated in general terms. Familiar concepts from the Theory of Linear Elasticity are introduced in a summary manner. For a fuller discussion of these, we refer to standard references (Love (1934), Sokolnikoff (1956), Green and Zerna (1968), Gurtin (1972)). Coleman and Noll (1961) have shown that the theory described here may be considered to be a limit, for infinitesimal deformations, of the general (non-linear) theory of materials with memory. 

Green, A. E., and Rivlin, R. S., "The Mechanics of Non-Linear Materials with Memory, Part One," Arch. Rat. Mech. 

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