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Part IV Elasticity and Thermodynamics of Reversible Processes

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Part IV Elasticity and Thermodynamics of Reversible Processes 175 Chapter 5 Elasticity: An Energy Approach Matter deforms when subjected to stresses. The English experimental physi- cist and inventor Robert Hooke (1635—1703), a contemporary of Isaac New- ton, rst stated formally this observation as an empirical relation, ‘ut tensio, sic vis’, which translates into our language as ‘extension is directly propor- tional to force’. In Engineering Mechanics, this observation translates into the theory of linear elasticity. Elasticity describes reversible material be- havior, and the notion of reversibility is a hallmark of another engineering science discipline: Thermodynamics. Thermodynamics is the study of en- ergy and energy transformations. Energy, which exists in many forms, such as heat, light, chemical energy, and electrical energy, is the ability to bring about change or to do work. Hence, an elastic, ie. reversible, behavior is readily recognized to be one where the energy provided to a material or structural system from the outside in form of work by forces and stresses, is stored as internal energy, which can be completely recovered at any time. Thermodynamics, therefore, provides a means to connect an observation to a fundamental physics law. The focus of this Chapter is to employ the laws of thermodynamics as a backbone for the engineering investigation of linear elastic behavior of materials and structures. 176 5.1. 1-D ENERGY APPROACH 177 0.4 d F0.35 0.3 0.25 0.2 KS 0.15 (x) 0.1 0.05 K x S 0 1 0 0.1 0.2x 0.3 0.4 0.5 F d Figure 5.1: Elasticity: 1-D Thought Model of Elasticity (left), and force— displacement curve with free energy (right). 5.1 1-D Energy Approach The 1-D spring system subjected to a force ononesideisthesimplest representation of an elastic behavior (Fig. 5.1). While equilibrium tells us that the externally supplied force is equal to the force in the spring, that is, = , Hooke’s observational law links to the extension at the point of load application through the proportionality constant (of dimension 1 2 []= [ ]= ): = = (5.1) Note that the elastic behavior of the spring may also be non-linear (Fig. 5.1). However, in all what follows we will restrict ourselves to linear elasticity. Let us now take another route to the problem, by evoking the First Law and the Second Law of Thermodynamics. 5.1.1 The First Law The First Law of Thermodynamics states that energy can be changed from one form to another, but that it cannot be created or destroyed. Energy is conserved. This is expressed by the internal energy U of a system (of dimension [U]=22). Specically the First Law informs us that the 178 CHAPTER 5. ELASTICITY: AN ENERGY APPROACH change in internal energy U of a system is due to work W and heat Q provided from the outside. Formally we write: U W Q (5.2) = + where we employ the symbol ‘ ( )’ to specify that the rate is not necessarily the time derivative of a function . For instance, the work rate W is generally not the total time derivative of the work W, ie. W =6 W .For instance, in the case of our 1-D spring system, the total work is W = , and the time derivative is W = + . In contrast, the work rate W is what the force realizes along the velocity = 0 at the point of load application, that is, W = . 5.1.2 The Second Law The Second Law of Thermodynamics species the direction of spontaneous change of energy, which is expressed by another physics quantity, the internal entropy S (of dimension [S]=221). The entropy is a measure of the quality of the energy. Specically, the Second Law informs us that the change of the internal entropy is always greater or equal to the entropy supplied to a system in heat form; that is: S Q (5.3) 0 where 0 0 stands for the absolute temperature, which is assumed to be constant in what follows. The dierence between the left and the right hand side is recognized as the fraction of entropy which is produced internally in aspontaneous(ie. uncontrolled) fashion, and which manifests itself as an internal heat source, D S Q = 0 0 (5.4) This internal heat source is called dissipation. More specically, if we substi- tute the First Law (5.2) into (5.4), we obtain: D W U S = ( 0 ) 0 (5.5) In this form, the Second Law appears as a balance law between the external work rate W and some form of recoverable energy (U 0S). This energy 5.1. 1-D ENERGY APPROACH 179 is called the free energy or Helmholtz Energy, which we denote by ,and which expresses the maximum internal capacity of a system to do work: =U 0S (5.6) By denition, an elastic behavior is one where the entire work rate supplied from the outside is stored as recoverable energy; that is: D W (5.7) Elastic: = =0 Hence, since W is known, it suces to determine the expression of the free energy of the spring. In the case of our 1-D spring system, the free energy is recognized to be the area below the curve (Fig. 5.1); thus in the 1 2 case of the linear behavior =( )= 2 . Use in (5.7) yields: μ ¶ =0 (5.8) Since this relation must hold for any rate we come to the same result as (5.1): ; = = (5.9) In contrast to (5.1) which was based on equilibrium and Hooke’s observational law, we here operate with energy states described by a state variable; here . If we add now one additional equilibrium condition to (5.9) we have instead of (5.1): ! = = = (5.10) The partial derivative of w.r.t. is called state equation, and species the constitutive law of the spring force in equilibrium with the external force . Finally, if the spring stiness is constant, we speak about linear elas- ticity. Inreturnif () depends on ,weneedtodealwithanon linear elasticity problem. On the other hand, it does not need much imagination to gure out that the spring stiness cannotbenegative,sinceitwould make not much sense that the spring extension ,inresponsetotheload , takes o in the opposite direction of the applied force. Behind this ob- servation, however, hides an important property of the free energy function 180 CHAPTER 5. ELASTICITY: AN ENERGY APPROACH x L 123 x z P 0 P Figure 5.2: 1-D Elasticity Example: Three-Truss System =(). In fact, the non-negativity of the spring constant is ensured provided that () is a convex function of its arguments; that is: 2 = 2 0 (5.11) We will see later that this convexity property is of critical importance for solving elasticity problems. 5.1.3 Example: Three-Truss System By way of example, consider the truss system displayed in gure 5.2. The structure is composed of three bars of same length held on the top and connected on the other side to a rigid beam device which is subjected to a load at midspan between two trusses. In a rst approach, such a truss system can be considered as a discrete system, in which each bar represents a spring of stiness . We want to evaluate the forces in the trusses (ie. the spring forces) and evaluate the displacement 0 at the point of load application. Ad hoc Approach To refresh our memory, let us start with equilibrium. Elementary force equilibrium and moment equilibrium (around the point of load application) yields: 1 + 2 + 3 = (5.12a) 31 + 2 3 =0 (5.12b) 5.1. 1-D ENERGY APPROACH 181 We readily recognize that the structure is statically indeterminate of degree 1; that is, there are only two equilibrium relations for the three unknowns, 12 and 3. Let us now introduce a linear elastic constitutive law for each truss of the form: = (5.13) where stands for the elongation of bar =1 3. We now need to ensure the kinematical compatibility between the bar elongations and the constraints on both the top (where the bars are xed) and the bottom where each bar is con- strained by the displacement of the rigid bars at the nodes. Compatibility requires: 1 = 1 (5.14a) 2 2 = = + ( ) (5.14b) 2 1 3 0 1 4 3 = = + ( ) (5.14c) 3 1 3 0 1 While there are apriorithree displacement unknowns in the problem, the geometrical compatibility allows us to reduce these unknowns to two. We here choose 1 and 0, the rst being the displacement of node 1 (Fig. 5.2), the second the displacement at the point of load application. Substitution of the constitutive law (5.13) and the geometrical compatibility conditions (5.14) in the equilibrium relations (5.12) now yields a system of two equations for the two unknowns: μ ( 1 +2 0¶)= (5.15a) 11 2 =0 (5.15b) 3 1 3 0 11 The linear elastic problem then is readily solved: 0 = 24 and 1 = 1 12 . The truss forces are given by: 7 1 = ; 2 = ; 3 = (5.16) 12 3 12 The problem nicely shows that the use of a constitutive law and geometrical compatibility conditions allows one to obtain a unique solution for a stati- cally indeterminate structure. In this approach, we start with equilibrium, substitute for forces a constitutive law that links forces to displacements, and nally ensure that the displacements are geometrically compatible. 182 CHAPTER 5. ELASTICITY: AN ENERGY APPROACH Energy Approach Let us now take the alternative energy route to the problem. Instead of equilibrium, we start out with the external work rate of the applied force : W = 0 (5.17) According to the Thermodynamics of reversible processes expressed by (5.7), this external work rate is equal to the change in free energy of the system.
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