MODELING MECHANICAL AND ELECTRICAL
2.4 ENERGY-BASED MODELING METHODS
The concepts of stress, strain, electric field, and electric displacement can be brought together under a single principle by considering the relationship between these state variables and the work and energy associated with a system. As discussed in Section 2.2, the product of mechanical force and mechanical displacement, or elec- tric field and electric displacement, results in work being done to a body. The work performed on a body can be related to the stored energy, as we shall see, and this rela- tionship will enable efficient methods of developing models of smart material systems.
Energy-based methods are based on the first law of thermodynamics, which states that a change in thetotal internal energyof a body is equal to the sum of thework performed on the bodyand theheat transfer. Denoting the total internal energy E (not to be confused with the electric field variable introduced earlier in the chapter), the work W, and the heat transfer Q, the first law is written as
dE=dW+dQ. (2.156)
In its most basic form, the first law of thermodynamics is a statement of the balance of energy. In this book we utilize this concept to develop equations that govern the deformation and motion of smart material systems.
In Section 2.2 we were introduced to the relationship between work and energy for mechanical and electrical systems. In that discussion we saw that the concepts of work and energy are strongly interrelated. If we have prescribed forces, mechanical or electrical, applied to a body, they will perform work on that body. In the same manner, if the forces have a functional relationship to the mechanical or electrical response, the body will store energy internally, and this stored energy is quantified by a potential functionV. The forces that produce this stored energy are related to the potential function through the gradient operator.
Another important result from Section 2.2 is that energy and work associated with electrical and mechanical systems are defined in terms of particularstate variables.
For example, work and energy associated with a mechanical system are defined in terms of force, stress, displacement, or strain. Mechanical work on a particle is quantified by a force acting through a distance, whereas the mechanical work (or stored energy) of an elastic body is defined in terms of strain and stress. Similarly, the work and energy of an electrical system are defined in terms of charge, electric field, and electric displacement.
For the moment, let us consider the first law, equation (2.156), applied to a system that does not exhibit heat transfer. In this case we can write that the change in internal energy is equivalent to the work done on the system,
dE=dW. (2.157)
Assume that the internal energy and work performed on the system are expressed in terms of a set ofgeneralized state variables. We denote the generalized state variables wi to highlight the fact that the variable can represent a mechanical state variable or
an electrical state variable. A critical feature of the generalized state variables is that they areindependent. Assuming that there areNindependent generalized states, then
dE(w1, . . . , wN)= ∂E
∂w1
dw1+ ∂E
∂w1
dw2+ · · · + ∂E
∂wN
dwN
(2.158) dW(w1, . . . , wN)= ∂W
∂w1
dw1+ ∂W
∂w1
dw2+ · · · + ∂W
∂wN
dwN.
Substituting equation (2.158) into equation (2.157) produces the equality
∂E
∂w1
dw1+ ∂E
∂w1
dw2+ · · · + ∂E
∂wN
dwN = ∂W
∂w1
dw1+∂W
∂w1
dw2+ · · · + ∂W
∂wN
dwN. (2.159) Equation (2.159) must hold forarbitarychanges in the state variables to maintain the energy balance stated by the first law. Assuming that the changes in the generalized states are arbitrary, and coupling this assumption to the fact that they are chosen to be independent of one another, means that all terms precedingdwion both sides of the equation must equal one another for the equation to be valid. Thus, use of the first law (ignoring heat transfer) results in the following set of equations:
∂E
∂w1 = ∂W
∂w1
∂E
∂w2 = ∂W
∂w2 (2.160)
...
∂E
∂wN
= ∂W
∂wN
.
The set of equations (2.160) must be satisfied for the energy balance specified by the first law to be maintained.
2.4.1 Variational Motion
The statement of energy balance in the first law leads to a set of equations that must be satisfied for differential changes in the generalized state variables. As introduced in Section 2.3, these changes in state variables are completely arbitrary. This definition of the change in the state variable, though, can be problematic if the change in the state variable violates any constraints associated with the system under examination.
For example, displacement on the boundary of an elastic body may be constrained to be zero, and it is important that the differential change in the state variables be chosen such that this constraint is not violated. The need to define a set of small changes in the state variables that are consistent with the constraints of the problem
f1 f1
f2
f3
Figure 2.16 Three-link system with applied forces, with a geometric constraint on one of the nodes.
leads us to the concept of avariationof the state variable. A variational change in the state variable is a differential change that is consistent with the geometric constraints of the problem. In this book we denote a variational change by the variableδ; thus, a variational change in the state variablewi is denotedδwi. As is the case with a differential, we can take the variation of a vector. An example would be the variation of the vector of generalized state variables,w, which is denotedδw.
To illustrate the concept of a variational motion, consider the system shown in Figure 2.16, consisting of three rigid links connected at three nodes. There is an applied force at each node that is performing work on the system. One of the nodes is constrained to move in thex2direction while the other two nodes are unconstrained.
Consider a differential motion of the node that lies against the frictionless constraint at the top of the figure. One potential differential motion is shown in Figure 2.17a. In this differential motion, we note that the node is moved through the constraint, and the connections between the node and the bars are not maintained.
Let’s consider a differential motion thatdoes maintain the geometric constraints of the problem. As shown in Figure 2.17b, we have a motion that is consistent with
du
f1
(a)
δu constraints
are maintained f1
(b)
Figure 2.17 (a) Potential differential motion and (b) variational motion of a node. Note that the variational displacement is consistent with the geometric constraints of the problem.
the boundary constraint of the frictionless surface, and the connection between the node and the bars is maintained. The differential motions in Figure 2.17a andbare both valid, but we see that only the differential motion in Figure 2.17bis consistent with the constraints of the problem.
Thevariational workassociated with a forcefis the dot product of the force and the variational displacement,δu,
δW=f·δu, (2.161)
which represents the amount of work performed when the article undergoes a differ- ential displacement that is consistent with the geometric constraints.
An important aspect of a variational motion is that constraint forces do not con- tribute to the variational work of a particle. Since all constraint forces are perpendicular to the motion, and the variational work is a dot product of the force and the variational displacement, constraint forces do not add to this function. This is beneficial to the development of equilibrium expressions for the system because the constraint forces need not be considered in the analysis of a problem.
The concept of variational motion also applies to the development of energy functions. Applying a variational displacement to a force that is dependent on u produces a variation in the energy, which is denoted
δU=f(u)·δu. (2.162)
2.5 VARIATIONAL PRINCIPLE OF SYSTEMS IN STATIC EQUILIBRIUM