tionship is two polynomials in the Laplace parameter s as follows: must be related by The coefficient term in equation(8.20)can also be written as a ratio of when t>0 and for long times when to,the mathematically exact rela- Value Theorem of Laplace transforms(see Problem 8.2)that for short times and it can be shown by using the Initial Value Theorem and the Final However,a usually good approximation is ically related by a simple inverse relationship.That is,in general, However,the corresponding time domain properties are not mathemat- 5(6C() creep compliance and the Laplace transform of the relaxation modulus ing to equation(8.19)and equation(8.20),the Laplace transform of the spondence Principle,which will be discussed later.Note also that accord- materials and is another building block in the Elastic-Viscoelastic Corre- the correspondence between the equations for elastic and viscoelastic Laplace transform of the relaxation modulus.This is another example of ality constants are the Laplace transform of the creep compliance and the forms of the stresses and strains are linearly related,and the proportion- as Hooke's law for linear elastic materials,except that the Laplace trans- Note that equation(8.19)and equation(8.20)are now of the same form Principles of Composite Material Mechanics (8.2) 图 (622) ment is of the form g=o/k,whereas the corresponding equation for the viscous dashpot is de/dt=o/u. where viscosity #Thus,the stress-strain relationship for the elastic spring ele- law and the dashpot is assumed to be filled with a Newtonian fluid of cous dashpot,where the spring of modulus k is assumed to follow Hooke's constructed from simple elements such as the elastic spring and the vis- As shown in figure 8.8 to figure 8.10,useful physical models can be derivatives of stress and strain. physical models of linear viscoelastic behavior that include various time behavior is the dependence of stress on strain rate;such strain rate effects can be modeled with equation(8.28).We now consider several simple (ie.,aoe=boo).Recall that one of the physical manifestations of viscoelastic case of equation(8.28)when all time derivatives of stress and strain vanish Note that the linear elastic material described by Hooke's law is a special differential equation as well as by the Boltzmann superposition integral. Thus,linear viscoelastic behavior may also be described by an ordinary 但w+3P of equation(8.26),we find that 被0个 d2o Making use of equation(8.27)and taking the inverse Laplace transform derivative of a function f()is But if we neglect the initial conditions,the Laplace transform of the nth Thus,we can write P(s)=ao+as+azs2+...+ans" Analysis of Viscoelastic and Dynamic Behavior (828 图 826