# Closed-Form Solution for Viscoelastic Creep

In any solid body subjected to external loading or displacement, the resulting stress and strain must simultaneously satisfy three basic equations: the equilib­rium equations, the kinematic equations, and the constitutive equations. Visco­elastic stress analysis problems are more difficult to solve than elasticity problems because time dependency requires the solution of the differential equations of the constitutive law. If the boundary conditions and the tempera­ture remain invariable, the time variable in the equations can be removed by transforming the equation into the Laplace transform domain as shown earlier.

In this section, the elastic solution shown in Eq 1 is used to determine the deflection of the thick plate. As shown below, for constant Poisson’s ratio, Eq 1 is transformed to the equivalent elastic solution with the Laplace domain variable s

TABLE 4—Shear and bulk relaxation modulus Prony series parameters at 5 °C used for modeling.

 Instantaneous modulus, MPa N Gi, MPa Ki, MPa t – s 79.81 1 1.605 4.757 10 2 42.09 39.92 44.62 3 29.21 82.91 200.60 4 6.598 20.54 26874.5

To make the conversion back from the Laplace domain to the time domain, Maple software (Maplesoft, Adept Scientific Nordic, DK-2600 Glostrup, Produk- tionsvej 26, Sweden) is used. The shear relaxation modulus, the corresponding relaxation times and the Poisson’s ratio are used and converted into the creep equation, Eq 16. The parameters of Eq 16 are shown in Table 5. The vertical deflection of the thick plate is calculated based on the analytic Eq 17

N

■ ^ t

D(t)=D0 +Y, A(1 _ e_Q (16)

i=1

where w0 is vertical deflection of the thick plate (mm).