Quasi-static analysis of multilayered domains with viscoelastic layer using incremental-layerwise finite element method

Abstract

This paper presents a layerwise finite element formulation for quasi-static analysis of laminated structures with embedded viscoelastic material whose constitutive behavior is represented by the Prony series. To account the time dependence of the constitutive relations of linear viscoelastic materials, the incremental formulation in the temporal domain is used. This approach avoids the use of relaxation functions and mathematical transformations. A computer code based on the presented formulation has been developed to provide the numerical results. The high accuracy of the method is exhibited by comparing the results with existing solutions in the literature and also with those obtained using the ABAQUS software. Finally, and as an application of the presented formulation, the effects of time and load rate on the quasi-static structural response of asphalt concrete (AC) pavements are studied.

Appendix A

The elements of the stiffness matrices[K _ij ], the mass matrices [M _ij ], the load vector increments and in Eq. (17) are:

$$\begin{aligned} {[} K_{uu} ] =& A_{p11}^{jr}D_{c}^{is} + B_{p13}^{rj}B_{c}^{is} + B_{p13}^{jr}B_{c}^{si} + D_{p33}^{jr}A_{c}^{is} \end{aligned}$$

(23a)

$$\begin{aligned} {[} K_{uw} ] =& B_{p12}^{rj}B_{c}^{is} + A_{p13}^{jr}D_{c}^{is} + D_{p23}^{jr}A_{c}^{is} + B_{p33}^{jr}B_{c}^{si} \end{aligned}$$

(23b)

$$\begin{aligned} {[} K_{wu} ] =& B_{p12}^{jr}B_{c}^{si} + D_{p23}^{jr}A_{c}^{is} + A_{p13}^{jr}D_{c}^{is} + B_{p33}^{rj}B_{c}^{is} \end{aligned}$$

(23c)

$$\begin{aligned} {[} K_{ww} ] =& D_{p22}^{jr}A_{c}^{is} + B_{p23}^{jr}B_{c}^{si} + B_{p23}^{rj}B_{c}^{is} + A_{p33}^{jr}D_{c}^{is} \end{aligned}$$

(23d)

$$\begin{aligned} & {[}M_{uu} ] = A_{c}^{is}I_{p}^{jr} \end{aligned}$$

(24a)

$$\begin{aligned} &{[} M_{ww} ] = A_{c}^{is}I_{p}^{jr} \end{aligned}$$

(24b)

$$ A_{pmn}^{jr} = \int_{0}^{h} C_{pmn} ( \Delta t_{n} ) \psi_{j} ( z ) \psi_{r} ( z )b\,dz $$

(25)

$$\begin{aligned} & B_{pmn}^{rj} = \int_{0}^{h} C_{pmn} ( \Delta t_{n} ) \psi_{j} ( z ) \frac{d\psi_{r} ( z )}{dz}b\,dz \end{aligned}$$

(26a)

$$\begin{aligned} &{} B_{pmn}^{jr} = \int_{0}^{h} C_{pmn} ( \Delta t_{n} ) \psi_{r} ( z ) \frac{d\psi_{j} ( z )}{dz}b\,dz \end{aligned}$$

(26b)

$$ D_{pmn}^{jr}= \int_{0}^{h} C_{pmn} ( \Delta t_{n} ) \frac{d\psi_{j} ( z )}{dz} \frac{d\psi_{r} ( z )}{dz}b\,dz $$

(27)

$$ A_{c}^{is} = \int_{0}^{L} \varphi_{i} ( x ) \varphi_{s} ( x )\,dx $$

(28)

$$\begin{aligned} B_{c}^{is} =& \int_{0}^{L} \frac{d\varphi_{i} ( x )}{dx} \varphi_{s} ( x )\,dx \end{aligned}$$

(29a)

$$\begin{aligned} B_{c}^{si} =& \int_{0}^{L} \frac{d\varphi_{s} ( x )}{dx} \varphi_{i} ( x )\,dx \end{aligned}$$

(29b)

$$ D_{c}^{is} = \int_{0}^{L} \frac{d\varphi_{i} ( x )}{dx} \frac{d\varphi_{s} ( x )}{dx}\,dx $$

(30)

$$ I_{p}^{jr} = \int_{0}^{h} \rho \psi_{j} ( z )\psi_{r} ( z )b\,dz $$

(31)

$$\begin{aligned} &\{ \Delta f_{u} \}_{t_{n}} = \Delta f_{j} ( t_{n} )\delta_{iNx} \end{aligned}$$

(32a)

$$\begin{aligned} &\{ \Delta f_{w} \}_{t_{n}} = \Delta f_{i} ( t_{n} ) \delta_{iNz} \end{aligned}$$

(32b)

$$\begin{aligned} &\Delta f_{i} ( t_{n} ) = \int_{0}^{L} \Delta q_{T} ( x,t_{n} ) \varphi_{i} ( x )\,dx \end{aligned}$$

(33a)

$$\begin{aligned} &\Delta f_{j} ( t_{n} ) = \int_{0}^{h} \Delta q_{L} ( z,t_{n} ) \psi_{j} ( z )\,dz \end{aligned}$$

(33b)

$$\begin{aligned} &\{ \tilde{f}_{u} \}_{t_{n - 1}} = \tilde{E}_{cxx}E_{c}^{i} + \tilde{f}_{cxz}^{j}f_{c}^{i} \end{aligned}$$

(34a)

$$\begin{aligned} & \{ \tilde{f}_{w} \}_{t_{n - 1}} = \tilde{f}_{czz}^{j}f_{c}^{i} + \tilde{E}_{cxz}^{j}E_{c}^{i} \end{aligned}$$

(34b)

$$\begin{aligned} &\tilde{E}_{cxx}^{j} = \int_{0}^{h} \psi_{j} ( z ) ( \tilde{\sigma}_{xx} )_{t_{n - 1}}b\,dz \end{aligned}$$

(35a)

$$\begin{aligned} &\tilde{E}_{cxz}^{j} = \int_{0}^{h} \psi_{j} ( z ) ( \tilde{\sigma}_{xz} )_{t_{n - 1}}b\,dz \end{aligned}$$

(35b)

$$\begin{aligned} &\tilde{f}_{cxz}^{j} = \int_{0}^{h} \frac{d\psi_{j} ( z )}{dz} ( \tilde{\sigma}_{xz} )_{t_{n - 1}}b\,dz \end{aligned}$$

(36a)

$$\begin{aligned} &\tilde{f}_{czz}^{j} = \int_{0}^{h} \frac{d\psi_{j} ( z )}{dz} ( \tilde{\sigma}_{zz} )_{t_{n - 1}}b\,dz \end{aligned}$$

(36b)

$$ f_{c}^{i} = \int_{0}^{L} \varphi_{i} ( x )\,dx $$

(37)

$$ E_{c}^{i} = \int_{0}^{L} \frac{d\varphi_{i} ( x )}{dx}\,dx = \varphi_{i} ( L ) - \varphi_{i} ( 0 ) $$

(38)

C _pmn (Δt _n ) is the mth row and nth column from viscoelastic constitutive matrix.