Math Tool

For the math tool reference should be made to LaplaceTransform for the development of the viscoelastic model.

Linear Viscoelasticity

The Boltzmann Superposition Principle states that the incremental strain at time t induced by the application of the stress \mathrm{d}\sigma at time \tau is

\mathrm{d}\varepsilon &=& J(t-\tau) \mathrm{d}\sigma

in tensor notation

\mathrm{d}\boldsymbol{\varepsilon}(t) &=& \boldsymbol{J}(t-\tau):\frac{\partial\boldsymbol{\sigma}}{\partial \tau} \mathrm{d}\tau

Hence integrating over the total load application time t

\boldsymbol{\varepsilon}(t) &=& \int_0^t \boldsymbol{J}(t-\tau):\frac{\partial\boldsymbol{\sigma}}{\partial \tau} \mathrm{d}\tau

Viscoelastic correspondence principle

Due to the presence of the stress time derivative it is algebraically more efficient the Laplace Carson transform (LCT) defined by:

G(s) &=& s \int_{\infty}^{t} g(t) \mathrm{e}^{-st} \mathrm{d} t

When definition is applied to the behaviour laws of previous section

\boldsymbol{\varepsilon}(s) &=& \boldsymbol{J}(s):\boldsymbol{\sigma(s)}

which is analogous to a linear elastic behaviour law. Consider two problems where the geometry, area of application and type of boundary conditions are identical, but in one case the material is linear elastic and in the other case, the material is linear viscoelastic. If the LCT is applied to the viscoelastic problem, it becomes similar to the elastic problem in the Laplace-Carson space. Therefore, if the solution is known for the elastic problem, the solution of the viscoelastic problem can be obtained in the Laplace–Carson space by replacing the elastic properties with the LCT of the viscoelastic properties and by replacing the loadings by their corresponding LCT.

Numerical Inversion for Homogenization

Approachs can be summarized as

• Direct Numerical Inversion of the Lapalce Transform
• Series Expantion of the Immaginary Modulus in term of known inverse transform functions
• Direct Time Integration of the Maxewell equation in the time domain

Direct Numerical Inversion of the Laplace-Carson Transform

This approach is the one used in 2 it is computaionally expensive and need some acceleration algorithm. Recent Algorithm are supplied in 3.

Here, Erik Cheever pages is described the core method for inversion with some code

Series Expantion in the Laplace Domain

Approach presented in 1

Direct Time integration of the Maxwell Equations

Approach presented in 4, 5, 6

Reference

[1] M. L\xE9vesque M.D. Gilchrist N. Bouleau K. Derrien D. Baptiste, "Numerical inversion of the Laplace–Carson transform applied to homogenization of randomly reinforced linear viscoelastic media", Comput Mech (2007) 40:771–789 DOI 10.1007/s00466-006-0138-6

[2] S. Scheiner and C. Hellmich, A.M.ASCE, Continuum Microviscoelasticity Model for Aging Basic Creep of Early-Age Concrete, DOI: 10.1061/͑ASCE͒0733-9399͑2009͒135:4͑307͒

[3] L DAMORE, G. LACCETTI, and A. MURLI, " Algorithm 796: A Fortran Software Package for the Numerical Inversion of the Laplace Transform Based on a Fourier Series Method", ACM Transactions on Mathematical Software, Vol. 25, No. 3, September 1999, Pages 306 –315.

[4] Sorvari, Joonas; H\xE4m\xE4l\xE4inen, Jari, "Time integration in linear viscoelasticity - a comparative study http://dx.doi.org/10.1007/s11043-010-9108-7

[5] H. POON AND M. FOUAD AHMAD, A FINITE ELEMENT CONSTITUTIVE UPDATE SCHEME FOR ANISOTROPIC, VISCOELASTIC SOLIDS EXHIBITING NON-LINEARITY OF THE SCHAPERY TYPE, Int. J. Numer. Meth. Engng. 46, 2027}2041 (1999)

[6] Andr\xB4 Schmidt, Lothar Gaul, "FE Implementation of Viscoelastic Constitutive Stress-Strain Relations Involving Fractional Time Derivatives"

-- RobertoBernetti - 13 Jan 2011