A Fourier transformation-based temporal integration scheme for viscoplastic solids subjected to fatigue deterioration

Kindrachuk, Vitaliy M.; Unger, Jörg F.

doi:10.1016/j.ijfatigue.2017.03.015

Cited by 9 publications

(13 citation statements)

References 28 publications

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“…Approximations provide very accurate solutions through almost the whole fatigue loading history. The limitations can be observed only close to the fatigue failure when the damage rate throughout the load cycle is no longer negligible …”

Section: Ftti Schemementioning

confidence: 99%

“…Extrapolation is completed once the converged solutions of and , referred to as

normalΔ {bold-italicu}^{false[q^{*} false]}

normalΔ {\overset{ε}{true}}^{eq false[q^{*} false]}

, are obtained at the q ∗ ‐th iteration

bold-italicu false(Υ + normalΔ Υ false) = bold-italicu false(Υ false) + {normalΔ bold-italicu}^{false[q^{*} false]},

{\overset{ε}{true}}^{eq} false(Υ + normalΔ Υ false) = {\overset{ε}{true}}^{eq} false(Υ false) + normalΔ {\overset{ε}{true}}^{eq false[q^{*} false]} .

The respective values of the history variables follow from , and

d false(Υ + normalΔ Υ false) = d false(Υ false) + normalΔ d + normalδ d^{false[q^{*} false]},

normalκ false(Υ + normalΔ Υ false) = Φ^{- 1} ((), d false(Υ + normalΔ Υ false)) .

A more accurate second‐order extrapolation of history variables has been often proposed in other works() and is implemented in the present study for the extrapolation of Δ d . Application of even higher‐order approximations can be also accomplished; however, they are not necessarily more advantageous regarding the performance of the FTTI scheme . Notice that an implicit determination of the microchronological contribution to damage Δ d would require the computation of derivatives of the cyclic damage rate Δ c d with respect to Δ u and

normalΔ \overset{ε}{true}

…”

Section: Ftti Schemementioning

confidence: 99%

“…They develop during a single load cycle once the global equilibrium of the domain is reached first at macrochronological time Υ and, afterward, at Υ +ΔΥ/2. Following the spirit of the FTTI method, the computation of the cyclic damages is a 2 stage procedure: (i) the calculation of the Fourier coefficients is performed under assumption of negligible cyclic damage growth, and then, (ii) the integration of the history variables is performed in quadrature points for prescribed nodal values. Determination of the Fourier coefficients is the goal of this section.…”

Section: Ftti Schemementioning

confidence: 99%

“…The number of problems is therefore determined by the number of Fourier modes required to accurately approximate the oscillatory boundary conditions. The displacement field can then be reconstructed by taking the solutions of , ie,

bold-italicu \approx true \sum_{k = - k^{*}}^{k^{*}} u_{k} false(Υ false) e^{i k normalω normalτ} .

Similarly, quasi‐linear approximations of the microchronological problem have been successfully realized in the works of Kindrachuk and Unger() and Crouch and Oskay. () Adopting such a mathematical treatment is however ineffective to get the microchronological solution for the nonlocal equivalent strain .…”

Section: Ftti Schemementioning

confidence: 99%

“…In a recent paper, a Fourier transformation‐based temporal integration (FTTI) has been incorporated into the FE framework for the simulation of fatigue in viscoplastic solids. The displacement field is expressed as a dual‐time Fourier series.…”

Section: Introductionmentioning

confidence: 99%

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A Fourier transformation‐based method for gradient‐enhanced modeling of fatigue

Kindrachuk

Titscher

Unger

2018

Numerical Meth Engineering

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Summary A key limitation of the most constitutive models that reproduce a degradation of quasi‐brittle materials is that they generally do not address issues related to fatigue. One reason is the huge computational costs to resolve each load cycle on the structural level. The goal of this paper is the development of a temporal integration scheme, which significantly increases the computational efficiency of the finite element method in comparison to conventional temporal integrations. The essential constituent of the fatigue model is an implicit gradient‐enhanced formulation of the damage rate. The evolution of the field variables is computed as a multiscale Fourier series in time. On a microchronological scale attributed to single cycles, the initial boundary value problem is approximated by linear BVPs with respect to the Fourier coefficients. Using the adaptive cycle jump concept, the obtained damage rates are transferred to a coarser macrochronological scale associated with the duration of material deterioration. The performance of the developed method is hence improved due to an efficient numerical treatment of the microchronological problem in combination with the cycle jump technique on the macrochronological scale. Validation examples demonstrate the convergence of the obtained solutions to the reference simulations while significantly reducing the computational costs.

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Section: Ftti Schemementioning

confidence: 99%

“…Extrapolation is completed once the converged solutions of and , referred to as