An approximate analytical 2D-solution for the stresses and strains in eigenstrained cubic materials

Dreyer, Wolfgang; Müller, Wolfgang; Olschewski, J.

doi:10.1007/bf01179256

Cited by 18 publications

(18 citation statements)

References 13 publications

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“…(2014) based on earlier work by Müller (1998) and Dreyer at al. (1999), in which derivatives are calculated by means of corresponding central finite difference expressions, to which discrete Fourier transforms are taken and the shift theorem is applied (see Berbenni et al (2014) for details).…”

Section: -2 Numerical Details Of the Non-local Theory Implementationmentioning

confidence: 98%

“…In these EVP-FFT implementations, a local formulation of crystal plasticity was adopted, rendering their predictions size-independent. While the potential for an efficient non-local plasticity FFT-based implementation has been recognized, overcoming the numerical challenges associated with the required computation and use in the constitutive relation of higher order derivatives of the plastic distortion field was only possible considering the alternative use of discrete Fourier transforms 5 (DFT) (Müller, 1998;Dreyer et al, 1999;Willot and Pellegrini, 2008;Willot, 2015) in lieu of the original Moulinec-Suquet continuous Fourier transform (CFT)-based algorithm, in particular drawing upon recent work by Berbenni et al (2014) devoted to the calculation of stress fields associated to the presence of disclinations in elastic media using the FFT-based approach.…”

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confidence: 99%

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Numerical implementation of non-local polycrystal plasticity using fast Fourier transforms

Lebensohn

Needleman

2016

Journal of the Mechanics and Physics of Solids

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Section: -2 Numerical Details Of the Non-local Theory Implementationmentioning

confidence: 98%

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confidence: 99%

Numerical implementation of non-local polycrystal plasticity using fast Fourier transforms

Lebensohn

Needleman

2016

Journal of the Mechanics and Physics of Solids

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“…Note that this behaviour is the opposite of the dilatoric case (cf. Dreyer et al [6]). Moreover, note that #/and therefore 22 is negative and also the self energies g'0 remain positive even for high degrees of anisotropy (e.g., silver).…”

Section: Resultsmentioning

confidence: 94%

“…For example, with respect to nickel superalloys, minimisation of interaction energies between regions of high order (known as 7' "rafts") and regions of low order (the matrix 7) can lead to an understanding of the mechanisms by which textures are formed (see Hazotte, Grosdidier and Denis [5]). This effect is visible in Figure 1.3, and has been investigated for the dilatoric transformation in the text by Dreyer et al [6] for the case of two dimensions, and by Mfiller and Neumann [7] for the case of three dimensions, using Fourier transform techniques.…”

Section: Introductionmentioning

confidence: 90%

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An approximate 2-D solution for the shear-induced strain fields in eigenstrained cubic materials

Brown

Müller

2001

Acta Mechanica

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Summary. An approximate analytical 2-D solution for the strain field components Ell , s and e22 occurring in a cubic material due to a coherently bonded shear eigenstrained inclusion of cylindrical geometry was obtained by means of Continuous Fourier Transforms (CFT). A Discrete Fourier Transform (DFT) based numerical model was used in order to test the validity of the results. For the case where the cylindrical inclusion and the surrounding media are elastically homogeneous and the orientation of their principal crystal axes are the same, a correlation between the analytical and numerical models is demonstrated, both for strongly and weakly anisotropic materials. Moreover, the strain fields within the inclusion are shown to be of homogeneous isotropic type. Finally, an expression for the closed-form strain energy of two cylindrical inclusions at arbitrary radius and angle was derived, and then used to determine the minimum energy configuration for the system.

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A Fast Fourier Transform-Based Approach for Generalized Disclination Mechanics Within a Couple Stress Theory

Berbenni

Taupin

Fressengeas

et al. 2016

Advanced Structured Materials

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Recently, a small-distortion theory of coupled plasticity and phase transformation accounting for the kinematics and thermodynamics of generalized defects called generalized disclinations (abbreviated gdisclinations) has been proposed [2, 3]. Then, a first numerical spectral approach has been developed to solve the elasto-static equations of field dislocation and g-disclination mechanics set out in this theory for periodic media and for linear elastic media using the classic Hooke's law [6]. Here, given a spatial distribution of generalized disclination density tensors in a homogenous linear higher order elastic media described, a couple stress theory with elastic incompatibilities of first and second orders is developed. The incompatible and compatible elastic second and first distortions are obtained from the solution of Poisson and Navier-type equations in the Fourier space. The efficient Fast Fourier Transform (FFT) algorithm [17] is used based on intrinsic Discrete Fourier Transforms (DFT) that are well adapted to the discrete grid to compute higher order partial derivatives in the Fourier space. Therefore, stress and couple stress fields can be calculated using the inverse FFT. The numerical examples are given for straight wedge disclinations and associated wedge disclination dipoles which are of importance to geometrically describe tilt grain boundaries at fine scales in polycrystalline solids.

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An approximate analytical 2D-solution for the stresses and strains in eigenstrained cubic materials

Cited by 18 publications

References 13 publications

Numerical implementation of non-local polycrystal plasticity using fast Fourier transforms

Numerical implementation of non-local polycrystal plasticity using fast Fourier transforms

An approximate 2-D solution for the shear-induced strain fields in eigenstrained cubic materials

A Fast Fourier Transform-Based Approach for Generalized Disclination Mechanics Within a Couple Stress Theory

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