Fourth-order tensors – tensor differentiation with applications to continuum mechanics. Part I: Classical tensor analysis

Kintzel, O.; Başar, Yavuz

doi:10.1002/zamm.200410242

Cited by 23 publications

(15 citation statements)

References 22 publications

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“…Equation (15) implies that F N i and F TÑ i are also eigenvectors of A and A T corresponding to λ i , respectively. Thus, there exist some scalars t i , i = 1, 2, 3 such that…”

Section: Theorem 1 If An Isotropic Tensor-valued Function F (A) Satismentioning

confidence: 98%

Basis-free expressions for derivatives of a subclass of nonsymmetric isotropic tensor functions

王志乔

兑关锁

2007

Appl Math Mech

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The present paper generalizes the method for solving the derivatives of symmetric isotropic tensor-valued functions proposed by Dui and Chen (2004) to a subclass of nonsymmetric tensor functions satisfying the commutative condition. This subclass of tensor functions is more general than those investigated by the existing methods. In the case of three distinct eigenvalues, the commutativity makes it possible to introduce two scalar functions, which will be used to construct the general nonsymmetric tensor functions and their derivatives. In the cases of repeated eigenvalues, the results are acquired by taking limits.

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“…Equation (15) implies that F N i and F TÑ i are also eigenvectors of A and A T corresponding to λ i , respectively. Thus, there exist some scalars t i , i = 1, 2, 3 such that…”

Section: Theorem 1 If An Isotropic Tensor-valued Function F (A) Satismentioning

confidence: 98%

Basis-free expressions for derivatives of a subclass of nonsymmetric isotropic tensor functions

王志乔

兑关锁

2007

Appl Math Mech

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“…The double contractions of fourth-order and second-order tensors are defined as [17,18] (A⊗B) : The double contractions of fourth-order tensors are as follows [18]:…”

Section: Tensor Calculationsmentioning

confidence: 99%

Two‐point constitutive equations and integration algorithms for isotropic‐hardening rate‐independent elastoplastic materials in large deformation

Wang

Dui

2008

Numerical Meth Engineering

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SUMMARYThis paper presents alternative forms of hyperelastic-plastic constitutive equations and their integration algorithms for isotropic-hardening materials at large strain, which are established in two-point tensor field, namely between the first Piola-Kirchhoff stress tensor and deformation gradient. The eigenvalue problems for symmetric and non-symmetric tensors are applied to kinematics of multiplicative plasticity, which imply the transformation relationships of eigenvectors in current, intermediate and initial configurations. Based on the principle of plastic maximum dissipation, the two-point hyperelastic stress-strain relationships and the evolution equations are achieved, in which it is considered that the plastic spin vanishes for isotropic plasticity. On the computational side, the exponential algorithm is used to integrate the plastic evolution equation. The return-mapping procedure in principal axes, with respect to logarithmic elastic strain, possesses the same structure as infinitesimal deformation theory. Then, the theory of derivatives of non-symmetric tensor functions is applied to derive the two-point closed-form consistent tangent modulus, which is useful for Newton's iterative solution of boundary value problem. Finally, the numerical simulation illustrates the application of the proposed formulations.

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“…However, this is not directly possible for variables that are only linear dependent on the base vectors like the deformation gradient. Thus, to incorporate the Green-Lagrange strains into the principle of virtual work, the required transformations between the fourth-order tangent moduli P ,F and S ,E are derived by considering rules of tensor differentiation presented in detail in [18].…”

Section: Introductionmentioning

confidence: 99%

“…According to the tensor differentiation rules proposed in Kintzel and Başar [18], the symmetry of the strain tensor E = E T remains a minor symmetry of the fourth-order tensor (E, F ) to = E, F with respect to the outer basis. By applying the contraction of S and E, F in (41), the well-known relation…”

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confidence: 99%

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Consistent micro–macro transitions at large objective strains in curvilinear convective coordinates

Grytz

Meschke

2007

Numerical Meth Engineering

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SUMMARYThis paper presents a computational homogenization scheme that is of particular interest for problems formulated in curvilinear coordinates. The main goal of this contribution is to generalize the computational homogenization scheme to a formulation of micro-macro transitions in curvilinear convective coordinates, where different physical spaces are considered at the homogenized macro-continuum and at the locally attached representative micro-structures. The deformation and the coordinate system of the micro-structure are assumed to be coupled with the local deformation and the local coordinate system at a corresponding point of the macro-continuum. For the consistent formulation of micro-macro transitions, the operations scale-up and scale-down are introduced, considering the rotated representation of tensor variables at the different physical reference frames of micro-and macro-structure. The second goal of this paper is to use objective strain measures like the Green-Lagrange strain tensor for the solution of boundary value problems on the micro-and macro-scale by providing the required transformations for the work-conjugate stress, strain and tangent tensors into variables admissible for the considered micro-macro transitions and satisfying the averaging theorem.

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Fourth-order tensors – tensor differentiation with applications to continuum mechanics. Part I: Classical tensor analysis

Cited by 23 publications

References 22 publications

Basis-free expressions for derivatives of a subclass of nonsymmetric isotropic tensor functions

Basis-free expressions for derivatives of a subclass of nonsymmetric isotropic tensor functions

Two‐point constitutive equations and integration algorithms for isotropic‐hardening rate‐independent elastoplastic materials in large deformation

Consistent micro–macro transitions at large objective strains in curvilinear convective coordinates

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