Positive definiteness in coupled strain gradient elasticity

Nazarenko, Lidiia; Glüge, Rainer; Altenbach, Holm

doi:10.1007/s00161-020-00949-2

Cited by 23 publications

(23 citation statements)

References 31 publications

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“…In the present paper we extend the results [38] for coupled strain gradient elasticity, where potential energy density was presented as uncoupled quadratic form of the strain and modified second gradient of displacement. This allows to evaluate explicit tensorial expression of the complementary energy in the coupled case.…”

supporting

confidence: 63%

“…The presence of the coupling tensor ℂ 5 significantly complicates the determination of conditions for positive definiteness of as well as the calculation of the compliance tensors 4 , 5 , 6 needed for the definition of the complementary energy density. It has been shown [38] that it is possible to decouple…”

Section: Block Diagonalizationsmentioning

confidence: 99%

“…To decouple the strain and strain gradient contributions, Equation ( 1) can be transformed by introducing a modified strain and a modified stiffness tensor of sixth-rank [38]:…”

Section: Variantmentioning

confidence: 99%

“…holds. It has been shown [38] that such a transformation simplifies the analysis of definiteness of and Equation (12) has been used to derive inequalities for the material parameters such that is positive definite, including the coupling parameter in ℂ 5 . In the next Section it will be shown that the modified form of the potential energy density allows also to obtain tensorial expressions for the compliance tensors 4 , 5 , 6 and further to calculate the complementary energy density.…”

Section: Variantmentioning

confidence: 99%

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Inverse Hooke's law and complementary strain energy in coupled strain gradient elasticity

2021

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The inverse Hooke's law and complementary strain energy density has been examined in the context of the theory of coupled gradient elasticity for second gradient materials. To this end, it was assumed that the potential energy density is a quadratic form of the strain and of the second gradient of displacement. Existence of the coupling term significantly complicates the problem. To avoid this complication the equation for the potential energy density was transformed in order to present it as an uncoupled quadratic form of a modified strain and the second gradient of displacement or of the strain and a modified second gradient of displacement. These transformations, which is in essence a block matrix diagonalization, lead to a decoupling of strains and strain gradient in the potential energy density and makes it possible to determine tensorial relations for the compliance tensors of fourth-, fifth-, and sixth-rank. Both modifications result in the same compliance tensors and are valid for an arbitrary material symmetry class. In the case of hemitropic materials, the compliance tensors have the same symmetry and the same form as the stiffness tensors and are characterized by eight independent constants, namely the two classical isotropic constants, five constants in the strain gradient part and one constant in the coupling term. Explicit expressions for these eight parameters are obtained from the tensorial relations for the compliance tensors and are compared with the direct solution of a linear system for the compliance's. All three solutions are identical, what we consider as a verification of the presented results.

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supporting

confidence: 63%

Section: Block Diagonalizationsmentioning

confidence: 99%

“…To decouple the strain and strain gradient contributions, Equation ( 1) can be transformed by introducing a modified strain and a modified stiffness tensor of sixth-rank [38]:…”

Section: Variantmentioning

confidence: 99%

Section: Variantmentioning

confidence: 99%

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Inverse Hooke's law and complementary strain energy in coupled strain gradient elasticity

2021

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“…Inequality constraints on first and on second gradient constitutive parameters in absence of coupling terms were also presented in Mindlin [36], dell'Isola et al [12]. In Nazarenko et al [39] results of Mindlin [36] and dell'Isola et al [12] have been extended for the case of coupled strain gradient elasticity. In this regard, it has been introduced a block diagonalization of the composite stiffness in strain gradient elasticity.…”

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confidence: 92%

Uniqueness theorem in coupled strain gradient elasticity with mixed boundary conditions

Nazarenko

Glüge

Altenbach

2021

Continuum Mech. Thermodyn.

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The equilibrium equations and the traction boundary conditions are evaluated on the basis of the condition of the stationarity of the Lagrangian for coupled strain gradient elasticity. The quadratic form of strain energy can be written as a function of the strain and the second gradient of displacement and contains a fourth-, a fifth- and a sixth-order stiffness tensor $${\mathbb {C}}_4$$ C 4 , $${\mathbb {C}}_5$$ C 5 and $${\mathbb {C}}_6$$ C 6 , respectively. Assuming invariance under rigid body motions the balance of linear and angular momentum is obtained. The uniqueness theorem (Kirchhoff) for the mixed boundary value problem is proved for the case of the coupled linear strain gradient elasticity (novel). To this end, the total potential energy is altered to be presented as an uncoupled quadratic form of the strain and the modified second gradient of displacement vector. Such a transformation leads to a decoupling of the equation of the potential energy density. The uniqueness of the solution is proved in the standard manner by considering the difference between two solutions.

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Verification of strain gradient elasticity computation by analytical solutions

et al. 2021

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As there are different computational methods for simulating problems in generalized mechanics, we present simple applications and their closed-form solutions for verifying a numerical implementation. For such a benchmark, we utilize these analytical solutions and examine three-dimensional numerical simulations by the finite element method (FEM) using IsoGeometric Analysis (IGA) with the aid of open source codes, called tIGAr, developed within the FEniCS platform. A study for the so-called wedge forces and double tractions help to comprehend their roles in the displacement solution as well as examine the significance by comparing to the closed form solutions for given boundary conditions. It is found that numerical results are in a good agreement with the analytical solutions if wedge forces and double tractions are considered. It is also presented how the wedge forces become necessary in order to maintain equilibrium in strain gradient materials.

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Positive definiteness in coupled strain gradient elasticity

Cited by 23 publications

References 31 publications

Inverse Hooke's law and complementary strain energy in coupled strain gradient elasticity

Inverse Hooke's law and complementary strain energy in coupled strain gradient elasticity

Uniqueness theorem in coupled strain gradient elasticity with mixed boundary conditions

Verification of strain gradient elasticity computation by analytical solutions

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