Extrema of Young’s modulus for cubic and transversely isotropic solids

Cazzani, Antonio; Rovati, Marco

doi:10.1016/s0020-7683(02)00668-6

Cited by 107 publications

(56 citation statements)

References 16 publications

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“…The data are from Musgrave (2003) The extreme values are also given by the coordinates in the region with ν min = ν 110 , ν max = ν 001 . The materials there † Data from Landolt and Bornstein (1992), see also Cazzani and Rovati (2003). Landolt and Bornstein (1992).…”

Section: (B) Application To Cubic Materialsmentioning

confidence: 99%

Poisson's ratio in cubic materials

Norris

2006

Proc. R. Soc. A.

100

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Expressions are given for the maximum and minimum values of Poisson's ratio ν for materials with cubic symmetry. Values less than −1 occur if and only if the maximum shear modulus is associated with the cube axis and is at least 25 times the value of the minimum shear modulus. Large values of |ν| occur in directions at which the Young's modulus is approximately equal to one half of its 111 value. Such directions, by their nature, are very close to 111. Application to data for cubic crystals indicates that certain Indium Thallium alloys simultaneously exhibit Poisson's ratio less than -1 and greater than +2.

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Section: (B) Application To Cubic Materialsmentioning

confidence: 99%

Poisson's ratio in cubic materials

Norris

2006

Proc. R. Soc. A.

100

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“…Numerical searching is practical and straightforward; thus Cazzani and Rovati provide a detailed analysis of the extrema of Young's modulus for cubic and transversely isotropic materials Keywords: Poisson's ratio, Young's modulus, shear modulus, anisotropic. [Cazzani and Rovati 2003] and for materials with tetragonal symmetry [Cazzani and Rovati 2005], with extensive illustrative examples. Boulanger and Hayes [1995] obtained analytic expressions related to extrema of Young's modulus.…”

Section: Introductionmentioning

confidence: 99%

Extreme values of Poisson’s ratio and other engineering moduli in anisotropic materials

Norris

2006

JOMMS

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Conditions for a maximum or minimum of Poisson's ratio of anisotropic elastic materials are derived. For a uniaxial stress in the 1-direction and Poisson's ratio ν defined by the contraction in the 2-direction, the following three quantities vanish at a stationary value: s 14 , [2νs 15 + s 25 ] and [(2ν − 1)s 16 + s 26 ], where s IJ are the components of the compliance tensor. Analogous conditions for stationary values of Young's modulus and the shear modulus are obtained, along with second derivatives of the three engineering moduli at the stationary values. The stationary conditions and the hessian matrices are presented in forms that are independent of the coordinates, which lead to simple search algorithms for extreme values. In each case the global extremes can be found by a simple search over the stretch direction n only. Simplifications for stretch directions in a plane of orthotropic symmetry are also presented, along with numerical examples for the extreme values of the three engineering constants in crystals of monoclinic symmetry.

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“…The advantage of using the group of independent elastic material constants from Kelvin notation is the reflection of mathematical essences of an elastic constitutive tensor, such as eigenvalues and tensor invariants. However, the independent elastic material constants based on the components of a standardized compliance matrix proposed in this paper have more straightforward physical meaning, since each constant represents the strain-stress ratio under uniaxial loading and it is obtained directly and easily in experiments [Cazzani and Rovati, 2003;Rovati and Taliercio, 2003]. We therefore recommend using the proposed elastic material constants, just like the Young's modulus and Poisson ratio for isotropic materials are widely used, but they are not elastic material constants from Kelvin notation.…”

Section: ∂Smentioning

confidence: 99%

Standardized Compliance Matrices for General Anisotropic Materials and a Simple Measure of Anisotropy Degree Based on Shear-Extension Coupling Coefficient

Zhao

Song

Liu

2016

Int. J. Appl. Mechanics

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The compliance matrix for a general anisotropic material is usually expressed in an arbitrarily chosen coordinate system, which brings some confusion or inconvenience in identifying independent elastic material constants and comparing elastic properties between different materials. In this paper, a unique stiffest orientation-based standardized compliance matrix is established, and 18 independent elastic material constants are clearly shown. During the searching process for the stiffest orientation, it is interesting to find from our theoretical analysis and an example that a material with isotropic tensile stiffness does not definitely possess isotropic elasticity. Therefore, the ratio between the maximum and minimum tensile stiffnesses, although widely used, is not a correct measure of anisotropy degree. Alternatively, a simple and correct measure of anisotropy degree based on the maximum shear-extension coupling coefficient in all orientations is proposed. However, for a two-dimensional constitutive relation, both the stiffness ratio and the shear-extension coupling coefficient can be adopted as proper measures of anisotropy degree.

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Extrema of Young’s modulus for cubic and transversely isotropic solids

Cited by 107 publications

References 16 publications

Poisson's ratio in cubic materials

Poisson's ratio in cubic materials

Extreme values of Poisson’s ratio and other engineering moduli in anisotropic materials

Standardized Compliance Matrices for General Anisotropic Materials and a Simple Measure of Anisotropy Degree Based on Shear-Extension Coupling Coefficient

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