Projector representation of isotropic linear elastic material laws for directed surfaces

Fiber orientation tensors represent averaged measures of fiber orientations inside a microstructure. Although, orientation-dependent material models are commonly used to describe the mechanical properties of representative microstructure, the influence of changing or differing microstructure on the material response is rarely investigated systematically for directional measures which are more precise than second-order fiber orientation tensors. For the special case of planar orientation distributions, a set of admissible fiber orientation tensors of fourth-order is known. Fiber orientation distributions reconstructed from given orientation tensors are of interest both for numerical averaging schemes in material models and visualization of the directional information itself. Focusing on the special case of planar orientations, this paper draws the geometric picture of fiber orientation distribution functions reconstructed from fourth-order fiber orientation tensors. The developed methodology can be adopted to study the dependence of material models on planar fourth-order fiber orientation tensors. Within the set of admissible fiber orientation tensors, a subset of distinct tensors is identified. Advantages and disadvantages of the description of planar orientation states in two- or three-dimensional tensor frameworks are demonstrated. Reconstruction of fiber orientation distributions is performed by truncated Fourier series and additionally by deploying a maximum entropy method. The combination of the set of admissible and distinct fiber orientation tensors and reconstruction methods leads to the variety of reconstructed fiber orientation distributions. This variety is visualized by arrangements of polar plots within the parameter space of fiber orientation tensors. This visualization shows the influence of averaged orientation measures on reconstructed orientation distributions and can be used to study any simulation method or quantity which is defined as a function of planar fourth-order fiber orientation tensors.

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“…following Blinowski et al [27, equation (2.3)] and Aßmus et al [31, equation (38)]. Deviator operators are defined by…”

Section: Fiber Orientation Distributions Based On Planar Fiber Orient...mentioning

confidence: 99%

Fiber orientation distributions based on planar fiber orientation tensors of fourth order

Bauer

Böhlke

2022

Mathematics and Mechanics of Solids

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“…By means of Equations (, , ), the stiffness tensors can be additively decomposed into their dilatoric and deviatoric components:

A = 2 B h {bold-scriptP}_{1}^{frakturS} + 2 G h {bold-scriptP}_{2}^{frakturS},

D = 2 B \frac{h^{3}}{12} {bold-scriptP}_{1}^{frakturS} + 2 G \frac{h^{3}}{12} {bold-scriptP}_{2}^{frakturS},

Z = κ G h P,

where the compression modulus of the surface

is a material property in terms of Poisson's ratio ν, and the Young's modulus Y . Furthermore, the transverse shear correction factor

0 < κ \leq 1

is introduced to compensate for the assumed uniform distribution of the shear stress.…”

Section: Duhamel‐neumann Equations For Planar Surfacesmentioning

confidence: 99%

“…First‐order shear deformation theory (FSDT) is the next widely‐used approach for the structural analysis of laminates. This theory is based upon the rigid director assumption for the cross‐section and it can be applied to beams and plates . Thermomechanical analysis of laminates using the theory of multi‐layered beams is also present in the literature .…”

Section: Introductionmentioning

confidence: 99%

On thermal strains and residual stresses in the linear theory of anti‐sandwiches

et al. 2019

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The current study aims to formulate the behaviour of anti‐sandwiches that considers residual stresses and thermal strains by means of a layer‐wise approach. To this end, the constitutive equations of classical linear elasticity are extended to form the Duhamel‐Neumann equations. The direct approach is adopted to project the three‐dimensional problem onto two‐dimensional surfaces, i.e., each layer of the anti‐sandwich is modelled as a single surface that represents the overall behaviour of the layer. The formulation of the problem begins with the decoupling of deformation into independent membrane, bending, twisting and transverse shear modes. The infinitesimal deformation measures are set up on the kinematics of the problem and related to the corresponding kinetic measures via the proposed stiffness relations. The resulting boundary value problem is converted into the equivalent weak form by means of a variational approach. The presented set of equations can be solved inexpensively using various computational approaches such as the finite element method.

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Theory of Planar Surface Continua

Aßmus

2018

Structural Mechanics of Anti-Sandwiches

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Projector representation of isotropic linear elastic material laws for directed surfaces

Cited by 9 publications

References 21 publications

Fiber orientation distributions based on planar fiber orientation tensors of fourth order

Fiber orientation distributions based on planar fiber orientation tensors of fourth order

On thermal strains and residual stresses in the linear theory of anti‐sandwiches

Theory of Planar Surface Continua

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