Anisotropic strain gradient thermoelasticity for cellular structures: Plate models, homogenization and isogeometric analysis

Bone-related defects that cannot heal without significant surgical intervention represent a significant challenge in the orthopedic field. The use of implants for these critical-sized bone defects is being explored to address the limitations of autograft and allograft options. Three-dimensional cellular structures, or bone scaffolds, provide mechanical support and promote bone tissue formation by acting as a template for bone growth. Stress shielding in bones is the reduction in bone density caused by the difference in stiffness between the scaffold and the surrounding bone tissue. This study aimed to reduce the stress shielding and introduce a cellular metal structure to replace defected bone by designing and producing a numerically optimized bone scaffold with an elastic modulus of 15 GPa, which matches the human’s cortical bone modulus. Cubic cell and diagonal cell designs were explored. Strut and cell dimensions were numerically optimized to achieve the desired structural modulus. The resulting scaffold designs were produced from stainless steel using laser powder bed fusion (LPBF). Finite element analysis (FEA) models were validated through compression testing of the printed scaffold designs. The structural configuration of the scaffolds was characterized with scanning electron microscopy (SEM). Cellular struts were found to have minimal internal porosity and rough surfaces. Strut dimensions of the printed scaffolds were found to have variations with the optimized computer-aided design (CAD) models. The experimental results, as expected, were slightly less than FEA results due to structural relative density variations in the scaffolds. Failure of the structures was stretch-dominated for the cubic scaffold and bending-dominated for the diagonal scaffold. The torsional and bending stiffnesses were numerically evaluated and showed higher bending and torsional moduli for the diagonal scaffold. The study successfully contributed to minimizing stress shielding in bone tissue engineering. The study also produced an innovative metal cellular structure that can replace large bone segments anywhere in the human body.

We here study the plate models as 2D extension of the beam models. We don’t start with the classical theory of the so-called Kirchhoff plates as 2D generalization of the Euler beam, often referred to as a ‘thin’ plate model that does not account for shear deformation. Such model requires the continuity of second derivatives of transverse displacement field that is not easy to achieve for distorted elements. Hence, we present an alternative model in terms of the Reissner-Mindlin plate as 2D generalization of the Timoshenko beam, yet referred to as a ‘thick’ plate model that can take into account the shear deformation. The variational formulation now requires the inter-element continuity of only the first derivatives of transverse displacement and (independent) rotations, but again lead to locking phenomena in finite element discrete approximations. We present several Reissner-Mindlin plate elements with non-conventional interpolations that can handle locking. We also show how to recover the discrete approximation of the Kirchhoff plate model in terms of the discrete Kirchhoff plate finite element. The plate models presented in this chapter are further used to illustrate typical goals of adaptivity procedure in structural engineering that pertain to model selection or validation, as well as to discrete approximation quality or verification.

Aluminum cellular structures have been fabricated by combining a two-dimensional [0°/90°]2 arrangement of square Al 6061-T6 alloy tubes with orthogonal tubes inserted in the out-of-plane direction. By varying the tube wall thickness, the resulting three-dimensional cellular structures had relative densities between 11 and 43%. The dynamic compressive response of the three-dimensional cellular structure, and the two-dimensional [0°/90°]2 array and out-of-plane tubes from which they were constructed, have been investigated using a combination of instrumented Kolsky bar impact experiments, high-speed video imaging, and finite element analysis. We find the compression rate has no effect upon the strength for compression strain rates up to 2000 s-1, despite a transition to higher-order buckling modes at high strain rates. The study confirms that a synergistic interaction between the colinear aligned and out-of-plane tubes, observed during quasistatic loading, extends to the dynamic regime. Finite element simulations, using a rate-dependent, piecewise linear strain hardening model with a von Mises yield surface and an equivalent plastic strain failure criterion, successfully predicted the buckling response of the structures, and confirmed the absence of strain-rate hardening in the three-dimensional cellular structure. The simulations also reveal that the ratio of the impact to back-face stress increased with strain rate and relative density, a result with significant implications for shock-load mitigation applications of these structures. © 2014 Mathematical Sciences Publishers.

Variational formulations and isogeometric analysis for the dynamics of anisotropic gradient-elastic Euler-Bernoulli and shear-deformable beams

Stiffness and strength evaluation of lattice-based mechanical metamaterials by decoupled two-scale analysis

A new non-classical model for circular Kirchhoff plates subjected to axisymmetric loading is developed using a modified couple stress theory, a surface elasticity theory and Hamilton’s principle. The equations of motion and boundary conditions are simultaneously obtained through a variational formulation. The new plate model contains a material length scale parameter to capture the microstructure effect and three surface elasticity constants to describe the surface energy effect. The current non-classical plate model includes the plate models considering the microstructure influence or the surface energy effect alone as special cases and recovers the classical elasticity-based Kirchhoff plate model when both the microstructure and surface energy effects are suppressed. To demonstrate the new model, the static bending problem of a clamped solid circular Kirchhoff plate subjected to a uniform normal load is analytically solved by directly applying the general formulas derived. The numerical results reveal that the deflection of the clamped circular plate predicted by the current Kirchhoff plate model is smaller than that predicted by the classical elasticity-based model, but the former approaches the latter with the increase of the plate thickness. It is also found that the difference between the plate deflection predicted by the model incorporating the surface energy effect and that predicted by the classical model is significant when the plate thickness is sufficiently small. However, the difference is diminishing as the plate thickness increases. These predicted size effects at the micron scale agree with the general trends observed in experiments.

A size-dependent third-order shear deformable plate model incorporating strain gradient effects for mechanical analysis of functionally graded circular/annular microplates

ABSTRACTFocusing on the homogenization of periodic composite materials, this study investigates computational methods based on volume integral equations. Such formulations are revisited from the standpoint of the preconditioning of the original cell problem by the introduction of a comparison material. This allows for to recovery of simple convergence criteria for iterative steepest‐descent and fixed‐point schemes for composites with general non‐linear behaviour. In the case of linear materials, the preconditioned volume integral formulation coincides with the well‐known Lippmann–Schwinger equation. The spectral properties of the featured linear integral operator, which is bounded and self‐adjoint, are investigated to shed light on the behaviour of conventional computational homogenization methods. The so‐called Lippmann–Schwinger spectrum is analyzed, with its bounds governing the convergence rate of iterative solution methods. The associated eigenvectors, which constitute the eigenstates of the composite material considered, are also described in detail to understand their role in constructing the solution to the cell problem and ultimately in computing the effective properties. Formulated in the continuous setting, this analysis is followed by the investigation of a discrete representation of the integral operator considered. A number of examples on synthetic microstructures are finally considered in the conductivity setting to illustrate the obtained theoretical results and highlight the role of the spectral properties in the operation of computational homogenization methods. This paves the way for the development of reduced models and more efficient computations.

In this contribution, we derive a consistent variational formulation for computational homogenization methods and show that traditional FE and IGA approaches are special discretization and solution techniques of this most general framework. This allows us to enhance dramatically the numerical analysis as well as the solution of the arising algebraic system. In particular, we expand the dimension of the continuous system, discretize the higher dimensional problem consistently and apply afterwards a discrete null‐space matrix to remove the additional dimensions. A benchmark problem, for which we can obtain an analytical solution, demonstrates the superiority of the chosen approach aiming to reduce the immense computational costs of traditional FE and IGA formulations to a fraction of the original requirements. Finally, we demonstrate a further reduction of the computational costs for the solution of general nonlinear problems.

On the derivation of an asymptotically correct shear correction factor for the Reissner-Mindlin plate model

The linear theory of coupled gradient elasticity has been considered for hemitropic second gradient materials, specifically the positive definiteness of the strain and strain gradient energy density, which is assumed to be a quadratic form of the strain and of the second gradient of the displacement. The existence of the mixed, fifth-rank coupling term significantly complicates the problem. To obtain inequalities for the positive definiteness including the coupling term, a diagonalization in terms of block matrices is given, such that the potential energy density is obtained in an uncoupled quadratic form of a modified strain and the second gradient of displacement. Using orthonormal bases for the second-rank strain tensor and third-rank strain gradient tensor results in matrix representations for the modified fourth-rank and the sixth-rank tensors, such that Sylvester’s formula and eigenvalue criteria can be applied to yield conditions for positive definiteness. Both criteria result in the same constraints on the constitutive parameters. A comparison with results available in the literature was possible only for the special case that the coupling term vanishes. These coincide with our results.

Size-independent strain gradient effective models based on homogenization methods: Applications to 3D composite materials, pantograph and thin walled lattices

A long‐standing problem in the modelling of polycrystal behavior from the knowledge of the constitutive equations of single crystals is the incorporation of grain size effects. Standard ho‐mogenization procedures efficiently take into account grain to grain interaction primarily induced by strain incompatibilities at grain boundary and grain morphology effects. Continuum crystal plasticity models are available that satisfactorily describe dislocation forest hardening associated with multiplication and dynamic recovery of dislocation populations. One strong limitation of these techniques is that the local crystal plasticity constitutive parameters at the grain scale are assumed to be known. Conversely, these material parameters can be identified from the macroscopic polycrystal response by an inverse approach involving the homogenization model. But then the found crystal plasticity model will be valid only for the considered grain size and the model can hardly be used for prediction of polycrystal behavior for another grain state of the material. Generalized continuum approaches arise as the suitable mechanical framework for formulating enhanced crystal plasticity models that intrinsically include grain size dependent local material responses. They incorporate the dislocation density tensor as a consitutive variable. Recent developments have shown the limitations of existing strain gradient models in the case of a two–phase laminate microstructure under single or double slip [1]. They plead for a theory even more general than the strain gradient or Cosserat approach, namely the micromorphic model presented here. The effect of the dislocation density tensor is introduced into the classical crystal plasticity framework by means of a micromor‐phic theory of single crystals. A computational homogenization strategy is presented in order to simulate the global and local responses of two–dimensional polycrystalline aggregates for grain sizes ranging from 1 to 200 microns. The model is shown to induce a size–dependent kinematic hardening component which is responsible for the observed strong size effects. The yield stress at a given averaged plastic deformation is shown to follow a power law scaling relation for grain sizes larger than a critical grain size. The field of plastic deformation is also strongly affected by grain size, micron–size grains leading to the formation of intense slip bands crossing several grains. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Scale-dependent nonlocal strain gradient isogeometric analysis of metal foam nanoscale plates with various porosity distributions

Research of combined hybrid method applied in the Reissner–Mindlin plate model