Showing papers on "Linear elasticity published in 2021"

On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects

Abstract: Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn–Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell–cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. In the overall model an equation of Cahn–Hilliard type is coupled to the system of linear elasticity and a reaction–diffusion equation for a nutrient concentration. The highly non-linear coupling between a fourth-order Cahn–Hilliard equation and the quasi-static elasticity system lead to new challenges which cannot be dealt within a gradient flow setting which was the method of choice for other elastic Cahn–Hilliard systems. We show existence, uniqueness and regularity results. In addition, several continuous dependence results with respect to different topologies are shown. Some of these results give uniqueness for weak solutions and other results will be helpful for optimal control problems.

Meshless physics-informed deep learning method for three-dimensional solid mechanics

Abstract: Deep learning and the collocation method are merged and used to solve partial differential equations describing structures' deformation. We have considered different types of materials: linear elasticity, hyperelasticity (neo-Hookean) with large deformation, and von Mises plasticity with isotropic and kinematic hardening. The performance of this deep collocation method (DCM) depends on the architecture of the neural network and the corresponding hyperparameters. The presented DCM is meshfree and avoids any spatial discretization, which is usually needed for the finite element method (FEM). We show that the DCM can capture the response qualitatively and quantitatively, without the need for any data generation using other numerical methods such as the FEM. Data generation usually is the main bottleneck in most data-driven models. The deep learning model is trained to learn the model's parameters yielding accurate approximate solutions. Once the model is properly trained, solutions can be obtained almost instantly at any point in the domain, given its spatial coordinates. Therefore, the deep collocation method is potentially a promising standalone technique to solve partial differential equations involved in the deformation of materials and structural systems as well as other physical phenomena.

Numerical evaluation of the representative volume element for random composites

Abstract: The Representative Volume Element (RVE) plays a central role in the homogenization of random heterogeneous microstructures, especially for composite and porous materials, with a view to predicting their effective properties. A quantitative evaluation of its size is proposed in this work in linear elasticity and linear thermal conductivity of random heterogeneous materials. A RVE can be associated with different physical and statistical properties of microstructures. The methodology is applied to specific two–phase microstructure–based random sets. Statistical parameters are introduced to study the variation in the RVE size versus volume fractions of components and the contrast in their properties. The key notion of the integral range is introduced to determine these variations. For a given desired precision, we can provide a minimal volume size for the computation of effective mechanical and thermal properties. Numerical simulations are performed to demonstrate that a volume exists which is statistically representative of random microstructures. This finding is an important component for homogenization–based multiscale modeling of materials.

Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase field fracture by nonlocal operator method

Abstract: The derivation of nonlocal strong forms for many physical problems remains cumbersome in traditional methods. In this paper, we apply the variational principle/weighted residual method based on nonlocal operator method for the derivation of nonlocal forms for elasticity, thin plate, gradient elasticity, electro-magneto-elasticity and phase-field fracture method. The nonlocal governing equations are expressed as an integral form on support and dual-support. The first example shows that the nonlocal elasticity has the same form as dual-horizon non-ordinary state-based peridynamics. The derivation is simple and general and it can convert efficiently many local physical models into their corresponding nonlocal forms. In addition, a criterion based on the instability of the nonlocal gradient is proposed for the fracture modelling in linear elasticity. Several numerical examples are presented to validate nonlocal elasticity and the nonlocal thin plate.

Foundations of the Theory of Elasticity, Plasticity, and Viscoelasticity

Abstract: The Theory of Stress-Strain State Physical Relations in the Elasticity Theory Statement and Problem-Solving Procedures of Elasticity Theory Variational Methods The Plane Elastic Problems Plate Bending Deformation of a Half-Space and Contact Problems Foundations of the Theory of Plasticity Linear Viscoelastic Continua Thermoviscoelastoplasticity Thermoviscoelastoplastic Characteristics of Materials Dynamic Problems of the Elasticity Theory

A simple Galerkin meshless method, the Fragile Points method using point stiffness matrices, for 2D linear elastic problems in complex domains with crack and rupture propagation

Abstract: The Fragile Points Method (FPM) is an elementarily simple Galerkin meshless method, employing Point-based discontinuous trial and test functions only, without using element-based trial and test functions. In this study, the algorithmic formulations of FPM for linear elasticity are given in detail, by exploring the concepts of point stiffness matrices and numerical flux corrections. Advantages of FPM for simulating the deformations of complex structures, and for simulating complex crack propagations and rupture developments, are also thoroughly discussed. Numerical examples of deformation and stress analyses of benchmark problems, as well as of realistic structures with complex geometries, demonstrate the accuracy, efficiency and robustness of the proposed FPM. Simulations of crack initiation and propagations are also given in this study, demonstrating the advantages of the present FPM in modeling complex rupture and fracture phenomena. The crack and rupture propagation modeling in FPM is achieved without remeshing or augmenting the trial functions as in standard, extended or generalized FEM. The simulation of impact, penetration and other extreme problems by FPM will be discussed in our future papers.

Lipschitz stability estimate and reconstruction of Lamé parameters in linear elasticity

Abstract: In this paper, we consider the inverse problem of recovering an isotropic elastic tensor from the Neumann-to-Dirichlet map. To this end, we prove a Lipschitz stability estimate for Lame parameters ...

Nonlinear Finite Element Analysis

Abstract: Safety assessment of structures can not be purely code based and certainly not based on linear elastic analyses. Hence, this chapter will cover key aspects of non-linear finite element analysis.

Linear elasticity of polymer gels in terms of negative energy elasticity

Abstract: We recently found that the energy contribution to the linear elasticity of polymer gels in the as-prepared state can be a significant negative value; the shear modulus is not proportional to the absolute temperature [1]. Our finding challenges the conventional notion that the polymer-gel elasticity is mainly determined by the entropy contribution. Existing molecular models of classical rubber elasticity theories, including the affine, phantom, and junction affine network models, cannot be used to estimate the structural parameters of polymer gels. In this focus review, we summarize the experimental studies on the linear elasticity of polymer gels in the as-prepared state using tetra-arm poly(ethylene glycol) (PEG) hydrogels with a homogenous polymer network. We also provide a unified formula for the linear elasticity of polymer gels with various network topologies and densities. Using the unified formula, we reconcile the past experimental results that seemed to be inconsistent with each other. Finally, we mention that there are still fundamental unresolved problems involving the linear elasticity of polymer gels.

Computational Simulation of 3D Fatigue Crack Growth under Mixed-Mode Loading

Abstract: The purpose of this research was to present a simulation modelling of a crack propagation trajectory in linear elastic material subjected to mixed-mode loadings and investigate the effects of the existence of a hole and geometrical thickness on fatigue crack growth and fatigue life under constant amplitude loading. For various geometry thickness, mixed-mode (I/II) fatigue crack growth studies were carried out to utilize a single edge cracked plate with three holes and compact tension shear specimens with various loading angles. Smart Crack Growth Technology, a new feature in ANSYS, was used in ANSYS Mechanical APDL 19.2 to predict the cracks’ propagation trajectory and their consequent fatigue life associated with evaluating the stress intensity factors. The maximum circumferential stress criterion is implemented as a direction criterion under linear elastic fracture mechanics (LEFM). According to the hole position, the results demonstrate that the fatigue crack grows towards the hole due to the unbalanced stresses on the hole induced crack tip. The results of this simulation are verified in terms of crack growth paths, stress intensity factors, and fatigue life under mixed-mode load conditions, with several crack growth studies published in the literature showing consistent results.

Functional mechanical metamaterial with independently tunable stiffness in the three spatial directions

Abstract: Mechanical metamaterials with variable stiffness recently gained a lot of research interest, as they allow for structures with complex boundary and load conditions. Herein, we highlight the design, additive manufacturing, and mechanical testing of a new kind of bending-dominated metamaterial. By advancing from well-established mechanical metamaterials, the proposed geometry allows for varying the stiffness in the three spatial directions independently. Therefore, structures with different orientational properties can be designed, ranging from isotropic to anisotropic structures, including orthotropic structures. The compression modulus can be varied in the range of several orders of magnitude. Gradual transitions from one unit cell to the next can be realized, enabling smooth transitions from soft to hard regions. Specimens have been additively manufactured with acrylic resins and polylactic acid using Digital Light Processing and Fused Filament Fabrication, respectively. Two different numerical models have been employed using ABAQUS to describe the mechanical properties of the structure and verified by the experiments. Compression tests were performed to investigate the linear elastic properties of isotropic structures. Numerical models, based on three-point-bending test data, have been employed to study orthotropic structures. Compression test results for orthotropic and anisotropic structures are shown to highlight the independent variability. The manufacturing of the structures is not limited to the presented techniques and materials but can be expanded to all available additive manufacturing techniques and their respective materials. For a video of the compression tests of a specimen with three different compression moduli along the spatial axes, see the Supplementary Data available online.

Nonlocal layerwise formulation for bending of multilayered/functionally graded nanobeams featuring weak bonding

Abstract: The size-dependent bending of perfectly/imperfectly bonded multilayered/stepwise functionally graded nanobeams, e.g. multiwalled carbon nanotubes with weak van der Waals forces, with any arbitrary numbers of layers, exhibiting different material, geometrical, and length-scale properties, is studied through a layerwise formulation of the stress-driven nonlocal theory of elasticity and the Bernoulli-Euler beam theory. The formulation is also valid for the continuously graded nanobeams, where the through-the-thickness material gradation with any arbitrary distribution is approximated in a stepwise manner through many layers. The size-dependency of each layer is accounted for through nonlocal constitutive relationships, which define the strains at each point as the output of integral convolutions in terms of the stresses in all the points of the layer and a kernel. Linear elastic uncoupled interfacial laws are implemented to model the mechanical response of the interfaces. The size-dependent system of equilibrium equations governing the deformations of the layers are derived and subjected to the variationally consistent edge boundary conditions and the constitutive boundary conditions associated with the stress-driven integral convolution. The formulation is applied to multilayered and sandwich nanobeams and the effects of the interfacial imperfections on the displacement fields and the interfacial displacement jumps are studied. It is found that the interfacial imperfections have greater impact on the field variables of multilayered nanobeams than that of the multilayered beams with the large-scale dimensions.