Identificación de la Aptitud de la Tierra

To demonstrate the performance of the implementation for full crack propagation for quasi- static loading, a numerical example is presented, which considers a thin plate subjected to the 3-point loading. The dimensions of the specimen are 0.5 x 0.5 x 0.02mm. The model is spatially discretised using 2nd order tetrahedrons. The vertical force with magnitude of 1000 N is applied in the middle top of the plate applied on 0.02 x 0.02mm area. The material density is prescribed by function of spatial coordinates: ρ(x, y, z) = 2x + 1. Similar to bones,

Young’s modulus depends on the density in a power-law model of the form: E = aρn_MPa,

where coefficients a and n are chosen to be 9200 and 2, respectively. Poisson ratio ν is equal

to 0.3 and critical Griffith energy release parameter gc= 1N/mm The crack is initialised in

the middle bottom of the plate with length of 0.025mm. The analysis is conducted for three different meshes consisting of 1340, 5145 and 10341 elements. Figures 6.10 and 6.11a) shows that configurational forces are driving the crack in the direction opposite to the density gradient.

It should be noted that the crack path is smooth even for a coarse mesh. The load-displacement curves in Figure 6.11b) demonstrate that the results for the consecutive refinements are converging. The presented value of displacement is known as the generalised displacement and does not represent a particular point on the structure, but its value is work conjugate to the

applied forces and is calculated as ug= 2Ψ/τf, where f = 1N is the reference force, and Ψ is

the total elastic energy integrated over the domain, τ is the arc-length load factor, and ugis the

generalised displacement. These results indicate the ability of the formulation to accurately and robustly predict crack paths for heterogeneous bodies without bias from the original mesh.

initial notch

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a) b)

Figure 6.10: a) Geometry and boundary conditions for plate example. b) Contours of crack evolution on a map of P11stress.

a) 0 0.2 0.4 0.6 0.8 1 1.2 ·10−3 0 0.5 1 1.5 2 ·10 −2 Displacement ug[mm] Load factor τ [-] Fine Mesh Coarse mesh Very fine mesh

b)

Figure 6.11: Heterogeneous 3D plate. a) model geometry with predicted crack, material distribution and coarse discretisation with 1340 tetrahedral elements. b) load-displacement response for three consecutive h-refinements.

6.4

Summary

This chapter presented a formulation for brittle fracture in elastic solids within the context of configurational mechanics that includes the influence of heterogeneous density distribution. Configurational forces are the driver for crack propagation, and it was shown that in order to evaluate these forces correctly forces at the crack front it is necessary to have a spatially smooth density field, with higher regularity than if the field is directly approximated on the finite element mesh. Therefore, density data is approximated as a smooth field using a Moving Weighted Least Squares method, that allows for the computation of higher-order derivatives. Numerical convergence was demonstrated for a simple finite plate, and the use of singularity elements was shown to further improve the rate of convergence. Furthermore, the calculated release energy rate was verified using centered FDM. The numerical example of plate heterogeneous material properties have been presented to demonstrate both the accuracy and robustness of the formulation.

In the next Chapter 7, the demonstrated configurational force driven approach is utilised for analysing release energy and crack propagation in adapted bones.

Chapter 7

Numerical investigations

The objective of this chapter is to assess the current state of development of the implemented framework in the form of numerical examples. In the first example, the remodelling of a proximal femur is considered. The developed model is utilised for assessment of the long- time response of the proximal femur bone to a hip replacement treatment. Subsequently, a comparative study of discrete and smeared approaches for approximating fracture is conducted. The presentation and discussion of the numerical results provide an insight into the potential of both methods for simulating large-scale crack propagation problems with homogeneous and heterogeneous material properties. Finally, a full framework for estimating bone fracture resistance is demonstrated. It combines all the advancements presented in Chapters 3, 4 and 6 to simulate training regime exerted on the equine MC3 bone and the following crack propagation.

7.1

Simulation of proximal femur adaptation

The proposed finite element framework has to be able to predict bone density distributions of horses undergoing specific trainings, as stated in Section 1.1. The accurate estimation of the bone stiffness is essential for calculating its resistance to fracture and possible injury prevention. In order to validate predictive capabilities of the used bone remodelling model, a well-studied hip-replacement procedure is considered and its long term influence on the density distribution. Total hip replacement is a common surgical procedure, where the damaged proximal end of the femur is replaced by a prosthesis, typically a metal ball attached to a stem inserted inside the bone. The method is very effective for fixing major fractures for elderly patients. However, long-term studies show that in active young people, such implantation can cause stress shielding in the bone (Kronick et al., 1997). When loads are carried by the

significantly more stiff metal stem, the surrounding bone tissue exhibits intense remodelling leading to the loss of the density and ultimately painful loosening of the implant. Therefore, an efficient and accurate FEM simulation of the long-term bone response can be an invaluable tool to support decision making for patient-specific treatments.