As alternatives to partial differential equations (PDEs), nonlocal continuum models given in integral forms avoid the explicit use of conventional spatial derivatives and allow solutions to exhibit desired singular behavior. As an application, peridynamic models are reformulations of classical continuum mechanics that allow a natural treatment of discontinuities by replacing spatial derivatives of stress tensor with integrals of force density functions. The thesis is concerned about the mathematical perspective of nonlocal modeling and local-nonlocal coupling for fracture mechanics both theoretically and numerically. To this end, the thesis studies nonlocal diffusion models associated with ``Neumann-type'' constraints (or ``traction conditions'' in mechanics), a nonlinear peridynamic model for fracture mechanics with bond-breaking rules, and a multi-scale model with local-nonlocal coupling. In the computational studies, it is of practical interest to develop robust numerical schemes not only for the numerical solution of nonlocal models, but also for the evaluation of suitably defined derivatives of solutions. This leads to a posteriori nonlocal stress analysis for structure mechanical models.

水準集結構拓撲優化方法同傳統的基於材料的拓撲優化方法相比具有明顯的優勢。由於採用了隱式的邊界表達，水準集方法能方便地處理結構形狀和拓撲的變化，且在優化過程中可以保持邊界的光滑。但這種動態結構邊界需要一種有限元分析方法可以適應其動態變化且能夠獲得足夠的計算精度。本文對傳統水準集結構拓撲優化中有限元分析存在的關鍵問題進行研究，同時針對應力約束下的結構拓撲優化，提出了一種新的拓撲優化方法。 / 首先， 擴展有限元法作為一種相對精確和高效的結構分析方法，本文將其引入到水準集結構拓撲優化中。引入擴展有限元法主要是為了處理優化過程中結構邊界上出現的材料的高度不連續情況，從而避免耗時的網格重新劃分。如果結構邊界從有限元單元內部通過，為了獲得足夠的計算精度，該單元內需要採用特殊的數值積分方法。常用的方法是將這個單元內被材料填充的區域劃分成小的子單元去適應單元內部的邊界，然後在各個子單元內採用高斯數值積分來獲得該單元的剛度矩陣。對於二維問題，如果結構邊界從一個單元內部通過，將單元分成幾部分，首先把單元內部的實體區域劃分成子三角形，然後計算出每個子三角形內的高斯積分點，最後單元剛度矩陣通過把所有子三角形的高斯積分點處的數值積分迭加得到。對於三維問題，則是將一個六面體單元分解為幾個四面體，然後在每一個四面體內部通過結構上定義的水準集函數值得到邊界，對於實體的部分劃分為子四面體，在每一個子四面體內計算出高斯積分點，此四面體的單元剛度矩陣為所有子四面體剛度矩陣的迭加，因此，該六面體的單元剛度矩陣為所劃分的四面體單元剛度矩陣的迭加。 / 其次，本文研究了提高擴展有限元法的計算精度和效率的方法。採用擴展有限元法進行結構分析時，如果被結構邊界剖分的有限元單元中實體部分體積比小到一定程度，將會影響到計算精度，本論文給出了處理擴展有限元中這種小單元情形的具體辦法。擴展有限元法作為一種結構分析計算方法，除了必須考慮精度外，效率也是一個重要的指標，尤其對於拓撲優化問題，因拓撲優化問題通常需要多步的反覆運算來獲取最優或局部最優解。為提高擴展有限元法的計算效率，相對於前面的基於剖分單元為子單元進行積分的辦法，本文提出了一種更高效的積分方法，即去除積分單元剖分，通過直接積分來計算被結構邊界剖分的單元的剛度矩陣。這種直接積分的方法不僅能保證結構分析的精度，更能顯著的提高計算效率，這對於水準集結構拓撲優化是非常有意義的。同時高階單元被用來從另一個角度分析擴展有限元法計算精度與效率之間的關係，換言之，可以用高階單元在相對粗的網格上來獲取同低階單元在相對密的網格上相同的分析結果精度，從而提高計算效率。但是這個問題需要找出計算精度在網格密度和單元階次之間的關係。 / 第三，本文以二維和三維結構的柔度最小化問題為例驗證了上述擴展有限元演算法在結構拓撲優化問題中應用的有效性。 / 最後，本文研究了基於應力約束的結構拓撲優化問題，並採用前面提出的擴展有限元法與水準集結合的拓撲優化方法。由於採用擴展有限元法進行結構分析可以獲得較準確的應力計算結果，特別是在結構邊界附近，這對於基於應力的拓撲優化問題有很大的優勢。而且，本文提出了一種形狀等效約束法來有效地控制局部應力約束，數值算例也證明擴展有限元法與形狀等效約束方法相結合對處理應力約束問題是一種非常有效的。同時，本文還提出了一種全新的通過拓撲優化來實現應力隔離結構設計的方法。通過在拓撲優化問題中不同區域施加不同的應力約束來有效地模擬這種應力隔離的問題。最終數值算例證明，該方法可以通過改變力的傳播途徑來達到有效地形成結構的應力隔離。 / Level set method is an elegant approach for structural shape and topology optimization, compared to the conventional material based topology optimization methods. The structural boundary is implicitly represented by a moving level set function. Thus, the shape and topology optimization can be processed simultaneously while maintaining a smooth boundary. The moving structural boundary demands a finite element analysis adaptable to the dynamic boundary changes and meeting required accuracy. In this thesis, the key issues of finite element methods of structural analysis for level set optimization method are investigated and an approach to stress-constrained topology optimization is presented. / Firstly, the extended finite element method (XFEM) is introduced into the level set method structural shape and topology optimization for obtaining a considerably accurate and efficient result of finite element analysis. In fact, the XFEM is employed as a structural analysis method to solve the problems of strong discontinuities between material and void domain during the level set optimization process in order to avoid the time cost remeshing. To achieve a reasonably accurate result of finite element analysis in the element intersected by structural boundary, special numerical integral schemes of XFEM are studied. The partition method is adopted to divide the integral domain into sub-cells, in which Gauss quadrature is utilized to calculate the element stiffness matrix. For two-dimensional (2D) problems, the integral domain is divided into sub-triangles, and the Gauss quadrature points in each sub-triangle are used to evaluate the element stiffness matrix which is the sum of all contributions of these sub-triangles. For three-dimensional (3D) problems, the hexahedral element is decomposed into multiple tetrahedra, and the integral domain in each tetrahedron is divided into sub-tetrahedra for obtaining the Gauss quadrature points. Therefore, the stiffness of each tetrahedron is obtained by summing all contributions of the sub-tetrahedra, which means the hexahedral element stiffness matrix is the accumulation of element stiffness matrixes with all these tetrahedra. / Secondly, the methods for improving the computational accuracy and efficiency of XFEM are studied. First of all, the practical solutions for dealing with the small volume fraction element of the proposed XFEM are provided since this kind of situation may result in the accuracy losing of finite element analysis. Besides computational accuracy of structural analysis, the efficiency is another sufficiently important issue of structural optimization problem. Therefore, a new XFEM integral scheme without quadrature sub-cells is developed for improving the computational efficiency of XFEM compared to the XFEM integral scheme with partition method, which can yield similar accuracy of structural analysis while prominently reducing the computational cost. Numerical experiments indicate that this performance is excellent for level set method shape and topology optimization. Moreover, XFEM with higher order elements are involved to improve the accuracy of structural analysis compared to the corresponding lower order element. Consequently, the computational cost is increased, therefore, the balance of computational cost between FE system scale and the order of element is discussed in this thesis. / Thirdly, the reliability and advantages of the proposed XFEM schemes are illustrated with several 2D and 3D mean compliance minimization examples that are widely employed in the recent literature of structural topology optimization. / Finally, the stress-based topology optimization problems with the proposed XFEM schemes are investigated. Due to the accuracy of structural analysis, XFEM schemes have natural advantages for solving the stress-based topology optimization problems using the level set method. Moreover, the shape equilibrium constraint approach is developed to effectively control the local stress constraint. Some numerical examples are solved to prove the high-performance of the proposed shape equilibrium constraint approach and XFEM schemes in the stress-constrained topology optimization problem. Meanwhile, a new approach of stress isolation design is presented through topology optimization. The stress isolation problem is modeled into a topology optimization problem with multiple stress constraints in different regions. Numerical experiments demonstrate that this approach can change the force transmission paths to successfully realize stress isolation in the structure. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Li, Li. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 113-123). / Abstract also in Chinese. / Abstract --- p.I / 摘要 --- p.IV / Acknowledgement --- p.VI / Contents --- p.VII / List of Figures --- p.XI / List of Tables --- p.XV / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- Related Works --- p.3 / Chapter 1.3 --- XFEM for Structural Optimization --- p.4 / Chapter 1.4 --- Topology Optimization with Stress Constraint --- p.7 / Chapter 1.5 --- Contributions and Organization of the Dissertation --- p.10 / Chapter 2 --- Level Set Method for Structural Optimization --- p.12 / Chapter 2.1 --- Structural Optimization Problem --- p.12 / Chapter 2.2 --- Implicit Level Set Representation --- p.14 / Chapter 2.3 --- Evolution of the Level Set Function --- p.15 / Chapter 2.4 --- Level Set Surface Reinitialization --- p.16 / Chapter 2.5 --- Velocity Extension --- p.17 / Chapter 3 --- Extended Finite Element Method (XFEM) --- p.19 / Chapter 3.1 --- Global Enrichment --- p.19 / Chapter 3.2 --- Local Enrichment --- p.20 / Chapter 3.3 --- Enrichment Function --- p.22 / Chapter 3.3.1 --- Enrichment for Strong Discontinuity --- p.22 / Chapter 3.3.2 --- Enrichment for Weak Discontinuity --- p.23 / Chapter 3.4 --- XFEM used in Structural Optimization --- p.23 / Chapter 4 --- Implementation of XFEM for Structural Optimization --- p.25 / Chapter 4.1 --- 2D XFEM Scheme --- p.26 / Chapter 4.1.1 --- Numerical Integral Scheme in 2D --- p.26 / Chapter 4.1.2 --- Evaluation of the 2D XFEM Scheme --- p.27 / Chapter 4.2 --- 3D XFEM Scheme --- p.30 / Chapter 4.2.1 --- Numerical Integral Scheme in 3D --- p.30 / Chapter 4.2.2 --- Evaluation of the 3D XFEM Scheme --- p.33 / Chapter 5 --- Computational Accuracy and Efficiency Aspects of XFEM --- p.36 / Chapter 5.1 --- XFEM Scheme for Small Volume Fraction Element --- p.38 / Chapter 5.1.1 --- Problem Definition --- p.39 / Chapter 5.1.2 --- Numerical Example --- p.41 / Chapter 5.2 --- Stress Smoothing in XFEM --- p.46 / Chapter 5.3 --- XFEM Integral Scheme without Quadrature Sub-cells --- p.50 / Chapter 5.3.1 --- 2D XFEM Integral Scheme without Quadrature Sub-cells --- p.50 / Chapter 5.3.2 --- 3D XFEM Integral Scheme without Quadrature Sub-cells --- p.53 / Chapter 5.4 --- Higher Order Elements with XFEM Scheme --- p.55 / Chapter 5.4.1 --- Higher Order Elements --- p.55 / Chapter 5.4.2 --- Numerical Example --- p.57 / Chapter 6 --- Minimum Compliance Optimization using XFEM --- p.64 / Chapter 6.1 --- Level Set Formulation of the Optimization Problem --- p.64 / Chapter 6.2 --- Finite Element Analysis with XFEM --- p.65 / Chapter 6.3 --- Shape Sensitivity Analysis --- p.65 / Chapter 6.4 --- Numerical Examples --- p.68 / Chapter 6.4.1 --- A 2D Short Cantilever Beam --- p.68 / Chapter 6.4.2 --- A 3D Short Cantilever Beam --- p.75 / Chapter 6.4.3 --- A Michell-type Structure in 3D --- p.77 / Chapter 7 --- Stress-Constrained Topology Optimization using XFEM --- p.81 / Chapter 7.1 --- Shape Equilibrium Approach to Stress Constraint --- p.81 / Chapter 7.1.1 --- Problem Formulation of Stress-Constrained Topology Optimization --- p.81 / Chapter 7.1.2 --- Shape Equilibrium Constraint Approach --- p.82 / Chapter 7.1.3 --- Material Derivatives of Stress Constraint --- p.83 / Chapter 7.1.4 --- Shape Sensitivity Analysis --- p.85 / Chapter 7.2 --- Finite Element Analysis with XFEM --- p.87 / Chapter 7.3 --- Minimal Weight Design with Stress Constraint --- p.88 / Chapter 7.3.1 --- Problem Definition --- p.88 / Chapter 7.3.2 --- Numerical Example --- p.89 / Chapter 7.4 --- Stress Isolation design --- p.94 / Chapter 7.4.1 --- Problem Definition --- p.94 / Chapter 7.4.2 --- Shape Sensitivity Analysis --- p.95 / Chapter 7.4.3 --- Numerical Examples --- p.97 / Chapter 8 --- Conclusions and Future Works --- p.109 / Chapter 8.1 --- Conclusions --- p.109 / Chapter 8.2 --- Future Works --- p.110 / Chapter 8.2.1 --- Adaptive XFEM --- p.111 / Chapter 8.2.2 --- Extend Shape Equilibrium Constraint Approach to 3D --- p.112 / Chapter 8.2.3 --- Extend the Stress Isolation Design Method into Industrial Applications --- p.112 / Bibliography --- p.113

Damaged cartilage can cause severe pain and restricted mobility, with few long term treatments available. The developing field of tissue engineering offers an alternative to the currently used full joint replacement. Restoring damaged cartilage through tissue engineering would enable an active lifestyle to be recovered and retained, without restrictions to joint mobility. This is increasingly important as the prevalence of osteoarthritis rises. Tissue engineering requires biomaterial scaffolds that mimic the function of the tissue while cells develop, and so the scaffold must provide the appropriate biological, chemical and mechanical stimuli. In this work, methods were developed to enable the design of scaffolds that mimic the microstructure and mechanical properties of articular cartilage. Electrospinning was investigated as a method to mimic the nanoscale collagen fibres within cartilage extracellular matrix. A parametric study was conducted to determine how changes to a gelatin solution affect the mechanical properties of the non-woven fibrous mesh. The solution properties had a clear impact on the morphology of the fibres, but the effect on the mesh mechanical properties was convoluted. The results demonstrated the need for greater understanding of the 3D morphology of electrospun meshes, to establish how these may be altered in order to design scaffolds with desirable mechanical properties. The fracture mechanics of soft materials are complex, and are generally overlooked when designing tissue engineering scaffolds. The complexities have led to a lack of standardised testing, making comparisons between studies impractical. In this work, fracture testing methods were compared, using a viscoelastic polymer to mimic some of the complexities of soft tissue mechanics. Mode III trouser tear tests and mode I pure shear tests were found to provide reliable measurements. Due to the ease of testing small samples, trouser tear testing was concluded to be the most advantageous for determining the fracture resistance of soft tissue engineering scaffolds. Finally, electrospun meshes were combined with hydrogels to create biomimetic scaffolds, which were characterised using tensile and trouser tear fracture tests. Fibre-reinforcement was shown to enhance the mechanical properties of a weak hydrogel, but diminished those of a strong, tough polyacrylamide (PAAm)-alginate hydrogel. The PAAm-alginate hydrogel exhibited mechanical properties close to those of natural articular cartilage, but without the microstructure that would enhance its suitability for use as a cartilage tissue engineering scaffold. An alternative method for reinforcing PAAm-alginate was proposed, which shows promise for producing a biocompatible scaffold that mimics both the mechanics and the microstructure of articular cartilage. Ultimately, this thesis aimed to improve the design of biomimetic scaffolds for cartilage tissue engineering, and advance mechanical characterisation techniques within this field.

Modeling fracture in geomaterials is essential to the understanding of many physical phenomenon which may posses natural hazards e.g. landslides, faults and iceberg calving or man-made processes e.g. hydraulic fracture and excavations. Continuum Damage Mechanics (CDM) models the crack as a solid region with a degraded stiffness. This continuum definition of cracks in CDM allows more feasible coupling with other forms of material non-linearity and eliminates the need to track complicated crack geometry. Using CDM to analyze fracture for the modeling of fracture in geomaterials encounters several challenges e.g.: 1) the need to model the multiple physical processes occurring in geomaterials, typically: coupled fluid flow and solid deformation, 2) the need to consider non-local damage and transport in order to capture the underlying long range interactions and achieve mesh-independent finite element solutions and 3) the elevated computational cost associated with non-linear mixed finite element formulations. The research presented in this thesis aims at improving the CDM formulations for modeling fracture geomaterials. This research can be divided into three main parts. The first is the introduction of a novel non-local damage transport formulation for modeling fracture in poroelastic media. The mathematical basis of the formulation are derived from thermodynamic equilibrium that considers non-local processes and homogenization principles. The non-local damage transport model leads to two additional regularization equations, one for non-local damage and the other for non-local transport which is reduced to non-local permeability. We consider two options for the implementation of the derived non-local transport damage model. The first option is the four-field formulation which extends the (u/P) formulation widely used in poroelasticity to include the non-local damage and transport phenomena. The second option is the three-field formulation, which is based on the coupling of the regularization equations under the assumptions of similar damage and permeability length scales and similar driving local stress/strain for the evolution of the damage and permeability. The three-field formulation is computationally cheaper but it degrades the physical modeling capabilities of the model. For each of these formulations, a non-linear mixed-finite element solution is developed and the Jacobian matrix is derived analytically. The developed formulations are used in the analysis of hydraulic fracture and consolidation examples. In the second part, a novel approach for CDM modeling of hydraulic fracture of glaciers is pretended. The presence of water-filled crevasses is known to increase the penetration depth of crevasses and this has been hypothesized to play an important role controlling iceberg calving rate. Here, we develop a continuum damage-based poro-mechanics formulation that enables the simulation of water-filled basal and/or surface crevasse propagation. The formulation incorporates a scalar isotropic damage variable into a Maxwell-type viscoelastic constitutive model for glacial ice and the effect of the water pressure on fracture propagation using the concept of effective solid stress. We illustrate the model by simulating quasi-static hydro-fracture in idealized rectangular slabs of ice in contact with the ocean. Our results indicate that water-filled basal crevasses only propagate when the water pressure is sufficiently large and that the interaction between simultaneously propagating water-filled surface and basal crevasses can have a mutually positive influence leading to deeper crevasse propagation which can critically affect glacial stability. In the third part, we propose a coupled Boundary Element Method (BEM) and Finite Element Method (FEM) for modeling localized damage growth in structures. BEM offers the flexibility of modeling large domains efficiently while the nonlinear damage growth is accurately accounted by a local FEM mesh. An integral-type nonlocal continuum damage mechanics with adapting FEM mesh is used to model multiple damage zones and follow their propagation in the structure. Strong form coupling, BEM hosted, is achieved using Lagrange multipliers. Since the non-linearity is isolated in the FEM part of the system of equations, the system size is reduced using Schur complement approach, then, the solution is obtained by a monolithic Newton method that is used to solve both domains simultaneously. The method is applied to multiple fractures growth benchmark problems and shows good agreement with the literature.

No description available.

This thesis is concerned with the characterisation of the parameters governing the tension-softening relations of the cohesive crack model. Parameter identification is an important area in fracture mechanics as it enables the use of a fracture model for the simulation of fracture processes in structures. Research, however, has shown that such a task is not trivial and continues to pose challenging problems to experimentalists and analysts alike. This dissertation presents general and efficient indirect methods for the characterisation of mode I fracture parameters defining the cohesive crack model. The identification problem is formulated as a special type of inverse problem. The formulation is in the form of a constrained optimisation problem known as a mathematical program with equilibrium constraints characterised, in the present instance, by complementarity conditions involving the orthogonality of two-sign constrained vectors. The solution of such a mathematical program is computationally challenging as it is disjunctive and nonconvex by nature. A number of nonlinear programming based approaches are proposed, after appropriate reformulation of the mathematical program as an equivalent nonlinear programming problem. Actual experimental data are used to validate and determine the most suitable algorithm for parameter identification. It was found that the smoothing-based method is by far superior than other schemes. As the problem is nonconvex and the nonlinear program can only guarantee a local or stationary point, global optimisation procedures are introduced in order to verify the accuracy of the solutions obtained by the algorithm. Two evolutionary search methods capable of finding the global optimum are implemented for parameter identification. The results generated by the evolutionary search techniques confirm the reliability of the solutions identified by the best nonlinear programming algorithm. All computations carried out in the thesis suggest the suitability and robustness of the selected algorithm for parameter identification.

School of Engineering Includes bibliographical references (leaves 248-256)

Introduction of a composition gradient at a join between two materials of different elastic properties should reduce the stress concentrating effect of the interfacial discontinuity. A crack oriented perpendicular to this elasticity gradient will experience mode-mixity, and possible subsequent crack deflection. Explicit analytical solutions for the stress state at the tip of an angled crack in a graded material of a given finite geometry do not exist, and ongoing crack path development in such a gradient has not been characterised. An infiltration processing technique is developed which allows two materials to be joined through a region of graded composition, of tailored width and composition profile. Composition discontinuities at layer interfaces in a stepped gradient can be tolerated due to the resulting interpenetrating network structured (INS) microstructure. Firing stresses were found to be a limitation of the processing technique, overcome by limiting the steepness of the elastic gradient. Alumina and epoxy resin graded composites were produced and tested under monotonic loading, resulting in stable crack path evolution. Stress-field asymmetry at the tip of a crack oriented perpendicular to an elastic gradient was demonstrated, followed by subsequent crack deflection. Stress intensity factor and deflection angle increase with increasing gradient steepness. Rising R-curve behaviour was demonstrated for all compositions of the INS composite, with initiation and plateau toughness decreasing with increasing epoxy content. Evidence of crack bridging by intact ligaments of the epoxy phase in the crack wake explains this behaviour. Crack deflection towards the epoxy region was anticipated and demonstrated for all gradient configurations. An increase in relative crack depth was seen to increase mode-mixity at the crack-tip and subsequent crack deflection, up to a relative depth of ~0.5. No conclusive evidence was found for the influence of crack bridging on crack deflection. Toughness was shown to increase with the inclusion of a microstructural gradient. Measured toughness within graded samples was shown to be controlled by both the local composition and the volume of bridging ligaments in the crack wake. The optimum gradient should ??? extend over the widest region practical, ??? encompass the widest composition range possible, and ??? demonstrate extrinsic crack extension toughening.

The main aim of this work is to examine fracture toughness improvement mechanisms of a composite material containing tough interlayers modified with large thermoplastic particles. ¶ Various vinyl-ester (VE)/ poly(acrylonitrile-butadiene-styrene) (ABS) blends were used for the interlayer-toughening of a VE/glass fibre composite to increase delamination resistance of the material under mode I and mode II loading. Dry ABS powder was mixed with the liquid resin in four different weight ratios: 3.5, 7, 11 and 15 phr (parts per hundred parts of resin) while the layer thickness was varied from 150 to 500um. Firstly, the tensile and mode I fracture toughness properties of the VE/ABS blends were assessed, and, by using the Raman Spectroscopy technique, a chemical reaction was discovered which occurred during ABS/VE mixing. This reaction consisted of butadiene dissolution from the ABS particles into the VE. Also, butadiene saturation within the VE was achieved at a composition of around 7% ABS particle content. Both mode I and mode II fracture toughness of the composite were significantly improved with the application of interlayers. Mode I fracture toughness GIc was found to be a function of interlayer thickness and ABS particle content variations, with the latter dominating GIc after the saturation point. Mode II fracture toughness was found to be independent of interlayer thickness and only moderately influenced by particle content. The toughening mechanisms that were the most influential within this interlayered material were plastic deformation and micro-cracking of the layer materials. Evidence of both mechanisms was found using optical and scanning electron microscopy (SEM). ¶ A numerical analysis was conducted, using the experimental results from this study, to further explain the basic toughening mechanisms and fracture behaviour in the materials. The aim of the analysis was to examine the influence of the particles on the plastic zone size that develops in front of the crack tip, and the interaction between the particles and the crack tip. For this purpose FEA elastic-plastic crack propagation models were employed. Good agreement with the experimental data was found.

Errata inserted / Bibliography: leaves 170-175 / iii, 175 leaves : ill ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, 1987