David Nicholson

Finite element analysis : thermomechanics of solids

Introduction To The Finite Element Method Introduction Overview of the Finite Element Method Mesh Development Mathematical Foundations: Vectors and Matrices Introduction Vectors Matrices Eigenvalues and Eigenvectors Coordinate Transformations Orthogonal Curvilinear Coordinates Gradient Operator in Orthogonal Coordinates Divergence and Curl of Vectors in Orthogonal Coordinates Appendix: Divergence and Curl of Vectors in Orthogonal Curvilinear Coordinates Mathematical Foundations: Tensors Tensors Divergence of a Tensor Invariants Positive Definiteness Polar Decomposition Theorem Kronecker Products of Tensors Examples Introduction to Variational Methods Introductory Notions Properties of the Variational Operator Example: Variational Equation for a Cantilevered Elastic Rod Higher Order Variations Examples Fundamental Notions of Linear Solid Mechanics The Displacement Vector The Linear Strain and Rotation Tensors Examples of Linear Strain and Rotation Tensors Traction and Stress Equilibrium Stress and Strain Transformations Principal Stresses and Strains Stress Strain Relations Principle of Virtual Work in Linear Elasticity Thermal and Thermomechanical Response Balance of Energy and Production of Entropy Classical Coupled Linear Thermoelasticity Thermal and Thermomechanical Analogs of the Principle of Virtual Work and Associated Finite Element Equations One-Dimensional Elastic Elements Interpolation Models for One Dimensional Elements Strain-Displacement Relations in One Dimensional Elements Stress-Strain Relations in One Dimensional Elements Element Mass and Stiffness Matrices from the Principle of Virtual Work Integral Evaluation by Gaussian Quadrature: Natural Coordinates Unconstrained Rod Elements Unconstrainted Elements for Beams and Beam-Columns Assemblage and Imposition of Constraints Damping in Rods and Beams General Discussion of Assemblage General Discussion of the Imposition of Constraints Inverse Variational Method Two- and Three-Dimensional Elements in Linear Elasticity and Linear Conductive Heat Transfer Two Dimensions Interpolation Models in Three Dimensions Strain Displacement Relations and Thermal Analogs Stress-Strain Relations Stiffness and Mass Matrices and Their Thermal Analogs Thermal Counterpart of the Principle of Virtual Work Conversion to Natural Coordinates in Two and Three Dimensions Assembly of Two and Three Dimensional Elements Solution Methods for Linear Problems - I Numerical Methods in FEA Time Integration: Stability and Accuracy Properties of the Trapezoidal Rule Integral Evaluation by Gaussian Quadrature Modal Analysis by FEA Solution Methods for Linear Problems -II Introduction Solution Method for an Inverse Problem Accelerated Eigenstructure Computation in FEA Fourth Order Time Integration Additional Topics in Linear Thermoelastic Systems Transient Conductive Heat Transfer in Linear Media Coupled Linear Thermoelasticity Incompressible Elastic Media Torsion of Prismatic Bars Buckling of Elastic Beams and Plates Introduction to Contact Problems Rotating and Unrestrained Elastic Bodies Finite Elements in Rotation Critical Speeds in Shaft-Rotor Shaft Finite Element Analysis for Unconstrained Elastic Bodies Appendix: Angular Velocity Vector in Spherical Coordinates Aspects on Nonlinear Continuum Thermomechanics Introduction Nonlinear Kinematics of Deformation Mechanical Equilibrium and the Principle of Virtual Work Principle of Virtual Work Under Large Deformation Nonlinear Stress-Strain-Temperature Relations: The Isothermal Tangent Modulus Tensor Introduction to Nonlinear FEA Introduction Types of Nonlinearlity Newton Iteration Combined Incremental and Iterative Methods: A Simple Example Finite Stretching of a Rubber Rod Under Gravity Newton Iteration Near a Critical Point Introduction to the Arc Length Method Incremental Principle of Virtual Work Incremental Kinematics Stress Increments Incremental Equation of Balance of Linear Momentum Incremental Principle of Virtual Work Incremental Finite Element Equation Contributions From Nonlinear Boundary Conditions Effect of Variable Contact Interpretation as Newton Iteration Buckling Tangent Modulus Tensors for Thermomechanical Response of Elastomers Introduction Compressible Elastomers Incompressible and Near-Incompressible Elastomers Stretch-Ration Based Models: Isothermal Conditions Extension to Thermohyperelastic Materials Thermomechanics of Damped Elastomers Constitutive Model in Thermoviscohyperelasticity Variational Principles and Finite Element Equations for A Thermoviscohyperelastic Material Tangent Modulus Tensors for Inelastic and Thermoinelastic Materials Plasticity Tangent Modulus Tensor in Small Strain Isothermal Plasticity Plasticity Under Finite Strain Thermoplasticity Tangent Modulus Tensor in Viscoplasticity Continuum Damage Mechanics Selected Advanced Numerical Methods in FEA Iterative Triangularization of Perturbed Matrices Stiff Arc Length Constraint in Nonlinear FEA Non-Iterative Solution of Finite Element Equations in Incompressible Solids References Index

Modal moment index for damage detection in beam structures

SummaryHealth monitoring and damage detection in structures such as airframes are crucial to their safety and performance. Damage-induced changes in natural frequencies, damping ratios and mode shapes can be detected using experimental modal analysis (EMA), which has recently enjoyed intense research for application in structural health monitoring. A main theme of the current research has been to formulate a parameter, calculated using the measured changes in eigenparameters, which exhibits a significant “jump” at the damage location and whose value is a direct measure of the severity of the damage. A damage-sensitive parameter thus serves to localize the damage and to assess its severity, and it should likewise minimize the likelihood of false indications or of missing damage. Here, using finite element analysis of the damaged cantilevered beam, we study several existing parameters and introduce a new parameter called the Modal Moment Index (MMI). MMI is shown to jump sharply at the damage site and to have a direct relationship to the damage level.

Extreme value probabilistic theory for mixed-mode brittle fracture

Abstract In this investigation, extreme value probabilistic methods are combined with Sihs mixed-mode fracture model to furnish strength distributions in plates of brittle materials with random cracks. The crack lengths are described by a two-parameter probability density function, their orientations follow a uniform distribution and the crack number follows a binomial distribution. Materials of interest are assumed to be isotropic and statistically homogeneous. A “weakest link” model, thought to be appropriate for brittle materials, is used in which catastrophic failure occurs if the dominant crack attains a critical condition. Extreme value distributions for strength of the plates are derived as a function of the size (crack number) of the plates, the parameters of the fracture model and the parameters of the crack length distribution. Numerical results are presented showing the effect of the normalized variance of the crack length distribution on the scale dependence of the mean and variance of the plate strength distribution.

Probabilistic theory for mixed mode fatigue crack growth in brittle plates with random cracks

Abstract A probabilistic fracture mechanics theory is proposed for fatigue life prediction in which a mixed-mode fatigue crack growth model is combined with extreme value probabilistic methods. The Sih–Barthelemy fatigue crack growth criterion based on strain energy intensity factor under uniaxial loading has been slightly modified and extended to multiaxial loading including biaxial loading and torsion. A probability density function is computed for the number of cycles to failure, and scaling relations are also computed. Crack number, length, and orientation at the onset of fatigue are treated as random varixsables whose probability distributions are known. Numerical results are presented to demonstrate dependence on plate size, on the variance of the crack length distribution, on the multiaxial loading factors, and on the parameters of the fatigue model.

Large deformation theory of coupled thermoplasticity including kinematic hardening

SummaryThermoplasticity is a topic central to important applications such as metalforming, ballistics and welding. The current investigation introduces a thermoplastic constitutive model accommodating the difficult issues of finite strain and kinematic hardening. Two potential functions are used. One is interpreted as the Helmholtz free energy. Its reversible portion describes elastic behavior, while its irreversible portion describes kinematic hardening. The second potential function describes dissipative effects and arises directly from the entropy production inequality. It is shown that the dissipation potential can be interpreted as a yield function. With two simplifying assumptions, the formulation leads to a simple energy equation, which is used to derive a rate variational principle. Together with the Principle of Virtual Work in rate form, finite element equations governing coupled thermal and mechanical effects are presented. Using a uniqueness argument, an inequality is derived which is interpreted as a finite strain thermoplastic counterpart to the classical inequality for “stability in the small”. A simple example is introduced using a von Mises yield function with linear kinematic hardening, linear isotropic hardening and linear thermal softening.

Numerical simulation of temperature-dependent, anisotropic tertiary creep damage

Directionally -solidified (DS) Ni -base superalloys are commonly applied as turbine materials to primarily withstand creep conditions manifested in either marine -, air - or land based gas turbines components. The thrust for increased efficiency of these syste ms, however, translates into the need for these materials to exhibit considerable strength and temperature resistance. Accurate prediction of crack initiation behavior of these hot gas path components is an on -going challenge for turbine designers. Aside from the spectrum of mechanical loading, blades and vanes are subjected to high temperature cycling and thermal gradients. Imposing repeated start -up and shut -down steps leads to creep and fatigue damage. The presence of s tress concentrations due to cooli ng holes , edges, and sites sustain foreign object damage must also be taken into account. These issues and the interaction thereof can be mitigated with the application of high fidelity constitutive models implemented to predict material response under giv en thermomechanical loading history. In the current study, the classical Kachanov -Rabotnov model for tertiary creep damage is implemented in a general -purpose finite element analysis (FEA) software. The evolution of damage is considered as a vector -valued quantity to account for orientation -dependent damage accumulation. Creep deformation and rupture experiments on samples from a representative DS Ni -base superalloys tested at temperatures between 649 and 982°C and three o rientations (longitudinally -, tran sversely -oriented , and intermediately oriented ). The damage model coefficients corresponding to secondary and tertiary creep constants are characterized for temperature and orientation dependence. This advanced formulation can be implemented for modeling f ull -scale parts containing temperature gradients .

On finite element analysis of an inverse problem in elasticity

This investigation concerns an inverse problem modelled by the finite element method. For a given mesh and set of physical properties, even though a well-posed direct problem possesses a unique solution in classical linear elasticity, variational arguments establish that a corresponding inverse problem may not. Furthermore, even when the inverse problem admits a unique solution when modelled ‘exactly’ using the classical linear theory of elasticity, an unfortunate mesh choice may cause the finite element model of the inverse problem to fail to do so. The current investigation introduces a simple matrix nonsingularity criterion assuring that the finite element model possesses a unique solution. An apparently new and readily computed numerical test is introduced to verify satisfaction of the criterion. Examples are given illustrating the effectiveness of the criterion and its application to mesh design.

TEMPERATURE AND ORIENTATION DEPENDENCE OF CREEP DAMAGE OF TWO NI-BASE SUPERALLOYS

Both polycrystalline (PC) and directionally-solidified (DS) Ni-base superalloys are commonly applied as turbine materials to primarily withstand creep conditions manifested in either marine-, air- or land-based gas turbines components. The thrust for increased efficiency of these systems, however, translates into the need for these materials to exhibit considerable strength and temperature resistance. This is critical for engine parts that are subjected to high temperature and stress conditions sustained for long periods of time, such as blades, vanes, and combustion pieces. Accurate estimates of stress and deformation histories at notches, curves, and other critical locations of such components are crucial for life prediction and calculation of service intervals. In the current study, the classical Kachanov-Rabotnov model for tertiary creep damage is implemented in a general-purpose finite element analysis (FEA) software. Creep deformation and rupture experiments on samples from two representative Ni-base superalloys (PC and DS) tested at temperatures between 649 and 982°C and two orientations (longitudinally- and transversely-oriented for the DS case only) are applied to extend this damage formulation. The damage model coefficients corresponding to secondary and tertiary creep constants are characterized for temperature and orientation dependence. This updated formulation can be implemented for modeling full-scale parts containing temperature distributions.© 2007 ASME

Inverse method for identifying the underlying crack distribution in plates with random strengths

SummaryIn the current investigation we seek to identify the underlying crack number and crack length distributions in brittle plates with a known strength distribution. The inverse problem in probabilistic fracture mechanics is defined, and the numerical procedure to solve the inverse problem is constructed. The simulation process of generating simulated plates containing simulated random cracks is elaborated. The maximum strain energy release rate criterion (Gmax) is applied to each simulated random crack to find the crack strength. The strength of the simulated plate is equated to the strength of the weakest simulated crack in the plate based on the weakest link notion. The underlying crack number and crack length distributions are obtained by minimizing the difference between the simulated plate strengths and the known plate strengths. The gamma, lognormal and two-parameter Weibull distributions are employed for the underlying crack length distribution, and are compared in order to identify the best choice. Numerical examples demonstrate that the three PDFs are all acceptable for reasons to be explained. In the appendix, the direct problem in probabilistic fracture mechanics is presented as part of the demonstration of a method for using the crack distribution identified in the inverse problem to predict the strength and the probability of fracture in a practical application.

Finite Element Analysis of Thermomechanical Contact of an Elastomeric O-Ring Seal

This study concerns the development of a finite element model to support design improvements in elastomeric seals subject to high temperature and pressure, such as in aircraft engines. Existing finite element codes familiar to the authors do not couple thermal and mechanical fields, nor do they implement thermomechanical contact models suitable for highly deformable materials. Recently, the authors have introduced a thermohyperelastic constitutive model for near-incompressible elastomers. In two subsequent studies, using the constitutive model, a method has been introduced for finite element analysis of coupled thermomechanical response, including boundary contributions due to large deformation and variable contact. A new thermomechanical contact model has also been introduced to accommodate the softness of elastomers. The method has been implemented in a special purpose code which concerns a seal compressed into a well. Several computations are used to validate the code. Simulations of a seal in an idealized geometry indicate rapid pressure increase with increasing compression and temperature.Copyright