URL study guide
https://tue.osiris-student.nl/onderwijscatalogus/extern/cursus?cursuscode=4MB00&collegejaar=2025&taal=enDescription
The functionality, strength and reliability of a mechanical component, product or device are controlled by the deformations or loads imposed through its service conditions. Deformations are caused by the (external) loads, giving rise to stresses within the material. In most engineering design problems, deformations and rotations remain small and reversible (elastic), whereby no energy is dissipated in the material. This regime is adequately described with the theory of linear elasticity, which is widely used as the basic tool for designing and optimizing products and defining their service limits. The modeling of complex products is inevitably a three-dimensional problem. Strains and stresses therefore must be characterized in a 3D-space, whereby the efficient use of vectors and tensors is indispensable.The course therefore starts with a chapter on vector and tensor calculus, which presents all mathematical ingredients for the 3D description of evolving continua. The 3D kinematics of a deformable solid is next discussed, whereby the most important deformation and strain tensors are introduced (deformation gradient tensor, right Cauchy-Green, Green-Lagrange strain tensor, etc.). In the engineering limit that deformations and rotations are small, linearization is possible, yielding the conventional linear infinitesimal strain tensor. Principal strain directions, principal strains and stretch ratios are important characteristic quantities. As a result of the applied deformation, stresses arise, which can be described by different stress tensors, out of which the Cauchy stress tensor is the most important one. Principal stresses and corresponding directions are also derived.
The deformed state is described with a set of partial differential equations, the equilibrium equations (balance laws), whereby the applied loads on the material constitute the boundary conditions. The unknown stresses appearing in the balance laws are related to the deformations through the constitutive equations describing the mechanical material behaviour. For small deformations this simplifies to linear elasticity, which defines a linear relationship between the stress and strain tensor, involving a several elastic material constants. Even though the mechanical behaviour is generally anisotropic, most attention will be given to isotropic behaviour, where the mechanics is not dependent on material directions. The limit of the elastic domain, often used as an engineering design criterion, is given full attention. Different elastic limits, or flow criteria, are presented out of which the Von Mises criterion is the most important one. The complete set of governing equations describes the material deformation behaviour controlled by linear elasticity. The resulting theory and equilibrium problem is used for a series of design problems. Pressurized cylinders (thin-walled, thick-walled, shrink-fit compound), rotating discs and stress concentrations resulting from holes in a thin plate are solved and analyzed accordingly. Analytical solutions can only be obtained for problems that are sufficiently simple in terms of geometry and boundary conditions. For all other practical cases, approximate numerical solutions are required. This calls for appropriate solution techniques for the system of partial differential equations, for which the finite element method (FEM) is widely used in mechanics. The foundations of this numerical solution method will be provided in the course 4MC10. In Solid Mechanics, the commercial FE package MSC.Marc/Mentat will be used to solve realistic problems and analyse the results based on the theory provided. The analysis of the mechanical solution allows to adapt the model and design, which is intrinsic for any optimization step.
Objectives
Solid Mechanics forms the basis of 3D mechanical design of any structures, parts, or their components, which are made of various materials, based on criteria for strength and stiffness. The learning objectives are:- Understand, use and exploit tensor/vector operations in the context of the course.
- Describe, analyse and compute the kinematics of solids (deformations, strains) for small deformation cases.
- Describe, analyse and compute the forces and stresses acting in a 3D continuum.
- Describe and understand the governing equilibrium equations.
- Analyse and apply the required boundary conditions for solving the equilibrium problem.
- Compute and analyse deformations, stresses and the elastic limits (yield criteria) for the design of products and structures.
- Design mechanically loaded components on the basis of 3D linear elasticity, by applying analytical and numerical (FEM) solution methods (making use of commercial FEM software).