# 4MB00 - Solid Mechanics

### Contents

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 its service limits.

If deformations tend to be larger, or deformation rates are significant, the material will not behave in a linear elastic manner anymore. The corresponding equations are non-linear and the modelling techniques rely on the deformation history and the time-dependent material behaviour. Such non-linear phenomena are essential for the analysis of forming and production processes, both for polymers and metals.

The course is introduced by 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 number 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 number of 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 number 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. A short introduction on the background of the FEM method is given, after which the MSC.Marc/Mentat software is used for model and analyse realistic 3D problems. Results are evaluated, allowing for different optimization steps in the model.

The second part of the course focuses on the non-linearities for large deformations or displacements, either of a physical (e.g. dissipative materials) or geometrical (e.g. large strains and rotations) nature. The essence of a variety of material models, e.g. non-linear elastic, elasto-plastic, linear visco-elastic, creep, is analyzed through simple one-dimensional loading configurations. This configuration is representative for the commonly used tensile test in material characterization.

The description of non-linearities requires additional strain measures (logarithmic strain) and stress tensors (Cauchy, Kirchfoff, first and second Piola-Kirchhoff). Energetic arguments are forwarded to couple stress and strain measures as to be used in material models.

Analytical solutions for the 1D configurations studied are easily obtained. Approximate solutions are constructed through the FEM method. The material models available in MSC.Marc/Mentat are used to solve the corresponding problems and to evaluate the obtained results in close comparison with the analytical results. Since the system of equations to be solved is now non-linear, an iterative solution procedure (Newton-Raphson) is used, for which the basics are given. The final chapter focuses on time-dependent material behaviour, for which the same analytical and numerical tools will be used.

### Learning objectives

Understanding and knowledge of

- Kinematics (deformations, strains)
- Forces and stresses
- Equilibrium equations, boundary conditions
- Various material models
- The analysis of deformations, stresses and the elastic limits (yield criteria) for the design of products and structures. Application of analytical and numerical (FEM) solution methods for simple engineering problems.
- Non-linear mechanical material behaviour and its mathematical modelling. Predict material behaviour in simple one-dimensional situations, both analytically and numerically.

Upon completing this course, students should be able to elaborate and solve specific engineering problems based on 3D linear elasticity, analyse and interpret the corresponding results.