SSG 516 Continuum Mechanics
Introduction
Textbooks
1. Bower, AF, Applied Mechanics of Solids, CRC Press,
2010, pp 711-764, 1-63, 65-93
2. Gurtin, ME, Eliot, F & Anand, L, The Mechanics and
Thermodynamics of Continua, Cambridge 2010, Part
I: Vector and Tensor Algebra, Part II: Vector and
Tensor Analysis.
3. Fakinlede, OA, Nonlinear Mechanics, PDF, 2011,
Solved problems pages 1-200
Course Materials
 Materials used in this course are largely based (not
restricted to) on the books shown below.
 Specific locations in the texts that you will need to
read are shown for your convenience.
 Worked examples in the third material will form the
bases of continuous assessment and examinations
What is Continuum Mechanics?
 At this stage in your degree program, you have
studied courses such as Fluid Mechanics, Strength of
Materials, Thermodynamics, etc. and applied them to
the behavior of materials without necessarily
perceiving the relationship between the different
issues.
 Here we take as our starting point, the balance laws
for mass, linear and angular momenta, energy and
entropy imbalance
Continuum Mechanics
 This collection of physical laws will be augmented by
specific constitutive models to fully describe the
response of material bodies when they are subjected
to Mechanical, Thermal or other kinds of external
influences or loading.
 All bodies are subject to exactly the same physical
laws. Material constitutions differ. The latter is usually
described by a model with specific parameters
evaluated by experimental data.
Mathematical Description
 The combination of physical laws and material
constitutive models form the mathematical
description of the body in question.
Solution Methods
 Only the simplest of these can be completely solved
by the present mathematical tools available.
 In typical problems, we are compelled to seek
approximate solutions.
 Several general algorithms have devised over the
years to do just that.
 With the advent of powerful computational tools,
some of these have grown in their range of
applicability and sophistication
Models and Simulations
 We are therefore able to model and simulate the
loading and responses in order to gain valuable
insight that can be useful in design, failure prediction
and prevention as well as using these materials in
new ways such as biological implants, and other
medical devices
Design Examples
 Civil Structures. Buildings, roads and bridges,
retaining walls and soil foundations.
 Mechanical Design. Load bearing components for
vehicles, engines, and other appliances. Selection of
materials, dimensions and shapes appropriate to
loading and service conditions
 Manufacturing processes such as metal and polymer
production, Computation of energy requirements and
process design.
Application Examples
 Biomechanics. Implants and medical devices for
prosthetics, analysis of loads generated by biological
processes such as heartbeats, blood flows and
extraneous loads due to medical conditions such as
blockages, etc
 Other examples of applications are in Geomechanics,
materials science, microelectronics, nanotechnology,
etc
Computations
 In your earlier courses such as strength of materials,
you were able to compute the stresses and
displacements in elastic materials subject to simple
loading conditions. Furthermore, computations of the
idealized “elastica” using the Euler-Bernoulli’s
bending theory was used to predict displacements of
members subjected to various kinds of loading and
support systems.
Computations
 Continuum Mechanics allow you to model under
more general conditions. Materials are not necessarily
elastic and may not be isotropic. Instead of focusing
on the elastica, computer simulators allow you to
model actual member dimensions directly. In
addition, other computations that can be done
include:
 The forces that can cause prescribed shape changes.
 Failure conditions due to multiaxial stresses; theories
of failure. Extrapolations from uniaxial material
properties
 Computation of natural frequencies of vibration for
dynamical problems; Avoidance of catastrophic
resonance conditions
 Critical fracture loads
 Failure under cyclic (fatigue) loading
Multiphysics
 Classical theoretical analysis model systems on
dominant physical causes.
 It is becoming increasing important to look at the
interaction of several physical phenomena
concurrently. The strains in a loaded material may be
caused by applied forces as well as thermal loading
concurrently.
 This consideration has give rise to several simulation
packages emphasizing the “multiphysics”
Constitutive Models
 The most successful constitutive model is that of
homogeneous linear isotropic elasticity.
 Several important materials are not accurately
modeled under these assumptions.
 The relaxation of any of these introduces great
complexity to the mathematical formulation.
 Even materials that are constitutively linear may have
nonlinear strains as a result of large displacements or
rotations.
Limitations
 The scope of Continuum Mechanics is wide
 In this course we will have the modest goals of
understanding the
1.
2.
3.
4.
theory of shape changes that define appropriate measure of
displacement - kinematics
Theory of stress – a mathematical description of the internal
forces in a solid.
Mechanical balance laws that every material is subjected to
Work and energy concepts and conjugate analysis