1. APPLIED MECHANICS Lecture 10 Slovak University of Technology
Faculty of Material Science and Technology in Trnava
APPLIED MECHANICS
Lecture 10

2. FUNDAMENTALS OF CONTINUUM MECHANICS
The fundamental equations of structural mechanics:
stress-strain relationship contains the material property information that must be evaluated by laboratory or field experiments,
the total structure, each element, and each infinitesimal particle within each element must be in force equilibrium in their deformed position,
displacement compatibility conditions must be satisfied.
If all three equations are satisfied at all points in time, other conditions will automatically be satisfied.
Example - at any point in time the total work done by the external loads must equal the kinetic and strain energy stored within the structural system plus any energy that has been dissipated by the system.
Virtual work and variational principles are of significant values in the mathematical derivation of certain equations.

3. FUNDAMENTALS OF CONTINUUM MECHANICS
The linear stress-strain relationships contain the material property constants, which can only be evaluated by laboratory or field experiments.
The mechanical material properties for most common material, such as steel, are well known and are defined in terms of three numbers:
modulus of elasticity E,
Poisson’s ratio n,
coefficient of thermal expansion a.
Simplification - materials are considered isotropic (equal properties in all directions) and homogeneous (the same properties at all points in the solid).
Real materials have anisotropic properties, which may be different in every element in a structure.

4. FUNDAMENTALS OF CONTINUUM MECHANICS
All stresses are by definition in units of force-per-unit-area.
In matrix notation, the six independent stresses:
From equilibrium of element:
The six corresponding engineering strains:

5. FUNDAMENTALS OF CONTINUUM MECHANICS MATERIAL PROPERTIES - Anisotropic materials
Anisotropic materials - The most general form of the three dimensional strain-stress relationship for linear structural materials subjected to both mechanical stresses and temperature change can be written in the following matrix form

6. FUNDAMENTALS OF CONTINUUM MECHANICS MATERIAL PROPERTIES - Anisotropic materials
In symbolic matrix form
d = Cf + DTa
C - compliance matrix - the most fundamental definition of the material
properties
DT - temperature increase - in reference to the temperature at zero stress,
a - matrix indicates the strains caused by a unit temperature increase.
Basic energy principles require that the C matrix for linear material be symmetrical. Hence,

7. FUNDAMENTALS OF CONTINUUM MECHANICS MATERIAL PROPERTIES - Orthotropic materials
Orthotropic materials - the shear stresses, acting in all three reference planes, cause no normal strains.
C - 9 independent material constants,
- 3 independent thermal expansion coefficients
This type of material property is very common - rocks, concrete, wood, many fiber reinforced materials exhibit orthotropic behaviour.

8. FUNDAMENTALS OF CONTINUUM MECHANICS MATERIAL PROPERTIES - Isotropic materials
Isotropic materials - equal properties in all directions, the most commonly used approximation to predict the behavior of linear elastic materials.
- shear modulus
For isotropic materials: Young's modulus E, Poisson's ratio n need to be defined.

9. FUNDAMENTALS OF CONTINUUM MECHANICS MATERIAL PROPERTIES – Plane strain isotropic materials
Plane strain isotropic materials - e1, g13, g23 and t13, t23 are zero,
matrix is reduced to a 33 array.
The stress-strain relationship
For the case of plane strain - the displacement and strain in the 3-direction are zero. Poisson's ratio n - approaches 0,5.
The normal stress in the 3-direction is

10. FUNDAMENTALS OF CONTINUUM MECHANICS MATERIAL PROPERTIES – Plane stress isotropic materials
Plane stress isotropic materials – s3, t13, t23 are zero, matrix is reduced to a 33 array.
The stress-strain relationship

11. FUNDAMENTALS OF CONTINUUM MECHANICS MATERIAL PROPERTIES – Fluid-like materials
Fluid-like materials - isotropic materials, which have a very low shear modulus compared to their bulk modulus, materials are referred to as nearly incompressible solids.
The pressure-volume relationship for a solid or a fluid
where l - bulk modulus of the material.
The volume change e is equal to e1 + e2 + e3, and the hydrostatic pressure s indicates equal stress in all directions.

12. FUNDAMENTALS OF CONTINUUM MECHANICS EQUILIBRIUM AND COMPATIBILITY
Equilibrium equations - set the externally applied loads equal to the sum of the internal element forces at all joints or node points of a structural system;
They are the most fundamental equations in structural analysis and design.
The exact solution for a problem in solid mechanics requires that the differential equations of equilibrium for all infinitesimal elements within the solid must be satisfied.

13. FUNDAMENTALS OF CONTINUUM MECHANICS EQUILIBRIUM AND COMPATIBILITY
Fundamental equilibrium equations
The 3D equilibrium of an infinitesimal element
The body force - Xi, is per unit of volume in the i-direction and represents gravitational forces or pure pressure gradients.

14. FUNDAMENTALS OF CONTINUUM MECHANICS EQUILIBRIUM AND COMPATIBILITY
Stress resultant – forces and moments
For a finite size element or joint a substructure or complete structural system the following six equilibrium equations must be satisfied
Compatibility requirements
For continuous solids - defined strains as displacements per unit length. To calculate absolute displacements at a point, we must integrate the strains with respect to a fixed boundary condition. A solution is compatible if the displacement at all points is not a function of the path. Therefore, a displacement compatible solution involves the existence of a uniquely defined displacement field.

15. FUNDAMENTALS OF CONTINUUM MECHANICS EQUILIBRIUM AND COMPATIBILITY
Strain displacement equations
The small displacement fields u1, u2 and u3 are specified.
The consistent strains can be calculated directly from the following well-known strain-displacement equations

16. FUNDAMENTALS OF CONTINUUM MECHANICS EQUILIBRIUM AND COMPATIBILITY
Definition of rotation
Rotation of a horizontal line may be different from the rotation of a vertical line - following mathematical equations are used to define rotation of the three axes

17. FUNDAMENTALS OF CONTINUUM MECHANICS EQUILIBRIUM AND COMPATIBILITY
Dynamic equilibrium
Real physical structures behave dynamically when subjected to loads or displacements -the additional inertia forces are introduced.
If the loads or displacements are applied very slowly, the inertia forces can be neglected and a static load analysis can be justified. Dynamic analysis - extension of static analysis.
Equation of motion can be expressed: