1. Constitutive Relations in Solids Elasticity
H. Garmestani, Professor
School of Materials Science and Engineering
Georgia Institute of Technology
Outline:
Materials Behavior
Tensile behavior …

2. The Elastic Solid and Elastic Boundary Value Problems
Constitutive equation is the relation between kinetics (stress, stress-rate) quantities and kinematics (strain, strain-rate) quantities for a specific material. It is a mathematical description of the actual behavior of a material. The same material may exhibit different behavior at different temperatures, rates of loading and duration of loading time.). Though researchers always attempt to widen the range of temperature, strain rate and time, every model has a given range of applicability.
Constitutive equations distinguish between solids and liquids; and between different solids.
In solids, we have: Metals, polymers, wood, ceramics, composites, concrete, soils…
In fluids we have: Water, oil air, reactive and inert gases

3. The Elastic Solid and Elastic Boundary Value Problems (cont.)
Load-displacement response

4. Examples of Materials Behavior
Uniaxial loading-unloading stress-strain curves for
(a) linear elastic;
(b) nonlinear elastic; and
(c) inelastic behavior.

5. Constitutive Equations: Elastic
Elastic behavior is characterized by the following two conditions:
(1) where the stress in a material () is a unique function of the strain (),
(2) where the material has the property for complete recovery to a “natural” shape upon removal of the applied forces
Elastic behavior may be Linear or non-linear

6. Constitutive Equation
The constitutive equation for elastic behavior in its most general form as
where
C is a symmetric tensor-valued function and e is a strain tensor we introduced earlier.
Linear elastic  = Ce
Nonlinear-elastic  = C(e) e

7. Equations of Infinitesimal Theory of Elasticity
 Boundary Value Problems
 we assume that the strain is small and there is no rigid body rotation. Further we assume that the material is governed by linear elastic isotropic material model.
Field Equations
(1)
(2) Stress Strain Relations
(3)Cauchy Traction Conditions (Cauchy Formula)
(4)

8. Equations of the Infinitesimal Theory of Elasticity (Cont'd)
In general, We know that
For small displacement
Thus

9. Equations of the Infinitesimal Theory of Elasticity (Cont'd)
Assume v << 1, then
For small displacement,
Thus for small displacement/rotation problem

10. Equations of the Infinitesimal Theory of Elasticity (Cont'd)
Consider a Hookean elastic solid, then
Thus, equation of equilibrium becomes

11. Equations of the Infinitesimal Theory of Elasticity (Cont'd)
For static Equilibrium Then
The above equations are called Navier's equations of motion.
In terms of displacement components

12. Plane Elasticity
In a number of engineering applications, the geometry of the body and loading allow us to model the problem using 2-D approximation. Such a study is called ''Plane elasticity''.  There are two categories of plane elasticity, plane stress and plane strain. After these, we will study two special case: simple extension and torsion of a circular cylinder.

13. Plane Strain &Plane Stress
For plane stress,
(a)    Thus equilibrium equation reduces to
(b)    Strain-displacement relations are
(c)    With the compatibility conditions,

14. Plane Strain &Plane Stress
(d)     Constitutive law becomes, Inverting the left relations,
Thus the equations in the matrix form become:
(e)    In terms of displacements (Navier's equation)

15. Plane Strain (b) (Cont'd)
(b)     Inverting the relations,               can be written as:
(c)      Navier's equation for displacement can be written as:

16. The Elastic Solid and Elastic Boundary Value Problems
 Relationship between kinetics (stress, stress rate) and kinematics (strain, strain-rate) determines constitutive properties of materials.
Internal constitution describes the material's response to external thermo-mechanical conditions. This is what distinguishes between fluids and solids, and between solids wood from platinum and plastics from ceramics.
Elastic solid
Uniaxial test:
The test often used to get the mechanical properties

17. Linear Elastic Solid
If      is Cauchy tensor and       is small strain tensor, then in general,
where         is a fourth order tensor, since T and E are second order tensors is called elasticity tensor. The values of these components with respect to the primed basis ei’ and the unprimed basis ei are related by the transformation law
However, we know that                   and                     then
We have          symmetric matrix with 36 constants, If elasticity is a unique scalar function of stress and strain, strain energy is given by

18. Linear Elastic Solid Show that if for a linearly elastic solid, then
Solution:
Since for linearly elastic solid , therefore
Thus from , we have
Now, since
Therefore,

19. Linear Elastic Solid (cont.)
Now consider that there is one plane of symmetry (monoclinic) material, then
    One plane of symmetry =>     13
If there are 3 planes of symmetry, it is called an ORTHOTROPIC material, then
    orthortropy =>  3 planes of symmetry =>   9
Where there is isotropy in a single plane, then
    Planar isotropy   =>      5
When the material is completely isotropic (no dependence on orientation)
    Isotropic  =>      2

20. Linear Elastic Solid (cont.)
Crystal structure
Rotational symmetry
Number of independent elastic constants
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Hexagonal
Cubic
Isotropic
None
1 twofold rotation
2 perpendicular twofold rotations
1 fourfold rotation
1 six fold rotation
4 threefold rotations 21 13 9 6 5 3 2

21. Linear Isotropic Solid
A material is isotropic if its mechanical properties are independent of direction
Isotropy means
Note that the isotropy of a tensor is equivalent to the isotropy of a material defined by the tensor.
Most general form of (Fourth order) is a function

22. Linear Isotropic Solid
Thus for isotropic material
and are called Lame's constants.
is also the shear modulus of the material (sometimes designated as G).

23. Relationship between Youngs Modulus EY, Poisson's Ratio g, Shear modulus m=G and Bulk Modulus k
We know that
So we have
Also, we have

24. Relationship between EY, g, m=G and k (Cont'd)
Note: Lame’s constants, the Young’s modulus, the shear modulus, the Poisson’s ratio and the bulk modulus are all interrelated. Only two of them are independent for a linear, elastic isotropic materials,