1. Concepts of stress and strain
Stress at a point FN F
positive side
Plane Q
area A Fs
negative side

2. Concepts of stress and strainz y
For the cube face in the
–x direction, we have
stresses:
Stress tensor
σxz
σ-x-x
σxy
σ-x-y
σ-x-x
σxx x

3. Concepts of stress and strain
In general, we can define the stress vector acting on an arbitrary plane with direction cosines la1, la2, la3 referred to
The x, y, z coordinate system. The arbitrary plane is defined in terms of its unit normal direction;
Stress vector on an arbitrary plane

4. Concepts of stress and strain
Transformation of stress components: suppose we have the stress components in
one coordinate system (x, y, z) and we want to get the components in another (say rotated)
coordinate system (x’, y’, z’):

5. Concepts of stress and strainx3
In testing single crystals in the form of wires or cylinders it is possible to apply a single normal component of stress, , as shown at left. The plastic deformation depends on a particular shear component of stress referred to a particular crystallographic direction on a particular crystallographic plane. Application of the transformation equation shows that the shear stress on the 3’ plane in the 2’ direction, , can be expressed in terms of the applied stress, , and the angles and
Slip plane normal
x3’
x2’
Slip plane
Slip direction
x1’ x2 x1

6. Concepts of stress and strain
Principal stresses and stress invariants
For any general state of stress at a point P, there exists 3 mutually perpendicular planes on which the shear stresses
Vanish. The resulting stresses on these planes are normal stresses and are called principal stresses (Eigen values) and
The normal directions defining these planes are called principal directions (Eigen vectors). These principal directions
can be referred to a coordinate system (x, y, z) in space.

7. Concepts of stress and strain
The condition for a non trivial solution to exist is given by the so-called characteristic equation:
This results in a cubic equation that can be solved for the 3 roots (sI, sII, sIII) corresponding to the 3 principal stresses. The principal directions can be solved by substituting separately each of the principal stresses into the set on linear equations and solving for the direction cosines subject to the condition that;

8. Concepts of stress and strain
Strain at a point (small displacements):
1-D strain A B A* B* P

9. Concepts of stress and strain
More generally, the displacement, u, in the x-direction will also depend on the
y and z coordinates of a point under a more general deformation, i.e.,
Similar expressions can be written for the displacement in the y-direction, v, and
z-direction w.

10. Concepts of stress and strain
The terms that look like
Can Involve not only a deformation strain
but also a rigid-body rotation.
Pure shear
Pure rotation
with no shear
simple shear

11. Concepts of stress and strain
The 9 relative displacement components can be decomposed into a symmetric part
defining the strain tensor and a rotation tensor, i.e.,
The strain tensor transforms the same way as the stress tensor and all 2nd order
Symmetric tensors.

12. Concepts of stress and strain
Very often, the shear strain is expressed as engineering shear strain, γ.
However, be careful. since this form of shear stain does not transform the
same way as a symmetric 2nd order tensor.

13. Elastic stress-strain relations
Since stress and strain are 2nd order tensors their coupling
Is by a 4th order tensor, C (stiffness);
Alternatively the strains can be written in terms of the stresses
Where the S is called compliance
The relation between a stiffness and a compliance component
involves a formal matrix inversion.

14. Elastic stress-strain relations
In general a 4th rank tensor has 34 components but owing to
symmetries in the stress & strain tensors the 81 components
reduce to 21 independent components for crystals of low
symmetry (Triclinic). For cubic crystals the 21 components can be further reduced to 3 independent components. An isotropic solid has only 2 independent elastic constants.
For everything that we will be doing in this class, we will be assuming that our solids elastically isotropic.

15. Elastic stress-strain relations
Isotropic elastic stress-strain relations

16. Elastic stress-strain relations
Isotropic elastic stress-strain relations:
There are numerous elastic constants for an isotropic solid but
they are all describable in terms of any 2 elastic constants.

17. Elastic strain energy F s
The work done in deforming an elastic solid is reversible and
Is stored in the solid as strain energy.
Uniaxial loading F s
Strain energy per unit volume

18. Elastic stress-strain relations
For general loading