Stress-Strain Relation

Stress at a Point

All of the 3D problems, where deformations are not homogeneous which means that stress varies from point to point.

It becomes difficult define stress in a 3D problem in terms of the uniaxial bar case.

The stress distribution on the plate with a hole is shown above. It is clear that the stress distribution is not equal, the material closer to the hole is more stressed than the material closer to the edges. We can also call this to be stress concentration effect due to the hole in the plate.

We need to define what would is called “Stress at a point”.

Components in stress in 3D are defined by the Cauchy stress tensor.

Where σ11, σ22, and σ33 are normal stresses, and σ12, σ13, σ21, σ23, σ31, and σ32 are shear stresses. The first index i indicates that the stress acts on a plane normal to the this axis, and the second index j denotes the direction in which the stress acts.

According to the principle of conservation of angular momentum, equilibrium requires that the summation of moments with respect to an arbitrary point is zero, which leads to the conclusion that the stress tensor is symmetric, thus having only six independent stress components, instead of the original nine.

Thus the stress tensor is symmetric.

Failure Theories/Yield Criteria

From the stress tensor we have 9 quantities of stress, which are in actual 6 as σij = σji. Now using these 6 quantities we need to find an equivalent stress which can be compared with the material yield stress to determine the failure if any of the part under the loading. This equivalent stress is called Mises stress.

Von Mises Criteria

In order to get the equivalent stresses we first need to find the principle stresses. Values of Principal stress in a 3 dimensional systems are given by solution of following equation.

Where I1, I2 & I3 are stress invariant.

I1, I2 & I3 can be easily substituted to solve the cubic equation and find the three values of σ.

These σ1, σ2 & σ3 are the principle stresses. They can also be defined as the Eigen values of the stress matrix.

Equivalent stresses can be calculated using the 6 quantities of the stress matrix or the principal stresses using the above equation.

Considering case of uniaxial tension on a bar, we calculate this equivalent stress and compare it with the yield stress.

Trescas Theory

Trescas theory is also known as maximum shear stress theory.

After calculation σ1, σ2 & σ3 are the principle stresses maximum shear stress can be calculated assuming the condition:        σ1> σ2 > σ3

τ = (σ1 – σ3)/2;           where τ = Maximum shear stress

τ is compared with the maximum shear stress in a component under uniaxial load i.e. (σy/2), where σy = yield stress.

Trescas theory is valid quite good, however there are some difficulties. Von Mises criteria is more predominantly used in the industry.

 

Strain

In case of a body under axial load which undergoes deformation, this deformation can be quantified as the change in length of the body. This change in length can be defined by a term called strain which is proportional to stress within the elastic limit of the material of the body.

Strain (ϵ) can be defined as the change in length (δ) divided by the original length (l).

ϵ = δ / l

Deformation can be defined as the non-uniformity in displacement of the particles that make up the body and this is what caused strain.

Infinitesimal or Small Strain

It’s a mathematical approach to define deformations in a solid body when the displacement of particles inside the body are infinitesimally small compared to any other dimension of the body such that its geometry and material properties are unchanged by the deformation.

This approach is used in FEA for linear analysis. From this we get the strain-displacement relationships define strain.

Strain Energy Density

A material is defined to be elastic when it has a strain energy density function. This strain density is a function of strain and is a single energy valued positive definitive function.

Isotropic materials

Materials which are considered elastic have a well-defined assumption that these materials are isotropic.

Isotropic material homogeneous means their properties are same in all directions. Most metals are isotropic.

Stress Strain Relationship

Finite element mesh generated by the FEA application can be 3D which requires stress strain relationship applicable to 3D case is given by:

2D FEA mesh can also be generated which simplifies the 3D case keeping some assumptions. 2D simplifications are as follows:

1. Plane Stress

Stress-Strain relationship for plane stress.

2. Plane Strain

Stress-Strain relationship for plane strain.

3. Axisymmetric Condition

In axisymmetric problems, the radial displacements develop a circumferential strain which induces stresses σr, σƟ, σz & τrz where r, Ɵ & z indicate the radial, circumferential and longitudinal directions respectively.

Because of symmetry about the z axis, the stresses are independent of the Ɵ coordinate.

Consider an axisymmetric ring element and its cross section to represent the general state of strain for an axisymmetric problem.

There is strain in the Ɵ direction but it is independent of Ɵ, but depends on the r.

Stress strain relationship for axisymmetric element is given by.