# Introduction

In modern industrial applications, polymer components are becoming ever more pervasive due to low cost and high strength-to-weight ratio as one factor among many. Very often, engineers utilize classical methods from strength of materials to assess the strength of both metal and polymer components. However, the applicability of such calculations is limited since the underlying assumptions of classical methods assume linearity of the material’s stress-strain curve and small strain.

One such often-used computation that assumes material linearity and small strain in ascertaining the strength of a metal component is the stress concentration factor (SCF). Using the theory of elasticity, the SCF for many geometries has been tabulated. But, the question becomes, to what degree can one use the SCF in the design of components made from nonlinear materials like elastomers, thermoplastics, and other types of polymers?

The purpose of this study is to explore the limitations of the SCF associated with the small-strain and material linearity assumptions.  To that end, we will simulate the classic plate with a central hole and compare the resulting SCF with the theoretical value using three different materials: structural steel as the baseline, an elastomer modeled using 3rd order Yeoh hyperelasticity, and a generic ABS using the Ansys Three Network Model (TNM).

# Theory

In this study, we simulate a finite rectangular plate with a central hole subject to a tensile force on its end faces, resulting in tensile stress,, where P is the applied tensile force, W is the width of the plate, and t is the thickness of the plate.  However, when working with finite plates with holes, one needs to define the nominal stress, , where d is the diameter of the hole. Using the nominal stress, the stress concentration factor is defined as , where the maximum stress is found as the maximum equivalent stress on the surface of the hole.

For the finite plate with a central hole, an empirical relationship for K given the ratio of hole diameter to plate width, , is

# Case Study

For the case study, we utilize a plate, having the following dimensions:

 Dimension Value [mm] W 50 t 2 d 5

Thus, the nominal stress area = 90 mm2 and .  Plugging these values into the empirical relation, we find the theoretical SCF, K = 2.72.

# Simulation

## Workbench Model and Geometry

The simulation model consists of three Static Structural systems from within Ansys Workbench for each of the three considered materials utilizing the same quarter-symmetry plate geometry.

## Material Properties

The material properties for each of the three cases are

1. Structural Steel using isotropic elasticity taken from Engineering Data in Workbench.
1. E = 200 GPa and Poisson ratio = 0.3
2. Sample elastomer taken from Engineering Data in Workbench, fit with 3rd order Yeoh hyperelasticity.
3. Generic ABS, utilizing data from and fit using MCalibration2 to the PolyUMod TNM model and then entered into Engineering Data as the Ansys TNM model.

## Mesh

The image here shows the mesh that is utilized in common for all cases.  The maximum stress is expected to be on the surface of the hole, theoretically, so the mesh is refined near the hole.  The final mesh shown below is a result of a mesh convergence study conducted for the steel material case.

## Loads and Boundary Conditions

The loads and boundary conditions are shown here for the Steel's Static Structural system.

Given the material properties detailed above, the applied load is different for each material.  For the elastomer and ABS, the applied force is selected in order to activate material nonlinearity and ensure model convergence.  The applied forces and nominal stresses are tabulated here, noting that the nominal stress area is halved due to model symmetry:

 Material Force [N] Snom [MPa] Steel 4,500 100 Elastomer 90 2 ABS 2,160 48

# Results

Below is a plot of the simulation results for steel.  Using von Mises equivalent stress, the resulting SCF from simulation is 2.77 which is in good agreement with theory.

For each considered material, a reference stress is chosen to normalize the nominal stress to facilitate direct A/B comparison, where for steel and ABS, the reference stress is chosen to indicate the extent of the linear region.  For the elastomer there is effectively no linear region, so the reference stress is chosen to be a desired maximum applied nominal stress.

 Material Reference Stress, Sref [MPa] Steel 100 Elastomer 1 ABS 35

Using the results from each of the three materials, this graph shows how the SCF varies with the ratio of nominal-to-reference stress.

Here we find that the SCF matches theory very well for steel within the elastic region and for ABS within approximately 40% of its elastic region, decreasing as the material is increasingly stress.  For the Elastomer, we see that the applicability region of the SCF is very minimal as the material does not actually have a linear region.

# Conclusions

The results indicate that the use of SCF is limited to stresses that reside with the linear region of stiff materials where the small strain approximation applies.  For materials that are softer where the small strain assumption is violated, the SCF has very limited applicability even in the linear region of the stress-strain curve.  Moreover, for materials that are hyperelastic, the SCF if effectively rendered inapplicable, thus suggesting that elementary stress calculations are prone to error and that simulation is required to accurately assess stress.

# Going Further

Apply the same methodology to your own materials and applications for which such SCF are tabulated.