# CAPTURING MATERIAL SIZE EFFECTS IN BEAMS WITH NONLOCAL MODELS

Parisa Khodabakhshi and J.N. Reddy

## INTRODUCTION

• Shortcomings of conventional local theories of continuum mechanics:
1. Dispersion of elastic waves, crack propagation in fracture mechanics, dislocations
2. Recent developments in the field of material science
• Categories of nonlocal theories:
2. Integral theories I will display → Eringen integral model (1983)

• Gradient theories exhibit stiffness enhancing effects, whereas, integral theories have a softening effect.
• Integral theories: stress at each point depends on strain at all points in the domain

• Existing research in the literature is based on the differential form of Eringen nonlocal model.
Special class of kernel function (1D)

where

• Paradox of a cantilever beam with point load: no nonlocal features are captured.

## OBJECTIVES

• Construct a finite element framework for the integral form of Eringen nonlocal model:
1. A different choice of kernel function

2. Two-phase Eringen nonlocal model (1987)

• Numerical Analysis of typical beams
• Assessment of the integral form of Eringen nonlocal model

## 1D KERNEL FUNCTION

• Condition to be satisfied

• Concept of compact support:

• Determination of the elements lying in the influence zone of each element

## EULER-BERNOULLI BEAM

• Constitutive relation:

1. Length scale parameter,
2. Phase parameter,

• Finite element formulation for a typical element:
•

• Global system of equations:
•

## CONCLUSIONS

• The integral form of Eringen nonlocal model doesn’t show the paradox of cantilever beams with point load. → nano-sized actuators and sensors
• Eringen nonlocal model in general has a softening effect, however, some stiffness enhancement can be seen in the case of the simply supported beam.
• Some error is introduced at the boundaries regarding the condition on the kernel function. A modified kernel function will be employed in the future.
• By integrating the gradient theories with Eringen nonlocal model a wide range of nonlocal features can be captured.

## REFERENCES

[1] P. Khodabakhshi, and J.N. Reddy, A unified integro-differential nonlocal model, International Journal of Engineering Science, 95:60-75, 2015.
[2] A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocations and surface waves, Journal of Applied Physcis, 54(9):4703-4710, 1983.

## ACKNOWLEDGEMENT

The authors gratefully acknowledge the support of the present research by the National Science Foundation (NSF) through Grant No. CMMI-1068181 and Oscar S. Wyatt Endowed Chair in the Department of Mechanical Engineering at Texas A&M University.