CAPTURING MATERIAL SIZE EFFECTS IN BEAMS WITH NONLOCAL MODELS
- 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:
1. Gradient 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)
- Paradox of a cantilever beam with point load: no nonlocal features are captured.
- 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
- Constitutive relation:
1. Length scale parameter,
2. Phase parameter,
- Finite element formulation for a typical element:
- Global system of equations:
- 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.
 P. Khodabakhshi, and J.N. Reddy, A unified integro-differential nonlocal model, International Journal of Engineering Science, 95:60-75, 2015.
 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.
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.