Current Research Projects

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:
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
$\boldsymbol{ \sigma } (\mathbf{x}) = \int_{V'} \alpha (\mathbf{x}, \mathbf{x}', \boldsymbol{ \tau }) \mathbf{C} : \boldsymbol{ \varepsilon } (\mathbf{x}') d V'$
Existing research in the literature is based on the differential form of Eringen nonlocal model.
Special class of kernel function (1D)

$\alpha (x,x', \tau ) = \frac{1}{l \sqrt{\pi \tau } } e^{- \frac{(x-x')^2}{l^2 \tau} } \rightarrow (1 - \tau^2 l^2 \nabla^2) \mathbf{t} = \boldsymbol{ \sigma }$

where

$\tau = \frac{e_0 a}{l} \rightarrow \tau l = e_0 a = l_c$
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

$\alpha (x,x', l_c ) = \alpha_0 e ^{- \frac{|x-x'|}{l_c} }$

2. Two-phase Eringen nonlocal model (1987)

$\boldsymbol{ \sigma } (\mathbf{x}) = \xi _1 \mathbf{C}: \boldsymbol{ \varepsilon }(\mathbf{x}) + \xi _2 \int_{V'} \alpha (\mathbf{x}, \mathbf{x}', \tau) \mathbf{C}: \boldsymbol{ \varepsilon }(\mathbf{x}') dV'$
Numerical Analysis of typical beams
Assessment of the integral form of Eringen nonlocal model

1D KERNEL FUNCTION

Condition to be satisfied
$\int_{- \infty }^{ \infty } \alpha_0 e ^{- \frac{|x-x'|}{l_c} } dx' = 1 \rightarrow \alpha_0 = \frac{1}{2 l_c}$
Concept of compact support:
$l_i = 6 \times l_c \rightarrow e^{- \frac{l_i}{l_c} } \approx 2.5 \times 10^{-3}$

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

EULER-BERNOULLI BEAM

Constitutive relation:
$\sigma (x) = \xi _1 E \varepsilon (x) + \int _0^L \xi_2 \frac{1}{2l_c} e^{- \frac{|x-x'|}{l_c}} E \varepsilon (x') dx', \quad \varepsilon (x) = \frac{d^2w}{d x^2}$

1. Length scale parameter, $l_c$
2. Phase parameter, $\xi _1$
Finite element formulation for a typical element:

$\xi _1 \mathbf{k}_e^l \Delta_e + \xi_2 \sum_{e'} \mathbf{k}_{ee'}^{nl} \Delta_{e'} = \mathbf{F}_e$

$(\mathbf{k}_e^l)_{ij} = \int_{x_1^e}^{x_2^e}EA_e \frac{d^2 \phi_i^e}{dx^2} \frac{d^2\phi_j^e}{dx^2} dx$

$(\mathbf{k}_{ee'}^{nl})_{ij} = \int_{x_1^e}^{x_2^e}\int_{x_1^{e'}}^{x_2^{e'}} \frac{1}{2l_c} e^{- \frac{|x-x'|}{l_c}} EA_e \frac{d^2 \phi_i^e}{dx^2} \frac{d^2\phi_j^{e'}}{dx^2} dx'dx$

$\Delta _e = \left \langle w _1 ^e \quad \left. -\frac{dw}{dx} \right|_{x=x_1^e} \quad w _2 ^e \quad \left. - \frac{dw}{dx} \right|_{x=x_2^e} \right \rangle ^T$

$(\mathbf{F}_e)_i = \int_{x_1^e}^{x_2^e} q(x) \phi_i^e dx + \sum _{j=1}^4 Q_j \phi_j ^e (x_i ^e)$

Global system of equations:

$(\xi_1 \mathbf{K}^l + \xi_2 \mathbf{K}^{nl} ) \mathbf{U} = \mathbf{F}$

NUMERICAL RESULTS

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.