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:

        \[ l_i = 6 \times l_c \rightarrow e^{-  \frac{l_i}{l_c} }  \approx 2.5 \times 10^{-3} \]

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

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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.