Solid State Physics

Surface Effects in Solid Mechanics: Models, Simulations and by Holm Altenbach, Nikita F Morozov

By Holm Altenbach, Nikita F Morozov

This booklet summarizes the particular state-of-the-art and destiny tendencies of floor results in strong mechanics. floor results are an increasing number of vital within the certain description of the habit of complex fabrics. one of many purposes for this can be the well known from the experiments proven fact that the mechanical homes are considerably prompted if the structural dimension is especially small like, for instance, nanostructures. during this e-book, numerous authors research the impression of floor results within the elasticity, plasticity, viscoelasticity. additionally, the authors speak about all vital diversified ways to version such results. those are in response to numerous theoretical frameworks comparable to continuum theories or molecular modeling. The publication additionally offers purposes of the modeling approaches.

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Sci. 34(6), 475–489 (1992) A Comparison of Atomistic and Surface Enhanced Continuum Approaches at Finite Temperature Denis Davydov, Ali Javili, Paul Steinmann and Andrew McBride Abstract The surface of a continuum body generally exhibits properties that differ from those of the bulk. Surface effects can play a significant role for nanomaterials, in particular, due to their large value of surface-to-volume ratio. The effect of solid surfaces at the nanoscale is generally investigated using either atomistic or enhanced continuum models based on surface elasticity theory.

2) as ⎞ ⎛ ⎞⎛ ⎞ ⎛ C11 C12 C13 ε11 σ11 ⎝ σ22 ⎠ = ⎝ C12 C22 C23 ⎠ ⎝ ε22 ⎠ (4) σ12 C13 C23 C33 2ε12 36 I. E. Berinskii and F. M. Borodich Graphene lattice has third order axial symmetry. This means that an in-plane rotation by the angle Θ = 2π/3 transfers the lattice into itself. This transformation is determined by transformation matrix A= cos Θ sin Θ − sin Θ cos Θ = √ 3/2 −1/2 √ − 3/2 −1/2 (5) One can see that detA = 1 and A−1 = A T , therefore, AA T = E, where E is a unity matrix. After rotation the stress matrix σ σ11 σ12 σ12 σ22 (6) σ˜ = Aσ A T .

Hence, one has ∂u 1 , ∂ x1 ∂u 2 = , ∂ x2 1 ∂u 2 ∂u 3 = + 2 ∂ x3 ∂ x2 1 ∂u 1 + 2 ∂ x2 1 ∂u 1 = + 2 ∂ x3 ∂u 3 = =0 ∂ x3 ε11 = ε12 = ε22 ε13 ε23 = 0, ε33 ∂u 2 ∂ x1 ∂u 3 ∂ x1 , = 0, (1) The classical Hooke’s law for 2D case may be presented in tensor form as 2 2 σi j = Ci jkl εkl , i, j = 1, 2 (2) k=1 l=1 One can represent this equation in matrix form using elasticity matrix C. In 3D case this matrix generally has a dimension 6×6 but in 2D case we can reduce it to 3×3. Let us construct it taking a symmetry of the elasticity tensor into account.

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