By Fowles G.R., Cassiday G.L.

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1 Boundary value problem in 1D elasticity: (a) initial conﬁguration (solid line) and deformed conﬁguration (dashed line) producing free-end displacement u(l) under uniform stress ﬁeld σ, (b) 1D model, (c) constitutive law of linear elasticity, (d) constitutive law of nonlinear elasticity. dx from the initial conﬁguration remains approximately the same, since the mapping into deformed conﬁguration is deﬁned according to ⎛ ⎞ 2 ⎜ ⎟ ⎜1 + du + 1 d u dx⎟ dx ≈ dx ⎝ ⎠ 2 dx 2 dx 1 ≈0 Therefore, one can keep the coordinates deﬁned in the initial conﬁguration for any operation where such an inﬁnitesimal segment would enter, such as diﬀerentiation or integration in the deformed conﬁguration.

Dx from the initial conﬁguration remains approximately the same, since the mapping into deformed conﬁguration is deﬁned according to ⎛ ⎞ 2 ⎜ ⎟ ⎜1 + du + 1 d u dx⎟ dx ≈ dx ⎝ ⎠ 2 dx 2 dx 1 ≈0 Therefore, one can keep the coordinates deﬁned in the initial conﬁguration for any operation where such an inﬁnitesimal segment would enter, such as diﬀerentiation or integration in the deformed conﬁguration. Furthermore, in this manner the inﬁnitesimal strain ﬁeld which is really applied in the deformed conﬁguration can be represented in the initial conﬁguration and so can the stress ﬁeld.

This means that two scales are suﬃciently separated to allow for the computational procedure to be split in two separate phases. We then present the strong coupling of the scales, where such a split in not allowed, and where one has to adapt the computational procedure to the constant communication between two scales. The key notion of the coupling between two scales is discussed in detail, showing both the displacement-based coupling (which provides the equivalent result to upper or Voigt bound in homogenization theory) and force-based coupling (equivalent to lower or Reuss bound).