PSI - Issue 2_B

A. Spagnoli et al. / Procedia Structural Integrity 2 (2016) 2667–2673

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A. Spagnoli et al. / Structural Integrity Procedia 00 (2016) 000–000

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where a central dot indicates the scalar product of vectors. Clearly, r in = r i · n i and u in = w i · n i . We intend to treat only problems with a known contact surface (complete contact) and therefore require that w in = 0 , r in ≥ 0 . (8) Coulomb’s law of friction reads: | r it | ≤ µ i r in (9) 0 < | r it | = µ i r in = ⇒ ˙ w it = − λ i r it , λ i ≥ 0 , (10) | r it | < µ i r in = ⇒ ˙ w it = 0 . (11) where µ i is the coe ffi cient of friction and a superposed dot denotes time derivative. This law may be said to have two ingredients. Firstly, there is a condition which states that the contact forces should belong to a set of admissible such forces, the so called Coulomb’s friction cone. Secondly, there is a condition which specifies when and how sliding takes place: sliding is opposite to the friction force. Note that r in ≥ 0 is obviously included in (9). 2.1. Residual state The residual state is defined by an elastic unloading ( F = 0 ), keeping p and w fixed (’welding’). The elastic strain in this state is denoted e R and the total strain is ε R , while the displacement is denoted u R . The residual stresses and contact forces are σ R and r R . The governing equations become: ε R = Bu R , (12) ε R = e R + p , (13) C T r R = D T σ R , (14) σ R = Ee R , (15) w = Cu R . (16) The residual state, defined by (12) to (16), is connected to the ’real’ state by an elastic process achieved by reintro ducing the force. Denoting this elastic response by index E it holds that: e E = Bu E , (17) ε = e R + e E + p , (18) F + C T r E = D T σ E , (19) σ E = Ee E , (20) 0 = Cu E , (21) u = u R + u E . (22) r = r R + r E . (23) Note that e = e R + e E so by (6), (15) and (20) we have

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σ = σ R + σ E .