1 MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS Version 2011-01-14 Stress tensor Definition of traction vector (1) Cauchy theorem (2) Equilibrium (3) Invariants (4) (5) (6) or, written in terms of principal stresses, (7) (8) (9) Coordinate transformation (10) or, inverted (11) 2 Stress deviator (12) Stress deviator invariants (13) Displacement and strain Definition of displacement (14) Definition of infinitesimal strain (15) ‘Engineering’ shear ; (16) Voigt notation of stress and strain Elastic anisotropy A material is symmetric with respect to the transformation → if (17) or, using Voigt notation (18) 3 where L is the transformation matrix for the transformation → in Voigt notation: (19) Stiffness (C) and ‘engineering’ compliance (S) matrices for some important classes of materials Linearly orthotropic material (20) and (21) In Eq. (21), the actual number of independent material constants is reduced to 9 by the relations (22) Transversely isotropic material (symmetry axis ) (23) and 4 (24) In Eq. (23), the actual number of independent material constants is reduced to 5 by the relation (25) Plasticity. Yield criteria Flow function and equivalent stress: von Mises (26) Flow function and equivalent stress: Tresca (27) Plasticity. Flow rules General (28) (29) (30) Perfect plasticity (31) 5 Perfect plasticity, von Mises (32) where the equivalent plastic strain increment d is defined as (33) Isotropic hardening, von Mises (34) or, in most cases, (35) where (36) (cf Eq. (31)). Flow rule: (37) or (38) Linear isotropic hardening, von Mises (39) Eqs. (37) and (38) can now be simplified into (40) and (41) (since during plastic flow). 6 In the uniaxial tensile test, during plastic flow Kinematic hardening, von Mises The hardening is described by a backstress (42) The flow rule is (43) Kinematic hardening, Prager/von Mises Prager’s linear hypothesis: (44) This leads to a simplified expression for the flow rule: (45) In the uniaxial tensile test, during plastic flow (Note the factor 3/2 in the denominator, which is a difference against the corresponding isotropic uniaxial test!) Plasticity. Computational aspects Continuum tangent stiffness matrix (46) where is the continuum tangent matrix. has the following principal structure: (47) One common way of writing it in detail is 7 (48) Viscoplasticity Additive decomposition: (49) Norton uniaxial creep law for stationary creep (50) Multiaxial creep laws (51) with Stationary creep (von Mises/Norton/Odqvist) (52) Multiplicative isotropic hardening (53) Perzyna overstress model (54) Viscoelasticity Maxwell material (55) (56) 8 (57) Kelvin material (58) (59) (60) Standard linear solid (61) (62) (63) Relaxation modulus The Laplace transform of the relaxation modulus can be computed from the Laplace transform of the creep compliance as (64) Hereditary integrals For a given stress history , the strain response can be computed as the hereditary integral (65) or, in the Laplace transform space, (66) For a given strain history , the stress response can be computed as the hereditary integral 9 (67) or, in the Laplace transform space, (68) Multiaxial hereditary integral (69) or, split into a deviator ( and a bulk ( part (70) (71) In analogy with the previous uniaxial hereditary integrals, these can also be written as Laplace- transformed equations [cf Eq. (68)]: (72) (73) (74) th (be careful with the notations here: is the full 4 order relaxation modulus tensor, while is the shear modulus; is the stress deviator, while is the Laplace space variable) These equations can be used together with ‘the viscoelastic correspondence principle’ for solving multiaxial problems. Damage Isotropic damage postulate (75) which replaces in the constitive laws. For instance, in linear elasticity: (76) 10 or, if (77) (78) Elastic damage: evolution law (79) ( is the maximum experienced value of the largest principal strain during the elastic history, is a fracture strain, and is a threshold strain.) Plastic damage: evolution lkaw (80) where is the critical damage Creep damage: Kachanov damage evolution law: (81) in which (82) where is the largest principal stress and is von Mises equivalent stress. .