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Linear operators of higher order, or higher-order tensors, are frequently employed in continuum mechanics. In this section we introduce some basic definitions and operations involving higher-order tensors. 53). The alternating tensor is a linear operator that maps vectors into skew symmetric tensors. 81) for arbitrary vectors a, b, c and d. The multiplication of the alternating tensor by a vector v yields the second-order tensor v = ijk vk ei ⊗ ej . 52) can be equivalently written in compact form as w = ( v) u.

121) and the product ‘∗’ is identified with the standard product between scalars. 114), we find that the linearisation of y about x0 in the present case reads l(u) ≡ y(x0 ) + ∇y(x0 )u = x20 + 2x0 u. 4. DERIVATIVES OF FUNCTIONS OF VECTOR AND TENSOR ARGUMENTS Let us now consider some simple illustrative examples of derivatives of functions whose arguments and/or values may be scalar, vectors or tensors. 1. Function derivative and linearisation. Scalar function of vector argument We start with the scalar function of a vector argument defined by y(x) ≡ x · x = xi xi .

Calligraphic majuscules X, Y, . ) • Typewriter style letters HYPLAS, SUVM, . : used exclusively to denote FORTRAN procedures and variable names, instructions, etc. 2. SOME IMPORTANT CHARACTERS The specific meaning of some important characters is listed below. We remark that some of these symbols may occasionally be used with a different connotation (which should be clear from the context). A Generic set of thermodynamical forces A Finite element assembly operator (note the large font) A First elasticity tensor a Spatial elasticity tensor B B Left Cauchy–Green strain tensor e Elastic left Cauchy–Green strain tensor B Discrete (finite element) symmetric gradient operator (strain-displacement matrix) B Generic body b ¯ b Body force C Right Cauchy–Green strain tensor c Cohesion D Damage internal variable Reference body force D Stretching tensor D e Elastic stretching D p Plastic stretching D Infinitesimal consistent tangent operator e Infinitesimal elasticity tensor ep D Infinitesimal elastoplastic consistent tangent operator D Consistent tangent matrix (array representation of D) De Elasticity matrix (array representation of De ) Dep Elastoplastic consistent tangent matrix (array representation of Dep ) E Young’s modulus Ei Eigenprojection of a symmetric tensor associated with the ith eigenvalue E E¯ Three-dimensional Euclidean space; elastic domain ei Generic base vector; unit eigenvector of a symmetric tensor associated with the ith eigenvalue D Set of plastically admissible stresses 10 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS F Deformation gradient F e Elastic deformation gradient F p Plastic deformation gradient f ext Global (finite element) external force vector ext f(e) int External force vector of element e f Global (finite element) internal force vector int f(e) Internal force vector of element e G Virtual work functional; shear modulus G Discrete (finite element) full gradient operator H Hardening modulus H Generalised hardening modulus I1 , I2 , I3 Principal invariants of a tensor I Fourth-order identity tensor: Iijkl = δik δjl IS Fourth-order symmetric identity tensor: Iijkl = 12 (δik δjl + δil δjk ) Id Deviatoric projection tensor: Id ≡ IS − 13 I ⊗ I I Second-order identity tensor IS Array representation of IS i Array representation of I J Jacobian of the deformation map: J ≡ det F J2 , J3 Stress deviator invariants J Generalised viscoplastic hardening constitutive function K Bulk modulus KT Global tangent stiffness matrix (e) KT Tangent stiffness matrix of element e K Set of kinematically admissible displacements L Velocity gradient L e Elastic velocity gradient L p Plastic velocity gradient m α Unit vector normal to the slip plane α of a single crystal N ¯ N Plastic flow vector O The orthogonal group O + ¯ ≡ N/ N Unit plastic flow vector: N The rotation (proper orthogonal) group INTRODUCTION 0 Zero tensor; zero array; zero generic entity o Zero vector P First Piola–Kirchhoff stress tensor p Generic material point p Cauchy or Kirchhoff hydrostatic pressure Q Generic orthogonal or rotation (proper orthogonal) tensor q von Mises (Cauchy or Kirchhoff) effective stress R R Rotation tensor obtained from the polar decomposition of F e Elastic rotation tensor R Real set r Global finite element residual (out-of-balance) force vector s Entropy s Cauchy or Kirchhoff stress tensor deviator α s Unit vector in the slip direction of slip system α of a single crystal t ¯t Surface traction Reference surface traction U Right stretch tensor e Elastic right stretch tensor Up Plastic right stretch tensor U Space of vectors in E u Generic displacement vector field u Global finite element nodal displacement vector U V Left stretch tensor V e Elastic left stretch tensor V p Plastic left stretch tensor V Space of virtual displacements v Generic velocity field W Spin tensor W e Elastic spin tensor W p Plastic spin tensor x Generic point in space αp Ogden hyperelastic constants (p = 1, .