By EA de Souza Neto, Prof. D Periæ, Prof. DRJ Owen

The topic of computational plasticity encapsulates the numerical tools used for the finite aspect simulation of the behaviour of a variety of engineering fabrics thought of to be plastic – i.e. those who suffer an enduring swap of form in accordance with an utilized strength. Computational equipment for Plasticity: conception and functions describes the idea of the linked numerical tools for the simulation of a variety of plastic engineering fabrics; from the best infinitesimal plasticity idea to extra advanced harm mechanics and finite pressure crystal plasticity types. it really is cut up into 3 elements - uncomplicated ideas, small lines and big lines. starting with straight forward conception and progressing to complicated, advanced thought and laptop implementation, it truly is appropriate to be used at either introductory and complex degrees. The book:

- Offers a self-contained textual content that enables the reader to profit computational plasticity idea and its implementation from one volume.
- Includes many numerical examples that illustrate the appliance of the methodologies described.
- Provides introductory fabric on similar disciplines and approaches equivalent to tensor research, continuum mechanics and finite components for non-linear reliable mechanics.
- Is observed via purpose-developed finite point software program that illustrates the various thoughts mentioned within the textual content, downloadable from the book’s significant other website.

This accomplished textual content will attract postgraduate and graduate scholars of civil, mechanical, aerospace and fabrics engineering in addition to utilized arithmetic and classes with computational mechanics parts. it is going to even be of curiosity to analyze engineers, scientists and software program builders operating within the box of computational good mechanics.

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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, .