Contents
Cover
Title Page
Copyright
Dedication
Preface
Nomenclature
Chapter 1: Introduction
1.1 Composite Structures
1.2 Failures of Composites
1.3 Crack Analysis
1.4 Analytical Solutions for Composites
1.5 Numerical Techniques
1.6 Scope of the Book
Chapter 2: Fracture Mechanics, A Review
2.1 Introduction
2.2 Basics of Elasticity
2.3 Basics of LEFM
2.4 Stress Intensity Factor, K
2.5 Classical Solution Procedures for K and G
2.6 Quarter Point Singular Elements
2.7 J Integral
2.8 Elastoplastic Fracture Mechanics (EPFM)
Chapter 3: Extended Finite Element Method
3.1 Introduction
3.2 Historic Development of XFEM
3.3 Enriched Approximations
3.4 XFEM Formulation
3.5 XFEM Strong Discontinuity Enrichments
3.6 XFEM Weak Discontinuity Enrichments
3.7 XFEM Crack-Tip Enrichments
3.8 Transition from Standard to Enriched Approximation
3.9 Tracking Moving Boundaries
3.10 Numerical Simulations
Chapter 4: Static Fracture Analysis of Composites
4.1 Introduction
4.2 Anisotropic Elasticity
4.3 Analytical Solutions for Near Crack Tip
4.4 Orthotropic Mixed Mode Fracture
4.5 Anisotropic XFEM
4.6 Numerical Simulations
Chapter 5: Dynamic Fracture Analysis of Composites
5.1 Introduction
5.2 Analytical Solutions for Near Crack Tips in Dynamic States
5.3 Dynamic Stress Intensity Factors
5.4 Dynamic XFEM
5.5 Numerical Simulations
Chapter 6: Fracture Analysis of Functionally Graded Materials (FGMs)
6.1 Introduction
6.2 Analytical Solution for Near a Crack Tip
6.3 Stress Intensity Factor
6.4 Crack Propagation in FGM Composites
6.5 Inhomogeneous XFEM
6.6 Numerical Examples
Chapter 7: Delamination/Interlaminar Crack Analysis
7.1 Introduction
7.2 Fracture Mechanics for Bimaterial Interface Cracks
7.3 Stress Intensity Factors for Interlaminar Cracks
7.4 Delamination Propagation
7.5 Bimaterial XFEM
7.6 Numerical Examples
Chapter 8: New Orthotropic Frontiers
8.1 Introduction
8.2 Orthotropic XIGA
8.3 Orthotropic Dislocation Dynamics
8.4 Other Anisotropic Applications
References
Index
This edition first published 2012
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Library of Congress Cataloging-in-Publication Data
Mohammadi, S. (Soheil)
XFEM fracture analysis of composites / Soheil Mohammadi.
pages cm
Includes bibliographical references and index.
ISBN 978-1-119-97406-2
1. Composite materials–Fracture. 2. Composite materials–Fatigue. 3. Fracture mechanics. 4. Finite element method. I. Title.
TA418.9.C6M64 2012
620.1′186–dc23
2012016776
A catalogue record for this book is available from the British Library.
ISBN: 978-1-119-97406-2
To: Mansoureh, Sogol & Soroush
Preface
A decade after its introduction, the extended finite element method (XFEM) has now become the primary numerical approach for analysis of a wide range of discontinuity applications, including crack propagation problems. The simplicity of the idea of enrichment for reproducing a singular/discontinuous nature of the field variable, the flexibility in handling several cracks and crack propagation patterns on a fixed mesh, and the level of accuracy with minimum additional degrees of freedom (DOFs) have transformed XFEM into the most efficient computational approach for handling various complex discontinuous problems. Concepts of XFEM are now even taught in a number of postgraduate courses, for instance advanced fracture mechanics and meshless methods, in major engineering departments, such as Civil, Mechanical, Material, Aerospace and so on, all over the world.
On the other hand, the highly flexible design of composites allows attractive prescribed tailoring of material properties, fitted to the engineering requirements for a wide range of engineering and industrial applications; from advanced aerospace and defence systems to traditional structural strengthening techniques, and from large scale turbines and power plants to nanoscale carbon nanotubes (CNTs) applications. Despite excellent characteristics, composites suffer from a number of shortcomings, mainly in the form of unstable cracking which can be initiated and propagated under different production imperfections and service circumstances. Therefore, the study of the crack stability and load bearing capacity of these types of structures, which directly affect the safety and economics of many important industries, has become one of the important topics of research for the computational mechanics community.
This text is dedicated to discussing various aspects of the application of the extended finite element method for fracture analysis of composites on the macroscopic scale. Nevertheless, most of the discussed subjects can be similarly used for fracture analysis of other materials, even on microscopic scales. The book is designed as a textbook, which provides all the necessary theoretical bases before discussing the numerical issues.
The book can be classified into four parts. The first part is dedicated to the basics. The introduction chapter provides a general overview of the problem in hand and summarily reviews available analytical and numerical techniques for fracture analysis of composites. The second chapter deals with the basics of the theory of elasticity, and is followed by discussions on asymptotic solutions for displacement and stress fields in different fracture modes, basic concepts of stress intensity factors, energy release rate, various forms of the J contour integral and mixed mode fracture criteria.
The second part, Chapter , is a redesigned and completed edition of the same chapter in my previous book, and presents a detailed discussion on the extended finite element method. After presenting the basic formulation, the chapter continues with three sections on available options for strong discontinuity enrichment functions, weak discontinuity enrichments for material interfaces, and a collection of several crack-tip enrichments for various engineering applications. It concludes with sample simulations of a wide range of problems, including classical in-plane mixed mode fracture mechanics, cracking in plates and shells, simulation of shear band creation and propagation, self-similar fault rupture, sliding contact, hydraulic fracture, and dislocation dynamics, to assess the accuracy, performance, robustness and efficiency of XFEM.
The main part of the book includes four comprehensive chapters dealing with various aspects of fracture in composite structures. Static crack analysis in orthotropic materials, dynamic fracture mechanics for stationary and moving cracks, inhomogneous functionally graded materials and bimaterial delamination analysis are discussed in detail. After a review of anisotropic and orthotropic elasticity, all chapters begin with a complete discussion of available analytical solutions for near crack-tip fields in the corresponding orthotropic problem, followed by orthotropic mixed mode fracture mechanics and associated forms of the interaction integral. XFEM enrichment functions for each class of orthotropic materials are obtained and numerical issues related to XFEM discretization are addressed. A number of illustrative numerical simulations are presented and discussed at the end of each chapter to assess the performance of XFEM compared to alternative analytical and numerical techniques.
The final part reviews a number of ongoing research topics for orthotropic materials. First, the orthotropic version of the extended isogeometric analysis (XIGA) is presented by briefly explaining the basic concepts of NURBS and IGA methodology and discussing a number of simple isotropic and orthotropic simulations. Then, the newly developed idea of plane strain anisotropic dislocation dynamics is briefly presented and related XFEM formulation and necessary enrichment functions for the self-stress state of edge dislocations are explained. The book concludes with two brief introductory sections on orthotropic biomaterial applications of XFEM and the piezoelectric problems.
I would like to express my sincere gratitude to Prof. T. Belytschko, for his valuable friendly comments and encouraging message after the publication of the first book on XFEM, and to Prof. A.R. Khoei, as a friend, a colleague and a referee with excellent comments and discussions on various subjects of computational mechanics, especially XFEM. In preparing the present book, particularly the first two parts, I have used the available contributions from brilliant research works by many others, and I have done my best to properly and explicitly acknowledge their achievements within the text, relevant figures, tables and formulae. I am much indebted to their outstanding works, and I apologize sincerely for any unintentional failure in appropriately acknowledging them.
My special thanks to many of my former and present M.Sc. and Ph.D. students who have endeavored many aspects of XFEM over the past decade. First, Dr A. Asadpoure, with whom we started to explore XFEM and new orthotropic enrichment functions in 2002. The results for the dynamic fracture of stationary and moving cracks were obtained by Dr D. Motamedi, and Ms S. Esna Ashari developed the orthotropic bimaterial enrichment functions and simulated all the delamination and interlaminar crack problems. Most of the results for FGM problems were prepared by Mr H. Bayesteh, who actively contributed in many other parts of the book. Mr S.S. Ghorashi and Mr N. Valizadeh skillfully developed the XIGA methodology and Mr S. Malekafzali implemented XFEM for anisotropic dislocation dynamics. Other results were obtained by my students: S.H. Ebrahimi, A. Daneshyar, M. Parchei, M. Goodarzi, S.N. Rezaei, S.N. Mahmoudi and M.M.R. Kabiri.
My acknowledgement is extended to John Wiley & Sons, Ltd., Publication for the excellent professional work that facilitated the whole process of publication of the book; in particular to Dr E.F. Kirkwood and E. Willner, D. Cox, R. Davies, L. Wingett, A. Hunt, C. Lim, S. Sharma and Dr R. Whitelock.
The inspiration and power for completing this work have been the love and understanding of my family, as they had to comply with all my commitments. After a life-time engagement in mathematics, physics and engineering, satisfaction is not obtained just in academic or professional progress, novelty and innovation; it should perhaps be sought in ethics, responsibility, love and freedom. This book has been completed at the twilight of a long hard winter, with a hope for a bright flourishing spring of prosperity and freedom to come. I would like to proudly dedicate this work to all spirited noble Iranian students who accomplish academic achievements while challenging for more DOFs!
Soheil Mohammadi
Spring 2012
Tehran, IRAN
Nomenclature
Parameters not shown in this nomenclature are temporary variables or known constants, defined immediately when cited in the text.
α | Curvilinear coordinate |
α | First Dundurs parameter |
α, β | Newmark parameters |
α, β, γ | FGM constants |
![]() |
Curvilinear coordinate α of an ellipse |
![]() |
Components of coordinate transformation tensor |
β | Curvilinear coordinate |
β | Second Dundurs parameter |
![]() |
Curvilinear coordinate β of an ellipse |
![]() ![]() |
Dilatational and shear wave functions |
γ | Wedge angle |
![]() |
Surface energy density |
![]() ![]() |
Dilatational and shear wave functions |
![]() |
Engineering shear strain |
δ | Plastic crack tip zone |
δ | Variation of a function |
![]() |
Dirac delta function |
![]() |
Kronecker delta function |
![]() ![]() |
Local displacements of crack edges |
![]() |
Strain tensor |
ε | Oscillation index |
![]() ![]() |
Strain components |
![]() |
Dimensionless angular geometric function |
![]() |
Auxiliary strain components |
![]() |
Applied displacement loading |
![]() |
Yield strain |
ξ | Local curvilinear (mapping) coordinate system |
![]() |
Knot i |
![]() |
Crack-tip position |
![]() |
Distance function |
![]() |
Gauss point position along the contour J |
η | Local curvilinear (mapping) coordinate system |
η | Equivalent inelastic strain |
θ | Crack propagation angle with respect to initial crack |
θ | Angular polar coordinate |
![]() |
Crack angle |
![]() ![]() |
Orthotropic angular functions |
![]() ![]() |
Dynamic distance functions |
κ, ![]() |
Material parameters |
![]() |
Effective material parameter |
λ | Lame modulus |
λ | Power of radial enrichment |
λ | Ratio of orthotropic Young modules E2/E1 |
λ, ![]() |
Roots of the characteristic equation |
μ, ![]() |
Isotropic and orthotropic shear modulus |
ν, ![]() |
Isotropic and orthotropic Poisson's ratios |
![]() |
Average orthotropic Poisson's ratios |
ρ | Radius of curvature |
ρ | Density |
![]() |
Stress tensor |
![]() |
Applied normal traction |
![]() |
Critical stress for cracking |
![]() |
von Mises effective stress |
![]() ![]() |
Stress components |
![]() |
Dimensionless angular geometric function |
![]() |
Auxiliary stress components |
![]() |
Yield stress |
![]() |
Hoop stress |
![]() |
Applied tangential traction |
![]() |
Decohesive shear stress |
![]() |
Level set function |
![]() |
Complex stress function |
![]() |
Angle of orthotropic axes |
![]() |
Crack angle |
![]() |
Ramp function for transition domain |
![]() |
Electric potential |
![]() |
Enrichment function for weak discontinuities |
![]() |
Complex stress function |
ψ | Friction coefficient |
ψ | Phase angle |
![]() |
Enrichment function |
![]() |
Level set function |
![]() |
Complex stress function |
![]() |
Boundary |
![]() |
Infinitesimally small internal contour |
![]() |
Crack boundary |
![]() |
Traction (natural) boundary |
![]() |
Displacement (essential) boundary |
![]() |
Finite variation of a function |
![]() |
Time-step |
![]() |
Crack length increment |
![]() |
Time interval shape functions |
![]() |
Knot vector |
Π | Potential energy |
Φ | Airy stress function |
![]() |
MLS shape functions |
![]() |
Complex functions |
Ω | Domain |
![]() ![]() |
Non-overlapping subdomains |
![]() |
Domain associated with the partition of unity |
![]() |
Dislocation glide enrichment |
(1, 2) | Material axes |
a | Crack length/half length |
a | Semi-major axis of ellipse |
![]() |
Effective crack length |
a(x) | Vector of unknown coefficients |
a, ah | Heaviside enrichment degrees of freedom |
ai, ak | Enrichment degrees of freedom |
aenr | Enrichment degrees of freedom |
a | Area associated with the domain J integral |
A1 | Area inside the infinitesimally small internal contour ![]() |
A+, A− | Area of the influence domain above and below the crack |
Ai, Aij | Coefficients |
b | Width of a plate |
b | Semi-minor axis of ellipse |
bk, blk | Crack tip enrichment degrees of freedom |
![]() |
Burgers vector for dislocation α |
![]() |
Magnitude of the Burgers vector for dislocation α |
bn | Series coefficients |
B | Matrix of derivatives of shape functions |
B12, B66 | Coefficients of characteristic equation |
![]() |
B-spline basis function of order p |
Bh | Matrix of derivatives of final shape functions |
Bri | Strain-displacement matrix (derivatives of shape functions) |
Bui | Matrix of derivatives of classical FE shape functions |
Bai | Matrix of derivatives of Heaviside enrichment shape functions |
Bbi | Matrix of derivatives of crack tip enrichment shape functions |
Bci | Matrix of derivatives of weak discontinuity enrichment shape functions |
Bci | Matrix of derivatives of transition shape functions |
c | Dugdale effective crack length |
cJ | Size of crack tip contour for J integral |
cij | Material compliance constants |
cm | Degrees of freedom for weak discontinuity enrichment |
cm | Degrees of freedom for transitional enrichment |
cd | Dilatational wave speed |
cL | Wave speed along the loading axis |
cR | Rayleigh speed |
cs | Shear wave speed |
c | Material constitutive matrix |
![]() |
4th order material compliance tensor |
Cijkl | Cartesian components of ![]() |
Cn | Coefficient |
![]() |
NURBS curve |
dij | Material modulus constants |
ddij | Dynamic material modulus constants |
dm | Degrees of freedom for transitional enrichment |
D | Dynamic function |
D | Two dimensional Material modulus matrix |
![]() |
4th order material elasticity modulus tensor |
![]() |
Cartesian components of ![]() |
![]() |
Components of D |
Di, Dx, Dy | Elastic displacement vector |
E, Ei | Isotropic and orthotropic Young's modules |
Ei, Ex, Ey | Electric field |
![]() |
Effective material parameter |
E0 | Reference Young modulus |
E12 | Equivalent bimaterial elastic modulus |
![]() |
Average orthotropic Young modules |
fk(x) | Set of PU functions |
f | Nodal force vector |
fri | Nodal force components (classic and enriched) |
fb | Body force vector |
ft | External traction vector |
fc | Cohesive crack traction vector |
fext | External force vector |
fI, fII | Functions of the crack-tip speed |
fdI, fdII | Universal functions of the crack-tip speed |
fIij, fIIij, fIIIij | Mode I, II and II angular functions |
fpuk | Set of PU functions |
Fl(x), ![]() |
Crack tip enrichment functions |
![]() |
Delamination function |
![]() |
Deformation gradient |
![]() |
Angular function for a crack-tip kink problem |
![]() ![]() |
Orthotropic crack-tip enrichment functions |
G | Shear modulus |
G, ![]() |
Fracture energy release rate |
Gc | Critical fracture energy release rate |
G1, G2 | Mode I and II fracture energy release rates |
h | Intrinsic shear band thickness |
ht | Characteristic thickness of the bonding layer |
h | Slope of linear softening curve |
![]() |
Intrinsic hardening coefficient |
![]() |
Heaviside function |
![]() |
Complex number ![]() |
I | 2nd order identity tensor |
![]() |
4th order symmetric identity tensor |
I(t) | Corresponding creep compliance |
J | Jacobian matrix |
J, Js | J integral |
Jact | Actual J integral |
Jaux | Auxiliary J integral |
J1, J2 | Components of the J vector |
Jdk | Dynamic J integral |
k0 | Dimensionless constant for the power hardening law |
K | Bulk modulus |
K | Stiffness matrix |
Krsij | Components of stiffness matrix |
K | Stress intensity factor |
K, ![]() |
Complex stress intensity factor |
Kc | Critical stress intensity factor |
K0 | Reference stress intensity factor |
KI, KII, KIII | Mode I, II and III stress intensity factors |
![]() ![]() |
Normalized mode I and mode II stress intensity factors |
KauxI, KauxII | Auxiliary mode I and mode II stress intensity factors |
KIc, KIIc | Critical mode I and mode II stress intensity factors |
K1Ic, K2Ic | Fracture toughnesses along the principal planes of elastic symmetry |
![]() |
Fracture toughness at propagation |
KdIc | Dynamic crack initiation toughness |
KdIc | Dynamic crack growth (propagation) toughness |
![]() |
Hoop stress intensity factor |
lij | Coefficient |
L | Length of the singular element |
L(vc) | Dynamic matrix for orthotropic materials |
m | Number of enrichment functions |
mt | Number of nodes to be enriched by crack-tip enrichment functions |
mh | Number of nodes to be enriched by Heaviside enrichment functions |
mf | Number of crack-tip enrichment functions |
mm | Number of weak discontinuity enrichment functions |
mst | Number of transition enrichment functions 1 |
m sh | Number of transition enrichment functions 2 |
mk | Roots of characteristic equation mk=mkx+imky |
m | Concentrated bending moment |
m | Interaction integral |
M(1), M(2) | Interaction integral associated with two modes I and II |
Md | Dynamic interaction integral |
M | Mass matrix |
Mij | Components of mass matrix |
n | Power number for the HKK plastic model |
n | Number of nodes for each finite element |
ngA | Number of gauss points inside contour area a |
![]() |
Number of gauss points on contour ![]() |
np | Number of independent domains of partition of unity |
nn, nnodes | Number of nodes in a finite element |
ne, nelem. | Number of finite elements |
ncp | Number of control points |
ncells | Number of background cells of EFG |
nDOFs | Number of degrees of freedom |
n | Normal vector |
Nj | Matrix of shape functions |
Nj | Shape function |
Nelements | Number of finite elements |
nnodes | Number of nodes |
Nenrich. | Number of enrichment functions |
nDOFs | Number of degrees of freedom |
![]() |
Hierarchical shape functions for the transition domain |
![]() |
New set of GFEM shape functions |
p(x) | Basis function |
p | A point on curvilinear coordinate system ![]() |
p, pk, ![]() |
Orthotropic parameters |
p | Basis function |
penr | Enrichment basis function |
plin | Linear basis function |
pk | k-th basis function |
pl(x) | l-order polynomial function |
p | Concentrated force |
p | External load vector |
Pcr | Critical load |
q | Arbitrary smoothing function |
q, qk, ![]() |
Orthotropic parameters |
qi | Nodal values of the arbitrary smoothing function |
![]() |
Local crack tip polar coordinates |
rJ | Radius of J integral contour |
rd, rs | Dilatational and shear distance functions |
rl, rs | Orthotropic distance functions |
rp, rp1, rp2 | Crack tip plastic zone |
R | Ramp function |
RK | Ratio of dynamic stress intensity factors |
![]() |
Ratio of opening to sliding displacements |
![]() |
NURBS function of order p |
s | Roots of characteristic equation s=s1+is2 |
sk | Roots of characteristic equation sk=skx+isky |
![]() |
Roots of characteristic equation ![]() |
sm | Roots of characteristic equation sm=sm1+ism2 |
![]() |
Slope of softening curve |
Sij | Material constants |
![]() |
NURBS surface |
t | Time |
t | Traction |
th | Unit vector for tangential direction |
tij | Material function |
t0 | Time for the wave to reach the crack tip |
tp | FRP thickness |
Ti(t) | Time shape functions |
Tj | Enriched time interval |
Tj | Transformation matrix |
Ti | Control points |
u | Displacement vector |
![]() |
Velocity vector |
![]() |
Prescribed displacement |
![]() |
Prescribed velocity |
![]() |
Acceleration vector |
ui | Displacement field component |
![]() |
Velocity field component |
![]() |
Acceleration field component |
uauxi | Auxiliary displacement field component |
![]() |
Auxiliary velocity field component |
uenr | Enriched displacement field |
uFE | Classical finite element displacement field |
uXFEM | XFEM displacement field |
utra | Transition enrichment part of the displacement field |
uH | Heaviside enrichment part of the displacement field |
utip | Crack-tip enrichment part of the displacement field |
umat | Weak discontinuity enrichment part of the displacement field |
uEnrtip(x, t) | Crack-tip part of the approximation |
uhn | Displacement at time n |
![]() |
Velocity at time n |
![]() |
Acceleration at time n |
uh, uh(x) | Approximated displacement field |
![]() |
Approximated velocity field |
![]() |
Approximated acceleration field |
![]() |
Nodal displacement vector |
![]() |
Displacement angular functions |
ux, uy, uz | x, y and z displacement components |
![]() ![]() |
Local displacements of the nodes along the crack in the singular element |
UI, UII | Symmetric and antisymmetric crack tip displacements |
Us | Strain energy |
Ues, Ups | Elastic and plastic strain energies |
![]() |
Surface energy |
vc | Crack-tip velocity |
v | Velocity vector |
vc | Classical velocity DOFs |
ve | Additional velocity DOFs |
V(t) | Vector of approximated velocity degrees of freedom |
V | Volume |
W | External work |
waux | Auxiliary work |
Wext | Virtual work of the external loading |
Wg | Gauss weight factor |
Wi | Weights associated with each control point i |
![]() |
Gauss weighting factor for contour ![]() |
WAg | Gauss weighting factor for area inside contour integral J |
Wint | Internal virtual work |
wM | Interaction work |
wd | Kinetic energy density |
ws | Strain energy density |
Wt(t) | Time weight function |
x, y, z | Cartesian coordinates |
(X, Y) | Global coordinate system |
(X1, X2) | Global coordinate system |
x | Position vector |
xc | Position of crack or discontinuity |
xtip | Position of the crack tip |
![]() |
Position of the projection point on an interface |
(X1, X2) | 2D coordinate system |
(X1, X2) | Material axes |
![]() |
Local crack tip coordinate axes |
z | Complex variable z=x+iy |
![]() |
Conjugate complex variable ![]() |
zi | Complex parameters |
![]() |
Time derivative |
![]() |
Material time derivative |
![]() ![]() |
The first and second temporal derivatives of a function |
![]() ![]() |
The first and second spatial derivatives of a function |
![]() ![]() |
The first and second integrals of a function |
![]() |
Nabla operator |
![]() |
Jump operator across an interface |
: | Inner product of two second order tensors |
![]() |
Tensor product of two vectors |
![]() |
Real part of a complex number |
![]() |
Imaginary part of a complex number |
![]() |
Proportional |
BEM | Boundary element method |
CAD | Computer aided design |
CNT | Composite nanotube |
COD | Crack opening displacement |
CTOD | Crack-tip opening displacement |
DCT | Displacement correlation technique |
DEM | Discrete element method |
DOF | Degree of freedom |
EDI | Equivalent domain integral |
EFG | Element free Galerkin |
ELM | Equilibrium on line |
EPFM | Elastic plastic fracture mechanics |
FDM | Finite difference method |
FE | Finite element |
FEM | Finite element method |
FGM | Functionally graded materials |
FMM | Fast marching method |
FPM | Finite point method |
FRP | Fibre reinforced polymer |
GFEM | Generalized finite element method |
GNpj | Generalized Newmark approximation of degree p for equations of order j |
HRR | Hutchinson-Rice-Rosengren |
IGA | Isogeometric analysis |
LEFM | Linear elastic fracture mechanics |
LSM | Level set method |
MCC | Modified crack closure |
MLPG | Meshless local Petrov Galerkin |
MLS | Moving least squares |
NURBS | Non-uniform rational B-splines |
OUM | Ordered upwind method |
PU | Partition of unity |
PUFEM | Partition of unity finite element Method |
RKPM | Reproducing kernel particle method |
SAR | Statically admissible stress recovery |
SIF | Stress intensity factor |
SPH | Smoothed particle hydrodynamics |
SSpj | Single step approximation of degree p for equations of order j |
STXFEM | Space time extended finite element method |
TXFEM | Time extended finite element method |
WLS | Weighted least squares |
XFEM, X-FEM | Extended finite element method |
XIGA | Extended isogeometric analysis |