Nonlinear Problems in Machine Design

Nonlinear Problems in Machine Design
اسم المؤلف
Eliahu Zahavi David Barlam
التاريخ
1 يناير 2020
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Nonlinear Problems in Machine Design
Eliahu Zahavi
David Barlam
Contents
Part I: Theoretical Fundamentals
Chapter 1
Basics of Solid Mechanics .1
1.1 Stress 1
1.2 Linear Strain .17
1.3 Stress–Strain Relationship .24
1.4 Variational Principles .34
1.5 Solution of the Boundary Value Problem 38
Chapter 2
Finite Element Method .49
2.1 Introduction to Finite Element Theory 49
2.2 Isoparametric Elements 60
2.3 Hierarchical Functions .70
2.4 Bending Elements: Beams and Plates .76
2.5 Accuracy of FE Solution .82
Chapter 3
Nonlinear Problems .89
3.1 Introduction 89
3.2 Example: Two-Spar Frame 89
3.3 Iterative Methods .94
Chapter 4
Plasticity 107
4.1 One-Dimensional Theory .107
4.2 Yield Criteria for Multi-axial Stresses 117
4.3 Constitutive Theories of Plasticity .122
4.4 Finite Element Implementation .137
Chapter 5
Large Displacements .141
5.1 Tensor Analysis of a Deformed Body .141
5.2 Deformation and Strain 148
5.3 Stress 167
5.4 Constitutive Equations .1725.5 Finite Element Implementation .180
Chapter 6
Contact Problems .185
6.1 Introduction 185
6.2 Penalty Method 190
6.3 Lagrange Multiplier Method 202
6.4 Critical Review .209
Chapter 7
Fatigue Failure Prediction Methods 211
7.1 Strain Method .211
7.2 Cumulative Damage .227
7.3 Fracture Mechanics 230
Part II: Design Cases
Chapter 8
Design of Machine Parts .247
8.1 Nonlinear Behavior of Machine Parts .247
8.2 Failure of Machine Parts under Static Load .249
8.3 Fatigue of Machine Parts under Fluctuating Load 253
Chapter 9
Leaf Springs .265
9.1 Introduction 265
9.2 Design Fundamentals .266
9.3 FE Analysis of Leaf Springs 273
9.4 Conclusions 285
Chapter 10
Threaded Fasteners .289
10.1 Introduction 289
10.2 Forces in Bolt Connection .292
10.3 Stresses .297
10.4 Nonlinear Analysis Using FE Method 302
10.5 Conclusions 330
Chapter 11
Flange Connection .333
11.1 Introduction 333
11.2 One-Dimensional Analysis 334
11.3 FE analysis .33911.4 Conclusions 347
Chapter 12
Fretting Fatigue in an Axle .351
12.1 Introduction 351
12.2 Case Study: Axle Failure Due to Fretting .352
12.3 Design Improvement 361
12.4 Conclusions 364
Appendix A 367
Appendix B 387
Appendix C 391
Index 401
Nomenclature
a, b, c vectors (Appendix A)
a half length of crack (Chapter 7)
a current slave node location (Chapter 6)
a
f crack length at failure (Chapter 7)
a
i factor of number of cycles (Chapter 7)
a
i initial crack length (Chapter 7)
a
i, bi,ci coefficients of element shape functions in constant-strain triangle
(Chapter 2)
a
k unknown parameters in Galerkin and Rayleigh-Ritz approximation
(Chapter 2)
a
mn, bmn, cmn components of matrices (Appendix B)
a
s, bs, , components of vectors (Appendix A)
a
0 previous slave node location (Chapter 6)
a
1, a2, a3 Cartesian coordinates in initial configuration (Chapter 5)
a1, a2, a3 Cartesian coordinates in initial configuration (Chapter 5)
b width of plate (Chapter 7)
b
0, b1, b2 coefficients in stress-strain correlation equation (Chapter 5)
c hardening modulus (Chapter 4)
c fatigue ductility exponent (Chapter 7)
d
i feasible direction (Chapter 3)
d
s principal values of a deviator (Appendix A)
determinant of matrix or tensor (Appendix A)
e vector of principal direction (Appendix A)
e engineering strain (Chapter 4)
e
h pointwise error for displacements (Chapter 2)
e
k components of principal direction vector (Chapter 1)
e
mnp permutation symbol (Appendix A)
e
x, ey, ez normal components of strain deviator (Chapter 1)
e
ε pointwise error for strain (Chapter 2)
e
σ pointwise error for stress (Chapter 2)
e
1, e2, e3 principal values of strain deviator (Chapter 1)
ek principal direction vectors (Appendix A)
components of principal direction vectors (Appendix A)
f vector of contact forces (Chapter 6)
f, , function and its derivatives (Chapter 3)
f function of yield criterion (Chapter 4)
f Paris crack propagation function (Chapter 7)
a
s′ bs′
det …
ek
i
f′ f″fI compliance function (Chapter 7)
fk(x) coordinate functions (Chapter 1)
fij function of stress distribution in the vicinity of crack (Chapter 7)
fn normal forces in contact zone (Chapter 6)
trial tangential force (Chapter 6)
corrected tangential force (Chapter 6)
∆ft correction of tangential force (Chapter 6)
g vector of penetration (Chapter 6)
gk covariant base vectors (Appendix A)
gk base vectors in initial configuration (Chapter 4)
gn distance between the bodies in contact (Chapter 6)
gsk components of metric tensor in initial configuration
gk contravariant (reciprocal) base vectors (Appendix A)
gk reciprocal base vectors in initial configuration (Chapter 5)
gsk contravariant components of metric tensor in initial configuration
(Chapter 5)
h height of cylinder in current configuration (Chapter 5)
h
0 height of cylinder in initial configuration (Chapter 5)
h
0 initial vertical position of a moving end of bar (Chapter 3)
i
k, unit direction vectors of Cartesian coordinate system (Appendix A)
k critical value of material (Chapter 4)
k spring rate (Chapter 9)
l length of bar (Chapters 3, 5)
l final length of specimen (Chapter 4)
l length of contact element (Chapter 6)
l half length of a leaf (Chapter 9)
l
c length of a bar in the current state (Chapter 3)
l
0 length of bar in initial configuration (Chapters 3, 5)
l
0 original length of specimen (Chapter 4)
l
1, l3 length of rod (Chapter 4)
m number of leaves (Chapter 9)
n unit normal vector (Chapter 1)
n hardening exponent (Chapter 4)
n´ cyclic strain hardening exponent (Chapter 7)
n
i number of cycles (Chapter 7)
p pressure (Chapter 11)
pn vector of traction (Chapter 1)
px, py, pz components of tractions (Chapter 1)
∆p pressure increment (Chapter 11)
q lateral distributed load (Chapter 2)
specific loads (Chapter 1)
r position vector in undeformed (initial) configuration (Chapter 5)
f t trial
f t n + 1
ik′
qkr radius of yield surface projection on the deviatoric plane
(Chapter 4)
r radius of cylinder in current configuration (Chapter 5)
r radius of a leaf (Chapter 9)
r residual (Chapter 1)
rk
covariant components of vector r (Appendix A)
rn
radius of yield surface at time step tn (Chapter 4)
r0
radius of cylinder in initial configuration (Chapter 5)
rk contravariant components of vector r (Appendix A)
s deviatoric stress tensor (Chapter 4)
sk nodal stresses (Chapter 2)
s
n deviatoric stress at time step tn (Chapter 4)
trial value of deviatoric stress (Chapter 4)
s
x, sy, sz normal components of stress deviator (Chapter 1)
s1, s2, s3 principal deviatoric stress components
s
12, s23 deflections of leaves (Chapter 9)
t unit tangential vector (Chapter 6)
t traction (Chapter 5)
t time (Chapter 4)
t, ti leaf thickness (Chapter 9)
tn
time step (Chapter 4)
t
(s) vector formed by components of tensor (Appendix A)
t
–1, t–2, t–3 tractions on the faces of elemental tetrahedron (Chapter 5)
u displacement vector (Chapter 1)
u exact solution (Chapter 2)
u, v, w components of displacement vector (Chapter 1)
u
e element nodal displacements (Chapter 2)
u
h finite element solution (Chapter 2)
u
i displacements of the nodes in contact zone (Chapter 6)
u
i displacement vector at iteration i (Chapter 3)
u
n vector of nodal displacements (Chapter 2)
u
n normal displacements in contact zone (Chapter 6)
u
s, s = 1,2,3 components of displacement vector (Chapter 5)
u
t tangential displacements in contact zone (Chapter 6)
u1, u2 displacements of master nodes (Chapter 6)
utot vector of total displacements (Chapter 6)
∆ u
i correction vector in Newton–Raphson procedure (Chapter 3)
∆u, ∆ui increment of displacement (Chapter 3)
v virtual displacements field (Chapter 2)
v displacement vector (Chapter 5)
v virtual displacement field (Chapter 2)
v Coulomb law of friction (Chapter 6)
v
n virtual nodal displacements (Chapter 2)
s
n + 1v
0 displacement vector of the end of bar (Chapter 5)
v0 vertical displacement of the end of bar (Chapter 5)
vol volume of parallelepiped based on base vectors (Appendix A)
w displacement of beam (Chapter 2)
w constraint function (Chapter 6)
w width of leaf (Chapter 9)
x vector of coordinates within an element (Chapter 2)
x, y, z components of position vector
x
e vector of nodal coordinates (Chapter 2)
x
i vector of location of the nodes in contact zone after loading
(Chapter 6)
x
S location of slave node after loading (Chapter 6)
x
S0 location of slave node before loading (Chapter 6)
x
1, x2 locations of master nodes (Chapter 6)
x
1, x2, x3 Cartesian components in current configuration (Chapter 5)
x1, x2, x3 Cartesian components in current configuration (Chapter 5)
yA1, yA2,
yB1, yB2 deflections of bearing points of leaves (Chapter 9)
A matrix that defines a nonlinear strain vector (Chapter 5)
A matrix of contact surface (Chapter 6)
A antisymmetric part of any tensor (Appendix A)
[A] matrix of transformation (Appendix A)
A, B, C matrices (Appendix B)
A cross-section area (Chapters 1, 4)
A area of triangle element (Chapter 2)
A
i matrix of contact surface at iteration i (Chapter 6)
A
k cross-section area of rod-element (Chapter 2)
A
0 cross-section area in initial configuration (Chapter 5)
AT transpose matrix (Appendix B)
A–1 inverse matrix (Appendix B)
∆A
c small area in current configuration (Chapter 5)
dA1, dA2, dA3 areas of faces of elemental tetrahedtron (Chapter 5)
B strain-displacement transformation matrix (Chapter 2)
B matrix derivative of contact matrix A (Chapter 6)
B minimum specimen stiffness (Chapter 7)
B
e element strain-displacement transformation matrix (Chapter 2)
B
L linear strain-displacement transformation matrix (Chapter 5)
B
N nonlinear strain-displacement transformation matrix (Chapter 5)
BHN Brinell hardness
[C] matrix (Chapter 2)
C empirical constant in Miner equation (Chapter 7)
C constant in crack propagation equation (Chapter 7)C
mn, m, n
= 1, ., 6 material constants in general elastic stress-strain correlation
(Chapter 1)
C1, C2 material constituents of Mooney–Rivlin material (Chapter 5)
D deformation tensor (Chapter 5)
D differentiation matrix operator (Chapter 2)
D fourth-order tensor of material properties (Appendix A)
D deviatoric part of a tensor (Appendix A)
D
i damaging effect (Chapter 7)
D
prsq fourth-order tensor of material properties (Appendix A)
D
s deviator of stress tensor (Chapter 1)
DA Almansy deformation tensor (Chapter 5)
DC Cauchy–Green deformation tensor (Chapter 5)
E Young modulus (Chapter 1)
Ee elastic stiffness tensor (Chapter 4)
Eep elasto-plastic stiffness tensor (Chapter 4)
E
1, E2, E3 principal values of Cauchy-Green or Almansy strain tensor
(Chapter 5)
EA Almansy strain tensor (Chapter 5)
EC Cauchy–Green strain tensor (Chapter 5)
EH Hencky strain tensor (Chapter 5)
, ,
components of Cauchy–Green strain tensor in Cartesian
coordinate system (Chapter 5)
F applied load (Chapter 3)
F yield surface (Chapter 4)
F load (Chapter 3)
F
e element nodal forces (Chapter 2)
F
n vector of nodal forces (Chapter 2)
Fn vector of nodal forces (Chapter 5)
Ftot vector of total loads (Chapter 6)
resistance of structure (Chapter 3)
F
∆ vector of hierarchical nodal forces (Chapter 2)
∆F, ∆Fi increment of force (Chapter 3)
G shear modulus (Chapter 1)
G energy release rate (Chapter 7)
GI
energy release rate of mode I (Chapter 7)
Gk base vectors in current configuration (Chapter 5)
Gsk covariant components of metric tensor in current configuration
(Chapter 5)
Gk reciprocal base vectors in current configuration (Chapter 5)
Gsk contravariant components of metric tensor in current configuration
(Chapter 5)
E
xx
C
E
yy
C
,…, ECzz
Fi rH tensor of simple shear (Chapter 5)
H slope of hardening function (Chapter 4)
Hi
Hermit polynomials (Chapter 2)
I identity tensor (Chapter 5)
I bending moment of inertia (Chapter 2)
I moment of inertia of a single leaf (Chapter 9)
I
1, I2, I3 invariants of tensor (Appendix A)
J Jacobian matrix (Chapter 2)
Ji
, Jc Jacobians of transformation (Chapter 5)
K stiffness matrix (Chapter 4)
K stress intensity factor (Chapter 7)
K stress intensity factor of mode I (Chapter 7)
K correction factor of spring leaves deflection (Chapter 9)
K
e element stiffness matrix (Chapter 2)
K
b stiffness matrix of bodies in contact (Chapter 6)
K
c stiffness matrix of whole contact surface (Chapter 6)
K´ cyclic strength coefficient (Chapter 7)
K
ic fracture toughness (Chapter 7)
K
ks stiffness factors (Chapter 1)
K
L, KR, Kσ constituents of stiffness matrix in nonlinear problems (Chapter 5)
K
max, Kmin maximum and minimum stress intensity factors (Chapter 7)
Kt
theoretical stress concentration factor (Chapter 7)
K
th threshold stress intensity factor (Chapter 7)

strain concentration factor (Chapter 7)

stress concentration factor (Chapter 7)
normal stiffness matrix of contact element (Chapter 6)
tangential stiffness matrix of contact element (Chapter 6)
stiffness matrix of contact element (Chapter 6)
Kt tangent stiffness matrix (Chapter 5)
Kt tangent stiffness (Chapter 3)
, ,
, constituents of hierarchical stiffness matrix (Chapter 2)
∆K range of stress intensity (Chapter 7)
L matrix of directional cosines (Chapter 2)
N matrix of shape functions (Chapter 2)
N axial force (Chapter 3)
Ni
number of cycles (Chapter 7)
Nf
number of load fluctuations up to failure (Chapter 7)
Nk interpolation functions (Chapter 2)
N
S matrix of normals (Chapter 6)
N0 matrix of normals (Chapter 6)
K
c n ,
e
K
c t ,
e
K
ec
K
e
uu K
ue

K
∆e
u K
e
∆∆N1, N2, N3 normal vectors in current configuration (Chapter 5)
N1, N2, N3 axial loads in rods (Chapter 4)
internal loads due to fictitious external load (Chapter 4)
internal residual loads (Chapter 4)
O orthogonal rotation tensor (Chapter 5)
P polynom of principal values (Appendix A)
P
k contact forces between leaves (Chapter 9)
P
lim limit load (Chapter 4)
P
lim limit value of tensile force of Signorini material in simple tension
(Chapter 5)
P
f fictitious load (Chapter 4)
P
p Legendre polynomial of an order p (Chapter 2)
P
yp load that corresponds to onset of yield (Chapter 4)
∆P
n concentrated force acting upon a small area (Chapter 1)
Q tensor (Appendix A)
Q concentrated load (Chapter 1)
Q plastic potential (Chapter 4)
R vector of body forces (Chapter 2)
R position vector in current configuration (Chapter 5)
R residual vector of unbalanced forces (Chapter 5)
R vector-derivative of potential over displacements vector
(Chapter 6)
R residual force (Chapter 3)
R stress ratio (Chapter 7)
R
i vector-derivative of potential over displacements vector at
iteration i (Chapter 6)
R
i residual vector (Chapter 3)
R
i residual (Chapter 3)
S stress tensor (Chapter 5)
S symmetric part of any tensor (Appendix A)
S nominal stress (Chapter 7)
SF stiffening factor (Chapter 9)
S
k,max maximum bending stresses in leaves (Chapter 9)
S
max, Smin maximum and minimum cyclic stress (Chapter 7)
S
u tensile strength (Chapter 9)
S
xx, Syy, Szz,
S
xy, Syz, Szx components of stress tensor (Chapter 5)
S
yp yield point (Chapter 9)
S1, S2, S3 components of stress tensor (Chapters 7, 8)
SC Cauchy stress tensor (Chapter 5)
SP first Piola stress tensor (Chapter 5)
second Piola stress tensor (Chapter 5)
T tensor (Appendix A)
N 1f, N 2f
N 1 r, N 2 r
S
pT symbol of transposition of vector, tensor, or matrix
T
sk, components of tensor (Appendix A)
TT transpose tensor (Appendix A)
T–1 inverse tensor (Appendix A)
T(3) tensor of third order
T(4) tensor of fourth order
strain tensor (Chapter 1)
T
σ stress tensor (Chapter 1)
matrix of tangential vectors (Chapter 6)
, , constituents of strain tensor (Chapter 1)
U left stretch tensor (Chapter 5)
U internal energy (Chapter 3)
stored elastic energy (Chapter 1)
Ua
released strain energy due to crack propagation (Chapter 7)
U0 stored elastic strain energy (Chapter 7)
V right stretch tensor (Chapter 5)
V1, V2, V3 principal stretches (Chapter 5)
W work or strain energy
W section modulus of a single leaf (Chapter 9)
We
external work
Wi internal work
specific internal work

required destruction work (Chapter 7)
W
yp critical value of distortion energy (Chapter 4)
internal dilatation (volumetric) specific work (Chapter 1)
internal distortion specific work (Chapter 1)
X, Y, Z volumetric forces (Chapter 1)
X
i vector of location of the nodes in contact zone before loading
(Chapter 6)
X1, X2, X3 global Cartesian coordinates (Chapter 5)
Y function of hardening (Chapter 4)
Y0
yield constant in perfectly plastic models (Chapter 4)
Y(f) slip surface (Chapter 6)
α tensor of backstress (Chapter 4)
α angle of rotation (Chapter 5)
α, β, γ directional angles of unit normal (Chapter 1)
α
mn coefficients of transformation of tensor components (Appendix A)
β angle between rods (Chapter 4)
β material constant of Mooney–Rivlin material (Chapter 5)
γ parameter of simple shear (Chapter 5)
γe
elastic specific surface energy (Chapter 7)
T sk′
T
ε
T S 0
T
1s
T
2s
T
3s
U i
W i
W i v
W i dγp
plastic specific surface energy (Chapter 7)
γxy, γyz, γxz shear strains (Chapter 1)
shear strains in cylindrical coordinate system (Chapter 1)
δ symbol of variation or increment of any quantity
δ1, δ2 elongations (Chapter 5)
δmn Kronecker symbol (components of identity tensor)
ε tensor of small deformation (Chapter 1)
ε vector of strains (Chapter 2)
εN nonlinear strain vector (Chapter 5)
ε full (logarithmic) strain (Chapter 4)
ε
e effective strain (Chapter 4)
εk axial strain of rod-element (Chapter 2)
ε
m mean strain
normal strains in cylindrical coordinate system (Chapter 1)
ε
x, εy, εz normal strains (Chapter 1)
ε1, ε2, ε3 principal strains (Chapter 1)
εe elastic component of total strain (Chapter 4)
εp plastic component of total strain (Chapter 4)
corrected plastic strain (Chapter 4)
δεi strain increment (Chapter 7)
elastic strain increment (Chapter 7)
plastic strain increment (Chapter 7)
∆ε cyclic strain amplitude (Chapter 7)
∆εe elastic strain amplitude (Chapter 7)
∆εp plastic strain amplitude (Chapter 7)
∆ε
eff effective strain amplitude (Chapter 7)
strain fracture limit (Chapter 7)
θ polar angle (Chapter 1)
θ angle of simple shear (Chapter 5)
θ
x, θy slopes of a plate (Chapter 2)
θ1, θ2 slopes at the ends of beam (Chapter 2)
θ1, θ2, θ3 curvilinear coordinates (Chapter 5)
volumetric change (Chapter 1)
λ, µ Lame constants (Chapter 1)
λ proportional factor in deformation plasticity (Chapter 4)
λ
i line-search parameter (Chapter 3)
λk principal values of tensor (Appendix A)
λ
n, λt Lagrange multipliers (Chapter 6)
dλ plastic flow scalar (Chapter 4)
µs static coefficient of friction (Chapter 6)
µd dynamic coefficient of friction (Chapter 6)
γ rθ, , γ θz γ zr
ε
r, , εθ εz
ε
n + 1
p
δεie
δεi p
ε
f ′
ϑdµ scalar in Ziegler’s stress correlation (Chapter 4)
κ matrix of penalties (Chapter 6)
κ hardening parameter (Chapter 4)
κn
normal penalty (Chapter 6)
κt
tangential penalty (Chapter 6)
ν Poisson’s ratio
ν lateral reduction (Chapter 5)
ρ radius of curvature (Chapter 9)
σ vector of stresses (Chapter 2)
interpolant of stress field (Chapter 2)
σ axial stress (Chapter 1)
σ
e effective stress
σh vector of stresses obtained by FE solution (Chapter 2)
σ
m mean stress
(σm)eff effective mean stress (Chapter 7)
σ
n normal component of traction (Chapter 1)
normal stress components in cylindrical coordinate system
(Chapter 1)
σ
x, σy, σz normal stresses (Chapter 1)
σ
yp yield point (Chapter 4)
σ1, σ2, σ3 principal stresses (Chapter 1)
σN equivalent stress amplitude (Chapter 7)
∆σ stress amplitude (Chapter 7)
stress fracture limit (Chapter 7)
τ
max critical value of shear stress (Chapter 4)
τ
n tangential component of traction (Chapter 1)
τ
oct octahedral shear stress (Chapter 1)
shear stresses in cylindrical coordinate system (Chapter 1)
τ
xy, τyz, τzx shear stresses (Chapter 1)
τ1, τ2, τ3 maximum shear stresses (Chapter 1)
φk integrals of Legendre polynomials (internal shape functions)
(Chapter 2)
ξk coordinates of nodes (Chapter 2)
χ0, χ1, χ2 coefficients in stress-strain correlation
π
c potential energy of contact element (Chapter 6)
η relative error (Chapter 2)
∆ determinant of a second order tensor (Appendix A)
∆ symbol of increment of any quantity
∆ζ scalar magnitude of correction term of tangential force (Chapter 6)

e vector of deviations (Chapter 2)

k coefficients of hierarchical approximation (Chapter 2)
Ω tensor of small rotation (Chapter 5)
Π total potential energy
ˆσ σ
r, , σθ σz
σ
f ′
τ
rθ, , τθz τzrΠ
b potential of bodies in contact (Chapter 6)
Π
c potential of contact zone (Chapter 6)
δΠ
c virtual work of contact forces (Chapter 6)
Φ minimized function (Chapter 3)
Λ vector of Lagrange multipliers (Chapter 6)
Λ
i vector of Lagrange multipliers at iteration i
gradient in initial configuration
gradient in current configuration
Brackets:
[ .] matrix brackets (Chapter 2)
( .) tensor brackets (Appendix A)
{ .} vector brackets (Chapter 2)i
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