Applications in Solid Mechanics Fifth deal.II Users and Developers - - PowerPoint PPT Presentation
Applications in Solid Mechanics Fifth deal.II Users and Developers Workshop Texas A&M University August 4, 2015 A McBride BD Reddy S Bartle, A de Villiers, B Grieshaber, M Hamed, J-P Pelteret, T Povall, N Richardson Centre for Research in
Fifth deal.II Users and Developers Workshop Texas A&M University August 4, 2015
A McBride BD Reddy S Bartle, A de Villiers, B Grieshaber, M Hamed, J-P Pelteret, T Povall, N Richardson
Centre for Research in Computational and Applied Mechanics, University of Cape Town, South Africa
fmuid mechanics. Areas of activity in solid
Contrary to initial expectations, numerical experiments with the IP methods show that bilinear quadrilateral elements, while performing very well for highly com- pressible materials, perform poorly for the nearly incompressible case, unlike their triangular counterparts. Figures 5.2 to 5.4 show sets of results that will be discussed
OR WHY IT’S GOOD NOT TO HAVE TETRAHEDRAL ELEMENTS IN DEAL.II
❖ Grieshaber (2013) PhD ❖ GRIESHABER, MCB, REDDY (ACCEPTED), SINUM div σ (u) = f
σ (u) := 2µε (u) + λtr ε (u) 1 = 2µε (u) + λ (r · u) 1,
λ = Eν (1 + ν) (1 2ν), µ = E 2 (1 + ν)As ν ! 1
2
∈ aUI
h (uh, v) = lUI h (v)
aUI
h (u, v) =
X
Ωe∈Th
Z
Ωe
σ (u) : ε (v) dx + θ 2µ X
E∈ΓiD
Z
E
b buc c: { {ε (v)} }ds + θ λ X
E∈ΓiD ngp
X
i=1
[b buc c: { {r · v1} }] |piwi 2µ X
E∈ΓiD
Z
E
{ {ε (u)} }: b bvc cds λ X
E∈ΓiD ngp
X
i=1
[{ {r · u1} }: b bvc c] |piwi + kµ µ X
E∈ΓiD
1 hE Z
E
b buc c: b bvc c ds + kλ λ X
E∈ΓiD
1 hE
ngp
X
i=1
[[[u]][[v]]] |piwi, lUI
h (v) =
X
Ωe∈Th
Z
Ωe
f · v dx + θ 2µ X
E∈ΓD
Z
E
(g ⌦ n) : ε (v) ds + θ λ X
E∈ΓD ngp
X
i=1
[(g ⌦ n) : (r · v 1)] |piwi + kµ µ X
E∈ΓD
1 hE Z
E
(g ⌦ n) : (v ⌦ n) ds + kλ λ X
E∈ΓD
1 hE
ngp
X
i=1
[(g · n) (v · n)] |piwi + X
E∈ΓN
Z
E
h · v ds.
to ngp = 2)
θ = 1 θ = −1 θ = 0
u = g
σ (u) n = h
❖ GRIESHABER, MCB, REDDY (ACCEPTED), SINUM
(a) NIPG (b) IIPG (c) SIPG
(a) NIPG (b) IIPG (c) SIPG
). ν = 0.49995.
, E = 250.
(a) Modified SIPG (b) Modified IIPG
neps SG SG Q2 NIPG — UI IIPG — UI SIPG — UI Skewed mesh 4 1.3855 5.99901 2.07077 4.67802 2.08011 4.6961 2.21427 4.72803 8 1.95749 7.06822 2.12496 6.13406 2.08011 4.6961 2.21427 4.72803 16 2.01279 7.49608 2.24957 7.14275 2.15449 7.14171 2.28465 7.14403 32 2.19539 7.64827 2.633127 7.54839 2.38855 7.54625 2.50211 7.54509 Unstructured mesh 32 3.054 7.66333 5.44538 7.61069 4.42923 7.60888 4.5467 7.60599
❖ GRIESHABER, MCB, REDDY (IN REVIEW)
(a) Exact solution (b) Modified NIPG (c) Modified IIPG (d) Modified SIPG
(a) Exact solution (b) NIPG (c) IIPG (d) SIPG
convergence deteriorates upon refinement
OR HOW TO TRICK DEAL.II TO HANDLE LOWER-DIMENSIONAL MANIFOLDS PRIOR TO CODIMENSION ONE AND MANIFOLDS
๏ Treat the 3D body as a 2D surface with a director at each point ๏ Formulate problem in terms of quantities averaged through the thickness (see SIMO & FOX) ๏ SRI through the thickness to prevent membrane locking ๏ account for transverse isotropy and incompressibility
❖ Bartle (2009) MSc
❖ de Villiers (current) PhD
❖ WICK (2011) FLUID-STRUCTURE INTERACTIONS USING
DIFFERENT MESH MOTION TECHNIQUES. COMPUTERS &
STRUCTURES ❖ WICK (2013). SOLVING MONOLITHIC FLUID-STRUCTURE
INTERACTION PROBLEMS IN ARBITRARY LAGRANGIAN
EULERIAN CO-ORDINATES WITH THE DEAL.II LIBRARY. ARCHIVE OF NUMERICAL SOFTWARE
Incompressible finite elasticity with transverse isotropy Turek flag
๏ generally restricted to linear theory
dA dV
surface energy bulk energy surface tension H20 carbon nanotubes ❖ JAVILI, MCB, STEINMANN & REDDY (2014), COMP MECH ❖ SURFACE ELASTICITY ON bitbucket.org ❖ Davydov, Javili, Steinmann, McB (2013), in Surface Effects in Solid Mechanics
b ϕ = ϕ|∂B0 b f · b F = b I =: I − N ⊗ N and b F · b f = b i =: i − n ⊗ n .
y F := ∂x/∂X
erse f := ∂X/∂x
b ations are denote re b F := ∂b x/∂c X
from differential y b f := ∂c X/∂b x
Strong form: balance of linear momentum balance of angular momentum
DivP + bp
0 = 0
, F · Pt = P · Ft in B0 d Divb P +b bp
0 − P · N = 0
, b F · b Pt = b P · b Ft
b b b b b Weak form: Z
B0
P : Gradδϕ dV + Z
S0
b P : d Gradδb ϕ dA − Z
B0
δϕ · bp
0 dV −
Z
SN
δb ϕ ·b bp
0 dA = 0
∀δϕ ∈ H1
0(B0) , ∀δb
ϕ ∈ H1
0(S0)
surface divergence theorem
Z
S0
d Div {b
Z
C0
{b
N dL − Z
S0
b C {b
b P · N = 0
superficial tensor b C = −d DivN twice mean surface curvature
J := DetF : Jacobian determinant µ , λ : Lam´ e constants Free energy Ψ(F) = 1
2 λ ln2 J + 1 2 µ [F : F − 3 − 2 ln J]
Piola stress P(F) = ∂Ψ ∂F = λ ln J f t + µ [F − f t]
b J := d Detb F : Surface Jacobian determinant b µ ,b λ : Surface Lam´ e constants b γ : Surface tension Surface Free energy b Ψ(b F) = 1
2 b
λ ln2 b J + 1
2 b
µ [b F : b F − 2 − 2 ln b J] +b γ b J Surface Piola stress b P(b F) = ∂b Ψ ∂b F = b λ ln b J b f t +b µ [b F − b f t] +b γ b J b f t
Th b Th M volume surface Ωe,0 b Ωe,0
(A)
Ψ(F) = 1
2 λ ln2 J + 1 2 µ [F : F − 3 − 2 ln J]
b b b b Ψ(b F) = 1
2 b
λ ln2 b J + 1
2 b
µ [b F : b F − 2 − 2 ln b J] +b γ b J
(B)
|u|
b γ
(A) (B) 5 10 2 7 0.1 0.2 0.3 0.4 0.5 0.6 0.637 midpoint deflection
b λ/λ = 0 b λ/λ = 1
|u|
| b P | |P |
❖ BARGMANN, WILNERS (TUHH, HELMHOLTZ GEESTHACHT)
layer in geological material
HIRSCHBERGER ET AL (2009)
❖ MCB, MERGHEIM, JAVILI, STEINMANN, BARGMANN (2012), JMPS
B
−
B
+
M X c M ∂B0 N M
+
M
−
I0 ∂I0 c X
b ϕ
B
+ t
B
− t
x It I
+ t
I
− t
ϕ m b x+
b x
b x−
F (X, t) = Gradϕ(X, t) b F := [ Gradb ϕ(c X, t)
deformation gradients
b I = b P0 = I − M ⊗ M [ Grad {b
I a b P0 · a c X
c M M
projection and surface gradient
{•} {•} {e
→
micro
b F F
X M
−
M
+
B
+
B
−
M
I
+
I0 I
−
J•K = •|I+
0 − •|I−
c X
๏ constitutive traction separation law*
*see e.g. NEEDLEMAN ET AL (1987,1993), MOSLER & SCHEIDER (2011)
JϕK e t0
constitutive law
๏ constitutive traction separation law replaced by micro-scale simulation to capture heterogeneities
microscale
JϕK e t0 b P = 2b F · ∂ b
C b
ψ0
๏ interface has own energetic structures (Helmholtz energy): ๏ account for in-plane deformation b P b F
microscale
JϕK e t0
B
−
B
+
B
+ t
B
− t
ϕ F I0 It
I
+ t
I
− t
c X
b x+ b x−
b x
e t0
b P
ϕ F x Bt ∂Bt b F B0 X
∂B0
B0 ⊂ R3
๏ boundary conditions imposed via linear constraints ๏ currently solving micro-scale problem assuming macroscopic kinematic data 1 1 0.5
B0 ∂Bint,+ ∂Bint,− B
−
B
+
I0
rve ∂Bpln,+ ∂Bpln,− c X
consider two scenarios: void and stiff inclusion neo-Hookean material
JϕK = [0.5 0 0.5]t
(e) b F = 2 4 1.5 1 3 5
kτk
(a)
JϕK = [0 0 0.5]t
(b)
JϕK = [0.5 0 0]t
(c)
JϕK = [0.5 0 0.5]t
b F = 2 4 1.5 1 3 5 (d) (f b F = 2 4 1 1 1.25 3 5
T E = ru S EY
EY Ee E = Ee + Ep
T T
Ep
❖ Richardson (2012), MSc ❖ POVALL (2013), MSC ❖ POVALL, MCB, REDDY (2014), COMP MAT SCI
sa ¼
sa r ma;
h l = 25 h l = 125 h l = 1.25
Constrained shear. Double slip system Joint work with Daya Reddy and Morton Gurtin
1
2h l = ∞ 2h l = 1.25
0.002 0.004 0.006 0.008 0.01 0.5 1 1.5 2 2.5 3
ϒ σ12 / µ × 10 2h/l = ∞ 2h/l = 1.25
Shear with elastic
slip system
2
lattice configuration
Fe Fp
L ¼ $xv ¼ @vi @xj ei ej ¼ Le þ FeLpFe1;
divr þ b ¼ 0;
Lp ¼ X
a
masa ma:
/aðsa; SaÞ ¼ jsaj Sa 6 0: ka P 0; /aðsa; SaÞ 6 0 and ka/aðsa; SaÞ ¼ 0: _ Sa ¼ X
b
habðSÞjmbj; Sajt¼0 ¼ S0;
habðSbÞ ¼ vabhðSbÞ |fflfflfflfflffl{zfflfflfflfflffl}
self hardening
þ q 1 vab
|fflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
latent hardening
;
where q > 0 is an interaction constant
reference configuration V0
X
x = ϕ(X, t)
F V x
❖ M Hamed (current) PhD (an attendee of the deal.II workshop in Cape Town)
J div τ + ρb = ρ˙ v ρ ˙ ψ = Jτ : d div q ˙ ηθ η ˙ θ
Balance linear momentum and energy Thermoplasticity
g(x, ¯ y(x)) 0 tN := n (¯ y(x)) · σ n (¯ y(x)) 0 tN g(x, ¯ y(x)) = 0
h(x, ¯ y(x)) := ktTk µtN 0 ξ 0 ξh(x, ¯ y(x)) = 0 Normal contact constraints Tangential contact constraints
τ = pJ1 + dev[τ] Φ = kdev[τ]k r 2 3[K 0(αn) + ˆ y(θ)] 0 λ 0 , λΦ = 0
Thermoplasticity
Z
Ω
n div
∗
u¯ pJ + r
∗
u · dev[τ]
− ⇢Z
Ω ∗
u · fdΩ + Z
Γ ∗
u · tdΓ
⇢Z
Γ ∗
g tNdΓ
Z
Ω
⇢∗ θ h c ˙ θ − Dmech + ¯ H i + r
∗
θ · q
− ⇢Z
Ω ∗
θ ¯ RdΩ + Z
Γ ∗
θ [−S] dΓ
Balance linear momentum and contact Balance energy
grain boundaries (finite gradient crystal plasticity)
❖ Gottschalk (2015) PhD (Hannover)
(b)
❖ McB, Bargmann, Reddy (2015), Comp Mech
gradient crystal thermoplasticity
C code user element F code