A Consistent Configuration Formulation Involving ContinuumDamage Mechanics Based Lagrangian Hydrodynamic
Computational Framework for 3D High- and HypervelocityImpact/Damage/Penetration Analysis
X. Zhou∗, D. Sha†, K. K. Tamma‡, and Y. Miller§
Abstract
A consistent configuration formulation involving continuum damage mechanics based Lagrangianhydrodynamic computational framework is proposed to model 3D high- and hyper-velocity im-pact/damage/penetration problems. The contributions of the study are: (i) consistent formulation ofthe dynamic equation, contact formulation, constitutive relation, and time integration with respect to thesame configuration; (ii) incorporate continuum damage mechanics into the hydrodynamic constitutive re-lations; and (iii) a robust time integration algorithm for the coupled equation of state and the internalenergy equation. As a consequence, the present framework is robust and accurate for finite deformationcontact/impact/damage/penetration applications, which is demonstrated by various numerical examples.
1. Introduction
Hydrocodes are important tools for survivability andlethality of armor/anti-armor applications, space-craft safety assessments and evaluations [1, 2], etc.The existing hydrocodes mostly utilize fracture me-chanics based failure models (the failure of the ma-terial is determined by the maximum accumulatedplastic strain) [3]. In contrast to the fracture me-chanics based failure models, which focus on whenthe material fails, the continuum damage mechanicsbased failure models (the failure of the material is de-termined by the evolution of an internal variable thatmeasures the micro-defect density) can further pro-vide an in-depth understanding of the physical mech-anisms of how the material fails [4]. Furthermore,since Lemaitre introduced an approach to incorpo-rate fracture mechanics based failure model into thecontinuum damage mechanics based isotropic dam-age failure models [5,6], the Lemaitre damage modelhas been successfully applied for the quasi-static fail-ure of steel [7] and high-velocity penetration [8]. Thepresent study aims to not only extend the Lemaitredamage model to the hydrodynamic type of con-stitutive model for hypervelocity penetration prob-lems, but also involves a novel computational frame-work that employs a consistent configuration for-mulation in handling high- and hypervelocity im-pact/damage/penetration analysis with finite defor-mation.
The high-velocity (the projectile velocity is in theorder of 102 m/s) and hypervelocity (the projectile
velocity is greater than 103 m/s) penetration is ahighly nonlinear phenomena involving finite defor-mation [9]. Although the Eulerian approach (mate-rial flows through a fixed mesh) has the advantageto handle the large deformation of the material [10]in contrast to the Lagrangian approach (the spacediscretization mesh/points deform following the ma-terial points), the Eulerian approach has the inher-ent difficulties to handle the traction-free and contactboundary conditions [11]. The Lagrangian approachcan be naturally incorporated with the constitutivemodel for solid mechanics, since the constitutive re-lation for solid mechanics is defined on the materialpoints. Amongst the various Lagrangian approaches,the fixed mesh Lagrangian finite element method hasthe least computational complexity. With regardsto the computational finite deformation phenomena,the mesh distortion and the reduction of the timestep size for explicit time operators has been re-peatly reported to be a deficiency associated with thefixed mesh Lagrangian finite element method. How-ever, our recent study shows that when the consistentconfiguration formulation (formulating the governingequations consistently with respect to a configura-tion close to the current configuration) is utilized, thefixed mesh Lagrangian finite element method is in-deed robust enough to handle the finite deformationof the material up to failure [12–14]. The decrease ofthe time step size for explicit time operators can bealso circumvented by selecting/designing appropri-ately the necessary time integration operators [15].
∗Research Associate, Army High Performance Computing Research Center (AHPCRC), [email protected]†Researcher, Army High Performance Computing Research Center (AHPCRC), [email protected]‡Professor, to receive correspondence, Department of Mechanical Engineering/Army High Performance Computing Research
Center (AHPCRC), [email protected]§Graduate Research Student, Department of Mechanical Engineering, University of Minnesota, Current Address: Cap-
tain Yvonne Miller, Department of Civil and Mechanical Engineering, United States Military Academy, West Point,[email protected]
The presentation of this exposition is outlinedas the follows. First the dynamic equilibrium equa-tion including frictional contact boundary conditionsin Eulerian form is formulated. Then the devia-toric portion of the hydrodynamic constitutive equa-tion incorporating the elasto-visco-thermo-plastic-damage is formulated. And, the hydrostatic portionof the hydrodynamic constitutive equation utilizingthe equation of state and internal energy equationis then discussed. Finally, some numerical examplesare performed to demonstrate the applicability androbustness of the present computational frameworkand formulation.
2. Dynamic Equilibrium Equation
Let the open set Ωt ⊂ ℜ3 be the domain of inter-est of the configuration at time t, with the bound-ary ∂Ωt = Γft ∪ Γdt , Γft ∩ Γdt = ∅, and the closureΩt = Ωt ∪ ∂Ωt. For finite strain dynamic problems,the dynamic equilibrium equation of the configura-tion at time t in the Eulerian form with Cauchy stressis described by
∂(ρtvt)
∂t+ ρtηvt −∇ · σt = bt in Ωt (1)
σtn = f on Γft (2)
xt = xd on Γdt (3)
where ρt is the material density of the configurationat time t, vt is the material particle velocity at timet, σt is the Cauchy stress of the configuration attime t, η is the viscous damping ratio in the senseof Rayleigh damping, bt is the body force, n is theboundary surface outward unit vector, xt is mate-rial particle coordinate, Γft is the boundary subjectto traction boundary condition, Γdt is the boundarysubject to prescribed displacement boundary condi-tion.
Let σn = σn = σnk = −niσijnjnk ∈ ℜ3
be the normal stress on the contact surface, n bethe normal direction vector on the contact surface,g ∈ ℜ3 be the relative gap vector between point pairson the contact surfaces. And, let n(x) be the out-ward normal direction at point x ∈ ℜ3 of the surfaceΓc ⊂ Γc, and n(x) be the outward normal directionat point x ∈ ℜ3 of the surface Γc ⊂ Γc. The initialgap vector g0 satisfies, g0 := g0 ∈ ℜ3 | g0 = x−x =minz(z − x), n(z) · n(x) = −1, x ∈ ℜ3 on Γc, x ∈ℜ3, z ∈ ℜ3 on Γc. Let gn and gτ be the normaland tangential directions of the gap vector g, theng = gn + gτ and gn · gτ = 0. And, the impenetra-ble constraint in the normal direction on the contactsurfaces is given as the following Kuhn-Tucker com-plementarity conditions, (τn, gn − λcnn)Γc(t) = 0,λcn ≥ 0, ψn(σn) = σ · n ≥ 0, λcnψn(σn) =0, ∀ τ n ∈ ψn(τn) ≥ 0, where λcn ≥ 0 representsthe impenetrability condition on the contact surface,and (•, •)Γ =
RΓ• : •ds represents the inner-product.
Let R = V | V ∈ ℜ3 × [0,∞), then a yieldfunction of the frictional stress vector in the tangen-
tial direction on the contact surface can be definedby ψτ (στ ) = µc | σn | − | στ |≥ 0 on Γc × [0,∞),where στ = σtk = −niσik − niσkjnjnk ∈ ℜ3 isthe tangential stress vector on the contact surface,and µc ∈ ℜ is the Coulomb frictional coefficient.
Let gτ ∈ ℜ3 be the the relative tangential ve-locity of the point pairs on the contact surfaces,the classical Coulomb friction law thus is repre-sented by (τ τ − στ , gτ ) ≤ 0, στ ∈ ψτ (στ ) ≥ 0,∀ τ τ ∈ ψτ (τ τ ) ≥ 0. The corresponding weakform employing the Lagrangian multiplier is givenas (τ τ , gτ − λcτ∇στψτ (στ ))Γc(t) = 0, ψτ (στ ) ≥ 0,λcτ ≥ 0, λcτψτ (στ ) = 0, ∀ τ τ ∈ ψτ (τ τ ) ≥ 0, where
∇στψτ (στ ) = ∂ψτ (στ )∂στ
.Define the following set of virtual displacement
field as, [H10 (Ωt)]
3 := w | w ∈ [H1(Ω)]3, w =0 on Γd, and the set of virtual velocity field as,[H1
v(Ωt)]3 := v | v ∈ [H1(Ω)]3, v = xdt on Γd,
where H1(Ω) is the first order differentiable Hilbertspace. The Eulerian weak form of the dynamic equi-librium equation subjected to the contact boundarycondition is given as the following.
and subject to the constraints λcn ≥ 0, ψn(σn) =σn ·n ≥ 0, λcnψn(σn) = 0, on Γc(t), and ψτ (στ ) ≥ 0,λcτ ≥ 0, λcτψτ (στ ) = 0, on Γc(t).
2. Hydrodynamic Constitutive Equation
Consider the isotropic damage hydrodynamic consti-tutive equation given by,
8>>><>>>:
eσ =σ
1 − ϕ= eσD + eσH
eσ∇D = 2GD
eD
eσH = −p(ρ, e)I
(7)
where eσ is the effective stress tensor, σ is the Cauchystress tensor, ϕ is the isotropic damage state vari-able, eσD is the deviatoric portion of the effectivestress tensor, eσH is the hydrostatic portion of theeffective stress tensor, G is the shear modulus, D
eD
is the elastic part for the deviatoric portion of thevelocity strain tensor, p is the hydrostatic pressure,ρ is the material density, e is the internal energy perunit mass, I is the rank two identity tensor, andthe symbol •∇ denotes to either the Jaunamm, theGreen-Naghdi, or Truesdell stress rate.
2.1 Damage Models for the Deviatoric
Stress
From a practical perspective, the Lemaitre plastic-damage model [16] is adapted for the deviatoricstress of the hydrodynamic constitutive equation.For the low- and high-velocity impact/penetration
2
American Institute of Aeronautics and Astronautics
problems, when the amount of dissipative workdone by the plasticity and damage in the systemdoes not raise the temperature significantly, for ex-ample, a projectile impacting on a thin structure,the Lemaitre plastic-damage model should be ap-plicable. However, when the amount of dissipa-tive work gives rise to a rapid increase of temper-ature for the material, for example, in the caseof hypervelocity penetration problems, a tempera-ture dependent thermo-visco-plastic model such asthe Johnson-Cook fracture model [17] needs to bemodified and incorporated into the Lemaitre plastic-damage model to account for the temperature andstrain rate effect. The Lemaitre plastic-damagemodel and the modified Johnson-Cook thermo-visco-plastic-damage model are discussed in the text tofollow.
2.1.1 Lemaitre Plastic-Damage Model for
Deviatoric Stress
Plastic-Damage Potential
Let the plastic-damage potential function be
Fpd(σ, R(r);Y, ϕ)
=fp(σ, R(r);ϕ) + fd(Y, ϕ)(8)
where R(r) is the isotropic hardening, r is the inter-nal state variable for isotropic hardening, Y is theenergy release rate density and is represented in theform
Y =eσ2eqRv
2E(9)
where E is the Young’s modulus, eσeq is the effectivevon Mises equivalent stress and is given by
eσeq =
q32σD : σD
1 − ϕ(10)
Rv is the triaxiality function. For hydrodynamics,the triaxiality function can be derived as the follow-ing,
Rv =3K
3K +G+
9G
3K +G
σH
σeq
!2
(11)
where σH is the hydrostatics pressure, G is the shearmodulus, K is the bulk modulus. For hydrodynamic,the bulk modulus K is derived from the equation ofstate and the derivation of the bulk modulus is dis-cussed in a later section.
For the associated plasticity model, the plasticyield function fp(σ, R(r);ϕ) is given as,
fp(σ, R(r);ϕ) = eσeq − σy −R(r) ≤ 0 (12)
where σy is the uniaxial tension yield stress. And,the damage potential Fd(Y, ϕ) is given as,
fd(Y, ϕ) =S
(s+ 1)(1 − ϕ)
Y
S
!s+1
(13)
where S and s are the material parameters.
Plastic-Damage Evolution
Let DD = DeD + D
pD, and the corresponding Levy-
Mises plastic flow rule is given by
DpD = λp
∂Fpd
∂σ=
λp
1 − ϕ
∂fp
∂eσ =3λp
2(1 − ϕ)
eσDeσeq
(14)
and subject to the Kuhn-Tucker conditions,
λp ≥ 0, λpfp = 0, λpfp = 0 (15)
And, also from the associated plastic flow rule, ityields,
r = λp∂Fpd
∂R= (1−ϕ)ε = (1−ϕ)
r2
3DpD : D
pD (16)
where ε is the accumulated effective plastic strainrate. And the damage evolution is given by,
ϕ = λp∂fd∂Y
=
σ2eqRv
2SE(1 − ϕ)2
!sε (17)
Assuming σy and Rv are constant, the integration ofthe damage evolution law yields,
ϕ =
σ2eqRv
2SE(1 − ϕ)2
!s(ε− εD) (18)
where εD is the effective plastic-damage thresholdstrain, and
ϕc =
σ2eqRv
2SE(1 − ϕ)2
!s(εF − εD) (19)
where ϕc is the critical damage state, and εF is the ef-fective plastic-damage fracture strain. Furthermore,in the uniaxial tension fracture, Rv = 1, εF = ǫF ,and εD = ǫD, where ǫF and ǫD are the uniaxial ten-sion fracture strain and the uniaxial tension damagethreshold strain, respectively. Therefore, it yields,
ϕc
ǫF − ǫD=
σ2eq
2SE(1 − ϕ)2
!s(20)
such that the damage evolution function becomes
ϕ =ϕcR
sv
ǫF − ǫDε (21)
Again, assuming that the triaxiality function is con-stant, the integral form of the damage evolution func-tion becomes
ϕ =ϕcR
sv
ǫF − ǫD< ε− εD > (22)
where < • > is the Macauley bracket. Subjectingthe integral form of the damage evolution functionto the critical damage state, yields
ϕc =ϕcR
sv
ǫF − ǫD< εF − εD > (23)
3
American Institute of Aeronautics and Astronautics
It leads to
εF =ǫF
Rsv(24)
εD =ǫD
Rsv(25)
Consistency Condition
With the potential functions and the evolution func-tions available, we now can derive the consistencycondition to determine the plastic-damage strainmagnitude. The consistency condition is given as,
Fpd =∂Fpd
∂σD: σ
∇D −
∂Fpd
∂rr +
∂Fpd
∂ϕϕ = 0 (26)
Let∂Fpd
∂rr = hε, yields
Fpd =(1 − ϕ)∂fp
∂σD: C : (DD − ε
∂fp
∂eσD) − hε+
∂fd
∂ϕϕ
=3σD
2σeq: C : (DD − ε
3σD
2σeq) − hε
+σeqϕc
(1 − ϕ)2(εF − εD)ε
=0
(27)
where C = 2GI is the elasticity tensor, and I is thefourth-order identity tensor. Therefore, the devia-toric part of the constitutive equation becomes,
eσ∇D =
σ∇D
1 − ϕ= C : (DD − εnpd) (28)
where
ε =npd : C : DD
npd : C : npd +Hpd(29)
npd =
3σD
2σeq (30)
Hpd = h−σeqϕc
(1 − ϕ)2(εF − εD)(31)
And finally, the constitutive equation in the consis-tent form can be written as
σ∇D = (1 − ϕ)
C −
C : npd ⊗ npd : C
npd : C : npd +Hpd
!: DD
(32)
2.1.2 Modified Johnson-Cook Thermo-
Visco-Plastic-Damage Model for Devia-
toric Stress
The von Mises tensile flow stress of the Johnson-Cook model [17] is given by,
σJC = (A+Bεn)(1 + C ln(ε
ε0))(1 − Tm) (33)
where A, B, C, n, and m are material constants,ε0 = 1.0s−1, T = (T − Troom)/(Tmelt − Troom),
Troom is the room temperature, Tmelt is the mate-rial melting temperature. And the fracture strain ofthe Johnson-Cook model [17] is given by
εF = (D1+D2eD3
σHσeq )(1+D4 ln(
ε
ε0))(1+D5T ) (34)
for σH
σeq< 1.5, where D1-D5 are material constants.
For σH
σeq≥ 1.5, the model becomes
εF = (D1+D2e1.5D3)(1+D4 ln(
ε
ε0))(1+D5T ) (35)
The Thermo-Visco-Plastic-Damage Potential
Let the thermo-visco-plastic-damage potential func-tion be
Ftvpd(σ, Rtvp(ε),Q−1(ε);Y, ϕ)
=ftvp(σ, Rtvp(ε),Q−1(ε);ϕ) + fd(Y, ϕ)
(36)
where
ftvp = eσeq −Rtvp(ε) −Q−1(ε) ≤ 0 (37)
Rtvp = (A+Bεn)(1 − Tm) (38)
Q−1(ε) = frp = CRtvp ln(ε
ε0) (39)
Q(frp) = ε = ε0e
frp
CRtvpH(frp) (40)
and H(•) is the Heaviside function.
Damage Evolution
Since the Johnson-Cook fracture strain has alreadyincluded the triaxiality effect, the damage evolutionequation can be given as,
ϕ =H(ε− εD)
εF − εDε (41)
where εD = γDεF , and γD can be determined fromexperiment. For example, γD = 0.7 for Weldox 460E steel has been observed in experiment [8].
Consistency Condition
The consistency condition is given by
0 =Ftvpd
=∂ftvp
∂σD: σ
∇D −
∂Rtvp
∂εε−
∂Q−1
∂εε+
∂fd
∂ϕϕ
(42)
Letting ε = ε∆t
, yields the constitutive equation inconsistent form as
σ∇D = (1−ϕ)
C −
C : ntvpd ⊗ ntvpd : C
ntvpd : C : ntvpd +Htvpd
!: DD
(43)where
Htvpd = h+ C(h+ε0Rtvp
ε∆t) ln(
ε
ε0)
−H(ε− εD)σeq
(1 − ϕ)2(εF − εD)(44)
h = Bnεn−1(1 − Tm) (45)
ntvp =
3σD
2σeq(46)
4
American Institute of Aeronautics and Astronautics
Temperature for Adiabatic Condition
During the high strain rate deformation, the tem-perature of the material rises due to the dissipativeplastic work. For the short duration of the mate-rial deformation under high strain rate, the thermalequilibrium of the material cannot be established.Therefore, the temperature raise can be computedby utilizing the adiabatic condition. And, the tem-perature equation is given as,
ρcT = ηeσ : (DD − DeD) (47)
where ρ is the material density, c is the material heatcapacity, and ηe = 0.9 is efficiency ratio [18].
2.2 Equation of State for the Hydrostatic
Pressure
The equation of state (EOS) along with the mate-rial internal energy equation governs the relation be-tween the pressure and density implicitly, and servesas the constitutive equations which express the statethat the material can achieve [19]. The rate of inter-nal energy per current unit mass is given as
e =1
ρ(σD : D − p tr(D)) (48)
where p is the hydrostatic pressure, and tr(D) is thetrace of the velocity strain. And, the Mie-Gruneisenequation of state [20] considered here is given by
p(ρ, e) = (1 −1
2γµ)pH + γρe (49)
where µ = η − 1, η = ρ/ρ0, ρ0 is the material den-sity at the initial reference state, γ is the Gruneisenparameter and is given as
γ =
8<:
γ0ρ0
ρη > 1
γ0 η ≤ 1
(50)
γ0 is the Gruneisen parameter at the initial referencestate, pH is the Hugoniot pressure at the density ρand is given as
pH =
(a0µ+ b0µ
2 + c0µ3 µ > 0
a0µ µ ≤ 0(51)
a0, b0, c0 are material parameters fitted from theHugoniot curves for the uniaxial strain shock waveconditions [20].
Let σt+∆tD be the deviatoric Cauchy stress at time
t + ∆t obtained from the deviatoric part of the hy-drodynamic constitutive relation, a time integrationalgorithm for solving the hydrostatic pressure fromthe equation of state is given as the following.
Algorithm 1
For the time increment from t to t + ∆t, the hy-
drostatic pressure can be integrated from following
algorithm.
pt+∆t =Aeos +Beose
t+∆tρt+∆t
1 + 3Beosβ∆E t+∆tH
(52)
where
Aeos = (1 −1
2γµ)pH (53)
Beos = γ (54)
et+∆t = et +1
ρt+∆t
h(ασ
tD + βσ
t+∆tD ) : ∆E
t+∆t
−3αpt∆E t+∆tH
i(55)
∆Et+∆t =
Z t+∆t
t
Ddt′ (56)
∆E t+∆tH =
Z t+∆t
t
DHdt′ (57)
DH =1
3trace(D) (58)
σD = (1 − ϕ)eσD (59)
β = 1 − α (60)
Bulk Modulus
The bulk modulus needs to be derived from the equa-tion of state for the triaxiality function. The bulkmodulus K is derived as the following.
K = −∂p
∂trace(D)=dAeosdµ
(µ+ 1) + p (61)
where
dAeosdµ
= (1 −1
2γµ)
dpHdµ
−1
2γpH (62)
and
dpHdµ
=
(a0 + 2b0µ+ 3c0µ
2 µ > 0
a0 µ ≤ 0(63)
Artificial Viscosity
For numerical shock wave computation, the artificialviscosity is needed to stabilize the numerical oscilla-tion [21]. The classical artificial viscosity is adaptedas the following.
q =
(α1∆x
2ρD2H + α2∆xρDH DH < 0
0 DH ≥ 0(64)
where α1 and α2 are the artificial viscosity parame-ters, ∆x is the element characteristic length. There-fore, the internal energy rate and the Cauchy stressis modified as
e = σ : D − 3(p+ q)DH (65)
σ = σD − (p+ q)I (66)
5
American Institute of Aeronautics and Astronautics
3. Consistent Configuration Formulation
When the deformable bodies are undergoing finitedeformation, the weak form of the equation of mo-tion is represented for the Eulerian configuration asshown in equation (4). Since the equation of motionis formulated for the Eulerian configuration, equa-tion (4) can not be directly integrated. Equation(4) needs to be mapped onto a stationary referenceconfiguration to perform the integration. However,the material constitutive equation is embedded in theequation of motion. In computational finite deforma-tion mechanics, the corotational stress rate such asthe Jaumann rate [22] or the Green-Naghdi rate [23]are used to account for the reference frame indif-ference of the constitutive model [24]. When thecorotational stress rates for the constitutive equationare employed, equation (4) cannot be consistentlymapped onto a fixed reference configuration consis-tently along with the constitutive equation. Thisinconsistency of the configuration mapping will notonly introduce error into the computation but alsomay cause the Jacobian of the element to be neg-ative. As a consequence, the severely distorted el-ements need to be eroded or converted into finiteparticles before the damage of the elements reachesthe failure or fracture threshold value.
Our recent study [13, 14] shows that whenthe Truesdell stress rate [25, 26], which is a non-corotational stress rate, is employed to account forthe reference frame indifference of the constitutivemodel [27], the equation of motion along with con-stitutive equation can be consistently mapped onto aarbitrary reference configuration. The result is sum-marized in the following theorem.
Theorem 1
Let ∆t = t − tα. Given the configuration at time
tα and time t as xtα and xt, and the Cauchy stress
at time tα as σtα , with the one parameter family
assumption for the arbitrary reference configuration
In the mapping relation of the governing equa-tions between the configurations as equations (67)and (68), the Truesdell stress rate for the stress
term is implied as a consequence of the definition ofthe stress-power-per-unit-mass-conjugate pair withrespect to the reference configuration regardless ofthe constitutive equation. As a consequence, whena stress rate other than the Truesdell stress rate isemployed for the constitutive equation, the resultingconstitutive equation is not consistent with the map-ping relation of the governing equation between con-figuration as described in Theorem 1. Further notethat in the mapping relation of the governing equa-tion between configurations as described in Theorem1, the mapping relation between the strain rate of dif-ferent configurations satisfy the form of the Cotter-Rivlin rate [28] which is conjugate with the Truesdellrate for stress. Therefore, when a stress rate otherthan the Truesdell stress rate is employed for theconstitutive equation, there exists a mismatch be-tween the strain employed in the constitutive equa-tion and the strain resulting from the mapping re-lation of the governing equation between configura-tions as described in Theorem 1. We believe thisis one of the causes that finite elements suffer fromsevere distortion for deformable bodies undergoingfinite deformation.
We term the use of the Truesdell stress rate forthe constitutive equation and along with the map-ping relation of the governing equation between con-figuration as described in Theorem 1 as the consis-tent configuration formulation (CCF) for finite defor-mation analysis. The consistent formulation betweenthe constitutive equation and the mapping relationof the governing equation with respect to the config-urations will prevent introducing an additional errorin the finite element analysis, and hence help the el-ement for retaining proper shape and subsequentlyenables extending finite element method to its appli-cable limit for finite deformation analysis.
4. Numerical Examples
Three numerical examples are selected to demon-strate the present formulation: (i) the numericalsimulation of two identical elasto-plastic rectangu-lar bars that impact in hypervelocity is selected todemonstrate the robustness of the present consistentconfiguration formulation; (ii) the numerical simu-lation for the penetration of a blunt nose cylindri-cal steel projectile on a steel disk is selected for thehigh-velocity penetration problems; (iii) the numer-ical simulation of a small steel cylinder impact on alarge steel cylinder is selected for the hypervelocitypenetration problems.
4.1 Hypervelocity Impact of Two Identical
Elasto-Plastic Bars
One rectangular elasto-plastic bar with the impact-velocity of 2000m/s impacts on another identicalrectangular elasto-plastic bar at rest. The dam-age and the equation of state is not considered for
6
American Institute of Aeronautics and Astronautics
this example. The initial and the final configu-rations are shown in Figure 1. The size of eachrectangular bar is 0.0025m×0.0025m×0.05m, and ismeshed by 5×5×50 hexadrahedral elements. Thematerial of the rectangular has the Young’s modulusof 0.31TPa, yield stress of 0.85GPa, plastic harden-ing of 0.55GPa, Poisson ration of 0.3, and densityof 7750kg/m3. The constitutive relation is modeledby Truesdell stress rate hypo-elasto-plasticity withisotropic hardening [13, 14]. The total simulationtime is 0.13ms. At the end of the simulation, the twobars separate apart and the impact ends form a rect-angular shape with a 45 degree shift of the originalrectangular shape. The objective of the example is todemonstrate the effectiveness of the present formula-tion which can handle extreme large deformation ofthe elements. As a consequence, in the subsequenthigh- and hyper velocity penetration simulations, noelements need to be eroded due to element distortion.
4.2 High-Velocity Penetration of Plugging
This numerical example simulates a cylindrical pro-jectile impact on a thin plate disk. The projectile isa steel cylinder with diameter of 0.02m and lengthof 0.08m. The material constants of the projectileutilize Young’s modulus of 0.204TPa, Poisson ra-tion of 0.33, density of 7,850kg/m3 , yield stress of1.9GPa, and plastic hardness of 1.5GPa. The tar-get is a Weldox 460 E steel disk plate with diameterof 0.5m and thickness of 0.008m. The material con-stants for the target are taken as Young’s modulus of0.2TPa, Poisson ration of 0.33, and density of 7,850kg/m3. The von Mises tensile flow stress constantsof the Johnson-Cook model utilize A as 0.49GPa,B as 0.807GPa, C as 0.0114, n as 0.73, m as 0.94,Tmelt as 1800K, Troom as 293K. The Johnson-Cookfracture strain constants utilize D1 as 0.0705, D2 as1.732, D3 as -0.54, D4 as -0.015, and D5 as 0. Thematerial constants for the Lemaitre continuum dam-age model are taken ϕc as 0.99, εD as 0.8εF . Theprojectile impacts on the target with an initial ve-locity of 298ms−1 . The simulation results are shownin Figure 2. The projectile has a residual velocity of245ms−1 from the simulation, while the experimentgives a residual velocity of 241.4 ms−1 resulting invery good agreement.
4.3 Hyper-Velocity Penetration
The numerical example simulates a cylindrical steelprojectile impact on a cylindrical aluminum targetwith impact velocity of 6km/s. The projectile hasa diameter of 0.0071m and length of 0.00747m. Thetarget has a diameter of 0.032m and length of 0.02m.The material properties are selected as in Table 1.The simulation results are shown in Figure 3. Al-though there are no deep-impact experimental re-sults to validate the numerical simulation, the pre-sented approach provides a viable tool to using ex-
isting constitutive relations for the assessment of thehyper-velocity deep-impact penetration situations.
Concluding Remarks
In this exposition, we propose a consistent con-figuration formulation involving continuum damagemechanics based Lagrangian hydrodynamic com-putational framework for 3D high- and hyper-velocity contact/impact/damage/penetration appli-cations. The contributions of the paper includedthree aspects: (i) a consistent configuration formula-tion for constitutive relation and the dynamic equa-tion; (ii) a hydrodynamic constitutive relation in-cluding continuum damage mechanics; (iii) a robusttime integration algorithm for the coupled equationof state and the internal energy equation. Several nu-merical examples were performed to demonstrate therobustness and the accuracy of the proposed frame-work for the contact/impact/damage/penetrationapplications.
ACKNOWLEDGMENTS
The authors are very pleased to acknowledge supportin part by the Army High Performance ComputingResearch Center (AHPCRC) under the auspices ofthe Department of the Army, Army Research Lab-oratory (ARL) under contract number DAAD19-01-2-0014. The content does not necessarily reflect theposition or the policy of the government, and no of-ficial endorsement should be inferred. Other relatedsupport in form of computer grants from the Min-nesota Supercomputer Institute (MSI), Minneapolis,Minnesota is also gratefully acknowledged.
References
[1] V. V. Bashurov, G. V. Bebenin, Y. N.Bukharev, V. I. Zhukov, A. G. Loilev, N. V.Lapichev, A. L. Mikhailov, and G. S. Smirnov.Experimental Modeling and Numerical Simula-tion of High- and hypervelocity Space DebrisImpact to Spacecraft Shield Protection. Inter-national Journal of Impact Engineering, 20:69–78, 1997.
[2] L. Chen, J. Y. Wu, C. G. Feng, and Z. Z. Gong.The Numerical Simulation of Spherical Projec-tile Impact Spacecraft Protecting Structures atHypervelocity. Journal of Astronautics, 24:625–628, 2003.
[3] J. A. Zukas. Introduction to Hydrocodes. Studiesin Applied Mechanics. Elsevier, P. O Box 211,1000 AE Amsterdam, The Netherlands, 2004.
[4] D. Krajcinovic. Damage Mechanics: Accom-plishments, Trends and Needs. InternationalJournal of Solids and Structures, 37:267–277,2000.
7
American Institute of Aeronautics and Astronautics
[5] J. Lemaitre. A Continuous Damage Mechan-ics Model for Ductile Fracture. Journal of En-gineering Materials and Technology, 107:83–89,1985.
[6] J. Lemaitre. Coupled Elasto Plasticity andDamage Constitutive Equations. ComputerMethods in Applied Mechanics and Engineering,51:31–49, 1985.
[7] D. J. Celentano, P. E. Tapia, and J. L.Chaboche. Experimental and Numerical Char-acterization of Damage Evolution in Steels.Technical Report 2004-229, ONERA: Tire aPart, 2004.
[8] T. Børvik, O. S. Hopperstad, T. Berstad, andM. Langseth. Numerical Simulation of PluggingFailure in Ballistic Penetration. InternationalJournal of Solids and Structures, 38:6241–6264,2001.
[9] R. Kinslow, editor. High-Velocity Impact Phe-nomena. Academic Press, New York, USA,1970.
[10] G. R. Anderson. An Overview of the Theoryof Hydrocodes. International Journal of ImpactEngineering, 5:33–59, 1987.
[11] G. T. Camacho and M. Ortiz. Adaptive La-grangian Modeling of Ballistic Penetration ofMetallic Targets. Computer Methods in AppliedMechanics and Engineering, 142:269–301, 1997.
[12] X. Zhou, D. Sha, and K. K. Tamma. A Con-sistent Configuration Formulation for 3D FiniteStrain Impact Problems. Computer Methods inApplied Mechanics and Engineering (in review),2005.
[13] X. Zhou, D. Sha, and K. K. Tamma. Onthe New Concept and Foundations of an Ar-bitrary Reference Configuration (ARC) Theoryand Fourmulation for Computational Finite De-formation Applications - Part I: Elasticity. In-ternational Journal for Computational Meth-ods in Engineering Science and Mechanics (InPress), 2006.
[14] X. Zhou, D. Sha, and K. K. Tamma. Onthe New Concept and Foundations of an Ar-bitrary Reference Configuration (ARC) The-ory and Fourmulation for Computational Fi-nite Deformation Applications - Part II: Elasto-Plasticity. International Journal for Computa-tional Methods in Engineering Science and Me-chanics (In Press), 2006.
[15] X. Zhou and K. K. Tamma. Algorithms by De-sign with Illustrations to Solid and StructuralMechanics/Dynamics. International Journal for
Numerical Methods in Engineering (in press),2006.
[16] J. Lemaitre. A Course on Damage Mechanics.Springer-Verlag, New York, 1996.
[17] G. R. Johnson and W. H. Cook. Fracture Char-acteristic of Three Metals Subjected to VariousStrains, Strain Rates, Temperatures and Pres-sures. Engineering Fracture Mechanics, 21:31–48, 1985.
[18] D. J. Bammann, M. L. Chiesa, M. F. Horste-meyer, and L. I. Weingarten. Structural Crash-worthiness and Failure, chapter Failure in Duc-tile Materials Using Finite Element Simulations,pages 1–54. Elsevier, Amsterdam, 1993.
[19] J. M. Walsh and R. H. Christian. Equationof State of Metals from Shock Wave Measure-ments. Physical Review, 97:1544–1556, 1955.
[20] J. M. Walsh, M. H. Rice, R. G. McQueen,and F. L. Yarger. Shock-Wave Compressionsof Twenty-Seven Metals. Equation of State ofMetals. Physical Review, 108:196–216, 1957.
[21] J. von Neumann and R. D. Richtmyer. AMethod for the Numerical Calculation of Hy-drodynamic Shocks. Journal of Applied Physics,21:232–237, 1950.
[22] G. Jaumann. Geschlossenes System Physikalis-cher und Chemischer Differentialgesefze. SitzSer. Akad. Wein, (IIa), 120:385–530, 1911.
[23] A. E. Green and P. M. Naghdi. A General The-ory of an Elastic-Plastic Continuum. Arch. Rat.Mech. Anal., 18:251–281, 1965.
[24] X. Zhou and K. K. Tamma. On the Applicabil-ity and Stress Update Formulation for Corota-tional Stress Rate Hypoelasticity ConstitutiveModels. Finite Elements in Analysis and De-sign, 39:783–816, 2003.
[25] C. Truesdell. The Mechanical Foundations ofElasticity and Fluid Dynamics. Journal of Ra-tional Mechanics and Analysis, 1:125–300, 1952.
[26] C. Truesdell. Corrections and Additions to ”TheMechanics Foundations of Elasticity and FluidDynamics”. Journal of Rational Mechanics andAnalysis, 2:593–616, 1953.
[27] S. H. Lo. Stress Evaluation Algorithms for RateConstitutive Equations in Finite DeformationAnalysis. International Journal for NumericalMethods in Engineering, 26:121–141, 1988.
[28] B. A. Cotter and R. S. Rivlin. Tensors Asso-ciated with Time-Dependent Stress. Q. Appl.Math., 13:177–182, 1955.
8
American Institute of Aeronautics and Astronautics
(a) Initial configuration of two bars impacting
(b) Final configuration, t=0.13ms (note the separation)
(c) Final configuration, t-0.13ms
Figure 1: Two identical elasto-plastic bar impact in hypervelocity and the results demonstrate the robustness
and accuracy of the computational framework.
9
American Institute of Aeronautics and Astronautics
(a) Mesh (b) Initial configuration (Zoom in)
(c) t=20µs (d) t=60µs
Figure 2: Simulation of high-velocity plugging penetration.
