VIII International Conference on Fracture Mechanics of Concrete and Concrete Structures FraMCoS-8 J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang (Eds) A VISCOELASTIC RETARDED DAMAGE MATERIAL LAW FOR CONCRETE STRUCTURES EXPOSED TO IMPACT AND EXPLOSIONS ULRICH H ¨ AUSSLER-COMBE AND TINO K ¨ UHN Institute of Concrete Structures Technische Universit¨ at Dresden, 01062 Dresden, Germany e-mail: Ulrich.Haeussler-Combe;[email protected], www.tu-dresden.de/biwitb/mbau/ Key words: Concrete, Retarded Damage, Strain-Rate Effect, Dynamic Strength Increase, Spallation Abstract. The paper introduces a novel material model which includes the effects of dynamic strength increase of concrete. The model is based on the physical assumption of combined vis- cous effects and a retarded damage approach. Furthermore, this model is implemented in a Finite- Element-Method with implicit time integration and applied to numerical investigations of concrete bars exposed to impulse loading and wave propagation, respectively. Particular cases of direct tensile wave propagation and furthermore spallation of bars due to tensile failure are investigated with special respect to the dynamic tensile strength increase, dynamic failure mechanisms and crack energy. 1 INTRODUCTION Strength of concrete may exceed quasistatic values in case of high strain-rates caused by high velocity loading such as impacts or ex- plosions. This effect was experimentally val- idated in a number of experimental investiga- tions [2], [9]. It may have a considerable influ- ence on the behavior of concrete structures and lead to an increased dynamic load bearing ca- pacity compared to the quasistatic case. While the so called lateral confinement may contribute to an increased compressive strength, the in- crease of tensile strength must be caused by physical mechanisms regarding the concrete’s material structure. Two different physical phe- nomena are involved according to the current state of knowledge. While low strain rate ef- fects are dominated by moisture and the move- ment of water in the different capillary sys- tems of concrete [12], the damage at high strain rates appears to be dominated by inertia effects of micro-cracking [11], [4]. The micro-cracks cannot propagate arbitrarily fast as a displace- ment of internal crack faces relative to their im- mediate surrounding is involved. This leads to a retardation of crack propagation or retarded damage, respectively. A suitable framework is given with continuum mechanics and concepts of elasticity, viscosity, damage and plasticity. In order to compute the behavior of structures a macroscopic approach is appropriate. A number of proposals have been published for stress-strain relations to incorporate the strain-rate effect. A majority modifies strength parameters as have been determined under qua- sistatic conditions by dynamic strength increase factors according to experimental results, see, e.g., [8]. These are phenomenological ap- proaches and do not consider physical mech- anisms. Approaches based on viscoelasticity or viscoplasticity have been proposed by, e.g. [1]. Direct modifications of damage parameters ruled by the strain-rate were proposed by, e.g., [14]. First concepts of retarded damage used in stress-strain relation were given by [4], [6]. The following paper bases upon the latter works and combines damaged viscoelasticity with a retardation of damage to develop a gen- 1
Transcript
VIII International Conference on Fracture Mechanics of Concrete and Concrete StructuresFraMCoS-8
J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang (Eds)
A VISCOELASTIC RETARDED DAMAGE MATERIAL LAW FORCONCRETE STRUCTURES EXPOSED TO IMPACT AND EXPLOSIONS
ULRICH HAUSSLER-COMBE AND TINO KUHN
Institute of Concrete StructuresTechnische Universitat Dresden, 01062 Dresden, Germany
Abstract. The paper introduces a novel material model which includes the effects of dynamicstrength increase of concrete. The model is based on the physical assumption of combined vis-cous effects and a retarded damage approach. Furthermore, this model is implemented in a Finite-Element-Method with implicit time integration and applied to numerical investigations of concretebars exposed to impulse loading and wave propagation, respectively. Particular cases of direct tensilewave propagation and furthermore spallation of bars due to tensile failure are investigated with specialrespect to the dynamic tensile strength increase, dynamic failure mechanisms and crack energy.
1 INTRODUCTION
Strength of concrete may exceed quasistaticvalues in case of high strain-rates caused byhigh velocity loading such as impacts or ex-plosions. This effect was experimentally val-idated in a number of experimental investiga-tions [2], [9]. It may have a considerable influ-ence on the behavior of concrete structures andlead to an increased dynamic load bearing ca-pacity compared to the quasistatic case. Whilethe so called lateral confinement may contributeto an increased compressive strength, the in-crease of tensile strength must be caused byphysical mechanisms regarding the concrete’smaterial structure. Two different physical phe-nomena are involved according to the currentstate of knowledge. While low strain rate ef-fects are dominated by moisture and the move-ment of water in the different capillary sys-tems of concrete [12], the damage at high strainrates appears to be dominated by inertia effectsof micro-cracking [11], [4]. The micro-crackscannot propagate arbitrarily fast as a displace-ment of internal crack faces relative to their im-
mediate surrounding is involved. This leads toa retardation of crack propagation or retardeddamage, respectively. A suitable framework isgiven with continuum mechanics and conceptsof elasticity, viscosity, damage and plasticity. Inorder to compute the behavior of structures amacroscopic approach is appropriate.
A number of proposals have been publishedfor stress-strain relations to incorporate thestrain-rate effect. A majority modifies strengthparameters as have been determined under qua-sistatic conditions by dynamic strength increasefactors according to experimental results, see,e.g., [8]. These are phenomenological ap-proaches and do not consider physical mech-anisms. Approaches based on viscoelasticityor viscoplasticity have been proposed by, e.g.[1]. Direct modifications of damage parametersruled by the strain-rate were proposed by, e.g.,[14]. First concepts of retarded damage used instress-strain relation were given by [4], [6].
The following paper bases upon the latterworks and combines damaged viscoelasticitywith a retardation of damage to develop a gen-
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Ulrich Haussler-Combe and Tino Kuhn
eral triaxial material law including the strain-rate effect. A key point is given with the reg-ularization of the softening material behaviorwith the gradient damage approach and its ex-tension with respect to retarded damage.
The paper is organized as follows: Section 2develops the material law as a an ordinary dif-ferential equation combining stress and strainand their rates. This is incorporated in a Finite-Element-Method combined with a Newmark-Method for temporal discretization as describedin Section 3. It is applied to two particularcases of wave propagation: tensile wave prop-agation with continuously increasing loading inSection 4 and spallation with a moderate com-pressive amplitude reflecting as tensile wave inSection 5. Finally, some conclusions are givenin Section 6
2 THE CONSTITUTIVE LAW2.1 Quasistatic part
The constitutive law bases upon isotropicdamaged elasticity
σ = (1−D)E · ε (1)
with the stress tensor σ, the strain tensor ε, thelinear isotropic elasticity tensor E with an ini-tial Young’s modulus E0 and Poisson’s ratio ν0
as parameters. Scalar damage is measured byD with a condition 0 ≤ D ≤ 1. Damagedepends on a loading history. A strain basedapproach is chosen relating the strain state εwith an equivalent damage strain κ by a re-lation F (ε, κ) = 0. Furthermore, a relationD = D(κ) connects damage D and the equiv-alent damage strain κ. Finally, Kuhn-Tuckerconditions F ≤ 0, D ≥ 0, D F = 0 withthe time derivative D of D distinguish load-ing from unloading states. The exact forms ofF (ε, κ), D(κ) are given in [5]. The approachintroduces several material parameters beneathE0, ν0. These parameters rule nonlinear uniax-ial stress-strain behavior and multiaxial strengthproperties.
The incremental form of Eq. (1) is given by
σ = (1−D)E · ε− Dσ0, σ0 = E · ε (2)
with the time derivatives σ, ε, D of σ, ε, D.This is splitted into volumetric and deviatoricparts
σ = σvol + σdev (3)
whereby
σvol = K0
[(1−D)εvol − D εvol
]σdev = 2G0
[(1−D)εdev − D εdev
] (4)
with the initial bulk modulus and shear modulus
K0 =E0
3(1− 2ν0), G0 =
E0
2(1 + ν0)(5)
Due to D = D(κ) the time derivative D is con-nected to the time derivative κ of the equivalentdamage strain, which in turn is connected to εby F = 0 in case of loading [5].
2.2 ViscosityThe strain rate effect in the lower strain rate
range is covered by a viscous approach. It isapplied to the deviatoric part of Eq. (4). Thegeneral form for three parameter viscoelasticityis given by [10]
σdev = q1 εdev + q0 ε
dev − p0 σdev (6)
The Maxwell model will be used in the follow-ing leading to coefficients
q1 = 2(G0 +G1), q0 =2G0G1
η1
, p0 =G1
η1
(7)
The shear modulus G0 corresponds to Eq. (5).The shear modulus G1 and the viscosity η1
come into effect with larger strain rates. A highstrain rate or high viscosity leads to a higher re-sulting shear stiffness temporarily approachingG0 + G1. The viscoelastic approach Eq. (6) isextended with
σdev = (1−D) q1εdev − D q1ε
dev
+(1−D)2q0εdev − (1−D)p0σ
dev (8)
to describe damage. This particular form is cho-sen to include the quasistatic form Eq. (3) as aspecial case. The superposition with σvol ac-cording to Eq. (4) is straightforward and leadsto a relation for σ depending on ε, ε, D, D andfurthermore on εdev,σdev.
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Ulrich Haussler-Combe and Tino Kuhn
Standard Kelvin Modell
2G0
2G1 2η1
Standard Maxwell Model
2G02G1
2η1
Figure 1: Three parameter models for viscoelasticity.
2.3 Gradient damageA damage material law given by Eq. (1) is
characterized by a maximum stress or strength,respetively, followed by a softening, i.e. de-creasing stresses with increasing strains. Thisleads to localization phenomena within struc-tures. Usage of such a law within numeri-cal methods requires a regularization to avoida fundamental mesh sensivity. The gradientdamage approach will be used in the following.Thus, a nonlocal equivalent damage strain κ isemployed within this setting which is relatedto the local equivalent strain κ by a differentialequation
κ(x)− cκ ∆κ(x) = κ(x), cκ =R2
2(9)
with the Laplace differential operator ∆ and acharacteristic length R. A given field κ(x) withhighly localized values in a narrow band willlead to a field κ(x) localized in another bandwhose width is controlled by the value of R.Regarding the stress-strain law Eq. (1) of a ma-terial point in a position x the value κ replacesκ upon deriving damage D.
The parameter R is a measure of the ma-terial’s heterogeneity and assumed as a mate-rial constant. An approximately linear relationcan be derived between R and the crack energyGf [6]. This is used to choose appropriate val-ues for R.
Retardation of damage is assumed as anothercontribution to the strain rate effect beneath vis-cosity. This is modeled by an extension ofEq. (9) with an inertial like part with the secondtime dervative of the nonlocal equivalent strain¨κ and an mass-like parameter mκ [6]
mκ ¨κ(x) + κ(x)− cκ ∆κ(x) = κ(x) (10)
A model is given with Fig. 2 with a row ofsprings in parallel each with stochastically vary-ing strength. This yields the uniaxial stressstrain behavior with limited strength and sub-sequent softening. It is extended with inertialmasses which sustain forces in a short time pe-riod in case of spring failure. This effect is ruledby the value of mκ which is assumed as anothermaterial parameter for the strain rate effect be-neath G2, η2.
Basic Damage Model Retarded Damage Model
Figure 2: Models for damage and retarded damage.
3 THE NUMERICAL METHODDynamic equilibrium of a structure is de-
scribed by the virtual work principle∫VδuT · u ρdV +
∫VδεT · σ dV
=∫VδuT · b dV +
∫AtδuT · t dA (11)
with the Cauchy stress σ, body forces b, spe-cific mass ρ, acceleration u, virtual displace-ments δu, corresponding virtual strains δε, sur-face tractions t, the body’s volume V and thatpart of surface At with prescribed tractions.Boundary conditions are defined as prescribeddisplacements u on surface part Au and as sur-face tractions t on a surface part At.
This has to be complemented with a weakform for the differential equation (10) relatingκ to κ. It is given by [6]∫Vδκ ¨κ mκdV +
∫Vδκ κ dV
+∫V∇δκ · ∇κ cκdV =
∫Vδκ κ dV (12)
with the nabla operator∇ and a virtual variationδκ. Boundary conditions for nonlocal fields arestill on open research issue. According to awidely accepted approach a zero normal deriva-tiven · ∇κ = 0 of the nonlocal equivalent dam-age strain is assumed.
Eqns (11,12) form a base to apply the Finite-Element-Method. The fields of displacements
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Ulrich Haussler-Combe and Tino Kuhn
u(x) and nonlocal equivalent damage strainsκ(x) are used as independent variables. Theyare spatially discretized with(
u(x)κ(x)
)= N (x) ·U (13)
with a matrix N (x) of shape functions and avector UI collecting nodal values of displace-ments and nonlocal equivalent strains. The spa-tial derivatives are given by(
ε(x)∇κ(x)
)= B(x) ·U (14)
with a matrix B(x) of nodal derivatives ofshape functions. Applying standard meth-ods of discretization on Eqns. (11,12) usingEqns. (13,14) leads to system of nonlinear ordi-nary differential equations of 2nd order depend-ing on time t
M · U + f(U) = p(t) (15)
with the generalized mass matrix M , the ac-celeration U of nodal variables, the generalizedinternal nodal forces f nonlinearily dependingon the nodal variables U and the nodal loads pdepending on time t.
Temporal discretization of this system isperformed with the implicit Newmark-Method.This requires the evaluation of the tangentialstiffness [6]
K =∂f
∂U(16)
and leads to scheme1
β∆t2M ·
[U i − U i
]+Ki · [U i −U i−1]
= pi − f i−1(17)
with a time step ∆t, a time ti = i∆t, further-more U i = U(ti), Ki = K(U i), pi = p(ti),f i−1 = f(ti−1) and
U i = U i−1 + ∆t U i−1 (18)
+∆t2
2(1− 2β) U i−1 (19)
U i =1
β∆t2
[U i − U i
](20)
U i = U i−1 + ∆t[γU i (21)
+(1− γ)U i−1]
(22)
and integration parameters chosen with β =1/4, γ = 1/2 as a necessary requirement for nu-merical stability. The system of algebraic equa-tions (17) nonlinearly depending on U i may besolved with a Newton-Raphson method whileproceeding time step by time step.
This completes the discretization. Applica-tion examples for wave propagation problemsand plane strain beams under impact actions aredescribed in [7]. Two special cases will be con-sidered in the following: propagation of a uni-axial tensile wave along a bar to discuss dy-namic strength increase factors and reflectionof a uniaxial compressive wave at a bar’s endas tensile wave leading to spallation to discussaspects of dynamic crack energy.
4 UNIAXIAL TENSILE WAVE PROPA-GATION
We consider a linear elastic bar with Young’smodulus E and a specific mass % exposed touniaxial wave propagation. Its left end is givenwith a coordinate x = 0 and a right end withx = L and L = 1m. A discretization with fornode axis symmetric plane elements with an el-ement length of Le = 3mm and a time step ac-cording to the wave speed within the element∆t = 0.7 · 10−3ms is chosen for the for thenumerical computations. Aspects of an appro-priate selection are discussed in [7].
A tensile stress wave with a constant strainrate ε0 is induced on the left end with prescrib-ing the left end displacement u0 depending ontime t
u0(t) = −1
2ε0 c · t2, t ≥ 0 (23)
with a uniaxial wave speed c =√E/% leading
to a left end stress
σ = E ε0 · t (24)
A sequence of stress waves along the bar forseveral times is shown in Fig. 3 for E =36000 MN/m2, % = 24 kN/m3 for a concretegrade C40 according to [3] and ε0 = 1 s−1. The
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Ulrich Haussler-Combe and Tino Kuhn
linear elastic stress waves are characterized by aconstant slope according to the prescribed strainrate ε0. The tensile strength of concrete willobiviously be reached after a short period.
The course of stress waves changes if a non-linear material behavior with limited strength isassumed according to Section 2. An example isgiven with Fig. 4 where the Visco-Elastic Re-tarded Damage approach (VERD) was used forthe same concrete grade as before with materialstrain rate parameters mκ = 1 · 10−12s2, E2 =E, η2 = 2.8 · 10−8.
0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0012345678 s t r a i n r a t e = 1 ( 1 / s )
This starts with the same behavior as inthe linear elastic case. The initial linear elas-tic state is followed by a nonlinear harden-ing state whereby tensile stresses considerably
exceed the quasi static tensile strength fct =5.5 MN/m2 due to viscosity and gradient re-tarded damage. The material achieves a max-imum dynamic tension and switches into thesoftening branch.
These basically applies to all material pointsalong the bar but at different times and with adifferent extent due to nonlinear wave propaga-tion. Furthermore, the actually achieved strainrates apart from the left bar end differ from theprescribed nominal value ε0 because of the non-linear material behavior. Finally, a strain local-ization occurs in the left end bar region withmaximum values at x = 0. This is mesh inde-pendent due to the regularization approach. Us-ing this setup a variety of associated values ofstress, strain and strain rate can be determined.
��� ������ ������ ������ ����� �����������
V E R D r a t e 1 e - 5 r a t e 0 . 0 1 r a t e 0 . 1 r a t e 1 r a t e 1 0
s t a t i c s t r e n g t h
str
ess (
MPa)
s t r a i n ( - )
q u a s i s t a t i c s t r a i n r a t e = 1 x 1 0 - 6
v i s c o e l a s t i c i n c r e a s e
s t r a i n r a t e = 1 0
s t r a i n r a t e = 1
s t r a i n r a t e = 0 . 1 s t r a i n r a t e = 0 . 0 1
s t o p p e d d u e t o c a l c u l a t i o n t i m e
Figure 5: Uniaxial stress-strain relations (VERD) at dif-ferent strain rates.
This leads to uniaxial stress-strain curveswhich are parametrized by the strain rate. Ex-amples are shown in Fig. 5 for the viscoelas-tic retarded damage approach (VERD). First ofall, a higher stress i.e. a higher strength exceed-ing the quasi static tensile strength is reachedfor higher strain rates. Furthermore, the initialYoung’s modulus also increases due to the ac-tivation of the additional stiffness E2 in case ofhigher strain rates, see Fig. 1 ‘Maxwell’.
For a comparison the computed uniaxialstress-strain curves for retarded damage with-out viscosity (ERD) are shown in Fig. 6. Theachieved stresses are lower for the same strain
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Ulrich Haussler-Combe and Tino Kuhn
rate compared to VERD but still considerablyexceed the quasi static tensile strength. On theother hand, the initial Young’s modulus remainsunchanged as no additional stiffness is activatedwith the material model according to Fig. 2alone. Experimental data which might validatethis particular effect are rare and show a largescatter.
��� ������ ������ ������ ����� ����������� E R D
r a t e 1 e - 5 r a t e 0 . 0 1 r a t e 0 . 1 r a t e 1 r a t e 1 0
s t a t i c s t r e n g t h
stres
s (MP
a)
s t r a i n ( - )
q u a s i s t a t i c s t r a i n r a t e = 1 x 1 0 - 6
s t r a i n r a t e = 1 0
s t r a i n r a t e = 1
s t r a i n r a t e = 0 . 1
s t r a i n r a t e = 0 . 0 1
c o n s t a n t e l a s t i c m o d u l u s
Figure 6: Uniaxial stress-strain relations (ERD) at differ-ent strain rates.
More credible experimental data are avail-able for the maximum achieved tensile stress ordynamic tensile strength, respectively, varyingwith the strain rate. This leads to the dynamicstrength increase factor (DIF) as the relation be-tween dynamic strength and quasistatic strengthdepending on the strain rate.
1 0 1 D I F e x p e r i m e n t a l M a l v a r C 4 0 C E B E D E R D V E D V E R D
dyna
mic/s
tatic s
treng
th (lo
g MPa
)
s t r a i n - r a t e ( l o g 1 / s )
C E B , C 4 0
m o d i f i e d C E B( M a l v a r 9 8 , C 4 0 )
E R D
V E D
v i s c o u s s t r e n g t h i n c r e a s e
E D
V E R D
C E B 3 0 x 1 0 - 6
M a l v a r1 x 1 0 - 6
M a l v a r1 x 1 0 0
C E B 3 0
C E B 3 0 0
Figure 7: Experimental and computed dynamic tensilestrength increase factors.
A comparison of experimental and computedDIF-data is shown in Fig. 7 in a double loga-rithmic scale. Approximately a bilinear courseis given. The dynamic material parameterswere chosen with the same values as before.This choice approximately reproduces the DIF-recommendations of the CEB-Modelcode [3]with the computed values. Other choices mightlead to a better approximation of experimentalvalues.
5 SPALLATION
5.1 Basic relations
Up to now the variations of strength andYoung’s modulus were discussed under highstrain rate conditions. Another issue concernsthe crack energy. Crack energy may be definedas energy dissipated in the process of macrocrack creation. This energy is widely acceptedas material constant under quasi static condi-tions.
An appropriate setup to determine the crackenergy under high strain rate condition is givenby the spallation experiment. A uniaxial com-pressive stress wave is induced on the left endof a bar of length L. It is assumed as half sineshaped and its maximum stress value is far be-low the compressive strength but considerablyabove the tensile strength. On the right end itis reflected as tensile wave. In case of linearelastic wave propagation the reflection processis characterized by stress shapes as shown inFig. 8, i.e. upon reaching the right end the stressamplitude reduces, goes through zero and in-creases to its original value with reversed sign.
During the reflection process the right end ofthe bar, which initially has a zero displacement,is moved to the right. The associated velocityis shown in Fig. 9 varying with the same timet. The time of the maximum velocity corre-sponds to the time when the stress wave passesthrough zero, see Fig. 8. Upon velocity reduc-tion the amplitude of the stress wave increasesas has been described before. Thus, a relationcan be derived between velocity reduction andstress amplitude.
r e f l e c t e d w a v e a t 2 xw a v e t r a n s i t i o n t i m e
w a v e a t w a v e t r a n s i t i o n t i m e
Figure 8: Reflection of compressive stress wave as tensilewave on free end.
Such a relation basically also holds in case ofa material with limited tensile strength. In casewhen the tensile stress wave amplitude reachesthe uniaxial tensile strength a fragment of thebar will break apart on the right end and willstart to fly away in the right direction. This oc-curs at a time t1 whose value is needed for latercalculations. The velocity of the right end ofthe fragment will than stop its decelaration andholds some remaining value, see Fig. 9.
v e l o c i t y w a v e a t w a v e t r a n s i t i o n t i m e
r e f l e c t i n g w a v e s
v e l o c i t y w a v e a t 2 x w a v e t r a n s i t i o n t i m e
Figure 9: Reflection of velocity wave on free end.
The absolute difference between this valueand the maximum velocity is called pull-back-velocity ∆vvp. A relation between the pull-back-velocity and the corresponding stress am-plitude or dynamic uniaxial tensile strength ft,dis given by
ft,d =1
2% c∆vvp (25)
with the specific mass % and the wave speed c.This relation serves to determine values of thedynamic tensile strength ft,d in spallation exper-iments.
Assuming a linear elastic tensile behavior upto the moment t1 of tensile failure for a givenstress wave and given value ∆vvp also allows tocalculate the spatial point x1 where the failureoccurs, i.e. the fragment length. Furthermore,the velocity v(x, t) along the fragment may becomputed for the time t1 leading to an impulse
I1 =
∫ L
x1
v(x, t1) %Adx
≈ 1
2%A(L− x1) [v(x1, t1) + v(L, t1)]
(26)
A analogous relation holds for a later time t2when a macro crack has fully developed in thecross section x1 leading to an impulse I2.
During the spallation process a stress σ istransmitted over the cracked cross section dueto the formation of a crack band and the soften-ing stress-strain behavior of the material. Thisis related to the difference of impuls by∫ t2
t1
σ Adt = I1 − I2 = ∆I (27)
The stress starts with the dynamic tensilestrength ft,d and ends up with zero after the for-mation of a macro crack. On the other hand,stress and crack energy are connected by
Gf,d =
∫ δ2
0
σ Adδ
=
∫ t2
0
σδ Adt ≈ δmean
∫ t2
0
σ Adt
(28)
with a variable crack width δ, a crack width δ2
at time t2 and the crack width velocity δ. Thisleads to
Gd,f = δmean ∆I (29)
Finally, the velocity of the crack width remainsto be determined. It is derived from the velocityof the fragment’s left end
δ(t) = v(x1, t)− v(x1, t1) (30)
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Ulrich Haussler-Combe and Tino Kuhn
The mean velocity may be approximated by
δ(t)mean ≈1
2
[δ(t2) + δ(t1)
]=
1
2δ(t2) (31)
This completes a method which allows the ex-perimental determination of the high strain ratecrack energy from measured values of veloci-ties [13]. It bases on the knowledge of the stresswave. This may be determined in a Hopkinson-Bar setup with the specimen connected to anincident bar only. Furthermore, a linear elas-tic behavior up to the point of tensile failure isassumed.
Whether this assumption is valid may tosome degree be controlled by comparing themeasured fragment length and the theoreticallength L − x1. Moreover, this method relieson correctly measured velocities which may beachieved with high speed cameras.
A somehow crucial point exists with the de-termination of the time t2, i.e. the time when thecrack has fully developed to become a macrocrack and does not transmit stresses anymore.
stresses which are not or not directly acces-sible to experimental measurement. But suchmethods at least require an assumption aboutmaterial models and the specification of theirparameters.
The method as it has been described in theprevious section may serve to link experimentalinvestigations and numerical simulations.
The proposed material formulation and thenumerical method are applied to a spallationtest simulation. An axisymmetric formulationis used to describe a long cylindrical specimenaccording to the experimental setup in [13].
The specimen of 0.250 m length and 0.075m diameter was discretized with 1485 axisym-metric four node square elements which leadsto approx. 2.5 mm mesh size.
The specimen was pressure loaded from theleft end with a half sine wave of amplitude 18MPa and 0.1 ms duration. It is free in motionand a wave reflection occurs at the right end.
The time step of the Newmark method waschosen according to explicit solution methodswith 0.8 of the maximum wave transition timewithin an element.
A C40 concrete parameter set was consid-ered according to Section 4 which leads to awave transition time through the specimen of 65µs and a time step size of 0.5 µs. The Poisson’sratio is assumed with 0 to avoid spurious sec-ondary waves.
Fig. 10 illustrates the stress distributionalong the specimen at different time steps forthe VERD material formulation. The pressurewave is travelling through the specimen and re-flects at the free end into a tensile wave movingback. (see section 5.1).
- 0 . 1 0 - 0 . 0 5 0 . 0 0 0 . 0 5 0 . 1 0- 2 0
- 1 5
- 1 0
- 5
0
5
stres
s (MP
a)
p o s i t i o n ( m )
p r i m a r y p r e s s u r e w a v et i m e o f r e a c h i n gt h e f r e e e n d
r e f l e c t e d p r e s s u r e w a v e
t i m e o f r e a c h i n g t h e m a x i m u m t e n s i l e s t r e n g t h
r e m a i n i n g d a m a g e dt e n s i l e w a v er e m a i n i n g d a m a g e d
p r e s s u r e w a v e
Figure 10: Stress wave distribution along the specimenscenter axis.
The tensile wave amplitude exceeds thestrength of the material and tensile damage oc-curs nearly at the center of the specimen accord-ing to the incoming impulse length.
The specimen than spalls at this position intotwo separate pieces while the remaining internalstress waves are still propagating and reflectingin both parts.
Fig. 11 shows the corresponding displace-ment distribution at different time steps. Thespallation time is reached at 110 µs with the ini-tiation of the separation.
The mean speed of the left part with 1.38m/sec is less than 1.89 m/sec for the right part atthis time and the secondary part will fly away.
8
Ulrich Haussler-Combe and Tino Kuhn
The softening process takes approximately30 µs, after this time the specimen is fully sep-arated with the remaining mean part velocitiesof 2.13 m/s and 1,96 m/s. This indicates a gapbetween both.
The corresponding local strain distributionis illustrated in Fig. 12 and shows a widelyspanned zone between both parts. At the posi-tion -0.04 m a strain peaks can be recognized asa strong starting localization in this area leadingto a ”separation” at the respective element. Thisis a well known numerical effect due to missingregularization and does not reflect some phys-ical behavior. To avoid this effect the damageformulation is coupled to a nonlocal strain vari-able as introduced in Section 2.3.
c r a c k i n i t i a t i n g s t r a i nd i s t r i b u t i o n
Figure 12: Local strain distribution.
The corresponding nonlocal strain distribu-
tion is shown in Fig. 13 at the same time steps.One can see that the regularization proceduresmoothes the maximum strain values. The max-imum strain can now be recognized at approxi-mately x=-0.03 m at the left side from the centerof Fig. 13.
c r a c k i n i t i a t i n g s t r a i nd i s t r i b u t i o n
s p a l l p l a n e
Figure 13: Nonlocal strain distribution.
The spallation plane is located at the posi-tion of maximum strain and leads to two partsof 84 mm and 166 mm length. Using Eq. (25)the tensile strength is calculated with 4.3 MPawith a pullback velocity ∆vvp = 0.78 m/sec.Furthermore, the knowledge of the separationplane and the masses of the remaining pieces al-lows for calculating the dynamic crack energyfrom the transferred impulse by Eq. (29). Inthis particular case the dynamic crack energy 27N/m is much less than the expected experimen-tal value according to [13] of approximately 150N/m. The difference is probably caused by theassumed viscous damage evolution and regu-larization parameters which are based on quasistatic experimental data. Especially this last is-sue needs more investigations.
A computed stress-strain relation is shown inFig. 14 for the most damaged element. It has theregimes of (1) the compressive load for the inci-dent pressure wave, followed by (2) the unload-ing branch due to the wave reflection with rever-sal to tension. It reaches the tensile strength at(3) followed by a softening part and is again (4)unloading the not fully damaged material. It un-dergoes further a secondary compressive load at
9
Ulrich Haussler-Combe and Tino Kuhn
(5) with reaching the maximum stress at (6) andagain softens at the compressive domain (7) tofull damage, which finally leads to (8) the largestraining part which indicates the gap.
Figure 14: Stress strain evolution of the most damagedelement of the VERD model.
The assumption of a constant homogeneouswave traveling through the specimen introduc-ing a ductile crack at a predefined position,underlies all experimental investigations. Thismight be insufficient as it is indicated by thenumerical simulations. Fig. 15 shows the evolu-tion of the strain-rate with time at different posi-tions. The strain-rate is not constant as assumedin experimental observations and rather showsvarying values during the cracking process.
s p u r i o u s v a l u e s a f t e r f u l l l o c a l d a m a g e
time o
f loca
l dam
age
t h e o r e t i c a l e l a s t i c w a v e t r a n s i t i o n t i m e
s t r a i n r a t e a t t h e s p e c i m e n s e n t r a n c e
c e n t e r p o s i t i o n
Figure 15: Strain rates distribution at different positions.
If the strain-rate varies with time and posi-tion the local strength will be different depend-
ing on both. The viscous and the retardationpart of the material model introduce this strain-rate variation leading to the inhomogeneousdamage within the specimen. The strain-ratefor this particular simulation with the VERDformulation spans from 18 1/s to 24 1/s duringthe main cracking process instead of 30 1/s forpurely linear elastic considerations.
These prior observations were made withthe VERD material formulation leading to timeand space dependent damage behavior as de-scribed before. Additionally, Fig. 16 comparesthe stress evolution at some representative po-sitions for the linear elastic (E) and the visco-elastic (VE) material model without damage.
e l a s t i c f o r m u l a t i o n v i s c o e l a s t i c f o r m u l a t i o n
t h e o r e t i c a l e l a s t i c w a v e t r a n s i t i o n t i m e
s t r e s s r e l a x a t i o n e f f e c t
p o s i t i o n l e f t s p e c i m e n c e n t e r
Figure 16: Stress distribution at 4 different positions forthe E and VE formulation.
The viscous part increases the elastic stiff-ness with increasing strain-rate. Thus, the wavespeed increases and the traveling time throughthe specimen reduces. The dynamic elasticmodulus increases from 36 GPa for the linearelastic case to 50 GPa for the visco-elastic caseand the wave speed increases to 4677 m/secas mean value of the specimen. Furthermore,it can be seen that the viscous stress evolu-tion varies with the position. The stress wavechanges in shape and speed. This probablyleads to some overlaid inhomogeneous bounc-ing with time and the observed oscillation.
The same effect can be recognized by intro-ducing the damage part according to the elasticdamage model (ED), the visco-elastic damage
10
Ulrich Haussler-Combe and Tino Kuhn
model (VED), the retarded damage model with-out (ERD) and with viscous part (VERD), seeFig. 17.
s t r e n g t h i n c r e a s ed u e t o r e t a r d a t i o n
stres
s (MP
a)
t i m e ( u s )
p r e s s u r e l o a d E D V E D E R D V E R D
t h e o r e t i c a l e l a s t i c w a v e t r a n s i t i o n t i m e
v i s c o - e l a s t i c s o u n d -s p e e d i n c r e a s e
c e n t e r
v i s c o - e l a s t i c s t r e n g t h i n c r e a s e
s t r e s s v a l u e s f o r t h e E D a n d E R D f o r m u l a t i o n a r e o v e r l a i d
Figure 17: Stress distribution at the center of the speci-men for the 4 damaged formulations.
The elastic (ED) formulation reacts with amaximum tensile strength equal to the static as-sumption of approx. 3.5 MPa while the viscous(VED) one increase to 4.3 MPa. The corre-sponding mean strain rate reduces in this casefrom 30 1/s to 11 1/s. Introducing the damageretardation part, the viscous influence remainsunchanged and the same kind of oscillation canbe found as has been described before.
������� ������� ��� ������ ���������
���
���
��
�
�
r e t a r d e d s o f t e n i n g
E D V E D E R D V E R D
t e n s i o n
c o m p r e s s i o n
s t a t i c s t r e n g t h
stres
s (MP
a)
s t r a i n ( - )
e l a s t i c s o f t e n i n g
V E D , V E R D
E D , E R D
Figure 18: Overlaid stress strain relation for the 4 dam-aged formulation.
The superimposed stress-strain relations inFig. 18 for the position with the largest stressvalues may explain this effect more clearly. The
none-viscous formulations react with a full sep-aration in the tensile domain which correspondsto a brittle behavior.
The viscous contributions on the other handlead to a more bearable strength in tension and anot fully damaged state during the first unload-ing branch. Full damage is finally reached in thenext loading cycle in the compressive domain.
V E R D s e c o n d a r y d a m a g ei n c o m p r e s s i o n
o s c i l l a t i o n d u e t o i n e r t i a e f f e c t s
Figure 19: Relative damage.
This behavior can be summarized with intro-ducing a global differential damage evolutionparameter stepwise holding the actual damageincrease. Fig. 19 compares this parameter forthe four damage formulations. It can be seen,that the non-viscous formulations react with ashort high damage rate and the viscous formu-lations with a lower, more ductile first part, fol-lowed by the secondary damage increase.
6 CONCLUSIONSA novel material model with viscoelastic re-
tarded damage is discussed in this paper. Theviscous part of the formulation leads to a mod-erate strength increase with increasing strainrate at lower values up to approx. 1 1/s. The re-tardation part assumes crack opening inertia ef-fects at higher rates and significantly increasesthe virtual strength in this domain. While theformulation of the stress-strain relations is fullytriaxial the model is applied to particular caseof uniaxial wave propagation. The three mate-rial parameters may be calibrated such that the
11
Ulrich Haussler-Combe and Tino Kuhn
typical course of dynamic strength increase fac-tors as given by recommendations and experi-mental investigations may be approximated to adesired degree. Numerical simulations of spal-lation experiments exhibit some discrepanciesto experimental investigations. On one handthese discrepancies have to be contributed to animprovable parameter calibration of the mate-rial model, on the other hand the simulationsreveal complex spallation mechanisms whichseem not to be adequately regarded in the setupand processing of spallation experiments.
REFERENCES[1] F. Barpi. Impact behaviour of concrete:
a computational approach. Engineer-ing Fracture Mechanics, 71:2197–2213,2004.
[2] P.H. Bischoff and S.H. Perry. Compres-sive behavior of concrete at high strainrates. Materials and Structures, 24:425–450, 1991.
[3] Comite Euro-International de Beton.CEB-FIP Model Code 1990. ThomasTelford, London, 1993.
[4] J. Eibl and B. Schmidt-Hurtienne. Strain-rate-sensitive constitutive law for con-crete. Journal of Engineering Mechanics,,125:1411–1420, 1999.
[5] U. Haussler-Combe and J. Hartig. Formu-lation and numerical implementation of aconstitutive law for concretewith strain-based damage and plasticity. Interna-tional Journal of Non-Linear Mechanics,43:399–415, 2008.
[6] U. Haussler-Combe and M. Kitzig. Mod-eling of concrete behavior under highstrain rates with inertially retardeddam-age. International Journal of Impact En-gineering, 36:1106–1115, 2009.
[7] U. Haussler-Combe and T. Kuehn. Model-ing of strain rate effects for concrete with
viscoelasticity and retardeddamage. Inter-national Journal of Impact Engineering,2012.
[8] T. Holmquist, G. Johnson, and W. Cook.A computational constitutive model forconcrete subjected to high strainrates andhigh pressures. In Proc. 14th Interna-tional Symposisum on Ballistics., pages591–600, Quebec, 1993.
[9] L. J. Malvar and C. A. Ross. Reviewof strain rate effects for concrete in ten-sion. ACI Materials Journal, 95:735–739,1998.
[10] L. E. Malvern. Introduction to theMechanics of a Continuous Medium.Prentice-Hall, Englewood Cliffs, New Jer-sey, 1. auflage edition, 1969.
[11] H. W. Reinhardt and J. Weerheijm. Tensilefracture of concrete at high loading ratestaking account of inertiaand crack velocityeffects. International Journal of Fracture,51:31–42, 1991.
[12] P. Rossi. A physical phenomenon whichcan explain the mechanical behaviour ofconcreteunder high strain rates. Materialsand Structures, 24:422–424, 1991.
[13] H. Schuler, Ch. Mayrhofer, and K. Thoma.Spall experiments for the measurement ofthe tensile strength and fractureenergy ofconcrete at high strain rates. InternationalJournal of Impact Engineering, 32:1635–1650, 2006.
[14] A. Suffis, T. A. A. Lubrecht, andA. Combescure. Damage model withdelay effect: Analytical and numericalstudies of theevolution of the characteris-tic damage length. International Journalof Solids and Structures, 40:3463–3476,2003.
