The strain rate effect of concrete

Page 1

The strain rate effect of concrete – Theory and a parameter study with the smoothed particle hydrodynamics (SPH) method

THE STRAIN RATE EFFECT OF CONCRETE

Franz Poelzl

KYUSHU UNIVERSITY DEPARTMENT OF CIVIL ENGINEERING STRUCTURAL ANALYSIS LABORATORY

INTERNSHIP 2016

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

Contents 1

Introduction .......................................................................................................................................... 2

2

Von Mieses yield criterion .................................................................................................................... 3

3

Drucker-Prager yield criterion............................................................................................................... 6 3.1

Drucker`s Postulate ....................................................................................................................... 9

3.2

Flow Rule ..................................................................................................................................... 10

3.3

Hardening Rule............................................................................................................................ 10

4

Introduction to the strain-rate effect ................................................................................................. 11

5

Description of the model .................................................................................................................... 12

6

Loading scenarios ................................................................................................................................ 13 6.1

Loading scenario 1, global strain rate 15.625 with v=3.125 [m/s].............................................. 14

6.2

Loading scenario 2, global strain rate 31.25 with v=6.25 [m/s].................................................. 16

6.3

Loading scenario 3, global strain rate 62.50 with v=12.5 [m/s].................................................. 18

7

Discussion of the results and outlook ................................................................................................. 20

8

Lists ..................................................................................................................................................... 21 8.1

9

List of Figures .............................................................................................................................. 21

References .......................................................................................................................................... 22

1

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

1 Introduction In order to describe plasticity, a yield criterion can be defined, which combines a spatial stress state with a material resistance value which gets obtained from uniaxial bending tests. Different yield criteria are fitting for different materials, for example the von Mieses yield criterion is in good accordance with steel whereas the Drucker-Prager criterion fits better for concrete due to a different ultimate tensile and compressive strength. It is also important to consider, that a yield criterion for a brittle material like glass is inappropriate. In that case, the principal stress is a more suitable parameter to assess the utilization rate of the material. Plasticity in von Mieses model is a pure shear process, which means that the shape of a body is undergoing a change but the volume stays constant during this process. If not, then it is not only a plastic process but also a hydrostatic process. A hydrostatic process changes the volume of a body but not it`s shape. Therefore, it is possible to draw the conclusion that a hydrostatic process can never cause yielding because yielding only can occur through a sufficient amount of shape change which is true for a isotropic hardening criteria. In the following paragraph, a short overview of the later on used tensor notation is given. Assuming Cartesian coordinates, the symmetric and quadratic 2nd order stress tensor is σ11 σ = σij = [σ21 σ31

σ12 σ22 σ32

σ13 σ11 σ23 ] = [σ12 σ33 σ13

σ12 σ22 σ23

σ13 σ23 ] σ33

Rotating σ in the Cartesian coordinate system must be done with the rotation matrix Q and the angle between the axes of the transformed and the untransformed coordinate system. cos⁡(x`x) cos⁡(x`, y) … Q = [cos⁡(y`, x) cos⁡(y`, y) …] … … … The transformation matrix Q has the following properties det(Q) = 1 QT = Q−1 Q ∗ QT = I In order to obtain σ`, which is any rotated stress tensor, the following operation must be done 2

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

σ` = Q ∗ σ ∗ QT = Q ∗ σ ∗ Q−1 The eigenvalues of the stress tensor, either σ` or σ are at the same time the principal stresses. The affiliated eigenvectors are pointing in the direction of the principal stresses. The maximum shear stress τmax which occurs, can be calculated directly from the maximum and minimum principal stress 1 τmax = (σmax − σmin ) 2

2 Von Mieses yield criterion The invariant J2 , which is the second invariant of the deviatoric part of the Cauchy stress tensor, must reach a certain value to fulfill the von Mieses yield criterion. The Cauchy stress tensor σij ⁡can be divided up into a deviatoric part sij ⁡and into a hydrostatic part πδij σij = sij + πδij Where π is the mean stress π=

σkk 1 = I1 3 3

where I1 is the first invariant of the Cauchy stress tensor. So the deviatoric part of the Cauchy stress tensor is

sij = σij −

σ11 − π sym σkk δij = [ σ21 ] σ22 − π 3 σ31 σ32 σ33 − π

As a side note, it is also important to see that τmax can be calculated from sij as well as from σij because 1 1 τmax = (σ11 − σ33 ) = (σ11 − π − σ33 + π) 2 2 The first invariant J1 of sij is J1 = skk = 0 This is because deviatoric stress is a state of pure shear where no volume changes are happening.

3

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

The second invariant J2 of sij is 1 J2 = sij sij 2 The third invariant J3 of sij is J3 = det⁡(sij ) The von Mieses stress, or effective stress is 1 3 σe = √3J2 = √3 ∗ sij sij = √ sij sij 2 2 2

2

2

3 1 1 1 σe = √ ((σ11 − (σ11 + σ22 + σ33 )) + 2 ∗ σ221 + 2 ∗ σ231 + (σ22 − (σ11 + σ22 + σ33 )) + 2 ∗ σ232 + (σ33 − (σ11 + σ22 + σ33 )) )⁡ 2 3 3 3

1 σe = √ ((σ11 − σ22 )2 + (σ22 − σ33 )2 + (σ33 − σ11 )2 + 6(σ221 + σ231 + σ232 )) 2 In the case of principal stresses, the von Mieses stress is 1 σe = √ (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 2 In the state of pure shear stress, every σkk and σ31 and σ23 must be zero which gives σv = √3σ12 If the yield strength σy of the considered material is equal to the effective stress σe , then yielding occurs. σy = σe It is important to note that σe is dependent from sij but not from⁡πδij , so a change in hydrostatic pressure does not affect the yield criterion. Or in other words, the yielding criterion of the von Mieses stress is J2 = k 2 Where k is the yield stress of the material in pure shear 4

Parameter study of the strain rate effect on concrete Kyushu University k=

Franz Poelzl Internship 2016

Ďƒy √3

Where Ďƒy is the yield strength of the material Ďƒv = Ďƒy = √3J2 So the relation is Ďƒy = √3J2 = √3k Or k 2 = J2 =

Ďƒ2y 3

The von Mieses yielding surface can be expressed as f = J2 − k 2 = 0 ‌ plastic f = J2 − k 2 < 0 ‌ đ?‘’đ?‘™đ?‘Žđ?‘ đ?‘Ąđ?‘–đ?‘? Because of Hooke`s law, a linear relationship between von Mieses stress Ďƒe â Ąand strain Îľ can be found pretty easily. The deviatoric part ξ̂ ij of the strain tensor Îľij is Îľij = Îľij − Ě‚

ξkk δ 3 ij

Alternatively it is possible to calculate the strain just from the Cauchy stress tensor Îľij =

1 ((1 + ν)Ďƒij − νδij Ďƒkk ) E

If ν = 0.5, which means the material is incompressible, then ξ̂ ij can be directly obtained, which is the deviatoric part of the strain tensor. The geometrical interpretation of the von Mieses yield criterion Ďƒy = Ďƒe leads to a so-called yield surface; in this case the “von Mieses yield surfaceâ€?, which has a shape like a smooth cylindrical surface. If the deviatoric stress vector is inside the von Mieses yield surface, then the material is in an elastic state, outside it is in a plastic state.

5

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

Figure 1; Geometrical interpretation of the von Mieses yield criterion [1]

3 Drucker-Prager yield criterion The Drucker-Prager yield criterion has the form of f = √J2 + αI1 − k = 0 1 f = √ sij sij + ασkk − k = 0 2 Where α, k ≥ 0 are material constants. Where f is the yield surface which is a function of pressure and J2 . The material constant α is also called “fitting coefficient”. The Drucker-Prager yield criterion becomes identical with the von Mieses yield criterion if ∝= 0 and k = σy √3

f = √J2 −

σy √3

=0

σy = √3J2

6

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

Considering a material like concrete, with a certain ultimate tensile strength σt and a certain ultimate compressive strength σc leads to the following two equations, where tension is positive and compression is negative f=

f=

σt √3 σc √3

+ ασt − k = 0 − ασc − k = 0

This gives subsequently α in terms of σc and σt σt √3

+ ασt − k =

σc √3

0 = −α(σt + σc ) +

α=

− ασt − k

1 √3

(σc − σt )

(σc − σt ) √3(σt + σc )

And also it gives k in terms of σc and σt k = σt (

1 √3

+

(σc − σt )

) √3(σc − σt )

Or k = σc (

1 √3

−

(σc − σt )

) √3(σc − σt )

Both expressions must be equivalent, which can be shown. For example, for σt = 3 and σc = 30, k = 3.1491 in both cases. So, the equation for the Drucker-Prager yield criterion can be expressed in terms of terms of σc and σt instead of α and k f = √J2 +

(σc − σt ) √3(σt + σc )

I1 − σt (

7

1 √3

+

(σc − σt )

)=0 √3(σc − σt )

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

Or f = √J2 +

(σc − σt ) √3(σt + σc )

I1 − σc (

1 √3

−

(σc − σt )

)=0 √3(σc − σt )

Usually, the material parameters α and k are obtained from material experiments. The Drucker-Prager model can be used for concrete and other pressure dependent materials.

Figure 2; Drucker-Prager yield function [2]

A different expression for the Drucker-Prager yield criterion with m =

f=

σc σt

is

m−1 m+1 1 √ ((σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 ) ∗ (σ1 + σ2 + σ3 ) + 2 2 2

If σc = σt , then

f=

0 2 1 1 ∗ (σ1 + σ2 + σ3 ) + √ (… ) = √ ((σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 ) 2 2 2 2

Which is nothing but the von Mieses yield criterion, so it demonstrates the convenience for materials with different ultimate tensile and compression strength, which cannot be expressed with the von Mieses yield criterion.

8

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

Figure 3; 3-Dimensional Drucker-Prager yield surface [3]

The Drucker-Prager model can be written in a more general way f(p, J2 ) = √J2 − α ∗ p − k(εp , εṗ , T, … ) = 0 where f is a function of the pressure p and J2 ⁡the second invariant of the deviatoric stress tensor whereas the pressure is always positive and is modified by the fitting coefficient α. The material parameter k can be a function of the plastic strain εp , of the plastic strain rate εṗ , the temperature T and various other parameter. For most engineering problems, neglecting all terms except from εp will lead to sufficient accurate results. Using the flow rule and looking at the pressure dependent part of the Drucker-Prager equation leads to ∂f ∂ 1 1 (−σkk )α = − (δik δjk )α = − δij α = ∂σij ∂σij 3 3 which basically means that the Drucker-Prager yield criteria is pressure dependent, because the term doesn’t vanish after deriving α ∗ p according to the plastic flow rule. Basically, this is dilatancy, which describes the phenomenon that plasticity is related to hydrostatic pressure which leads to plastic volume change which can be described with a so-called associated flow rule.

3.1 Drucker`s Postulate Two conditions must be fulfilled to meet Drucker`s Postulate. For a stable work hardening material, the net work must be positive

9

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

(σi − σ∗ )dεpi > 0 And the strain vector and the stress vector must point in the same direction dσdεp > 0

3.2 Flow Rule In general, assuming a plastic potential surface g which is independent from the yield surface f, the flow rule is p

dε = dλ

∂g = dλ∇g ∂σ

where dλ > 0 is a positive scalar which defines the vector length. By making the assumption, that g = f, the plastic flow rule now is compulsory the same and therefore independent from the yield surface  this is the so-called associated flow rule which is widely used. By the nature of the equation, it demands a smooth and convex yield surface. p

dε = dλ

∂f = dλ∇f ∂σ

By applying the associated flow rule, the associated von Mieses flow rule and the associated DruckerPrager flow rule can be obtained.

3.3 Hardening Rule A hardening rule is required in order to be able to describe the material behavior in plastic state. There are various hardening rule models out there, for example the     

perfect plasticity model isotropic hardening model kinematic hardening model mixed isotropic-kinematic hardening model slip theory and various others

For perfect plasticity, the yield surface does not change at all. Therefore, it is physically impossible for a material to go beyond its yield surface. For isotropic hardening, the shape of the yield surface is maintained, but it increases uniformly. That implies that with increasing tensile stress, the compression stress does so as well, which in general is not

10

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

true which can be cumbersome for complex loading paths. In fact, the Bauschinger effect cannot be taken into account.

4 Introduction to the strain-rate effect The strain rate effect considers the rate of strain which highly depends on the loading speed. In general, with increasing strain rate, the material strength increases as well, which means that the dynamic strength of a material is higher than its quasi-static strength. The increase of dynamic strength is different for each material type and depends significantly on the strain rate [1]. In the case of concrete, Bischoff and Perry assembled a huge quantity of testing results and derived from them an equation which can predict the dynamic strength increase in concrete. The strain-rate effect occurs due to an inertial confinement which means, there is a delay for the outer particles which have to be accelerated first, in order to exhibit the Poisson’s effect [2]. In Figure 4, the dynamic increase factor (DIF) according to the equation of Bischoff and Perry is shown for different concrete types. It can be seen, that the higher the static compressive strength đ?‘“̂′ đ?‘? is, the lower is the DIF. It should be noted, that above of a strain rate of đ?œ€đ?‘‘̇ >30đ?‘ −1 , the DIF increases much faster, which is especially the case for the tensile strength of concrete. đ??ˇđ??źđ??š =

đ?‘“̂′ đ?‘? đ?‘“ ′đ?‘?

đ?œ€đ?‘‘̇ 1.026đ?›ź =( ) đ?œ€Ě‡0

đ?‘“đ?‘œđ?‘&#x;â Ą|đ?œ€đ?‘‘̇ | ≤ 30đ?‘ −1 đ??ˇđ??źđ??š =

đ?‘“̂′ đ?‘? đ?‘“ ′đ?‘?

đ?œ€đ?‘‘̇ 1.026đ?›ź = đ?›žđ?‘ â Ą( ) đ?œ€Ě‡0

đ?‘“đ?‘œđ?‘&#x;â Ą|đ?œ€đ?‘‘̇ | > 30đ?‘ −1 đ?›ź = (5 + 9 ∗

đ?‘“ ′đ?‘? đ?‘“ ′0

−1

)

đ?›ž = đ?‘’ (6.156đ?›źâˆ’2) đ?‘“ ′ đ?‘? = 10â Ąđ?‘€đ?‘ƒđ?‘Žâ Ą Figure 4; DIF for the compressive strength of concrete according to CEB equation

11

đ?œ€Ě‡0 = 30 ∗ 10−6 â Ąđ?‘ −1

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

5 Description of the model For the parameter study, a concrete cube gets considered. The geometrical dimensions of the numerical specimen are constant during the parameter study whereas the length/width/height is 100/100/200 in millimeter. The vertical axis is z and is parallel to the vertical edges of the specimen. The concrete was chosen according to the European standard EN 1992-1, is called đ??ś40 and has an average statistical compression strength of đ?‘“đ?‘?,đ?‘š = 48.0[đ?‘€đ?‘ƒđ?‘Ž] whereas the tension strength đ?‘“đ?‘?đ?‘Ąđ?‘˜,0.95 = 4.6[đ?‘€đ?‘ƒđ?‘Ž]. The average statistical Young’s modulus is đ??¸đ?‘?đ?‘š = 35.0[đ??şđ?‘ƒđ?‘Ž] and the Poisson’s ration is assumed to be đ?œ? = 0.20[−] [3]. For the parameter study, the pink colored surface of the concrete specimen, which is depicted in Figure 5, is moving in the negative z direction with a constant velocity đ?‘Ł whereas different velocities thus also different global strain rates are calculated. In Figure 5, which illustrates the numerical model of the concrete specimen, the pink surface moves in –z direction with a constant speed đ?‘Ł. The bottom of the specimen is fixed in z direction which, the x and y direction are not fixed in order to avoid an overestimation in strength increase due to lateral strain confinement. Name width length height Compr.str. Tens.str. Young’s M. Poisson’s R.

Figure 5; The numerical concrete specimen

Abbr. w l h fcm fctk,0.95 Ecm ν

Value 100 100 200 48.0 4.6 35.0 0.20

Unit [mm] [mm] [mm] [MPa] [MPa] [GPa] [-]

Table 1; Overview of the specimen setup

The parameter study gets carried out with the aid of the SPH (smoothed particle hydrodynamics) method which in contrary to the classical FEM (finite element method) is a mesh-free method which has the big advantage over FEM that big distortions or cracks can be simulated in a much more stable way; which on the other hand can also lead to very unrealistic results. Moreover, the corner and edge zones

12

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

of the model do not have a good numerical quality because they have an impaired support area there. For more insight in the SPH method, an overview is given for example in [4].

6 Loading scenarios For the parameter study, different constant velocities đ?‘Ł are applied in negative z direction onto the whole surface of the specimen, for more insight details please refer to Chapter 5. Concrete has a high dynamic stiffness and can therefore withstand higher compression and tension stress under higher strain rates than under static loading scenarios. This aforementioned relationship of strain rate and stiffness is called strain rate effect, an outline of the strain rate effect is given in Chapter 0. Loading scenario 1 2 3

Constant loading velocity v [m/s] 3.125 6.250 12.500

Global strain rate sr [1/s] 1.5632E+01 3.1250E+01 6.2500E+01

Time step ts [s] 2.0E-09 1.0E-09 5.0E-10

Table 2; Loading scenarios for different velocities

In the subchapters, all loading scenarios which are described in Table 2 will be calculated. The data analysis will consider -for each time step- the maximum and minimum principal stress and strain in the model, but the values on the corners and the top surface and bottom surface of the SPH model are ignored. Furthermore, the principal stress and strain from the very center particle of the model which lies at � � ℎ 2 2 2

position đ?‘? = [ ; ; ] is analyzed as well. Sequences of principal stresses and strains will provide a quantitative overview and are helping to understand the highly nonlinear stress distribution over time while assessing the strain-stress diagrams. In the last chapter, a conclusive statement and a quantitative judgment about this parameter study is provided.

13

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

6.1 Loading scenario 1, global strain rate 15.625 with v=3.125 [m/s]

Figure 6;Global principal stress-strain extrema in the SPH model for loading scenario 1

Figure 7; Local principal stress-strain curve in the SPH model for SPH-particle 17225 for loading scenario 1

14

Parameter study of the strain rate effect on concrete Kyushu University ts=100000

Franz Poelzl Internship 2016 ts=100000 σ2

σ1

ts=100000 σ3

[MPa]

[MPa]

[MPa]

4

4

-61

-1

-2

-80

-8

-8-

-98

-14

-14

-116

-20

-21

-135

ts=75000 σ1

ts=75000 σ2

ts=75000 σ3

[MPa]

[MPa]

[MPa]

4

3

-48

-2

-2

-65

-7

-8

-82

-13

-14

-98

-19

-20

-115

ts=50000 σ1

ts=50000 σ2

ts=50000 σ3

[MPa]

[MPa]

[MPa]

6

6

-28

2

-2

-38

-1

-2

-48

-5

-5

-58

-9

-9

-68

ts=25000 σ1

ts=25000 σ2

ts=25000 σ3

[MPa]

[MPa]

[MPa]

6

5

-12

3

2

-18

0

0

-24

-3

-4

-30

-6

-7

-36

Figure 8; Vertical cut-through sequence of states of principal stress in the specimen for loading scenario 1

15

Parameter study of the strain rate effect on concrete Kyushu University

6.2 Loading scenario 2, global strain rate 31.25 with v=6.25 [m/s]

Figure 9;Global principal stress-strain extrema in the SPH model for loading scenario 2

Figure 10; Local principal stress-strain curve in the SPH model for SPH-particle 17225 for loading scenario 2

16

Franz Poelzl Internship 2016

Parameter study of the strain rate effect on concrete Kyushu University ts=100000

Franz Poelzl Internship 2016 ts=100000

ts=100000

σ1

σ2

σ3

[MPa]

[MPa]

[MPa]

12

11

-57

5

4

-77

-3

-3

-97

10

-11

-117

-17

-18

-136

ts=75000

ts=75000

ts=75000

σ1

σ2

σ3

[MPa]

[MPa]

[MPa]

19

18

-33

11

10

-60

4

3

-87

-4

-4

-114

-11

-12

-141

ts=50000

ts=50000

ts=50000

σ1

σ2

σ3

[MPa]

[MPa]

[MPa]

12

11

-24

6

5

-36

0

-1

-48

-6

-7

-60

-12

-13

-72

ts=25000

ts=25000

ts=25000

σ1

σ2

σ3

[MPa]

[MPa]

[MPa]

5

3

0

1

-3

-18

-3

-4

-37

-6

-8

-55

-10

-12

-73

Figure 11; Vertical cut-through sequence of states of principal stress in the specimen for loading scenario 2

17

Parameter study of the strain rate effect on concrete Kyushu University

6.3 Loading scenario 3, global strain rate 62.50 with v=12.5 [m/s]

Figure 12; Global principal stress-strain extrema in the SPH model for loading scenario 3

Figure 13; Local principal stress-strain curve in the SPH model for SPH-particle 17225 for loading scenario 3

18

Franz Poelzl Internship 2016

Parameter study of the strain rate effect on concrete Kyushu University ts=100000

Franz Poelzl Internship 2016 ts=100000 σ2

σ1

ts=100000 σ3

[MPa]

[MPa]

[MPa]

25

21

-49

12

9

-73

0

-3

-97

-12

-15

-121

-25

-27

-145

ts=75000 σ1

ts=75000 σ2

ts=75000 σ3

[MPa]

[MPa]

[MPa]

22

21

-2

12

10

-40

1

-2

-76

-10

-13

-114

-21

-25

-151

ts=50000 σ1

ts=50000 σ2

ts=50000 σ3

[MPa]

[MPa]

[MPa]

10

7

-1

2

-1

-37

-5

-8

-74

-13

-16

-111

-21

-23

-148

ts=25000 σ1

ts=25000 σ2

ts=25000 σ3

[MPa]

[MPa]

[MPa]

5

5

0

-4

-4

-34

-13

-13

-68

-22

-22

-101

-31

-31

-135

Figure 14; Vertical cut-through sequence of states of principal stress in the specimen for loading scenario 3

19

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

7 Discussion of the results and outlook In Table 3, a quantitative overview of the outcome of the parameter study is shown where đ?‘“đ?‘?,3đ??¸âˆ’3 is the compression stress at a global strain of 3E-3, đ?‘“đ?‘?,đ?‘šđ?‘Žđ?‘Ľ is the global maximum of compressive stress in the analysis, đ?‘“đ?‘Ą,1đ??¸âˆ’3 is the tension stress at a global strain of 1E-3 and đ?‘“đ?‘Ą,đ?‘šđ?‘Žđ?‘Ľ is the global maximum of tensile stress in the analysis. These values are set in relation to the “realâ€? and in the analysis occurring DIF and the theoretical CEB DIF respectively. The deviations from the analytical CEB factors are rather significant, especially for an increasing global strain rate. Further, they underestimate the tensile strength for lower strain rates. The reason could be, that the local strain rates are different from the global strain rates.

Table 3; Quantitative judgement of the parameter study

A strain rate effect can be observed from the parameter study, if looking at đ?‘“đ?‘?,đ?‘šđ?‘Žđ?‘Ľ which follows with a rather significant deviation the DIF pattern of the CEB equation. Looking at đ?‘“đ?‘Ą,đ?‘šđ?‘Žđ?‘Ľ , the DIF increases sharply in the numerical analysis from loading scenario 1 to loading scenario 3 which also is predicted by the CEB equation for a strain rate above â Ą|đ?œ€đ?‘‘̇ | > 30đ?‘ −1 .

20

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

8 Lists 8.1 List of Figures Figure 1; Geometrical interpretation of the von Mieses yield criterion [1] .................................................................6 Figure 2; Drucker-Prager yield function [2] ...................................................................................................................8 Figure 3; 3-Dimensional Drucker-Prager yield surface [3] .............................................................................................9 Figure 4; DIF for the compressive strength of concrete according to CEB equation ...................................................11 Figure 5; The numerical concrete specimen ................................................................................................................12 Figure 6;Global principal stress-strain extrema in the SPH model for loading scenario 1 ..........................................14 Figure 7; Local principal stress-strain curve in the SPH model for SPH-particle 17225 for loading scenario 1 ...........14 Figure 8; Vertical cut-through sequence of states of principal stress in the specimen for loading scenario 1 ...........15 Figure 9;Global principal stress-strain extrema in the SPH model for loading scenario 2 ..........................................16 Figure 10; Local principal stress-strain curve in the SPH model for SPH-particle 17225 for loading scenario 2 .........16 Figure 11; Vertical cut-through sequence of states of principal stress in the specimen for loading scenario 2 .........17 Figure 12; Global principal stress-strain extrema in the SPH model for loading scenario 3........................................18 Figure 13; Local principal stress-strain curve in the SPH model for SPH-particle 17225 for loading scenario 3 .........18 Figure 14; Vertical cut-through sequence of states of principal stress in the specimen for loading scenario 3 .........19

8.2 List of Tables Table 1; Overview of the specimen setup ...................................................................................................................12 Table 2; Loading scenarios for different velocities ......................................................................................................13 Table 3; Quantitative judgement of the parameter study ..........................................................................................20

21

Parameter study of the strain rate effect on concrete Kyushu University

Franz Poelzl Internship 2016

9 References

[1] "Wikipedia," [Online]. Available: https://commons.wikimedia.org/wiki/File:Yield_surfaces.svg . [Accessed 3 8 2016]. [2] "Wikipedia," [Online]. Available: http://download.autodesk.com/us/algor/userguides/mergedProjects/setting_up_the_analysis/No nlinear/Materials/Theoretical_Description_of_the_3-D_Drucker-Prager_Material_Model.htm. [Accessed 5 8 2016]. [3] "Wikipedia," [Online]. Available: https://en.wikipedia.org/wiki/Drucker%E2%80%93Prager_yield_criterion#/media/File:Drucker_Pr ager_Yield_Surface_3Da.png . [Accessed 2 8 2016]. [4] Y. Shui-sheng, L. Yu-bin and C. Yong, "The strain-rate effect of engineering materials and its unified model," Latin American Journal of Solids and Structures, 2013. [5] L. E. Schwer, "Strain Rate Induced Strength Enhancement in Concrete: Much ado about Nothing`?," 7th European LS-DZNA Conference, Windsor CA, 2009. [6] E. Standard, "EN 1992-1-1," Brussels , 2004. [7] J. Monaghan, "An introduction to SPH," Computer Physics Communications 48, Amsterdam, 1988.

22

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.