Dynamic Buckling of Functionally Graded Timoshenko Beam under Axial Load

Dynamic compressive strengths of concrete-like materials are usually obtained by conducting laboratory tests such as split Hopkinson pressure bar (SHPB) tests^[1-5].Many empirical formulae for dynamic increase factor (DIF) based on laboratory test data were reported in the related literature Ref.[6-9].

It has to be mentioned here that the SHPB test data available for concrete materials in the literature were very scattered^[7-10] due to the combined effects of strain rate, inertia and specimen size.More comprehensive investigation and discussion of the possible influencing factors can be also obtained in Ref.[1, 2, 7-11].The inertia (confinement) effect has been widely investigated experimentally^[12], theoretically^[13-14], numerically^{[5, 15]} and an empirical equation for the dynamic increase factor due to inertia (confinement) effect was suggested in Ref.[16], which took no account of the effect of concrete specimen size.As strain rate effects on the compressive strengths of concrete-like materials play an important part in the construction of material constitutive models which, in turn, exert a great deal of influence on the numerical simulations of concrete structures subjected to intense dynamic loadings, it is, therefore necessary to obtain the pure strain rate effect data by eliminating the data due to the effects of inertia (confinement) and specimen size from SHPB tests.

In the present study, numerical simulations with a rate-independent material model are carried out on the influence of specimen size in SHPB tests on concrete and a new empirical equation for the dynamic increase factors due to inertia (confinement) effect is proposed which takes account of specimen size effect through its volume.Comparisons are made between the results from the numerical simulation and those from the new empirical formula and discussed.

1. Material Model

The computational constitutive model for concrete developed in Ref.[1] is used to simulate the SHPB tests on concrete in the present study.This material model consists of equation of state (EOS), strength model including Lode effect, damage criteria and strain rate effects, etc.

1.1 Equation of State

The porous equation of state is used in the present study and can be expressed as^[17]

where μ=ρ/ρ₀-1 is the volumetric strain, in which ρ and ρ₀ are respectively the current and initial densities; α=ρ_s/ρ and α₀=ρ_s0/ρ₀ are the current and initial porosities in which ρ_s and ρ_s0 are the current and initial densities of solid (fully-compacted) material.

For μ > 0, concrete material is under compression condition

where K₁, K₂ and K₃ are the bulk moduli for fully compacted concrete material, p_crush is the pressure at which pore collapse occurs, p_lock is the pressure beyond which concrete material is fully compacted, n is the compaction exponent.

For μ < 0, concrete material is under tension condition.Hence, the pressure is

1.2 Strength Model

The strength surface of concrete can be written in the following form^[1]

where f_cc=f_c'δ_{m_t}η_c and f_tt=f_tδ_tη_t in which f_c' and f_t are the static compressive and tensile strength, δ_{m_t} and δ_t^[2] are the dynamic increase factors due to strain rate effects only in compression and tension, η_c and η_t^[18-20] are shape functions which represent shear damage and tensile softening of concrete respectively; B and N are empirical constants; r(θ, e)^[21] is the Lode effect with θ and e being the Lode angle, the ratio of the tensile meridian to the compression meridian, respectively.

By setting f_tt=0 and f_cc=f_c'r (residual strength), the residual strength surface for concrete can be obtained from Eq.(4)

Other information about shear, tensile damage, Lode effect and strain rate effect can be found in Ref.[1-2].

2. Specimen Size Effect

In this section, the commercial software LS-DYNA3D with user-defined subroutines is used to carry out the numerical study on the influence of specimen size in SHPB tests on concrete using a recently developed constitutive model for concrete^[1] by setting the material DIF (δ) equal to 1.The inertia effect dynamic enhancement factor (R_c) can be obtained by dividing the strength results of simulation by quasi-static strength.The values of various parameters used in the material model are listed in Table 1^[1].In this case, the numerically obtained strength increment of the studied specimen is attributed to the inertia (confinement) effect only.

Table  1.  Material parameters of concrete^[1]

Parameters of EOS									Parameters of constitutive model
ρ0/(kg·m-3)	ρs0/(kg·m-3)	pcrush/MPa	plock/GPa	n	K1/GPa	K2/GPa	K3/GPa		Strength surface						Shear damage					Tensile damage				Lode effect
									fc'/MPa	ft/MPa	B	N	G/GPa		λs	l	r	λm		c1	c2	εfrac		e1	e2	e3
2 400	2 680	15.2	3	3	13.9	30	10		45.6	3.8	1.7	0.7	10.5		4.6	0.45	0.3	0.3		3	6.93	0.007		0.65	0.01	5

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An SHPB system contains incident and transmitter pressure bars with a short specimen between them, as shown in Fig. 1.A stress pulse of trapezium shape is applied to the incident pressure bar as shown schematically in Fig. 2 in which the incident stress starts from 0, quickly rises to the peak value of p_peak at t₁, keeps as a constant for t₂, then drops back to 0 after t₃ and are summarized in Table 2.

Figure  1.  Finite element model of SHPB system

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Figure  2.  Loading function used in finite element analysis

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Table  2.  Load function parameters for direct compression analyses

t1/μs	t2/μs	t3/μs	ppeak/MPa
25	200	25	Varies

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Friction is an important factor which needs to be considered.However, in SHPB tests on concrete the effect of friction is negligibly small^[22] in terms of its contribution to the total dynamic increase factor and is usually ignored in numerical simulations^[23].Moreover, in SHPB tests measures are usually taken (i.e. by applying lubricant to both ends of concrete samples) to further reduce the effect of friction.Hence, in the present study the effect of friction is also ignored in the numerical simulations.

In this study, the dynamic strength increase factor is employed to reflect the effect of specimen size in SHPB tests on concrete.If there is no size effect there will be no difference for concrete samples with different sizes or volumes in terms of the dynamic increase factors at the same strain rate.Otherwise there will be size effect in SHPB tests on concrete.

Fig. 3 shows the numerical results for the dynamic increase factor due to inertia (confinement) effect R_c for concrete specimens with the same volume of different length/diameter ratios (i.e., Ø51 mm×51 mm, Ø64 mm×32 mm, Ø74 mm×24 mm) whilst Fig. 4 shows the numerical results for concrete samples with different volume of the same length/diameter ratios (i.e., Ø64 mm×32 mm, Ø80 mm×40 mm).It can be seen from Fig. 3 that the inertia effect is mainly related to the specimen volume and is insensitive to the length/diameter ratio when the volume and the material parameters of the specimens are kept the same.It also can be seen from Fig. 4 that the dynamic increase factor due to inertia (confinement) effect increases with the increase of the concrete specimen volume, which indicates that size effect does exist in SHPB tests on concrete as described above.

Figure  3.  Comparison of Eq.(6) with numerical results fordynamic increase factor due to inertia (confinement)effect for concrete specimens with the same volumeof different length/diameter ratios

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Figure  4.  Comparison of Eq.(6) with numerical results fordynamic increase factor due to inertia (confinement)effect for concrete specimens with different volumeof the same length/diameter ratio

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3. A New Empirical Equation

On the basis of the numerical results discussed in the previous section, a new empirical equation which takes consideration of the specimen size effect in SHPB tests on concrete is suggested here to describe the dynamic increase factor due to inertia (confinement) effect R_c by using the monotonically increasing and continuous properties of exponential functions in a simple and easy to use form, namely

where S, W, β are the constants to be determined numerically, ${\dot \varepsilon }$ is the strain rate, ${{{\dot \varepsilon }_0}}$ is the reference strain rate for a reference specimen usually taken to be ${{{\dot \varepsilon }_0}}$=1.0 s^-1, V is the volume of a concrete specimen under investigation, V₀ is the volume of a concrete sample with a reference size, say, Ø51 mm×51 mm.Set V=V₀ in Eq.(6), one obtains the dynamic increase factor due to inertia effect for the reference concrete specimen R_c0.

In the following, first, one determines the values of S and W in Eq.(6) using the numerical results for the concrete specimen with the same volume (i.e. V=V₀) as presented in Fig. 3 and, then, the value of β in Eq.(6) using the numerical results for the concrete samples with different sizes/volumes as given in Fig. 4.

Fig. 3 shows the comparison between the numerical results and Eq.(6) with S=6, W=2.8 and V=V₀ as indicated by the dashed line.It can be seen from Fig. 3 that reasonable agreement is obtained.Further examination reveals that the value of S in Eq.(6) should not be a constant but a function of strain rate.The slope of the curve is too small at the low strain rate, and too large at the high strain rate.By using the centrosymmetric properties of the hyperbolic tangent and the x-axis paralleled, S is taken as the following form to increase the slope of the curve at the low strain rate and decrease the slope at the high strain rate, namely

where F_i, W_i, S_i, G_i are the constants determined using the numerical results for concrete samples with constant volume as presented in Fig. 3, namely, F_i=6.0, W_i=2.8, S_i=0.8, G_i=8.5.The solid line in Fig. 3 is predicted from Eq.(6) with V=V₀ together with Eq.(7).It is clear from Fig. 3 that good agreement is obtained.

Fig. 4 shows the comparison between Eq.(6) with β=2.7 and the numerical results obtained for the concrete samples with different sizes/volumes.The values of all the other parameters in Eq.(6) are listed in Table 3.It is evident from Fig. 4 that the present model agrees well with the numerical simulations.

Table  3.  Values of various parameters in Eq.(6) and Eq.(7)

${{{\dot \varepsilon }_0}}$/s-1	Fi	Si	Wi	Gi	W	β
1.0	6.0	0.8	2.8	8.5	2.8	2.7

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In order to verify the validity of Eq.(6) more numerical simulations are performed on concrete specimens with different sizes. The numerical results are also presented in Fig. 4 and comparisons are also made between Eq.(6) and the numerical results.It can be seen from Fig. 4 that good agreement is obtained.

Fig. 5 shows the normalization of all the numerical results for concrete specimens with different sizes as given in Fig. 3 and Fig. 4 with respect to those for the reference concrete sample, namely, Ø51 mm×51 mm.It is clear from Fig. 5 that all the numerical results collapse into one line.It lends further support to the validity of Eq.(6) for the dynamic increase factor due to inertia (confinement) effect which takes into consideration the influence of specimen size.

Figure  5.  Variation of normalized numericallyobtained dynamic increase factor due to inertia(confinement) effect with strain rate

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4. Conclusions

The influence of specimen size in SHPB tests on concrete has been investigated numerically using a rate-independent material model.A new empirical equation for the dynamic increase factor due to inertia (confinement) effect has also been proposed which takes into account the specimen size effect through its volume.It is demonstrated that the new empirical formula agrees well with the numerical results for SHPB tests on concrete with different specimen sizes.