Modeling of concrete behavior under high strain rates with inertially retarded damage

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Abstract

The paper proposes a novel approach to model the influence of high strain rates on the behavior of quasi-brittle materials like concrete. It is based on gradient continuum damage, where the gradient part is extended with an inertia of damage. This causes a retardation of damage due to the fact that micro-cracks cannot spread out arbitrarily fast. The application is demonstrated with uniaxial tensile wave propagation and for a plane stress case. Increasing strain rates lead to an expansion of the linear stress–strain behavior with stresses exceeding the quasistatic material strength.

Introduction

The increase of concrete strength under high strain rates is important for extraordinary design situations, e.g. impact of vehicles and airplanes or blast waves from explosions or contact detonations upon concrete structures like bridges, offshore structures, tanks, chemical factories, power plants. A number of experimental investigations have been performed to study this effect, which can be observed for the compressive strength [1], and more pronounced for the tensile strength [2], see Fig. 1. Even in experiments it may become difficult to distinguish material behavior from structural system behavior, especially in the high dynamic range. Thus, inertial lateral confinement has been argued as a reason for strength increase. But this particular influence seems to play a role only for extremely high strain rates larger than 200 s−1 [4].

Generally, experiments should measure material properties and the influence of the experimental setup should be minimized as far as possible. Due to its heterogeneous structure this requirement is difficult to fulfill for concrete. A widely accepted experimental method is given with the Split-Hopkinson-bar [5], which up to now seems to be the most reliable measurement technique for material behavior under strain rates up to a range of 103 s−1 [6]. Extensive SHB investigations for concrete were performed, e.g. by Ref. [7], [8], [9], [10], [11], [12], [13]. Reliable results from experimental investigations are the basis for constitutive laws. A wide range of models have been proposed for concrete, which can be classified as microscopic, mesoscopic and macroscopic in a first approach. While microscopic and mesoscopic models distinguish the concrete constituents in different orders of resolution, macroscopic models assume a homogeneous material. This allows the application of the methods of classical continuum mechanics and makes macroscopic models suitable for calculations of whole structures. The macroscopic approach will be used in the following. Constitutive laws for high strain rates are generally formulated as extensions of laws for the quasistatic case. Following basic concepts have been proposed:

  • Quasistatic failure surfaces are enlarged depending on the strain rate [14], [15]. The enlargement factor is calibrated according to results of experimental investigations. This proposal is empirical and does not include a physical background.

  • Elastoplastic stress–strain relations are extended with rate dependent viscous parts, see, for instance [16], [17], which temporarily leads to stresses beyond quasistatic failure conditions. This can physically be motivated by the resistance of a rapid movement of fluid phases within the microstructure of concrete. Beneath describing strain rate influence, this approach also leads to a problem regularization in the softening range of the material behavior.

  • Consideration of the damage rate in damage evolution laws [18]. This leads to a delay effect for damage. The influence of this approach on strains and stresses has not been investigated in detail up to now.

All these approaches are directly coupled to rates of strains or stresses, i.e. a potential dynamic stress increase vanishes in the instant of strain or stress maximum values. This particular model behavior seems not to be fully reasonable. An alternative bases on the assumption that the activation of damage is retarded by inertial effects arising with micro-cracking. Ref. [19] implement this basic approach with a local dynamic relaxation for damage, which is derived from rheological models including micro-mass elements. This decouples stresses from strain rates to some extent and leads to increasing dynamic stresses also for vanishing strain rates, but some complexity arises with the selection of the relaxation function and its parameters. The basic concept is also used in this paper, but will be simplified to a large extent. The formulation uses isotropic strain-based damage combined with a gradient part to include nonlocal damage. This serves for two purposes, (1) a problem regularization can be achieved and the hyperbolicity of the dynamic problem is preserved [20], and (2) the tight relation between damage and strain is resolved, which is used to assign damage with some type of inertia as a novel approach. This inertial part retards damage under high strain rates and temporarily leads to higher stresses compared to the quasistatic case.

In Section 2, a triaxial isotropic damage law is defined with a strain-based formulation. Regarding regularization, this law is extended with nonlocal damage in Section 3. This is performed with gradient continuum damage, and additionally extended with a damage inertia part. Thus, nonlocal damage is introduced as a variable on the system level leading to a specific dynamic finite element formulation, which is described in Section 4. The properties of gradient damage are discussed for the uniaxial tension bar under quasistatic loading in Section 5. Altogether, the basis is prepared for the investigation of wave propagation problems. This is at first performed for the uniaxial tension bar in Section 6, with load histories corresponding to constant strain rates in a range of 0.5–50 s−1. Especially the influence of the damage inertia parameter will be discussed. A two-dimensional application problem is demonstrated in Section 7. The paper is concluded in Section 8 by pointing out the potential for applications and further developments.

Section snippets

A constitutive law for concrete based on damage

A constitutive law based on isotropic damageσ=(1D)E·εis chosen for the following, with a scalar damage measure D, the stress vector σ, the strain vector ε and the elasticity matrixE=E(1ν)(1+ν)(12ν)[1ν1νν1ν000ν1ν1ν1ν000ν1νν1ν100000012ν2(1ν)00000012ν2(1ν)00000012ν2(1ν)],with Young's modulus E and Poisson's ratio ν. The values E, ν are constant, while the damage D depends on the loading history and has a range 0  D  1. A widely accepted approach for damage evolution of quasi-brittle

Gradient based damage extended with inertia

Under ongoing loading concrete shows a quasi-brittle behavior due to its heterogeneous structure, i.e. a development of micro-cracks evolving into macro-cracks within a so-called process zone. The final formation of macro-cracks consumes a considerable amount of energy, which leads to a size effect and may contribute to a ductile behavior of whole structures. The size of the process zone or the measure of crack energy corresponds to the extent of the material heterogeneity. Regarding concrete,

Discretization

The dynamically extended gradient damage approach shall be incorporated in the finite element method. To begin with, Eq. (15) has to be transformed into a weak form. The standard way starts withVδκ¯[κmκκ¯¨κ¯+cΔκ¯]V=Vδκ¯κVVmκδκ¯κ¯¨VVδκ¯κ¯V+Vcδκ¯Δκ¯V=0with a test function δκ¯. The product rule of differentiation leads toδκ¯Δκ¯=div(δκ¯κ¯)δκ¯·κ¯with the scalar product ·, the divergence operator div and the nabla operator ∇. Using the Gauss theorem we haveVdiv(δκ¯κ¯)V=Aδκ¯n·κ¯

The uniaxial tension bar under quasistatic loading

It remains to determine the value of the interaction range R. We consider that localization ends up in macro-cracking and dissipation of crack energy. With a continuum approach crack energy for uniaxial tension results fromGf=0dwg(ε)wwith the localization zone width dw, its variable w and a specific crack energyg(ε)=εctεσ(ε)ε,εεctwhere the integration starts from concrete tensile strength with a strain εct and σ(ε) is given by Eq. (11). The crack energy Gf is assumed to be a material

Application for uniaxial wave propagation

For the linear elastic case uniaxial wave propagation is described byE2ux2=ϱ2ut2with the displacement u, Young's modulus E and the specific mass ϱ. A bar is considered, which is loaded from its left side x = 0. A solution of Eq. (35) is then given byu(x,t)=f(z),z=ctx,c=Eϱwith an arbitrary function f(z) and Mc-Auley brackets :a=aifa>0,a=0 otherwise. Eq. (36) describes a wave starting at the left side for t = 0 and moving to the right side with a speed c. A constant strain rate ε˙0 is

Application for a plane stress problem

A simple beam under impact loading is numerically investigated in the following. The geometry, boundary conditions and loading are shown in Fig. 15. Plane stress conditions are assumed. The load shape is given with a half-sine, whereby the duration is fixed with 10−4 s and the magnitude P is variable. A concrete grade C 40 is chosen for this problem with an initial modulus of elasticity E = 36,000 MN/m2 and a specific mass ϱ=2.4×103MNs2/m4. The largest natural period according to the beam theory

Summary and conclusions

The continuum based damage approach generally has proven to be suitable for the description of concrete behavior. A major characteristic of this approach is given with softening, which is connected with localization phenomena. Thus, continuum models have to be regularized, which can be done with gradient continuum damage. This introduces nonlocal damage as a further variable beneath displacements or strains, respectively. The relation between nonlocal damage and strains is ruled by a

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