A possible failure mechanism for concrete subjected to blast loading is spalling. Spalling is defined as material getting broken off from the side opposite to the one where the impact occurs. This is made possible in concrete due to the difference in compression and tensile strength. A valuable tool for understanding the phenomenon is the one dimensional stress wave theory. When a body is not in equilibrium, a stress wave will propagate through the material. There are two basic elastic stress waves; longitudinal waves and shear waves, but longitudinal waves will be discussed here. The following descriptions are given according to the compendium ”Støt og Energiopptak” by Langseth and Clausen [23].
When longitudinal stress waves move through a long stationary bar, they are divided into compression and tensional waves. For a compression stress wave, the individual particles move in the same direction as the stress wave, while for a tensional stress wave they move in the opposite direction. By assuming that the Poisson’s ratio effect is negligible and that the Hooks law is valid, the one dimensional wave equation can be derived as:
δ2u
δt2 =c2·δ2u
δx2 (2.18)
where u is the displacement in x-direction, and c is the wave speed given by:
c= s
E
ρ (2.19)
(a) Long stationary bar (b) Infinitesimal element Figure 2.15: Dynamic equilibrium of a bar.
CHAPTER 2. THEORY Equation (2.18) is valid for an infinitesimal bar element with thickness dx (see Figure 2.15).
N is the normal force, M is mass and a is the acceleration. A general solution to the one dimensional wave equation is:
u(x, t) = f(x−ct) +g(x+ct) (2.20) where f represents a wave moving in the positive direction andg in the negative. When the wave hits the end of the bar, it will be reflected. For a free end, the net stress must be zero and a compression wave is reflected as a tension wave and vice versa. Figure 2.16 illustrates the reflection of a triangular stress wave at a free end. The left side of the illustration shows the behaviour of the wave, while the right side shows the total stress. Please refer to the compendium [23] for the complete derivations of the listed equations.
Figure 2.16: Compression stress wave reflection at a free end [24].
CHAPTER 2. THEORY
Chapter 3 Materials
This chapter presents a qualitative description of concrete together with selected material models. A short description of possible failure mechanisms for concrete plates subjected to dynamical loads are also presented.
3.1 Qualitative description of concrete
Concrete behaves differently from other common construction materials like steel or alu-minium. Steel and aluminium are often assumed to be homogeneous isotropic and have the same mechanical properties in compression and tension. For concrete, the situation is more complex. The complexity derives mostly from the concrete’s non-homogeneous material be-haviour. Concrete is a composite material created by two main building blocks, aggregate and mortar. Both the interaction between them and the material properties to each of them influence the material behaviour. It is possible to create a concrete with a specific strength and properties suitable for specific situations. Adding harder aggregate will provide a higher concrete strength and a distribution of aggregate can create strength gradients over the ma-terial [25]. The tensile strength is dependent on the bond between the aggregate and the mortar [26]. The aggregate and the mortar also decide the level of micro-cracks present in the material before any load is applied, and the interaction between them causes new crack to appear and existing cracks to propagate when a load is applied. To fully understanding all aspects of concrete’s behaviour, knowledge on a micro-scale is necessary. This is however out of the scope of this thesis and will therefore not be presented. In the following sections, the general characteristics of concrete behaviour will be discussed.
CHAPTER 3. MATERIALS
3.1.1 Compression response
The behaviour of concrete in compression can be illustrated with a uniaxial compression test as shown in Figure 3.1, whereσ is the stress,is the strain andfcis the uniaxial compressive strength. When concrete is loaded in compression, its strain-stress relationship is assumed linear until the micro-cracks between the aggregate and the mortar begin to increase in size.
This happens at a load of approximately 30% of the ultimate strength [27]. From this point, the stress-strain relationship behaves non-linearly and the stiffness will decrease. When the concrete approaches its ultimate strength, the micro-cracks start to connect with each other forms continuous cracks throughout the concrete. The stress-strain relationship after the post-peak is generally assumed to be a softening phase. This is however disputed in the literature. Some clearly state that the softening phase exists and is influenced mainly by the principal tension strain [28]. Others state that the reason for the softening phase is not due to material behaviour, but the result of the interaction between the loading plates and the test specimen [29]. It is believed that the friction between the loading plates and the specimen limit the specimen from expanding near the loading plate [30]. This creates a complex tri-axial stress-state from which the softening phase develops. When the friction is reduced, the softening phase is reduced [31].
Figure 3.1: Compression response of concrete [27].
CHAPTER 3. MATERIALS
3.1.2 Tensile response
A major characteristic of concrete is that the compression and tension strength are not equal.
Due to its non-homogeneous behaviour, the tensile strength of concrete is approximately 5-10% of the compressive strength [27]. The tensile response of concrete can be observed by conducting both a load controlled test and a displacement controlled test. In the load controlled test, the concrete behaves almost linearly up to about approximately 70% of its ultimate strength (see Figure 3.2a). By unloading from this state, no irreversible deformations other than some creep-related displacement can be observed [26]. Above this elastic limit, the concrete behaves non-linearly until it reaches its ultimate strength and fails. However, concrete has a softening phase in the tensile response. To observe this effect, a displacement controlled test must be performed (see Figure 3.2b). Such a test is difficult to perform due to the brittle behaviour of concrete. There are three different phases of the tensile response where the two first are the same as in the load controlled test, and the last is a softening phase. This softening is a result of a continuous growth of micro-cracks which will finally form a discrete crack. The softening phase can influence structural components, but for ordinary design of concrete structures, this softening phase has no or little importance. This is because concrete’s tensile strength is assumed to be zero and the reinforcement is assumed to carry all tensile forces.
(a) Load controlled (b) Displacement controlled Figure 3.2: Tensile response of concrete [26].
CHAPTER 3. MATERIALS
3.1.3 Pressure dependency
Concrete is a pressure dependent material and both concrete strength and stiffness are known to be sensitive to multiaxial stress conditions [32]. For example, in biaxial stress conditions, the compressive strength increases with an average of 30% with regards to the uniaxial com-pression strength fc. Research shows that the cylinders subjected to lateral confinement in combination with axial load can reach a compression strength of 3.5 times the uniaxial strength (see Figure 3.3) [33]. The strength and stiffness increase is due to microscopic cracks being inhibited from propagating because of lateral confining stresses [27]. This also affects the ductility of the concrete, as shown in Figure 3.3, where concrete is able to withstand larger strains with an increased lateral confining pressure. For lateral tension stresses, the strength of the concrete will decrease.
Figure 3.3: Pressure dependency [27]. Figure 3.4: Volumetric response [27].
3.1.4 Volumetric expansion
Concrete is a porous material and it will compact under compression, until it reaches a value of approximately 80% of the ultimate strength. At this point, the minimal volume is reached and the volume will start to increase because micro-cracks will propagate through the
CHAPTER 3. MATERIALS material and form crack patterns (see Figure 3.4). Further loading will eventually cause the concrete to expand past its initial volume. As areas with high compressive stresses expand, tensile stresses can induce to neighbouring areas, which will result in crack formation [27].
3.1.5 Rate dependency
When dealing with fast transient dynamic loads, it is important to have knowledge of the material’s rate dependency. Studies have shown that concrete has a non-linear rate depen-dency, which can be separated into two regions. The location of the transition region is somewhat disputed due to a difference in experimental methods [34], but a general view is that the transition happens between a strain rate of 1s−1 to 20s−1 [32]. In the first region, the strength of the concrete increases almost linearly with the strain rate up to a dynamic in-crease factor of approximately 1.3. After the transition zone, the concretes strength inin-creases almost exponential with increasing strain rate.
Figure 3.5: Increase of concrete strength under high strain rates [32].
CHAPTER 3. MATERIALS
As shown in Figure 3.5, the dynamic increase factor is greater in tension than in compression.
Experiments show that the maximum dynamic strength can be 3.5 times the quasi-static strength in compression and the dynamic strength can be up to 13 times the quasi-static strength in tension [34].