Mechanical Behavior of Concrete E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 546

Veröffentlichungsjahr: 2013

Beschreibung

This title provides a comprehensive overview of all aspects of the mechanical behavior of concrete, including such features as its elastoplasticity, its compressive and tensile strength, its behavior over time (including creep and shrinkage, cracking and fatigue) as well as modeling techniques and its response to various stimuli. As such, it will be required reading for anyone wishing to increase their knowledge in this area.

Leseprobe

Table of Contents

Foreword

PART 1. INSTANTANEOUSOR TIME-INDEPENDENT MODELSFOR CONCRETE

Chapter 1. Test Techniques and Experimental Characterization

1.1. Introduction

1.2. Experimental specificities related to concrete material

1.3. Extensometers and experimental conditions

1.4. Behavior of concrete under uniaxial stress: classical tests

1.5. Concrete under multiaxial stresses

1.6. Conclusions regarding the experimental characterization of the multiaxial behavior of concrete

1.7. Bibliography

Chapter 2. Modeling the Macroscopic Behavior of Concrete

2.1. Introduction

2.2. The discrete approach

2.3. Continuous approach

2.4. Conclusion

2.5. Bibliography

Chapter 3. Failure and Size Effect of Structural Concrete

3.1. Introduction

3.2. Probabilistic structural size effect

3.3. Deterministic size effect

3.4. Fractality and size effect

3.5. Size effect and calibration of non-local models

3.6. Conclusions

3.7. Acknowledgement

3.8. Bibliography

PART 2. CONCRETE UNDER CYCLICAND DYNAMIC LOADING

Chapter 4. Cyclic Behavior of Concrete and Reinforced Concrete

4.1. Characterization tests of the cyclic behavior

4.2. Modeling the reinforced concrete cyclic behavior

4.3. Modeling of the cyclic behavior of concrete

4.4. Conclusions

4.5. Bibliography

Chapter 5. Cyclic and Dynamic Loading Fatigue of Structural Concrete

5.1. Introduction

5.2. The mechanisms of concrete fatigue

5.3. The fatigue strength under uniaxial compression or traction

5.4. Extension to Aas-Jakobsen’s formula

5.5. Fatigue under multiaxial loading

5.6. Fatigue under high-level cyclic loading

5.7. Fatigue strength under variable level cyclic loadings

5.8. Bibliography

Chapter 6. Rate-Dependent Behavior and Modeling for Transient Analyses

6.1. Introduction

6.2. Experimental behavior

6.3. Behavior modeling of concrete in dynamics

6.4. Bibliography

PART 3. TIME-DEPENDENT RESPONSEOF CONCRETE

Chapter 7. Concrete at an Early Age: the Major Parameters

7.1. Introduction

7.2. Influence of the composition of concrete

7.3. Consequences of boundary conditions

7.4. Conclusion

7.5. Bibliography

Chapter 8. Modeling Concrete at Early Age

8.1. Introduction

8.2. The coupled thermo-chemo-mechanical problem

8.3. Data collection and experimental methods

8.4. Conclusion

8.5. Bibliography

Chapter 9. Delayed Effects – Creep and Shrinkage

9.1. Introduction

9.2. Definitions and mechanisms

9.3. Experimental approach

9.4. Delayed response modeling

9.5. Codified models

9.6. Conclusion

9.7. Bibliography

Closing Remarks: New Concretes, New Techniques, and Future Models

List of Authors

Index

First published 2004 and 2005 in France by Hermes Science/Lavoisier in two volumes entitled: Comportement du béton au jeune âge and Comportement mécanique du béton © LAVOISIER 2004, 2005 First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2012

The rights of Jean-Michel Torrenti, Gilles Pijaudier-Cabot and Jean-Marie Reynouard to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Comportement du béton au jeune âge and comportement mécanique du béton . English

Mechanical behavior of concrete / edited by Jean-Michel Torrenti, Jean-Marie Reynouard, Gilles

Pijaudier-Cabot.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-178-0

1. Concrete--Analysis. 2. Concrete--Curing. I. Torrenti, Jean-Michel. II. Reynouard, Jean-Marie. III

Pijaudier-Cabot, Gilles. IV. Title.

TP882.3.C6613 2010

624.1'834--dc22

2009049963

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-178-0

Foreword1

Because it is a versatile, durable, and cost-effective building material, concrete is still the most used construction material today: we cast an average of 1 ton of concrete per year per inhabitant in the world. Therefore, it is important to understand the mechanical response of this material and to model it as accurately as possible. Depending on the required use and geographic location, concrete structures are subjected to different loading histories on different time scales. The aim of this book is to discuss the behavior of concrete under these different conditions. It results from the merging of two volumes published in French dealing with the constitutive behavior of concrete [REY 05; ACK 04]. Its content is directed toward masters students, doctoral students, engineers, and researchers who wish to have a concise overview of modern modeling of concrete and concrete structures.

The first part of the book concerns instantaneous or time-independent models. Chapter 1 is a comprehensive presentation of the advanced experimental techniques and experimental characterization that is required before modeling is undertaken. Chapter 2 details two classical types of constitutive models for concrete, elastic plastic models and continuum damage-based models, and their combinations. The implementation of these models in the framework of the finite element method calls for robust numerical techniques to integrate non-linear constitutive laws step by step. Therefore a section in Chapter 2 is dedicated to this aspect of modeling, which is so often neglected. The subsequent chapter deals with an original aspect of the behavior of concrete: structural size effect. Concrete is a quasi-brittle material. When cracking occurs, a very large microcracked area appears at the tip of the crack. The interaction between this area, denoted the “fracture process zone”, and the rest of the structure is the source of a pronounced size effect. The fractal nature of the cracks could also explain this effect. Whatever the physical explanation of the structural size effect, experimental results show that the nominal stress at rupture of a structural element could be decreased by 25% when the size of the structure is increased four times. A more accurate model and reduced safety coefficients in the design codes imply a consistent appraisal of this effect, which is still frequently concealed in safety factors today.

In the second part of the book, cyclic and dynamic loadings are considered. In the case of exceptional loads, such as earthquakes, complex models have to be implemented in order to describe the response of concrete. Chapter 4 is devoted to the cyclic behavior of concrete and shows how, in a structural calculation, complex aspects such as closure of cracks, and the existence of a hysteresis during loadingunloading cycles may be captured. Chapter 5 concerns the response of concrete to fatigue loads. The process of fatigue of concrete is directly related to the propagation of microcracks inside this material. Microcracking due to fatigue constitutes an irreversible phenomenon that contributes to the slow, but progressive, deterioration of the inner structure of the material. In the case of repeated loadings these effects must be taken into account and a review of existing models for the fatigue behavior of concrete is presented. Chapter 6 deals with the rate-dependant behavior of concrete and models for transient analyses. Rate dependency and the influence of a high level of hydrostatic pressure are examined regarding experimental, theoretical, and modeling.

Because concrete is a continually developing material, the last part is devoted to the evolution of the properties of concrete with time and to delayed strains. Concrete undergoes many phenomena during the first dozen hours that follow batching and setup when fresh, and it progressively hardens due to hydration. Concrete may be subjected to different thermal (thermal curing, ambient temperature), hydric (drying, wet curing), or mechanical stresses. In this context, its behavior will strongly depend on its composition and on the environmental loads during its early age. The aim of Chapter 7 is to describe these two groups of influential parameters. Chapter 8 presents an example of the modeling of concrete at early age. The modeling element is completed by the experimental determination of the parameters of the behavior of concrete at early age. Finally, Chapter 9 deals with two important phenomena: shrinkage and creep. Delayed strain is still subject of intensive research, from the structural scale down to the nanoscale. These efforts contribute then to delimit the range of validity of existing simplified models and to elaborate more advanced constitutive relations, e.g. based on multiple-scale analyses of concrete viewed as a cementitious composite. It is in this spirit that outstanding and future issues are outlined in the concluding remarks of this book.

Bibliography

[ACK 04] ACKER P., TORRENTI J.-M., ULM F.-J., Comportement du béton au jeune âge, Traité MIM, Hermes, 2004.

[REY 05] REYNOUARD J.-M., PIJAUDIER-CABOT G., Comportement mécanique du béton, Traité MIM, Hermes, 2005.

1 Written by Jean Michel TORRENTI, Gilles PIJAUDIER-CABOT, Jean-Marie REYNOUARD.

PART 1

Instantaneous or Time-Independent Models for Concrete

Chapter 1

Test Techniques and Experimental Characterization1

1.1. Introduction

Detailed experimental analysis of concrete material has always been a complex task. This complexity is due to the variability of the material, which depends on its composition, its conditioning, and on the mechanical, thermal, or chemical loads it is undergoing. For precise experimental characterization it is necessary to know the composition and “history” of the material. Accuracy and rigor, as always, are required for an experimental study on this type of material.

Moreover, the complexity of concrete characterization has amplified the need for experimental studies. The goal is to understand, explain, and model the behavior of concrete. Experimental studies of concrete will always have at least one of the following aims: characterization, modeling, or validation. Indeed it is necessary to characterize the behavior of material under different loadings experimentally. An understanding of the physical phenomena involved is then possible. Phenomenological or micro-mechanical modeling is improved using the observations made during tests, and the obtained models have to be validated via an experiment on a specimen or a structure.

In order to be concise, we will focus here on the experimental characterization of concrete under mutli-axial mechanical loading. Moreover, we will limit ourselves to the study of quasi-static loads, that is for deformation rates ranging from 10-5 to 10- 2 s-1. We can therefore suppose that long-term effects (mainly shrinkage and creep) and dynamic effects are negligible. We will consider the conditions to be isothermal and at room temperature. Indeed, thermal strains on concrete material leads to various physical and physicochemical processes (internal pressure due to water evaporation, aggregates setting off, phase changing, Calcium Silicate Hydrate CSH dehydration, etc.) which change the microstructure of the material, and therefore, its mechanical behavior (more information can be found in the following papers: [BAZ 96; HEI 98; NEV 00; NEC 00]). In addition, concretes undergoing chemical attack will not be considered in this study as microstructural modifications influence their mechanical behavior (see as examples [BAR 92; CAR 96; GER 96; TOR 99a]). The concretes and mortars studied herein are archetypal; specifically, their composition, assembly, and conservation modes are common in civil engineering, which allows us to show the general characteristics of their mechanical behaviors.

The initial focus of the chapter will be ways to consider the specificities of the materials towards a reliable and adequate experimental analysis. The interpretation of an experimental result of a concrete or a mortar cannot be performed out of this study’s context. Boundary conditions, curing conditions, or the maturation of the material have to be specified. In the second part, classical extensometry will be described. The influence of the boundary conditions will also be related to extensometry which can only be performed on the external surfaces of the specimen. Moreover, we will see that it is not always possible to obtain a “material” test, i.e. where all the characteristics and the applied stresses are homogenous. Depending on the observation scale, concrete can be considered as either a homogenous material (which we will assume in most of the interpretations) or as a composite or heterogenous material (implying an additional difficulty regarding the analysis of the physical phenomena explaining these types of behavior). In this chapter, we will consider the material as homogenous, even if microstructural observations cannot support our reasoning. The last section of this chapter will focus on the study of the main test techniques used to test a concrete material under multiaxial stresses. The results obtained for mortar or concrete will be given in parallel. Of course, these different techniques can be used to identify a mechanical behavior or validate a model.

1.2. Experimental specificities related to concrete material

1.2.1. Composition and variability of the material

“Classical” concretes (see [BAR 96a; DRE 98] for more information) that will be studied are made of a granular skeleton of a sand-lime type, with constant grain size distribution and with common cleanness [BER 96]. The binding agent is a commercially available cement without any particular properties [BAR 96b]. Workability is standard and concrete implementation is done as usual. We will not cover high-performance concretes (more information can be found in the following papers: [MAL 92; DEL 92; LAP 93; BAR 94; BEL 96; AIT 01]). Once mixed, concrete hardens and its mechanical performances increase with time. It then becomes a porous material, made of aggregates and of hydrated cement, which contains a fluid interlayer. The influences of this liquid interlayer can be delayed: it can modify the short-term behavior of concrete via evaporation; for example, leading to desiccation cracking, and therefore, the existence of defects located at the surface. Interactions between the microstructure, the fluid interlayer and the environment can also modify the mechanical behavior of concrete. The action of the environment can become a major factor and sometimes leads to a change of the poromechanical properties of concrete [COU 95].

Moreover, the aggregates and cement used to make the material are often of local origin, which implies a significant range from one production site to another. It should be noted that the experimental results obtained for a particular concrete or mortar are a function of all ingredients entering into the mix. Therefore, the comparison of two results must be done carefully. Nevertheless, it is possible to provide general characteristics of concrete behavior as long as it is assumed that concrete, even if made of sand, gravel, cement, and water, is a homogenous material at the considered mechanical test scale. In addition, a reliable comparison of various results can be performed by using a similar cement and Leucate’s normalized sand (complying with the EN 196-1 and ISO 679 rules), in the case of a mortar. This choice of cement and similar sand whatever the test, under constant conditions, allows the variability to be reduced. In the case of a concrete, it is commonly accepted that the mineralogical and geometrical nature of the aggregates used are kept the same.

In the same way, the behavioral variability of one specimen to another has to be considered. Even if the specimen is of the right size, concrete material is still a heterogenous material. This implies that the mechanical characteristics of the different components are not exactly identical. Under the action of mechanical loads, the stress and deformation fields within the specimen are also heterogenous [TOR 87], and vary from one specimen to another as the granular arrangement is not the same for a series of specimens assumed to be identical. A variation of the mechanical properties of the material (elasticity modulus, uniaxial tensile strength, uniaxial compressive strength, etc.) is then observed. These are more pronounced as the tensile stresses or the tensile deformations evolve in the specimen. Locally, those tensile stresses lead to microcracking, by exceeding the strength threshold, which is not constant. As a result, a dispersion of the mechanical characteristics appears when the applied stresses allow some extension (in the case of uniaxial tension or uniaxial compression). Nevertheless, when those extensions are not possible or are limited, experimental dispersion is decreased (in the case of a triaxial compression test). To conclude, at least three identical tests are performed to correctly quantify the mechanical behavior of mortar and concrete materials.

1.2.2. Specimen preparation

1.2.2.1. Specimen characteristics

Regarding all the mechanical tests we are about to present, it is necessary to use, or in some cases to design, a specimen that allows a number of rules to be respected: first, this specimen (therefore, its shape and its dimensions) have to respect the basic condition of the test of a material, i.e. it must be a representative elementary volume (usually called REV). This REV must comply with the assumption of homogeneity of the concrete; therefore, it should be large with respect to the heterogeneities within the material and must not be too large in order to limit the structural effects. Moreover, a size effect exists regarding concrete (see section 1.2.3), leading to the constant use of samples with the same dimensions. Finally, it must allow the application of mechanical effects.

Two types of samples are usually used: cylindrical specimens for uniaxial and triaxial tests; cubic specimens for biaxial and triaxial real tests.

1.2.2.2. How to obtain specimens

There are two ways in which specimens can be obtained. The first is molding. In France, the measurement of simple compressive strength is done on specimens cast using steel, cardboard, or polypropylene cylindrical molds with a height to diameter ratio of 2. Their diameter is chosen as a function of the diameter of the largest aggregate. The diameter of the specimen has to be at least three-times longer than the largest aggregate’s diameter in order to minimize the wall effects. These effects are due to an over-concentration of the fine elements in the mix near the mold’s wall. For specimens of large dimension with respect to the aggregate size, this effect is negligible but has to be considered for the smallest ones. The dimensions of the molds are normalized (NF P 18-400), the most commonly used have a 160 mm diameter and a 320 mm height. In the central zone of the specimen (effective zone) compressive stress may be considered as uniaxial and uniform (see section 1.3.2).

The second way of obtaining specimens is saw-cutting or core drilling (NF P 18- 405 codes). This consists of drilling the specimens from a large block of the material. With this technique, any specimen geometry can be obtained and wall effects are avoided. Drilling can lead to a superficial cracking of the material, which is an intrinsic bias of this technique. Figure 1.1 shows three specimens obtained from molding, core drilling and saw-cutting. Regarding molding (picture on the left side), the obtained surface state does not permit the aggregates to be seen. It should be noted that bubbles can be hidden by the cement paste at the surface and be a source of error if the strain is measured nearby. Therefore, it is necessary to prepare the surface before gluing strain gages in order to avoid this potential problem (see section 1.3.1) Conversely, the saw-cut and drilled specimens show the aggregates. The deformation gages stuck to their surfaces have to be large enough to “average” the differential deformations between the cement paste and aggregates. Indeed, those two constituents do not usually have the same elasticity modulus, which implies that on the scale of an aggregate, there is a heterogenous deformation.

Figure 1.1.Concrete specimens obtained using molding, core drilling and saw-cutting

1.2.2.3. Initial anisotropy

Mechanical behavior of concrete material is quite complex as it can be influenced by an initial anisotropy due to different phenomena, mainly related to its formation and its mode of implementation. In particular, the direction of casting has a non-negligible importance. Under the action of gravity, compaction of the granular skeleton occurs in a privileged way. Moreover, during the set up of the material into its mold, the rise of air bubbles dragged along the formworks and inside the material (Figure 1.2) can be limited to the presence of aggregates that block this entrapped air. Initial and oriented defects will then be present “under” the aggregates. Those initial defects will then influence the initiation and propagation of microcracking.

The initial anisotropy on the uniaxial tensile behavior is characterized by a decrease of the peak strength when the tension direction is parallel to the casting direction, compared with the one measured in the perpendicular direction [HUG 70]. Similar results have been obtained in uniaxial compression [HUG 70; MIE 84]: Figure 1.3 shows deformation curves for a compressive stress parallel to casting direction and along a perpendicular direction, the maximal stresses are logically the same, whereas peak deformation is more important than when loading is parallel to casting direction. The elastic properties, measured in both directions, are also different. It is necessary to consider this phenomenon in the characterization of concrete material, especially when the water to cement ratio is high (due to the excess of water and significant porosity).

Figure 1.2.Origin of the initial anisotropy induced by teeming direction [MIE 84]

Other sources of initial anisotropy can be identified: the hydric gradient existing between the surroundings and the core of a specimen; the effects of hydration heat of the cements (leading to differential dilatations); or also the segregation of the aggregates. As an example, in Figure 1.4 the deformations measured on three perpendicular sides of a concrete cube (40 mm side) under hydrostatic pressure are presented. This concrete cube has been obtained from sawing the central part of a prism of 40×40×160 mm3 dimensions. Regarding hydric gradient, the cube was placed for a month in water (in order to avoid desiccation) before any mechanical tests, and then stored under a relative humidity of 60% for a year, in order to enable the drying and desiccation shrinkage of the concrete to evolve. This physical phenomenon leads to microcracking of the material, oriented along the main hydric direction of the gradients, i.e. perpendicularly to the longitudinal direction of the prism. After sawing, four faces of the cube underwent drying, and therefore, significant microcracking. As the behavior of concrete is elastoplastic, there is an initial anisotropy due to drying, the elastic properties at the start of loading being different. In addition, this initial anisotropy influences the non-linear behavior, in particular, the irreversible deformations that evolve differently depending on their measurement on dried faces or faces prevented from drying.

Figure 1.3.Influence of initial anisotropy induced by the direction of casting on the uniaxial compressive behavior [MIE 84]

1.2.3. Preservation and curing conditions

Concrete is a “living” material, or more exactly, a material whose physicochemical and mechanical properties evolve with time as a function of its initial composition, its implementation, the environment in which it is being preserved, and the applied mechanical stresses (for example, creep of concretes will be the more important if loading is applied directly after the setting). Among them, the main factors modifying the mechanical behavior of the material in time are the temperature and the surrounding humidity. We can list other factors that will not be considered here, for example, the action of chemicals (leaching [CAR 96; GER 96; LEB 01; HEU 01], attack by sulfates and chlorides [BAR 92; NEV 00], etc.) or the effects of delayed deformations (shrinkage and creep, for a general picture see [BAZ 82; ULM 99; BAR 00; ULM 01]).

Figure 1.4.Influence of the initial anisotropy induced by the drying of concrete on the hydrostatic compressive behavior: hydrostatic pressure/deformations curves [BUR 01a]

The effects of temperature, resulting from the heat release during the hydration reaction of cement, on the maturation of concrete have been widely studied (see [NEV 00]), both from a kinetics standpoint, resulting in increasing strength, and from a thermal shrinkage standpoint. It should be noted that these effects do not modify the characteristics of the material. In the same way, the surrounding humidity in which the specimens are preserved has an influence on the maturation of the material and its delayed deformations [BAZ 82; ULM 99; NEV 00].

It is important to define “stable” curing and conservation conditions in order to precisely characterize the mechanical behavior of a concrete or a mortar. Usually, a 28 day humid curing (a shorter duration is obviously possible to investigate the variations of the mechanical behavior during this period), then conservation at room temperature equal to 20 ± 1°C and relative humidity of 60 ± 5% are commonly used conditions in research laboratories. Ideally, specimens have to be preserved in water or under a hygrometry higher than 90% until the test is carried out. Unfortunately, due to experimental reasons (gage sticking, surface preparation, etc.), this condition is barely observed in the literature. Due to the dependence of the mechanical behavior of concrete on its maturation conditions, for every mechanical characterization test, the following points (proposed by the TC14-CPC technical committee of The International Union of Laboratories and Experts in Construction Materials, Systems and Structures (RILEM) [RIL 72; 73]) have to be stated as a minimum:

– type and dimensions of the specimen;

– composition of concrete;

– how to implement concrete;

– how to obtain specimens;

– curing conditions;

– conservation conditions;

– number of identical tests performed or experimental scattering of the results.

These points contribute to the quality and traceability of the published results.

1.2.4. Size effect

A particular phenomenon involving materials with a cementitious matrix is the size effect; specifically, the measured mechanical behavior, in particular the maximal strength, depends on the size of the specimen used during the test (see [BAZ 98; VLI 00] for more information). Overall, uniaxial tensile strength and uniaxial compressive strength decrease when the size of the specimen increases. In addition, this relation is not linear.

This size effect, which will be described in more detail in Chapter 3, can be explained e.g. using the theory of the weakest link: the bigger the volume of the specimen, the more important the probability of finding a defect is (i.e. the more defects there are), and the more the probability of the decrease of the maximal strength increases. This size effect is less sensitive when there is a decreased possibility to get tractions in the specimen, which is the case of triaxial compression with a strong confinement. More generally, the size effect decreases with the increase of hydrostatic pressure [ROS 97]. Depending on the tests, this phenomenon has to be considered, especially when comparing different experimental results between each other.

Another important factor influencing the mechanical behavior of concrete is that, regarding temperature and hygrometry, the specimen is actually a “structure” whose two parameters evolve with time as a function of the conservation conditions and the sample size. This induces some size effect due to the ratio of the size of the specimen to the characteristic length/times involved in thermal and hydric analyses. For example, the maximal stress in uniaxial compression increases with the drying of concrete (all the other properties being constant), and the elasticity modulus

decreases with regard to the microcracking induced during desiccation [TER 80; BUR 00; YUR 03]. The hydric and thermal state of the material will be a function of the date the test was performed, but also of the size of the specimen (hydric and thermal equilibriums will be reached more or less rapidly). We then have to be aware of the current state of the material that is undergoing the mechanical test.

1.3. Extensometers and experimental conditions

Before characterizing the behavior of concrete under multiaxial stresses, we will detail some of the classical techniques of strain and displacement measurement and the boundary conditions on a concrete specimen. It is necessary in order to provide some consistent modeling of the experiments, and thus consistent interpretation of test data. The experimental issues are as follows:

– stress measurement is not possible, as it is a theoretical concept: it can only be deduced from the surface strain measurement or from a pressure measurement of a specimen. Stress determination is still the result of a calculation from other measured values or reverse methods, which initially postulate a constitutive law;

– deformation measurement in the core of a specimen is difficult and the global deformation field is usually extrapolated from surface measurements;

– metrology must not modify the deformation or stress field where the measurements are being carried out;

– boundary conditions in the experiment have to match, almost exactly, the desired distribution of stress or strain, for instance homogenous distributions if the constitutive law is to be directly calibrated.

1.3.1. Classical extensometry on a concrete specimen

Regarding concrete, the physical quantities usually measured for a multiaxial quasi-static test are local displacements of the sample, deformations, and applied forces (force, couple, or pressure). It should be noted that measurements of temperature, relative humidity, or acoustic emissions are also performed, but will not be dealt with in this chapter.

Sensors can be the cause of errors, i.e. there is an intrinsic difference between the measured value and the value given by the sensor. The sensitivity of the sensor is the ratio between the variation of physical effects (displacement, deformation, temperature variation, etc.) and the value (usually electrical) variation given by the sensor. A sensor works on a measurement range, which has to be considered. When a sensor is used it is then necessary to consider the following: accuracy describing the aptitude of the sensor in giving results close to the real value and the value required: the more sensitive a sensor is, the more accurate it is; fidelity describing the reproducibility of the performed measurement. Accuracy and fidelity both make a sensor ideal. To keep this accuracy, calibration is necessary on a regular basis.

Three main groups of sensors are used in civil engineering: “active” sensors which turn intrinsically physical effects into electronic data that can be used directly: a piezo-electrical sensor directly turns its applied stress into electricity; “passive” sensors whose parameters are sensitive to the physical value to be measured indirectly: i.e. dynamometric rings that are deformed (this is what is being measured) under a mechanical load (this is what we want to know); or strain gages where what is really being measured is the electrical resistance of a wire stuck to the specimen, caused by its deformation.

Figure 1.5.Photograph of a Vishay-Micromesures® gage stuck to a regular mortar

1.3.1.1. Strain gages

The principle of strain gages stuck on the surface of a specimen undergoing deformation is to measure the variation of electrical resistance of a thin wire whose cross-section diminishes upon stretching (due to Poisson’s effect). After multiplying by the gage factor, we obtain the local deformation of the specimen. The data acquisition to be processed is relatively simple [AVR 84]. The most common ones have a resistance of 120 Ω, those used to get more precision are of about 350 Ω. Due to the reduced size of gages, deformation measurement is performed locally in the direction of the grid. Rosettes (made of three gages at 45° or 60°) allow the whole deformation tensor to be locally determined. The size and type of gage is a function of the size of the specimen and of the aggregate size. The dimension of the gage has to be large enough so that it is not affected by local heterogenities. The drawbacks of gages are their fragility and often complex implementation (in the case of humid or damaged surfaces). The choice of the adequate glue can also be difficult, in particular, for really porous materials.

The measurable strain range, with an accuracy of 0.1%, goes from 5×10-6 to 10-1. For significant deformations (from 1 to 10%), one should be extremely careful with the choice of the glue, which has to accommodate those deformations. Numerous factors, such as temperature, humidity, test duration, and strain amplitude during the test, have to be considered when the “gage+glue” complex is to be determined [AVR 84]. Some particular gages are used to detect and monitor cracking propagation (gage with brittle wire), or pressure measurement (known as pressure gages, which is used for dynamic tests). Finally, it should be noted that gages are sensitive to temperature and other factors, such as hydrostatic pressure or transversal deformation of the grid. Appropriate corrections have to be made as a function of those different parameters.

1.3.1.2. Displacement sensors

Regarding displacement measurement, the tools usually used are “LVDT” (Figure 1.6b) and gage displacement sensors. A LVDT (linear variable differential transducer) is a linear displacement sensor based on variations of the magnetic properties of a ferromagnetic bar moving in a winding, which can easily be turned into electrical data. This sensor measures the displacement of a point at the surface of the specimen in the desired direction; it needs specific electronic conditioning, and the contact between the sensor and specimen must always be maintained (using glue, for example). Displacement sensors with gages have the advantage of using the same electronic data acquisition as is used by a single gage. Turning displacement into electrical data is done via measuring the 350 Ω strain gages stuck to a string’s blade, which get deformed when the measuring rod goes into the sensor’s body. It is worth noting that a small effort is then applied to the specimen (lower than 4.4 N in the case of a Vishay® sensor), which insures a glue-free constant contact.

Figure 1.6.Photograph of a) a gage displacement sensor, and b) a LVDT

Figure 1.7.Photograph of a cone deformeter (Controlab®)

The displacement measurements performed with these sensors are global. Analyses of specimen dilation or shrinkage, during bending, for example, can also be done with displacement sensors. For instance, a cone deformeter is presented in Figure 1.7, which is used to measure a deformation. The idea is to use a displacement sensor to measure the variation of the distance between two cones stuck on the surface of the specimen, the cones being present to ensure a perfect relative position between the deformeter and the measured point.

Figure 1.8.Compression extensometer [DUP 95]

Other displacement sensors such as vibrating strings are used in civil engineering. The vibration frequency of a string submitted to excitation depends on its tension. By fixing the two extreme points of a metallic string to a specimen, we can monitor its frequency variation, and thus the variation of length between two points of the specimen.

The uniaxial compression test is common on concrete (see section 1.4.5), and axial deformation measurement is easily carried out. For this reason a specific compression extensometer has been developed [BOU 80]. Its principle is similar to a cone deformeter as the measured displacement corresponds to the connection of two circular rings fixed on a concrete specimen using some punches (Figure 1.8). With the help of three displacement sensors located at 120°, and data acquisition, this sensor directly provides the average axial deformation of the specimen during loading. Its handling is simple and the accuracy of the obtained results is excellent [BOU 99].

To obtain the lateral deformation of the specimen during a uniaxial compression test, the use of a chain sensor can be considered (Figure 1.9). The principle is based on the measurement of the relative displacement of the two units of the roll chain which monitor the inflation of the specimen, and gives a global measure of the lateral deformation.

Finally, other types of measurements are possible, such as acoustical or optical measurements. In the case of acoustical measurements, the celerity of elastic waves in concrete are used to determine the elasticity modulus of the specimen. Optical measurements present the advantage that there is no contact between the measuring device and the sample. This can be interesting in the case of a specimen at high temperature or during the localization of deformation, when strain field is very inhomogenous. With image analysis, the relative displacements of a grid or a speckle pattern printed on the specimen before the test are measured optically. Recent studies on concrete undergoing bi-traction have produced interesting results [AST 01]. This principle corresponds to field measurements: some laser or image analysis based devices are real “equivalent” displacement sensors and present similar characteristics to classical sensors but without any contact.

Figure 1.9.Lateral deformation chain sensor (MTS®)

1.3.1.3. Force sensors

A force sensor is chosen by default and belongs to the test machine. Nevertheless, the precision of this sensor, its measurement range, and its rigidity should be carefully considered. The working principle of a force sensor is to measure the deformation of a probe body (Figure 1.10), precisely known as a function of the applied force (determined in a factory using calibration). Depending on the boundary conditions imposed to the specimen during the test, the rigidity of the sensor has to be analyzed.

Figure 1.10.Photograph of a force sensor (Béta®)

1.3.2. Boundary conditions and experimental set-up

Application of the desired boundary conditions is always done using a testing machine and supports on which the specimen is placed. Even if those supports can seem to be ideal, the specimen-machine interface influences the experimental result, and the testing frame as well. For example, during a compression loading test on a high-strength concrete, a brittle failure is observed (usually with the explosion of the specimen), mainly due to the elastic unloading of the test frame during the post-peak loading phase. The stiffer the frame is, the weaker its elastic deformation and the weaker the “explosion” of the specimen will be. Concerning the plates or the supports of the specimen, the ideal case would be that for contacts between the specimen and the test machine, the experimental apparatus exactly follows the deformations of the specimen (under the Poisson effect, for example). This is difficult to obtain, but it is possible to get close to it. Friction between the specimen and the plates of the machine can lead to experimental bias such as spurious bracing.

Therefore, preparation of the surfaces of the specimen should be done extremely cautiously. It is essential that that force transmission between the test machine (assumed to be as perfect as possible) and the specimen is done without any imperfections. In order to guarantee this, surfaces in contact with the plates of the machine have to be perfectly flat (general case) or, in general, that the platespecimen contact is homogenous (in the case of brush platen supports, Figure 1.14). In addition, parallelism between the outermost surfaces of a uniaxial or triaxial compressive specimen has to be as perfect as possible in order not to introduce a bending moment, which could alter the homogeneity of the compressive stress.

Along the same line, perpendicularity between the axis of the specimen (cylinder or cube) and the support surfaces has to be guaranteed. To overcome these two defects, a ball joint and fixed support system can be used (see Figure 1.11 for a uniaxial compression test). The fixed support enables the reference frame of the specimen to be aligned with that of the machine, whereas the ball joint mostly erases the perpendicularity and parallelism defects. Other types of boundary conditions can be used as a function of the relative importance of those defects regarding the mechanical behavior of the material, for example in the case of uniaxial deformation where the homogeneity of the strain field within the specimen is extremely sensitive to the boundary conditions. Figure 1.12 shows an example of a behavior obtained in uniaxial compression in terms of axial stress as a function of the axial deformations measured by two gages located at both ends of the specimen in its center. When the parallelism of the faces of the cylinder, and also the perpendicularity of its generator are not perfect regarding the machine, there might initially be some induced tensile strain in the specimen due to the bending moment, which obviously can lead to spurious damage of the material. Moreover, the obtained behaviors present total and irreversible different deformations, whereas a real material test should give an identical response. Figure 1.12 shows the stress versus (axial and radial) deformation curves obtained on a specimen similar to the previous one, with ball joint to support boundary conditions, which gives much better results. Finally, the choice of ball joint is a function of the size of the specimen so as to minimize the friction forces (within the ball joint) induced by the specimen regarding the ball joint effects.

Figure 1.11.Classical supports of a test machine: ball joint and fixed. Fracture mode of concrete under uniaxial compression [DUP 95]

Figure 1.13 shows the consequence of boundary conditions and the size of the specimens on the uniaxial tensile behavior of sandstone [MIE 96] (whose tensile behavior is similar to the behavior of a mortar). In the case of blocked supports in rotation (Figure 1.13a), the obtained behavior depends on the size of the specimen (different maximal value, different ductility). When the supports are free to rotate (Figure 1.13b), the obtained experimental responses are independent of the size of the specimen. This difference can be explained by the fact that in the case of fixed rotating supports, right at the initiation of a microcracking on the edge of the specimen, a bending moment occurs which is not avoided by the experimental set up. This moment is more important as the diameter of the specimen increases, which leads to lower apparent strengths for larger sizes (without correction for this bending moment). If the supports are ball jointed, this bending moment is balanced using the rotation supports.

Figure 1.13.Effect of the supports on the mechanical behavior of sandstone in traction: a) fixed rotating supports; b) ball jointed supports [MIE 96]

Friction between the specimen and the test machine is often unavoidable, and leads to the bracing of the edges of the specimen (Figure 1.11). In the case of uniaxial compression, bracing cones appear if friction is not avoided, resulting from a triaxial compression state at the ends of the specimen. The obtained cracking facies is then dependant on friction (Figure 1.11b). If there is very little friction between the plates of the machine and the specimen (Figure 1.11c), the compressive stress is uniform within the specimen and leads to vertical cracks. Reduction of friction is provided by anti-bracing systems, for example, grease, brush platens (Figure 1.14a), sand boxes (Figure 1.15), deformable supports, aluminum foil covered with talc, or a Teflon® plate. In general, those systems follow the specimen during its deformations perpendicular to the stress direction (usually due to the Poisson effect within the pane of the supports), while ensuring that no spurious force appears. Regarding uniaxial compressive tests, one of the first ideas was to lubricate the plate-specimen contact using grease. This works well for small applied loads, but for higher applied loads grease is driven away. Talc and stacked aluminum foils can also be placed between the specimen and the machine. It is also possible to use the plates of a material with a low friction coefficient between steel and concrete, such as Teflon®. Finally, a sand box can also be used [BOU 89] (Figure 1.15), whereby the sand grains act like small marbles allowing the lateral motion of the specimen surface under the Poisson effect.

Figure 1.14.a) Example of a brush platen; b) brush rotation effects [MIE 84]

These devices are suitable for uniaxial compression. In the case of multiaxial stresses, brush platens are more suitable (Figure 1.14a). The idea is to have a brush applying a compressive force on the specimen. The bending stiffness of the brush is as small as possible in order to avoid bracing effects (Figure 1.14b). The brushes are steel barrettes maintained together. The design of those special supports is based on the buckling loads of the brushes.

In general, the specific type of support used in the experiments induces some consequences on the experimental results, after the peak load is reached especially [KOT 83; MIE 84; TOR 87; VON 89; MIE 91; TOR 91; VLI 96; MIE 97a]. We can sum them up regarding uniaxial compression as follows [TOR 96]:

– elastic behavior is virtually the same, whatever the boundary conditions are;

– post-peak behavior of the specimens is strongly dependant on the boundary conditions of the specimen;

– localization of the deformations is observed (generally occurring just before the peak), and its appearance is maximal at the stress peak, leading to a macrocracking being parallel to the compression axis (Figure 1.11c);

– this localization depends on the boundary conditions;

– friction effects are more important when the specimen is small, a test without any anti-friction system leads to an overestimated ultimate strength;

– characterization of the post-peak behavior of the material is very difficult and problematic (see section 1.4.5.3).

Figure 1.15.Sand box [BOU 89]

An alternative to the use of specific supports is the inter-position of material interfaces between the specimen and the test machine, which bear transverse deformations similar to the deformation of the concrete tested: the ratios of the Poisson coefficient with Young’s modulus have to be close for the specimen and the interface. Aluminum [TOU 95] or reactive powder concretes [TOR 99a] may be used as interface materials.

1.4. Behavior of concrete under uniaxial stress: classical tests

In the first section of this chapter, we introduced most of specificities of experimental analysis on concrete. With what follows, we will describe classical tests. The first part will deal in more details with the uniaxial tests that are the most used, as most of the stress states in civil engineering are uniaxial. In the second part, multiaxial tests, necessary for particular applications, will be presented.

1.4.1. Direct uniaxial tension

During this test, a homogenous tensile stress is applied on the specimen using steel or aluminum heads stuck to the ends of the specimen (Figure 1.16). A double ball jointing of the ends of the specimen is used to make sure the tensile strain is homogenous within the specimen (see section 1.3.2). In general, the test is carried out at an imposed rate. This is one of the hardest test to perform as the brittle behavior of concrete under this type of stress only provides the first part of the curve, that is to say until the loading peak. At that point, local rupture of the material occurs causing the propagation of a through crack within the specimen. Localization of deformation comes along with cracking and causes an elastic unloading of the other parts of the specimen. Subsequent deformations are not homogenous within the specimen, the zone where they monotonically increase is called the “zone of deformation localization”. Elastic unloading may cause the loss of stability of the specimen and brutal failure. It is then difficult to obtain the behavior of the softening phase. In order to obtain the post peak response of the specimen, it is often necessary to implement some local displacement control during the test (local relative displacement measured on the specimen) [TER 80, GUO 87, VIS 94].

Figure 1.16.Principle of direct traction test

In order to better control post-peak behavior, in other words, to control the localization of the deformations, some authors rely on notched specimens (Figure 1.16b; [MIE 93]) or “diabolo”-shaped specimens (Figure 1.16c; [BEL 96]) allowing the localization zone to be known and then equipped with displacement transducers. Typical responses are shown in Figure 1.17a: initially, the material is almost elastic and linear until the peak, for stresses of about 2 to 5 MPa regarding usual concretes, and a softening phase follows the loading peak. During softening the material becomes damaged, and its elasticity modulus drops sharply, due to significant cracking. Some irreversible deformations are then observed if the material is unloaded. Visser and van Mier have shown that the saturation degree of concrete has a small influence on tensile behavior (Figure 1.17b; [VIS 94]). Nevertheless, the ratio between the peak stress of a dry mortar and of a humid mortar (saturated in water) in direct traction is always higher or equal to 1, and varies as a function of the nature of the aggregates [TER 80]. Starting at the onset of localization of the deformations and after, the curve σ – ε has no real intrinsic meaning if the strain ε is not the strain measured inside the localization zone (where the strain is assumed to remain almost constant).

Figure 1.17.Behavior of a strained concrete in direct tension: a) case of loading-unloading [TER 80]; b) influence of the water content [VIS 94]

1.4.2. Indirect uniaxial tension

Due to the experimental difficulty of carrying out direct traction tests, different tests are more commonly used, relying on the dissymmetry of the compressive and tensile strengths of the concretes. In some specific experimental situations, it is possible to obtain locally a tensile fracture of the specimen being loaded in compression. The most commonly used test is called the “Brazilian” or splitting test which is the diametral compression of a disk (Figure 1.18; NF P 18-408 rule). The distribution of the normal stress is sketched in Figure 1.18b. A quasi-uniform tensile stress develops along the diameter of the specimen. This tensile stress is proportional to the applied compressive force. Only the maximal force is measured, this test is only useful to determine the maximal tensile strength. It is not useful to determine the tensile response of concrete. The stress distribution is far from being homogenous over the specimen and the determination of the tensile response would require some structural analysis and then be determined indirectly.

Figure 1.18.Indirect traction tests with splitting and stress distribution

Figure 1.19.Indirect traction tests through three-point bending

Another type of indirect traction test is the three-point bending test on a concrete specimen that may or may not be notched (Figure 1.19, RILEM recommendation [RIL 73]) or the four-point bending test. The principle is to develop a moment within the beam, and therefore, to call upon the tensile lower fibers, the higher fibers being elastic due to dissymmetry behavior. The boundary conditions used are roll supports at the ends of the beam, to enable shrinkage during loading. The applied force and the deflection at the center of the beam are measured. It is possible to stick deformation gages on the beam in order to get local information, or cracking recording gages above the notch. The test control is achieved through an imposed displacement or an imposed force.

Because the tensile strain is not homogenous within the body of the specimen, same as in the splitting tests, the interpretation of the obtained results is difficult [SAO 88]. Nevertheless, with a three-point bending test it is possible to determine the tensile response of the material using a reverse analysis because of the simplicity of the distribution of the normal stress within the median section of the beam. Moreover, a notch is generally placed in this median cross section in order to introduce a defect minimizing the resistant section, and therefore to enable better control of the cracking and collapse process. It is then possible to control the test as a function of the crack propagation, which allows the tensile softening response of concrete to be determined because loss of stability of the loading process is avoided. This test is most often used for the determination of rupture parameters, or the characterization of the efficiency of reinforcement in concrete (reinforced concrete, fiber-reinforced concrete, adhesive composite material supports).

Figure 1.20.Typical force: deflection result with three-point bending (without any initial notches): a) without any particular precautions and b) with precautions

The three-point bending test requires some caution. Crushing of concrete at the supports often occurs (Figure 1.30a). This leads to a variation of the absolute position of the center fiber: therefore, the reference of the deflection has to be independent from possible crushing. This is performed by associating the concrete beam with a bar supported by two points belonging to the center fiber and using it as a measurement base. If those precautions are taken, we obtain a reliable representative response, from which it is possible to back calculate the tensile response of the material.

Figure 1.21.a) Principle of the PIED test; b) axial stress-axial deformation curve of the obtained concrete [RAM 90]

Figure 1.22.Principle of bi-tube traction test [LUO 86]

1.4.3. Diffuse tension test

This diffuse tension test called PIED (acronym in French for “Pour Identifier l’Endommagement Diffus”, Figure 1.21a) has been developed to “stabilize” the test, by controlling the appearance of one or several micro-cracks [PIJ 87; BAZ 89; MAZ 89; RAM 90; BER 91].

The idea is to apply the tensile force on concrete using aluminum rods stuck to the specimen. Those rods are elastically deformed, and impose a homogenously distributed deformation on concrete. Analysis of the results is done by uncoupling the behaviors of aluminum (the elastic behavior is well known) and the behavior of the concrete, same as in reinforced concrete. The results are similar to those obtained for a direct tension test. The process has been improved in order to achieve cyclic loads [RAM 90] (results are presented in Figure 1.21b) or to allow the material permeability to be measured upon microcracking (BIPEDE test [GER 96]).

Figure 1.23.Principle of simple compressive test

1.4.4. Bi-tube tension test

The principle behind this tension test is again to use the tensile/compressive dissymmetric behavior of the material. Luong had the idea to design a “specimenstructure” subjected to compression in order to induce tension in the material (Figure 1.22; [LUO 86]). The test consists of compressing the center part of the specimen whilst simply supporting the outer part. The intermediate part is then subjected to tesion, on a quasi-homogenous way. Both the center and outer parts of the specimen remain elastic during compression. The advantage of this method is that setting up the test is as simple as a regular simple compression test (see the following section).

1.4.5. Uniaxial compression

1.4.5.1. Behavior and rupture

This is the most commonly test (Figure 1.23). It is carried out on cylinders or cubes of concrete. In general, the normalized test is controlled at an imposed stress rate, but an imposed displacement allows the post-peak regime of the response to be obtained. Regular concretes are slightly more ductile than rocks. Their maximal strengths vary from 20 to 800 MPa (from ordinary concretes to reactive powder concrete now called Ductal®).

Figure 1.24.Typical results of simple compression test on concrete [RAM 90]: axial ( ε1) and radial ( ε2) deformations as a function of the axial stress

During loading, deformations perpendicular to the principal compressive stress appear, creating micro-cracks as the tensile deformation threshold is being exceeded. Micro-cracks coalescence leads to the collapse of the specimen. Moreover the elastic characteristics of the material evolve; the elasticity modulus decreases during the loading whereby the material becomes damaged due to micro-cracking. Some irreversible deformations appear. As we have already seen, the boundary conditions of the specimen play an important role on the characterization of the behavior of the material during simple compression. Due to friction, bracing cones appear at failure. Just the central part of the specimen is subjected to a uniaxial compression stress. Various “anti-bracing” devices have been developed (see section 1.3.2). Figure 1.24 shows a typical compressive uniaxial response of concrete. After the peak load, the Poisson coefficient suddenly increases. In the same way, damage growth is occurring more rapidly. The various stages of damage during a uniaxial compression test are presented in Figure 1.25 [MAZ 84].

In addition, it can be seen on Figure 1.24 that the material is slightly visco-elastic during compression. This is illustrated by the curved shape of the unloading curves as well as the hysteresis during loading-unloading cycles. Visco-elasticity is a function of the water content of concrete and decreases with the drying of the material.

Figure 1.25.Results comparing different analyses of concrete deterioration under uniaxial compression [MAZ 84]

1.4.5.1.1. Water content influence

Interlayer water within the material influences its visco-elastic behavior. Several other phenomena are related to the water content and the presence of hard inclusions (aggregates) within a cementitious matrix, which retracts as the water leaves. Concrete and mortar strengths increase when free water leaves [TER 80; POP 86; BAL 94; BUR 00; YUR 04], because of the capillary effect and increased suction within the partially saturated porous medium. At the same time, micro-cracking induced by desiccation shrinkage leads to a decrease of the initial elasticity modulus of the material [TER 80; TOU 95; BUR 00; YUR 04], thus a decrease of Young’s modulus [YUR 04]. This micro-cracking is due to the structural effect induced by the shrinkage difference between the core and the surface, and also to the effect of rigid inclusions within a retracting matrix, which leads to micro-cracking. The latter effect is the more important if the size of the aggregate is small [BIS 01], it will then be more sensitive within concretes than within mortars.

1.4.5.2. Measurements of the elastic constants of the material

Concrete can be considered as an initially isotropic material. The elastic parameters of the material are Young’s modulus (E) and Poisson coefficient (ν). Regarding common concretes, those usual values of the parameters are 30,000 MPa and 0.2, respectively, and are used in numerous constitutive laws and numerical calculations for concrete structures, as well as for the determination of the delayed (time dependent) deformations of concrete. Therefore it is important that the measurement of these parameters be accurate [BOU 99; TOU 99] following the rules of RILEM [RIL 72] or specific procedures such as the one proposed by the laboratoire central des ponts et chaussées [TOR 99b], in order to avoid any errors and misinterpretations due to the visco-elastic behavior of concrete. Moreover, it is also necessary to use an adequate extensometer [BOU 99] or deformation gages stuck at the center of the specimen. Figure 1.36 provides an example. After three cycles of loading-unloading, which can reach 9 MPa (about 25% of the compression strength), the elasticity modulus (E0) is measured as the slope of the line passing by the vertexes of the third hysteresis loop (if this loop is really occurring, otherwise, regular measurement is being carried out). The Poisson coefficient is equal to the ratio of the transversal modulus (Et) and the longitudinal modulus (E0), multiplied by (–1). Concretes present an elastic behavior up to 60% of their peak strength.

1.4.5.3. Characterization of post-peak behavior

The peak corresponds to the maximal value reached by the compression stress. In general, at this state, we observe the formation of macro-cracking parallel to the direction of compression [TOR 93a]. Experimentally, it is difficult to obtain the softening response because redistribution inside the specimen occurs and the strain distribution is no longer homogenous over the specimen. Moreover, the testing frame will also unload. The induced stress drop can be unstable with respect to the control mode of the test. In the case of imposed force, instability occurs at the force peak. In the case of imposed displacement, instability will occur when acceptable displacement increments are negative. We then talk about snap-back.

The post-peak response of concrete is necessary, for example, in studies related to the durability of the material as transport properties are very sensitive to the cracking of the material (see [BAZ 82; ACK 88; BAR 92] for general information). Displacement control of the plates of a test machine, which is very rigid, provides the post-peak response of a surfaced concrete specimen experimentally. Figure 1.27 shows an example axial stress-strain curve. Stress is calculated from the force of the machine and strain is global, that is to say computed from the variation of distance between the supports of the specimen. This type of curve cannot be used for the absolute measurement of elastic parameters or behavior, but allows comparisons (peak stress, peak deformation, softening, ductility, damage evolution, irreversible deformation evolution, etc.) between the tests performed under the same conditions.

Figure 1.26.Characteristics of post-peak behavior

Figure 1.27.Simple compressive behavior (σ–ε) curve of a concrete (water:cement=0.63) saturated in water, as measured by the compressive plates

Another possibility to obtain the complete curve of the uniaxial compressive mechanical behavior of concrete is to control the test as a function of an increasing parameter, for instance, the circumferential lateral deformation of the specimen [SHA1 87; GET 96; TOR1 99]. Indeed, this quantity always increases, even during the softening phase. It is also possible to “turn” the classical plane σ-ε, in order to control the test within a plane for which “turned deformation” will always be increasing: the “turned stress” axis is a tangent to the curve of the behavior of concrete at the origin [MIE2 97; JAN 97], the rotation angle of these axis being that between the stress axis and the elastic part of the behavior of the material.

1.4.6. Uniaxial torsion