CIVL 111 CONSTRUCTION MATERIALS

2.1.1 Introduction
       Under a small load, all materials behave in an elastic manner. When stress is applied,
strain will be resulted. Once the stress is removed, the material returns to its original state
(i.e., the strain goes back to zero). Moreover, before a certain critical state is reached, the
stress and strain are linearly dependent on one another. Linear elastic behaviour is the basic
assumption of many engineering analyses. In this section, we are going to investigate the
physical basis of linear elastic behavior, define the Young‟s modulus and describe how it
affects structural design. Then, we will study the elastic behaviour of composite materials
(i.e., when two or more materials are used in combination to carry the load). This will
provide a foundation for the understanding of reinforced concrete design, which will be
taught in another class.
2.1.2 Physical Basis of Elastic Behaviour
       To most of us, the first manifestation of elastic behaviour is perhaps the behavior of a
spring. On loading, the deformation is clearly visible and on unloading, the spring returns to
its original length. The elastic behaviour of all materials can be explained in terms of the
simple spring. As we all know, materials consist of atoms that are held in equilibrium
positions by bonds. Since the bond force is linearly dependent on interatomic distance, bonds
can be considered as small springs placed between atoms (Fig.2.1). When loading is applied,
the atoms will move relative to one another. The movement will stop once the bond force
(resulting from change in interatomic distance) is in balance with the applied force.    Once
the loading is removed, the atoms will move back to the original equilibrium positions. The
behaviour of the whole material is therefore linear elastic.
       In Fig.2.1, for simplicity, we have only considered the presence of bonds along the
direction of the loading. In reality, there are interactions between neighbouring atoms at all
directions. (Actually, the interaction may extend beyond the first layer of neighbours, but
this is beyond the scope of our discussion here). Fig.2.2 shows an example of such kind of
interaction. Again, if we consider the bond to be a spring, when loading is applied in the
direction shown in Fig.2.2, the stretching of the spring results in forces with two components,
one parallel to the applied load and the other perpendicular to it. As a result, under loading,

materials not only deform parallel to the loading direction, but will also deform perpendicular
to it. This deformation is called the Poisson‟s effect and is important in understanding the
effect of confinement on material behaviour, a topic of importance in reinforced concrete
column design.

                                                   Bond behaves like spring

                                                     Displacement of atom as
                                                     bond stretch under load

                  Figure 2.1 Displacement of atoms under applied loading

                                        Vertical component of spring force
                                        pulls atoms away from one another

                                      Horizontal component of spring force
                                      moves atoms closer to one another

                        Figure 2.2 Illustration of the Poisson‟s Effect

2.1.3 Young‟s Modulus: Definition, Typical Values and Significance to Structural Design
        In the linear elastic regime, the stress () and strain () are directly proportional to one
another. The Young‟s modulus, E, is defined as:
        E =  / 
       The Poisson‟s ratio  is defined as:
         = - (strain perpendicular to loading direction) / (strain along loading direction)
        Note the negative sign in the equation for : with this definition, the situation
illustrated in Fig.2.2 will result in a positive value of . As examples,  = 0.28 for steel and
range from 0.14 to 0.20 for concrete.
        In some applications, which involve the shear deformation of materials, the shear
modulus G is used. For an isotropic material, which is defined as a material with equal
properties in all directions, G is related to E and  through:
        G = E / 2(1+)
        A high value of E means that a high level of stress is required to produce a given
strain. The material is then said to have a high stiffness. Physically, the magnitude of E is
governed by the intensity of the bond between atoms. Primary bonds such as covalent, ionic
and metallic bonds result in high E values. Secondary bonds, including hydrogen bond and
van der Waal‟s force, give rise to low values of E. The Young‟s moduli for common
materials are given in Table 2.1. The covalent bonded diamond is the stiffest material in the
world and included here as a reference. Steel, which is held by metallic bonds, has a high
modulus of around 200 GPa. Wood and polymers have a low modulus of 16 GPa or below.
Both wood and polymers are built up of long chains of carbon atoms. Along the chain, the
atoms are covalent bonded and therefore the stiffness is very high. However, individual
chains are held together by either the weak secondary bonds or occasional cross-links. It is
therefore quite easy for the chains to slide relative to one another, resulting in the low E
values. The modulus of concrete, depending on mix proportions, is around 20 to 40 GPa.
Concrete consists of many different chemical phases, fine and coarse aggregates as well as
pores. Its modulus is affected by many factors and we postpone the discussion to the chapter
on concrete.
        It is important for us not to confuse between the two concepts of strength and
stiffness. For example, aluminum and glass both have a modulus of 69 GPa. That is, they
have comparable stiffness.      However, their strength and failure modes are completely

                 Table 2.1 Reference E values (in GPa) for common materials
Diamond                  1,000                    Wood ( // grain )        9 - 16
Steel                    190 - 210                Wood ( | grain)          0.6-1
Aluminum                 69                       Polyesters               1-5
Glass                    69                       Epoxies                  3
Concrete                 20 - 40                  Ice (H2O)                9.1

        The Young‟s modulus is the material parameter governing the deformation of a
structure (Note: deformation is of course also affected by the member size). When material
of a lower E is used to replace one with a higher E (e.g. aluminum is employed to replace
steel to reduce environmental corrosion), the deflection should be checked to make sure that
it is not excessive. A reduction of E will also increase the likelihood of buckling failure,
which is the sudden lateral deflection of a slender member under compression (This can be
easily illustrated by compressing a thin plastic ruler). It should be noted that when a material
is damaged, its E value is always reduced. Since the speed of stress wave propagation in a
material is proportional to the square root of E, the wave speed in a damaged material will
also be reduced. By sending a wave into a structural member and measure the time for it to
travel between two points, the damage condition of a structure can be assessed in a non-
destructive manner.
2.1.4 Modulus of Composite Materials: Application to Reinforced Concrete Column
        In this section, we will consider the elastic behavior of composite materials. This is of
relevance because: (i) the same concepts apply to the analysis of reinforced concrete
members, (ii) there is increasing interest in the use of fiber reinforced composites in civil
engineering applications. Our discussions will be limited to composites with two phases, but
they can be easily extended to more general cases. Only two simple cases will be considered.
In both cases, the two phases are considered to be planar and aligned in the same direction
(Fig.2.3). In case 1, the loading is applied parallel to the aligned direction, and in case 2, the
loading is applied perpendicularly. The analysis of each case is given below.
Case 1: Loading along aligned direction
Assume a total load carrying area of A, and an average applied stress of . The total applied
force is therefore given by A. Let Va be the volume fraction of phase A, and Vb (= 1 - Va)
be the volume fraction of phase B. Phases A and B are bonded together. When loaded in
parallel, they must deform by the same amount (otherwise, they will not be fitted together

any more). In other words, the strain in each phase (along the loading direction) must be the
same, i.e., a = b = . The stress in the phases are then given by:
        a = Eaa= Ea
        b = Ebb = Eb
        The force carried by each phase is equal to the stress in each phase multiplied by the
        Fa = VaAa = VaA Ea
        Fb = VbAb = VbA Eb
        Force equilibrium requires F = Fa + Fb, which gives:
         = (VaA Ea + VbA Eb) 
or,     E =  = (VaEa + VbEb)
        This is called the parallel model, which states that the composite modulus is simply
the weighted average of the phase moduli, with the corresponding volume fraction used as
the weight.


                      CASE 1
                                                                      CASE 2
                                                             Loading perpendicular
        Loading along aligned direction
                                                               to aligned direction

              Figure 2.3 Calculation of composite modulus for two different cases

Case 2: Loading perpendicular to the aligned direction
        In this case, since the two phases are loaded in series, the stress on each phase must be
the same in order for equilibrium to be satisfied. That is, a = b = , where  is the applied
stress. The strain in each of the two phases are:
        a =  / Ea
        b =  / Eb
        Assuming the thickness of composite to be H.
        The total extension in phases A and B are given by:

         ea = a Va H
         eb = b Vb H
         For the composite, the total extension is given by e = H. Since e = ea + b,
          =  ( Va/Ea + Vb/Eb )
or       E =  = ( Va/Ea + Vb/Eb )-1
         It should be noted that cases 1 and 2 give the upper and lower bound for the elastic
modulus of a composite system. In a composite with any arbitrary arrangement of the two
phases, the modulus will always lie between the values given by the two expressions derived
Example: Elastic Behavior of a Reinforced Concrete Column
A 200 mm x 400 mm rectangular concrete column is reinforced with six 25 mm diameter
bars. The length of the column is 3 m. An axial load of 1000 kN is applied. How much
would the column be shortened and what are the stresses in the concrete and the steel? (Take
Es =200 GPa, Ec = 26.7 GPa)
The volume fraction of steel is given by: 6(12.5)2/(200x400) = 0.0368
Since steel and concrete are loaded in parallel, the effective modulus is given by:
         E = (0.0368) (200) + (1-0.0368) (26.7) = 33.08 GPa
Shortening of the column
= [1000 x 103 (N) / (200 x 400 (mm2)x 33.08 x 103 (N/mm2) )] x 3000 (mm)
= 3.78 x 10-4 x 3000 (mm) = 1.134 mm
Stress in steel
= 3.78 x 10-4 x 200 x 103 (N/mm2) = 75.6 N/mm2 (or 75.6 MPa)
Stress in concrete
= 3.78 x 10-4 x 26.7 x 103 = 10.1 MPa

NOTE: When reinforced concrete members are subjected to bending, the strain varies over
the depth of the member. To analyze its behavior, we make a similar assumption: the strain
in the steel is equal to that in the adjacent concrete. Details will be left to the class on
reinforced concrete design.

2.2.1 Phenomenon of Plastic Yielding
        When loading on a piece of metal is continuously increased, yielding will eventually
occur. Plastic yielding is indicated by a significant increase in deformation with a relatively
small increase in load (or, in the case of mild steel, with no increase in load). The ultimate
load carrying capacity, however, is higher than the load at which yielding first occurs. As a
result, the large deformation after yielding serves as warning before final failure occurs.
Materials that yield are often referred to as ductile materials, and ductility is a desirable
feature for all structures.
        The stress-strain behaviour of a typical metal is shown in Fig.2.4 below. From the
figure, the following can be defined:
        y : yield strength, the point where a sudden change in slope occurs
        0.1% : the 0.1% proof stress, obtained from a line starting from 0.1%  and
                extending parallel to the initial slope. This is often used when an abrupt
                change in slope is difficult to identify. One can also report the proof
                stress at other strain levels (0.2%, 0.3%) based on the same concept.
        TS: ultimate strength of the material
        f : tensile ductility or the plastic strain after failure

                                      Hardening          Onset of Necking
                       TS
                      
                       y                                                  Fracture
                                              Unloading line
                                               initial slope

                                                                f      

                         Figure 2.4 Stress-Strain Curve for most Metals

       For most metals, once the yield strength is reached, continuous straining will be
accompanied by a more gradual increase in stress. This stress increase is referred to as
hardening. During the hardening state, the deformation is still uniform along the member
(i.e., the strain is the same at all points). Once the ultimate strength is reached, localization of
deformation starts to occur at a particular section. On further straining, the area of this
section becomes smaller and smaller.          This phenomenon is called „necking‟ as strain
localization leads to the formation of a „neck‟ along the loaded member (see illustration in
Fig.2.4). Final failure is due to plastic fracture, and the member breaks into two parts. When
the two parts are fitted back together, the length (L) is often significantly above the original
length (Lo). The ratio (L - Lo)/Lo is called the tensile ductility. For steel, this value can range
from 0.1 to 0.6, depending on the grade.
       The stress strain curve shown in Fig.2.4 is applicable to most metals. However, it
does not represent the behaviour of mild steel, a material widely used in construction. The
stress strain curve of mild steel is shown in Fig.2.5. The curve exhibits an upper yield point
and a lower yield point. With further straining beyond the lower yield point, a constant stress
region (called the yield plateau) is observed before the material starts to harden like other
metals. It should be noted that the upper yield point depends on loading speed and specimen
type, while the lower yield point stays constant. As a result, the lower yield stress is taken to
be the yield strength of mild steel and is the value used in structural design.

                                 Upper and Lower
                                    Yield Points


                                           Unloading line
                                            initial slope

                           Figure 2.5 Stress Strain Curve for Mild Steel

2.2.2 Physical Basis of Plastic Behaviour
          The physical origin of plasticity has remained a puzzle for many years. Considering
the simple spring model in Fig.2.1, plastic behaviour will be resulted if each spring behaves
in a plastic manner. In other words, the bond force has to vary with distance in a way similar
to the stress strain curve. Analysis of inter-atomic interactions based on electrostatic theory
indicates that this is not the case. Moreover, analysis shows that the stress required to break
the bond is about one order of magnitude lower than the Young‟s modulus. For most metals,
however, the yield strength (and tensile strength) is about two to three orders of magnitude
lower. How can this be explained? The answer to this interesting question also provides a
basis for explaining plastic behaviour.
          A key in understanding plastic behaviour is the observation that atomic bonds do not
all break at the same time. Since only a small fraction of bonds are broken at a given time,
the applied stress at which bond breakage occurs is a lot lower than the theoretical value.
When forces are applied along bonds to pull them apart (as in Fig.2.1), there is no mechanism
for bonds to break one by one. However, if shear stress is applied, bonds can easily break
and re-

           A      C                          A       C                       A       C   E

           B      D                      B       D                       B       D       F

                                         A       C           E   G           A       C       E       G

                                     B       D           F       H   B       D       F       H   b

               Fig.2.6 Illustration of Plastic Deformation due to Dislocation Movement

form one after another. This process is difficult to explain in words but is illustrated in
Fig.2.6. One can observe from the figure that shearing leads to the stretching of bonds
between two horizontal layers (between atoms A and B, C and D). Eventually, these bonds
will be broken and replaced by a new bond between atoms A and D. The breaking and re-
formation of bonds (between different pair of atoms) result in an extra plane of atoms in the
atomic lattice. Such an extra plane is called a „dislocation‟. On further shearing, additional
bond breaking and re-formation move the dislocation from one side to the other side of the
atomic block. A „step‟ equal to the atomic spacing is then created. Continuous application
of the shear stress can result in another series of bond breaking and re-formation, leading to
additional displacement between the upper and lower blocks. The relative sliding between
the two blocks is the cause of the large strain after yielding.
       In Fig.2.6, for illustration, we show the formation of a dislocation under applied
stress. In real materials, dislocations pre-exist everywhere inside the atomic lattice. When
shear stress of a sufficient magnitude is applied, dislocation movements occur all over the
material. The material can then be considered as consisting of many separate „blocks‟ trying
to slide relative to one another. The strain in the material is therefore made up of two parts:
the elastic stretch of bonds within each block, and the relative sliding between the blocks.
The elastic part of the strain is completely recoverable on unloading, while the sliding leads
to the irrecoverable plastic strain. This is the reason the unloading line after yielding is
parallel to the loading line. Since dislocations can move in different directions, the blocks
also tend to slide in different directions and can eventually get into the way of one another,
making movement more difficult. This explains the existence of the hardening regime, where
increased stress is required to continue the yielding process.
       The dislocation theory described above implies that yielding is resulted from shear
stresses. When there is no shear stress (e.g., equal tension is applied in all directions),
yielding will not occur. As a result, a criterion for yielding should be based on the shear
stress. A discussion of common yield criterion will be postponed to the chapter on steel. To
increase the yield strength (or, to „harden‟ the material), the resistance to dislocation
movement has to be improved. Since many hardening techniques can lead to a reduction in
ductility of the material, the potential compromise between strength and ductility of metals
should be kept in mind. Specific hardening methods for steel and their effect on ductility will
also be presented in a later chapter.

2.2.3 Modelling of Plastic Behaviour
       Fig.2.7 shows two models for plastic behaviour.              In structural analysis, the
elastic/perfectly plastic model is often used for two reasons. First, it is a good approximation
for mild steel, which is commonly used in construction. Second, for a hardening material, it
simplifies the analysis and provides conservative results. The rigid perfectly plastic model is
employed for the calculation of ultimate collapse load for a ductile structure. Since collapse
occurs at a strain much higher than the yield strain, the initial elastic part can be neglected in
such an analysis.

                               Rigid/Perfectly Plastic Model

                                    Elastic/Perfectly Plastic Model

                           Figure 2.7 Models for Plastic Behaviour
2.2.4 Illustration of Plastic Behaviour with a Parallel System
       To illustrate plastic behaviour of structures, we will consider the parallel system
shown in Fig.2.8 below. The simple system consists of several members working together to
carry the applied load. One of the members will yield before the others. Real structures
consists of many members to carry the load (e.g., a building with many columns plus an
internal core to carry wind load, a bridge with multiple spans, each contributing to carrying
the traffic load on other spans). Each individual member will generally yield at a different
load. The qualitative behaviour of the simple system in Fig.2.8 therefore resembles that of a
real structure. By studying the simple system, post-yielding behaviour of real structures can
be understood.

                       1        3
             2L                         Rigid

                                                          y                      

               u : applied displacement
                   Figure 2.8 A Parallel System to Illustrate Plastic Behaviour
       In Fig.2.8, let‟s assume all members to have the same cross sectional area A. The
objective is to find the load (F) corresponding to a given displacement (u).
       The members do not yield at the same displacement.
For members 1 and 3, yielding occurs when: u = 2Ly
For member 2, yielding occurs when:             u = Ly
Elastic Stage: 0 < u/L < y
       F = AEu/L + 2AEu/(2L) = 2(AE/L)u
The first term is the contribution from member 2 and the second term is from 1 and 3.
Bar 2‟s yielding
       Bar 2‟s yielding occurs when: u = uy = Ly
       Fy = 2AEy
After First Yielding: y < u/L < 2y
       F = AEy + 2AEu/(2L)
          = AEy + 2AEy/2 + 2AE(u-Ly)/(2L)
          = Fy + (AE/L) (u - uy)
Before bar 2‟s yielding, the load increment is proportional to 2AE/L. After yielding, the
increment is proportional to AE/L (Fig.2.9). With bar 2‟s yielding, the structure becomes
more flexible on further loading. This is a general feature observed in any structure with
plastic members.

                          F          u - uy



                                   uy                               u
 Fig.2.9 Load vs Displacement for a Parallel system with Elastic/Perfectly Plastic Members
Whole system‟s yielding
       When the deformation of the whole system reaches 2 Ly , bar1 and 3 will yield. Thus
the whole system is yielded. The ultimate load capacity of the system is reached when the
whole system yields. The ultimate load is given by 3AEy. Let‟s compare this with the
ultimate load of a brittle member system. The strength and failure strain of the brittle material
are taken to be y and y. Since the material is brittle, once y is exceeded, the stress will drop
to zero immediately. At  = y, failure occurs in member 2. The load at this moment is F =
2AEy. After failure, member 2 cannot carry any load. The remaining load carrying capacity
of members 1 and 3 is also 2AEy. In other words, once the middle member fails, the system
cannot carry additional load and the whole system collapses.             This illustrates another
advantage of ductile materials over brittle materials. Even if the materials possess the same
strength (y in this case), the ultimate load carrying capacity of the ductile system can be
significantly higher than that of the brittle system.

2.3.1 Phenomenon of Time Dependent Behaviour
       For many materials (e.g. polymers, wood, concrete), the response to stress or strain
has a time dependent component. For example, when a fixed stress is applied, after an
instantaneous elastic response, the strain will continue to increase with time.               This
phenomenon is called creep and is illustrated in Fig.2.10(a). On the other hand, when a fixed
strain is applied (e.g., by stretching a member and then fixing its ends), the stress in the
member will decrease with time (Fig.2.10(b)). This phenomenon is called relaxation. If
creep and relaxation are linear (e.g., if the stress is doubled, the strain at a particular time is
also doubled), we can define the following two parameters:
       Creep compliance, J(t) = (t)/
       Relaxation modulus, Er(t) = (t)/
       The creep compliance can be obtained from a test with a fixed load applied to a
specimen.       Knowing J(t), the time dependent behaviour of the material under arbitrary
loading history can be obtained from superposition (see section 2.3.5).

                       (a)                                       (b)
                                                   

                                  elastic strain

                                        Time                                     Time

                     Fig.2.10 (a) Creep Behaviour, (b) Relaxation Behaviour
       For materials exhibiting creep behaviour, when a stress is applied, the strain will
increase with time. If stress is applied at a slower rate (i.e. over a longer period of time), the
resulted strain will be more than that due to a stress applied at a rapid rate. Fig.2.11 shows
the loading/unloading behaviour for two general cases. For creeping materials, the loading
and unloading curve do not overlap with one another. The area between the two curves
(called the hysteresis loop) reflects the energy absorped by the material over a
loading/unloading cycle. This energy absorption varies with loading rate, and is highest at
intermediate loading rate.

                           High Loading rate
                                   Intermediate Loading Rate

                                              Low Loading Rate

             Fig.11 Hysteresis Behaviour under High and Low Loading Rates
2.3.2 Implications to Structural Design
       When materials exhibit time dependent behaviour, it will affect structural behaviour
in a number of ways. The important effects of time behaviour are summarized below:
(1)    Due to creeping effects, the long term deformation of structures may be significantly
       above the short term deflection. Therefore, we should provide enough allowance
       between panels and other attachments to the primary structure. For large structure,
       the long term differential creep in different parts of a structure needs to be checked to
       ensure no problems will be caused.
(2)    The hysteresis loop as shown in Fig.2.11 indicates that energy can be absorbed during
       cyclic loading. The energy absorption results in damping of a structure as it is set
       under vibration (e.g., during an earthquake or typhoon). Note that the damping is
       frequency dependent, although this is often not considered in civil engineering
       designs, as damping is difficult to quantify in practice.
(3)    In prestressed concrete design, the creeping of concrete and relaxation of steel can
       lead to the loss of prestress. This has to be accounted for in design, and in some
       cases, re-stressing of the prestressed tendon has to be carried out.
(4)    Relaxation of a restrained member may lead to stress reversal. This is best illustrated
       by Fig.2.12, which shows a beam between two very stiff walls. As temperature
       increases, the beam tends to expand but its expansion is restrained by the walls. With
       this restraint, the beam is put under compression. Keeping the beam at the high
       temperature, it will eventually relax to a lower stress level. On cooling, the walls
       prevent the beam from contracting. As a result, tension is introduced.


                                         Stiff Walls

                         Fig. 2.12 A Beam constrained by Stiff Walls
2.3.3 Physical Basis of Time Dependent Behaviour
       Time dependent behaviour is due to the need of time for atoms or molecules to re-
arrange themselves under load. For example, when a polymer is under stress, the polymeric
chains tend to slide relative to one another. A finite time is required for the chains to go from
one state (i.e., a given arrangement) to another. When loaded for a longer time, more
movement will be resulted, leading to the phenomenon of creep. Relaxation can be explained
in a similar way. When a strain is suddenly applied, there is little time for the polymer chains
to move relative to one another. Most of the strain is then carried by the polymer chains
themselves, resulting in high stress. As time progresses, the relative movement between the
chains allows them to relax. The stress will then be significantly reduced.
       The rate at which molecular re-arrangement can occur depends on the thermal energy
of the molecules. As temperature increases, the energy also increases and creeping (or
relaxation) occurs at a higher rate. For metals and ceramics, the significance of creeping can
be assessed by looking at the homologous temperature defined by T/T m, where T is the
current temperature and Tm is the melting point of the material. Both temperatures should be
measured in the absolute scale (i.e., degree C plus 273). If T/Tm > 0.3 - 0.4 for metals, or if
T/Tm > 0.4 - 0.5 for ceramics, creeping will start to become important. For polymers, the
melting point is not well defined. Creeping becomes significant when the temperature goes
above the glass transition temperature (TG) of the material. Physically, this is the temperature
above which the van der Waal‟s forces between polymer chains start to break. In other
words, the chains start to slide after TG is exceeded. Creep and relaxation behaviour then
comes into the scene. At room temperature, common materials that may creep significantly
include concrete, wood, most polymers as well as lead, tin and glass.
       In general, the creep rate (i.e., the rate of strain increase under a given stress)
increases with applied stress. Creep behaviour is not necessarily linear. For many metals and
ceramics, the creep rate at high temperature is proportional to the stress raised to a high

power. However, at room temperature and working stress levels, the creep strain of many
common materials (polymers, wood, concrete) are linearly dependent on stress. In such a
case, material behaviour can be described by models combining springs and dashpots. The
studying of these models will constitute the subject matter of the next section.
2.3.4 Modelling of Creep at Low Temperature (Viscoelastic Models)
       Models with spring and dashpots can be used to describe linear creep behaviour. The
spring (Fig.2.13(a)) is a linear elastic element with direct proportionality between stress and
strain. For the dashpot (Fig.2.13(b)), the rate of straining is directly proportional to the
applied stress. This is similar to the behaviour of viscous liquid, the strain rate of which is
directly proportional to the applied shear stress. Since the material can be considered as a
combination of linear elastic and viscous elements, it is called a linear viscoelastic material.

          (a) Spring                                        (b) Dashpot

                 = /E                                       d/dt = 
          Fig. 2.13 Spring and Dashpot for the Modelling of Viscoelastic Behaviour
       Using one spring and one dashpot, two different models can be created by putting the
elements either in series or in parallel. The behaviour of each of these simple models will be
studied below.
(I) Maxwell Model: (Spring and Dashpot in Series)

                                      E                    
                                    Fig.2.14 Maxwell Model
       In the Maxwell model, the material is considered to be made up of two parts in series.
The elastic (time independent) part is represented by a spring with modulus E, and the
viscous (time dependent) part is represented by a dashpot of viscosity . The equation
relating stress and strain (as well as time) for this model is dervied below.
       Under an applied stress , the strain in the spring (1) and the strain rate are given by:
       1 = /E ; d1/dt = (1/E) d/dt

        The strain rate in the dashpot is given by:
        d2/dt = /
       The total strain, , is the sum of strain in the elastic and viscous parts.
         = 1 + 2 ; d/dt = d1/dt + d2/dt
        which gives:
        (1/E) d/dt + /= d/dt
        as the governing equation for the material.
(1) Creep Behaviour under constant stress applied from 0 < t < t1
        Under constant stress, d/dt = 0
        d/dt = /; = (/t+ (0)
        It takes finite time for the dashpot to respond to loading. Therefore, at t = 0, (0)= 0
and the dashpot acts as if it is rigid. The initial strain is then resulted from the spring alone.
        (0) = /E ; = /E + (/t
        At t = t1, the load is completely removed. The spring shortens by an amount equal to
/E. The remaining strain is (/t1. After load removal,  = d/dt = 0, implying d/dt = 0.
The strain will stay constant for t > t1. The stress and strain are plotted against time in
Fig.2.15 below.

                                                       

                                                  t                                   t
                                      t1                                  t1

           Fig.2.15 Creep Behaviour under Constant Stress for the Maxwell Model
(2) Relaxation Behaviour (Constant Strain applied at t = 0)
        Under constant strain, d/dt = 0. The governing equation gives:
        (1/E) d/dt = - /
        Integrating both sides, and noting that (0)= E (dashpot stays undeformed) at t = 0,
we have:
         = E exp(-Et/)

                                                          

                                                   t                               t
                     Fig.2.16 Relaxation Behaviour of the Maxwell Model
(II) Kelvin-Voigt Model (Spring and Dashpot in parallel)


                                  Fig.2.17 Kelvin-Voigt Model
       For this model, the spring and dashpot are put under the same strain .
       Stress in the spring: 1 = E
       Stress in the dashpot: 2 = d/dt
       The total stress (), which is the sum of 1 and 2, is related to  through:
           d/dt + E = 
       The above is the governing equation for the Kelvin-Voigt model.
(1) Creep Behaviour under constant stress applied from 0 < t < t1
       The governing equation is a first order differential equation, which can be solved by
the following procedure:
       Multiplying each side of the governing equation by exp(Et/), we have
       d/dt exp(Et/) + (E/) exp(Et/)  = () exp(Et/),
       The left hand side of the equation can be written as: d [ exp(Et/)]/dt.
       Carrying out the integration, and noting that (0) = 0 (because the dashpot takes a
finite time to respond), the strain is given by:
        = (/E) [1 - exp(-Et/)]
       If the stress is removed at t = t1. Then = 0 for t > t1. The governing equation
       d/dt = - E
      Integrating, with (t1) at t = t1 as the initial condition, gives:

       (t1) exp[-E( t - t1)/]
       The behaviour is illustrated in Fig.2.18

                                                  

                                           t                                        t
                                  t1                                t1

                      Fig.2.18 Relaxation Behaviour of the Kelvin-Voigt Model
(2) Relaxation Behaviour (Constant Strain applied at t = 0)
       At t = 0, dashpot is theoretically rigid. In other words, the strain should be zero. To
force the strain to reach a finite value, infinite stress is required. For t > 0, the strain is
constant, implying d/dt = 0. The governing equation gives  = E. The relaxation response
is shown in Fig.2.19.

                                                         Infinite stress
                                                           at t = 0

                                               t                                t
                      Fig.2.19 Relaxation Behaviour of the Kelvin-Voigt Model
       In describing the creep/relaxation behaviour of real materials, each of the two models
above has its own shortcomings. For the Maxwell model, the strain rate is constant, and after
stress is removed, there is no time-dependent gradual strain recovery. For the Kelvin-Voigt
model, no instantaneous material response is allowed, thus producing the artifact of infinite
stress when a finite strain is suddenly applied. For real materials, applied stress is always
accompanied by an instantaneous response. Subsequently, the strain will increase with time
but at a decreasing rate. After stress is removed, part of the strain is recovered immediately,
while another part will be slowly recovered after a period of time. To describe the behaviour
of real materials, the two simple models can be combined as in Fig.2.20. This combined
model is called the Burger‟s body and can be used to describe the time-dependent behaviour
of both concrete and wood.

                               E1            1

                   Instantaneous          Steady               Transient
                     Response           State Creep            Response

                                Transient Steady State
                               Response     Creep

                Fig.2.20 The Burger‟s Body and its Response to Constant Stress
2.3.5 Strain Response under Arbitrary Stress History - Superposition
       The creep strain for a unit stress, or creep compliance J(t), can be obtained
experimentally from a single test (under constant stress). Once the compliance is known, the
creep behaviour under a non-constant stress can be obtained by superposition as illustrated in
Fig. 2.21. To apply superposition, any increase in stress level is replaced by a new constant
stress applied at the time when stress change takes place. Decrease in stress level is replaced
by the removal of a constant stress. In the figure, the stress is shown to increase by discrete
amounts. For a continuously changing stress, the stress history can be approximated with
discrete stress increments occurring over very small time steps. This is the same principle
behind numerical integration.


                      2                                                   1
                                        3                                     1+2-3

                                         t                                      t

                       Fig.2.21 Illustration of the superposition principle


2.4.1 Introduction

       Fracture is the failure of materials due to the propagation of a crack. In the discussion
of ductile behaviour, we have mentioned that the final failure in tension is due to fracture
after the onset of necking. In such a case, a large elongation can be observed before final
fracture takes place. We are therefore given plenty of warning. In many materials, however,
failure occurs by fast fracture. When loading increases, the material (and hence the structure)
behaves in an elastic manner. Then, fracture suddenly occurs without any warning. This
failure mode, referred to as brittle failure, is exhibited by many common materials such as
glass, rock and plain concrete. (Note: due to the presence of aggregates that act as bridges in
the crack, fracture in concrete is often considered “quasi-brittle”.      This will be further
discussed in a later chapter.) Moreover, even for materials that normally behave in a ductile
manner (e.g., steel, especially high strength steel), fast failure may occur under some
circumstances. Indeed, the unexpected fracture failure of tanks, bridges and ships earlier in
this century has resulted in a series of investigations leading to the present day understanding
of fracture processes.
       Fast fracture is due to the sudden propagation of a crack inside a structural component
at a load level below that required for yielding of a complete cross section. (Otherwise,
yielding will occur instead.) Cracks pre-exist in many materials due to a number of reasons.
When solid materials are formed, the densification may not be perfect. Trapped air in the
molten or liquid state can turn into pores and cracks in the solidified material. During
handling of materials, such as transportation of structural components and their installation,
surface damages can be introduced. One common example is the scratch on the surface of
window glass. When members are welded together, cracks may form around the weld due to
residual stresses and phase changes in the material (this will be further discussed in the Steel
chapter). Also, under repeated loading, cracks may nucleate in materials and grow larger
with each loading cycle. The slow growth of crack in this manner is referred to as fatigue
crack growth.
       In the following, the physical basis of fracture and a simple way to model fast fracture
are first described, followed by a discussion of the various parameters affecting the change in
failure mode. Then, our focus will turn to the studying of fatigue. Material fatigue leads to
the gradual weakening of structural members and can often convert the failure mode from

ductile to brittle. An understanding of fatigue is therefore essential in the assessment of long-
term safety of structures.
2.4.2 Fast Fracture: Physical Basis and Modelling
       When loading is applied to a material, the stress at the tip of a sharp crack is infinite if
elastic behaviour is assumed. Since no real material can stay elastic at very high stress levels,
an inelastic zone will always be present in front of the crack tip (see Fig.2.22). In metals,
inelastic behaviour is due to material yielding. For concrete, this is due to the formation of
micro-cracks in front of the main crack. When the load is increased, the inelastic zone grows
in size. Ultimately, the crack will propagate suddenly at a very high speed. An important
question to address is: what is the criterion for fast fracture to occur?

                                                                  Loading Direction



                                                    Distance from
                          Inelastic                   Crack Tip

             Fig.2.22 Stress Distribution and Inelastic Zone in front of a Sharp Crack
       To answer the above question, we need to consider the energy balance of a system
when crack propagation occurs. Crack propagation requires energy. When a crack extends
by a small amount (a) (Fig.2.23), new surfaces are formed. Since atoms on the material
surface contain more energy than those in the bulk, surface energy needs to be provided.
Also, when the crack propagates, energy is required to extend the inelastic region (see

                                                             Extension of
                                                            Inelastic Zone

                             additional Crack Area formed
                             due to Crack Propagation

             Fig.2.23 Energy Absorbing Mechanisms during Crack Propagation

        To see where the required energy may come from, we can look at the structural
component after the crack has propagated by a small amount (Fig.2.24). With a slightly
larger crack, the component becomes less stiff (i.e., it is easier to deform). If it is fixed on
both ends, the stress will decrease all over the member and energy is released (Fig.2.24a). If
it is under fixed load, the displacement at the loading points will increase (Fig2.24b).
Additional work is therefore done on the member. Part of the work is converted into
additional strain energy stored in the member (Note: in this case, the strain in the uncracked
part of the member increases and more energy is stored). The rest is available for extending
the crack. In summary, for the two limiting cases described above, and all other cases in
between, crack propagation will be accompanied by a release of energy from the system. If
this energy is sufficient for the formation of new surfaces and the extension of the inelastic
zone, fracture will occur.

 (a)    Load        Before                     (b)   Load                  
                                                        P                        Work Done
                                                                                   = P
                                     Energy                 Before
                             After Crack                                            = P
                             Propagation                         After

                     Displacement                               Displacement
                        Fig.2.24 Energy Changes as Crack Propagation occurs

        Under general conditions, the mathematical modelling of the fracture process is very
difficult. However, if the inelastic zone is much smaller in size than the specimen, we can
define a parameter called critical energy release rate (Gc) and use the following fracture
        Fracture occurs when: G = Gc,

where G is the energy release rate of the system under a given loading condition, which can
be obtained in terms of the applied load and the specimen geometry. In many cases, instead
of calculating G, we compute a parameter K defined by K = (EG)1/2. The fracture criterion is
then converted into the form:
       Fracture occurs when: K = Kc = (EGc)1/2
K is a parameter of physical significance because it characterizes the stress concentration in
front of the crack tip (Fig.2.25).      Along the crack direction, the tensile stress (acting
perpendicular to the crack) is given by:
       = K/(2r)1/2
where r is the distance from the crack tip. K is called the stress intensity factor. K c is called
the critical stress intensity factor or simply the fracture toughness of the material.

                                                               Loading Direction


                                                       Distance from
                                                        Crack Tip, r

                        Fig.2.25 Variation of Stress in front of Crack Tip

       Values of K have been obtained for many common loading configurations. Some
examples are given in Fig. 2.26. Kc is a material parameter (i.e., it stays the same for the
same material) and can be obtained from standard specimens in the laboratory.

                                                         P/2             S         P/2

         K = a (sec
                             a )
                                                              [ 2.9(a/W)1/2 - 4.6(a/W)3/2 + 21.8(a/W)5/2
                             W                   BW
                                                                 - 37.6(a/W)7/2 + 38.7(a/W)9/2 ]

                                  K=             1/2   [ 29.6(a/W)1/2 - 185.5(a/W)3/2 + 655.7(a/W)5/2
             a                           BW
                                                          - 1017(a/W)7/2 + 63.9(a/W)9/2 ]


             Fig.2.26 Stress Intensity Factor for a few Common Loading Configurations

2.4.3 Failure of Metal: Ductile or Brittle
       As we mention earlier, metals can sometimes fail in a brittle manner. In this section,
we will look at the various factors affecting the failure mode of metals. (Note: similar
arguments can be made for other materials though the analysis is often more complicated.)
Consider a steel plate with a small internal crack of size 2a, under uniform tensile stress. If
the width W is very large compared with 2a, sec(2a/W) is very close to unity. The stress
intensity factor is then given by (see Fig.2.26):
       K = (a)1/2
       Fast fracture occurs when K=Kc, or at an applied stress of:
       = F = Kc/(a)1/2
       Material failure can also occur in a ductile manner when  = y. Gross yielding will
then occur over the whole cross section of the material. Whether failure will be brittle or
ductile depends on the relative magnitude of F and y. Since F decreases with increasing
crack size, for a member with a relatively large crack, failure will be brittle. For a member

with a small crack, ductile failure will occur. This is illustrated in Fig.2.27. The transition
crack size (aT) is given by:
       aT = (1/Kc/y)2

                                 Gross        Brittle
                                Yielding     Fracture
                                                   y

                                                               = K c / a

                                        aT                    Crack Size, a

                Fig.2.27 Transition between Brittle and Ductile Failure Modes

       The transition crack size is a function of the fracture toughness and yield strength of
the material. For a tough metal with low yield strength, the failure can stay ductile at a larger
crack size. For strong metals with low toughness, failure is brittle even if only a small crack
exists. The transition crack size depends on temperature and strain rate. Yielding is due to
the movement of dislocations in the atomic lattice, which requires a finite amount of time.
When loading is very rapid, or when temperature is very low (so the atoms are at very low
energy levels), it is more difficult for dislocations to move and the yield strength tends to
increase. With less yielding, the inelastic zone in front of the crack tip becomes smaller. As
the crack propagates, the newly formed inelastic region (shaded area in Fig.2.23) will also be
smaller in size. The energy required for crack propagation then also decreases, leading to a
lower material toughness. With higher yield strength and reduced fracture toughness, the
transition crack size can decrease significantly. In other words, for a given structural member
with a fixed crack size, brittle fracture is more likely to occur at low temperatures or under
impact loading. With an understanding of the relation between yield strength and fracture
toughness, we can also explain why the strengthening of metals (through alloying, for
example) will often reduce the fracture toughness. When very high strength metals are used
in structures, extra care should be taken to make sure that failure will not occur in a brittle
manner (e.g., the structure can be inspected for crack size to ensure that the maximum crack
size is below the transition value).
2.4.4 Fatigue - Phenomenon and Empirical Expressions

       When cyclic loading is applied to a material, failure may occur at a stress much lower
than the strength under static loading. This apparent weakening of the material is called
fatigue. The strength reduction with the number of load cycles is illustrated in Fig.2.28 with
the S-N diagram.     This diagram, which can be obtained experimentally, is useful for
component design. Once we know the number of load cycles a structural component is
expected to endure over its life span, the appropriate strength value can be chosen. It should
be noted that the strength approaches a constant value for very large N. This value is called
the fatigue threshold. If applied stress is kept below this value, the material can sustain an
infinite number of cycles.

                                            Number of Cycles, N

                       Fig.2.28 The S-N Curve under Cyclic Loading
       Besides reading off the strength from the S-N curve, various empirical expressions
have been proposed to relate the magnitude of cyclic loading or stress range () to the
number of cycles to failure (Nf). Here,  is defined as the difference between the maximum
applied stress (max) and the minimum applied stress (min). Also, we can define m to be the
mean stress during load application. These definitions are illustrated in Fig.2.29. In equation
form, they are:
        = max - min
       m = (max + min)/2

                        max
                             m                    
                        min


                       Fig.2.29 Definition of terms for Cyclic Loading

In the study of fatigue behaviour, we have to distinguish between high cycle and low cycle
fatigue. In high cycle fatigue, the applied stress is often low so neither max nor min are high
enough to cause gross yielding (in tension and compression respectively). The number of
cycles to failure is then high. In low cycle fatigue, one or both of max and min exceed the
yield strength. Failure will then occur after a small number of cycles. For high cycle and low
cycle fatigue under zero mean stress (m = 0), the following expressions have been proposed:
Basquin‟s Law for high cycle fatigue:
        (Nf)a = C1
where C1 and „a‟ are constants. For common materials, „a‟ range from about 1/8 to 1/15.
Coffin-Manson Law for low cycle fatigue:
       PL (Nf)b = C2
where C2 and b are constants, with „b‟ varying between 0.5 and 0.6. After yielding has
occurred, the variation of plastic strain (PL), rather than the stress variation itself, is found
to govern fatigue behaviour.
       When the mean stress is non-zero, the number of cycles to failure will decrease for a
given stress range. To keep the same number of cycles to failure, the stress range  needs
to be reduced to m in accordance with Goodman‟s Rule:
       m =  ( 1 - |m| / TS )
       In the equation, |m| is the absolute value of the mean stress and TS is the ultimate
strength in tension.
       When the magnitude of cyclic loading is not constant, the lifetime can be predicted
with Miner‟s rule:
       (Ni/Nfi) =1
where Ni is the number of cycles under  and Nfi is the number of cycles to failure under
. Physically, the equation means that every time load cycles are applied, a part of the
structural life is used. Eventually, when all the lifetime is used up, failure will occur.
       It should be noted that the expressions presented in this section are all based on
experimental data. The constants in Basquin‟s law and Coffin-Manson law are dependent on
loading configuration (e.g., direct tension-compression or bending) as well as specimen type
(e.g., rod vs plate). These empirical laws are useful when a component is tested in the same
way as it will be used in practice. For example, the fatigue results for a steel reinforcing bar
under direct tension and compression can be used directly in practice because this is the way
a reinforcing bar will be loaded in a real structure. In more general cases, the prediction of

fatigue failure requires an understanding of the fatigue process. This will be the focus of the
next section.
2.4.5 Physical Basis of Fatigue and K-Based Modelling
        For an initially uncracked component, the formation of cracks under cyclic loading is
illustrated in Fig.2.30(a). At locations of stress concentrations (e.g. bends, corners), local
yielding may occur. Dislocation movements lead to sliding of materials at an angle to the
applied stress. In some locations, sliding will lead to the formation of extrusions on the
surface while in some other locations, intrusions will be present. An intrusion will act like a
sharp notch to initiate the propagation of a crack. The crack, which is originally parallel to
the sliding direction, eventually orients itself perpendicular to the applied stress. Fig.2.30(b)
illustrates the growth of an existing crack. When the maximum stress is applied, the crack
opens. Yielding and the associated material sliding result in the formation of new surface at
the crack tip. When stress decreases, the crack starts to close (fully or partially) and the new
surface folds forward to extend the crack. Note that the area that can fold forward depends
on the difference in crack openings at maximum and minimum stress. This explains the
significance of stress range in fatigue behaviour. Slow crack growth under cyclic loading
leads to the gradual weakening of a structural component, because the fracture stress
decreases with crack size. Also, as the crack grows larger, the failure mode may change from
ductile to brittle.

         (a)                                         (b)

                      Extrusions                         K min
                          Preferred Sliding
                                                                        New Surface
                          Intrusions produces            K max
                             a Sharp Notch

                                                                        New Surface
                                                                       Folds Forward
                            Crack Initiated              K min
                            from Notch Tip

   Fig.2.30 Mechanisms for (a) Crack Initiation and (b) Crack Propagation during Fatigue

       The modelling of crack initiation (Fig.2.30a) is very difficult. However, for many
large structures used in civil engineering (such as bridges, storage tanks, pressure vessels,
etc), cracks are almost always present at the joints, especially when welding is carried out.
We can therefore focus on the modelling of crack propagation. The design strategy is as
follows. For a structural component, we calculate (using the approach described in 2.4.3) the
critical crack size before fast fracture occurs. Then, after the structure is built, important
components are inspected for the initial crack size. If no cracks are found, we use the
resolution of inspection as the initial crack size. For example, if the inspection can only
reveal cracks 1mm in size or larger, 1 mm is used as a conservative estimate of initial crack
size. Knowing the initial and critical crack sizes, the number of cycles to failure can be
calculated with the help of Paris’ law, which states:
       da/dN = A(K)n
       da/dN is the growth in crack size per unit cycle
       A and n are constants obtained from experiments
       K = Kmax - Kmin is the change in stress intensity factor during the loading cycle.
       Note that K is a function of crack size „a‟. The equation can be re-arranged in the
following form:
       dN = da / [A(K)n]
       N goes from zero to Nf when the crack grows from the initial size to the critical. Nf
can therefore be obtained through direct integration.

        Nf  
              a i [A(K ) ]

        where ai is the initial crack size and ac is the critical crack size when fast occur occurs
under the applied load.


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