Perfectly plastic material 3.6.1 Perfectly elastic material

BEHAVIOUR OF MATERIALS UNDER DIFFERENT LOADING REGIMES 3.1 General

3.6.2 Perfectly plastic material 3.6.1 Perfectly elastic material

Figures 3.13 (a), (b) and (c) are the illustrations of strain response of perfectly elastic material for stepped loading, sustained plus cyclic and sustained plus short-term loading under MSA, respectively. In Fig 3.13 (a), the material assumes an elastic strain at to, point A to B, due to the load Plas in Fig 3.2 (a). This strain sustains until point C, at tl, corresponding to the sustained load PI. where the material acquires additional elastic strain, point C to D due to another additional load P2. Thus total strain due to the sustained loads PI and P2 remains to be there until again, another load is imposed upon at t2and so on.

Tn Fig 3.13 (b), the material acquires an elastic strain at to, point A to B that sustains corresponding to the sustained load P, as in Fig 3.2 (b). From tl to t2, the strain increases gradually as the cyclic load (traffic load) P, is applied gradually, point C to D and reaches the peak during the loading phase. This strain is recovered in the same fashion as the load P, is released gradually from t2 to t3, point D to E during the unloading phase, as the strain being purely elastic.

Similarly, in .Fig 3.13 (c), the material with an elastic strain at to, point A to B, sustains it along with the sustained load P" point B to C. At tl, it acquires additional elastic strain due to application of the short-term load P,hort.""'"point C to D, sustains as long as the load is imposed, point D to E and reduces to the pre-event strain level, at t2. point E to F. That is, whole of the strain is recovered upon withdrawal of the short-term load, since the material being purely elastic.

Figures 3.14 (a), (b) and (c) are, respectively, the illustrations of the strain responses of stepped, sustained plus cyclic loading and sustained plus short-term loading under MSA. In Fig 3.14(a), the material assumes the strain gradually, to to tl, point A to B, as long as the load PI is sustained. At tI, when an additional load P2 is applied, the strain is added up from tl to t2, point B to C, and so on for further additional loads.

3.6.3 Elasto-plastic material

In Fig 3.14 (b), the material acquires the plastic strain gradually from to to tl, point A to B as the load is sustained. From tl to t2, the strain due to the increasing traffic load P^t is added upon to that due to the sustained load P, gradually, point B to C, during the loading phase. During the unloading phase, point C to D, the rate of increment in strain is reduced due to the diminishing traffic load PI, t2 to t]. Nevertheless, the increment is there during the unloading phase as well, due to the sustained load.

Similar accumulation of plastic strain is illustrated for further cycles in the same figure.

The similar pattern as in (a) is shown in the Fig 3.14 (c). The plastic strain increases gradually from to to tI, point A to B due to the sustained load P,. From t] to h, the strain due to application of the short-term load P'hort-tenn is added gradually to that due to the sustained load P" point B to C. No strain is recovered even after recession of the short-term load P'hort-tcrm, at t2, since the material is perfectly plastic. There is a further increase in the plastic strain beyond C due to the sustained load and there is no recovery of the strain at all.

Figures 3.15 (a), (b) and (c) are the diagrams portraying the strain response of elasto- plastic material for stepped, sustained plus cyclic and sustained plus short-term loading under MSA, respectively. In the case of stepped loading, Fig 3.15 (a), there is an acquirement of elastic strain at to, point A to B, with the application of the sustained load PI, followed by the plastic strain from to to t], point B to C as long as the load is sustained. At t], there is again an assumption of additional elastic strain, point C to D, followed by the plastic strain from t] to t2, point D to E, due to another additional sustained load P2. Similar pattern of strain follows for other additional loads as portrayed in the same figure.

In Fig 3.15 (b), at to, there is an assumption of elastic strain followed by the plastic strain from to to t], point B to C, due to the sustained load P,. At tI, the cyclic load P, is applied which induces a gradual increase in strain from t] to t2, point C to D during the loading phase. The elastic but the plastic part of this strain is recovered upon

3.6.4 Elasto-visco-plastic (EVP) material

release of the traffic load Pt gradually, from t2 to tJ, point D to E, during the unloading phase. Similarly it follows for the other cycles.

In the case of sustained plus short-term loading as in Fig 3.15 (c), at to and from toto tl, the strain pattern due to the sustained load Ps can be observed to be similar as in the case of sustained plus cyclic loading. At tl, an additional elastic strain is induced, point C to D as the short-term load Pshort.termis applied, followed by the plastic increase from tl to t2, point D to E, as long as the load Pshort-termacts. At t2, when the load Pshort-termis released the elastic part of the strain is recovered leaving the plastic strain as locked in, point E to F. Thus the material acquires an additional locked in strain after the subsidence of the short- term loading.

In Fig 3.16 (a), which is an illustration of the strain response ofEVP material due to stepped loading under MSA, the material assumes an elastic strain at to, point A to B, as the sustained load PI is applied. This is followed by the gradual induction of viscous and plastic strain from to to tI, point B to C, until the load is sustained. An additional load P2 is imposed onto the material at tl, that induces the additional elastic strain, point C to D, followed by the increase in plastic plus viscous strain due to sustenance of the load P2, from tl to t2, point D to E. Similar strain pattern is illustrated for another additional sustained load PJ in the figure.

Figure 3.16 (b) is an illustration of the strain response of EVP material due to sustained plus cyclic load under MSA. At to, the material acquires an instantaneous strain due to application of a sustained load Ps, point A to B, followed by the gradual increase in viscous plus plastic strain, from toto tj, point B to C. A traffic load Pt, is applied at tj, that induces an additional strain increasing gradually and reaching the peak at t2, corresponding to the load PI, from tl to t2, point C to D during the loading phase. During unloading phase, from t2 to tJ, Point D to E, the strain is reduced gradually as the traffic load P, subsides. Thus there is an increasing locked in strain in the material after every cycle.

3.7 ER - EL plots for EVP materials subjected to different Multi-stage actions (MSA)

Similarly, in Fig 3.16 (c), which is the illustration of the strain response of the EVP material due to sustained plus short-tenn load under MSA, the strain pattern due to the sustained load P^t is identical to that in the above case. At tj, a short-tenn load Pshort-termis applied which induces an additional elastic strain, point C to D, followed by the viscous plus plastic strain increase, from tlto tz, Point D to E. The elastic strain is recovered as the load Pshort-tennis released at tz, point E to F. Thus the material is left with a certain amount oflocked in strain after occurrence of the event (short-tenn load).

Figures 3.17 (a), (b) and (c) are the plots ofERand ELfor EVP materials subjected to different multi -stage loading viz. stepped, sustained plus cyclic loading and sustained plus short-tenn loading, respectively. In the case of stepped loading, Fig 3.17 (a), the material achieves a recoverable strain ER upon application of the sustained load PI at to, point A to B. This recoverable strain ERremains constant but the locked in strain ELincreases from zero as long as the load PI is sustained, from to to tj, point B to D. Similar pattern of additional recoverable strain ERand locked in strain ELare illustrated in the figure due to another additional sustained load P2, from tl to

tz.

And so on and so forth for further stepped loads.

Fig 3.17 (b) is an illustration of ER- ELplot for sustained plus cyclic case where the material acquires an instantaneous recoverable strain ER at to that remains constant whereas the locked in strain CLis induced from zero onward, from toto tj, point B to

C,

due to the sustained load P,. From tl to tz, as the traffic load Pt increases during loading phase, both the recoverable CRand locked in CLstrains increase gradually.

During the unloading phase, from

tz

to tJ, the whole of the recoverable strain ER is recovered while the locked in strain ELin part with the release of the traffic load Pt.

That is, after every cycle the locked in strain CLgets accumulated. Similarly in the

further cycles, the recoverable strain ER and locked in strain EL increase during loading phase and reduce during unloading phase of the traffic load

P(,

gradually.

In the same manner, Fig 3.17 (c), which is the illustration for the case of sustained plus short-term load, the recoverable strain ^ER is induced at ^!D, point A to B, which remains constant followed by the increase in the locked in strain EL from toto t], point B to C, due to the sustained load Ps. At tl, the recoverable strain ER increases instantaneously upon the application of the short-term load Pshort-term, point C to D, from where onward, the locked in strain ^EL is induced that increases until the sustenance of the short-term load Pshort-term, from t1 to t2, point D to E. This.

recoverable strain ^ER retards back to the previous value at t2, point E to F, while the locked in strain ELdoes not, and continues to grow. .

t

t

c

B

c

B A

to

(a) Sustained

A D

t1 t2

(c) Impact/short term p

p

Figure 3.1 Examples of single stage actions

P,+ P2 +PJ, J

p]+ P2~ P^J P,

P

2

P,

~ t^j t2 t~

(a) Stepped loading

p

t

t~

P, •

•

P, P,

~;

1

~ t] ~

P

(c) Sustained plus short term loading

Figure 3.2 Examples of multi stage actions (b) Sustained plus cyclic loading

p"

t

t~

t

BE =

Elastic Strain

BR

=Recoverable Strain

..E

, ,E

~

c

, C =E

&R E

B

_~

~ C

A ,B

A

~

(a) Sustained loading

(b) Cyclic loading

(b) Short-term loading

Figure 3.3 Strain response of perfectly elastic material under SSA

&p = Plastic Strain

&L=Locked-in Strain

t

t

t

CID

(a) Sustained loading

(b) Cyclic loading

(c) Short-term loading

&p

Figure 3.4 Strain response of perfectly plastic material under SSA

t

t

t

,.. Ep(t=t ) = Ep(,~,) 3

Er=EE+Ep

=ER + EL

c

Er=EE+Ep

=ER + EL