• Introduction:
• Program
• Examples of Application

Monotonic Plastic Zone Size for Plane Stress and Plane Strain

by: Jorge Alberto Rodrguez Durn, D.Sc.

Pontifical Catholic University of Rio de Janeiro, PUC-Rio

e-mail: duran@mec.puc-rio.br

2002 Jorge Alberto Rodrguez Durn

NOTE: This worksheet demonstrates the use of Maple for calculate plastic zone shapes and sizes for plane stress and plane strain for cracks under static loading.

Introduction:

The sizes of plastic zones were obtained by equaling VonMises criteria (using Irwin's elastic solutions), to monotonic yield strength of material (Sy). For this reason, the shape and size of plastic zones showed here do not take into account the strain hardening of material within plastic zone. Sizes in polar coordinates r and theta are functions of quadratic relation between elastic stress intensity factor K and Sy (K/Sy)^2. For a established angle theta, the graphic shows the estimated sizes of plastic zones (coordinate r, in meters) for different relations of (K/Sy)^2. Parameters of procedure "plastic_zones" are n, Sy, Kini, Kfin and material, where n is the number of frames, Sy is the yield strength of material (in MPa), Kini and Kfin are the initial and final stress intensity factors (in MPa*m^1/2), respectively, which define the view frame.

>

restart:

>	with(plots):

Warning, the name changecoords has been redefined

Program

>	plastic_zones:=proc(n,Sy,Kini,Kfin,material)

>	local sigmax,sigmay,sigma1,sigma2,sigma3,u,K,r,txy,vonmisesT,vonmisesD,zpT,zpD,i,J,TU,x,Y,P,H,D,zplT,zplD,cx,cy,m,o,c,JJ,JD,JM,mat,JMM;

>

u:=0.33;

>	sigmax:=K/(2*Pi*r)^0.5*cos(theta/2)*(1-sin(theta/2)*sin(theta*1.5)):

>	sigmay:=K/(2*Pi*r)^0.5*cos(theta/2)*(1+sin(theta/2)*sin(theta*1.5)):

>	txy:=K/(2*Pi*r)^0.5*cos(theta/2)*sin(theta/2)*cos(1.5*theta):

>	sigma1:=(sigmax+sigmay)/2+.5*((sigmay-sigmax)^2+4*(txy)^2)^.5:

>	sigma2:=(sigmax+sigmay)/2-.5*((sigmay-sigmax)^2+4*(txy)^2)^.5:

>	sigma3:=u*(sigma1+sigma2):

>	vonmisesT:=1/sqrt(2)*sqrt((sigma1-sigma2)^2+sigma2^2+sigma1^2):

>	vonmisesD:=1/sqrt(2)*sqrt((sigma1-sigma2)^2+(sigma2-sigma3)^2+(sigma3-sigma1)^2):

>	zpT:=solve({vonmisesT=Se},{r}):

>	zplT:=eval(r,zpT[1])*Se^2/K^2:

>	zpD:=solve({vonmisesD=Se},{r}):

>	zplD:=eval(r,zpD[1])*Se^2/K^2:

>	m:=evalf(((Kfin/Sy)^2-(Kini/Sy)^2)/n,2); o:=evalf(((Kfin/Sy)^2-(Kini/Sy)^2)/m,2);

>	c:=[evalf(seq(m*x+(Kini/Sy)^2,x=0..o+1),3)];

>	mat:=convert(material,string):

>	for i from 1 to n+1 do

>	Y[i]:=polarplot(zplT*c[i],theta=0..2*Pi,thickness=2,color=red,title="Monotonic Plastic Zones \n Red = Plane Stress, Blue = Plane Strain");

>	P[i]:=polarplot(zplD*c[i],theta=0..2*Pi,thickness=2,color=blue);

>

od:

>	cx:=evalf(eval(zplT,theta=0),4)*c[n+1];

>	cy:=evalf(eval(zplT,theta=Pi/2),4)*c[n+1];

>	J:=textplot([cx,cy,"(K/Sy)^2 = "],'align=LEFT'):

>	JJ:=textplot([cx,0,"r [m]"],'align={RIGHT,BELOW}',font=[TIMES,BOLD,14]):

>	JD:=textplot([0,cy,"r [m]"],'align={LEFT,ABOVE}',font=[TIMES,BOLD,14]):

>	JM:=textplot([cx,-cy,"Material = "],'align=LEFT'):

>	JMM:=textplot([cx,-cy,mat],'align=RIGHT'):

>	for i from 1 to n+1 do

>	TU:=convert(c[i],string):

>	H[i]:=textplot([cx,cy,TU],align=RIGHT):

>

od:

>	for i from 1 to n+1 do

>	D[i]:=display([Y[i],P[i],J,JJ,JD,JM,JMM,H[i]]);

>

od:

>	display([seq(D[i],i=1..n+1)],insequence=true,scaling=constrained);

>

end:

Examples of Application

The following are two examples of application of procedure plastic_zones. The value of yield strength Sy and toughness (to establish the range of K) were experimentally determined at PUC-Rio facilities. The first example shows the size of plastic zones in plane stress and plane strain for a low alloy steel (API 5L X-60, Sy = 457 MPa, Kini = 20 MPa.m^1/2 and Kfin = 70 MPa.m^1/2) and the second one treats of a high strength aluminum alloy (7075 T-6, Sy = 498 MPa, Kini = 5 MPa.m^1/2 and Kfin = 20 MPa.m^1/2). While both materials have similar values of yield strength, low alloy steel presents sizes of plastic zones bigger than those for the aluminum alloy, which can be explained for the low plane strain fracture toughness of the Aluminum 7075 T-6 (KIc = 25.4 MPa.m^1/2).

>	plastic_zones(40,457,20,70,API_5L_X60_Steel);

>	plastic_zones(40,498,5,20,Aluminum_7075_T-6);

>

Disclaimer:  While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the contributors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material.