[學習]DSGZ模型 ¹

一種基於唯象理論的材料本構模型，包含了四種模型（Johnson-Cook模型，G’Shell-Jonas模型，Matsuoka模型和Brooks模型）。該模型是關於應變，應變率和溫度的函數，可以描述玻璃化和半結晶化聚合物的形變特性，尤其是內在應變軟化和後續的硬化階段。該模型通過五個應力應變點獲得本構模型的八個參數.
三個應力-應變點在同一曲線上，分別爲上屈服點，下屈服點（軟化點），硬化點；第四個點爲不同應變率下，同一溫度下的上屈服點；第五個點爲同一應變率下，不同溫度下的上屈服點。

基本模型

$\sigma(\varepsilon,\dot {\varepsilon},T) = K \{ f(\varepsilon)+[\frac {\varepsilon\times e^{(1-\frac {\varepsilon} {C_3 \times h(\dot \varepsilon,T)})}} {C_3\times h(\dot \varepsilon,T)}-f(\varepsilon)] \times e^{[ln(g(\dot \varepsilon,T)-C_4)] \times \varepsilon} \} \times h(\dot \varepsilon,T)$

這裏，
$f(\varepsilon)=(e^{-C_1 \times \varepsilon}+\epsilon^{C_2})(1-e^{-\alpha \times \varepsilon})$
$h(\dot \varepsilon,T)=(\dot \varepsilon)^m e^{\frac a T}$

參數說明

$C_1$ 和 $C_2$
$C_1$ 和 $C_2$ 確定基準應力應變曲線的走勢，由方程求解獲得：

$\left\{ \begin{array} {lr} \frac {e^{-C_1 \times \varepsilon_1}+\varepsilon^{C_2}} {e^{-C_1 \times \varepsilon_2}+\varepsilon^{C_2}}=\frac {\sigma_1}{\sigma_2} &\\ \frac {e^{-C_1 \times \varepsilon_1}+\varepsilon^{C_2}} {e^{-C_1 \times \varepsilon_3}+\varepsilon^{C_2}}=\frac {\sigma_1}{\sigma_3} \end{array} \right.$

K
比例係數。
m
描述同意溫度下，兩個不同應變率曲線之間的差異。
$(\frac {\dot \varepsilon_1} {\dot \varepsilon_2})^m=\frac {\sigma_1} {\sigma_2}$
a
描述同一應變率下，兩個不同溫度曲線之間的差異。
$a = \frac {ln(\frac {\sigma_1} {\sigma_2})} {\frac {1} {T_1} - \frac {1} {T_2}}$
$C_3$
描述不同應變利率和溫度下上屈服點的差異。
注意： $C_3 \times h(\dot \varepsilon,T)=\varepsilon_y$ , $\varepsilon_y$ 和 $\sigma_y$ 爲軟化前的最大應邊和應力，即材料的上屈服點。
$\alpha$ 和 $C_4$
兩個參數都是用來擬合應力應變曲線的初始階段（硬化前）。
$1-e^{-\alpha \times\varepsilon}=0.97$ ， $\varepsilon$ 應選區軟化過程的終點（下屈服點）。
$C_4$ 針對不同聚合物材料選取不同的參數。
玻璃化聚合物：
$C_4 = 7+ln(\dot \varepsilon^m \times e^{\frac \alpha T})$
半結晶化聚合物：
$C_4 = 200+ln(\dot \varepsilon^m \times e^{\frac \alpha T})$

參數的影響規律

爲了便於對模型個部分的理解：
令
$j=\frac {\varepsilon\times e^{(1-\frac {\varepsilon} {C_3 \times h(\dot \varepsilon,T)})}} {C_3\times h(\dot \varepsilon,T)}$

$l=e^{[ln(g(\dot \varepsilon,T)-C_4)] \times \varepsilon}$
繪製出這兩個函數隨應變的變化趨勢。

$j$ 和 $l$ 兩個函數都是0~1的函數。
$l$ 函數爲隨 $\epsilon$ 衰減的函數，其影響範圍爲應力應變曲線的初始階段。
$j$ 函數爲應力應變的前期函數，其峯值表明上屈服點的位置以及軟化的速度。

測試參數

測試參數：

strain = [0.1,0.3,1.0];
stress =[112.,95.,160.,105.,80.]
strainrate = [0.001,0.0005]
T = [296.,323]
K = 4.5

	$c_1$	$c_2$	$m$	$a$	$K$	$c_3$	$c_4$	$\alpha$
論文	1.91	1.49	0.064	1191	3.9	0.0029	11	11.7
測試	1.35	2.09	0.093	1191.5	4.5	0.003398	10.38	11.69

代碼（Python）

# -*- coding: utf-8 -*-
"""
Author:Yujin.Wang
Date:2019.04.07
The lib. is used to translate the F-D curve to MAT24 effective plastic strain - true stress.
Usage:
return TrueStrain,TrueStress,[PeakStrain,PeakStress]
################################################################################
Parameters:
    A - float, Specimen Crosssection area
	l0 - float, Gage lengrh
    FDfile - Filename, Force-Disp curve
    strain_rate - list, Strain rate
    curvenum - int, Curve number in .key file
    ratio - float, Display ratio
#################################User Define###################################
    Ymould_user - float, Young modulus
    yieldpoint - float, Yield point
    pointnum - int, the number of output points
    strainturnlist - list, Turn strain 
    scaleturnlist - list,  Scale of the turn strain 
    curvescalelist - list, Curve scale equalent to SFO
    alignstrain - float, align the strain to user defined value(>max(strainturnlist))
    extend - float, Extend strain to a user defined value
    scaleextend - float, Scale of the extend strain 
"""
import time
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import interpolate,optimize
import scipy.signal as signal
from sympy import symbols,nsolve

def f_strain(strain,c1,c2,a):
    return (np.power(np.e,-c1*strain) + np.power(strain,c2))*(1-np.power(np.e,-a*strain))

def h_strainrate_T(strainrate,m,T,alpha):
    return np.power(strainrate,m)*np.power(np.e,alpha/T)

def DGSZ(strain,strainrate,T,K,c1,c2,c3,c4,a,m,alpha):
    f = f_strain(strain,c1,c2,a)
    h = h_strainrate_T(strainrate,m,T,alpha)
    j = (strain*np.power(np.e,(1.-strain/(c3*h)))/(c3*h))
    l =  np.power(np.e,(np.log(h)-c4)*strain)
    plt.figure(2)
    plt.plot(strain,f)
    plt.title('f')
    plt.figure(3)
    plt.title('j')
    plt.plot(strain,j)
    plt.figure(4)
    plt.title('j-f')
    plt.plot(strain,j-f)
    plt.figure(5)
    plt.title('l')
    plt.plot(strain,l)
    plt.figure(6)
    plt.title('(j-f)*l')
    plt.plot(strain,((j-f)*l))
    plt.figure(7)
    plt.title('(f+(j-f)*l)')
    plt.plot(strain,(f+(j-f)*l))
    plt.figure(8)
    plt.plot(strain,K*(f+(j-f)*l)*h)
    plt.figure(1)
    value = K* ( f + (j-f)*l )*h
    return value

def DGSZ_c1c2(strain,stress):
    c1,c2 = symbols('c1,c2')
    equations = [(np.power(np.e,-c1*strain[0]) + np.power(strain[0],c2))     \
                 /(np.power(np.e,-c1*strain[1]) + np.power(strain[1],c2)) - stress[0]/stress[1], \
                 (np.power(np.e,-c1*strain[2]) + np.power(strain[2],c2))     \
                 /(np.power(np.e,-c1*strain[1]) + np.power(strain[1],c2)) - stress[2]/stress[1]]
    result = nsolve(equations,[c1,c2],[1,1],verify=True,prec=150)
    return result
    
def DGSZ_m(strainrate,stress):
    return np.log(stress[0]/stress[1]) / np.log(strainrate[0]/strainrate[1])

def DGSZ_alpha(T,stress):
    return np.log(stress[0]/stress[1])/(1./T[0]-1./T[1])

def DGSZ_K(stress,strain,strainrate,m,T,alpha,c1,c2):
    h = h_strainrate_T(strainrate,m,T,alpha)
    return stress/(h*np.power(np.e,-c1*strain)+np.power(strain,-c2))

def DGSZ_c3(strain,strainrate,m,T,alpha):
    h = h_strainrate_T(strainrate,m,T,alpha)
    print (strain)
    return strain/h
    
def DGSZ_c4(strainrate,m,alpha,T):
    return 7 + np.log(np.power(strainrate,m)*np.power(np.e,alpha/T))

def DSGZ_a(strain):
    return -np.log(0.03)/strain

def CowperSymondsFunc(x,c,p):
    rate,ES0 = x
    return ES0*(1+np.power((rate/c),1./p))

def ConditionRemove(data,condition):
    a =[]
    isflag = 1
    string ="if "+ condition +":\n"
    string +="                     for k in range(len(data)):\n   \
                        data[k].pop(i+1)\n   \
                  isflag = 1\n"
    while isflag == 1:
        a.extend(data)
        len0 = len(a[0])
        try:
            for i in range(len(a[0])):
                exec(string)
        except:
            if len(data[0]) == len0:
                break
    return data
    
def MAT89LCSS(fdfile,A,l0):
    '''fdfile -- Force-Displacement Curve
       A -- Area of the Cross section of specimen
       l -- Length of the specimen (gage length or grip length or equalent length)
       Output : TrueStrain,TrueStress,[PeakStrain,PeakStress]
     '''
    data = pd.read_csv(fdfile)
    x = data.Disp
    y = data.Force
    EngStrain = x/l0
    EngStress = y/A
    EngStressFilt =  pd.Series(signal.medfilt(EngStress,201))
    PeakStress = max(EngStressFilt[:800])
    PeakStrain = EngStrain[EngStressFilt[:800].idxmax()]
    print ("PeakStress:%f\tPeakStrain:%f\t Elastic modulus:%f" %(PeakStress,PeakStrain,PeakStress/PeakStrain))
    plt.figure(1)
    plt.grid("on")
    plt.scatter(PeakStrain,PeakStress, marker='o')
    plt.title("Engineering Strain VS. Engineering Stress Curve")
    plt.xlabel("Engineering Strain[-]")
    plt.ylabel("Engineering Stress[GPa]")
    plt.legend(["0.01mm/ms","0.1mm/ms","1mm/ms","10mm/ms"])
    TrueStrain =  np.log(1+EngStrain)
    TrueStress = EngStressFilt*(1+EngStrain)
    plt.figure(2)
    plt.grid("on")
    plt.title("True Strain VS. True Stress Curve")
    plt.xlabel("True Strain[-]")
    plt.ylabel("True Stress[GPa]")
    plt.plot(TrueStrain,TrueStress)
    plt.legend(["0.01mm/ms","0.1mm/ms","1mm/ms","10mm/ms"])
    return TrueStrain,TrueStress,[PeakStrain,PeakStress]

def CurveKey1(x,y,strainrate_list,curvenum):
    # x,y = CowperSymondsCurve(FDfile[0],A,l0,Ymould_user,80,ratelist,x_p,y_p)
    fout = open("Curve.key",'w')
    fout.write('*KEYWORD\n')
    fout.write( "$%10s%10s%10s%10s%10s%10s%10s%10s\n" %('c1','c2','m','a','K','c3','c4','alpha'))
    fout.write( "$%10.4f%10.4f%10.4f%10.4f%10.4f%10.4f%10.4f%10.4f\n" %(c1,c2,m,a,K,c3,c4,alpha))
    fout.write('$ Created: ' + time.strftime("%d.%m.%Y %H:%M:%S") + '\n')
    fout.write("*DEFINE_TABLE\n%d\n" %(curvenum))
    for i in strainrate_list:
        fout.write("%f\n" %(np.log(i)))
    for index,strainrate in enumerate(strainrate_list):
        res = [ x[index], y[index]]
        # condition = "(res[0][i]<res[0][i+1] or res[1][i]>res[1][i+1])"
        # x[index], y[index]= ConditionRemove(res,condition)
        curvenum += 1
        flag = 0 
        fout.write('*DEFINE_CURVE_TITLE\nRate %.5f\n' %(strainrate_list[index]))
        fout.write('$     LCID      SIDR       SFA       SFO      OFFA      OFFO    DATTYP\n')
        fout.write('      %d         0    1.0000    1.0000    0.0000    0.0000\n' %(curvenum))
        plt.figure(5)
#        plt.plot(x[index],y[index])
        plt.grid('on')
        for i in range(len(x[index])):
            if x[index][i] > 0:
                if flag == 0 :
                    fout.write("%f,%f\n" %(0,y[index][i]))
                    flag = 1
                if flag != 0 :
                    fout.write("%f,%f\n" %(x[index][i],y[index][i]))
    fout.write("*END\n")
    fout.close()
    plt.savefig('curve')
    return

if __name__ == '__main__':
    Ymould_user = 1.5 #定義彈性模量
    curvenum = 2350 #key文件中曲線的編號起始編號
    pointnum = 80#
    #############################################################
    strain = [0.1,0.3,1.0];
    stress =[112.,95.,160.,105.,80.]
    
    strainrate = [0.001,0.0005]
    T = [296.,323]
    c1,c2 = DGSZ_c1c2(strain[:3],stress[:3])
    m = DGSZ_m(strainrate,[stress[0],stress[3]])
    alpha = DGSZ_alpha(T,[stress[0],stress[4]])
#    K = DGSZ_K(stress[0],strain[0],strainrate[0],m,T[0],alpha,c1,c2)
    K = 4.5
    c3 = DGSZ_c3(strain[0],strainrate[0],m,T[0],alpha)
    c4 = DGSZ_c4(strainrate[0],m,alpha,T[0])
    a = DSGZ_a(strain[1])
    plt.scatter(strain[:3],stress[:3])
    res_stress = [];res_strain = []
    res_stress_89 = [];res_strain_89 = []
    print ("c1\tc2\tm\ta\tK\tc3\tc4\talpha")
    print ("%f \t %f \t %f \t %f \t %f \t %f \t %f \t %f" %(c1,c2,m,alpha,K,c3,c4,a))
        #
    strain_curve = np.linspace(0.0001,1.,pointnum)
    strainrate_output = [0.0001,0.0005,0.001]
    for index,sr in enumerate(strainrate_output):
        stress = DGSZ(strain_curve,sr,T[0],K,c1,c2,c3,c4,a,m,alpha)
        plt.plot(strain_curve,stress)
        strain_temp_list = [];stress_temp_list = []
        for i in range(pointnum):
            try:
                if stress[i] == stress[i-1]:
                   stress[i] = stress[i]*2.
            except:
                pass
            strain_temp = strain_curve[i] - stress[i]/Ymould_user
            if strain_temp > 0.2:
                if strain_temp_list == []:
                   strain_temp_list.append(0)
                   stress_temp_list.append(stress[i]/3.)
                strain_temp_list.append(strain_curve[i])
                stress_temp_list.append(stress[i])
        res_strain.append(strain_temp_list)
        res_stress.append(stress_temp_list)
        res_strain_89.append(strain_curve)
        res_stress_89.append(stress)   
        plt.grid('on')

特定溫度和應變率下的三維圖形。

# -*- coding: utf-8 -*-
import time
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy import interpolate,optimize
import scipy.signal as signal
from sympy import symbols,nsolve
from mpl_toolkits.mplot3d import Axes3D

def f_strain(strain,c1,c2,a):
    return (np.power(np.e,-c1*strain) + np.power(strain,c2))*(1-np.power(np.e,-a*strain))

def h_strainrate_T(strainrate,m,T,alpha):
    return np.power(strainrate,m)*np.power(np.e,alpha/T)

def DGSZ(strain,strainrate,T,K,c1,c2,c3,c4,a,m,alpha):
    f = f_strain(strain,c1,c2,a)
    h = h_strainrate_T(strainrate,m,T,alpha)
    print ('h:\n',strainrate)
    print (h)
    j = (strain*np.power(np.e,(1.-strain/(c3*h)))/(c3*h))
    l =  np.power(np.e,(np.log(h)-c4)*strain)
    value = K* ( f + (j-f)*l )*h
    return value

def DGSZ_c1c2(strain,stress):
    c1,c2 = symbols('c1,c2')
    equations = [(np.power(np.e,-c1*strain[0]) + np.power(strain[0],c2))     \
                 /(np.power(np.e,-c1*strain[1]) + np.power(strain[1],c2)) - stress[0]/stress[1], \
                 (np.power(np.e,-c1*strain[2]) + np.power(strain[2],c2))     \
                 /(np.power(np.e,-c1*strain[1]) + np.power(strain[1],c2)) - stress[2]/stress[1]]
    result = nsolve(equations,[c1,c2],[1,1],verify=True,prec=150)
    return result
    
def DGSZ_m(strainrate,stress):
    return np.log(stress[0]/stress[1]) / np.log(strainrate[0]/strainrate[1])

def DGSZ_alpha(T,stress):
    return np.log(stress[0]/stress[1])/(1./T[0]-1./T[1])

def DGSZ_K(stress,strain,strainrate,m,T,alpha,c1,c2):
    h = h_strainrate_T(strainrate,m,T,alpha)
    return stress/(h*np.power(np.e,-c1*strain)+np.power(strain,-c2))

def DGSZ_c3(strain,strainrate,m,T,alpha):
    h = h_strainrate_T(strainrate,m,T,alpha)
    return strain/h
    
def DGSZ_c4(strainrate,m,alpha,T):
    return 7 + np.log(np.power(strainrate,m)*np.power(np.e,alpha/T))

def DSGZ_a(strain):
    return -np.log(0.03)/strain

    



if __name__ == '__main__':
    Ymould_user = 1.5 #定義彈性模量
    curvenum = 2350 #key文件中曲線的編號起始編號
    pointnum = 20#
    #############################################################
    strain = [0.1,0.3,1.0];
    stress =[112.,95.,160.,105.,80.]
    
    strainrate = [0.001,0.0005]
    T = [296.,323]
    c1,c2 = DGSZ_c1c2(strain[:3],stress[:3])
    c1 = np.float(c1)
    c2 = np.float(c2)
    m = DGSZ_m(strainrate,[stress[0],stress[3]])
    alpha = DGSZ_alpha(T,[stress[0],stress[4]])
#    K = DGSZ_K(stress[0],strain[0],strainrate[0],m,T[0],alpha,c1,c2)
    K = 4.5
    c3 = DGSZ_c3(strain[0],strainrate[0],m,T[0],alpha)
    c4 = DGSZ_c4(strainrate[0],m,alpha,T[0])
    a = DSGZ_a(strain[1])
    plt.scatter(strain[:3],stress[:3])
    res_stress = [];res_strain = []
    res_stress_89 = [];res_strain_89 = []
    print ("c1\tc2\tm\ta\tK\tc3\tc4\talpha")
    print ("%f \t %f \t %f \t %f \t %f \t %f \t %f \t %f" %(c1,c2,m,alpha,K,c3,c4,a))
        #
    x = np.linspace(0.0001,1.,pointnum)
    y1 = np.linspace(0.1,10.,pointnum)
    y2 = np.linspace(238.,358.,pointnum)
    
    
    X,Y = np.meshgrid(x,y1)
    
    # strainrate,temperature = np.meshgrid(x,y)
    stress = DGSZ(X,Y,273.,K,c1,c2,c3,c4,a,m,alpha)    
    fig = plt.figure(1)
    ax = Axes3D(fig) 
    ax.plot_surface(X,Y, stress, rstride = 1, cstride = 1, cmap = plt.get_cmap('ocean'),alpha=1)
    ax.set_xlabel("Strain")
    ax.set_ylabel("Strainrate")
    ax.set_zlabel("Stress")    
    
    X,Y = np.meshgrid(x,y2)
    strainrate,temperature = np.meshgrid(x,y2)
    stress = DGSZ(X,0.1,Y,K,c1,c2,c3,c4,a,m,alpha)
    fig = plt.figure(2)
    ax = Axes3D(fig) 
    ax.plot_surface(X,Y, stress, rstride = 1, cstride = 1, cmap = plt.get_cmap('ocean'),alpha=1)
    ax.set_xlabel("Strain")
    ax.set_ylabel("Temperature")
    ax.set_zlabel("Stress")    
    plt.show()

Y. Duan, A.Sagal, R. Greif. A uniform Phenomenological Constitutive Model for Glassy and Semicrystalline Polymers. Polymer Engineering and Science,vol.41.2001.08 ↩︎

DSGZ模型

[學習]DSGZ模型 ¹

基本模型

參數說明

參數的影響規律

測試參數

代碼（Python）

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DSGZ模型

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基本模型

參數說明

參數的影響規律

測試參數

代碼（Python）

[學習]DSGZ模型 ¹