ME330(lab2)

setwd("/Users/joshuachang/desktop/ME330_Project")

# Steel
steel_1018CR_data = read.csv("M12_A_1018_CR_lab2.csv", header = TRUE)
steel_1045CR_data = read.csv("M12_C_1045_CR_lab2.csv", header = TRUE)
steel_1045NM_data = read.csv("M12_A_1045_NM_lab2.csv", header = TRUE)
steel_304_data = read.csv("M12_B_304_lab2.csv", header = TRUE)

#Aluminum 
al_2024_data = read.csv("M12_A_2024_lab2.csv", header = TRUE)
al_7075_data = read.csv("M12_B_7075_lab2.csv", header = TRUE)

# Others
brass_data = read.csv("M12_A_Brass_lab2.csv", header = TRUE)
pmma_data = read.csv("M12_B_PMMA_lab2.csv", header = TRUE)

# lab 1 Compression Data
steel_1018_Compression_data = read.csv("M12_A_1018_lab1.csv", header = TRUE)
al_2024_Compression_data = read.csv("M12_A_7075_lab1.csv", header = TRUE)

# --- Steel 1018 Compression (Lab 1) ---
dis_1018_comp  <- as.numeric(steel_1018_Compression_data$Displacement)
F_1018_comp    <- as.numeric(steel_1018_Compression_data$Force)
eps_1018_comp  <- as.numeric(steel_1018_Compression_data$Composite.strain)
gage_D_1018_comp   <- 12.74     # [mm]
max_F_1018_comp    <- 78.32    # [kN] — used only for reference
max_eps_1018_comp  <- 0.482    # max strain — used only for reference
A_1018_comp        <- pi/4 * gage_D_1018_comp^2  # cross-sectional area [mm^2]
sigma_eng_1018_comp  <- F_1018_comp / A_1018_comp * 1000               # [MPa]
sigma_true_1018_comp <- sigma_eng_1018_comp * (1 + eps_1018_comp)     # [MPa]
eps_true_1018_comp   <- log(1 + eps_1018_comp)
ultimate_1018_comp <- max(sigma_eng_1018_comp, na.rm = TRUE)         # [MPa]
fracture_idx_1018_comp <- length(sigma_eng_1018_comp) # last index


# --- Aluminum 2024 Compression (Lab 1) ---
dis_2024_comp  <- as.numeric(al_2024_Compression_data$Displacement)
F_2024_comp    <- as.numeric(al_2024_Compression_data$Force)
eps_2024_comp  <- as.numeric(al_2024_Compression_data$Composite.strain)
gage_D_2024_comp   <- 12.84
max_F_2024_comp    <- max(F_2024_comp, na.rm = TRUE)    # [kN] 
max_eps_2024_comp  <- max(eps_2024_comp, na.rm = TRUE) 
A_2024_comp        <- pi/4 * gage_D_2024_comp^2         # [mm^2]
sigma_eng_2024_comp  <- F_2024_comp / A_2024_comp * 1000               # [MPa]
sigma_true_2024_comp <- sigma_eng_2024_comp * (1 + eps_2024_comp)     # [MPa]
eps_true_2024_comp   <- log(1 + eps_2024_comp)
ultimate_2024_comp       <- max(sigma_eng_2024_comp, na.rm = TRUE)     # [MPa]
fracture_idx_2024_comp   <- length(sigma_eng_2024_comp)                # assumed last point

# --- Steel 1018_CR ---
dis_1018 <- as.numeric(steel_1018CR_data$Displacement)
F_1018   <- as.numeric(steel_1018CR_data$Force)
eps_1018 <- as.numeric(steel_1018CR_data$Strain.1)
gage_D_1018 <- 7.26  # [mm]
max_F_1018  <- 28.15 # [kN]
max_eps_1018 <- 17.22
A_1018 <- pi/4 * gage_D_1018^2
sigma_eng_1018 <- F_1018 / A_1018 * 1000
sigma_true_1018 <- sigma_eng_1018 * (1 + eps_1018)
eps_true_1018 <- log(1 + eps_1018)
ultimate_1018 <- max(sigma_eng_1018, na.rm = TRUE)
fracture_1018 <- length(sigma_eng_1018)

# --- Steel 1045_CR ---
dis_1045_CR <- as.numeric(steel_1045CR_data$Displacement)
F_1045_CR   <- as.numeric(steel_1045CR_data$Force)
eps_1045_CR <- as.numeric(steel_1045CR_data$Strain.1)
gage_D_1045_CR <- 7.18
max_F_1045_CR <- 41.15
max_eps_1045_CR <- 9.768
A_1045_CR <- pi/4 * gage_D_1045_CR^2
sigma_eng_1045_CR <- F_1045_CR / A_1045_CR * 1000
sigma_true_1045_CR <- sigma_eng_1045_CR * (1 + eps_1045_CR)
eps_true_1045_CR <- log(1 + eps_1045_CR)
ultimate_1045_CR <- max(sigma_eng_1045_CR, na.rm = TRUE)
fracture_1045_CR <- length(sigma_eng_1045_CR)

# --- Steel 1045_NM ---
dis_1045_NM <- as.numeric(steel_1045NM_data$Displacement)
F_1045_NM   <- as.numeric(steel_1045NM_data$Force)
eps_1045_NM <- as.numeric(steel_1045NM_data$Strain.1)
gage_D_1045_NM <- 7.21
max_F_1045_NM <- 29
max_eps_1045_NM <- 28.01
A_1045_NM <- pi/4 * gage_D_1045_NM^2
sigma_eng_1045_NM <- F_1045_NM / A_1045_NM * 1000
sigma_true_1045_NM <- sigma_eng_1045_NM * (1 + eps_1045_NM)
eps_true_1045_NM <- log(1 + eps_1045_NM)
ultimate_1045_NM <- max(sigma_eng_1045_NM, na.rm = TRUE)
fracture_1045_NM <- length(sigma_eng_1045_NM)

# --- Steel 304 ---
dis_304 <- as.numeric(steel_304_data$Displacement)
F_304   <- as.numeric(steel_304_data$Force)
eps_304 <- as.numeric(steel_304_data$Strain.1)
gage_D_304 <- 7.18
max_F_304 <- 28.57
max_eps_304 <- 57.92
A_304 <- pi/4 * gage_D_304^2
sigma_eng_304 <- F_304 / A_304 * 1000
sigma_true_304 <- sigma_eng_304 * (1 + eps_304)
eps_true_304 <- log(1 + eps_304)
ultimate_304 <- max(sigma_eng_304, na.rm = TRUE)
fracture_304 <- length(sigma_eng_304)

# --- Aluminum 2024 ---
dis_2024 <- as.numeric(al_2024_data$Displacement)
F_2024   <- as.numeric(al_2024_data$Force)
eps_2024 <- as.numeric(al_2024_data$Strain.1)
gage_D_2024 <- 7.22
max_F_2024 <- 19.42
max_eps_2024 <- 20.1
A_2024 <- pi/4 * gage_D_2024^2
sigma_eng_2024 <- F_2024 / A_2024 * 1000
sigma_true_2024 <- sigma_eng_2024 * (1 + eps_2024)
eps_true_2024 <- log(1 + eps_2024)
ultimate_2024 <- max(sigma_eng_2024, na.rm = TRUE)
fracture_2024 <- length(sigma_eng_2024)

# --- Aluminum 7075 ---
dis_7075 <- as.numeric(al_7075_data$Displacement)
F_7075   <- as.numeric(al_7075_data$Force)
eps_7075 <- as.numeric(al_7075_data$Strain.1)
gage_D_7075 <- 6.11
max_F_7075 <- 22.6
max_eps_7075 <- 16.48
A_7075 <- pi/4 * gage_D_7075^2
sigma_eng_7075 <- F_7075 / A_7075 * 1000
sigma_true_7075 <- sigma_eng_7075 * (1 + eps_7075)
eps_true_7075 <- log(1 + eps_7075)
ultimate_7075 <- max(sigma_eng_7075, na.rm = TRUE)
fracture_7075 <- length(sigma_eng_7075)

# --- Brass ---
dis_brass <- as.numeric(brass_data$Displacement)
F_brass   <- as.numeric(brass_data$Force)
eps_brass <- as.numeric(brass_data$Strain.1)
gage_D_brass <- 7.26
max_F_brass <- 14.47
max_eps_brass <- 33.82
A_brass <- pi/4 * gage_D_brass^2
sigma_eng_brass <- F_brass / A_brass * 1000
sigma_true_brass <- sigma_eng_brass * (1 + eps_brass)
eps_true_brass <- log(1 + eps_brass)
ultimate_brass <- max(sigma_eng_brass, na.rm = TRUE)
fracture_brass <- length(sigma_eng_brass)

# --- PMMA ---
dis_pmma <- as.numeric(pmma_data$Displacement)
F_pmma   <- as.numeric(pmma_data$Force)
eps_pmma <- as.numeric(pmma_data$Strain.1)
gage_D_pmma <- 7
max_F_pmma <- 3.471
max_eps_pmma <- 3.397
A_pmma <- pi/4 * gage_D_pmma^2
sigma_eng_pmma <- F_pmma / A_pmma * 1000
sigma_true_pmma <- sigma_eng_pmma * (1 + eps_pmma)
eps_true_pmma <- log(1 + eps_pmma)
ultimate_pmma <- max(sigma_eng_pmma, na.rm = TRUE)
fracture_pmma <- length(sigma_eng_pmma)

colors <- c("firebrick", "darkorange", "steelblue", "black", 
            "green4", "purple", "goldenrod", "deeppink")
labels <- c("1018 Steel", "1045 Steel (CR)", "1045 Steel (NM)", "304 Steel",
            "2024 Aluminum", "7075 Aluminum", "Brass", "PMMA")
# Global x/y limits
xmax <- max(1)
ymax <- max(1000)

# Plot base curve
par(mfrow = c(1, 1))
plot(eps_1018, sigma_eng_1018, type = "l", col = colors[1], lwd = 2,
     xlim = c(0, xmax), ylim = c(0, ymax),
     xlab = "Engineering Strain (-)", ylab = "Engineering Stress (MPa)",
     main = "Engineering Stress-Strain Curves for All Materials")

# Overlay all other materials
lines(eps_1045_CR,   sigma_eng_1045_CR,   col = colors[2], lwd = 2)

lines(eps_1045_NM,   sigma_eng_1045_NM,   col = colors[3], lwd = 2)
lines(eps_304,       sigma_eng_304,       col = colors[4], lwd = 2)

lines(eps_2024,      sigma_eng_2024,      col = colors[5], lwd = 2)
lines(eps_7075,      sigma_eng_7075,      col = colors[6], lwd = 2)

lines(eps_brass,     sigma_eng_brass,     col = colors[7], lwd = 2)
lines(eps_pmma,      sigma_eng_pmma,      col = colors[8], lwd = 2)

# Add legend
legend("topright", legend = labels, col = colors, lwd = 2, bty = "n")
grid()

Engineering Stress-Strain vs. True Stress-Strain

par(mfrow = c(1, 2)) 
### 1018 Steel
plot(eps_1018, sigma_eng_1018, type = "n",
     xlab = "Strain (mm/mm)", ylab = "Stress (MPa)",
     main = "1018 Steel: Engineering vs True",
     ylim = c(0, 1200)) 
grid()

points(eps_1018, sigma_eng_1018, col = "royalblue", pch = 16)
points(eps_true_1018, sigma_true_1018, col = "darkorange", pch = 16)

legend("bottomright", legend = c("Engineering σ–ε", "True σ–ε"),
       col = c("royalblue", "darkorange"), pch = 16, bty = "n")

### 1045 Steel (Cold Rolled)
plot(eps_1045_CR, sigma_eng_1045_CR, type = "n",
     xlab = "Strain (mm/mm)", ylab = "Stress (MPa)",
     main = "1045 Steel (CR): Engineering vs True",
     ylim = c(0, 1200)) 

grid()
points(eps_1045_CR, sigma_eng_1045_CR, col = "royalblue", pch = 16)

points(eps_true_1045_CR, sigma_true_1045_CR, col = "darkorange", pch = 16)
legend("bottomright", legend = c("Engineering σ–ε", "True σ–ε"),
       col = c("royalblue", "darkorange"), pch = 16, bty = "n")

### 1045 Steel (Normalized)
plot(eps_1045_NM, sigma_eng_1045_NM, type = "n",
     xlab = "Strain (mm/mm)", ylab = "Stress (MPa)",
     main = "1045 Steel (NM): Engineering vs True",
     ylim = c(0, 1200)) 
grid()

points(eps_1045_NM, sigma_eng_1045_NM, col = "royalblue", pch = 16)
points(eps_true_1045_NM, sigma_true_1045_NM, col = "darkorange", pch = 16)

legend("bottomright", legend = c("Engineering σ–ε", "True σ–ε"),
       col = c("royalblue", "darkorange"), pch = 16, bty = "n")

### 304 Stainless Steel 
plot(eps_304, sigma_eng_304, type = "n",
     xlab = "Strain (mm/mm)", ylab = "Stress (MPa)",
     main = "304 Stainless Steel: Engineering vs True",
     ylim = c(0, 1200))  

grid()
points(eps_304, sigma_eng_304, col = "royalblue", pch = 16)

points(eps_true_304, sigma_true_304, col = "darkorange", pch = 16)
legend("bottomright",
       legend = c("Engineering σ–ε", "True σ–ε"),
       col = c("royalblue", "darkorange"),
       pch = 16, bty = "n")

### 2024 Aluminum
plot(eps_2024, sigma_eng_2024, type = "n",
     xlab = "Strain (mm/mm)", ylab = "Stress (MPa)",
     main = "2024 Aluminum: Engineering vs True",
     ylim = c(0, 1200)) 
grid()

points(eps_2024, sigma_eng_2024, col = "royalblue", pch = 16)
points(eps_true_2024, sigma_true_2024, col = "darkorange", pch = 16)

legend("bottomright", legend = c("Engineering σ–ε", "True σ–ε"),
       col = c("royalblue", "darkorange"), pch = 16, bty = "n")

### 7075 Aluminum
plot(eps_7075, sigma_eng_7075, type = "n",
     xlab = "Strain (mm/mm)", ylab = "Stress (MPa)",
     main = "7075 Aluminum: Engineering vs True",
     ylim = c(0, 1200))  

grid()
points(eps_7075, sigma_eng_7075, col = "royalblue", pch = 16)

points(eps_true_7075, sigma_true_7075, col = "darkorange", pch = 16)
legend("bottomright", legend = c("Engineering σ–ε", "True σ–ε"),
       col = c("royalblue", "darkorange"), pch = 16, bty = "n")

### Brass (Annealed)
plot(eps_brass, sigma_eng_brass, type = "n",
     xlab = "Strain (mm/mm)", ylab = "Stress (MPa)",
     main = "Annealed Brass: Engineering vs True",
     ylim = c(0, 1200))  
grid()

points(eps_brass, sigma_eng_brass, col = "royalblue", pch = 16)
points(eps_true_brass, sigma_true_brass, col = "darkorange", pch = 16)

legend("topright", legend = c("Engineering σ–ε", "True σ–ε"),
       col = c("royalblue", "darkorange"), pch = 16, bty = "n")

### PMMA
plot(eps_pmma, sigma_eng_pmma, type = "n",
     xlab = "Strain (mm/mm)", ylab = "Stress (MPa)",
     main = "PMMA: Engineering vs True",
     ylim = c(0, 1200))  

grid()
points(eps_pmma, sigma_eng_pmma, col = "royalblue", pch = 16)

points(eps_true_pmma, sigma_true_pmma, col = "darkorange", pch = 16)
legend("topright", legend = c("Engineering σ–ε", "True σ–ε"),
       col = c("royalblue", "darkorange"), pch = 16, bty = "n")

For the yield strain values for each materials below, I have eyeball the approximate value, find the young’s modulus, and the 0.2%. Update/adjust the yield strain based on the alignment of each graph

— 1018 Steel —

eps_yield_1018 <- 0.0026
# Fit linear elastic region up to yield (0 intercept)
fit_1018 <- lm(sigma_eng_1018[eps_1018 <= eps_yield_1018] ~ 0 + eps_1018[eps_1018 <= eps_yield_1018])
E_1018 <- coef(fit_1018)[1]

# Calculate 0.2% offset line: σ = E(ε − 0.002)
sigma_offset_1018 <- E_1018 * (eps_1018 - 0.002)
diff_1018 <- sigma_eng_1018 - sigma_offset_1018
i_1018 <- which(diff(sign(diff_1018)) != 0)[1]
yield_strain_1018 <- eps_1018[i_1018]
yield_stress_1018 <- sigma_eng_1018[i_1018]

# --- Plot ---
plot(eps_1018, sigma_eng_1018,
     type = "l", col = "steelblue", lwd = 2,
     xlab = "Strain (-)", ylab = "Stress (MPa)",
     main = "1018 Steel", ylim = c(0, 1200))
grid()

points(eps_1018, sigma_eng_1018, col = "steelblue", pch = 16)
abline(a = 0, b = E_1018, col = "purple", lwd = 2, lty = 2)                # Original elastic line

abline(a = -E_1018 * 0.002, b = E_1018, col = "darkgreen", lwd = 2, lty = 3) # 0.2% offset line
points(yield_strain_1018, yield_stress_1018, pch = 19, col = "red", cex = 1.5)

text(x = yield_strain_1018,
     y = yield_stress_1018,
     labels = paste0(round(yield_stress_1018, 1), " MPa"),
     pos = 3, col = "red", cex = 1.2)

cat("Elastic Modulus (E)         =", round(E_1018, 2), "MPa\n")

Elastic Modulus (E)         = 202558.9 MPa

cat("0.2% Offset Yield Strain    =", round(yield_strain_1018, 5), "\n")

0.2% Offset Yield Strain    = 0.00519 

cat("0.2% Offset Yield Stress    =", round(yield_stress_1018, 2), "MPa\n")

0.2% Offset Yield Stress    = 660.2 MPa

— 1045 Steel (Cold Rolled) —

eps_yield_1045_CR <- 0.0033
fit_1045_CR <- lm(sigma_eng_1045_CR[eps_1045_CR <= eps_yield_1045_CR] ~ 0 + eps_1045_CR[eps_1045_CR <= eps_yield_1045_CR])
E_1045_CR <- coef(fit_1045_CR)[1]
sigma_offset_1045_CR <- E_1045_CR * (eps_1045_CR - 0.002)
i_1045_CR <- which(diff(sign(sigma_eng_1045_CR - sigma_offset_1045_CR)) != 0)[1]
yield_strain_1045_CR <- eps_1045_CR[i_1045_CR]
yield_stress_1045_CR <- sigma_eng_1045_CR[i_1045_CR]

plot(eps_1045_CR, sigma_eng_1045_CR, type = "l", col = "firebrick", lwd = 2,
     xlab = "Strain (-)", ylab = "Stress (MPa)", main = "1045 Steel (CR)", ylim = c(0, 1200))
grid()

abline(a = 0, b = E_1045_CR, col = "purple", lwd = 2, lty = 2)
abline(a = -E_1045_CR * 0.002, b = E_1045_CR, col = "darkgreen", lwd = 2, lty = 3)

points(yield_strain_1045_CR, yield_stress_1045_CR, col = "red", pch = 19, cex = 1.5)
text(yield_strain_1045_CR, yield_stress_1045_CR, labels = paste0(round(yield_stress_1045_CR, 1), " MPa"), pos = 3, col = "red")

cat("Elastic Modulus (E)         =", round(E_1018, 2), "MPa\n")

Elastic Modulus (E)         = 202558.9 MPa

cat("0.2% Offset Yield Strain    =", round(yield_strain_1045_CR, 5), "\n")

0.2% Offset Yield Strain    = 0.0046 

cat("0.2% Offset Yield Stress    =", round(yield_stress_1045_CR, 2), "MPa\n")

0.2% Offset Yield Stress    = 964.95 MPa

—1045 Steel (Normalized) —

eps_yield_1045_NM <- 0.0035
fit_1045_NM <- lm(sigma_eng_1045_NM[eps_1045_NM <= eps_yield_1045_NM] ~ 0 + eps_1045_NM[eps_1045_NM <= eps_yield_1045_NM])
E_1045_NM <- coef(fit_1045_NM)[1]
sigma_offset_1045_NM <- E_1045_NM * (eps_1045_NM - 0.002)
i_1045_NM <- which(diff(sign(sigma_eng_1045_NM - sigma_offset_1045_NM)) != 0)[1]
yield_strain_1045_NM <- eps_1045_NM[i_1045_NM]
yield_stress_1045_NM <- sigma_eng_1045_NM[i_1045_NM]

plot(eps_1045_NM, sigma_eng_1045_NM, type = "l", col = "indianred", lwd = 2,
     xlab = "Strain (-)", ylab = "Stress (MPa)", main = "1045 Steel (NM)", ylim = c(0, 1200))
grid()

abline(a = 0, b = E_1045_NM, col = "purple", lwd = 2, lty = 2)
abline(a = -E_1045_NM * 0.002, b = E_1045_NM, col = "darkgreen", lwd = 2, lty = 3)

points(yield_strain_1045_NM, yield_stress_1045_NM, col = "red", pch = 19, cex = 1.5)
text(yield_strain_1045_NM, yield_stress_1045_NM, labels = paste0(round(yield_stress_1045_NM, 1), " MPa"), pos = 3, col = "red")

cat("Elastic Modulus (E)         =", round(E_1018, 2), "MPa\n")

Elastic Modulus (E)         = 202558.9 MPa

cat("0.2% Offset Yield Strain    =", round(yield_strain_1045_NM, 5), "\n")

0.2% Offset Yield Strain    = 0.00354 

cat("0.2% Offset Yield Stress    =", round(yield_stress_1045_NM, 2), "MPa\n")

0.2% Offset Yield Stress    = 462.92 MPa

— 304 Stainless Steel —

eps_yield_304 <- 0.0030
fit_304 <- lm(sigma_eng_304[eps_304 <= eps_yield_304] ~ 0 + eps_304[eps_304 <= eps_yield_304])
E_304 <- coef(fit_304)[1]
sigma_offset_304 <- E_304 * (eps_304 - 0.002)
i_304 <- which(diff(sign(sigma_eng_304 - sigma_offset_304)) != 0)[1]
yield_strain_304 <- eps_304[i_304]
yield_stress_304 <- sigma_eng_304[i_304]

plot(eps_304, sigma_eng_304, type = "l", col = "forestgreen", lwd = 2,
     xlab = "Strain (-)", ylab = "Stress (MPa)", main = "304 Stainless Steel", ylim = c(0, 1200))
grid()

abline(a = 0, b = E_304, col = "purple", lwd = 2, lty = 2)
abline(a = -E_304 * 0.002, b = E_304, col = "darkgreen", lwd = 2, lty = 3)

points(yield_strain_304, yield_stress_304, col = "red", pch = 19, cex = 1.5)
text(yield_strain_304, yield_stress_304, labels = paste0(round(yield_stress_304, 1), " MPa"), pos = 3, col = "red")

cat("Elastic Modulus (E)         =", round(E_1018, 2), "MPa\n")

Elastic Modulus (E)         = 202558.9 MPa

cat("0.2% Offset Yield Strain    =", round(yield_strain_304, 5), "\n")

0.2% Offset Yield Strain    = 0.00537 

cat("0.2% Offset Yield Stress    =", round(yield_stress_304, 2), "MPa\n")

0.2% Offset Yield Stress    = 473.95 MPa

—2024 Aluminum —

eps_yield_2024 <- 0.005
fit_2024 <- lm(sigma_eng_2024[eps_2024 <= eps_yield_2024] ~ 0 + eps_2024[eps_2024 <= eps_yield_2024])
E_2024 <- coef(fit_2024)[1]
sigma_offset_2024 <- E_2024 * (eps_2024 - 0.002)
i_2024 <- which(diff(sign(sigma_eng_2024 - sigma_offset_2024)) != 0)[1]
yield_strain_2024 <- eps_2024[i_2024]
yield_stress_2024 <- sigma_eng_2024[i_2024]

plot(eps_2024, sigma_eng_2024, type = "l", col = "dodgerblue3", lwd = 2,
     xlab = "Strain (-)", ylab = "Stress (MPa)", main = "2024 Aluminum", ylim = c(0, 1200))
grid()

abline(a = 0, b = E_2024, col = "purple", lwd = 2, lty = 2)
abline(a = -E_2024 * 0.002, b = E_2024, col = "darkgreen", lwd = 2, lty = 3)

points(yield_strain_2024, yield_stress_2024, col = "red", pch = 19, cex = 1.5)
text(yield_strain_2024, yield_stress_2024, labels = paste0(round(yield_stress_2024, 1), " MPa"), pos = 3, col = "red")

cat("Elastic Modulus (E)         =", round(E_1018, 2), "MPa\n")

Elastic Modulus (E)         = 202558.9 MPa

cat("0.2% Offset Yield Strain    =", round(yield_strain_2024, 5), "\n")

0.2% Offset Yield Strain    = 0.00638 

cat("0.2% Offset Yield Stress    =", round(yield_stress_2024, 2), "MPa\n")

0.2% Offset Yield Stress    = 362.22 MPa

— 7075 Aluminum —

eps_yield_7075 <- 0.004
fit_7075 <- lm(sigma_eng_7075[eps_7075 <= eps_yield_7075] ~ 0 + eps_7075[eps_7075 <= eps_yield_7075])
E_7075 <- coef(fit_7075)[1]
sigma_offset_7075 <- E_7075 * (eps_7075 - 0.002)
i_7075 <- which(diff(sign(sigma_eng_7075 - sigma_offset_7075)) != 0)[1]
yield_strain_7075 <- eps_7075[i_7075]
yield_stress_7075 <- sigma_eng_7075[i_7075]

plot(eps_7075, sigma_eng_7075, type = "l", col = "darkorange", lwd = 2,
     xlab = "Strain (-)", ylab = "Stress (MPa)", main = "7075 Aluminum", ylim = c(0, 1200))
grid()

abline(a = 0, b = E_7075, col = "purple", lwd = 2, lty = 2)
abline(a = -E_7075 * 0.002, b = E_7075, col = "darkgreen", lwd = 2, lty = 3)

points(yield_strain_7075, yield_stress_7075, col = "red", pch = 19, cex = 1.5)
text(yield_strain_7075, yield_stress_7075, labels = paste0(round(yield_stress_7075, 1), " MPa"), pos = 3, col = "red")

cat("Elastic Modulus (E)         =", round(E_1018, 2), "MPa\n")

Elastic Modulus (E)         = 202558.9 MPa

cat("0.2% Offset Yield Strain    =", round(yield_strain_7075, 5), "\n")

0.2% Offset Yield Strain    = 0.00885 

cat("0.2% Offset Yield Stress    =", round(yield_stress_7075, 2), "MPa\n")

0.2% Offset Yield Stress    = 694.39 MPa

—Annealed Brass—

eps_yield_brass <- 0.007
fit_brass <- lm(sigma_eng_brass[eps_brass <= eps_yield_brass] ~ 0 + eps_brass[eps_brass <= eps_yield_brass])
E_brass <- coef(fit_brass)[1]
sigma_offset_brass <- E_brass * (eps_brass - 0.002)
i_brass <- which(diff(sign(sigma_eng_brass - sigma_offset_brass)) != 0)[1]
yield_strain_brass <- eps_brass[i_brass]
yield_stress_brass <- sigma_eng_brass[i_brass]

plot(eps_brass, sigma_eng_brass, type = "l", col = "goldenrod", lwd = 2,
     xlab = "Strain (-)", ylab = "Stress (MPa)", main = "Annealed Brass", ylim = c(0, 1200))
grid()

abline(a = 0, b = E_brass, col = "purple", lwd = 2, lty = 2)
abline(a = -E_brass * 0.002, b = E_brass, col = "darkgreen", lwd = 2, lty = 3)

points(yield_strain_brass, yield_stress_brass, col = "red", pch = 19, cex = 1.5)
text(yield_strain_brass, yield_stress_brass, labels = paste0(round(yield_stress_brass, 1), " MPa"), pos = 3, col = "red")

cat("Elastic Modulus (E)         =", round(E_1018, 2), "MPa\n")

Elastic Modulus (E)         = 202558.9 MPa

cat("0.2% Offset Yield Strain    =", round(yield_strain_brass, 5), "\n")

0.2% Offset Yield Strain    = 0.0063 

cat("0.2% Offset Yield Stress    =", round(yield_stress_brass, 2), "MPa\n")

0.2% Offset Yield Stress    = 238.93 MPa

—PMMA—

eps_yield_pmma <- 0.007
fit_pmma <- lm(sigma_eng_pmma[eps_pmma <= eps_yield_pmma] ~ 0 + eps_pmma[eps_pmma <= eps_yield_pmma])
E_pmma <- coef(fit_pmma)[1]
sigma_offset_pmma <- E_pmma * (eps_pmma - 0.002)
i_pmma <- which(diff(sign(sigma_eng_pmma - sigma_offset_pmma)) != 0)[1]
yield_strain_pmma <- eps_pmma[i_pmma]
yield_stress_pmma <- sigma_eng_pmma[i_pmma]

plot(eps_pmma, sigma_eng_pmma, type = "l", col = "black", lwd = 2,
     xlab = "Strain (-)", ylab = "Stress (MPa)", main = "PMMA", ylim = c(0, 1200))
grid()

abline(a = 0, b = E_pmma, col = "purple", lwd = 2, lty = 2)
abline(a = -E_pmma * 0.002, b = E_pmma, col = "darkgreen", lwd = 2, lty = 3)

points(yield_strain_pmma, yield_stress_pmma, col = "red", pch = 19, cex = 1.5)
text(yield_strain_pmma, yield_stress_pmma, labels = paste0(round(yield_stress_pmma, 1), " MPa"), pos = 3, col = "red")

cat("Elastic Modulus (E)         =", round(E_1018, 2), "MPa\n")

Elastic Modulus (E)         = 202558.9 MPa

cat("0.2% Offset Yield Strain    =", round(yield_strain_pmma, 5), "\n")

0.2% Offset Yield Strain    = 0.01378 

cat("0.2% Offset Yield Stress    =", round(yield_stress_pmma, 2), "MPa\n")

0.2% Offset Yield Stress    = 51.94 MPa

Elastic Region (Up to Yield Strain)

# Truncate to each material's elastic region (up to yield)
truncate_to_yield <- function(eps, sigma, strain_limit) {
  idx <- which(eps <= strain_limit)
  return(list(eps = eps[idx], sigma = sigma[idx]))
}

# Truncated curves
s1018     <- truncate_to_yield(eps_1018,     sigma_eng_1018,     eps_yield_1018)
s1045_CR  <- truncate_to_yield(eps_1045_CR,  sigma_eng_1045_CR,  eps_yield_1045_CR)
s1045_NM  <- truncate_to_yield(eps_1045_NM,  sigma_eng_1045_NM,  eps_yield_1045_NM)
s304      <- truncate_to_yield(eps_304,      sigma_eng_304,      eps_yield_304)
s2024     <- truncate_to_yield(eps_2024,     sigma_eng_2024,     eps_yield_2024)
s7075     <- truncate_to_yield(eps_7075,     sigma_eng_7075,     eps_yield_7075)
sbrass    <- truncate_to_yield(eps_brass,    sigma_eng_brass,    eps_yield_brass)
spmma     <- truncate_to_yield(eps_pmma,     sigma_eng_pmma,     eps_yield_pmma)

# Plot
plot(0, 0, type = "n", xlim = c(0, 0.015), ylim = c(0, 1200),
     xlab = "Strain (mm/mm)", ylab = "Stress (MPa)",
     main = "Elastic Region (Up to Yield Strain)")
grid()

# Add lines
lines(s1018$eps,    s1018$sigma,    col = "steelblue",   lwd = 3)
lines(s1045_CR$eps,s1045_CR$sigma, col = "gray40",      lwd = 3)

lines(s1045_NM$eps,s1045_NM$sigma, col = "darkorange",  lwd = 3)
lines(s304$eps,     s304$sigma,    col = "gold",        lwd = 3)

lines(s2024$eps,    s2024$sigma,   col = "skyblue",     lwd = 3)
lines(s7075$eps,    s7075$sigma,   col = "dodgerblue",  lwd = 3)

lines(sbrass$eps,   sbrass$sigma,  col = "chocolate",   lwd = 3)
lines(spmma$eps,    spmma$sigma,   col = "forestgreen", lwd = 3)

# Legend
legend("bottomright",
       legend = c("1018 CR", "1045 CR", "1045 NM", "304 SS",
                  "2024 Aluminum", "7075 Aluminum", "Brass", "PMMA"),
       col = c("steelblue", "gray40", "darkorange", "gold",
               "skyblue", "dodgerblue", "chocolate", "forestgreen"),
       lwd = 3, bty = "n", cex = 0.9)

1018CR Steel: Tension vs Compression (Engineering Stress-Strain)

plot(abs(eps_1018), abs(sigma_eng_1018), type = "l", lwd = 2, col = "blue",
     xlab = "Strain (abs)", ylab = "Engineering Stress [MPa]",
     main = "1018 Cold-Rolled Steel: Tension vs Compression")
lines(abs(eps_1018_comp), abs(sigma_eng_1018_comp), col = "red", lwd = 2, lty = 2)

legend("topleft", legend = c("Tension", "Compression"),
       col = c("blue", "red"), lty = c(1, 2), lwd = 2)
grid()

2024 Aluminum: Tension vs Compression (Engineering Stress-Strain)

plot(abs(eps_2024), abs(sigma_eng_2024), type = "l", lwd = 2, col = "blue",
     xlab = "Strain (abs)", ylab = "Engineering Stress [MPa]",
     main = "2024 Aluminum Alloy: Tension vs Compression")
lines(abs(eps_2024_comp), abs(sigma_eng_2024_comp), col = "red", lwd = 2, lty = 2)

legend("topleft", legend = c("Tension", "Compression"),
       col = c("blue", "red"), lty = c(1, 2), lwd = 2)
grid()

Table Report Information

cat("Elastic Properties Summary (E in GPa, Stress in MPa):\n\n")

Elastic Properties Summary (E in GPa, Stress in MPa):

cat(sprintf("%-18s | E = %9.2f | Yield Strain = %.5f | Yield Stress = %7.2f | Ultimate Strength = %7.2f\n",
    "1018 Steel CR",     E_1018 / 1000,     yield_strain_1018,     yield_stress_1018,     ultimate_1018))

1018 Steel CR      | E =    202.56 | Yield Strain = 0.00519 | Yield Stress =  660.20 | Ultimate Strength =  680.01

cat(sprintf("%-18s | E = %9.2f | Yield Strain = %.5f | Yield Stress = %7.2f | Ultimate Strength = %7.2f\n",
    "1045 Steel CR",     E_1045_CR / 1000,  yield_strain_1045_CR,  yield_stress_1045_CR,  ultimate_1045_CR))

1045 Steel CR      | E =    354.83 | Yield Strain = 0.00460 | Yield Stress =  964.95 | Ultimate Strength = 1016.32

cat(sprintf("%-18s | E = %9.2f | Yield Strain = %.5f | Yield Stress = %7.2f | Ultimate Strength = %7.2f\n",
    "1045 Steel NM",     E_1045_NM / 1000,  yield_strain_1045_NM,  yield_stress_1045_NM,  ultimate_1045_NM))

1045 Steel NM      | E =    189.27 | Yield Strain = 0.00354 | Yield Stress =  462.92 | Ultimate Strength =  710.29

cat(sprintf("%-18s | E = %9.2f | Yield Strain = %.5f | Yield Stress = %7.2f | Ultimate Strength = %7.2f\n",
    "304 SS",            E_304 / 1000,      yield_strain_304,      yield_stress_304,      ultimate_304))

304 SS             | E =    136.35 | Yield Strain = 0.00537 | Yield Stress =  473.95 | Ultimate Strength =  705.62

cat(sprintf("%-18s | E = %9.2f | Yield Strain = %.5f | Yield Stress = %7.2f | Ultimate Strength = %7.2f\n",
    "2024 Aluminum",     E_2024 / 1000,     yield_strain_2024,     yield_stress_2024,     ultimate_2024))

2024 Aluminum      | E =     75.30 | Yield Strain = 0.00638 | Yield Stress =  362.22 | Ultimate Strength =  474.33

cat(sprintf("%-18s | E = %9.2f | Yield Strain = %.5f | Yield Stress = %7.2f | Ultimate Strength = %7.2f\n",
    "7075 Aluminum",     E_7075 / 1000,     yield_strain_7075,     yield_stress_7075,     ultimate_7075))

7075 Aluminum      | E =     97.05 | Yield Strain = 0.00885 | Yield Stress =  694.39 | Ultimate Strength =  770.79

cat(sprintf("%-18s | E = %9.2f | Yield Strain = %.5f | Yield Stress = %7.2f | Ultimate Strength = %7.2f\n",
    "Annealed Brass",    E_brass / 1000,    yield_strain_brass,    yield_stress_brass,    ultimate_brass))

Annealed Brass     | E =     52.82 | Yield Strain = 0.00630 | Yield Stress =  238.93 | Ultimate Strength =  349.55

cat(sprintf("%-18s | E = %9.2f | Yield Strain = %.5f | Yield Stress = %7.2f | Ultimate Strength = %7.2f\n",
    "PMMA",              E_pmma / 1000,     yield_strain_pmma,     yield_stress_pmma,     ultimate_pmma))

PMMA               | E =      4.36 | Yield Strain = 0.01378 | Yield Stress =   51.94 | Ultimate Strength =   90.19

Helper Function: Compute K, n from true stress/true plastic strain (Ramberg–Osgood)

fit_powerlaw = function(eps_eng, eps_true, sigma_true, E_MPa,
                         yield_strain, sigma_eng,
                         min_ep = 1e-5) {

  # Index Interval from yield to UTS (pre-necking)
  i_start <- which(eps_eng >= yield_strain)[1]
  i_end   <- which.max(sigma_eng)
  # i_start: first index where engineering strain reaches yield → beginning of plastic deformation
  # i_end: where engineering stress reaches its maximum → necking usually starts here
  
  ## pre-caution check
  if (is.na(i_start)) i_start <- 1
  if (is.na(i_end) || i_end < i_start) i_end <- length(eps_true)

  # stress hardening phase index range
  idx <- i_start:i_end

  # True plastic strain{strain-hardening interval}
  eps_p_true <- eps_true[idx] - sigma_true[idx] / E_MPa
  sigma_t    <- sigma_true[idx]

  # pre-caution check, only keep valid and positive plastic strain
  keep <- is.finite(eps_p_true) & is.finite(sigma_t) &
          eps_p_true > min_ep & sigma_t > 0
  eps_p_true <- eps_p_true[keep]
  sigma_t    <- sigma_t[keep]

  # Linear fit in log-log space: log σ = log K + n log ε_p
  fit <- lm(log(sigma_t) ~ log(eps_p_true))
  n   <- unname(coef(fit)[2])
  K   <- unname(exp(coef(fit)[1])) # MPa

  list(K = K, n = n,
       idx_range = c(i_start, i_end),
       fit = fit,
       eps_p_true = eps_p_true, sigma_t = sigma_t)
}

## ---- 7075 Aluminum ----
res_7075 <- fit_powerlaw(
  eps_eng    = eps_7075,
  eps_true   = eps_true_7075,
  sigma_true = sigma_true_7075,
  E_MPa      = E_7075,             # E in MPa
  yield_strain = yield_strain_7075,
  sigma_eng  = sigma_eng_7075
)

## ---- Annealed Brass ----
res_brass <- fit_powerlaw(
  eps_eng    = eps_brass,
  eps_true   = eps_true_brass,
  sigma_true = sigma_true_brass,
  E_MPa      = E_brass,            # E in MPa
  yield_strain = yield_strain_brass,
  sigma_eng  = sigma_eng_brass
)

## --- Plot: Annealed Brass log–log fit ---
# Scatter plot of log(true plastic strain) vs. log(true stress)
# This shows the experimental data in log-log space
plot(log(res_brass$eps_p_true), log(res_brass$sigma_t),
     xlab = "log(True plastic strain)",
     ylab = "log(True stress) [MPa]",
     main = "Annealed Brass: log–log fit",
     pch = 16)

# Add dashed line: linear regression fit to the log-log data
# This line represents log(σ_t) = log(K) + n * log(ε_p)
# The slope = n (strain hardening exponent), intercept = log(K)
abline(res_brass$fit, lwd = 2, lty = 2)

## --- Plot: 7075 Aluminum log–log fit ---
plot(log(res_7075$eps_p_true), log(res_7075$sigma_t),
     xlab = "log(True plastic strain)",
     ylab = "log(True stress) [MPa]",
     main = "7075 Aluminum: log–log fit",
     pch = 16)
abline(res_7075$fit, lwd = 2, lty = 2)

## Data Report
cat("\nRamberg–Osgood parameters (σ_true = K * ε_p_true^n):\n")

Ramberg–Osgood parameters (σ_true = K * ε_p_true^n):

cat(sprintf("  Annealed Brass : K = %.2f MPa, n = %.4f\n", res_brass$K, res_brass$n))

  Annealed Brass : K = 595.62 MPa, n = 0.2890

cat(sprintf("  7075 Aluminum  : K = %.2f MPa, n = %.4f\n", res_7075$K, res_7075$n))

  7075 Aluminum  : K = 946.06 MPa, n = 0.0535