A typical stress strain curve is shown in figure 3-6. The slope of the curve increases from a finite value at strain equals zero. As the strain increases the slope increases until it reaches a constant value. This constant value is the initial modulus used in this study. The initial non-linear portion is mostly due to seating error because the two upper and lower surface of the specimen are not totally parallel. The value of the strain where the slope starts to be constant is around 0.3%. The more parallel the upper and lower surfaces the less the seating error. By subtracting this amount of strain and shifting the whole stress strain curve towards the vertical axis such that the linear portion can be extended to the origin, will give the corrected curve.

The corrected stress strain curve is used in this study. The corrected stress strain curve can be simply divided into three portions; the initial linear portion, the post yield linear portion and a connecting curve between them. Six main items can be derived from the corrected stress strain curve; the initial slope, the post yield slope, the stress at 10% strain, the stress at 5% strain, the stress at 1% strain and the strain at the end of the initial slope. The term yield in this study is given to that end of the linear portion point.

Figure 3‑1 Typical EPS Uniaxial Compression Stress Strain Curve

As was mentioned in chapter two one of the factors that will affect the stress strain curve under monotonic compression is the loading rate. Figures 3-7 through 3-37 are the results of testing 0.05m cube specimens of different densities under five loading rates. The tests are conducted in a displacement control mode. Loading rates of 50mm/min, 5mm/min, 0.5mm/min, 0.05mm/min and 0.005mm/min, which are equivalent to strain rates of 100%/min, 10%/min, 1%/min, 0.1%/min and 0.01%/min are used in this study. The tests were run using the MTS system.

            Figures 3-7 through 3-11 show the variation of the strength at 1%, 5% and 10% strain with density, for the practical range of density for the five strain rates. Twelve tests were done for each curve. The straight lines are best fits for the experimental results. The equations for each line are shown in the figures with the “x” variable refer to the density in kg/m³ and the “y” variable refers to the strength. The correlation coefficient is shown with each equation. For all five curves it can be noticed that as the density increases the strength increases as was mentioned earlier in chapter two. Thus the effect of the density on strength becomes more significant with strain. The slope of the best-fit straight lines increases with % strain.

From the figures it can be found that the strength at 5% strain is around 90% of the strength at 10% strain. The ratio between the two strengths at 5 and 10% strain is least for the slowest loading rate of 0.01%/min and increases for the large loading of 100%. For a 20 kg/m³ density the ratio between the strength at 5% strain and the strength at 10% strain equals to 88% at a loading rate of 100%strain/min while at a loading rate of 0.01%/min the ratio is equal to 92%. Also it can be found that the strength at 1% strain is around 45% of the strength at 10% strain. The ratio between the two strengths of 1 and 10% strain is least for the slow loading rate of 0.01%/min and increases for the large loading of 100%/min. For a 20 kg/m³ density the ratio between the strength at 1% strain and the strength at 10% strain equals to 38% at a loading rate of 100%strain/min while at a loading rate of 0.01%/min the ratio is equal to 50%. In general the difference in the strength for the three strain level values increases with density for all loading rates.

Figure 3‑2 Stresses at Different Strain Levels for a 100%strain/min

Figure 3‑3 Stresses at Different Strain Levels for a 10%strain/min

Figure 3‑4 Stresses at Different Strain Levels for a 1%strain/min

Figure 3‑5 Stresses at Different Strain Levels for a 0.10%strain/min

Figure 3‑6 Stresses at Different Strain Levels for a 0.01%strain/min

Figure 3-12 shows the strength at 10% strain for two strain rates 0.01%/min and 100%/min. The higher the loading rate the higher is the strength. The difference in the strength for the two loading rates is equal to 50 kPa at 20kg/m³. The difference in the strength for the two loading rates increases as the density increases.

Figure 3-13 shows the strength at 5% strain for two strain rates 0.01%/min and 100%/min. The higher the loading rate the higher is the strength. The difference in the strength for the two loading rates is equal to 40 kPa at 20kg/m³. The difference in the strength for the two loading rates increases as the density increases.

Figure 3‑7 Stresses at 10% Strain for Different Strain Rates

Figure 3‑8 Stresses at 5% Strain for Different Strain Rates

Figure 3-14 shows the strength at 1% strain for two strain rates 0.01%/min and 100%/min. The higher the loading rate the higher is the strength. The difference in the strength for the two loading rates is equal to 10 kPa at 20kg/m3. The difference in the strength for the two loading rates increases as the density increases. Figures 3-15, 3-16 and 3-17 show the effect of the strain rate on the strength at 1, 5 and 10% strain in a percentage form.

Figures 3-18, 3-19, 3-20 and 3-21 show the strength at 10% strain, 5% strain and 1% strain for different strain rates for EPS geofoam types XI, I, VIII and IX respectively. Natural logarithmic equations fit well the relation between the strength and the strain rate. The equations are shown in the figures where the symbol “x” refers to the strain rate in %/min, while the symbol “y” refers to the strength in kPa. The correlation coefficient is written below each equation. These equations are suitable for calculating the stress if the loading rate is known.

Three equations for the stresses at 1, 5 and 10% strain are derived from the previous results as shown:

s_10% = 7.3*R^0.04 * D –35                                                        Equation 3‑1

s_5% = 6.6*R^0.04 *D –35                                                          Equation 3‑2

s_1% = 3.5*R^0.01 *D –22                                                          Equation 3‑3

Where:

s_1% is the strength at 1% strain in kPa

s_5% is the strength at 5% strain in kPa

s_10% is the strength at 10% strain in kPa

R is the strain rate in %/min

D is the density in kg/m³

Figure 3‑9 Stresses at 1% Strain for Different Strain Rates

Figure 3‑10 Effect of Strain Rate on the Strength at 1% Strain

Figure 3‑11 Effect of Strain Rate on the Strength at 5% Strain

Figure 3‑12 Effect of Strain Rate on the Strength at 10% Strain

Figure 3‑13 Type XI Stresses for Different Strain Rates

Figure 3‑14 Type I Stresses for Different Strain Rates

Figure 3‑15 Type VIII Stresses for Different Strain Rates

Figure 3‑16 Type IX Stresses for Different Strain Rates

Figures 3-22 through 3-26 show the variation of the initial and post yield modulus with density for the practical range of density for the five strain rates. Twelve tests were done for each curve. The equations for each best-fit line are shown in the figures with the “x” variable as density in kg/m³ and the “y” variable as modulus in MPa. The correlation coefficient is shown below each equation. For all five figures it can be noticed that as the density increases the modulus increases. The effect of density on initial modulus is greater than on the post yield modulus. This can be observed from the slope of the straight lines.

Figure 3‑17 Initial and Post Yield Modulus for a 100%strain/min

Figure 3‑18 Initial and Post Yield Modulus for a 10%strain/min

Figure 3‑19 Initial and Post Yield Modulus for a 1%strain/min

Figure 3‑20 Initial and Post Yield Modulus for a 0.1%strain/min

Figure 3‑21 Initial and Post Yield Modulus for a 0.01%strain/min

Figure 3-27 shows the initial modulus for two strain rates 0.01%/min and 100%/min. The higher the loading rate the higher is the modulus. The difference in the modulus for the two loading rates is equal to 1 MPa at 20kg/m³. The difference in the initial modulus for the two loading rates increases as the density increases.

Figure 3-28 shows the post yield modulus for two strain rates 0.01%/min and 100%/min. The higher the loading rate the higher is the modulus. The difference in the modulus for the two loading rates is equal to 0.18 MPa at 20kg/m³. The difference in the initial modulus for the two loading rates increases as the density increases.

Figure 3‑22 Initial Modulus for Different Strain Rates

Figure 3‑23 Post Yield Modulus for Different Strain Rates

Figures 3-29, 3-30, 3-31 and 3-32 show the initial and the post yield modulus for different strain rates for EPS geofoam types XI, I, VIII and IX respectively. The figures were obtained by calculating the moduli from the equations shown in figures 3-22 through 3-26 using the minimum density. Natural logarithmic equations fit well the relation between the strength and the strain rate. The equations are shown in the figures where the symbol “x” refers to the strain rate in %/min, while the symbol “y” refers to the moduli in MPa. The correlation coefficient is written below each equation. These equations are useful in calculating stresses in displacement control problems.

Figure 3‑24 Type XI Initial and Post Yield Modulus for Different Strain Rates

Figure 3‑25 Type I Initial and Post Yield Modulus for Various Strain Rates

Figure 3‑26 Type VIII Initial and Post Yield Modulus for Different Strain Rates

Figure 3‑27 Type IX Initial and Post Yield Modulus for Different Strain Rates

Two equations for the initial and the post yield modulus are derived from the previous results as shown:

E _initial = 0.35*R^0.01 * D –2.2                                                   Equation 3‑4

E _{post yield} = 0.007*R^0.04 *D                                                      Equation 3‑5

Where:

E _initial is the initial modulus in MPa

E _{post yield} is the post yield modulus in MPa

R is the strain rate in %/min

D is the density in kg/m³

Figures 3-33 through 3-37 show the strain at the end of the initial linear portion of the stress strain curve for the commonly used densities subjected to various loading rates. For the five figures, none of the data results reached less than 1% strain. The results are scattered and can be best correlated to a straight line. The general trend is that as the density increases the linear initial portion end strain slightly decreases.

Figure 3‑28 End of Linear Portion for a 0.01%strain/min

Figure 3‑29 End of Linear Portion for a 0.1%strain/min

Figure 3‑30 End of Linear Portion for a 1%strain/min

Figure 3‑31 End of Linear Portion for a 10%Strain/min

Figure 3‑32 End of Linear Portion for a 100%Strain/min