This file describes the hyperbolic constitutive model based on Duncan Chang's theory. Prepared by Mostafa Abedi

1. Constitutive Modelling of Soil
(Duncan-Chang Model)
Mostafa Abedi
‫رحیم‬ ‫یا‬ ‫رحمان‬ ‫یا‬
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2. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Comparison Duncan-Chang with other hyperbolic
models
Shear Modulus (G):
Determination G (Shear Modulus) relative to  (shear starin)
2
Ramberg-Osgood Model (1992)
Hardin-Drnevich Model (1972)
Davidenkov Model (1953)
and etc.

3. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Comparison Duncan-Chang with other hyperbolic
models
Tangent Modulus (Et):
Determination Et (tangent Modulus) relative to - (axial stress-strain)
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Duncan-Chang(1960)

4. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Duncan-Chang constitutive equations
Kondner and et al. (1963) have shown that the nonlinear stress-strain
curves of both clay and sand may be approximated by hyperbolae with a
high degrss of accuracy.
The hyperbolic equation propose by Kondner was:
𝜎1 − 𝜎3 =
𝜀
𝑎 + 𝑏. 𝜀
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5. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Duncan-Chang constitutive equations
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E0 or Ei Et
Esec

6. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Duncan-Chang constitutive equations
𝑞 = 𝜎1 − 𝜎3 =
𝜀𝑎
𝑎 + 𝑏. 𝜀𝑎
→ lim
𝜀𝑎→∞
𝜀𝑎
𝑎 + 𝑏. 𝜀𝑎
=
1
𝑏
= 𝑞𝑢𝑙𝑡 = (𝜎1 − 𝜎3) 𝑢𝑙𝑡
𝐸 =
𝜕𝑞
𝜕𝜀𝑎
=
𝜕(𝜎1 − 𝜎3)
𝜕𝜀𝑎
=
𝑎
𝑎 + 𝑏. 𝜀𝑎
2
→ lim
𝜀𝑎→0
𝑎
𝑎 + 𝑏. 𝜀𝑎
2
=
1
𝑎
= 𝐸𝑖 = 𝐸0
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7. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Duncan-Chang constitutive equations
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8. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
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The a and b values
should be calculate
in 70% to 95%
( - /1-3) chart

9. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Duncan-Chang constitutive equations
Failure ratio (Rf):
𝑅𝑓 =
𝜎1 − 𝜎3 𝑓
𝜎1 − 𝜎3 𝑢𝑙𝑡
≤ 1.0
independent of confining pressure
Rf = 0.75  1.0
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10. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Duncan-Chang constitutive equations
If the parameters a and b are expressed in terms of E0, Rf, and (1-3):
𝜎1 − 𝜎3 =
𝜀
1
𝐸0
+
𝜀. 𝑅𝑓
𝜎1 − 𝜎3 𝑓
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11. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Duncan-Chang constitutive equations
Janbu (1963):
𝐸0 = 𝑘. 𝑝𝑎
𝜎3
𝑝𝑎
𝑛
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12. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Duncan-Chang constitutive equations
Mohr-Coulomb failure criterion:
𝜎1 − 𝜎3 𝑓 =
2𝑐. 𝑐𝑜𝑠𝜑 + 2𝜎3. 𝑠𝑖𝑛𝜑
1 − 𝑠𝑖𝑛𝜑
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13. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Duncan-Chang constitutive equations
• Tangent modulus values (Et):
𝐸𝑡 = 1 − 𝑅𝑓
(1 − 𝑠𝑖𝑛𝜑)(𝜎1 − 𝜎3)
2(𝑐. 𝑐𝑜𝑠𝜑 + 𝜎3. 𝑠𝑖𝑛𝜑)
2
𝑘.
𝜎3
𝑝𝑎
𝑛
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14. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Duncan-Chang constitutive equations
• Tangent modulus values (Et):
𝐸𝑡 = 1 − 𝑅𝑓
(1 − 𝑠𝑖𝑛𝜑)(𝜎1 − 𝜎3)
2(𝑐. 𝑐𝑜𝑠𝜑 + 𝜎3. 𝑠𝑖𝑛𝜑)
2
𝑘.
𝜎3
𝑝𝑎
𝑛
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15. Constitutive Modelling of Soil (Duncan-Chang Model)
Mostafa Abedi
/15
Duncan-Chang constitutive equations
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The incremental models described
cannot account for the strain
softening behavior after peak
strength (e.g., for dense sands and
overconsolidated clays).