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Experimental identification of thermo-mechanical cohesive zone models for complex loading
Tarik Madani, Yann Monerie, Stéphane Pagano, Céline Pelissou, Bertrand Wattrisse
To cite this version:
Tarik Madani, Yann Monerie, Stéphane Pagano, Céline Pelissou, Bertrand Wattrisse. Experimental identification of thermo-mechanical cohesive zone models for complex loading. Workshop MIST 2015 : Friction, Fracture, Failure , Oct 2015, Montpellier, France. �hal-01273991�
T.MADANI
1,2Y. MONERIE
2,3S. PAGANO
2,3C. PELISSOU
1,2B. WATTRISSE
2,31 Institut de Radioprotection et de Sûreté Nucléaire, B.P. 3, 13115 Saint-Paul-lez-Durance Cedex, France
2 Laboratoire de Micromécanique et Intégrité des Structures, IRSN-CNRS-Université de Montpellier, France
3 Laboratoire de Mécanique et Génie Civil, Université de Montpellier, CC 048, 34095 Montpellier Cedex, France
Workshop MIST 12-15 octobre 2015 à Montpellier IRSN/PSN-RES/SEMIA/LPTM
Motivation
Objectives:
Identification of cohesive zone models for heterogeneous materials
Methodology:
Inverse method
Measurements:
Thermomechanical imaging techniques
Constitutive Equation Gap Method (CEGM)
Determine the mechanical properties of a material by minimizing the
energy difference between the
"measured" 𝑈𝑚 and a “computed"
𝑈𝑐 displacement fields.
𝑼𝒄 → 𝜺𝒄 → 𝝈𝒄 and
𝒑 = {𝑬, 𝒗, 𝑮, 𝝈𝟎, 𝒌}
𝑚𝑖𝑛 𝐸 𝑈
𝑐, 𝐵(𝑝)
𝑬 𝑼𝒄, 𝑩(𝒑) =
𝟏
𝟐𝑻 𝑩(𝒑): 𝜺 𝑼𝒄 − 𝑩(𝒑): 𝜺 𝑼𝒎 : 𝑩(𝒑)−𝟏: 𝑩(𝒑): 𝜺 𝑼𝒄 − 𝑩(𝒑): 𝜺 𝑼𝒎 𝒅𝑽𝒅𝒕
Ω 𝒕 𝟎
Finite element analysis
{𝑈
𝑚, 𝑅
𝑖}
Full-field measurements
{𝑈
𝑚, 𝑅
𝑖}
R2
R1
Displacement field
Methodology
𝑆 𝑒𝑓𝑓
cohesive zone
𝜎
𝑖𝑛𝑐𝜎
Step 2: summarize the « volume » damage as a « surface » damage.
Damageable elasto-plasticity
“ structural”
𝑅𝑐
[𝑢]
“+”
bulk
“=”
𝜀 𝜀
𝐵Isochoric elasto-
plasticity Step 1: identify the stress field from the strain field.
Damage
𝐹 𝐹
𝑙 𝑐 𝜀
𝐹 𝐹
𝑙 𝑐 𝜀 𝐵
𝑙 𝑐 𝜀
𝑆 𝑖𝑛𝑐
𝛿
• 𝑩 𝒑 independent of loading, Explicit estimate of the elastic parameters (cubic):
Elastic identification
1
𝐸(𝑖) = 1 2
𝜀𝑤 𝑚𝑥𝑥 − 𝜀𝑚𝑦𝑦 2𝑑𝑉𝑑𝑡
𝑡 𝑖
𝜎𝑤 𝑐𝑥𝑥 − 𝜎𝑐𝑦𝑦 2𝑑𝑉
𝑡 𝑖 𝑑𝑡 + 𝜀𝑤 𝑚𝑥𝑥 + 𝜀𝑚𝑦𝑦 ²𝑑𝑉𝑑𝑡
𝑡 𝑖
𝜎𝑤 𝑐𝑥𝑥 + 𝜎𝑐𝑦𝑦 ²𝑑𝑉
𝑖 𝑑𝑡
𝑡
𝑣(𝑖) = 𝐸 𝑖 2
𝜀𝑤 𝑚𝑥𝑥 − 𝜀𝑚𝑦𝑦 2𝑑𝑉𝑑𝑡
𝑡 𝑖
𝜎𝑤 𝑐𝑥𝑥 − 𝜎𝑐𝑦𝑦 2𝑑𝑉
𝑡 𝑖 𝑑𝑡 − 𝜀𝑤 𝑚𝑥𝑥 + 𝜀𝑚𝑦𝑦 ²𝑑𝑉𝑑𝑡
𝑡 𝑖
𝜎𝑤 𝑐𝑥𝑥 + 𝜎𝑐𝑦𝑦 ²𝑑𝑉
𝑖 𝑑𝑡
𝑡
𝐺(𝑖) = 1 2
𝜎𝑤 𝑐𝑥𝑦 2𝑑𝑉𝑑𝑡
𝑡 𝑖
𝜀𝑤 𝑚𝑥𝑦 2𝑑𝑉
𝑖 𝑑𝑡
𝑡
𝐾 = 𝑎 ∆𝜀𝑝 𝑏 + ∆𝜀𝑝 𝑤𝑖𝑡ℎ 𝑎 = 1
2𝑘 𝑎𝑛𝑑 𝑏 = 𝜎0 𝑘
Plastic identification
• 𝑩(𝒑) depends on the Von Mises stress 𝝈𝒏𝒆𝒒:
𝐵 𝑝 =
𝐸(1 + 2𝐾𝐸)
3𝐾²𝐸² − 2𝐾𝐸 𝜈 − 2 + 1 − 𝜈²
𝐸(𝜈 + 𝐾𝐸)
3𝐾²𝐸² − 2𝐾𝐸 𝜈 − 2 + 1 − 𝜈² 0 𝐸(𝜈 + 𝐾𝐸)
3𝐾²𝐸² − 2𝐾𝐸 𝜈 − 2 + 1 − 𝜈²
𝐸(1 + 2𝐾𝐸)
3𝐾²𝐸² − 2𝐾𝐸 𝜈 − 2 + 1 − 𝜈² 0
0 0 𝐺
1 + 6𝐾𝐺 𝐾 = ∆𝛾
3 + 2 ∗ ∆𝛾
∆𝛾 = 3 2𝑘
3 2
𝛼
𝜎0 − 1 𝛼² = 𝜎 − 𝑋 𝑇𝑃 𝜎 − 𝑋
• Resolution of the non-linear system obtained by the stationarity condition with respect to « 𝑎 » and « 𝑏 »,
• Computation of 𝐾 𝑎𝑛𝑑 𝛼,
• Determination of 𝜎0 𝑎𝑛𝑑 𝑘 by a linear fit of the data
3
2 𝛼 = 𝑓(2 3
2 𝐾α) 𝑘 is the slope of the curve and 𝜎0 is the intersection of the curve with the y axis.
Rewrite 𝑲
𝜎
𝑦𝑦𝑐𝜎
𝑦𝑦𝑚299 𝑀𝑃𝑎 507 𝑀𝑃𝑎
299 𝑀𝑃𝑎 507 𝑀𝑃𝑎
𝜎
𝑦𝑦𝑐𝜎
𝑦𝑦𝑚341 𝑀𝑃𝑎 421 𝑀𝑃𝑎
341 𝑀𝑃𝑎 421 𝑀𝑃𝑎
Results: Polycrystalline structure
Distribution of transversal stress fields, mesh doesn’t match the material heterogeneity:Distribution of transversal stress fields, mesh
perfectly consistent with the material heterogeneity:
Experimental identification of thermo-mechanical
cohesive zone models for complex loading
Conclusions :
Identification of heterogeneous fields (stress, …) for elasto-plastic behavior: linear and non-linear hardening.
Application of this method to real full-field measurements.
Prospect :
Extend the constitutive equation gap method to softening behaviors.
Identification of Cohesive Zone Models.
Introduction of a calorimetric gap in the identification functional.
