PhD Position

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A new convex yield function to model the ductile damage of structures under extreme loading conditions

Supervisor: Professor Guadalupe Vadillo

The derivation of the Gurson yield function relies on spherical and cylindrical voids with the simplifying assumption that the matrix surrounding the void is perfectly plastic. The model was afterwards modified by introducing two parameters, q1 and q2, to adjust the predictions to numerical results of cylindrical and spherical voids in strain hardening materials. The Gurson-Tvergaard-Needleman (GTN) model is given in terms of a macroscopic yield criterion and the evolution law of a single microscopic variable, the void volume fraction. Despite the apparent success and wide popularity of the GTN model in predicting ductile damage, it still suffers from several limitations such as its inaccuracy to properly model material effects as strain rate hardening and thermal softening or its inapplicability to model ductile damage under low stress triaxiality (shear dominated deformations). While considerable improvements have been made on the study of void growth and coalescence in porous viscoplastic solids, further research still need to be done in order to resolve the interplay of strain rate hardening and thermal softening on the damage evolution of ductile materials, specifically under shear dominated problems.

Research outputs:

An original yield function which, as distinctive feature, includes strain rate hardening and temperature dependences of the material flow stress. The yield function will be implemented into a Finite Element code (ABAQUS) to predict practical cases of damage development in aerospace and defense structural materials.

Multidisciplinary/ intersectoral research approach:

The ESR will develop at the University Carlos III of Madrid the main issues of the mathematical formulation of a new Gurson-type yield function which shall include the effects of material strain rate hardening and thermal softening. The ESR will conduct a secondment at the TECHNION where she/he will perform experiments using rate and temperature sensitive ductile metals for validation of the new yield function that will be developed. Additionally, the ESR will conduct a secondment in France and visit the University of Lorraine and CIMULEC. At the University of Lorraine he/she will incorporate strain rate hardening effects in the new yield function to be proposed. At CIMULEC he/she will analyze and model (real) practical applications in which ductile damage endangers the integrity of printed circuit boards.

Training activities:

The successful candidate will have access to the PhD program of the University Carlos III of Madrid as well as to the training activities organized within the OUTCOME consortium (see here for more information).


The successful candidate will be employed for 3 years and receive a financial package plus an additional mobility and family allowance according to the rules for Early Stage Researchers (ESRs) in an EU Marie Skłodowska-Curie Actions Innovative Training Networks (ITN):

  • Living allowance – 3035.36€ (per month).
  • Mobility allowance – 600€ (per month).
  • Family allowance – 500€ (per month – if applicable).

This amount is a gross contribution to the salary costs. Net salary will result from deducting all compulsory social security/direct taxes from the gross salary according to the law applicable to the agreement concluded with the ESR.

Additional information about the funding provided by the ITN  projects can be found here.

Key publications:

Tvergaard, V. Influence of voids on shear band instabilities under plane strain conditions. International Journal of Fracture. 1981; 17: 389-407.
Tvergaard, V. On localization in ductile materials containing spherical voids. International Journal of Fracture. 1982; 18: 237-252.
Gurson, A.L. Continuum theory of ductile rupture by void nucleation and growth part I. yield criteria and flow rules for porous ductile media. Journal of Engineering Materials and Technology. 1977; 99: 2-15.
Vadillo, G., Zaera, R., Fernández-Sáez, J. Consistent integration of the constitutive equations of Gurson materials under adiabatic conditions. Computer Methods in Applied Mechanics and Engineering, 2008; 197:1280-1295
Vadillo, G., Fernández-Sáez, J. An analysis of Gurson model with parameters dependent on triaxiality based on unitary cells. European Journal of Mechanics A/Solids. 2009; 28: 417-427
Vadillo, G., Reboul, J., Fernández-Sáez, J. A modified Gurson model to account for the influence of the Lode parameter at high triaxialities. European Journal of Mechanics A/Solids. 2016; 56: 31-44


We are looking for highly motivated candidates who want to pursue a scientific career in mechanical engineering (academic or industrial). An ideal candidate would have a background in mechanical or civil engineering, physics or mathematics and a strong interest in collaborating with a team working attitude in an interdisciplinary project. Good communication in English is required.
To apply for the proposed Thesis, and in order to meet the specific requirements of the Marie Skłodowska-Curie funded PhDs, the candidates must not have resided or carried out their main activity (work, studies…) in Spain for more than 12 months in the last 3 years. Candidates must have a MSc degree or obtain a MSc degree by August 2016.


The candidates must provide a letter of motivation where they clearly state why, under their point of view, they should be enrolled in OUTCOME.

At least, one recommendation letter from the scientist/s who mentored the candidate during his/her master studies is required. The letter must clearly expose the profile of the candidate with emphasis in the qualities which make him/her suitable for being recruited in OUTCOME. Additional recommendation letters from any other professor/professional will be welcomed.

We are committed to provide flexible hours and home working conditions for researchers having family obligations. The following web-site contains relevant information related to the EU equal opportunities policy.

Moreover, the web-site  facilitates geographic mobility by providing help to find a job for an accompanying partner.


The application period is closed.
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