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Covariant and contravariant stress tensors: difference between the Cauchy stress and the 2nd Piola-Kirrchoff stress

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I am currently reading a book about "Material Inhomogeneities in Elasticity" written by Gérard Maugin.

He brings up the covariancy and contravariancy of some well-known stress tensors and in particular that of the Cauchy stress tensor. I was not very familiar with the concept of covariance, so i read up on this very specific tensorial notion. 

I have been able to demonstrate easily that the deformation gradient F is a mixed tensor (covariant and contravariant).

jfmolinari's picture

Faculty Position at EPFL

Please find below a link to an open position (Assistant Professor, tenure track) at EPFL (Ecole Polytechnique Federale Lausanne, Switzerland).

Linearization procedure in nonlinear FEM


As far as I know, there are two ways to derive the linearized equations (employed in the Newton-Raphson method) in nonlinear finite element methods. 

In the first approach, starting with the weak formulation, the FE approximations are substituted into the weak form, you then get the semi-discrete equations. After that, the linearization is applied on this discrete equation to get the tangent matrix. The book of Prof. Ted Belytschko followed this approach.

Fatigue Analysis

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Is it possible to carryout the Fatigue Analysis using LS Dyna.

If it is not possible then what will be the best software to carry out the analysis.

Predicting the Young’s modulus of nanowires from first-principles calculations on their surface and bulk materials

Prof. Xiaodong Li and I are collaborating in identifying those physical mechanisms that govern the mechanical properties of nanomaterials. Recently, we have a manuscript accepted by the Journal of Applied Physics and would like to share it with you.

Zhigang Suo's picture

Mechanics thrives whenever it participates in advancing a new technology

This morning I was looking at the book Mechatronic Reliability by Wei Yang.  The forword I wrote for the book brought back memories of days he and I spent in Santa Barbara and Princeton.  The text of the forword follows.

2008 Timoshenko Medal Acceptance Speech by Sia Nemat-Nasser

Sia Nemat-Nasser

Delivered at the Annual Dinner of the Applied Mechanics Division, in the Back Bay Ballroom, Sheraton Hotel, Boston, in the evening of 4 November 2008, the Election Day of the United States of America 

Internship available in FEA of elastomers

Internship Announcement  

Title : FEA of Elastomers

Category: Internship in Industry

Employer: Schlumberger Technology Corporation

Location: Sugar Land, TX, USA

Working on wave propagation on pipe/cylinder

I am working on pipeline crack detection simualtion using Ansys. since wave propagation is a transient analysis and I have never used Ansys for this kind of simulation. Help me out if you know any source where I can take it as a reference and start my simulation.


Damage mechanics with UMAT

Do anybody know any things or have any experience about implementation of damage mechanics formulation for concrete in UMAT?

Pablo Zavattieri's picture

MRS Symposium KK: "Structure-Property Relationships in Biomineralized and Biomimetic Composites". Call for Abstracts

Deadline for Abstract submission: November 3. See  A symposium entitled "Structure-Property Relationships in Biomineralized and Biomimetic Composites " will be organized at the MRS-spring meeting in San Francisco next year (April 13-17, 2009).

Problem in importing 3D modle from Pro E to ANSYS

Hi all

 I have done 3D modeling in Pro E. When i am trying to import it in ANSYS its taking alot of time. almost 1 hour has passed and modle is still not imported in ANSYS. I have converted my modle in IGES format.

Is there any way to import the modle easily from Pro E to ANSYS??? 

MRS symposium LL "Architectured Multifunctional Materials", Call for Abstracts

Deadline for Abstract submission: November 3. See  A symposium entitled "Architectured Multifunctional Materials" will be organized at the MRS-spring meeting in San Francisco next year (April 13-17, 2009). For details see below and attached file.

Huapei Wan's picture

Pressure on circular area of a semi-infinite plane: typos in textbooks "contact mechanics" "surface mechanics" "elasticity"

This is supposed to be a very classic analytical solution in applied elasticity. Forgive me if this is a repost or some correction has been made before. I had a hard time with it.

Problem statement:  uniform pressure is applied on cirular area (radius=r) on top of a semi-infinite plane. Find the vertical deflection.

We got discrepancy between FEA solution and analytical solution which I found from different textbooks (famous and non-famous): Deflection within the circle is good, however, no good outside the circle.

We found the correct solution is attached.

ANSYS: melting using element death--how to transfer convection BC from deactivated elements?

I am modelling melting by element death.  I have a cold (melting) rectangle with a hot fluid convecting at the exterior.  I am using a do loop so that time increases at each loop.  I define a "melting" temperature, so that elements are killed off when they achieve a specified temperature.  

Interpretting results in ANSYS

I  have done modal analysis and when ANSYS asked to input the start and  end frequency i typed 150 as my starting frequency and 2000 as my ending frequency. No of modes to extract 10 and no of modes to expand 10. Due to no of modes ANSYS has only displyed 10 results. When i entered in postproc and results summary only 10 results were displayed and starting frequency was 150.01 to 150. 39. Now if i want to see all the frequencies how is this possible???Should i maximise the no of modes or is their any other option to view the results???

Modelling crack in Ansys with interface elements

Hi Every body

I am Dhaneshwar Mishra, a PhD student working in the area of fracture mechanics, basically fracture failure of interfaces of bimaterial, composite layers etc. I am regular reader of IMechanica and got lots of insights by reading different postings on it. I am trying to model the interface in ANSYS 10.0 with their Interface element INTER204. While searching this element in Ansys element library, i could not find it. Will some body guide me to model interface of composite layers using the concept of cohesive zone modelling?

Thank you

ANSYS shell99

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Does anyone knows what's the theory of the element shell99 in Ansys? I tried to search in ANSYS Help, but didn't find anything. Is it kirchoff classical plates theory?

Finance and Mechanics


Recent events about the financial crisis and generally spread turmoils on Economics and Financial Environments  lead me to these conclusions.

Critical Speed:  informations, a key issue on finance, reached a critical speed. Maybe this is the "first" critical speed of the system. Speed of informations has reached the same order of magnitude of people's judgement (executives and investors more than others) so there's a sort of over-reaction about actual events.

Modeling of large strain deformation in Abaqus

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Hi everybody,

I'm trying to model large strain deformations in Abaqus by writing a user defined material code in order to implement the governing constitutive equations of the material. As far as I know, Cauchy stress tensor should be used in the model to presume non-linear finite elements, but I don't know how one can implement Cauchy stress tensor in a UMAT used by Abaqus.



stop hole for arresting fatigue crack

how can we analyse fatigue crack by using ansys and find the optimum hole for arresting cracks.

thanks to all.


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