# Total and updated Lagrangian formulations

We know that for non-linear finite element analysis, we generally have 2 formulations:

1)Total Lagrangian formulation

2)Updated Lagrangian formulation

In the Total Lagrangian the refernce frame at t=0 is used i.e. all the integrations are carried out using the original volume and surface area and the sterss measures generally used include the Second Piolla and Green Lagrangian strain

Whereas, in updated Lagrangian formulation, the reference configuration at time 't' is used (the equation of principle of virtual work being solved at time t+delta_t).

So, all the integrations are carried out using volumes / surface areas at time 't'.

My question is:

I have read in the book by Bathe, that, in problems involving material non linearity + large displacements  + large strains - the updated Lagrangian formulation should be used.

Whereas, in rpoblems involving material non linearity + large dispalcements + small starins the Total Lagrangian formulatino may be used.

IS this correct? (I mean Bathe's book cannot be wrong so, am I interprating it correctly)?

Why is he distinguishin the formulation with material non-linearity?

Thanks,

Kajal

### Hi,

Hi,

both formulations are equvalent and can be used in both cases.

Bafty

### large strain with material non linearity

Thanks. I too did think so.

Can I then ask how does a formulation for large strain and large displacements differ from a formulation with large displacement and small strain?

I had been thinking that since the updated Lagrangian formulation, uses the voume and surface area at time 't' (whereas total lagrangian uses volume and surface area at time '0' i.e. the initial configuration), updated lagrangian is one which would take into account large strains and large displacements

### The small-strain stress

The small-strain stress-strain law can be used for small-strain large rotation formulation. So I think that only diference is in a constitutive equation you choose.

### No, you are not correct

When we use the terminology 'constitutive', you are talking about material laws.

A geometric non linear problem can:

1) large displacement  and small strain problem , or

2) large displacement and large strain problem.

My question is what is so special about 2) , what is so special that needs to be done to within a total lagrangian formualtion or updated lagrangian formulation to amke it work for large displacement and large strain?

### There is nothing special

There is nothing special about the second formulation. The total lagrangian formulation can be easily used for large strain and large deformation analysis as has been done in many papers. The simplest large strain models are hyperelastic materials such as neo-hookean and so on. In the large-rotation small-strain one just assume that the strains are small thus standard small-strain constitutive law can be used.

### thank you, then?

Thank you.

do you mean to say if I derived my finite element equations using the total Lagrangian  formulation(or, updated Lagrangian formulation), then, if I encounter a problem where we have large strains (in general), then these large starins should be captured through the finite element code?

### To include large strains, you

To include large strains, you need to change your B matrix to compute strain. You can work with deformation gradient for example, then you need to compute B matrix related to displacement gradient, not only to its symmetric part. Then you get F and form F you compute First Piola Kirchhoff stress through constitutive law and from that internal forces. You can also work with second Green-Lagrangian strain and second Piola-Kicrchoff stress or another one.

### thanks

Thanks Bafty...I will be coming into coding inclusing large strains in the next 3 to 4 months and this discussion and your insight above will be very useful.

Kajal 