# Produce large displacement under small displacement approximation?

I have managed to make my model converge fairly well and achieve large displacements and deformations. It exhibits stress and strain good continuity by solving it under small displacement and deformation assumption. In other terms, engineering strain is computed instead of Green Lagrange strain, stress tensor is Cauchy and not Piola Kirchhoff, and material frame is associated to spatial frame.

I know the small displacement/deformation assumption is only for no more than 5% strain, but actually, the strain reaches 50% ; it moves 1cm into an 8cm solid box.

I know also that every single FEM can only be an approximation of reality.

What can I expect about the validity of the model ? Does it mean absolutely nothing because it is quite nonsense (but the only thing is that the error is highered) ?

• The way I see it is, if you're using a model under certain assumptions and yet those assumptions aren't met, then you should be expected more error than the error the FEM would give you alone. It might be very wrong or only slightly incorrect, depending on your problem. Based on how large the strain is relative to the strain assumption of the model, I would suspect the results aren't really a good predictor for reality. Jul 21 '16 at 16:33
• If you change the length of a 8cm box by 1cm that would be a 12.5% strain rather than 50% strain wouldn't it? Even so, the error in the solution based on a small-strain assumption is likely to be very large. Beyond the error due to the small-strain kinematic relations, there will also be significant errors due to the incorrect modeling of the material behavior (you did not state the material of your part). Jul 21 '16 at 17:17
• @BillGreene You're right, it's actually 1mm into something like 4-5. My material is assumed to be linear viscoelastic. Do you have an idea how can I at least measure the error ? Jul 21 '16 at 18:58
• @Blue_Elephant, solve a problem you know the answer to using the Method of Manufactured Solutions or an analytical solution. Jul 21 '16 at 19:40
• Without knowing more, it is difficult to say. How is that you happened to be using a FE formulation that seems to be so ill-suited to the particular problem you are trying to solve? Is this an assignment where you have to implement your own FE code? If you are a student, do you have access to any commercial FE solvers e.g. Abaqus? Jul 21 '16 at 23:59

I know also that every single FEM can only be an approximation of reality.

By the FEM you just obtain an approximation to an (hopefully) well posed problem.

Logical steps are as follows.

1. You first set a problem by choosing appropriate equations (congruence, constitutive equations, balance), a geometric domain, material constitutive parameters, and appropriate boundary and initial conditions. If the problem is well-posed this defines a unique solution $L$.
2. By the FEM you obtain an approximation $L_\text{FEM} \approx L$.

The small strain/displacement assumption affect the equations that define $L$ (in the sense that you end up with a set of linear congruence and balance equations), and presumably you have a well-defined problem, and therefore are able to easily compute a converged solution $L_\text{FEM}$.

The real question is if $L$ is physically meaningful or not, and of course this question has nothing to do with the particular method you used to compute an approximation to it. This is a question about the equations you are going to solve: are they able to model the physics of your problem?

It is impossible to answer this question without analysing your specific problem and your solution. However in general it is very dangerous to apply continuum mechanics equations outside their domain of validity: why a solution $L$ defined under small displacement assumptions should be physically valid for finite displacements? In my own experience this is almost never the case, apart from some trivial problems.

The equations of linear mechanics can be presented as the linearisation of more general non-linear equations. Intuitively this means that you search for a solution on the tangent to a curve, instead of for a solution on the curve itself: you can make a strong point about the fact that you will end very far from the correct solution.

When you mention the convergence of the model, do you mean that you have used some form of iterative method? In that case, it is likely that you were using some non-linear model (unadvertely?).

The following is a list of possible non-linearities one can encounter based on the ubiquitous "Governing Equations":

• System's internal balance equations (Statics or dynamics)

   Nonlinear damping

• Constitutive equations (Material)

   Plasticity
Viscosity
Creeping
Hiperelasticity

• Kinematic equations (Geometry)

   Large deformations with small strains
Large strains
Buckling
Instability

• Boundary conditions

   Forces
Pressure 