Measurement of deformation. Measurement of elastic force. Constitutive law. Finite element method

Transcription

1 Deformable Bodies

2 Deformation x p(x) Given a rest shape x and its deformed configuration p(x), how large is the internal restoring force f(p)? To answer this question, we need a way to measure deformation

3 Measurement of deformation Measurement of elastic force Constitutive law Finite element method

4 Displacement field Displacement field directly measures the difference between the rest shape and the deformed shape It s not rigid-motion invariant. For example, a pure translation p = x + 1 results in nonzero displacement field u = 1

5 Displacement gradient Displacement gradient is a matrix field Need to compute deformation gradient Both displacement gradient and deformation gradient are translation invariant but rotation variant

6 Green s strain Green s strain can be defined as Green s strain is rigid-motion invariant (both translation and rotation invariant) rp T rp I =(RS) T RS I = S T R T RS I = S T S I

7 Cauchy s strain When the deformation is small, Cauchy s strain is a good approximation of Green s strain Is Cauchy s strain rigid motion invariant? Consider a point at rest shape x = (x, y, z) T and its deformed shape p = (-y, x, z) T, what is the Cauchy s strain for this deformation?

8 Measurement of deformation Measurement of elastic force Constitutive law Finite element method

9 Elastic force Strain measures deformation, but how do we measure elastic force due to a deformation? Stress measures force per area acting on an arbitrary imaginary plane passing through an internal point of a deformable body Like strain, there are many formula to measure stress, such as Cauchy s stress, first Piola-Kirchhoff stress, second Piola- Kirchhoff stress, etc

10 Stress Stress is represented as a 3 by 3 matrix, which relation to force can be expressed as da is the infinitesimal area of the imaginary plane upon which the stress acts on n is the outward normal of the imaginary plane.

11 Cauchy s stress All quantities (i.e. f, da and n) are defined in deformed configuration Consider this example, what is the force per area at the rightmost plane?

12 Cauchy s stress The internal force per area at the right most plane is σ11 measures force normal to the plane (normal stress) σ21 and σ31 measure force parallel to the plane (shear stress)

13 Measurement of deformation Measurement of elastic force Constitutive law Finite element method

14 Constitutive law Constitutive law is the formula that gives the mathematical relationship between stress and strain In 1D, we have Hooke s law Constitutive law is analogous to Hooke s law in 3D, but it is not as simple as it looks

15 Constitutive law 2 4 " 11 " 12 " 13 " 21 " 22 " 23 " 31 " 32 " What is the dimension of C?

16 Materials For a homogeneous isotropic elastic material, two independent parameters are enough to characterize the relationship between stress and strain E is the Young s modulus, which characterize how stiff the material is ν is the Poisson ratio, ranging from 0 to 0.5, which describe whether material preserves its volume under deformation

17 Measurement of deformation Measurement of elastic force Constitutive law Finite element method

18 Finite element method So far we view deformable body as a continuum, but in practice we discretize it into a finite number of elements The elements have finite size and cover the entire domain without overlaps Within each element, the vector field is described by an analytical formula that depends on positions of vertices belonging to the element

19 Tetrahedron Rest shape of a tetrahedron is represented by x0, x1, x2, x3 Deformed shape is represented by p0, p1, p2, p3 Any point x inside the tetrahedron in the rest shape can be expressed using the barycentric coordinate

20 Barycentric coordinates FEM assumes that deformed shape is linearly related to rest shape within each tetrahedron Therefore, p(x) can be interpolated using the same barycentric coordinates of x p(x) can also be computed as

21 Elastic force To simulate each vertex on a tetrahedra mesh, we need to compute elastic force applied to vertex Based on p(x), compute current strain of each tetrahedron Use constitutive law to compute stress For each face of tetrahedron, calculate internal force: A is the area of the face and n is the outward face normal Distribute the force on each face to its vertices

22 Linear FEM Assuming deformation is small around rest shape, it is valid to use Cauchy s strain and calculate face normal and area using rest shape Simplified relationship between internal force and deformation K can be pre-computed and maintain constant over time

23 Corotational FEM When object undergoes rotation, the assumption of small deformation is invalid because Cauchy s strain is not rotation invariant Corotational FEM is an effective method to eliminate the artifact due to rotation first extract rotation R from the transformation rotate the deformed tetrahedron to the unrotated frame R T p calculate the internal force K(R T p x) rotate it back to the deformed frame: f = RK(R T p x)

24 Corotational FEM

25

26