The Stiffness Matrix | Library builder

From Eq. 11.24, it appears that the stiffness matrix is a derivative of the internal forces:

n Fint n

FLhK] = — F — = — *ljBLjdvj (11.25)

	1	2
1	Derivative of problem 1 nodal forces with respect to problem 1 nodal unknowns	Derivative of problem 1 nodal forces with respect to problem 2 nodal unknowns
2	Derivative of problem 2 nodal forces with respect to problem 1 nodal unknowns	Derivative of problem 2 nodal forces with respect to problem 2 nodal unknowns

Fig. 11.2 Illustrative layout of stiffness matrix

Two contributions will be obtained (Fig. 11.2). On the one hand, one has to derive the stress state with respect to the strain field, itself depending on the displacement field. On the other hand, the integral is performed on the volume, and the B matrix depends on the geometry. If we are concerned with large strains and if we are using the Cauchy’s stresses, geometry is defined in the current configuration, which is changing from step to step, and even from one iteration to the other. These two contributions, the material one, issued from the constitutive model, and the geometric one, have to be accurately computed in order to guarantee the quadratic convergence rate.

A similar discussion may be given for diffusive problems. However, the geometry is not modified for pure diffuse problems, so only the material term is to be considered.

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