The theory of large strain computations can be addressed by defining a few basic physical quantities (motion and deformation) and the corresponding mathematical relationship. The applied loads acting on a body make it move from one position to another. This motion can be defined by studying a position vector in the “deformed” and “undeformed” configuration. Say the position vectors in the “deformed” and “undeformed” state are represented by {x} and {X} respectively, then the motion (displacement) vector {u} is computed by (see Figure 3.1: Position Vectors and Motion of a Deforming Body):

(3–1)

Figure 3.1:  Position Vectors and Motion of a Deforming Body

The deformation gradient is defined as:

(3–2)

which can be written in terms of the displacement of the point via Equation 3–1 as:

(3–3)

where:

The information contained in the deformation gradient [F] includes the volume change, the rotation and the shape change of the deforming body. The volume change at a point is

(3–4)

where:

Vo = original volume

V = current volume

det [ ] = determinant of the matrix

The deformation gradient can be separated into a rotation and a shape change using the right polar decomposition theorem:

(3–5)

where:

Once the stretch matrix is known, a logarithmic or Hencky strain measure is defined as:

(3–6)

([ε] is in tensor (matrix) form here, as opposed to the usual vector form {ε}). Since [U] is a 2nd order tensor (matrix), Equation 3–6 is determined through the spectral decomposition of [U]:

(3–7)

where:

The polar decomposition theorem (Equation 3–5) extracts a rotation [R] that represents the average rotation of the material at a point. Material lines initially orthogonal will not, in general, be orthogonal after deformation (because of shearing), see Figure 3.2: Polar Decomposition of a Shearing Deformation. The polar decomposition of this deformation, however, will indicate that they will remain orthogonal (lines x-y' in Figure 3.2: Polar Decomposition of a Shearing Deformation). For this reason, non-isotropic behavior (e.g. orthotropic elasticity or kinematic hardening plasticity) should be used with care with large strains, especially if large shearing deformation occurs.

Figure 3.2:  Polar Decomposition of a Shearing Deformation

Computationally, the evaluation of Equation 3–6 is performed by one of two methods using the incremental approximation (since, in an elastic-plastic analysis, we are using an incremental solution procedure):

(3–8)

with

(3–9)

where [ΔU_n] is the increment of the stretch matrix computed from the incremental deformation gradient:

(3–10)

where [ΔF_n] is:

(3–11)

[F_n] is the deformation gradient at the current time step and [F_n-1] is at the previous time step.

(Hughes([156])) uses the approximate 2nd order accurate calculation for evaluating Equation 3–9:

(3–12)

where [R_1/2] is the rotation matrix computed from the polar decomposition of the deformation gradient evaluated at the midpoint configuration:

(3–13)

where [F_1/2] is (using Equation 3–3):

(3–14)

and the midpoint displacement is:

(3–15)

{u_n} is the current displacement and {u_n-1} is the displacement at the previous time step. [Δε_n] is the “rotation-neutralized” strain increment over the time step. The strain increment is also computed from the midpoint configuration:

(3–16)

{Δu_n} is the displacement increment over the time step and [B_1/2] is the strain-displacement relationship evaluated at the midpoint geometry:

(3–17)

This method is an excellent approximation to the logarithmic strain if the strain steps are less than ~10%. This method is used by the standard 2-D and 3-D solid and shell elements.

The computed strain increment [Δε_n] (or equivalently {Δε_n}) can then be added to the previous strain {ε_n-1} to obtain the current total Hencky strain:

(3–18)

This strain can then be used in the stress updating procedures, see Rate-Independent Plasticity and Rate-Dependent Plasticity (Including Creep and Viscoplasticity) for discussions of the rate-independent and rate-dependent procedures respectively.

According to Callen([244]), the coefficient of thermal expansion is defined as the fractional increase in the length per unit increase in the temperature. Mathematically,

(3–19)

where:

α = coefficient of thermal expansion

= current length

T = temperature

Rearranging Equation 3–19 gives:

(3–20)

On the other hand, the logarithmic strain is defined as:

(3–21)

where:

= logarithmic strain

o = initial length

Differential of Equation 3–21 yields:

(3–22)

Comparison of Equation 3–20 and Equation 3–22 gives:

(3–23)

Integration of Equation 3–23 yields:

(3–24)

where:

= initial (reference) strain at temperature To

To = reference temperature

In the absence of initial strain ( ), then Equation 3–24 reduces to:

(3–25)

The thermal strain corresponds to the logarithmic strain. As an example problem, consider a line element of a material with a constant coefficient of thermal expansion α. If the length of the line is _o at temperature T_o, then the length after the temperature increases to T is:

(3–26)

Now if one interpreted the thermal strain as the engineering (or nominal) strain, then the final length would be different.

(3–27)

where:

The final length is then:

(3–28)

However, the difference should be very small as long as:

(3–29)

because

(3–30)

The element matrices and load vectors are derived using an updated Lagrangian formulation. This produces equations of the form:

(3–31)

where the tangent matrix has the form:

(3–32)

[K_i] is the usual stiffness matrix:

(3–33)

[B_i] is the strain-displacement matrix in terms of the current geometry {X_n} and [D_i] is the current stress-strain matrix.

[S_i] is the stress stiffness (or geometric stiffness) contribution, written symbolically as:

(3–34)

where [G_i] is a matrix of shape function derivatives and [τ_i] is a matrix of the current Cauchy (true) stresses {σ_i} in the global Cartesian system. The Newton-Raphson restoring force is:

(3–35)

Some of the plane stress and shell elements account for the thickness changes due to the out-of-plane strain ε_z using an approach similar to that of Hughes and Carnoy ([157]). Shells, however, do not update their reference plane (as might be required in a large strain out-of-plane bending deformation); the thickness change is assumed to be constant through the thickness. General element formulations using finite deformation are developed in General Element Formulations and apply to current-technology elements only.

NLGEOM,ON activates large strain computations in those elements which support it. NLGEOM,ON also activates the stress-stiffening contribution to the tangent matrix.

For elements which have large strain capability, stresses (output as S) are true (Cauchy) stresses in the rotated element coordinate system (the element coordinate system follows the material as it rotates). Strains (output as EPEL, EPPL, etc.) are the logarithmic or Hencky strains, also in the rotated element coordinate system.

An exception is for the hyperelastic elements. For these elements, stress and strain components maintain their original orientations and some of these elements use other strain measures.