Some Fundamentals of Solid Mechanics
2.2 Lagrangian Description of Motion
2.2.1 General
The motion of a body in space is usually described by the Lagrangiandescription of motion, also referred to as the material description. In this context a body consists of an infinite set of material particles occupying a region in space. The simultaneous position of this particle set at a given time is called a configuration of the body. A sequence of such configurations defines the motion of the body.
Different optional Lagrangian formulations are available depending on the refer-ence configuration used for the kinematic and static variables involved. The most
common forms are [3]:
∙ Total Lagrangian (TL) description based on the initial configuration 𝐶𝑜 as reference.
∙ Updated Lagrangian1 (UL) description based on the current deformed con-figuration 𝐶𝑛 as reference.
∙ Corotated Lagrangian(CL) description based on a moving undeformed con-figuration 𝐶𝑜𝑛 (positioned ‘close’ to 𝐶𝑛) as reference. 𝐶𝑜𝑛 is often termed the
‘ghost’ reference configuration.
CL-description is in fact a modification of TL in the sense that both are referring to an undeformed state. However, for large displacement analysis it is easier to introduce simplifications in the CL-formulation since rigid body motions there are extracted before computing strains and stresses. The CL-concept has thus been successfully adopted in several works in the recent years, e.g. by Nygård [4] and Mathisen [5], and it also serves as basis for the large displacement formulation in the general purpose finite element program FENRIS [6].
The selected formulation for this work will be the CL-description of motion. At this general level however, it is no need to distinguish clearly between the CL- and TL-formulations. Thus, the expressions presented in the sequel are equally applicable to both formulations as long as the correct reference configuration is employed. The UL-formulation on the other hand, will not be covered further.
2.2.2 Displacements and Strains
The displacement vector field 𝑢 of a deformed body is given by
𝑢 = 𝑥 − 𝑋 (2.1)
where 𝑥 is the deformed position of a material particle and 𝑋 is the corresponding position of the particle in the reference configuration. The deformation gradient tensor 𝐹 of the body is defined as
𝑑𝑥 = 𝐹 ·𝑑𝑋 (2.2)
Using Eq.(2.1), its Cartesian components become 𝐹𝑖𝑗 = 𝜕𝑥𝑖
𝜕𝑋𝑗 = 𝛿𝑖𝑗 + 𝜕𝑢𝑖
𝜕𝑋𝑗 (2.3)
where 𝛿𝑖𝑗 is the Kronecker delta [1].
1In some literature also termed Eulerian description.
The strains are measured in terms of the Green strain tensor 𝐸 which is defined through
𝑑𝑠2 − 𝑑𝑆2 = 2𝑑𝑋·𝐸·𝑑𝑋 (2.4)
where 𝑑𝑆 and 𝑑𝑠 are the lengths of the infinitesimal material vectors 𝑑𝑋 and 𝑑𝑥, respectively. 𝐸 can be expressed in terms 𝐹 by
𝐸 = 1
2(𝐹𝑇 ·𝐹 − 1) (2.5)
where 1 is the unit tensor whose components are given by 𝛿𝑖𝑗. In terms of displace-ments, 𝐸 reads on indicial form
𝐸𝑖𝑗 = 1 2(𝜕𝑢𝑖
𝜕𝑋𝑗 + 𝜕𝑢𝑗
𝜕𝑋𝑖 + 𝜕𝑢𝑘
𝜕𝑋𝑖
𝜕𝑢𝑘
𝜕𝑋𝑗) (2.6)
From the last expression it is seen that 𝐸 is a symmetric tensor, i.e.
𝐸𝑖𝑗 = 𝐸𝑗𝑖 (2.7)
Thus, the number of independent components in the tensor reduces from nine to six.
The Green strain tensor has further the properties that it vanishes for rigid body motions and reduces to infinitesimal strains (the linear terms in Eq.(2.6)) when both displacements and rotations are ‘small’.
The strain increments when going from a deformed configuration 𝐶𝑛 to𝐶𝑛+1, are given as the difference between Green strains in the two configurations
Δ𝐸 = 𝑛+1𝐸 − 𝑛𝐸 (2.8)
Note that the strains in this expression are all referring to the same reference config-uration. Using Eq.(2.6), the Cartesian components become
Δ𝐸𝑖𝑗 = 1
2(𝜕Δ𝑢𝑖
𝜕𝑋𝑗
+ 𝜕Δ𝑢𝑗
𝜕𝑋𝑖
+ 𝜕𝑢𝑘
𝜕𝑋𝑖
𝜕Δ𝑢𝑘
𝜕𝑋𝑗
+ 𝜕Δ𝑢𝑘
𝜕𝑋𝑖
𝜕𝑢𝑘
𝜕𝑋𝑗
) (2.9)
where quadratic incremental terms have been left out.
It may also be proved [4] that Green strain components referring to configuration 𝐶𝑜 and configuration 𝐶𝑜𝑛 (with corotated base vectors) become identical. In a CL-formulation this implies that the strain components for a deformed state can be carried over from one reference configuration to the next without transformation.
In succeeding chapters strains will be referred to a column vector of the six in-dependent strain components rather than using a full tensor representation. The off-diagonal terms in 𝐸 are first added to retain a full shear strain characteristic.
Using Eq.(2.7) gives
𝐺𝑖𝑗 = 𝐸𝑖𝑗 + 𝐸𝑗𝑖 = 2𝐸𝑖𝑗 ; 𝑖 > 𝑗 (2.10) The column vector of independent strain components may then be symbolized by
𝑒 =
⎧
⎪⎨
⎪⎩
𝐸𝑖𝑖
𝐺𝑖𝑗
⎫
⎪⎬
⎪⎭
(2.11)
2.2.3 Stresses and Equilibrium
The traction or stress vector 𝑡 at a point referred to the deformed configuration of a body is defined by
𝑡 = 𝑑𝑓
𝑑𝐴 (2.12)
where 𝑑𝑓 is the infinitesimal force vector that acts on the infinitesimal area 𝑑𝐴 at the surface or section. Note that 𝑡 is not a vector field since it depends not only on the position, but also on the direction of the outward unit normal vector 𝑛 to the area 𝑑𝐴.
Equilibrium at the point of an infinitesimal tetrahedron whose faces are normal to the Cartesian base vectors and to 𝑛, leads to the following expression involving the Cauchy stress tensor𝜎𝜎𝜎𝜎
𝑡 = 𝑛·𝜎𝜎𝜎𝜎 (2.13)
or on component form
𝑡𝑖 = 𝑛𝑗𝜎𝑗𝑖 (2.14)
Force equilibrium in the deformed configuration gives the static version of Cauchy’s equation of motion
𝜕𝜎𝑗𝑖
𝜕𝑥𝑗 + 𝑓𝑖 = 0 (2.15)
where 𝑓𝑖 is the body force per unit deformed volume. Furthermore, moment equilib-rium reveals the symmetry of the Cauchy stress tensor, i.e.
𝜎𝑖𝑗 = 𝜎𝑗𝑖 (2.16)
Thus, the number of independent components in the tensor reduces from nine to six.
Since the Cauchy stress tensor refers to the deformed configuration, it will not be energy-conjugate to the Green strain tensor. Thus, there is a need for an alternative stress measure that has the undeformed configuration as reference. Such a stress measure can be derived in the following way:
Instead of the real force𝑑𝑓 acting on𝑑𝐴in the deformed configuration, a ‘pseudo’-force𝑑𝑓* acting on 𝑑𝐴𝑜 in the undeformed configuration is constructed through the relation
𝑑𝑓* = 𝐹−1·𝑑𝑓 (2.17)
Thus,𝑑𝑓* relates to𝑑𝑓 in the same way as𝑑𝑋 relates to 𝑑𝑥according to the inverse of Eq.(2.2). Analogous to Eqs.(2.12-2.14), a ‘pseudo’-traction 𝑡* is defined
𝑡* = 𝑑𝑓*
𝑑𝐴𝑜 (2.18)
together with a ‘pseudo’-stress tensor 𝑆
𝑡* = 𝑁 ·𝑆 (2.19)
or on component form
𝑡*𝑖 = 𝑁𝑗𝑆𝑗𝑖 (2.20)
Here 𝑁 is the outward unit normal vector to the area 𝑑𝐴𝑜, and 𝑆 is commonly referred to as the 2nd Piola-Kirchhoff stress tensor. The latter can be related to the Cauchy stress tensor by use of Nanson’s formula [1], which gives
𝑆 = 𝜌𝑜
𝜌 𝐹−1·𝜎𝜎𝜎𝜎·(︁𝐹−1)︁𝑇 (2.21) with Cartesian components
𝑆𝑖𝑗 = 𝜌𝑜 𝜌
𝜕𝑋𝑖
𝜕𝑥𝑘 𝜎𝑘𝑙
𝜕𝑋𝑗
𝜕𝑥𝑙 = 𝜌𝑜 𝜌
(︃
𝛿𝑖𝑘− 𝜕𝑢𝑖
𝜕𝑥𝑘
)︃
𝜎𝑘𝑙
(︃
𝛿𝑗𝑙− 𝜕𝑢𝑗
𝜕𝑥𝑙
)︃
(2.22) where 𝜌 and 𝜌𝑜 are the mass densities in deformed and undeformed configurations, respectively2. The above stress relationship represents a symmetric transformation, and since the Cauchy stress tensor in itself is symmetric, so becomes also the 2nd Piola-Kirchhoff stress tensor, i.e.
𝑆𝑖𝑗 = 𝑆𝑗𝑖 (2.23)
Thus, the number of independent components in 𝑆 reduces from nine to six. When both displacements and rotations are ‘small’, the 2nd Piola-Kirchhoff stress ap-proaches the Cauchy stress. As for the Green strain components, it may also be proved that the components of𝑆 referring to configuration𝐶𝑜 and configuration𝐶𝑜𝑛 (with corotated base vectors) become identical. Consequently, the stress components for a deformed state using a CL-formulation can be carried over from one reference configuration to the next without transformation.
By inserting the inverse relationship of Eq.(2.22(first part)) into Eq.(2.15), the static equilibrium in deformed configuration expressed in terms of the 2nd Piola-Kirchhoff stress, takes the form
𝜕
𝜕𝑋𝑗
(︃
𝑆𝑗𝑘 𝜕𝑥𝑖
𝜕𝑋𝑘
)︃
+ 𝑓𝑜𝑖 = 0 (2.24)
where
𝑓𝑜𝑖 = 𝜌𝑜
𝜌 𝑓𝑖 (2.25)
which is the body force intensity in deformed configuration scaled to a unit volume of the undeformed state (𝑑𝑉 /𝑑𝑉𝑜=𝜌𝑜/𝜌).
Like strains, stresses will in succeeding chapters also be referred to a column vector of the six independent components rather than using a full tensor representation.
Denoting the off-diagonal terms of the symmetric 𝑆-tensor by
𝑇𝑖𝑗 = 𝑆𝑖𝑗 ; 𝑖 > 𝑗 (2.26)
2Usually the ratio of mass densities will be close to unity.
the column vector of independent stress components may then be symbolized 𝑠 =
⎧
⎪⎨
⎪⎩
𝑆𝑖𝑖
𝑇𝑖𝑗
⎫
⎪⎬
⎪⎭
(2.27)
2.2.4 Constitutive Relations
To give an expression in its most general form for computing total current stresses 𝑆 is a difficult task. However, a functional relationship that fits within the frame of this work, may read
𝑆 = 𝑆(𝐸,Δ𝑇, 𝑚𝑖, 𝜅𝑖) (2.28)
where𝐸represents the current strains3 and Δ𝑇 is the current change in temperature from the reference state. Furthermore,𝑚𝑖 signifies material properties that in return may depend on previous histories of time, temperature, humidity, stresses and strains.
Finally, 𝜅𝑖 symbolizes effects related to internal state variables which again may depend on previous histories of stresses and strains.
In nonlinear analysis a need typically arises for an incremental material law. Dif-ferentiation of Eq.(2.28) with respect to 𝐸 yields
Δ𝑆 = 𝐶𝑇 : Δ𝐸 (2.29)
where Δ𝑆 and Δ𝐸 denote small but finite increments of stresses and strains, and 𝐶𝑇 is the incremental or tangential constitutive tensor of 4th order. The Cartesian components become
Δ𝑆𝑖𝑗 = 𝐶𝑇 𝑖𝑗𝑘𝑙Δ𝐸𝑘𝑙 (2.30)
In principle, 𝐶𝑇 may be expressed by a similar functional relationship as 𝑆, thus 𝐶𝑇 = 𝐶𝑇(𝐸,Δ𝑇, 𝑚𝑖, 𝜅𝑖) (2.31) With the column vector representation of the six independent components of stresses and strains, i.e. 𝑠and 𝑒, the incremental material law takes the form
Δ𝑠 = 𝐶𝑡Δ𝑒 (2.32)
where now𝐶𝑡 is termed the incremental ortangential constitutive matrix. This is the adopted form of the incremental material law for succeeding chapters of this work.
Indeed, the components of𝐶𝑡are obtained by direct differentiation of the stress-strain relationships, i.e.
𝐶𝑡 = 𝜕𝑠
𝜕𝑒 (2.33)
Since𝐶𝑡 in general not becomes symmetric, it will consist of 36 independent compo-nents for the 3D case.
3Effect of (‘high’) current strain rates is left out since this work deals with static conditions.
2.2.5 Boundary Conditions
Boundary conditions are divided into two kinds. The displacement boundary condi-tions are given by
𝑢 = 𝑢ˇ on 𝜕𝑉𝑜𝑢 (2.34)
or on component form
𝑢𝑖 = ˇ𝑢𝑖 on 𝜕𝑉𝑜𝑢 (2.35)
where 𝑢ˇ is the prescribed displacement vector on the boundary surface 𝜕𝑉𝑜𝑢.
The stress or traction boundary conditions are specified on the complementary boundary surface 𝜕𝑉𝑜𝑡. From Eqs.(2.19, 2.20) the compact and component forms become
𝑁 ·𝑆 = ˇ𝑡* on 𝜕𝑉𝑜𝑡 (2.36)
𝑁𝑗𝑆𝑗𝑖 = ˇ𝑡*𝑖 on 𝜕𝑉𝑜𝑡 (2.37)
whereˇ𝑡*is the prescribed surface ‘pseudo’-traction vector that acts in the undeformed configuration. The correspondence to the actual prescribed surface traction ˇ𝑡 in the deformed configuration can be obtained by combining Eqs.(2.12,2.17,2.18). Thus
𝑡ˇ* = 𝑑(𝜕𝑉)
𝑑(𝜕𝑉𝑜)𝐹−1·ˇ𝑡 = 𝐹−1·ˇ𝑡𝑜 (2.38) or in terms of Cartesian components
𝑡ˇ*𝑖 = 𝑑(𝜕𝑉) 𝑑(𝜕𝑉𝑜)
𝜕𝑋𝑖
𝜕𝑥𝑗 ˇ𝑡𝑗 = 𝜕𝑋𝑖
𝜕𝑥𝑗 ˇ𝑡𝑜𝑗 (2.39)
where𝑡ˇ𝑜 is the prescribed surface traction in deformed configuration scaled to a unit surface of the undeformed state, i.e.
𝑡ˇ𝑜 = 𝑑(𝜕𝑉)
𝑑(𝜕𝑉𝑜) ˇ𝑡 (2.40)