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1 Kinematics

Kinematics deals with the mathematical description of deformations, displacements and rotations.

These rigid body motions or deformations are typically due to the forces which are applied to the considered continuum. In this chapter, the kinematics in a material point will be described, assuming that the related external loads that caused it are yet unknown.

The chapter is organized as follows. We start with the general aspects of deformation: continuity, the displacement vector and the deformation gradient tensor F\T{F}, together with the two special cases of rigid body motion and homogeneous deformation. Because F\T{F} itself still contains rigid rotation, we then isolate the true deformation through the right and left Cauchy-Green tensors C\T{C} and B\T{B}, and split F\T{F} into stretch and rotation by means of the right and left polar decompositions.

Next, we convert stretch into strain, introducing the linear, logarithmic and Green-Lagrange strain measures and their tensorial counterparts. Finally, we turn to time derivatives: the material and spatial time derivatives, the velocity gradient tensor L\T{L}, and its split into the rate-of-deformation tensor D\T{D} and the spin tensor Ω\T{\Omega}.

1.1General aspects of deformation

Upon deformation of a material, a fundamental distinction is made between the material (or Lagrangian) framework and the spatial (Eulerian) framework. The governing kinematic relations in continuum mechanics are based on the one-to-one correspondence between these two frameworks.

1.1.1Continuity

The continuity of the displacements is one of the basic assumptions in classical continuum mechanics. This assumption implies that the material is described in an average fashion, without considering the physical discrete microstructure or a material (e.g. molecular or crystalline structure). The neighborhood of a material point is considered to be dense and continuous. The kinematic and mechanical quantities that ensue from such a theory are therefore considered as average values over the real microstructure. If a material deforms, changes in distance between “dense” material points will occur. The transformation from the undeformed state to the deformed state is a one-to-one mapping, where each material point in the deformed state is associated with one and only material point from the undeformed state. The continuity of the deformation implies that neighborhoods of a material point P0P_0 in the initial region R0R_0 remain neighborhoods of the transformed material point PP in the transformed region RR, see Figure 1.1.

Deformation of a continuum.

Figure 1.1:Deformation of a continuum.

Fracture and damage can therefore not be treated trivially in this framework.

1.1.2The displacement vector

Upon deformation a material point changes position from x⃗0\V{x}_0 to x⃗\V{x}. This change in position is characterized by the displacement vector, which is given by

u⃗  =  x⃗−x⃗0\V{u} \;=\; \V{x} - \V{x}_0
(1.1)

The components of the displacement vector u⃗\V{u} with respect to an Euclidean basis are (u1,u2,u3)(u_1,u_2,u_3), also denoted by uiu_i in index notation.

1.1.3The deformation gradient tensor

Let’s consider two material points that are infinitely close, i.e. one material point is currently in the position x⃗\V{x}, while its neighbour is in position x⃗+dx⃗\V{x} + d\V{x}. The principle of continuity tells us that these two points were also neighbours in the initial undeformed state. Their positions in that initial state are respectively given by x⃗0\V{x}_{0} and x⃗0+dx⃗0\V{x}_0 + d\V{x}_0. The vector dx⃗0d\V{x}_0 that initially interconnected these two points is thus transformed or deformed into a new vector dx⃗d\V{x}, see Figure 1.2.

The linearized deformation map - the deformation tensor \T{F}

Figure 1.2:The linearized deformation map - the deformation tensor F\T{F}

The linear transformation of the vector dx⃗0d\V{x}_0 into the vector dx⃗d\V{x} is described by a second-order tensor, called the deformation gradient tensor F\T{F}. From the nonlinear deformation map ϕ\phi it appears that

dx⃗  =  (x⃗+dx⃗)−(x⃗)  =  ϕ(x⃗0+dx⃗0)−ϕ(x⃗0)  =  ∂ϕ∂x⃗0⋅dx⃗0  =  F⋅dx⃗0\begin{align} d\V{x} \;&=\; (\V{x} + d\V{x}) - (\V{x}) \notag \\ \;&=\; \phi(\V{x}_0 + d\V{x}_0) - \phi(\V{x}_0) \notag \\ \;&=\; \parder{\phi}{\V{x}_0}\inp d\V{x}_0 \notag \\ \;&=\; \T{F}\inp d\V{x}_0 \end{align}
(1.2)

From the definition of the gradient operator, equation (A.118), it can be noticed that

dx⃗  =  dx⃗0⋅∇⃗0x⃗  =  (∇⃗0x⃗)T⋅dx⃗0\begin{align} d\V{x} \;&=\; d\V{x}_0 \inp \grad_0 \V{x} \\ \;&=\; (\grad_0 \V{x})^T \inp d\V{x}_0 \end{align}
(1.3)

The deformation gradient tensor F\T{F} is thus given by

F  =  ∂ϕ∂x⃗0  =  (∇⃗0x⃗)T\begin{align} \T{F} \;&=\; \parderf{\phi}{\V{x}_0} \\ \;&=\; (\grad_0 \V{x})^T \end{align}
(1.4)

The second-order tensor F\T{F} is thus the spatial derivative of the nonlinear deformation map in the considered material point, which leads to a locally linear deformation map. The tensor is the transposed Lagrangian gradient of the current position vector. The tensor thus expresses the deformation with respect to the initial undeformed state. The gradient symbol ∇⃗0\grad_0 clearly indicates this with its subscript 0. Since F\T{F} is a second-order tensor, it is fully identified by means of the image (i.c. the deformation) of three (i.c. infinitesimal) vectors, which span an infinitesimally volume as shown in Figure 1.3.

The deformation gradient tensor \T{F}

Figure 1.3:The deformation gradient tensor F\T{F}

The matrix notation F‾\M{F} of this tensor F\T{F} with respect to an Euclidean basis is given by

F‾  =  [∂x1∂X1∂x1∂X2∂x1∂X3∂x2∂X1∂x2∂X2∂x2∂X3∂x3∂X1∂x3∂X2∂x3∂X3] \M{F} \;=\; \begin{bmatrix} \parder{x_1}{X_1} & \parder{x_1}{X_2} & \parder{x_1}{X_3} \\[1mm] \parder{x_2}{X_1} & \parder{x_2}{X_2} & \parder{x_2}{X_3} \\[1mm] \parder{x_3}{X_1} & \parder{x_3}{X_2} & \parder{x_3}{X_3} \end{bmatrix}
(1.5)

or in index format

FiJ  =  xi,JF_{_{iJ}} \;=\; x_{_{i,J}}
(1.6)

The Lagrangian gradient of the displacement also leads to a second order tensor, called the displacement gradient tensor u\T{\disp}. Its formal definition sounds

u  =  (∇⃗0u⃗)T \T{\disp} \;=\; (\grad_0\V{u})^T
(1.7)

Like in equation (1.3) (i.e. the definition of the Lagrangian gradient of a vector), this tensor maps the initial interconnection vector dx⃗0 d\V{x}_0 on the relative displacement vector between the two considered adjacent material points.

du⃗  =  dx⃗0⋅(∇⃗0u⃗)  =  dx⃗0⋅uT  =  uT⋅dx⃗0\begin{align} d\V{u} \;&=\; d\V{x}_0 \inp (\grad_0 \V{u}) \\ \;&=\; d\V{x}_0 \inp \T{\disp}^T \\ \;&=\; \T{\disp}^T \inp d\V{x}_0 \end{align}
(1.8)

The relative displacement vector du⃗d\V{u} measures the displacement that occurred between these two adjacent points with respect to their original relative positions

du⃗  =  dx⃗−dx⃗0d\V{u} \;=\; d\V{x} - d\V{x}_0
(1.9)

Using an Euclidean basis, the components of u\T{\disp} are commonly stored in a 3×33 \times 3 matrix u‾\M{\disp}

u‾  =  [∂u1∂X1∂u1∂X2∂u1∂X3∂u2∂X1∂u2∂X2∂u2∂X3∂u3∂X1∂u3∂X2∂u3∂X3]\M{\disp} \;=\; \begin{bmatrix} \parder{u_1}{X_1} & \parder{u_1}{X_2} & \parder{u_1}{X_3} \\[1mm] \parder{u_2}{X_1} & \parder{u_2}{X_2} & \parder{u_2}{X_3} \\[1mm] \parder{u_3}{X_1} & \parder{u_3}{X_2} & \parder{u_3}{X_3} \end{bmatrix}
(1.10)

In index notation this relation trivially reads

uiJ  =  ui,J\disp_{_{iJ}} \;=\; \disp_{_{i,J}}
(1.11)

Evidently, the tensors F\T{F} and u\T{\disp} are closely related

F  =  (∇⃗0x⃗)T  =  (∇⃗0[x⃗0+u⃗])T  =  (∇⃗0x⃗0)T⏟I+(∇⃗0u⃗)T⏟u  =  I+u\begin{align} \T{F} \;&=\; (\grad_0 \V{x})^T \notag \\ \;&=\; (\grad_0 [\V{x}_0 + \V{u}])^T \notag \\ \;&=\; \underset{\T{I}}{\underbrace{(\grad_0 \V{x}_0)^T}} + \underset{\T{\disp}}{\underbrace{(\grad_0 \V{u})^T}} \notag \\ \;&=\; \T{I} + \T{\disp} \end{align}
(1.12)

or

FiJ  =  δiJ+uiJF_{_{iJ}} \;=\; \delta_{_{iJ}} + \disp_{_{iJ}}
(1.13)

Since F\T{F} and u\T{\disp} are both second-order tensors, they describe linear transformations. This means that the deformation along three independent vectors in (3D) material point suffices to fully characterize these tensors. An initially cubic volume element will always be deformed by F\T{F} into a parallelepiped.

Consider a volume element that is spanned by three non-coplanar infinitesimal material vectors dX⃗1=dx⃗01d\V{X}_{1}=d\V{x}_{01}, dX⃗2=dx⃗02d\V{X}_{2}=d\V{x}_{02} and dX⃗3=dx⃗03d\V{X}_{3}=d\V{x}_{03}. After deformation these vectors are transformed into new vectors dx⃗1d\V{x}_{1}, dx⃗2d\V{x}_{2} and dx⃗3d\V{x}_{3}. The deformation tensor F\T{F} gives the relation between the two sets of vectors

dx⃗1=F⋅dX⃗1dx⃗2=F⋅dX⃗2dx⃗3=F⋅dX⃗3\begin{split} d\V{x}_1 &= \T{F} \inp d\V{X}_{1} \\ d\V{x}_2 &= \T{F} \inp d\V{X}_{2} \\ d\V{x}_3 &= \T{F} \inp d\V{X}_{3} \end{split}
(1.14)

If both sets of vectors are known, F\T{F} can be fully determined. The volume dV0dV_0 of the parallelepiped spanned by the three vectors in the initial situation equals:

dV0=dX⃗1×dX⃗2⋅dX⃗3,dV_0 = d\V{X}_{1} \cross d\V{X}_{2} \inp d\V{X}_{3} ,
(1.15)

whereas the volume dVdV in the deformed situation equals:

dV=dx⃗1×dx⃗2⋅dx⃗3=(F⋅dX⃗1)×(F⋅dX⃗2)⋅(F⋅dX⃗3)=det⁡(F)(dX⃗1×dX⃗2⋅dX⃗3)=det⁡(F)dV0\begin{align} dV &= d\V{x}_1 \cross d\V{x}_2 \inp d\V{x}_3 \notag \\ &= (\T{F} \inp d\V{X}_{1}) \cross (\T{F} \inp d\V{X}_{2}) \inp (\T{F} \inp d\V{X}_{3}) \notag \\ &= \det(\T{F}) (d\V{X}_{1} \cross d\V{X}_{2} \inp d\V{X}_{3} ) \\ &= \det(\T{F}) dV_0 \end{align}
(1.16)

Thus we may interpret det⁡(F)\det(\T{F}), which is often denoted by JJ, as the volume ratio[1] of the continuum:

J  =  det⁡(F)=dVdV0J \;=\; \det (\T{F}) = \tfrac{dV}{dV_0}
(1.17)

Since, based on physical arguments, it holds that:

J=det⁡(F)>0J = \det(\T{F}) > 0
(1.18)

we can conclude that F\T{F} is a regular tensor. Thus the inverse tensor F−1\T{F}^{-1}, describing the transformation from the deformed to the undeformed state, exists:

dX⃗=dx⃗0=F−1⋅dx⃗d\V{X} = d\V{x}_0 = \T{F}^{-1} \inp d\V{x}
(1.19)

with:

F−1=(∇⃗x⃗0)T\T{F}^{-1} = (\grad \V{x}_0 )^T
(1.20)

The gradient operators ∇⃗0\grado and ∇⃗\grad are mutually related by:

∇⃗=F−T⋅∇⃗0\grad = \T{F}^{-T} \inp \grado
(1.21)

as follows from:

I=F−T⋅FT=F−T⋅(∇⃗0x⃗)=∇⃗x⃗\T{I} = \T{F}^{-T} \inp \T{F}^T = \T{F}^{-T} \inp ( \grado \V{x} ) = \grad \V{x}
(1.22)

where we used the identity ∇⃗x⃗=I\grad\V{x} = \T{I}.

1.1.4Special deformations

1.1.4.1Rigid body motion

The response of a continuum on an external load may be a pure rigid body motion, consisting of translation and rotation. In this case the mutual displacements of the material points of the body are zero, see Figure 1.4.

Rigid body motion of a continuum.

Figure 1.4:Rigid body motion of a continuum.

We may consider the transformation of the initial to the actual situation as a pure rotation Q\T{Q} of the continuum, followed by a pure translation t⃗\V{t}. If the centre of rotation is located at a material point PP with initial position x⃗0\V{x}_{0}, the actual position y⃗\V{y} of an arbitrary point in the continuum with initial position y⃗0\V{y}_0 is given by:

y⃗=x⃗0+t⃗+Q⋅(y⃗0−x⃗0)\V{y} = \V{x}_0 + \V{t} + \T{Q}\inp(\V{y}_0 - \V{x}_{0})
(1.23)

1.1.4.2Homogeneous deformation

Another special case arises when the deformation tensor is independent of the initial position x⃗0\V{x}_0 in the continuum. Then the equation:

F=(∇⃗0x⃗)T\T{F} = (\grad_0 \V{x})^T
(1.24)

can be rewritten as:

∇⃗0x⃗=FTordx⃗=F⋅dx⃗0\grad_0 \V{x} = \T{F}^T \quad\text{or}\quad d\V{x} = \T{F}\inp d\V{x}_0
(1.25)

which is easily integrated to yield:

x⃗=x⃗0⋅FT+t⃗=F⋅x⃗0+t⃗\V{x} = \V{x}_0 \inp \T{F}^T + \V{t} = \T{F}\inp\V{x}_0 + \V{t}
(1.26)

where t⃗\V{t} represents a rigid translation. This special case is called homogeneous deformation.

1.1.5The right Cauchy-Green deformation tensor

The true deformation of a continuum consists of lengthening of material vectors and change of angle between material vectors, see Figure 1.5.

Change of length and orientation of an infinitesimal vector d\V{x}_0.

Figure 1.5:Change of length and orientation of an infinitesimal vector dx⃗0d\V{x}_0.

The stretch or stretch ratio[2] λ\lambda of a material vector dx⃗0d\V{x}_0 that is transformed into dx⃗d\V{x} is defined as:

λ=∥dx⃗∥∥dx⃗0∥\lambda = \tfrac{\|d\V{x}\|}{\|d\V{x}_0\|}
(1.27)

It can be related to the deformation tensor F\T{F} as follows:

λ=∥dx⃗∥∥dx⃗0∥=dx⃗⋅dx⃗dx⃗0⋅dx⃗0=dx⃗0⋅FT⋅F⋅dx⃗0dx⃗0⋅dx⃗0=∣∣dx⃗0∣∣∣∣dx⃗0∣∣e⃗0⋅FT⋅F⋅e⃗0=e⃗0⋅FT⋅F⋅e⃗0\begin{align} \lambda &= \tfrac{\|d\V{x}\|}{\|d\V{x}_0\|} \notag \\ &= \tfrac{\sqrt{d\V{x} \inp d\V{x}}}{\sqrt{d\V{x}_0 \inp d\V{x}_0}} \notag \\ &= \tfrac{\sqrt{d\V{x}_0 \inp \T{F}^T \inp \T{F} \inp d\V{x}_0}} {\sqrt{d\V{x}_0 \inp d\V{x}_0}} \notag \\ &= \tfrac{||d\V{x}_0||}{||d\V{x}_0||} \sqrt{\V{e}_0 \inp \T{F}^T \inp \T{F} \inp \V{e}_0} \notag \\ &= \sqrt{\V{e}_0 \inp \T{F}^T \inp \T{F} \inp \V{e}_0} \end{align}
(1.28)

Now let us consider the change of the angle between two material vectors, see Figure 1.6.

Shear deformation between two infinitesimal vectors d\V{x}_{01} and d\V{x}_{02}.

Figure 1.6:Shear deformation between two infinitesimal vectors dx⃗01d\V{x}_{01} and dx⃗02d\V{x}_{02}.

For simplicity we consider two vectors dx⃗01d\V{x}_{01} and dx⃗02d\V{x}_{02} that are mutually perpendicular in the initial situation. For the angle θ\theta between the transformed vectors it holds that:

cos⁡(θ)=dx⃗1⋅dx⃗2dx⃗1⋅dx⃗1dx⃗2⋅dx⃗2=dx⃗01⋅FT⋅F⋅dx⃗02dx⃗01⋅FT⋅F⋅dx⃗01dx⃗02⋅FT⋅F⋅dx⃗02=e⃗01⋅FT⋅F⋅e⃗02e⃗01⋅FT⋅F⋅e⃗01e⃗02⋅FT⋅F⋅e⃗02\begin{align} \cos(\theta) &= \tfrac{d\V{x}_1 \inp d\V{x}_2} {\sqrt{d\V{x}_1 \inp d\V{x}_1} \sqrt{d\V{x}_2 \inp d\V{x}_2}} \notag \\ &= \tfrac{d\V{x}_{01} \inp \T{F}^T \inp \T{F} \inp d\V{x}_{02}} {\sqrt{d\V{x}_{01} \inp \T{F}^T \inp \T{F} \inp d\V{x}_{01}} \sqrt{d\V{x}_{02} \inp \T{F}^T \inp \T{F} \inp d\V{x}_{02}}} \notag \\ &= \tfrac{\V{e}_{01} \inp \T{F}^T \inp \T{F} \inp \V{e}_{02}} {\sqrt{\V{e}_{01} \inp \T{F}^T \inp \T{F} \inp \V{e}_{01}} \sqrt{\V{e}_{02} \inp \T{F}^T \inp \T{F} \inp \V{e}_{02}}} \end{align}
(1.29)

A change of angle between two vectors is called shear deformation[3]. The expressions for stretch (1.27) and shear (1.29) show that the true deformation is not described by the deformation tensor F\T{F} itself, but by the so-called right Cauchy-Green deformation tensor[4] C\T{C}, defined as:

C=FT⋅F\T{C} = \T{F}^T \inp \T{F}
(1.30)

The right Cauchy-Green deformation tensor is symmetric:

CT=(FT⋅F)T=C\T{C}^T = (\T{F}^T \inp \T{F} )^T = \T{C}
(1.31)

and positive-definite:

a⃗⋅C⋅a⃗ = a⃗⋅FT⋅F⋅a⃗ = (F⋅a⃗)⋅(F⋅a⃗) > 0sinceF⋅a⃗≠0⃗ if a⃗≠0⃗\begin{alignat}{2} \V{a} \inp \T{C} \inp \V{a} &\:=\: &&\V{a} \inp \T{F}^T \inp \T{F} \inp \V{a} \notag \\ &\:=\: &&( \T{F} \inp \V{a} ) \inp ( \T{F} \inp \V{a} ) \notag \\ &\:>\: && 0 \qquad {\rm since} \qquad \T{F} \inp \V{a} \neq \V{0} \quad \text{ if } \V{a} \neq \V{0} \end{alignat}
(1.32)

As a consequence of these properties, the eigenvectors m⃗\V{m} and eigenvalues μ\mu of C\T{C} are real, the eigenvalues are positive and the eigenvectors are mutually perpendicular (or may be chosen so, see section Section A.3.6.5). With respect to the basis of eigenvectors {m⃗1,m⃗2,m⃗3}\{ \V{m}_1, \V{m}_2, \V{m}_3 \}, C\T{C} can be written as:

C=μ1m⃗1m⃗1+μ2m⃗2m⃗2+μ3m⃗3m⃗3\T{C} = \mu_1 \V{m}_1 \V{m}_1 + \mu_2 \V{m}_2 \V{m}_2 + \mu_3 \V{m}_3 \V{m}_3
(1.33)

If, in equation (1.29), we choose the directions of the material vectors e⃗01\V{e}_{01} and e⃗02\V{e}_{02} along two different eigenvectors of C\T{C}, it is obvious that:

cos⁡(θ(m⃗i,m⃗j))=0ifi≠j\cos(\theta(\V{m}_i,\V{m}_j)) = 0 \qquad {\rm if} \qquad i\neq j
(1.34)

Thus we can interpret the directions of the eigenvectors m⃗i\V{m}_i of C\T{C} as three mutually orthogonal directions in the initial state, that remain mutually orthogonal in the deformed state. These directions are called principal directions[5], and denoted by n⃗0i\V{n}_{0i}, where n⃗0i=m⃗i\V{n}_{0i} = \V{m}_i. The stretch ratio λ\lambda in a principal direction n⃗0i\V{n}_{0i} is given by:

λ(n⃗0i)=n⃗0i⋅C⋅n⃗0i=n⃗0i⋅(μ1n⃗01n⃗01+μ2n⃗02n⃗02+μ3n⃗03n⃗03)⋅n⃗0i=μi=λi\begin{align} \lambda(\V{n}_{0i}) &= \sqrt{ \V{n}_{0i} \inp \T{C} \inp \V{n}_{0i} } \notag \\ &= \sqrt{ \V{n}_{0i} \inp ( \mu_1 \V{n}_{01} \V{n}_{01} + \mu_2 \V{n}_{02} \V{n}_{02} + \mu_3 \V{n}_{03} \V{n}_{03} ) \inp \V{n}_{0i} } \notag \\ &= \sqrt{ \mu_i } \notag \\ &= \lambda_i \end{align}
(1.35)

The strain ratios λ1,λ2,λ3\lambda_1, \lambda_2, \lambda_3 are called principal strain ratios[6]. Now we can write the spectral form of C\T{C} in terms of principal strain ratios λi\lambda_i and principal directions n⃗0i\V{n}_{0i}:

C=λ12n⃗01n⃗01+λ22n⃗02n⃗02+λ32n⃗03n⃗03\T{C} = \lambda_1^2 \V{n}_{01} \V{n}_{01} + \lambda_2^2 \V{n}_{02} \V{n}_{02} + \lambda_3^2 \V{n}_{03} \V{n}_{03}
(1.36)
Example 1.1 (Simple shear)

Consider the homogeneous deformation x1=X1+γX2x_1 = X_1 + \gamma X_2, x2=X2x_2 = X_2, x3=X3x_3 = X_3, with γ\gamma the amount of shear. From the definition of F\T{F}, the matrix representation with respect to the Cartesian basis {e⃗x,e⃗y,e⃗z}\xyzbasis is

F‾  =  [1γ0010001]sodet⁡(F)=1\M{F} \;=\; \begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \qquad\text{so}\qquad \det(\T{F}) = 1
(1.37)

The determinant equals one, so simple shear is isochoric: it changes no volumes. The right Cauchy-Green tensor follows as

C‾  =  F‾TF‾  =  [1γ0γ1+γ20001]\M{C} \;=\; \M{F}^T \M{F} \;=\; \begin{bmatrix} 1 & \gamma & 0 \\ \gamma & 1 + \gamma^2 & 0 \\ 0 & 0 & 1 \end{bmatrix}
(1.38)

Applying λ(e⃗0)=e⃗0⋅C⋅e⃗0\lambda(\V{e}_0) = \sqrt{ \V{e}_0 \inp \T{C} \inp \V{e}_0 } to the two in-plane basis directions gives λ(e⃗1)=1\lambda(\V{e}_1) = 1 and λ(e⃗2)=1+γ2\lambda(\V{e}_2) = \sqrt{1+\gamma^2}: material lines along e⃗1\V{e}_1 keep their length, while those along e⃗2\V{e}_2 are stretched. The angle θ\theta between the deformed images of e⃗1\V{e}_1 and e⃗2\V{e}_2 follows from (1.29):

cos⁡(θ)  =  γ1+γ2\cos(\theta) \;=\; \frac{\gamma}{\sqrt{1+\gamma^2}}
(1.39)

which is nonzero for any γ≠0\gamma \neq 0 - the two directions no longer meet at a right angle. Note that C≠I\T{C} \neq \T{I} even though the volume is unchanged, confirming that C\T{C}, and not det⁡(F)\det(\T{F}), is what measures the true deformation.

1.1.6Right polar decomposition of the deformation tensor

We now define the right stretch tensor U\T{U} as:

U=C=λ1n⃗01n⃗01+λ2n⃗02n⃗02+λ3n⃗03n⃗03\T{U} = \sqrt{ \T{C} } = \lambda_1 \V{n}_{01} \V{n}_{01} + \lambda_2 \V{n}_{02} \V{n}_{02} + \lambda_3 \V{n}_{03} \V{n}_{03}
(1.40)

The tensor U\T{U} is symmetric:

UT=U\T{U}^T = \T{U}
(1.41)

positive-definit:

a⃗⋅U⋅a⃗>0∀  a⃗>0\V{a} \inp \T{U} \inp \V{a} > 0 \qquad \forall\; \V{a}>0
(1.42)

and regular, i.e. the inverse U−1\T{U}^{-1} exists:

U−1=1λ1n⃗01n⃗01+1λ2n⃗02n⃗02+1λ3n⃗03n⃗03\T{U}^{-1} = \tfrac{1}{\lambda_1} \V{n}_{01} \V{n}_{01} + \tfrac{1}{\lambda_2} \V{n}_{02} \V{n}_{02} + \tfrac{1}{\lambda_3} \V{n}_{03} \V{n}_{03}
(1.43)

Moreover,

det⁡(U)=det⁡(F)=λ1λ2λ3=J\det(\T{U}) = \det(\T{F}) = \lambda_1 \lambda_2 \lambda_3 = J
(1.44)

since:

det⁡(U⋅U)=det⁡2U=det⁡(C)=det⁡(FT⋅F)=det⁡2(F)\det(\T{U} \inp \T{U}) = {\det}^2 \T{U} = \det(\T{C}) = \det(\T{F}^T \inp \T{F}) = {\det}^2(\T{F})
(1.45)

Now consider a block of material, spanned by material vectors dx⃗01=∥dx⃗01∥n⃗01d\V{x}_{01} = \| d\V{x}_{01} \| \V{n}_{01}, dx⃗02=∥dx⃗02∥n⃗02d\V{x}_{02} = \| d\V{x}_{02} \| \V{n}_{02} and dx⃗03=∥dx⃗03∥n⃗03d\V{x}_{03} = \| d\V{x}_{03} \| \V{n}_{03}, see Figure 1.7.

Pure stretch of a block, spanned by eigenvectors of \T{U}, after deformation by \T{U}.

Figure 1.7:Pure stretch of a block, spanned by eigenvectors of U\T{U}, after deformation by U\T{U}.

This material block is transformed by the tensor U\T{U} into a block, spanned by the vectors dx⃗01∗d\V{x}_{01}^*, dx⃗02∗d\V{x}_{02}^* and dx⃗03∗d\V{x}_{03}^*, with:

dx⃗01∗=U⋅dx⃗01=U⋅∥dx⃗01∥n⃗01=λ1∥dx⃗01∥n⃗01=∥dx⃗01∗∥n⃗01dx⃗02∗=λ2∥dx⃗02∥n⃗02=∥dx⃗02∗∥n⃗02dx⃗03∗=λ3∥dx⃗03∥n⃗03=∥dx⃗03∗∥n⃗03\begin{align} d\V{x}_{01}^* &= \T{U} \inp d\V{x}_{01} = \T{U} \inp \| d\V{x}_{01} \| \V{n}_{01}\\ &= \lambda_1 \| d\V{x}_{01} \| \V{n}_{01} = \| d\V{x}_{01}^* \| \V{n}_{01} \\ d\V{x}_{02}^* &= \lambda_2 \| d\V{x}_{02} \| \V{n}_{02} = \| d\V{x}_{02}^* \| \V{n}_{02} \\ d\V{x}_{03}^* &= \lambda_3 \| d\V{x}_{03} \| \V{n}_{03} = \| d\V{x}_{03}^* \| \V{n}_{03} \end{align}
(1.46)

Thus, this special block of material, spanned by material vectors along the eigenvectors of U\T{U}, is transformed into a block with identical orientation, but with changed dimensions.

Simultaneous stretch and shear of a arbitrary block, after deformation by \T{U}.

Figure 1.8:Simultaneous stretch and shear of a arbitrary block, after deformation by U\T{U}.

Now we define a tensor R\T{R} that relates the deformation gradient tensor F\T{F} to the right stretch tensor U\T{U} according to:

R=F⋅U−1\T{R} = \T{F} \inp \T{U}^{-1}
(1.47)

Since:

RT⋅R=U−T⋅FT⋅F⋅U−1=U−T⋅U⋅U⋅U−1=U−T⋅UT⋅U⋅U−1=I\begin{align} \T{R}^T \inp \T{R} &= \T{U}^{-T} \inp \T{F}^T \inp \T{F} \inp \T{U}^{-1} \notag \\ &= \T{U}^{-T} \inp \T{U} \inp \T{U} \inp \T{U}^{-1} \notag \\ &= \T{U}^{-T} \inp \T{U}^T \inp \T{U} \inp \T{U}^{-1} \notag \\ &= \T{I} \end{align}
(1.48)

the tensor R\T{R} is orthogonal. Moreover, since:

det⁡(R)=det⁡(F⋅U−1)=det⁡(U)det⁡(U−1)=det⁡(U⋅U−1)=det⁡(I)=1\begin{align} \det(\T{R}) &= \det(\T{F} \inp \T{U}^{-1}) \notag \\ &= \det(\T{U}) \det(\T{U}^{-1}) \notag \\ &= \det( \T{U} \inp \T{U}^{-1} ) \notag \\ &= \det(\T{I}) = 1 \end{align}
(1.49)

R\T{R} is a rotation tensor.

Right polar decomposition of the deformation \T{F}, into stretch and shear, described by \T{U} and a rotation, described by \T{R}.

Figure 1.9:Right polar decomposition of the deformation F\T{F}, into stretch and shear, described by U\T{U} and a rotation, described by R\T{R}.

In fact, as Figure 1.9 illustrates, we now decomposed the total deformation, described by F\T{F} into two processes: first the material is deformed according to the right stretch tensor U\T{U}, next it is rotated according to the rotation tensor R\T{R}. This decomposition is called right polar decomposition[7]:

F=R⋅U\T{F} = \T{R} \inp \T{U}
(1.50)

This polar decomposition is unique.

1.1.7Finger or left Cauchy-Green tensor.

The true deformation of a continuum can also be expressed with respect to the current deformed configuration. The stretch of a material vector dx⃗0d\V{x}_0 that is transformed into dx⃗d\V{x} is then given by:

λ=∥dx⃗∥∥dx⃗0∥=dx⃗⋅dx⃗dx⃗0⋅dx⃗0=dx⃗⋅dx⃗F−1⋅dx⃗⋅F−1⋅dx⃗=∣∣dx⃗∣∣∣∣dx⃗∣∣1e⃗⋅F−T⋅F−1⋅e⃗=1e⃗⋅F−T⋅F−1⋅e⃗\begin{align} \lambda &= \tfrac{\|d\V{x}\|}{\|d\V{x}_0\|} \notag \\ &= \tfrac{\sqrt{d\V{x} \inp d\V{x}}}{\sqrt{d\V{x}_0 \inp d\V{x}_0}} \notag \\ &= \tfrac{\sqrt{d\V{x} \inp d\V{x}}} {\sqrt{\T{F}^{-1}\inp d\V{x} \inp \T{F}^{-1}\inp d\V{x}}} \notag \\ &= \tfrac{||d\V{x}||}{||d\V{x}||} \frac{1}{\sqrt{\V{e} \inp \T{F}^{-T}\inp \T{F}^{-1} \inp \V{e}}} \notag \\ &= \frac{1}{\sqrt{\V{e} \inp \T{F}^{-T} \inp \T{F}^{-1} \inp \V{e}}} \end{align}
(1.51)

The stretch is thus related to the deformation tensor F\T{F} in the current configuration by:

λ  =  1e⃗⋅B−1⋅e⃗\lambda \;=\; \frac{1}{\sqrt{\V{e} \inp \T{B}^{-1} \inp \V{e}}}
(1.52)

where the tensor B\T{B} is called the left Cauchy-Green deformation tensor or Finger tensor, given by

B  =  F⋅FT\T{B} \;=\; \T{F} \inp \T{F}^T
(1.53)

Now let us reconsider the change of the angle between two material vectors, where we attempt to express this change with respect to the current configuration, see Figure 1.10:

Shear deformation towards two infinitesimal vectors d\V{x}_{1} and d\V{x}_{2}.

Figure 1.10:Shear deformation towards two infinitesimal vectors dx⃗1d\V{x}_{1} and dx⃗2d\V{x}_{2}.

For simplicity we now consider two deformed vectors dx⃗1d\V{x}_{1} and dx⃗2d\V{x}_{2} that are mutually perpendicular in the deformed configuration. The angle ϕ\phi between these two vectors in the initial (undeformed) configuration (i.e. dx⃗01d\V{x}_{01} and dx⃗02d\V{x}_{02}) is given by:

cos⁡(ϕ)=dx⃗01⋅dx⃗02dx⃗01⋅dx⃗01dx⃗02⋅dx⃗02=dx⃗1⋅F−T⋅F−1⋅dx⃗2dx⃗1⋅F−T⋅F−1⋅dx⃗1dx⃗2⋅F−T⋅F−1⋅dx⃗2=e⃗1⋅F−T⋅F−1⋅e⃗2e⃗1⋅F−T⋅F−1⋅e⃗1e⃗2⋅F−T⋅F−1⋅e⃗2=e⃗1⋅B−1⋅e⃗2e⃗1⋅B−1⋅e⃗1e⃗2⋅B−1⋅e⃗2\begin{align} \cos(\phi) &= \tfrac{d\V{x}_{01} \inp d\V{x}_{02}} {\sqrt{d\V{x}_{01} \inp d\V{x}_{01}} \sqrt{d\V{x}_{02} \inp d\V{x}_{02}}} \notag \\ &= \tfrac{d\V{x}_{1} \inp \T{F}^{-T} \inp \T{F}^{-1} \inp d\V{x}_{2}} {\sqrt{d\V{x}_{1} \inp \T{F}^{-T} \inp \T{F}^{-1} \inp d\V{x}_{1}} \sqrt{d\V{x}_{2} \inp \T{F}^{-T} \inp \T{F}^{-1} \inp d\V{x}_{2}}} \notag \\ &= \tfrac{\V{e}_{1} \inp \T{F}^{-T} \inp \T{F}^{-1} \inp \V{e}_{2}} {\sqrt{\V{e}_{1} \inp \T{F}^{-T} \inp \T{F}^{-1} \inp \V{e}_{1}} \sqrt{\V{e}_{2} \inp \T{F}^{-T} \inp \T{F}^{-1} \inp \V{e}_{2}}} \notag \\ &= \tfrac{\V{e}_{1} \inp \T{B}^{-1} \inp \V{e}_{2}} {\sqrt{\V{e}_{1} \inp \T{B}^{-1} \inp \V{e}_{1}} \sqrt{\V{e}_{2} \inp \T{B}^{-1} \inp \V{e}_{2}}} \end{align}
(1.54)

Clearly, with respect to the current configuration, the shear between two vectors is characterized by the inverse of the left Cauchy-Green deformation tensor B\T{B}.

The left Cauchy-Green deformation tensor is symmetric:

BT=(F⋅FT)T=B\T{B}^T = (\T{F} \inp \T{F}^T )^T = \T{B}
(1.55)

and positive-definite:

a⃗⋅B⋅a⃗=a⃗⋅F⋅FT⋅a⃗=(FT⋅a⃗)⋅(FT⋅a⃗)>0sinceFT⋅a⃗≠0⃗ if a⃗≠0⃗\begin{aligned} \V{a} \inp \T{B} \inp \V{a} &= &&\V{a} \inp \T{F} \inp \T{F}^T \inp \V{a} \\ &= &&( \T{F}^T \inp \V{a} ) \inp ( \T{F}^T \inp \V{a} ) \\ &> && 0 \qquad {\rm since} \qquad \T{F}^T \inp \V{a} \neq \V{0} \quad \text{ if } \V{a} \neq \V{0} \end{aligned}
(1.56)

which also holds for its inverse B−1\T{B}^{-1}.

As a consequence of these properties, the eigenvectors n⃗\V{n} and eigenvalues μ\mu of B\T{B} are real, the eigenvalues are positive and the eigenvectors are mutually perpendicular (or may be chosen so, see section Section A.3.6.5). With respect to the basis of eigenvectors {n⃗1,n⃗2,n⃗3}\{ \V{n}_1, \V{n}_2, \V{n}_3 \}, B\T{B} can be written in its spectral form by:

B=μ1n⃗1n⃗1+μ2n⃗2n⃗2+μ3n⃗3n⃗3\T{B} = \mu_1 \V{n}_1 \V{n}_1 + \mu_2 \V{n}_2 \V{n}_2 + \mu_3 \V{n}_3 \V{n}_3
(1.57)

which trivially leads to the spectral form of B−1\T{B}^{-1} as well

B−1=1μ1n⃗1n⃗1+1μ2n⃗2n⃗2+1μ3n⃗3n⃗3\T{B}^{-1} = \frac{1}{\mu_1} \V{n}_1 \V{n}_1 + \frac{1}{\mu_2} \V{n}_2 \V{n}_2 + \frac{1}{\mu_3} \V{n}_3 \V{n}_3
(1.58)

If, in equation (1.29), we choose the directions of the material vectors e⃗1\V{e}_{1} and e⃗2\V{e}_{2} along two different eigenvectors of B−1\T{B}^{-1} (or B\T{B}), it is obvious that:

cos⁡(θ(n⃗i,n⃗j))=0ifi≠j\cos(\theta(\V{n}_i,\V{n}_j)) = 0 \qquad {\rm if} \qquad i\neq j
(1.59)

Thus we can interpret the directions of the eigenvectors n⃗i\V{n}_i of B\T{B} as three mutually orthogonal directions in the current deformed state, that were already mutually orthogonal in the initial undeformed state. These directions are called the principal directions in the current state[8].

The stretch ratio λ\lambda in a principal direction n⃗i\V{n}_{i} is given by:

λ(n⃗i)=1n⃗i⋅B−1⋅n⃗i=1n⃗i⋅(1μ1n⃗1n⃗1+1μ2n⃗2n⃗2+1μ3n⃗3n⃗3)⋅n⃗i=11μi = μi=λi\begin{align} \lambda(\V{n}_{i}) &= \frac{1}{\sqrt{ \V{n}_{i} \inp \T{B}^{-1} \inp \V{n}_{i} }} \notag \\ &= \frac{1}{\sqrt{ \V{n}_{i} \inp ( \frac{1}{\mu_1} \V{n}_{1} \V{n}_{1} + \frac{1}{\mu_2} \V{n}_{2} \V{n}_{2} + \frac{1}{\mu_3} \V{n}_{3} \V{n}_{3} ) \inp \V{n}_{i}}} \notag \\ &= \frac{1}{\sqrt{\frac{1}{\mu_i}}} \:=\: \sqrt{\mu_i} \notag \\ &= \lambda_i \end{align}
(1.60)

Clearly, the eigenvalues of B\T{B} are equal to the eigenvalues of C\T{C}, which well justifies the name of the so-called principal strain ratios λ1,λ2,λ3\lambda_1, \lambda_2, \lambda_3.

Finally, the spectral form of B\T{B} in terms of the principal strain ratios λi\lambda_i and principal directions n⃗i\V{n}_{i} is given by

B=λ12n⃗1n⃗1+λ22n⃗2n⃗2+λ32n⃗3n⃗3\T{B} = \lambda_1^2 \V{n}_{1} \V{n}_{1} + \lambda_2^2 \V{n}_{2} \V{n}_{2} + \lambda_3^2 \V{n}_{3} \V{n}_{3}
(1.61)

1.1.8Left polar decomposition of the deformation tensor

We now define the left stretch tensor V\T{V} as:

V=B=λ1n⃗1n⃗1+λ2n⃗2n⃗2+λ3n⃗3n⃗3\T{V} = \sqrt{ \T{B} } = \lambda_1 \V{n}_{1} \V{n}_{1} + \lambda_2 \V{n}_{2} \V{n}_{2} + \lambda_3 \V{n}_{3} \V{n}_{3}
(1.62)

The tensor V\T{V} is symmetric:

VT=V\T{V}^T = \T{V}
(1.63)

positive-definit:

a⃗⋅V⋅a⃗>0∀  a⃗>0\V{a} \inp \T{V} \inp \V{a} > 0 \qquad \forall\; \V{a}>0
(1.64)

and regular, i.e. the inverse V−1\T{V}^{-1} exists:

V−1=1λ1n⃗1n⃗1+1λ2n⃗2n⃗2+1λ3n⃗3n⃗3\T{V}^{-1} = \tfrac{1}{\lambda_1} \V{n}_{1} \V{n}_{1} + \tfrac{1}{\lambda_2} \V{n}_{2} \V{n}_{2} + \tfrac{1}{\lambda_3} \V{n}_{3} \V{n}_{3}
(1.65)

Moreover,

det⁡(V)=det⁡(F)=λ1λ2λ3=J\det(\T{V}) = \det(\T{F}) = \lambda_1 \lambda_2 \lambda_3 = J
(1.66)

since:

det⁡(V⋅V)=det⁡2V=det⁡(B)=det⁡(F⋅FT)=det⁡2(F)\det(\T{V} \inp \T{V}) = {\det}^2 \T{V} = \det(\T{B}) = \det(\T{F} \inp \T{F}^T) = {\det}^2(\T{F})
(1.67)

Now consider a block of material in the deformed configuration, spanned by material vectors dx⃗1=∥dx⃗1∥n⃗1d\V{x}_{1} = \| d\V{x}_{1} \| \V{n}_{1}, dx⃗2=∥dx⃗2∥n⃗2d\V{x}_{2} = \| d\V{x}_{2} \| \V{n}_{2} and dx⃗3=∥dx⃗3∥n⃗3d\V{x}_{3} = \| d\V{x}_{3} \| \V{n}_{3}, see Figure 1.11.

Pure stretch of a block, spanned by eigenvectors of \T{V}, before deformation by \T{V}.

Figure 1.11:Pure stretch of a block, spanned by eigenvectors of V\T{V}, before deformation by V\T{V}.

This material block is the result of the transformation by the tensor V\T{V} from a block, spanned by the vectors dx⃗1∗d\V{x}_{1}^*, dx⃗2∗d\V{x}_{2}^* and dx⃗3∗d\V{x}_{3}^*, with:

dx⃗1∗=V−1⋅dx⃗1=V−1⋅∥dx⃗1∥n⃗1=1λ1∥dx⃗1∥n⃗1=∥dx⃗1∗∥n⃗1dx⃗2∗=1λ2∥dx⃗2∥n⃗2=∥dx⃗2∗∥n⃗2dx⃗3∗=1λ3∥dx⃗3∥n⃗3=∥dx⃗3∗∥n⃗3\begin{align} d\V{x}_{1}^* &= \T{V}^{-1} \inp d\V{x}_{1} = \T{V}^{-1} \inp \| d\V{x}_{1} \| \V{n}_{1}\\ &= \frac{1}{\lambda_1} \| d\V{x}_{1} \| \V{n}_{1} = \| d\V{x}_{1}^* \| \V{n}_{1} \\ d\V{x}_{2}^* &= \frac{1}{\lambda_2} \| d\V{x}_{2} \| \V{n}_{2} = \| d\V{x}_{2}^* \| \V{n}_{2} \\ d\V{x}_{3}^* &= \frac{1}{\lambda_3} \| d\V{x}_{3} \| \V{n}_{3} = \| d\V{x}_{3}^* \| \V{n}_{3} \end{align}
(1.68)

Thus, this special block of material, spanned by material vectors along the eigenvectors of V\T{V}, was transformed from a block with identical orientation, but with changed dimensions.

Simultaneous stretch and shear of a arbitrary block, after deformation by \T{V}.

Figure 1.12:Simultaneous stretch and shear of a arbitrary block, after deformation by V\T{V}.

The tensor that relates the deformation gradient tensor F\T{F} to the left stretch tensor V\T{V} is nothing else than the already defined rotation tensor R\T{R}

R=V−1⋅F\T{R} = \T{V}^{-1} \inp \T{F}
(1.69)
Left polar decomposition of the deformation \T{F}, into a rotation \T{R} and a stretch and shear part, described by \T{V}.

Figure 1.13:Left polar decomposition of the deformation F\T{F}, into a rotation R\T{R} and a stretch and shear part, described by V\T{V}.

Again, as shown in Figure 1.13, the total deformation described by F\T{F} is decomposed into two processes: first the material is rotated according to the rotation tensor R\T{R} and next deformed according to the left stretch tensor V\T{V}. This decomposition is called left polar decomposition[9]:

F=V⋅R\T{F} = \T{V} \inp \T{R}
(1.70)

This polar decomposition is unique as well.

1.2Strains and strain tensors

1.2.1Strains

We have introduced the stretch ratio λ\lambda as a measure for the change of length of a material vector. If there is no deformation, the stretch ratio equals one. Often it is more convenient to describe change of length with a quantity that equals zero when there is no deformation. Such a quantity is called a strain[10], often denoted by ϵ\epsilon. In general, it is a function of the stretch ratio λ\lambda: ϵ=f(λ)\epsilon = f(\lambda). The function f(λ)f(\lambda) should meet the following requirements:

Three often used strain measures, that comply with these requirements are the

linear strain ϵl\epsilon_l:

ϵl=λ−1\epsilon_l = \lambda -1
(1.71)

the logarithmic strain ϵln\epsilon_{ln}:

ϵln=ln⁡(λ)\epsilon_{ln} = \ln(\lambda)
(1.72)

the Green-Lagrange strain ϵgl\epsilon_{gl}:

ϵgl=12(λ2−1)\epsilon_{gl} = \half (\lambda^2 -1)
(1.73)

Figure 1.14 compares these measures. They all coincide with λ−1\lambda - 1 near λ=1\lambda = 1, but diverge strongly at large stretches, which is exactly why the choice of strain measure matters once deformations are no longer small.

Comparison of strain definitions as a function of the stretch \lambda.

Figure 1.14:Comparison of strain definitions as a function of the stretch λ\lambda.

1.2.2Strain tensors

The stretch ratio of an arbitrary material vector with orientation e⃗0\V{e}_0 in the initial state can be calculated using the right Cauchy-Green deformation tensor:

λ(e⃗0)=e⃗0⋅C⋅e⃗0\lambda(\V{e}_0) = \sqrt{ \V{e}_0 \inp \T{C} \inp \V{e}_0 }
(1.74)

We will now look for a strain tensor ϵ\T{\epsilon}, that enables calculation of the strain E(e⃗0){\mathcal{E}}(\V{e}_0) in a similar way according to:

ϵ(e⃗0)=e⃗0⋅ϵ⋅e⃗0=f(λ)\epsilon(\V{e}_0) = \V{e}_0 \inp \T{\epsilon} \inp \V{e}_0 = f(\lambda)
(1.75)

1.2.2.1The linear strain tensor

In case of the linear strain, we may define the linear strain tensor[11] E\T{\mathcal E} as:

E=U−I\T{\mathcal E} = \T{U} - \T{I}
(1.76)

This linear strain tensor is also valid for large rotations, which is not the case for the infinitesimal linear strain tensor which is defined by

ε=12[F+FT]−I\T{\strn} = \half\left[\T{F} + \T{F}^T\right] - \T{I}
(1.77)

If the tensor E\T{\mathcal E} is applied to a principal direction n⃗0i\V{n}_{0i} of U\T{U}, we get:

n⃗0i⋅(U−I)⋅n⃗0i=n⃗0i⋅(U)⋅n⃗0i−1=λ(n⃗0i)−1=ϵl(n⃗0i)\begin{align} \V{n}_{0i} \inp (\T{U} -\T{I}) \inp \V{n}_{0i} &= \V{n}_{0i} \inp (\T{U}) \inp \V{n}_{0i} -1 \notag \\ &= \lambda(\V{n}_{0i}) - 1 \notag \\ &= \epsilon_l(\V{n}_{0i}) \end{align}
(1.78)

Thus, for this special case the condition ϵl(e⃗0)=e⃗0⋅E⋅e⃗0\epsilon_l(\V{e}_0) = \V{e}_0 \inp \T{\mathcal E} \inp \V{e}_0 gives indeed the linear strain. However, application to an arbitrary direction e⃗0\V{e}_0, yields that:

e⃗0⋅E⋅e⃗0≠λ(e⃗0)−1\V{e}_0 \inp \T{\mathcal E} \inp \V{e}_0 \neq \lambda(\V{e}_0) - 1
(1.79)

i.e. linear strain cannot generally be found using the strain tensor U−I\T{U} - \T{I}.

1.2.2.2The logarithmic strain tensor

In case of the logaritmic strain a logarithmic strain tensor[12] Λ\T{\Lambda} may be defined as:

Λ=ln⁡(U)\T{\Lambda} = \ln(\T{U})
(1.80)

However, it can be shown that:

e⃗0⋅Λ⋅e⃗0  =  ϵln(e⃗0)only ife⃗0=n⃗0i\V{e}_0 \inp \T{\Lambda} \inp \V{e}_0 \;=\; \epsilon_{ln}(\V{e}_0) \qquad \text{only if} \quad \V{e}_0 = \V{n}_{0i}
(1.81)

Thus, similar to the case of linear strain, the logarithmic strain cannot generally be found using the logarithmic strain tensor.

1.2.2.3The Green-Lagrange strain tensor

The Green-Lagrange strain tensor[13] is defined as:

E  =  12(C−I)\T{E} \;=\; \half ( \T{C} - \T{I} )
(1.82)

If we calculate the strain in an arbitrary direction e⃗0\V{e}_0, using the Green-Lagrange strain tensor E\T{E}, we get:

e⃗0⋅12(C−I)⋅e⃗0=e⃗0⋅12C⋅e⃗0−12=12(λ2(e⃗0)−1)=ϵgl(e⃗0)\begin{align} \V{e}_0 \inp \half ( \T{C} - \T{I} ) \inp \V{e}_0 &= \V{e}_0 \inp \half \T{C} \inp \V{e}_0 - \half \notag \\ &= \half ( \lambda^2(\V{e}_0) - 1) \notag \\ &= \epsilon_{gl}(\V{e}_0) \end{align}
(1.83)

Thus, in this case the strain calculated with the strain tensor is in agreement with the strain definition. Therefore, the Green-Lagrange strain is frequently used.

An alternative form of E\T{E} is obtained if we start from the following form of the deformation tensor F\T{F}:

F=(∇⃗0x⃗)T={∇⃗0(x⃗0+u⃗)}T=I+(∇⃗0u⃗)T\T{F} = ( \grad_0 \V{x} )^T = \{ \grad_0 ( \V{x}_0 + \V{u} ) \}^T = \T{I} + ( \grad_0 \V{u} )^T
(1.84)

Substitution of this expression in the definition of E\T{E} yields:

E=12(FT⋅F−I)=12[(∇⃗0u⃗)+(∇⃗0u⃗)T+(∇⃗0u⃗)T⋅(∇⃗0u⃗)]\begin{aligned} \T{E} &= \half ( \T{F}^T \inp \T{F} - \T{I} ) \\ &= \half [ (\grad_0 \V{u}) + (\grad_0 \V{u})^T + (\grad_0 \V{u})^T \inp (\grad_0 \V{u}) ] \end{aligned}
(1.85)

1.3Time derivatives

In this section we will consider the dependence of quantities on time. Let us consider a scalar quantity aa in a continuum, for example temperature, that depends on space and time. The variation of aa throughout the continuum can be considered a function of the {\em material coordinates} ξ∼\C{\xi} of the material points:

a=α(ξ∼,t)a = \alpha(\C{\xi}, t)
(1.86)

or of the initial position x⃗0\V{x}_0 of the material points:

a=A(x⃗0,t)a = A(\V{x}_0, t)
(1.87)

or of the actual position x⃗\V{x} of the material points:

a=A(x⃗,t)a = {\mathcal A}(\V{x}, t)
(1.88)

Depending on the choice, several temporal derivatives are defined.

1.3.1Material time derivative

The material time derivative[14] is the time derivative observed for a specific material point in the continuum. Since we can label a point by its material coordinates or by its initial position, we get:

a˙=dadt=lim⁡Δt→01Δt[α(ξ∼,t+Δt)−α(ξ∼,t)]=∂α∂t∣ξ∼\begin{align} \dot{a} &= \tparder{a}{t} \notag \\ &= \lim_{\Delta t \rightarrow 0} \tfrac{1}{\Delta t} [\alpha(\C{\xi}, t + \Delta t) - \alpha(\C{\xi}, t) ] \notag \\ &= \left. \parder{\alpha}{t} \right|_{\C{\xi} } \end{align}
(1.89)

or

a˙=dadt=lim⁡Δt→01Δt[A(x⃗0,t+Δt)−A(x⃗0,t)]=∂A∂t∣x⃗0\begin{align} \dot{a} &= \tparder{a}{t} \notag \\ &= \lim_{\Delta t \rightarrow 0} \tfrac{1}{\Delta t} [ A(\V{x}_0, t + \Delta t) - A(\V{x}_0, t) ] \notag \\ &= \left. \parder{A}{t} \right|_{\V{x}_0 } \end{align}
(1.90)

As an example we define the velocity of a material point as the rate of change of the position of the particle:

v⃗=∂α∂t∣ξ∼=∂A∂t∣x⃗0=x⃗˙\V{v} = \left. \parder{\alpha}{t} \right|_{\C{\xi}} = \left. \parder{A}{t} \right|_{\V{x}_0} = \dot{\V{x}}
(1.91)

The material time derivative is often used in solid mechanics, where we are interested in phenomena at points, fixed in the material or construction.

1.3.2Spatial time derivative

The spatial time derivative[15] is the time derivative observed for a specific point in space. Thus we get:

δaδt=lim⁡Δt→01Δt[A(x⃗,t+Δt)−A(x⃗,t)]=∂A∂t∣x⃗\begin{align} \tfrac{\delta a}{\delta t} &= \lim_{\Delta t \rightarrow 0} \tfrac{1}{\Delta t} [{\mathcal A}(\V{x}, t + \Delta t) - {\mathcal A}(\V{x}, t) ] \notag \\ &= \left. \parder{{\mathcal A}}{t} \right|_{\V{x}} \end{align}
(1.92)

The spatial time derivative is often used in fluid mechanics, where we are interested in phenomena at points, fixed in the space.

Consider a material point in the undeformed reference configuration with a position vector that is given by x⃗0\V{x}_0. The actual position of that material point in the current configuration at time tt is given by x⃗\V{x}, while at time t+Δtt+\Delta t the point has moved to a new current position given by x⃗+dx⃗\V{x} + d\V{x}. The relation between the material and the spatial time derivative can be derived by:

a˙=lim⁡Δt→01Δt{A(x⃗0,t+Δt)−A(x⃗0,t)}=lim⁡Δt→01Δt{A(x⃗+dx⃗,t+Δt)−A(x⃗,t)}=lim⁡Δt→01Δt{A(x⃗+dx⃗,t+Δt)−A(x⃗,t+Δt)+A(x⃗,t+Δt)−A(x⃗,t)}=lim⁡Δt→01Δt{dx⃗⋅(∇⃗a)∣t+Δt+A(x⃗,t+Δt)−A(x⃗,t)}=lim⁡Δt→0{dx⃗Δt⋅(∇⃗a)∣t+Δt}+lim⁡Δt→01Δt{A(x⃗,t+Δt)−A(x⃗,t)}=v⃗⋅(∇⃗a)+δaδt\begin{alignat}{2} \dot{a} &= &&\lim_{\Delta t \rightarrow 0} \tfrac{1}{\Delta t} \{ A(\V{x}_0, t + \Delta t) - A(\V{x}_0, t) \} \notag \\ &= &&\lim_{\Delta t \rightarrow 0} \tfrac{1}{\Delta t} \{ {\mathcal A}(\V{x} + d\V{x}, t + \Delta t) - {\mathcal A}(\V{x}, t) \} \notag \\ &= &&\lim_{\Delta t \rightarrow 0} \tfrac{1}{\Delta t} \{ {\mathcal A}(\V{x} + d\V{x}, t + \Delta t) - {\mathcal A}(\V{x}, t + \Delta t) + \notag \\ & &&{\mathcal A}(\V{x}, t + \Delta t) - {\mathcal A}(\V{x}, t) \} \notag \\ &= &&\lim_{\Delta t \rightarrow 0} \tfrac{1}{\Delta t} \{ d\V{x} \inp \left.(\grad a) \right|_{t + \Delta t} + {\mathcal A}(\V{x}, t + \Delta t) - {\mathcal A}(\V{x}, t) \} \notag \\ &= &&\lim_{\Delta t \rightarrow 0} \{ \tfrac{d\V{x}}{\Delta t} \inp \left. (\grad a) \right|_{t + \Delta t} \} \notag + \\ & &&\lim_{\Delta t \rightarrow 0} \tfrac{1}{\Delta t} \{ {\mathcal A}(\V{x}, t + \Delta t) - {\mathcal A}(\V{x}, t) \} \notag \\&=&& \V{v} \inp (\grad a) + \tfrac{\delta a}{\delta t} \end{alignat}
(1.93)

The first term on the right hand side is called the convective time derivative[16]. Often the difference between the value of a quantity aa and the functions α\alpha, AA, and A\mathcal A by which this value can be calculated is neglected. Then, summarising, we can write for the spatial time derivative:

δaδt=∂a∂t∣x⃗\tfrac{\delta a}{\delta t} = \left. \parder{a}{t} \right|_{\V{x}}
(1.94)

and for the material time derivative:

a˙=dadt=∂a∂t∣x⃗0\dot{a} = \tparder{a}{t} = \left. \parder{a}{t} \right|_{\V{x}_0}
(1.95)

The relation between the derivatives is:

a˙=δaδt+v⃗⋅(∇⃗a)\dot{a} = \tfrac{\delta a}{\delta t} + \V{v} \inp (\grad a)
(1.96)

For a vector a⃗\V{a}, this relation equals:

a⃗˙=δa⃗δt+v⃗⋅(∇⃗a⃗)\dot{\V{a}} = \tfrac{\delta \V{a}}{\delta t} + \V{v} \inp (\grad \V{a})
(1.97)

1.3.3Rate of deformation

Consider a material vector dx⃗d\V{x}. The rate of change of this vector dx⃗˙d\dot{\V{x}}, i.e. the change of the length and the orientation of the vector per unit of time, equals:

dx⃗˙=F˙⋅dx⃗0=F˙⋅F−1⋅dx⃗=L⋅dx⃗\begin{align} d\dot{\V{x}} &= \dot{\T{F}} \inp d\V{x}_0 \notag \\ &= \dot{\T{F}} \inp \T{F}^{-1} \inp d\V{x} \\ &= \T{L} \inp d\V{x} \end{align}
(1.98)

where the velocity gradient tensor[17] L\T{L} is introduced, which may also be written as

L=F˙⋅F−1=(∇⃗v⃗)T\begin{align} \begin{split} \T{L} & = \dot{\T{F}} \inp \T{F}^{-1} \\ & = (\V{\nabla} \V{v})^T \end{split} \end{align}
(1.99)

where v⃗\V{v} denotes the velocity. Using ∇⃗=F−T⋅∇⃗0\grad = \T{F}^{-T}\inp\grad_0, this relation is derived from

(F˙⋅F−1)T=F−T⋅F˙T=F−T⋅(∇⃗0v⃗)=F−T⋅FT⋅(∇⃗v⃗)=∇⃗v⃗\begin{align} (\dot{\T{F}} \inp \T{F}^{-1})^T &= \T{F}^{-T} \inp \dot{\T{F}}^T \notag \\ &= \T{F}^{-T} \inp ( \grad_0 \V{v} ) \notag \\ &= \T{F}^{-T} \inp \T{F}^T \inp (\grad \V{v}) \notag \\ &= \grad \V{v} \end{align}
(1.100)

Now any tensor A\T{A} can be written as the sum of a symmetric and a skew-symmetric tensor:

A=12(A+AT)+12(A−AT)∀  A\T{A} = \half(\T{A} + \T{A}^T) + \half(\T{A} - \T{A}^T) \qquad \forall\; \T{A}
(1.101)

Thus we can split the tensor F˙⋅F−1\dot{\T{F}} \inp \T{F}^{-1} into a symmetric (spatial) rate-of-deformation tensor[18] D\T{D} and a skew-symmetric spin tensor[19] (or vorticity tensor) Ω\T{\Omega}, according to:

D=12(F˙⋅F−1+(F˙⋅F−1)T)\T{D} = \half ( \dot{\T{F}} \inp \T{F}^{-1} + (\dot{\T{F}} \inp \T{F}^{-1})^T )
(1.102)
Ω=12(F˙⋅F−1−(F˙⋅F−1)T)\T{\Omega} = \half ( \dot{\T{F}} \inp \T{F}^{-1} - (\dot{\T{F}} \inp \T{F}^{-1})^T )
(1.103)

Using the velocity gradient tensor L\T{L} these relations may also be written as

D=12(L+LT)=12((∇⃗v⃗)T+∇⃗v⃗)\begin{aligned} \T{D} & = \half ( \T{L} + \T{L}^T) \\ & = \half ( (\grad \V{v})^T + \grad \V{v} ) \end{aligned}
(1.104)

and

Ω=12(L−LT)=12((∇⃗v⃗)T−∇⃗v⃗)\begin{aligned} \T{\Omega} & = \half ( \T{L} - \T{L}^T) \\ & = \half ( (\grad \V{v})^T - \grad \V{v} ) \end{aligned}
(1.105)

Consequently, the rate of change of the material vector dx⃗d\V{x} is given by:

dx⃗˙=D⋅dx⃗+Ω⋅dx⃗d\dot{\V{x}} = \T{D} \inp d\V{x} + \T{\Omega} \inp d\V{x}
(1.106)

1.3.3.1The spin tensor

Since the spin tensor Ω\T{\Omega} is skew-symmetric, a vector ω⃗\V{\omega} exists such that:

Ω⋅dx⃗=ω⃗×dx⃗\T{\Omega} \inp d\V{x} = \V{\omega} \cross d\V{x}
(1.107)

We conclude that the part of the rate of change of a material vector dx⃗d\V{x}, described by Ω\T{\Omega}, is oriented perpendicular to the material vector itself. The vector ω⃗\V{\omega} can be interpreted as the angular velocity vector[20], see Figure 1.15.

Rotation of a material vector d\V{x}.

Figure 1.15:Rotation of a material vector dx⃗d\V{x}.

The rate-of-rotation tensor, denoted by W\T{W}, is also a skew-symmetric tensor that equals

W  =  R˙⋅R−1  =  R˙⋅RT\begin{aligned} \T{W} \;&=\; \dot{\T{R}}\inp\T{R}^{-1} \\ \;&=\; \dot{\T{R}}\inp\T{R}^T \end{aligned}
(1.108)

From this relation, it can be easily shown that the relation between the spin tensor Ω\T{\Omega} and the rate-of-rotation tensor W\T{W} is given by

Ω  =  W+R⋅skew[U˙⋅U−1]⋅RT\T{\Omega} \;=\; \T{W} + \T{R}\inp skew[\dot{\T{U}}\inp\T{U}^{-1}]\inp \T{R}^T
(1.109)

where skew[.]skew[.] stands for the skew-symmetric part of the second-order tensor [.][.].

1.3.3.2The rate-of-deformation tensor

Since the rate-of-deformation tensor D\T{D} is symmetric, its eigenvalues νi\nu_i are real and positive, and its eigenvectors η⃗i\V{\eta}_i are mutually orthogonal, or can be chosen so. If we write D\T{D} in its spectral form, we get:

D=ν1η⃗1η⃗1+ν2η⃗2η⃗2+ν3η⃗3η⃗3\T{D} = \nu_1 \V{\eta}_1 \V{\eta}_1 + \nu_2 \V{\eta}_2 \V{\eta}_2 + \nu_3 \V{\eta}_3 \V{\eta}_3
(1.110)

If we apply D\T{D} to a special vector dx⃗=dxiη⃗id\V{x} = dx_i \V{\eta}_i, oriented along an eigenvector η⃗i\V{\eta}_i of D\T{D}, we obtain:

D⋅dx⃗=dxiD⋅η⃗i=dxiνiη⃗i=νidx⃗\T{D} \inp d\V{x} = dx_i \T{D} \inp \V{\eta}_i = dx_i \nu_i \V{\eta}_i = \nu_i d\V{x}
(1.111)

Thus, in this special case D\T{D} yields rate of change of the length of the vector, and not of its orientation, see Figure 1.16.

Pure lengthening of a material vector oriented along an eigenvector \V{\eta}_i of \T{D}, after application of \T{D}.

Figure 1.16:Pure lengthening of a material vector oriented along an eigenvector η⃗i\V{\eta}_i of D\T{D}, after application of D\T{D}.

However, for an arbitrary vector dx⃗=dx1η⃗1+dx2η⃗2+dx3η⃗3d\V{x} = dx_1 \V{\eta}_1 + dx_2 \V{\eta}_2 + dx_3 \V{\eta}_3 we get:

D⋅dx⃗=D⋅(dx1η⃗1+dx2η⃗2+dx3η⃗3)=dx1ν1η⃗1+dx2ν2η⃗2+dx3ν3η⃗3\begin{aligned} \T{D} \inp d\V{x} &= \T{D} \inp ( dx_1 \V{\eta}_1 + dx_2 \V{\eta}_2 + dx_3 \V{\eta}_3 ) \\ &= dx_1 \nu_1 \V{\eta}_1 + dx_2 \nu_2 \V{\eta}_2 + dx_3 \nu_3 \V{\eta}_3 \end{aligned}
(1.112)

showing that in this case application of D\T{D} yields a rate of change of both the length and the orientation of the vector, see Figure 1.17.

Change of length and orientation of an arbitrary material vector, after application of \T{D}.

Figure 1.17:Change of length and orientation of an arbitrary material vector, after application of D\T{D}.

Using the rate-of-deformation tensor, we can calculate the rate of change of the length of a material vector and the volume of a material element. Let us consider a material vector dx⃗0=∥dx⃗0∥e0⃗d\V{x}_0 = \|d\V{x}_0\| \V{e_0}, that is transformed into a vector dx⃗=∥dx⃗∥e⃗d\V{x} = \|d\V{x}\| \V{e} in the deformed situation. We have found before that the stretch λ\lambda of the vector is given by:

λ2=e⃗0⋅C⋅e⃗0\lambda^2 = \V{e}_0 \inp \T{C} \inp \V{e}_0
(1.113)

The material time derivative of this expression is given by:

DDt(λ2)=DDt(e⃗0⋅C⋅e⃗0)\tfrac{D}{Dt} ( \lambda^2 ) = \tfrac{D}{Dt} ( \V{e}_0 \inp \T{C} \inp \V{e}_0 )
(1.114)

which can be elaborated as:

2λλ˙=e⃗0⋅DDt(C)⋅e⃗0=e⃗0⋅DDt(FT⋅F)⋅e⃗0=e⃗0⋅{F˙T⋅F+FT⋅F˙}⋅e⃗0=e⃗0⋅FT⋅{F−T⋅F˙T+F˙⋅F−1}⋅F⋅e⃗0=(F⋅e⃗0)⋅{(F˙⋅F−1)T+F˙⋅F−1}⋅(F⋅e⃗0)=(λe⃗)⋅(2 D)⋅(λe⃗)\begin{aligned} 2 \lambda \dot{\lambda} &= \V{e}_0 \inp \tfrac{D}{Dt}(\T{C}) \inp \V{e}_0 \\ &= \V{e}_0 \inp \tfrac{D}{Dt}(\T{F}^T \inp \T{F}) \inp \V{e}_0 \\ &= \V{e}_0 \inp \{ \dot{\T{F}}^T \inp \T{F} + \T{F}^T \inp \dot{\T{F}} \} \inp \V{e}_0 \\ &= \V{e}_0 \inp \T{F}^T \inp \{ \T{F}^{-T} \inp \dot{\T{F}}^T + \dot{\T{F}} \inp \T{F}^{-1} \} \inp \T{F} \inp \V{e}_0 \\ &= (\T{F} \inp \V{e}_0) \inp \{ (\dot{\T{F}} \inp \T{F}^{-1})^T + \dot{\T{F}} \inp \T{F}^{-1} \} \inp (\T{F} \inp \V{e}_0) \\ &= (\lambda \V{e}) \inp (2\,\T{D}) \inp (\lambda \V{e}) \end{aligned}
(1.115)

Thus the rate of change of the length of a material vector, oriented in the direction e⃗\V{e}, normalised by its actual stretch equals:

λ˙λ  =  e⃗⋅D⋅e⃗\tfrac{\dot{\lambda}}{\lambda} \;=\; \V{e} \inp \T{D} \inp \V{e}
(1.116)

Now let us consider three material vectors, that in the refererence state were oriented along the orthogonal eigenvectors n⃗0i\V{n}_{0i} of C\T{C}. In the actual state, these material vectors are oriented along the directions n⃗i\V{n}_i, which are still orthogonal. Now we can choose these vectors as a basis, with respect to which we calculate the first invariant, or the trace, of the rate-of-deformation tensor D\T{D}:

tr(D)=n⃗1⋅D⋅n⃗1+n⃗2⋅D⋅n⃗2+n⃗3⋅D⋅n⃗3=λ˙1λ1+λ˙2λ2+λ˙3λ3=DDt{ln⁡(λ1)+ln⁡(λ2)+ln⁡(λ3)}=DDtln⁡(λ1λ2λ3)=DDtln⁡{det⁡(U)}=DDtln⁡{det⁡(F)}=DDt{ln⁡(J)}=J˙J\begin{align} \tr(\T{D}) &= \V{n}_1 \inp \T{D} \inp \V{n}_1 + \V{n}_2 \inp \T{D} \inp \V{n}_2 + \V{n}_3 \inp \T{D} \inp \V{n}_3 \notag \\ &= \tfrac{\dot{\lambda}_1}{\lambda_1} + \tfrac{\dot{\lambda}_2}{\lambda_2} + \tfrac{\dot{\lambda}_3}{\lambda_3} \notag \\ &= \tfrac{D}{Dt} \{ \ln(\lambda_1) + \ln(\lambda_2) + \ln(\lambda_3) \} \notag \\ &= \tfrac{D}{Dt} \ln(\lambda_1 \lambda_2 \lambda_3) \notag \\ &= \tfrac{D}{Dt} \ln \{ \det(\T{U}) \} \notag \\ &= \tfrac{D}{Dt} \ln \{ \det(\T{F}) \} \notag \\ &= \tfrac{D}{Dt} \{ \ln(J) \} \notag \\ &= \tfrac{\dot{J}}{J} \end{align}
(1.117)

Thus the rate of change of the volume[21] of the continuum equals:

J˙=Jtr(D)\dot{J} = J \tr(\T{D})
(1.118)

Finally, for applications in fluid mechanics it is useful to rewrite the expressions for D\T{D} and Ω\T{\Omega} in terms of the momentary velocity of the material.

Substitution of (F˙⋅F−1)T=∇⃗v⃗(\dot{\T{F}} \inp \T{F}^{-1})^T = \grad \V{v} into the tensors D\T{D} and Ω\T{\Omega} yields:

D  =  12{(∇⃗v⃗)T+(∇⃗v⃗)}\T{D} \;=\; \half \left\{ (\grad\V{v})^T + (\grad\V{v}) \right\}
(1.119)
Ω  =  12{(∇⃗v⃗)T−(∇⃗v⃗)}\T{\Omega} \;=\; \half \left\{ (\grad\V{v})^T - (\grad\V{v}) \right\}
(1.120)

Moreover, the rate of change of the volume of the continuum becomes:

J˙=J(∇⃗⋅v⃗)\dot{J} = J (\grad \inp \V{v})
(1.121)

This relation is derived from:

J˙  =  J  tr(D)=  J  tr[12{(∇⃗v⃗)T+(∇⃗v⃗)}]=  J  tr(∇⃗v⃗)=  J  (∇⃗v)⃗:I=  J  ∇⃗⋅v⃗\begin{align} \dot{J} \;&=\; J \; \tr(\T{D}) \notag \\ &=\; J \; \tr \left[ \half \left\{ (\grad\V{v})^T + (\grad\V{v}) \right\} \right] \notag \\ &=\; J \; \tr(\grad \V{v})\\ &=\; J \; (\grad \V{v)} : \T{I} \\ &=\; J \; \grad \inp \V{v} \end{align}
(1.122)
Footnotes

  1. volumeveranderingsfaktor

    ↩

  2. verlengingsfaktor

    ↩

  3. afschuiving

    ↩

  4. rechtse Cauchy-Green deformatietensor

    ↩

  5. hoofdrekrichtingen

    ↩

  6. hoofdverlengingsfaktoren

    ↩

  7. rechtse polaire decompositie

    ↩

  8. hoofdrekrichtingen

    ↩

  9. linkse polaire decompositie

    ↩

  10. rek

    ↩

  11. lineaire rektensor

    ↩

  12. logarithmische rektensor

    ↩

  13. Green-Lagrange rektensor

    ↩

  14. materiële tijdsafgeleide

    ↩

  15. ruimtelijke tijdsafgeleide

    ↩

  16. convectieve tijdsafgeleide

    ↩

  17. snelheidsgradiënttensor

    ↩

  18. deformatiesnelheidstensor

    ↩

  19. spintensor

    ↩

  20. hoeksnelheidsvector

    ↩

  21. volumeveranderingssnelheid

    ↩
Strength and Structure
Preface
Strength and Structure
Force and stress