HOME

TheInfoList



OR:

In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in
infinitesimal strain theory In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological
soft tissue Soft tissue is all the tissue in the body that is not hardened by the processes of ossification or calcification such as bones and teeth. Soft tissue connects, surrounds or supports internal organs and bones, and includes muscle, tendons, ...
.


Displacement

The displacement of a body has two components: a rigid-body displacement and a deformation. * A rigid-body displacement consists of a simultaneous
translation (physics) In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every ...
and rotation of the body without changing its shape or size. * Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration \kappa_0(\mathcal B) to a current or deformed configuration \kappa_t(\mathcal B) (Figure 1). A change in the configuration of a continuum body can be described by a displacement field. A ''displacement field'' is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. The distance between any two particles changes if and only if deformation has occurred. If displacement occurs without deformation, then it is a rigid-body displacement.


Material coordinates (Lagrangian description)

The displacement of particles indexed by variable may be expressed as follows. The vector joining the positions of a particle in the undeformed configuration P_i and deformed configuration p_i is called the
displacement vector In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
. Using \mathbf in place of P_i and \mathbf in place of p_i\,\!, both of which are vectors from the origin of the coordinate system to each respective point, we have the Lagrangian description of the displacement vector: \mathbf u(\mathbf X,t) = u_i \mathbf e_i where \mathbf e_i are the orthonormal
unit vectors In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vec ...
that define the
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
of the spatial (lab-frame) coordinate system. Expressed in terms of the material coordinates, i.e. \mathbf u as a function of \mathbf X, the displacement field is: \mathbf u(\mathbf X, t) = \mathbf b(t)+\mathbf x(\mathbf X,t) - \mathbf X \qquad \text\qquad u_i = \alpha_ b_J + x_i - \alpha_ X_J where \mathbf b(t) is the displacement vector representing rigid-body translation. The partial derivative of the displacement vector with respect to the material coordinates yields the ''material displacement gradient tensor'' \nabla_ \mathbf u\,\!. Thus we have, \nabla_\mathbf u = \nabla_\mathbf x - \mathbf R = \mathbf F - \mathbf R \qquad \text \qquad \frac = \frac - \alpha_ = F_ - \alpha_ where \mathbf F is the ''deformation gradient tensor''.


Spatial coordinates (Eulerian description)

In the Eulerian description, the vector extending from a particle P in the undeformed configuration to its location in the deformed configuration is called the
displacement vector In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
: \mathbf U(\mathbf x,t) = U_J\mathbf E_J where \mathbf E_i are the unit vectors that define the basis of the material (body-frame) coordinate system. Expressed in terms of spatial coordinates, i.e. \mathbf U as a function of \mathbf x, the displacement field is: \mathbf U(\mathbf x, t) = \mathbf b(t) + \mathbf x - \mathbf X(\mathbf x,t) \qquad \text\qquad U_J = b_J + \alpha_ x_i - X_J The partial derivative of the displacement vector with respect to the spatial coordinates yields the ''spatial displacement gradient tensor'' \nabla_ \mathbf U\,\!. Thus we have, \nabla_\mathbf U = \mathbf R^ - \nabla_\mathbf X = \mathbf R^ -\mathbf F^ \qquad \text \qquad \frac = \alpha_ - \frac = \alpha_ - F^_ \,.


Relationship between the material and spatial coordinate systems

\alpha_ are the direction cosines between the material and spatial coordinate systems with unit vectors \mathbf E_J and \mathbf e_i\,\!, respectively. Thus \mathbf E_J \cdot \mathbf e_i = \alpha_ = \alpha_ The relationship between u_i and U_J is then given by u_i=\alpha_U_J \qquad \text \qquad U_J=\alpha_ u_i Knowing that \mathbf e_i = \alpha_ \mathbf E_J then \mathbf u(\mathbf X, t) = u_i\mathbf e_i = u_i(\alpha_\mathbf E_J) = U_J \mathbf E_J = \mathbf U(\mathbf x, t)


Combining the coordinate systems of deformed and undeformed configurations

It is common to superimpose the coordinate systems for the deformed and undeformed configurations, which results in \mathbf b = 0\,\!, and the direction cosines become
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
s, i.e., \mathbf E_J \cdot \mathbf e_i = \delta_ = \delta_ Thus in material (undeformed) coordinates, the displacement may be expressed as: \mathbf u(\mathbf X, t) = \mathbf x(\mathbf X,t) - \mathbf X \qquad \text\qquad u_i = x_i - \delta_ X_J And in spatial (deformed) coordinates, the displacement may be expressed as: \mathbf U(\mathbf x, t) = \mathbf x - \mathbf X(\mathbf x,t) \qquad \text\qquad U_J = \delta_ x_i - X_J


Deformation gradient tensor

The deformation gradient tensor \mathbf F(\mathbf X,t) = F_ \mathbf e_j \otimes \mathbf I_K is related to both the reference and current configuration, as seen by the unit vectors \mathbf e_j and \mathbf I_K\,\!, therefore it is a ''
two-point tensor Two-point tensors, or double vectors, are tensor-like quantities which transform as Euclidean vectors with respect to each of their indices. They are used in continuum mechanics to transform between reference ("material") and present ("configurat ...
''. Due to the assumption of continuity of \chi(\mathbf X,t)\,\!, \mathbf F has the inverse \mathbf H = \mathbf F^\,\!, where \mathbf H is the ''spatial deformation gradient tensor''. Then, by the implicit function theorem, the Jacobian determinant J(\mathbf X,t) must be
nonsingular In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplica ...
, i.e. J(\mathbf X,t) = \det \mathbf F(\mathbf X,t) \neq 0 The ''material deformation gradient tensor'' \mathbf F(\mathbf X,t) = F_ \mathbf e_j\otimes\mathbf I_K is a second-order tensor that represents the gradient of the mapping function or functional relation \chi(\mathbf X,t)\,\!, which describes the motion of a continuum. The material deformation gradient tensor characterizes the local deformation at a material point with position vector \mathbf X\,\!, i.e., deformation at neighbouring points, by transforming (
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
) a material line element emanating from that point from the reference configuration to the current or deformed configuration, assuming continuity in the mapping function \chi(\mathbf X,t)\,\!, i.e.
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
of \mathbf and time t\,\!, which implies that cracks and voids do not open or close during the deformation. Thus we have, \begin d\mathbf &= \frac \,d\mathbf \qquad &\text& \qquad dx_j =\frac\,dX_K \\ &= \nabla \chi(\mathbf X,t) \,d\mathbf \qquad &\text& \qquad dx_j =F_\,dX_K \,. \\ & = \mathbf F(\mathbf X,t) \,d\mathbf \end


Relative displacement vector

Consider a particle or material point P with position vector \mathbf X = X_I \mathbf I_I in the undeformed configuration (Figure 2). After a displacement of the body, the new position of the particle indicated by p in the new configuration is given by the vector position \mathbf = x_i \mathbf e_i\,\!. The coordinate systems for the undeformed and deformed configuration can be superimposed for convenience. Consider now a material point Q neighboring P\,\!, with position vector \mathbf+ \Delta \mathbf = (X_I+\Delta X_I) \mathbf I_I\,\!. In the deformed configuration this particle has a new position q given by the position vector \mathbf+ \Delta \mathbf\,\!. Assuming that the line segments \Delta X and \Delta \mathbf x joining the particles P and Q in both the undeformed and deformed configuration, respectively, to be very small, then we can express them as d\mathbf X and d\mathbf x\,\!. Thus from Figure 2 we have \begin \mathbf+ d\mathbf&= \mathbf+d\mathbf+\mathbf(\mathbf+d\mathbf) \\ d\mathbf &= \mathbf-\mathbf+d\mathbf+ \mathbf(\mathbf+d\mathbf) \\ &= d\mathbf+\mathbf(\mathbf+d\mathbf)- \mathbf(\mathbf) \\ &= d\mathbf+d\mathbf \\ \end where \mathbf is the relative displacement vector, which represents the relative displacement of Q with respect to P in the deformed configuration.


Taylor approximation

For an infinitesimal element d\mathbf X\,\!, and assuming continuity on the displacement field, it is possible to use a
Taylor series expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
around point P\,\!, neglecting higher-order terms, to approximate the components of the relative displacement vector for the neighboring particle Q as \begin \mathbf(\mathbf+d\mathbf)&=\mathbf(\mathbf)+d\mathbf \quad & \text & \quad u_i^* = u_i+du_i \\ &\approx \mathbf(\mathbf)+\nabla_\mathbf u\cdot d\mathbf X \quad & \text & \quad u_i^* \approx u_i + \fracdX_J \,. \end Thus, the previous equation d\mathbf x = d\mathbf + d\mathbf can be written as \begin d\mathbf x&=d\mathbf X+d\mathbf u \\ &=d\mathbf X+\nabla_\mathbf u\cdot d\mathbf X\\ &=\left(\mathbf I + \nabla_\mathbf u\right)d\mathbf X\\ &=\mathbf F d\mathbf X \end


Time-derivative of the deformation gradient

Calculations that involve the time-dependent deformation of a body often require a time derivative of the deformation gradient to be calculated. A geometrically consistent definition of such a derivative requires an excursion into differential geometryA. Yavari, J.E. Marsden, and M. Ortiz
On spatial and material covariant balance laws in elasticity
Journal of Mathematical Physics, 47, 2006, 042903; pp. 1–53.
but we avoid those issues in this article. The time derivative of \mathbf is \dot = \frac = \frac \left frac\right= \frac\left frac\right= \frac\left mathbf(\mathbf, t)\right where \mathbf is the (material) velocity. The derivative on the right hand side represents a material velocity gradient. It is common to convert that into a spatial gradient by applying the chain rule for derivatives, i.e., \dot = \frac\left mathbf(\mathbf, t)\right= \frac\left mathbf(\mathbf(\mathbf, t),t)\right= \left.\frac\left mathbf(\mathbf,t)\right_ \cdot \frac = \boldsymbol\cdot\mathbf where \boldsymbol is the spatial velocity gradient and where \mathbf(\mathbf,t) = \mathbf(\mathbf,t) is the spatial (Eulerian) velocity at \mathbf = \mathbf(\mathbf, t). If the spatial velocity gradient is constant in time, the above equation can be solved exactly to give \mathbf = e^ assuming \mathbf = \mathbf at t = 0. There are several methods of computing the
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
above. Related quantities often used in continuum mechanics are the rate of deformation tensor and the spin tensor defined, respectively, as: \boldsymbol = \tfrac \left(\boldsymbol + \boldsymbol^T\right) \,,~~ \boldsymbol = \tfrac \left(\boldsymbol - \boldsymbol^T\right) \,. The rate of deformation tensor gives the rate of stretching of line elements while the spin tensor indicates the rate of rotation or
vorticity In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...
of the motion. The material time derivative of the inverse of the deformation gradient (keeping the reference configuration fixed) is often required in analyses that involve finite strains. This derivative is \frac \left(\mathbf^\right) = - \mathbf^ \cdot \dot \cdot \mathbf^ \,. The above relation can be verified by taking the material time derivative of \mathbf^ \cdot d\mathbf = d\mathbf and noting that \dot = 0.


Transformation of a surface and volume element

To transform quantities that are defined with respect to areas in a deformed configuration to those relative to areas in a reference configuration, and vice versa, we use Nanson's relation, expressed as da~\mathbf = J~dA ~\mathbf^ \cdot \mathbf where da is an area of a region in the deformed configuration, dA is the same area in the reference configuration, and \mathbf is the outward normal to the area element in the current configuration while \mathbf is the outward normal in the reference configuration, \mathbf is the
deformation gradient In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
, and J = \det\mathbf\,\!. The corresponding formula for the transformation of the volume element is dv = J~dV


Polar decomposition of the deformation gradient tensor

The deformation gradient \mathbf\,\!, like any invertible second-order tensor, can be decomposed, using the
polar decomposition In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive se ...
theorem, into a product of two second-order tensors (Truesdell and Noll, 1965): an orthogonal tensor and a positive definite symmetric tensor, i.e., \mathbf = \mathbf \mathbf = \mathbf \mathbf where the tensor \mathbf is a
proper orthogonal tensor Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
, i.e., \mathbf R^ = \mathbf R^T and \det \mathbf R = +1\,\!, representing a rotation; the tensor \mathbf is the ''right stretch tensor''; and \mathbf the ''left stretch tensor''. The terms ''right'' and ''left'' means that they are to the right and left of the rotation tensor \mathbf\,\!, respectively. \mathbf and \mathbf are both positive definite, i.e. \mathbf x \cdot \mathbf U \cdot \mathbf x > 0 and \mathbf x\cdot\mathbf V \cdot \mathbf x > 0 for all non-zero \mathbf x \in \R^3, and
symmetric tensor In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of orde ...
s, i.e. \mathbf U = \mathbf U^T and \mathbf V = \mathbf V^T\,\!, of second order. This decomposition implies that the deformation of a line element d\mathbf X in the undeformed configuration onto d\mathbf x in the deformed configuration, i.e., d\mathbf x = \mathbf F \,d\mathbf X\,\!, may be obtained either by first stretching the element by \mathbf U\,\!, i.e. d\mathbf x' = \mathbf U \,d\mathbf X\,\!, followed by a rotation \mathbf R\,\!, i.e., d\mathbf x' = \mathbf R \,d\mathbf x\,\!; or equivalently, by applying a rigid rotation \mathbf R first, i.e., d\mathbf x' = \mathbf R \, d\mathbf X\,\!, followed later by a stretching \mathbf V\,\!, i.e., d\mathbf x' = \mathbf V \, d\mathbf x (See Figure 3). Due to the orthogonality of \mathbf R \mathbf V = \mathbf R \cdot \mathbf U \cdot \mathbf R^T so that \mathbf U and \mathbf V have the same eigenvalues or ''principal stretches'', but different
eigenvectors In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
or ''principal directions'' \mathbf_i and \mathbf_i\,\!, respectively. The principal directions are related by \mathbf_i = \mathbf \mathbf_i. This polar decomposition, which is unique as \mathbf F is invertible with a positive determinant, is a corrolary of the
singular-value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is relate ...
.


Deformation tensors

Several rotation-independent deformation tensors are used in mechanics. In solid mechanics, the most popular of these are the right and left Cauchy–Green deformation tensors. Since a pure rotation should not induce any strains in a deformable body, it is often convenient to use rotation-independent measures of deformation in continuum mechanics. As a rotation followed by its inverse rotation leads to no change (\mathbf\mathbf^T=\mathbf^T\mathbf=\mathbf\,\!) we can exclude the rotation by multiplying \mathbf by its
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
.


The right Cauchy–Green deformation tensor

In 1839, George Green introduced a deformation tensor known as the ''right Cauchy–Green deformation tensor'' or ''Green's deformation tensor'', defined as:The
IUPAC The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
recommends that this tensor be called the Cauchy strain tensor.
\mathbf C=\mathbf F^T\mathbf F=\mathbf U^2 \qquad \text \qquad C_=F_~F_ = \frac \frac . Physically, the Cauchy–Green tensor gives us the square of local change in distances due to deformation, i.e. d\mathbf x^2=d\mathbf X\cdot\mathbf C \cdot d\mathbf X Invariants of \mathbf are often used in the expressions for strain energy density functions. The most commonly used invariants are \begin I_1^C & := \text(\mathbf) = C_ = \lambda_1^2 + \lambda_2^2 + \lambda_3^2 \\ I_2^C & := \tfrac\left \text~\mathbf)^2 - \text(\mathbf^2) \right = \tfrac\left C_)^2 - C_C_\right= \lambda_1^2\lambda_2^2 + \lambda_2^2\lambda_3^2 + \lambda_3^2\lambda_1^2 \\ I_3^C & := \det(\mathbf) = J^2 = \lambda_1^2\lambda_2^2\lambda_3^2. \end where J:=\det\mathbf is the determinant of the deformation gradient \mathbf and \lambda_i are stretch ratios for the unit fibers that are initially oriented along the eigenvector directions of the right (reference) stretch tensor (these are not generally aligned with the three axis of the coordinate systems).


The Finger deformation tensor

The
IUPAC The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
recommends that the inverse of the right Cauchy–Green deformation tensor (called the Cauchy tensor in that document), i. e., \mathbf C^, be called the Finger tensor. However, that nomenclature is not universally accepted in applied mechanics. \mathbf=\mathbf C^=\mathbf F^\mathbf F^ \qquad \text \qquad f_=\frac \frac


The left Cauchy–Green or Finger deformation tensor

Reversing the order of multiplication in the formula for the right Green–Cauchy deformation tensor leads to the ''left Cauchy–Green deformation tensor'' which is defined as: \mathbf B = \mathbf F\mathbf F^T = \mathbf V^2 \qquad \text \qquad B_ = \frac \frac The left Cauchy–Green deformation tensor is often called the ''Finger deformation tensor'', named after Josef Finger (1894).The
IUPAC The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
recommends that this tensor be called the Green strain tensor.
Invariants of \mathbf are also used in the expressions for strain energy density functions. The conventional invariants are defined as \begin I_1 & := \text(\mathbf) = B_ = \lambda_1^2 + \lambda_2^2 + \lambda_3^2\\ I_2 & := \tfrac\left \text~\mathbf)^2 - \text(\mathbf^2)\right = \tfrac\left(B_^2 - B_B_\right) = \lambda_1^2\lambda_2^2 + \lambda_2^2\lambda_3^2 + \lambda_3^2\lambda_1^2 \\ I_3 & := \det\mathbf = J^2 = \lambda_1^2\lambda_2^2\lambda_3^2 \end where J:=\det\mathbf is the determinant of the deformation gradient. For compressible materials, a slightly different set of invariants is used: (\bar_1 := J^ I_1 ~;~~ \bar_2 := J^ I_2 ~;~~ J\neq 1) ~.


The Cauchy deformation tensor

Earlier in 1828, Augustin-Louis Cauchy introduced a deformation tensor defined as the inverse of the left Cauchy–Green deformation tensor, \mathbf B^\,\!. This tensor has also been called the Piola tensor and the Finger tensor in the rheology and fluid dynamics literature. \mathbf=\mathbf B^=\mathbf F^\mathbf F^ \qquad \text \qquad c_=\frac \frac


Spectral representation

If there are three distinct principal stretches \lambda_i \,\!, the spectral decompositions of \mathbf and \mathbf is given by \mathbf = \sum_^3 \lambda_i^2 \mathbf_i \otimes \mathbf_i \qquad \text \qquad \mathbf = \sum_^3 \lambda_i^2 \mathbf_i \otimes \mathbf_i Furthermore, \mathbf U = \sum_^3 \lambda_i \mathbf N_i \otimes \mathbf N_i ~;~~ \mathbf V = \sum_^3 \lambda_i \mathbf n_i \otimes \mathbf n_i \mathbf R = \sum_^3 \mathbf n_i \otimes \mathbf N_i ~;~~ \mathbf F = \sum_^3 \lambda_i \mathbf n_i \otimes \mathbf N_i Observe that \mathbf = \mathbf~\mathbf~\mathbf^T = \sum_^3 \lambda_i~\mathbf~(\mathbf_i\otimes\mathbf_i)~\mathbf^T = \sum_^3 \lambda_i~(\mathbf~\mathbf_i)\otimes(\mathbf~\mathbf_i) Therefore, the uniqueness of the spectral decomposition also implies that \mathbf_i = \mathbf~\mathbf_i \,\!. The left stretch (\mathbf\,\!) is also called the ''spatial stretch tensor'' while the right stretch (\mathbf\,\!) is called the ''material stretch tensor''. The effect of \mathbf acting on \mathbf_i is to stretch the vector by \lambda_i and to rotate it to the new orientation \mathbf_i\,\!, i.e., \mathbf~\mathbf_i = \lambda_i~(\mathbf~\mathbf_i) = \lambda_i~\mathbf_i In a similar vein, \mathbf^~\mathbf_i = \cfrac~\mathbf_i ~;~~ \mathbf^T~\mathbf_i = \lambda_i~\mathbf_i ~;~~ \mathbf^~\mathbf_i = \cfrac~\mathbf_i ~.


Examples

; Uniaxial extension of an incompressible material : This is the case where a specimen is stretched in 1-direction with a stretch ratio of \mathbf\,\!. If the volume remains constant, the contraction in the other two directions is such that \mathbf or \mathbf\,\!. Then: \mathbf=\begin \alpha & 0 & 0 \\ 0 & \alpha^ & 0 \\ 0 & 0 & \alpha^ \end \mathbf = \mathbf = \begin \alpha^2 & 0 & 0 \\ 0 & \alpha^ & 0 \\ 0 & 0 & \alpha^ \end ; Simple shear :\mathbf=\begin 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end \mathbf = \begin 1+\gamma^2 & \gamma & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end \mathbf = \begin 1 & \gamma & 0 \\ \gamma & 1+\gamma^2 & 0 \\ 0 & 0 & 1 \end ; Rigid body rotation :\mathbf = \begin \cos \theta & \sin \theta & 0 \\ - \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end \mathbf = \mathbf = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end = \mathbf


Derivatives of stretch

Derivatives of the stretch with respect to the right Cauchy–Green deformation tensor are used to derive the stress-strain relations of many solids, particularly
hyperelastic material A hyperelastic or Green elastic materialR.W. Ogden, 1984, ''Non-Linear Elastic Deformations'', , Dover. is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density f ...
s. These derivatives are \cfrac = \cfrac~\mathbf_i\otimes\mathbf_i = \cfrac~\mathbf^T~(\mathbf_i\otimes\mathbf_i)~\mathbf ~;~~ i=1,2,3 and follow from the observations that \mathbf:(\mathbf_i\otimes\mathbf_i) = \lambda_i^2 ~;~~~~\cfrac = \mathsf^ ~;~~~~ \mathsf^:(\mathbf_i\otimes\mathbf_i)=\mathbf_i\otimes\mathbf_i.


Physical interpretation of deformation tensors

Let \mathbf = X^i~\boldsymbol_i be a Cartesian coordinate system defined on the undeformed body and let \mathbf = x^i~\boldsymbol_i be another system defined on the deformed body. Let a curve \mathbf(s) in the undeformed body be parametrized using s \in ,1/math>. Its image in the deformed body is \mathbf(\mathbf(s)). The undeformed length of the curve is given by l_X = \int_0^1 \left, \cfrac \~ds = \int_0^1 \sqrt~ds = \int_0^1 \sqrt~ds After deformation, the length becomes \begin l_x & = \int_0^1 \left, \cfrac \~ds = \int_0^1 \sqrt~ds = \int_0^1 \sqrt~ds \\ & = \int_0^1 \sqrt~ds \end Note that the right Cauchy–Green deformation tensor is defined as \boldsymbol := \boldsymbol^T\cdot\boldsymbol = \left(\cfrac\right)^T\cdot \cfrac Hence, l_x = \int_0^1 \sqrt~ds which indicates that changes in length are characterized by \boldsymbol.


Finite strain tensors

The concept of ''strain'' is used to evaluate how much a given displacement differs locally from a rigid body displacement. One of such strains for large deformations is the ''Lagrangian finite strain tensor'', also called the ''Green-Lagrangian strain tensor'' or ''Green – St-Venant strain tensor'', defined as \mathbf E=\frac(\mathbf C - \mathbf I)\qquad \text \qquad E_=\frac\left( \frac\frac-\delta_\right) or as a function of the displacement gradient tensor \mathbf E =\frac\left (\nabla_\mathbf u)^T + \nabla_\mathbf u + (\nabla_\mathbf u)^T \cdot\nabla_\mathbf u\right/math> or E_=\frac\left(\frac+\frac+\frac\frac\right) The Green-Lagrangian strain tensor is a measure of how much \mathbf C differs from \mathbf I\,\!. The ''Eulerian-Almansi finite strain tensor'', referenced to the deformed configuration, i.e. Eulerian description, is defined as \mathbf e=\frac(\mathbf I - \mathbf c)=\frac(\mathbf I - \mathbf B ^) \qquad \text \qquad e_ = \frac \left(\delta_ - \frac \frac\right) or as a function of the displacement gradients we have e_ = \frac \left(\frac + \frac - \frac \frac\right) d\mathbf=\mathbf F^ \, d\mathbf=\mathbf \,d\mathbf \qquad \text \qquad dX_M=\frac\, dx_n where \frac are the components of the ''spatial deformation gradient tensor'', \mathbf\,\!. Thus we have \begin d\mathbf^2 &= d\mathbf X \cdot d\mathbf X \\ &= \mathbf F^ \cdot d\mathbf x \cdot \mathbf F^ \cdot d\mathbf x \\ &= d\mathbf x \cdot \mathbf F^\mathbf F^ \cdot d\mathbf x \\ &= d\mathbf x\cdot\mathbf c\cdot d\mathbf x \end \qquad \text \qquad \begin (dX)^2&=dX_M\,dX_M \\ &= \frac\frac\,dx_r\,dx_s \\ &= c_\,dx_r\,dx_s \\ \end where the second order tensor c_ is called ''Cauchy's deformation tensor'', \mathbf c=\mathbf F^\mathbf F^\,\!. Then we have, \begin d\mathbf^2 - d\mathbf^2 &= d\mathbf x\cdot d\mathbf x-d\mathbf x\cdot\mathbf c\cdot d\mathbf x \\ &=d\mathbf x\cdot (\mathbf I - \mathbf c)\cdot d\mathbf x \\ &= d\mathbf x \cdot 2\mathbf e \cdot d\mathbf x \\ \end or \begin (dx)^2 - (dX)^2 &= dx_j\,dx_j-\frac\frac\,dx_r\,dx_s \\ &= \left(\delta_ - \frac\frac \right)\,dx_r\,dx_s \\ &=2e_\,dx_r\,dx_s \end where e_\,\!, are the components of a second-order tensor called the ''Eulerian-Almansi finite strain tensor'', \mathbf e=\frac(\mathbf I - \mathbf c) \qquad \text \qquad e_=\frac\left(\delta_ - \frac\frac \right) Both the Lagrangian and Eulerian finite strain tensors can be conveniently expressed in terms of the ''displacement gradient tensor''. For the Lagrangian strain tensor, first we differentiate the displacement vector \mathbf u(\mathbf X, t) with respect to the material coordinates X_M to obtain the ''material displacement gradient tensor'', \nabla_\mathbf u \begin \mathbf u(\mathbf X,t) &= \mathbf x(\mathbf X,t) - \mathbf X \\ \nabla_\mathbf u &= \mathbf F - \mathbf I \\ \mathbf F &= \nabla_\mathbf u + \mathbf I \\ \end \qquad \text \qquad \begin u_i& = x_i-\delta_X_J \\ \delta_U_J &= x_i-\delta_X_J \\ x_i&=\delta_\left(U_J+X_J\right) \\ \frac&=\delta_\left(\frac+\delta_\right) \\ \end Replacing this equation into the expression for the Lagrangian finite strain tensor we have \begin \mathbf E &= \frac\left(\mathbf F^T\mathbf F-\mathbf I\right) \\ &=\frac\left \left\\left( \nabla_\mathbf u+\mathbf I\right)-\mathbf I\right\\ &=\frac\left (\nabla_\mathbf u)^T + \nabla_\mathbf u + (\nabla_\mathbf u)^T \cdot\nabla_\mathbf u\right\\ \end or \begin E_&=\frac\left( \frac \frac - \delta_\right) \\ &=\frac \left delta_\left(\frac+\delta_\right)\delta_\left(\frac+\delta_\right)-\delta_\right\\ &=\frac\left delta_\left(\frac+\delta_\right)\left(\frac+\delta_\right)-\delta_\right\\ &=\frac\left left(\frac+\delta_\right)\left(\frac+\delta_\right)-\delta_\right\\ &=\frac\left(\frac +\frac +\frac \frac\right) \end Similarly, the Eulerian-Almansi finite strain tensor can be expressed as e_ = \frac \left(\frac +\frac - \frac \frac\right)


Seth–Hill family of generalized strain tensors

B. R. Seth from the
Indian Institute of Technology Kharagpur Indian Institute of Technology Kharagpur (IIT Kharagpur) is a public institute of technology established by the Government of India in Kharagpur, West Bengal, India. Established in 1951, the institute is the first of the IITs to be established ...
was the first to show that the Green and Almansi strain tensors are special cases of a more general strain measure. The idea was further expanded upon by
Rodney Hill Rodney Hill FRS (11 June 1921 – 2 February 2011) was an applied mathematician and a former Professor of Mechanics of Solids at Gonville and Caius College, Cambridge. Career In 1953 he was appointed Professor of Applied Mathematics at the ...
in 1968. The Seth–Hill family of strain measures (also called Doyle-Ericksen tensors)T.C. Doyle and J.L. Eriksen (1956). "Non-linear elasticity." ''Advances in Applied Mechanics'' 4, 53–115. can be expressed as \mathbf E_=\frac(\mathbf U^- \mathbf I) = \frac\left mathbf^ - \mathbf\right/math> For different values of m we have: * Green-Lagrangian strain tensor \mathbf E_ = \frac (\mathbf U^- \mathbf I) = \frac (\mathbf-\mathbf) * Biot strain tensor \mathbf E_ = (\mathbf U - \mathbf I) = \mathbf^-\mathbf * Logarithmic strain, Natural strain, True strain, or Hencky strain \mathbf E_ = \ln \mathbf U = \frac\,\ln\mathbf * Almansi strain \mathbf_ = \frac\left mathbf-\mathbf^\right/math> The second-order approximation of these tensors is \mathbf_ = \boldsymbol + (\nabla\mathbf)^T\cdot\nabla\mathbf - (1 - m) \boldsymbol^T\cdot\boldsymbol where \boldsymbol is the infinitesimal strain tensor. Many other different definitions of tensors \mathbf are admissible, provided that they all satisfy the conditions that:Z.P. Bažant and L. Cedolin (1991). ''Stability of Structures. Elastic, Inelastic, Fracture and Damage Theories.'' Oxford Univ. Press, New York (2nd ed. Dover Publ., New York 2003; 3rd ed., World Scientific 2010). * \mathbf vanishes for all rigid-body motions * the dependence of \mathbf on the displacement gradient tensor \nabla\mathbf is continuous, continuously differentiable and monotonic * it is also desired that \mathbf reduces to the infinitesimal strain tensor \boldsymbol as the norm , \nabla\mathbf, \to 0 An example is the set of tensors \mathbf^ = \left(^n - ^\right)/2n which do not belong to the Seth–Hill class, but have the same 2nd-order approximation as the Seth–Hill measures at m=0 for any value of n.Z.P. Bažant (1998).
Easy-to-compute tensors with symmetric inverse approximating Hencky finite strain and its rate
" ''Journal of Materials of Technology ASME'', 120 (April), 131–136.


Stretch ratio

The stretch ratio is a measure of the extensional or normal strain of a differential line element, which can be defined at either the undeformed configuration or the deformed configuration. The stretch ratio for the differential element d\mathbf X = dX \mathbf N (Figure) in the direction of the unit vector \mathbf N at the material point P\,\!, in the undeformed configuration, is defined as \Lambda_ = \frac where dx is the deformed magnitude of the differential element d\mathbf X\,\!. Similarly, the stretch ratio for the differential element d\mathbf x = dx\mathbf n (Figure), in the direction of the unit vector \mathbf n at the material point p\,\!, in the deformed configuration, is defined as \frac = \frac. The normal strain e_ in any direction \mathbf N can be expressed as a function of the stretch ratio, e_= \frac=\Lambda_- 1. This equation implies that the normal strain is zero, i.e. no deformation, when the stretch is equal to unity. Some materials, such as elastometers can sustain stretch ratios of 3 or 4 before they fail, whereas traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios, perhaps of the order of 1.1 (reference?)


Physical interpretation of the finite strain tensor

The diagonal components E_ of the Lagrangian finite strain tensor are related to the normal strain, e.g. E_=e_+\frac e_^2 where e_ is the normal strain or engineering strain in the direction \mathbf I_1\,\!. The off-diagonal components E_ of the Lagrangian finite strain tensor are related to shear strain, e.g. E_=\frac\sqrt\sqrt\sin\phi_ where \phi_ is the change in the angle between two line elements that were originally perpendicular with directions \mathbf I_1 and \mathbf I_2\,\!, respectively. Under certain circumstances, i.e. small displacements and small displacement rates, the components of the Lagrangian finite strain tensor may be approximated by the components of the infinitesimal strain tensor = \frac The square of the stretch ratio is defined as \Lambda_^2=\left (\frac\right )^2 Knowing that (dx)^2=C_dX_KdX_L we have \Lambda_^2 = C_ N_K N_L where N_K and N_L are unit vectors. The normal strain or engineering strain e_ in any direction \mathbf N can be expressed as a function of the stretch ratio, e_=\frac=\Lambda_-1 Thus, the normal strain in the direction \mathbf I_1 at the material point P may be expressed in terms of the stretch ratio as \begin e_=\frac&=\Lambda_-1\\ &=\sqrt -1=\sqrt-1\\ &=\sqrt-1 \end solving for E_ we have \begin 2E_&= \frac \\ E_&= \left(\frac\right)+ \frac \left(\frac\right)^2 \\ &=e_+\frace_^2 \end The ''shear strain'', or change in angle between two line elements d\mathbf X_1 and d\mathbf X_2 initially perpendicular, and oriented in the principal directions \mathbf I_1 and \mathbf I_2\,\!, respectively, can also be expressed as a function of the stretch ratio. From the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
between the deformed lines d\mathbf x_1 and d\mathbf x_2 we have \begin d\mathbf x_1 \cdot d\mathbf x_2 &=dx_1 dx_2 \cos\theta_ \\ \mathbf F \cdot d\mathbf X_1\cdot \mathbf F\cdot d\mathbf X_2&= \sqrt \cdot \sqrt \cos\theta_ \\ \frac &=\frac \cos\theta_\\ \mathbf I_1 \cdot \mathbf C \cdot \mathbf I_2&= \Lambda_\Lambda_\cos\theta_ \end where \theta_ is the angle between the lines d\mathbf x_1 and d\mathbf x_2 in the deformed configuration. Defining \phi_ as the shear strain or reduction in the angle between two line elements that were originally perpendicular, we have \phi_=\frac-\theta_ thus, \cos\theta_=\sin\phi_ then \mathbf I_1 \cdot \mathbf C \cdot \mathbf I_2= \Lambda_ \Lambda_\sin\phi_ or \begin C_&=\sqrt\sqrt\sin\phi_\\ 2E_+\delta_&=\sqrt\sqrt\sin\phi_\\ E_&=\frac\sqrt\sqrt\sin\phi_ \end


Deformation tensors in convected curvilinear coordinates

A representation of deformation tensors in
curvilinear coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
is useful for many problems in continuum mechanics such as nonlinear shell theories and large plastic deformations. Let \mathbf = \mathbf(\xi^1,\xi^2,\xi^3) denote the function by which a position vector in space is constructed from coordinates (\xi^1,\xi^2,\xi^3). The coordinates are said to be "convected" if they correspond to a one-to-one mapping to and from Lagrangian particles in a continuum body. If the coordinate grid is "painted" on the body in its initial configuration, then this grid will deform and flow with the motion of material to remain painted on the same material particles in the deformed configuration so that grid lines intersect at the same material particle in either configuration. The tangent vector to the deformed coordinate grid line curve \xi^i at \mathbf is given by \mathbf_i = \frac The three tangent vectors at \mathbf form a local basis. These vectors are related the reciprocal basis vectors by \mathbf_i\cdot\mathbf^j = \delta_i^j Let us define a second-order tensor field \boldsymbol (also called the metric tensor) with components g_ := \frac\cdot\frac = \mathbf_i\cdot\mathbf_j The Christoffel symbols of the first kind can be expressed as \Gamma_ = \tfrac \mathbf_i\cdot\mathbf_k)_ + (\mathbf_j\cdot\mathbf_k)_ - (\mathbf_i\cdot\mathbf_j)_ To see how the Christoffel symbols are related to the Right Cauchy–Green deformation tensor let us similarly define two bases, the already mentioned one that is tangent to deformed grid lines and another that is tangent to the undeformed grid lines. Namely, \mathbf_i := \frac ~;~~ \mathbf_i\cdot\mathbf^j = \delta_i^j ~;~~ \mathbf_i := \frac ~;~~ \mathbf_i\cdot\mathbf^j = \delta_i^j


The deformation gradient in curvilinear coordinates

Using the definition of the gradient of a vector field in curvilinear coordinates, the deformation gradient can be written as \boldsymbol = \boldsymbol_\mathbf = \frac\otimes\mathbf^i = \mathbf_i\otimes\mathbf^i


The right Cauchy–Green tensor in curvilinear coordinates

The right Cauchy–Green deformation tensor is given by \boldsymbol = \boldsymbol^T\cdot\boldsymbol = (\mathbf^i\otimes\mathbf_i)\cdot(\mathbf_j\otimes\mathbf^j) = (\mathbf_i\cdot\mathbf_j)(\mathbf^i\otimes\mathbf^j) If we express \boldsymbol in terms of components with respect to the basis we have \boldsymbol = C_~\mathbf^i\otimes\mathbf^j Therefore, C_ = \mathbf_i\cdot\mathbf_j = g_ and the corresponding Christoffel symbol of the first kind may be written in the following form. \Gamma_ = \tfrac _ + C_ - C_ = \tfrac \mathbf_i\cdot\boldsymbol\cdot\mathbf_k)_ + (\mathbf_j\cdot\boldsymbol\cdot\mathbf_k)_ - (\mathbf_i\cdot\boldsymbol\cdot\mathbf_j)_


Some relations between deformation measures and Christoffel symbols

Consider a one-to-one mapping from \mathbf = \ to \mathbf = \ and let us assume that there exist two positive-definite, symmetric second-order tensor fields \boldsymbol and \boldsymbol that satisfy G_ = \frac~\frac~g_ Then, \frac = \left(\frac~\frac + \frac~\frac\right)~g_ + \frac~\frac~\frac Noting that \frac = \frac~\frac and g_ = g_ we have \begin \frac & = \left(\frac~\frac + \frac~\frac\right)~g_ + \frac~\frac~\frac~\frac \\ \frac & = \left(\frac~\frac + \frac~\frac\right)~g_ + \frac~\frac~\frac~\frac \\ \frac & = \left(\frac~\frac + \frac~\frac\right)~g_ + \frac~\frac~\frac~\frac \end Define \begin _\Gamma_ & := \frac\left(\frac + \frac - \frac\right) \\ _\Gamma_ & := \frac\left(\frac + \frac - \frac\right) \\ \end Hence _\Gamma_ = \frac~\frac~\frac \,_\Gamma_ + \frac~\frac~g_ Define ^= _ ~;~~ ^= _ Then G^ = \frac~\frac~g^ Define the Christoffel symbols of the second kind as _\Gamma^m_ := G^ \,_\Gamma_ ~;~~ _\Gamma^\nu_ := g^ \,_\Gamma_ Then \begin _\Gamma^m_ & = G^~\frac~\frac~\frac \,_\Gamma_ + G^~\frac~\frac~g_ \\ & = \frac~\frac~g^~\frac~\frac~\frac \,_\Gamma_ + \frac~\frac~g^~\frac~\frac~g_ \\ & = \frac~\delta^\gamma_\rho~g^~\frac~\frac \,_\Gamma_ + \frac~\delta^\beta_\rho~g^~\frac~g_ \\ & = \frac~g^~\frac~\frac \,_\Gamma_ + \frac~g^~\frac~g_ \\ & = \frac~\frac~\frac \,_\Gamma^\nu_ + \frac~\delta^_~\frac \end Therefore, _\Gamma^m_ = \frac~\frac~\frac \,_\Gamma^\nu_ + \frac~\frac The invertibility of the mapping implies that \begin \frac\,_\Gamma^m_ & = \frac~\frac~\frac~\frac \,_\Gamma^\nu_ + \frac~\frac~\frac \\ & = \delta^\mu_\nu~\frac~\frac \,_\Gamma^\nu_ + \delta^\mu_\alpha~\frac \\ & = \frac~\frac \,_\Gamma^\mu_ + \frac \end We can also formulate a similar result in terms of derivatives with respect to x. Therefore, \begin \frac & = \frac\,_\Gamma^m_ - \frac~\frac \,_\Gamma^\mu_ \\ \frac & = \frac\,_\Gamma^\mu_ - \frac~\frac \,_\Gamma^m_ \end


Compatibility conditions

The problem of compatibility in continuum mechanics involves the determination of allowable single-valued continuous fields on bodies. These allowable conditions leave the body without unphysical gaps or overlaps after a deformation. Most such conditions apply to simply-connected bodies. Additional conditions are required for the internal boundaries of multiply connected bodies.


Compatibility of the deformation gradient

The necessary and sufficient conditions for the existence of a compatible \boldsymbol field over a simply connected body are \boldsymbol\times\boldsymbol = \boldsymbol


Compatibility of the right Cauchy–Green deformation tensor

The necessary and sufficient conditions for the existence of a compatible \boldsymbol field over a simply connected body are R^\gamma_ := \frac ,_\Gamma^\gamma_- \frac ,_\Gamma^\gamma_+ \,_\Gamma^\gamma_\,_\Gamma^\mu_ - \,_\Gamma^\gamma_\,_\Gamma^\mu_ = 0 We can show these are the mixed components of the Riemann–Christoffel curvature tensor. Therefore, the necessary conditions for \boldsymbol-compatibility are that the Riemann–Christoffel curvature of the deformation is zero.


Compatibility of the left Cauchy–Green deformation tensor

No general sufficiency conditions are known for the left Cauchy–Green deformation tensor in three-dimensions. Compatibility conditions for two-dimensional \boldsymbol fields have been found by Janet Blume.


See also

*
Infinitesimal strain In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimall ...
*
Compatibility (mechanics) In continuum mechanics, a compatible deformation (or strain) tensor field in a body is that ''unique'' tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. Compatibility is the study of ...
*
Curvilinear coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
* Piola–Kirchhoff stress tensor, the stress tensor for finite deformations. *
Stress measures In continuum mechanics, the most commonly used measure of stress is the Cauchy stress tensor, often called simply ''the'' stress tensor or "true stress". However, several alternative measures of stress can be defined: #The Kirchhoff stress (\boldsy ...
* Strain partitioning


References


Further reading

* * * * * * * * *{{Cite book , last = Rees , first = David , title = Basic Engineering Plasticity – An Introduction with Engineering and Manufacturing Applications , publisher = Butterworth-Heinemann , year = 2006 , url = https://books.google.com/books?id=4KWbmn_1hcYC , isbn = 0-7506-8025-3


External links


Prof. Amit Acharya's notes on compatibility on iMechanica
Tensors Continuum mechanics Elasticity (physics) Non-Newtonian fluids Solid mechanics