The skewsymmetrical part of the spatial velocity gradient, the vortex , spin or rotational speed tensor (symbol w or W ) has a dual vector, the angular velocity , which is proportional to the vortex vector or the vortex strength , which plays an important role in liquid and gas flows.
${\ displaystyle {\ vec {\ omega}} \ ,,}$
Definition and modes of representation
Material and spatial coordinates and the velocity field
The movement of a material point (fluid element) becomes mathematical with the movement function
 ${\ displaystyle {\ vec {x}} = {\ vec {\ chi}} ({\ vec {X}}, t)}$
described. The vector is the current position of the material point at the moment in the current configuration (lower case letters). The position of the material point under consideration is more precise in the initial or reference configuration of the body at a past time (capital letters). If the material point is fixed , the movement function reproduces its trajectory through space and if the spatial point is fixed , the stroke line through the point under consideration reproduces . In the Cartesian coordinate system with the standard basis , the point in space has the componentwise representation
${\ displaystyle {\ vec {x}}}$${\ displaystyle {\ vec {X}}}$${\ displaystyle t}$${\ displaystyle {\ vec {X}}}$${\ displaystyle t_ {0} \ leq t}$${\ displaystyle {\ vec {X}}}$${\ displaystyle {\ vec {x}}}$${\ displaystyle {\ vec {X}} = {\ vec {\ chi}} ^ { 1} ({\ vec {x}}, t)}$ ${\ displaystyle {\ hat {e}} _ {1,2,3}}$${\ displaystyle {\ vec {x}}}$
 ${\ displaystyle {\ vec {x}} = \ sum _ {i = 1} ^ {3} x_ {i} {\ hat {e}} _ {i} = \ sum _ {i = 1} ^ {3 } \ chi _ {i} ({\ vec {X}}, t) {\ hat {e}} _ {i}}$
and applies accordingly . The numbers are called spatial coordinates because they mark a point in space, and they are called material coordinates because they are attached to a material point. The movement function can be inverted at any time at any location
${\ displaystyle {\ vec {X}} = \ sum _ {i = 1} ^ {3} {X} _ {i} {\ hat {e}} _ {i}}$${\ displaystyle x_ {1,2,3}}$${\ displaystyle X_ {1,2,3}}$
 ${\ displaystyle {\ vec {x}} = {\ vec {\ chi}} ({\ vec {X}}, t) \ quad \ leftrightarrow \ quad {\ vec {X}} = {\ vec {\ chi }} ^ { 1} ({\ vec {x}}, t) \ ,,}$
because at one point in space there can only ever be one material point and one material point can only be in one place at a time. The derivative of the motion function with respect to time provides the velocity field:
 ${\ displaystyle {\ vec {x}} = {\ vec {\ chi}} ({\ vec {X}}, t) \ quad \ rightarrow \ quad {\ vec {v}} ({\ vec {x} }, t) = \ sum _ {i = 1} ^ {3} v_ {i} ({\ vec {x}}, t) {\ hat {e}} _ {i} = \ sum _ {i = 1} ^ {3} {\ frac {\ mathrm {D} \ chi _ {i} ({\ vec {X}}, t)} {\ mathrm {D} t}} {\ hat {e}} _ {i} = \ sum _ {i = 1} ^ {3} {\ dot {\ chi}} _ {i} ({\ vec {X}}, t) {\ hat {e}} _ {i} = {\ dot {\ vec {\ chi}}} ({\ vec {X}}, t) \ ,.}$
The material coordinates belong to the particle that is at the location at time t and whose speed is at that time . The velocity field is usually understood in terms of space, which is why it is only referred to here in the spatial representation with (for English velocity "speed"). On the far right is the material velocity field, which is calculated with the substantial time derivative of the motion function. The dot notation is used here exclusively for the substantial time derivative.
${\ displaystyle {\ vec {X}} = {\ vec {\ chi}} ^ { 1} ({\ vec {x}}, t)}$${\ displaystyle {\ vec {x}}}$${\ displaystyle {\ vec {v}} ({\ vec {x}}, t) = {\ dot {\ vec {\ chi}}} ({\ vec {X}}, t)}$${\ displaystyle {\ vec {v}}}$
Speed gradient and deformation gradient
The deformation gradient is the derivation of the movement according to the material coordinates:
 ${\ displaystyle \ mathbf {F}: = \ operatorname {GRAD} \ left ({\ vec {\ chi}} ({\ vec {X}}, t) \ right): = \ sum _ {i, j = 1} ^ {3} {\ frac {\ mathrm {d} \ chi _ {i} ({\ vec {X}}, t)} {\ mathrm {d} X_ {j}}} {\ hat {e }} _ {i} \ otimes {\ hat {e}} _ {j} =: {\ frac {\ mathrm {d} {\ vec {x}}} {\ mathrm {d} {\ vec {X} }}}}$
The arithmetic symbol " " forms the dyadic product and "GRAD" the material gradient with derivatives according to the material coordinates. The speed gradients arise from the substantial time derivative of the deformation gradient:
${\ displaystyle \ otimes}$
 ${\ displaystyle {\ begin {aligned} {\ dot {\ mathbf {F}}} = & {\ frac {\ mathrm {D}} {\ mathrm {D} t}} \ sum _ {i, j = 1 } ^ {3} {\ frac {\ mathrm {d} \ chi _ {i} ({\ vec {X}}, t)} {\ mathrm {d} X_ {j}}} {\ hat {e} } _ {i} \ otimes {\ hat {e}} _ {j} = \ sum _ {i, j = 1} ^ {3} {\ frac {\ mathrm {d} {\ dot {\ chi}} _ {i} ({\ vec {X}}, t)} {\ mathrm {d} X_ {j}}} {\ hat {e}} _ {i} \ otimes {\ hat {e}} _ { j} \\ = & \ sum _ {i, j = 1} ^ {3} {\ frac {\ mathrm {d} v_ {i} ({\ vec {x}}, t)} {\ mathrm {d } X_ {j}}} {\ hat {e}} _ {i} \ otimes {\ hat {e}} _ {j} = \ sum _ {i, j, k = 1} ^ {3} {\ frac {\ mathrm {d} v_ {i} ({\ vec {x}}, t)} {\ mathrm {d} x_ {k}}} {\ frac {\ mathrm {d} \ chi _ {k} ({\ vec {X}}, t)} {\ mathrm {d} X_ {j}}} {\ hat {e}} _ {i} \ otimes {\ hat {e}} _ {j} = \ underbrace {\ sum _ {i, k = 1} ^ {3} {\ frac {\ mathrm {d} v_ {i} ({\ vec {x}}, t)} {\ mathrm {d} x_ {k }}} {\ hat {e}} _ {i} \ otimes {\ hat {e}} _ {k}} _ {\ operatorname {grad} {\ vec {v}}} \ cdot \ underbrace {\ sum _ {j, l = 1} ^ {3} {\ frac {\ mathrm {d} \ chi _ {l} ({\ vec {X}}, t)} {\ mathrm {d} X_ {j}} } {\ hat {e}} _ {l} \ otimes {\ hat {e}} _ {j}} _ {= \ operatorname {GRAD} {\ vec {\ chi}}} \\ = & \ operatorname { degree} {\ b igl (} {\ vec {v}} ({\ vec {x}}, t) {\ bigl)} \ cdot \ operatorname {GRAD} {\ bigl (} {\ vec {\ chi}} ({\ vec {X}}, t) {\ bigl)} =: \ mathbf {l \ cdot F} \,. \ End {aligned}}}$
The arithmetic symbol “ ” forms the dyadic product , “grad” the spatial and “GRAD” the material gradient with derivatives according to the spatial or material coordinates. The material speed gradient is the time derivative of the deformation gradient or  because the order of the derivatives can be exchanged  the material derivative of the speed according to the material coordinates:
${\ displaystyle \ otimes}$ ${\ displaystyle {\ dot {\ mathbf {F}}}}$
 ${\ displaystyle {\ dot {\ mathbf {F}}} ({\ vec {X}}, t): = \ operatorname {GRAD} \ left ({\ dot {\ vec {\ chi}}} ({\ vec {X}}, t) \ right) = \ sum _ {i, j = 1} ^ {3} {\ frac {\ mathrm {d} {\ dot {\ chi}} _ {i}} {\ mathrm {d} X_ {j}}} {\ hat {e}} _ {i} \ otimes {\ hat {e}} _ {j} =: {\ frac {\ mathrm {d} {\ dot {\ vec {\ chi}}}} {\ mathrm {d} {\ vec {X}}}} = {\ frac {\ mathrm {d} {\ vec {v}}} {\ mathrm {d} {\ vec {x}}}} \ cdot {\ frac {\ mathrm {d} {\ vec {x}}} {\ mathrm {d} {\ vec {X}}}} = \ mathbf {l \ cdot F} \ ,.}$
Spatial velocity gradient
The spatial velocity gradient is the spatial derivative of the velocity according to the spatial coordinates:
${\ displaystyle \ mathbf {l}}$
 ${\ displaystyle \ mathbf {l} ({\ vec {x}}, t): = \ operatorname {grad} {\ bigl (} {\ vec {v}} ({\ vec {x}}, t) { \ bigr)}: = \ sum _ {i, j = 1} ^ {3} {\ frac {\ mathrm {d} v_ {i}} {\ mathrm {d} x_ {j}}} {\ hat { e}} _ {i} \ otimes {\ hat {e}} _ {j} =: {\ frac {\ mathrm {d} {\ vec {v}}} {\ mathrm {d} {\ vec {x }}}} = {\ frac {\ mathrm {d} {\ dot {\ vec {\ chi}}}} {\ mathrm {d} {\ vec {X}}}} \ cdot {\ frac {\ mathrm {d} {\ vec {X}}} {\ mathrm {d} {\ vec {x}}}} = {\ dot {\ mathbf {F}}} \ cdot \ mathbf {F} ^ { 1} \ ,.}$
The speed field is mostly represented in three dimensions, which is why the term “speed gradient” usually means the spatial speed gradient. In continuum mechanics, material quantities are usually capitalized and spatial variables are small, which is why the lowercase of the spatial velocity gradient is also used here. Its symmetrical part
 ${\ displaystyle \ mathbf {d}: = {\ frac {1} {2}} (\ mathbf {l + l} ^ {\ top}) = {\ frac {1} {2}} {\ begin {pmatrix } 2 {\ frac {\ partial v_ {x}} {\ partial x}} & {\ frac {\ partial v_ {x}} {\ partial y}} + {\ frac {\ partial v_ {y}} { \ partial x}} & {\ frac {\ partial v_ {x}} {\ partial z}} + {\ frac {\ partial v_ {z}} {\ partial x}} \\ {\ frac {\ partial v_ {y}} {\ partial x}} + {\ frac {\ partial v_ {x}} {\ partial y}} & 2 {\ frac {\ partial v_ {y}} {\ partial y}} & {\ frac {\ partial v_ {y}} {\ partial z}} + {\ frac {\ partial v_ {z}} {\ partial y}} \\ {\ frac {\ partial v_ {z}} {\ partial x} } + {\ frac {\ partial v_ {x}} {\ partial z}} & {\ frac {\ partial v_ {z}} {\ partial y}} + {\ frac {\ partial v_ {y}} { \ partial z}} & 2 {\ frac {\ partial v_ {z}} {\ partial z}} \ end {pmatrix}}}$
is the (spatial) strain rate tensor and its skewsymmetric part
 ${\ displaystyle \ mathbf {w}: = {\ frac {1} {2}} (\ mathbf {ll} ^ {\ top}) = {\ frac {1} {2}} {\ begin {pmatrix} 0 & {\ frac {\ partial v_ {x}} {\ partial y}}  {\ frac {\ partial v_ {y}} {\ partial x}} & {\ frac {\ partial v_ {x}} {\ partial z}}  {\ frac {\ partial v_ {z}} {\ partial x}} \\ {\ frac {\ partial v_ {y}} {\ partial x}}  {\ frac {\ partial v_ {x }} {\ partial y}} & 0 & {\ frac {\ partial v_ {y}} {\ partial z}}  {\ frac {\ partial v_ {z}} {\ partial y}} \\ {\ frac { \ partial v_ {z}} {\ partial x}}  {\ frac {\ partial v_ {x}} {\ partial z}} & {\ frac {\ partial v_ {z}} {\ partial y}}  {\ frac {\ partial v_ {y}} {\ partial z}} & 0 \ end {pmatrix}}}$
is the (spatial) spin, eddy or rotational speed tensor. The superscript marks the transposition . In the matrix representations, the speed components relate to a Cartesian coordinate system with x, y and z directions.
${\ displaystyle \ top}$${\ displaystyle v_ {x, y, z}}$
The angular velocity or vortex strength
The vortex tensor, because it is skew symmetric , can be a dual vector with the property
${\ displaystyle {\ vec {\ omega}} \ ,,}$
 ${\ displaystyle {\ vec {\ omega}} \ times {\ vec {u}} = \ mathbf {w} \ cdot {\ vec {u}} \ quad {\ text {for all}} \ quad {\ vec {u}} \ quad \ Leftrightarrow \ quad \ mathbf {w} = {\ vec {\ omega}} \ times \ mathbf {1} = \ sum _ {i = 1} ^ {3} {\ vec {\ omega }} \ times {\ hat {e}} _ {i} \ otimes {\ hat {e}} _ {i} \ ,,}$
be assigned. The tensor 1 is the unit tensor , “ ” the dyadic and “×” the cross product . In the case of the vortex tensor, the dual vector is the angular velocity , which is the rotational velocity vector in the case of rigid body movements, as the section of the same name explains below. The angular velocity is calculated with the Nabla operator${\ displaystyle \ otimes}$
 ${\ displaystyle \ nabla: = \ sum _ {k = 1} ^ {3} {\ hat {e}} _ {k} {\ frac {\ partial} {\ partial x_ {k}}}}$
according to the regulation
 ${\ displaystyle {\ vec {\ omega}} =  {\ frac {1} {2}} \ mathbf {1 \ cdot \! \! \ times w} =  {\ frac {1} {2}} \ mathbf {1} \ cdot \! \ times {\ frac {1} {2}} {\ bigl [} \ overbrace {(\ nabla \ otimes {\ vec {v}}) ^ {\ top}} ^ {= \ operatorname {grad} {\ vec {v}} = \ mathbf {l}}  \ overbrace {\ nabla \ otimes {\ vec {v}}} ^ {\ mathbf {l} ^ {\ top}} {\ bigr]}: =  {\ frac {1} {4}} ( \ nabla \ times {\ vec {v}}  \ nabla \ times {\ vec {v}}) = {\ frac {1} { 2}} \ nabla \ times {\ vec {v}} = {\ frac {1} {2}} \ operatorname {red} ({\ vec {v}}) \ ,,}$
because the scalar cross product “ ” of the unit tensor with a dyad exchanges the dyadic product with the cross product. The differential operator "red" stands for the rotation of the velocity field.
${\ displaystyle \ cdot \! \ times}$
The angular velocity is proportional to the vortex strength , which is of particular importance in liquid and gas flows.
Representation in cylinder and spherical coordinates
In axially symmetrical flows it is advisable to use a cylindrical or spherical coordinate system. In cylindrical coordinates {ρ, φ, z} with basis vectors it gets the form:
${\ displaystyle {\ hat {e}} _ {\ rho, \ varphi, z}}$
 ${\ displaystyle {\ begin {aligned} \ operatorname {grad} {\ vec {v}} = & {\ hat {e}} _ {\ rho} \ otimes (\ operatorname {grad} v _ {\ rho}) + {\ frac {v _ {\ rho}} {\ rho}} {\ hat {e}} _ {\ varphi} \ otimes {\ hat {e}} _ {\ varphi} + {\ hat {e}} _ {\ varphi} \ otimes (\ operatorname {grad} v _ {\ varphi})  {\ frac {v _ {\ varphi}} {\ rho}} {\ hat {e}} _ {\ rho} \ otimes {\ hat {e}} _ {\ varphi} + {\ hat {e}} _ {z} \ otimes (\ operatorname {grad} v_ {z}) \\ = & {\ begin {pmatrix} {\ frac {\ partial v _ {\ rho}} {\ partial \ rho}} & {\ frac {1} {\ rho}} {\ frac {\ partial v _ {\ rho}} {\ partial \ varphi}}  {\ frac { v _ {\ varphi}} {\ rho}} & {\ frac {\ partial v _ {\ rho}} {\ partial z}} \\ {\ frac {\ partial v _ {\ varphi}} {\ partial \ rho} } & {\ frac {v _ {\ rho}} {\ rho}} + {\ frac {1} {\ rho}} {\ frac {\ partial v _ {\ varphi}} {\ partial \ varphi}} & { \ frac {\ partial v _ {\ varphi}} {\ partial z}} \\ {\ frac {\ partial v_ {z}} {\ partial \ rho}} & {\ frac {1} {\ rho}} { \ frac {\ partial v_ {z}} {\ partial \ varphi}} & {\ frac {\ partial v_ {z}} {\ partial z}} \ end {pmatrix}} _ {{\ hat {e}} _ {\ rho, \ varphi, z} \ otimes {\ hat {e}} _ {\ rho, \ varph i, z}} \\ {\ text {with}} \ quad \ operatorname {grad} f = & {\ frac {\ partial f} {\ partial \ rho}} {\ hat {e}} _ {\ rho } + {\ frac {1} {\ rho}} {\ frac {\ partial f} {\ partial \ varphi}} {\ hat {e}} _ {\ varphi} + {\ frac {\ partial f} { \ partial z}} {\ hat {e}} _ {z} \,. \ end {aligned}}}$
In spherical coordinates {r, θ, φ} with basis vectors , he writes:
${\ displaystyle {\ hat {e}} _ {r, \ theta, \ varphi}}$
 ${\ displaystyle {\ begin {aligned} \ operatorname {grad} {\ vec {v}} = & {\ hat {e}} _ {r} \ otimes (\ operatorname {grad} v_ {r}) + {\ frac {v_ {r}} {r}} \ mathbf {1}  {\ frac {v_ {r}} {r}} {\ hat {e}} _ {r} \ otimes {\ hat {e}} _ {r} + {\ hat {e}} _ {\ theta} \ otimes (\ operatorname {grad} v _ {\ theta}) + {\ frac {v _ {\ theta}} {r \ tan \ theta}} {\ hat {e}} _ {\ varphi} \ otimes {\ hat {e}} _ {\ varphi}  {\ frac {v _ {\ theta}} {r}} {\ hat {e}} _ { r} \ otimes {\ hat {e}} _ {\ theta} \\ & + {\ hat {e}} _ {\ varphi} \ otimes (\ operatorname {grad} v _ {\ varphi})  {\ frac {v _ {\ varphi}} {r \ tan \ theta}} {\ hat {e}} _ {\ theta} \ otimes {\ hat {e}} _ {\ varphi}  {\ frac {v _ {\ varphi }} {r}} {\ hat {e}} _ {r} \ otimes {\ hat {e}} _ {\ varphi} \\ = & {\ begin {pmatrix} {\ frac {\ partial v_ {r }} {\ partial r}} & {\ frac {1} {r}} {\ frac {\ partial v_ {r}} {\ partial \ theta}}  {\ frac {v _ {\ theta}} {r }} & {\ frac {1} {r \ sin \ theta}} {\ frac {\ partial v_ {r}} {\ partial \ varphi}}  {\ frac {v _ {\ varphi}} {r}} \\ {\ frac {\ partial v _ {\ theta}} {\ partial r}} & {\ frac {1} {r}} {\ frac {\ partial v _ {\ theta}} {\ partial \ theta}} + {\ frac {v_ {r}} {r} } & {\ frac {1} {r \ sin \ theta}} {\ frac {\ partial v _ {\ theta}} {\ partial \ varphi}}  {\ frac {v _ {\ varphi}} {r \ tan \ theta}} \\ {\ frac {\ partial v _ {\ varphi}} {\ partial r}} & {\ frac {1} {r}} {\ frac {\ partial v _ {\ varphi}} {\ partial \ theta}} & {\ frac {1} {r \ sin \ theta}} {\ frac {\ partial v _ {\ varphi}} {\ partial \ varphi}} + {\ frac {v_ {r}} {r }} + {\ frac {v _ {\ theta}} {r \ tan \ theta}} \ end {pmatrix}} _ {{\ hat {e}} _ {r, \ theta, \ varphi} \ otimes {\ hat {e}} _ {r, \ theta, \ varphi}} \\ {\ text {with}} \ quad \ operatorname {grad} f = & {\ frac {\ partial f} {\ partial r}} { \ hat {e}} _ {r} + {\ frac {1} {r}} {\ frac {\ partial f} {\ partial \ theta}} {\ hat {e}} _ {\ theta} + { \ frac {1} {r \ sin \ theta}} {\ frac {\ partial f} {\ partial \ varphi}} {\ hat {e}} _ {\ varphi} \,. \ end {aligned}}}$
Representation in convective coordinates
Coordinate lines applied to a body follow the deformations of the body
Convective coordinates are curvilinear coordinate systems that are bound to a body and are carried along by all deformations that the body experiences, see picture. Convective coordinate systems are used in the kinematics of slim or thinwalled structures (e.g. rods or shells ). Material preferred directions of nonisotropic materials, such as B. of wood, can be described in convective coordinates. Expressed in convective coordinates, the velocity gradients have particularly simple representations.
Each material point is assigned onetoone convective coordinates via a reference configuration . The tangent vectors
${\ displaystyle {\ vec {X}}}$ ${\ displaystyle {\ vec {\ Theta}} = (\ Theta _ {1}, \ Theta _ {2}, \ Theta _ {3})}$
 ${\ displaystyle {\ vec {G}} _ {i}: = {\ dfrac {\ mathrm {d} {\ vec {X}} ({\ vec {\ Theta}})} {\ mathrm {d} \ Theta _ {i}}} \ quad {\ textsf {or}} \ quad {\ vec {g}} _ {i}: = {\ dfrac {\ mathrm {d} {\ vec {\ chi}} \ left ({\ vec {X}} ({\ vec {\ Theta}}), t \ right)} {\ mathrm {d} \ Theta _ {i}}} = {\ dfrac {\ mathrm {d} { \ vec {\ chi}}} {\ mathrm {d} {\ vec {X}}}} \ cdot {\ dfrac {\ mathrm {d} {\ vec {X}}} {\ mathrm {d} \ Theta _ {i}}} = \ mathbf {F} \ cdot {\ vec {G}} _ {i}}$
then form covariant bases in point or . The gradients of the convective coordinates
${\ displaystyle {\ vec {X}}}$${\ displaystyle {\ vec {x}} = {\ vec {\ chi}} ({\ vec {X}}, t)}$
 ${\ displaystyle {\ vec {G}} ^ {i}: = \ operatorname {GRAD} \ Theta _ {i}: = \ sum _ {j = 1} ^ {3} {\ dfrac {\ mathrm {d} \ Theta _ {i}} {\ mathrm {d} X_ {j}}} {\ hat {e}} _ {j} =: {\ frac {\ mathrm {d} \ Theta _ {i}} {\ mathrm {d} {\ vec {X}}}}}$
or.
 ${\ displaystyle {\ vec {g}} ^ {i}: = \ operatorname {grad} \ Theta _ {i}: = \ sum _ {j = 1} ^ {3} {\ frac {\ mathrm {d} \ Theta _ {i}} {\ mathrm {d} x_ {j}}} {\ hat {e}} _ {j} =: {\ frac {\ mathrm {d} \ Theta _ {i}} {\ mathrm {d} {\ vec {x}}}} = {\ frac {\ mathrm {d} \ Theta _ {i}} {\ mathrm {d} {\ vec {X}}}} \ cdot {\ frac {\ mathrm {d} {\ vec {X}}} {\ mathrm {d} {\ vec {x}}}} = {\ vec {G}} ^ {i} \ cdot \ mathbf {F} ^ { 1} = \ mathbf {F} ^ {\ top 1} \ cdot {\ vec {G}} ^ {i}}$
form the contravariant bases that are dual to the covariants . Expressed in these basic systems, the deformation gradient takes on the particularly simple form
 ${\ displaystyle \ mathbf {F} = \ sum _ {i = 1} ^ {3} {\ vec {g}} _ {i} \ otimes {\ vec {G}} ^ {i} \ quad \ rightarrow \ quad \ mathbf {F} ^ { 1} = \ sum _ {i = 1} ^ {3} {\ vec {G}} _ {i} \ otimes {\ vec {g}} ^ {i} \, .}$
The time derivative of the deformation gradient and the time derivative of the inverse result
 ${\ displaystyle {\ dot {\ mathbf {F}}} = \ sum _ {i = 1} ^ {3} {\ dot {\ vec {g}}} _ {i} \ otimes {\ vec {G} } ^ {i} \ ,, \ quad (\ mathbf {F} ^ { 1}) {\ dot {}} = \ sum _ {i = 1} ^ {3} {\ vec {G}} _ { i} \ otimes {\ dot {\ vec {g}}} ^ {i} \ ,,}$
because the initial configuration and the basis vectors defined in it do not depend on time. With these results, the spatial velocity gradient is written:
 ${\ displaystyle {\ begin {aligned} \ mathbf {l} = & {\ dot {\ mathbf {F}}} \ cdot \ mathbf {F} ^ { 1} = \ sum _ {i = 1} ^ { 3} {\ dot {\ vec {g}}} _ {i} \ otimes {\ vec {g}} ^ {i} \\ = & \ sum _ {i = 1} ^ {3} {\ dot { \ vec {g}}} _ {i} \ otimes {\ vec {g}} ^ {i}  \ underbrace {{\ frac {\ mathrm {D}} {\ mathrm {D} t}} \ left ( \ sum _ {i = 1} ^ {3} {\ vec {g}} _ {i} \ otimes {\ vec {g}} ^ {i} \ right)} _ {= {\ dot {\ mathbf { 1}}} = \ mathbf {0}} = \ sum _ {i = 1} ^ {3} ({\ dot {\ vec {g}}} _ {i} \ otimes {\ vec {g}} ^ {i}  {\ dot {\ vec {g}}} _ {i} \ otimes {\ vec {g}} ^ {i}  {\ vec {g}} _ {i} \ otimes {\ dot { \ vec {g}}} ^ {i}) =  \ sum _ {i = 1} ^ {3} {\ vec {g}} _ {i} \ otimes {\ dot {\ vec {g}}} ^ {i} \ ,, \ end {aligned}}}$
wherein the disappearance of the time derivative of the unit tensor 1 was exploited. The velocity gradients map the basis vectors to their rates:
 ${\ displaystyle {\ begin {aligned} {\ dot {\ vec {g}}} _ {i} = & {\ dot {\ mathbf {F}}} \ cdot {\ vec {G}} _ {i} \ ,, \ quad {\ dot {\ vec {g}}} ^ {i} = (\ mathbf {F} ^ {\ top 1}) {\ dot {}} \ cdot {\ vec {G}} ^ {i} \\ {\ dot {\ vec {g}}} _ {i} = & \ mathbf {l} \ cdot {\ vec {g}} _ {i} \ ,, \ quad {\ dot { \ vec {g}}} ^ {i} =  \ mathbf {l} ^ {\ top} \ cdot {\ vec {g}} ^ {i}. \ end {aligned}}}$
The symmetrical part of the spatial velocity gradient is the strain velocity tensor:
 ${\ displaystyle {\ begin {aligned} \ mathbf {d} = {\ dfrac {1} {2}} (\ mathbf {l + l} ^ {\ top}) = & {\ frac {1} {2} } \ sum _ {j = 1} ^ {3} {\ dot {\ vec {g}}} _ {j} \ otimes {\ vec {g}} ^ {j} + {\ frac {1} {2 }} \ sum _ {i = 1} ^ {3} {\ vec {g}} ^ {i} \ otimes {\ dot {\ vec {g}}} _ {i} = {\ frac {1} { 2}} \ sum _ {i, j = 1} ^ {3} \ left ({\ vec {g}} _ {i} \ cdot {\ dot {\ vec {g}}} _ {j} + { \ dot {\ vec {g}}} _ {i} \ cdot {\ vec {g}} _ {j} \ right) {\ vec {g}} ^ {i} \ otimes {\ vec {g}} ^ {j} \\ = &  {\ frac {1} {2}} \ sum _ {i = 1} ^ {3} {\ vec {g}} _ {i} \ otimes {\ dot {\ vec {g}}} ^ {i}  {\ frac {1} {2}} \ sum _ {j = 1} ^ {3} {\ dot {\ vec {g}}} ^ {j} \ otimes { \ vec {g}} _ {j} =  {\ frac {1} {2}} \ sum _ {i, j = 1} ^ {3} \ left ({\ dot {\ vec {g}}} ^ {i} \ cdot {\ vec {g}} ^ {j} + {\ vec {g}} ^ {i} \ cdot {\ dot {\ vec {g}}} ^ {j} \ right) { \ vec {g}} _ {i} \ otimes {\ vec {g}} _ {j} \,. \ end {aligned}}}$
With the metric coefficients and as well as the product rule , this is written:
${\ displaystyle g_ {ij}: = {\ vec {g}} _ {i} \ cdot {\ vec {g}} _ {j}}$${\ displaystyle g ^ {ij}: = {\ vec {g}} ^ {i} \ cdot {\ vec {g}} ^ {j}}$
 ${\ displaystyle \ mathbf {d} = {\ frac {1} {2}} \ sum _ {i, j = 1} ^ {3} {\ dot {g}} _ {ij} {\ vec {g} } ^ {i} \ otimes {\ vec {g}} ^ {j} =  {\ frac {1} {2}} \ sum _ {i, j = 1} ^ {3} {\ dot {g} } ^ {ij} {\ vec {g}} _ {i} \ otimes {\ vec {g}} _ {j} \ ,.}$
The Frobenius scalar products remain unchanged with a rotation or translation, which is why the strain velocity tensor vanishes precisely then, namely with rigid body movements.
${\ displaystyle {\ vec {g}} _ {i} \ cdot {\ vec {g}} _ {j}}$
Geometric linearization
In solid mechanics, only small deformations occur in many areas of application. In this case, the equations of continuum mechanics experience a considerable simplification through geometric linearization. For this purpose, the displacements that a material point experiences in the course of its movement are considered. Because the current position is the point that had the position in the initial configuration , the displacement is the difference
${\ displaystyle {\ vec {u}} ({\ vec {X}}, t)}$${\ displaystyle {\ vec {X}}}$${\ displaystyle {\ vec {x}} = {\ vec {\ chi}} ({\ vec {X}}, t)}$${\ displaystyle {\ vec {X}}}$
 ${\ displaystyle {\ vec {u}} = {\ vec {\ chi}} ({\ vec {X}}, t)  {\ vec {X}} \ ,.}$
The material gradient of the displacements is the tensor
 ${\ displaystyle \ mathbf {H} = \ mathrm {GRAD} \, {\ vec {u}} = \ mathrm {GRAD} \, {\ vec {\ chi}}  \ mathrm {GRAD} \, {\ vec {X}} = \ mathbf {F}  \ mathbf {1} \ ,.}$
and is called the displacement gradient. It differs from the deformation gradient only through the unit tensor 1 . If is a characteristic dimension of the body, then both and and here are required for small displacements , so that all terms that contain higher powers of or can be neglected. Then:
${\ displaystyle L_ {0}}$${\ displaystyle  {\ vec {u}}  \ ll L_ {0}}$${\ displaystyle \ parallel \ mathbf {H} \ parallel \ ll 1}$${\ displaystyle \ parallel {\ dot {\ mathbf {H}}} \ parallel \ ll 1 / s}$${\ displaystyle {\ vec {u}}, \, \ mathbf {H}}$${\ displaystyle {\ dot {\ mathbf {H}}}}$
 ${\ displaystyle {\ begin {aligned} \ mathbf {l} = & {\ dot {\ mathbf {F}}} \ cdot \ mathbf {F} ^ { 1} \ approx {\ dot {\ mathbf {H} }} \ cdot (\ mathbf {1H}) \ approx {\ dot {\ mathbf {H}}} = {\ dot {\ mathbf {F}}} \\\ mathbf {d} \ approx & {\ frac {1} {2}} ({\ dot {\ mathbf {H}}} + {\ dot {\ mathbf {H}}} ^ {\ top}) = {\ dot {\ boldsymbol {\ varepsilon}} } \\\ mathbf {w} \ approx & {\ frac {1} {2}} ({\ dot {\ mathbf {H}}}  {\ dot {\ mathbf {H}}} ^ {\ top} ) = {\ dot {\ mathbf {R}}} _ {L} \,. \ end {aligned}}}$
The tensor is the linearized strain tensor and R _{L} is the linearized rotation tensor . A distinction between the material and spatial velocity gradients is therefore not necessary for small deformations.
${\ displaystyle {\ boldsymbol {\ varepsilon}}}$_{}
Transformation properties
Line, area and volume elements
The spatial velocity gradient transforms the line ,
surface and volume elements into their rates in the current configuration :
 ${\ displaystyle {\ begin {aligned} (\ mathrm {d} {\ vec {x}}) {\ dot {}} & = \ mathbf {l} \ cdot \ mathrm {d} {\ vec {x}} \\ (\ mathrm {d} {\ vec {a}}) {\ dot {}} & = (\ operatorname {Sp} (\ mathbf {l}) \ mathbf {1}  \ mathbf {l} ^ { \ top}) \ cdot \ mathrm {d} {\ vec {a}} \\ (\ mathrm {d} v) {\ dot {}} & = \ operatorname {Sp} (\ mathbf {l}) \ mathrm {d} v = \ operatorname {div} ({\ vec {v}}) \, \ mathrm {d} v \,. \ end {aligned}}}$
In it is (for English area “surface”) the vector surface element and (for English volume “volume”) the volume element. The operator calculates the trace of his argument, which in the case of the velocity gradient is the divergence of the velocity field:
${\ displaystyle \ mathrm {d} {\ vec {a}}}$ ${\ displaystyle \ mathrm {d} v}$ ${\ displaystyle \ operatorname {Sp}}$
 ${\ displaystyle \ operatorname {Sp} (\ mathbf {l}) = \ operatorname {Sp} (\ mathbf {d}) = \ operatorname {div} ({\ vec {v}}) \ ,.}$
proof

The deformation gradient F transforms the line, surface and volume elements from the reference configuration to the current configuration:
${\ displaystyle {\ begin {aligned} \ mathrm {d} {\ vec {x}} = & \ mathbf {F} \ cdot \ mathrm {d} {\ vec {X}} \\\ mathrm {d} { \ vec {a}} = & \ operatorname {det} (\ mathbf {F}) \ mathbf {F} ^ {\ top 1} \ cdot \ mathrm {d} {\ vec {A}} \\\ mathrm {d} v = & \ operatorname {det} (\ mathbf {F}) \; \ mathrm {d} V \,. \ end {aligned}}}$
The operator forms the determinant and the transposed inverse . The surface of the body in the reference configuration has the surface element , i.e. H. the normal of the patch multiplied by the patch, and the same applies to the spatial surface element on the surface of the body in the current configuration. Material time derivative (with retained particles) provides for the line element:
The material time derivative of the volume element results from the derivation of the determinant from
The colon ":" stands for the Frobenius scalar product of tensors, which is defined for two tensors A and B via . The gradient of a vector field is defined with the Nabla operator and the dyadic product “ ”: The trace of a dyadic product is the scalar product of its factors: because the transposition has no influence on the trace. The scalar product of the Nabla operator with the velocity field is its divergence: So the trace of the velocity gradient is equal to the divergence of the velocity field.
In conclusion, nor the material time derivative of the surface element calculated using the product rule to
where the constancy of was Einheitstensors used:${\ displaystyle \ operatorname {det} (\ cdot)}$${\ displaystyle (\ cdot) ^ {\ top 1}}$ ${\ displaystyle \ mathrm {d} {\ vec {A}}}$${\ displaystyle \ mathrm {d} A}$${\ displaystyle {\ vec {N}}}$${\ displaystyle \ mathrm {d} {\ vec {a}}}$
${\ displaystyle {\ frac {\ mathrm {D}} {\ mathrm {D} t}} (\ mathrm {d} {\ vec {x}}) = {\ frac {\ mathrm {D}} {\ mathrm {D} t}} (\ mathbf {F} \ cdot \ mathrm {d} {\ vec {X}}) = {\ dot {\ mathbf {F}}} \ cdot \ mathrm {d} {\ vec { X}} = {\ dot {\ mathbf {F}}} \ cdot \ mathbf {F} ^ { 1} \ cdot \ mathrm {d} {\ vec {x}} = \ mathbf {l} \ cdot \ mathrm {d} {\ vec {x}} \ ,.}$
${\ displaystyle {\ begin {aligned} {\ frac {\ mathrm {D}} {\ mathrm {D} t}} (\ mathrm {d} v) = & {\ frac {\ mathrm {D}} {\ mathrm {D} t}} [\ operatorname {det} (\ mathbf {F}) \ mathrm {d} V] = \ operatorname {det} (\ mathbf {F}) (\ mathbf {F} ^ {\ top 1}: {\ dot {\ mathbf {F}}}) \ mathrm {d} V = \ operatorname {Sp} (\ mathbf {F} ^ { 1} \ cdot {\ dot {\ mathbf {F} }}) \ mathrm {d} v = \ operatorname {Sp} ({\ dot {\ mathbf {F}}} \ cdot \ mathbf {F} ^ { 1}) \ mathrm {d} v = \ operatorname { Sp} (\ mathbf {l}) \ mathrm {d} v \,. \ End {aligned}}}$ ${\ displaystyle \ mathbf {A: B}: = \ operatorname {Sp} (\ mathbf {A ^ {\ top} \ cdot B})}$${\ displaystyle \ otimes}$${\ displaystyle \ operatorname {grad} {\ vec {v}}: = (\ nabla \ otimes {\ vec {v}}) ^ {\ top} \ ,.}$${\ displaystyle \ operatorname {Sp (grad} {\ vec {v}}) = \ nabla \ cdot {\ vec {v}} \ ,,}$${\ displaystyle \ operatorname {div} {\ vec {v}}: = \ nabla \ cdot {\ vec {v}} \ ,.}$
${\ displaystyle {\ begin {aligned} {\ frac {\ mathrm {D}} {\ mathrm {D} t}} (\ mathrm {d} {\ vec {a}}) = & {\ frac {\ mathrm {D}} {\ mathrm {D} t}} (\ operatorname {det} (\ mathbf {F}) \ mathbf {F} ^ {\ top 1} \ cdot \ mathrm {d} {\ vec {A }}) = [\ operatorname {det} (\ mathbf {F}) (\ mathbf {F} ^ {\ top 1}: {\ dot {\ mathbf {F}}}) \ mathbf {F} ^ { \ top 1}  \ operatorname {det} (\ mathbf {F}) \ mathbf {F} ^ {\ top 1} \ cdot {\ dot {\ mathbf {F}}} ^ {\ top} \ cdot \ mathbf {F} ^ {\ top 1}] \ cdot \ mathrm {d} {\ vec {A}} \\ = & [\ operatorname {Sp} (\ mathbf {F} ^ { 1} \ cdot {\ dot {\ mathbf {F}}}) \ mathbf {1}  \ mathbf {F} ^ {\ top 1} \ cdot {\ dot {\ mathbf {F}}} ^ {\ top}] \ cdot \ operatorname {det} (\ mathbf {F}) \ mathbf {F} ^ {\ top 1} \ cdot \ mathrm {d} {\ vec {A}} = [\ operatorname {Sp} ({\ dot {\ mathbf {F}}} \ cdot \ mathbf {F} ^ { 1}) \ mathbf {1}  ({\ dot {\ mathbf {F}}} \ cdot \ mathbf {F} ^ { 1 }) ^ {\ top}] \ cdot \ mathrm {d} {\ vec {a}} \\ = & [\ operatorname {Sp} (\ mathbf {l}) \ mathbf {1}  \ mathbf {l} ^ {\ top}] \ cdot \ mathrm {d} {\ vec {a}} \,. \ end {aligned}}}$
${\ displaystyle \ mathbf {0} = {\ frac {\ mathrm {D}} {\ mathrm {D} t}} \ mathbf {1} = {\ frac {\ mathrm {D}} {\ mathrm {D} t}} (\ mathbf {F} ^ {\ top 1} \ cdot \ mathbf {F} ^ {\ top}) = {\ frac {\ mathrm {D}} {\ mathrm {D} t}} ( \ mathbf {F} ^ {\ top 1}) \ cdot \ mathbf {F} ^ {\ top} + \ mathbf {F} ^ {\ top 1} \ cdot {\ dot {\ mathbf {F}} } ^ {\ top} \ quad \ rightarrow \ quad {\ frac {\ mathrm {D}} {\ mathrm {D} t}} (\ mathbf {F} ^ {\ top 1}) =  \ mathbf { F} ^ {\ top 1} \ cdot {\ dot {\ mathbf {F}}} ^ {\ top} \ cdot \ mathbf {F} ^ {\ top 1} \ ,.}$

When the trace of the spatial velocity gradient l or  equivalent  the spatial distortion velocity tensor d or the divergence of the velocity field disappears, then the movement is locally volumepreserving. In the case of a rigid body movement, as shown below, Sp ( l ) = Sp ( w ) = 0, which confirms the constancy of the volume with such a movement. A positive divergence means expansion, which gives the divergence its name ( Latin divergere " striving apart") and which in reality is associated with a decrease in density .
Stretch and Shear Rates
Stretching and twisting of the tangents (red and blue) on material lines (black) in the course of a deformation
When a body is deformed, the distances between its particles and / or the angles between connecting lines between its particles change in the deformed areas. Mathematically, the tangent vectors on such connecting lines are considered, see figure on the right. If these tangent vectors change their lengths or the angles to one another, which happens to the same extent as the connecting lines are stretched or sheared, then their scalar products change and there are deformations. The rate of change of these scalar products is measured by the spatial distortion velocity tensor d :
 ${\ displaystyle {\ frac {\ mathrm {D}} {\ mathrm {D} t}} (\ mathrm {d} {\ vec {x}} \ cdot \ mathrm {d} {\ vec {y}}) = (\ mathbf {l} \ cdot \ mathrm {d} {\ vec {x}}) \ cdot \ mathrm {d} {\ vec {y}} + \ mathrm {d} {\ vec {x}} \ cdot (\ mathbf {l} \ cdot \ mathrm {d} {\ vec {y}}) = 2 \ mathrm {d} {\ vec {x}} \ cdot \ mathbf {d} \ cdot \ mathrm {d} {\ vec {y}}.}$
The rate of expansion in a certain direction is calculated from:
${\ displaystyle {\ hat {e}} = \ mathrm {d} {\ vec {x}} /  \ mathrm {d} {\ vec {x}} }$
 ${\ displaystyle {\ dot {\ varepsilon}} = {\ frac { \ mathrm {d} {\ vec {x}}  {\ dot {}}} { \ mathrm {d} {\ vec {x} } }} = {\ frac {{\ frac {\ mathrm {d} {\ vec {x}}} { \ mathrm {d} {\ vec {x}} }} \ cdot (\ mathrm {d } {\ vec {x}}) {\ dot {}}} { \ mathrm {d} {\ vec {x}} }} = {\ frac {\ mathrm {d} {\ vec {x}} \ cdot \ mathbf {l} \ cdot \ mathrm {d} {\ vec {x}}} { \ mathrm {d} {\ vec {x}}  ^ {2}}} = {\ hat {e} } \ cdot (\ mathbf {d + w}) \ cdot {\ hat {e}} = {\ hat {e}} \ cdot \ mathbf {d} \ cdot {\ hat {e}} = {\ frac { \ partial v} {\ partial x}} \ ,,}$
where the speed v and the coordinate x count in the direction. The shear rate results in the state from
${\ displaystyle {\ hat {e}}}$${\ displaystyle \ gamma = 0}$
 ${\ displaystyle {\ begin {aligned} {\ frac {\ mathrm {D}} {\ mathrm {D} t}} (\ mathrm {d} {\ vec {x}} \ cdot \ mathrm {d} {\ vec {y}}) = & {\ frac {\ mathrm {D}} {\ mathrm {D} t}} ( \ mathrm {d} {\ vec {x}}  \,  \ mathrm {d} {\ vec {y}}  \ sin (\ gamma)) = {\ frac {\ mathrm {D}} {\ mathrm {D} t}} ( \ mathrm {d} {\ vec {x}}  \,  \ mathrm {d} {\ vec {y}} ) \ sin (\ gamma) +  \ mathrm {d} {\ vec {x}}  \,  \ mathrm {d} {\ vec { y}}  \ cos (\ gamma) {\ dot {\ gamma}} \\\ gamma = 0 \ rightarrow & {\ frac {\ mathrm {D}} {\ mathrm {D} t}} (\ mathrm { d} {\ vec {x}} \ cdot \ mathrm {d} {\ vec {y}}) = 2 \ mathrm {d} {\ vec {x}} \ cdot \ mathbf {d} \ cdot \ mathrm { d} {\ vec {y}} =  \ mathrm {d} {\ vec {x}}  \,  \ mathrm {d} {\ vec {y}}  \, {\ dot {\ gamma}} \\\ rightarrow {\ dot {\ gamma}} = & 2 {\ frac {\ mathrm {d} {\ vec {x}}} { \ mathrm {d} {\ vec {x}} }} \ cdot \ mathbf {d} \ cdot {\ frac {\ mathrm {d} {\ vec {y}}} { \ mathrm {d} {\ vec {y}} }} = {\ frac {\ partial v_ { x}} {\ partial y}} + {\ frac {\ partial v_ {y}} {\ partial x}} \,. \ end {aligned}}}$
The speed and the x coordinate in the direction and the speed and the y coordinate in the direction count here .
${\ displaystyle v_ {x}}$${\ displaystyle \ mathrm {d} {\ vec {x}}}$${\ displaystyle v_ {y}}$${\ displaystyle \ mathrm {d} {\ vec {y}}}$
The strain rate tensor d thus determines the strain and shear rates in the current configuration.
Eigenvectors
If the tangent vectors considered in the previous section are eigenvectors of the velocity gradient or of the distortion velocity tensor, then this has remarkable consequences. For such an eigenvector of the velocity gradient, the following applies:
 ${\ displaystyle \ mathbf {l} \ cdot {\ hat {e}} = \ lambda {\ hat {e}} \ ,.}$
The factor is the eigenvalue belonging to the eigenvector . The Frobenius norm of the eigenvectors is indeterminate, which is why its amount is fixed here to one, which is expressed in the hat above the e. The time derivative of a tangent vector of length one in the instantaneous configuration provides
${\ displaystyle \ lambda}$${\ displaystyle {\ hat {e}}}$${\ displaystyle {\ hat {e}} = \ mathrm {d} {\ vec {x}} /  \ mathrm {d} {\ vec {x}} }$
 ${\ displaystyle {\ dot {\ hat {e}}} = {\ frac {(\ mathrm {d} {\ vec {x}}) {\ dot {}}} { \ mathrm {d} {\ vec {x}} }}  {\ frac {\ mathrm {d} {\ vec {x}}} { \ mathrm {d} {\ vec {x}}  ^ {2}}} {\ frac { \ mathrm {d} {\ vec {x}} \ cdot (\ mathrm {d} {\ vec {x}}) {\ dot {}}} { \ mathrm {d} {\ vec {x}}  }} = \ mathbf {l} \ cdot {\ frac {\ mathrm {d} {\ vec {x}}} { \ mathrm {d} {\ vec {x}} }}  \ left ({\ frac {\ mathrm {d} {\ vec {x}}} { \ mathrm {d} {\ vec {x}} }} \ cdot \ mathbf {l} \ cdot {\ frac {\ mathrm {d} {\ vec {x}}} { \ mathrm {d} {\ vec {x}} }} \ right) {\ frac {\ mathrm {d} {\ vec {x}}} { \ mathrm { d} {\ vec {x}} }} = \ mathbf {l} \ cdot {\ hat {e}}  ({\ hat {e}} \ cdot \ mathbf {l} \ cdot {\ hat {e }}) {\ hat {e}}}$
This rate disappears in the direction of the eigenvectors of the spatial velocity gradient. Inserting the strain velocity tensor and the vortex tensor also yields:
 ${\ displaystyle {\ dot {\ hat {e}}} = (\ mathbf {d + w}) \ cdot {\ hat {e}}  ({\ hat {e}} \ cdot (\ mathbf {d + w}) \ cdot {\ hat {e}}) {\ hat {e}} = \ mathbf {w} \ cdot {\ hat {e}} + \ mathbf {d} \ cdot {\ hat {e}}  ({\ hat {e}} \ cdot \ mathbf {d} \ cdot {\ hat {e}}) {\ hat {e}} \ ,.}$
Let be the eigenvector of d . Then is and therefore the time derivative is
${\ displaystyle {\ hat {e}}}$${\ displaystyle \ mathbf {d} \ cdot {\ hat {e}}  ({\ hat {e}} \ cdot \ mathbf {d} \ cdot {\ hat {e}}) {\ hat {e}} = 0}$
 ${\ displaystyle {\ dot {\ hat {e}}} = \ mathbf {w} \ cdot {\ hat {e}} = {\ vec {\ omega}} \ times {\ hat {e}} \ ,. }$
In combination with the above result
 ${\ displaystyle {\ dot {\ varepsilon}} = {\ hat {e}} \ cdot \ mathbf {d} \ cdot {\ hat {e}} \ quad \ rightarrow \ quad \ mathbf {d} \ cdot {\ hat {e}} = {\ dot {\ varepsilon}} {\ hat {e}}}$
is shown for eigenvectors of d :
 ${\ displaystyle \ mathbf {l} \ cdot {\ hat {e}} = (\ mathbf {d + w}) \ cdot {\ hat {e}} = {\ dot {\ varepsilon}} {\ hat {e }} + {\ vec {\ omega}} \ times {\ hat {e}} \ ,.}$
The polar decomposition of the deformation gradient into a rotation and a rotationfree stretching corresponds with the spatial speed gradient to the additive decomposition into the expansion rate and rotation speed.
kinematics
Substantial acceleration
The second Newton Law states that a force a material body in the direction of the force accelerates . At the local level, the material points are then driven by an external acceleration vector:
${\ displaystyle {\ vec {b}}}$
 ${\ displaystyle {\ dot {\ vec {v}}} ({\ vec {x}}, t) = {\ vec {b}} ({\ vec {x}}, t) \ ,.}$
But because in classical mechanics a point in space cannot be accelerated, but only a material point, the material time derivative of the speed must be formed on the left side of the equation , which  as usual  is noted with a point:
 ${\ displaystyle {\ dot {\ vec {v}}} ({\ vec {x}}, t): = {\ frac {\ mathrm {D}} {\ mathrm {D} t}} {\ vec { v}} ({\ vec {\ chi}} ({\ vec {X}}, t), t) = \ underbrace {\ frac {\ partial {\ vec {v}} ({\ vec {x}} , t)} {\ partial {\ vec {x}}}} _ {= \ operatorname {grad} {\ vec {v}} = \ mathbf {l}} \ cdot \ underbrace {\ frac {\ mathrm {D } {\ vec {\ chi}} ({\ vec {X}}, t)} {\ mathrm {D} t}} _ {= {\ vec {v}}} + {\ frac {\ partial {\ vec {v}} ({\ vec {x}}, t)} {\ partial t}} = {\ frac {\ partial {\ vec {v}}} {\ partial t}} + (\ operatorname {grad } {\ vec {v}}) \ cdot {\ vec {v}} = {\ frac {\ partial {\ vec {v}}} {\ partial t}} + \ mathbf {l} \ cdot {\ vec {v}} \ ,.}$
In it, the recorded vector belongs to the accelerated particle that is currently at the location and is its speed at time t. The last term in the above equation is a convective component, which causes the kinematic nonlinearity of the momentum balance in Euler's approach .
${\ displaystyle {\ vec {X}} = {\ vec {\ chi}} ^ { 1} ({\ vec {x}}, t)}$${\ displaystyle t}$${\ displaystyle {\ vec {x}}}$${\ displaystyle {\ vec {v}} ({\ vec {x}}, t) = {\ dot {\ vec {\ chi}}} ({\ vec {X}}, t)}$
In the geometrically linear case, the quadratic convective part is omitted and the following applies:
 ${\ displaystyle {\ dot {\ vec {v}}} ({\ vec {x}}, t) \ approx {\ frac {\ partial {\ vec {v}} ({\ vec {x}}, t )} {\ partial t}} \ ,.}$
Rigid body motion
The speed field (black) of a rigid body (gray) along its path (light blue) is made up of the speed of the center of gravity (blue) and the speed of rotation (red)
Every rigid body movement can be broken down into a translation and a rotation. Any stationary or moving point and the center of gravity of the body are suitable as a center of rotation, see figure on the right. Let be the timefixed difference vector between a particle of the rigid body and its center of gravity at a point in time . The translation of the body can then be represented with its movement of the center of gravity (with ) and its rotation with an orthogonal tensor that is dependent on time but not on location (with ). Translation and rotation taken together define the motion function and the material velocity field:
${\ displaystyle {\ vec {r}} ({\ vec {X}}) = {\ vec {X}}  {\ vec {S}}}$${\ displaystyle {\ vec {X}}}$${\ displaystyle {\ vec {S}}}$${\ displaystyle {t} _ {0}}$${\ displaystyle {\ vec {s}} (t)}$${\ displaystyle {\ vec {s}} (t_ {0}) = {\ vec {S}}}$ ${\ displaystyle \ mathbf {Q} (t)}$${\ displaystyle \ mathbf {Q} (t) \ cdot \ mathbf {Q} (t) ^ {\ top} = \ mathbf {1} \ ,, \; \ operatorname {det} (\ mathbf {Q} (t )) = + 1 \ ,, \; \ mathbf {Q} ({t} _ {0}) = \ mathbf {1}}$
 ${\ displaystyle {\ begin {aligned} {\ vec {\ chi}} ({\ vec {X}}, t) = & {\ vec {s}} (t) + \ mathbf {Q} (t) \ cdot ({\ vec {X}}  {\ vec {S}}) = {\ vec {x}} \; \ rightarrow \; \ mathbf {F} (t) = \ mathbf {Q} (t) \ \\ rightarrow {\ dot {\ vec {\ chi}}} ({\ vec {X}}, t) = & {\ dot {\ vec {s}}} (t) + {\ dot {\ mathbf { Q}}} (t) \ cdot ({\ vec {X}}  {\ vec {S}}) \; \ rightarrow \; {\ dot {\ mathbf {F}}} (t) = {\ dot {\ mathbf {Q}}} (t) \,. \ end {aligned}}}$
The uniform velocity of the center of gravity no longer appears in the material velocity gradient. The spatial speed field is created by the replacement in the material speed field:
${\ displaystyle {\ vec {X}}  {\ vec {S}} = \ mathbf {Q} ^ {\ top} (t) \ cdot {\ bigl (} {\ vec {x}}  {\ vec {s}} (t) {\ bigr)}}$
 ${\ displaystyle {\ begin {aligned} {\ dot {\ vec {\ chi}}} ({\ vec {X}}, t) = & {\ dot {\ vec {s}}} (t) + { \ dot {\ mathbf {Q}}} (t) \ cdot ({\ vec {X}}  {\ vec {S}}) = {\ dot {\ vec {s}}} (t) + {\ dot {\ mathbf {Q}}} (t) \ cdot \ mathbf {Q} ^ {\ top} (t) \ cdot {\ bigl (} {\ vec {x}}  {\ vec {s}} ( t) {\ bigr)} \\\ rightarrow {\ vec {v}} ({\ vec {x}}, t) = & {\ dot {\ vec {s}}} (t) + {\ dot { \ mathbf {Q}}} (t) \ cdot \ mathbf {Q} ^ {\ top} (t) \ cdot {\ bigl (} {\ vec {x}}  {\ vec {s}} (t) {\ bigr)} \,. \ end {aligned}}}$
from which the spatial velocity gradient , which is also independent of the location and the uniform velocity of the center of gravity, follows. The spatial velocity gradient is skew symmetrical here
${\ displaystyle \ mathbf {l} (t) = {\ dot {\ mathbf {Q}}} \ cdot \ mathbf {Q} ^ {\ top}}$
 ${\ displaystyle \ mathbf {l} + \ mathbf {l} ^ {\ top} = {\ dot {\ mathbf {Q}}} \ cdot \ mathbf {Q} ^ {\ top} + \ mathbf {Q} \ cdot {\ dot {\ mathbf {Q}}} ^ {\ top} = {\ frac {\ text {d}} {{\ text {d}} t}} (\ mathbf {Q \ cdot Q} ^ { \ top}) = {\ dot {\ mathbf {1}}} = \ mathbf {0} \ rightarrow \ mathbf {l} ^ {\ top} =  \ mathbf {l}}$
and therefore identical to its vortex tensor ( l = w ) which confirms that the symmetrical strain velocity tensor d vanishes for rigid body motions. The axial dual vortex vector of the vortex tensor is inserted into the velocity field
 ${\ displaystyle {\ vec {v}} ({\ vec {x}}, t) = {\ dot {\ vec {s}}} (t) + {\ vec {\ omega}} (t) \ times {\ bigl (} {\ vec {x}}  {\ vec {s}} (t) {\ bigr)} \ ,,}$
which no longer contains a visible tensor. Only in the cross product, which corresponds to a tensor transformation, there is still an indication of the vortex tensor.
The axis of rotation is an eigenvector of the velocity gradient (with its eigenvalue zero), which is why its time derivative disappears at all times (see above), which is also noticeable in that all points whose distance is measured in multiples of the velocity vector have the same velocity: for all . If the axis of rotation were linked to these particles, it should at most move parallel but not incline, as it would lead to believe. As a geometric object, the parameter of the movement is the “axis of rotation”, which is derived from the given orthogonal tensor Q , but is not bound to any particles and can even lie outside the rigid body. There are therefore no restrictions whatsoever on the angular acceleration at this point.
${\ displaystyle {\ hat {e}}: = {\ vec {\ omega}} /  {\ vec {\ omega}} }$${\ displaystyle {\ dot {\ hat {e}}}}$${\ displaystyle {\ vec {v}} ({\ vec {x}} + \ lambda {\ vec {\ omega}}, t) = {\ vec {v}} ({\ vec {x}}, t )}$${\ displaystyle {\ vec {x}} \ in \ mathbb {R} ^ {3}, \, \ lambda \ in \ mathbb {R}}$${\ displaystyle {\ dot {\ hat {e}}} = {\ vec {0}}}$ ${\ displaystyle {\ dot {\ vec {\ omega}}}}$
From the local time derivative of the velocity field (with a fixed point in space )
${\ displaystyle {\ vec {x}}}$
 ${\ displaystyle {\ begin {aligned} {\ frac {\ partial} {\ partial t}} {\ vec {v}} ({\ vec {x}}, t) = & {\ ddot {\ vec {s }}} (t) + {\ dot {\ vec {\ omega}}} (t) \ times {\ bigl (} {\ vec {x}}  {\ vec {s}} (t) {\ bigr )}  {\ vec {\ omega}} (t) \ times {\ dot {\ vec {s}}} (t) \\ [1ex] = & {\ ddot {\ vec {s}}} (t ) + {\ dot {\ vec {\ omega}}} (t) \ times {\ bigl (} {\ vec {x}}  {\ vec {s}} (t) {\ bigr)}  {\ vec {\ omega}} (t) \ times \ left [{\ vec {v}} ({\ vec {x}}, t)  {\ vec {\ omega}} (t) \ times {\ bigl ( } {\ vec {x}}  {\ vec {s}} (t) {\ bigr)} \ right] \ ,, \ end {aligned}}}$
what together with the material time derivative of the velocity field
 ${\ displaystyle {\ dot {\ vec {v}}} ({\ vec {x}}, t) = {\ frac {\ partial} {\ partial t}} {\ vec {v}} ({\ vec {x}}, t) + \ mathbf {l} \ cdot {\ vec {v}} ({\ vec {x}}, t) = {\ frac {\ partial} {\ partial t}} {\ vec {v}} ({\ vec {x}}, t) + {\ vec {\ omega}} (t) \ times {\ vec {v}} ({\ vec {x}}, t)}$
in the acceleration field (for English acceleration "acceleration") leads to a rigid body movement:
${\ displaystyle {\ vec {a}}}$
 ${\ displaystyle {\ vec {a}} ({\ vec {x}}, t): = {\ dot {\ vec {v}}} ({\ vec {x}}, t) = {\ ddot { \ vec {s}}} (t) + {\ dot {\ vec {\ omega}}} (t) \ times {\ bigl (} {\ vec {x}}  {\ vec {s}} (t ) {\ bigr)} + {\ vec {\ omega}} (t) \ times \ left [{\ vec {\ omega}} (t) \ times {\ bigl (} {\ vec {x}}  { \ vec {s}} (t) {\ bigr)} \ right] \ ,.}$
This derivation illuminates the local and material derivation of time and its characteristics in a rigid body movement.
Potential vortex
Potential vortex with streamlines (blue) and fluid elements (turquoise)
The potential vortex or free vortex is a classic example of a rotationfree potential flow , see picture on the right. Large eddies in low viscosity fluids are well described with this model. Examples of a potential vortex are the bathtub drain far from the discharge, but also, to a good approximation, a tornado . The velocity field of the potential vortex is given in cylindrical coordinates with the distance ρ from the vortex center by:
 ${\ displaystyle {\ vec {v}} = v _ {\ varphi} {\ hat {e}} _ {\ varphi} \ quad {\ text {with}} \ quad v _ {\ varphi}: = {\ frac { \ Gamma _ {0}} {2 \ pi \ rho}} \ ,.}$
The parameter controls the flow velocity and the velocity gradient results
${\ displaystyle \ Gamma _ {0}}$
 ${\ displaystyle {\ begin {aligned} \ operatorname {grad} {\ vec {v}} = & {\ hat {e}} _ {\ varphi} \ otimes \ left ( {\ frac {\ Gamma _ {0 }} {2 \ pi \ rho ^ {2}}} \ right) {\ hat {e}} _ {\ rho}  {\ frac {\ Gamma _ {0}} {2 \ pi \ rho ^ {2 }}} {\ hat {e}} _ {\ rho} \ otimes {\ hat {e}} _ {\ varphi} =  {\ frac {\ Gamma _ {0}} {2 \ pi \ rho ^ { 2}}} ({\ hat {e}} _ {\ varphi} \ otimes {\ hat {e}} _ {\ rho} + {\ hat {e}} _ {\ rho} \ otimes {\ hat { e}} _ {\ varphi}) = \ mathbf {d} + \ mathbf {w} \\\ rightarrow \ mathbf {d} = &  {\ frac {\ Gamma _ {0}} {2 \ pi \ rho ^ {2}}} ({\ hat {e}} _ {\ varphi} \ otimes {\ hat {e}} _ {\ rho} + {\ hat {e}} _ {\ rho} \ otimes {\ hat {e}} _ {\ varphi}) \ quad {\ text {and}} \ quad \ mathbf {w} = \ mathbf {0} \,. \ end {aligned}}}$
The rotational speed of the fluid elements around themselves disappears because w = 0 and as a result of this the movement is volumepreserving. When approaching the vortex center, the shear rate increases due to
${\ displaystyle \ operatorname {div} {\ vec {v}} = \ operatorname {Sp} (\ mathbf {l}) = \ operatorname {Sp} (\ mathbf {d}) = 0}$
 ${\ displaystyle {\ dot {\ gamma}} _ {\ rho \ varphi} = 2 {\ hat {e}} _ {\ rho} \ cdot \ mathbf {d} \ cdot {\ hat {e}} _ { \ varphi} =  {\ frac {\ Gamma _ {0}} {\ pi \ rho ^ {2}}}}$
over all limits, which cannot occur in real currents, because the viscosity , which is always present but neglected here, prevents this, as in the HamelOseen vortex .
Change of the reference system
Two observers, who analyze the deformation of a body, can exchange information about the body's field of motion and speed. Both observers will agree on the deformation gradient because it is an objective variable. Just as the occupant of a moving train assesses the speed of a bird flying past differently than a pedestrian in the vicinity, observers with different movements will  as mentioned at the beginning  measure different speed fields and speed gradients. The speed field and the speed gradient are not objective . For the proof of objectivity  or the opposite  the rotational movement of the reference system of the observer is decisive. The rotation of the moving observer relative to the material body is determined with an orthogonal tensor from the special orthogonal group${\ displaystyle \ mathbf {Q}}$
 ${\ displaystyle {\ mathcal {SO}} = \ {\ mathbf {Q} \ in {\ mathcal {L}}  \ mathbf {Q} ^ { 1} = \ mathbf {Q} ^ {\ top} \ ; \ wedge \; \ det (\ mathbf {Q}) = + 1 \}}$
described. The set contains all tensors (second level), denotes the transposition , the inverse and “det” the determinant . The tensors from this group perform rotations without reflection and are called "actually orthogonal".
${\ displaystyle {\ mathcal {L}}}$${\ displaystyle (\ cdot) ^ {\ top}}$${\ displaystyle (\ cdot) ^ { 1}}$
There are three types of objective tensors that behave in different ways in an Euclidean transformation:
Bodyrelated objective, material, onefield tensors

${\ displaystyle \ mathbf {T} '= \ mathbf {T}}$

for all ${\ displaystyle \ mathbf {Q} \ in {\ mathcal {SO}}}$

Objective, spatial, onefield tensors

${\ displaystyle \ mathbf {T} '= \ mathbf {Q \ cdot T \ cdot Q} ^ {\ top}}$

Objective twofield tensors such as the deformation gradient

${\ displaystyle \ mathbf {F} '= \ mathbf {Q \ cdot F}}$

If the observer, who is at rest relative to the body, determines the deformation gradient in a material point , then the moving observer measures using the Euclidean transformation
${\ displaystyle \ mathbf {F}}$
 ${\ displaystyle \ mathbf {F} '= \ mathbf {Q \ cdot F} \ quad \ rightarrow \ quad {\ dot {\ mathbf {F}}}' = {\ dot {\ mathbf {Q}}} \ cdot \ mathbf {F} + \ mathbf {Q} \ cdot {\ dot {\ mathbf {F}}} \ neq \ mathbf {Q} \ cdot {\ dot {\ mathbf {F}}} \ ,.}$
The material velocity gradient is therefore not objective. The spatial velocity gradient of the moving observer can also be calculated
${\ displaystyle {\ dot {\ mathbf {F}}}}$
 ${\ displaystyle {\ begin {aligned} \ mathbf {l} '= & {\ dot {\ mathbf {F}}}' \ cdot {\ mathbf {F} '} ^ { 1} = ({\ dot { \ mathbf {Q}}} \ cdot \ mathbf {F} + \ mathbf {Q} \ cdot {\ dot {\ mathbf {F}}}) \ cdot {\ mathbf {F}} ^ { 1} \ cdot \ mathbf {Q} ^ { 1} = {\ dot {\ mathbf {Q}}} \ cdot \ mathbf {F} \ cdot {\ mathbf {F}} ^ { 1} \ cdot \ mathbf {Q} ^ {\ top} + \ mathbf {Q} \ cdot {\ dot {\ mathbf {F}}} \ cdot {\ mathbf {F}} ^ { 1} \ cdot \ mathbf {Q} ^ {\ top} \\\ rightarrow \ mathbf {l} '= & \ mathbf {Q \ cdot l \ cdot Q} ^ {\ top} + {\ dot {\ mathbf {Q}}} \ cdot \ mathbf {Q} ^ {\ top} \ ,, \ end {aligned}}}$
which is therefore also not objective. The last term in the last equation is because of
 ${\ displaystyle {\ dot {\ mathbf {Q}}} \ cdot \ mathbf {Q} ^ {\ top} + ({\ dot {\ mathbf {Q}}} \ cdot \ mathbf {Q} ^ {\ top }) ^ {\ top} = {\ dot {\ mathbf {Q}}} \ cdot \ mathbf {Q} ^ {\ top} + \ mathbf {Q} \ cdot {\ dot {\ mathbf {Q}}} ^ {\ top} = (\ mathbf {Q \ cdot Q} ^ {\ top}) {\ dot {}} = {\ dot {\ mathbf {1}}} = \ mathbf {0}}$
skewsymmetric and cancels out for the symmetric strain rate tensor:
 ${\ displaystyle {\ begin {aligned} 2 \ mathbf {d} '= & \ mathbf {l' + l '} ^ {\ top} = \ mathbf {Q \ cdot l \ cdot Q} ^ {\ top} + {\ dot {\ mathbf {Q}}} \ cdot \ mathbf {Q} ^ {\ top} + \ mathbf {Q \ cdot l ^ {\ top} \ cdot Q} ^ {\ top} + \ mathbf {Q } \ cdot {\ dot {\ mathbf {Q}}} ^ {\ top} = \ mathbf {Q \ cdot (l + l ^ {\ top}) \ cdot Q} ^ {\ top} \\\ rightarrow \ mathbf {d} '= & \ mathbf {Q \ cdot d \ cdot Q} ^ {\ top} \ quad {\ text {for all}} \ quad \ mathbf {Q} \ in {\ mathcal {SO}} \ ,. \ end {aligned}}}$
The strain rate tensor is therefore objective, because it transforms like an objective, spatial, onefield tensor. The difference shows that the vortex tensor is again not objective:
${\ displaystyle \ mathbf {w = ld}}$
 ${\ displaystyle {\ begin {aligned} \ mathbf {w} '= & \ mathbf {l'd'} = \ mathbf {Q \ cdot l \ cdot Q} ^ {\ top} + {\ dot {\ mathbf {Q}}} \ cdot \ mathbf {Q} ^ {\ top}  {\ frac {1} {2}} \ mathbf {Q \ cdot (l + l ^ {\ top}) \ cdot Q} ^ { \ top} = {\ frac {1} {2}} \ mathbf {Q \ cdot (ll ^ {\ top}) \ cdot Q} ^ {\ top} + {\ dot {\ mathbf {Q}}} \ cdot \ mathbf {Q} ^ {\ top} \\\ rightarrow \ mathbf {w} '= & \ mathbf {Q \ cdot w \ cdot Q} ^ {\ top} + {\ dot {\ mathbf {Q}} } \ cdot \ mathbf {Q} ^ {\ top} \,. \ end {aligned}}}$
Objective time derivations
For the formulation of ratedependent material models, objective time derivatives are required for constitutive variables in the spatial approach , because it does not correspond to the experience that an observer in motion measures a different material behavior than a stationary one. Thus, the material models must be formulated with objective time derivatives. Just as the speed and its gradient are not objective  see the # description above  the time derivatives of other quantities transposed by the fluid are not objective either. However, there are several reference systeminvariant rates that are also objective for objective quantities and are formulated with the help of the velocity gradient, including:
Zaremba Jaumann derivation:${\ displaystyle {\ stackrel {\ circ} {\ mathbf {T}}}: = {\ dot {\ mathbf {T}}} + \ mathbf {T \ cdot w}  \ mathbf {w \ cdot T}}$
Covariant Oldroyd derivation:${\ displaystyle {\ stackrel {\ triangle} {\ mathbf {T}}}: = {\ dot {\ mathbf {T}}} + \ mathbf {T \ cdot l} + \ mathbf {l} ^ {\ top } \ cdot \ mathbf {T} = {\ stackrel {\ circ} {\ mathbf {T}}} + \ mathbf {T \ cdot d} + \ mathbf {d \ cdot T}}$
Contra variant Oldroyd derivation :${\ displaystyle {\ stackrel {\ nabla} {\ mathbf {T}}}: = {\ dot {\ mathbf {T}}}  \ mathbf {l \ cdot T}  \ mathbf {T} \ cdot \ mathbf {l} ^ {\ top} = {\ stackrel {\ circ} {\ mathbf {T}}}  \ mathbf {T \ cdot d}  \ mathbf {d \ cdot T}}$
Cauchy derivative: ${\ displaystyle {\ stackrel {\ diamond} {\ mathbf {T}}} = {\ dot {\ mathbf {T}}} + \ operatorname {Sp} (\ mathbf {l}) \ mathbf {T}  \ mathbf {l \ cdot T}  \ mathbf {T \ cdot l} ^ {\ top} \ ,.}$
For an objective vector, these are time derivatives
${\ displaystyle {\ vec {v}}}$
 ${\ displaystyle {\ begin {array} {rclcl} {\ stackrel {\ circ} {\ vec {v}}} & = & {\ dot {\ vec {v}}}  \ mathbf {w} \ cdot { \ vec {v}} \\ {\ stackrel {\ Delta} {\ vec {v}}} & = & {\ dot {\ vec {v}}} + \ mathbf {l} ^ {\ top} \ cdot {\ vec {v}} & = & {\ stackrel {\ circ} {\ vec {v}}} + \ mathbf {d} \ cdot {\ vec {v}} \\ {\ stackrel {\ nabla} { \ vec {v}}} & = & {\ dot {\ vec {v}}}  \ mathbf {l} \ cdot {\ vec {v}} & = & {\ stackrel {\ circ} {\ vec { v}}}  \ mathbf {d} \ cdot {\ vec {v}} \ end {array}}}$
lens. For more information, see the main article.
example
Shear of the unit square (blue) into a parallelogram (yellow)
A unit square made of a viscoelastic liquid is deformed into a parallelogram at constant shear rate, see figure on the right. The reference configuration is the unit square
 ${\ displaystyle {\ begin {pmatrix} X \\ Y \ end {pmatrix}} \ in [0,1] ^ {2}}$
In the current configuration, the points of the square have the spatial coordinates
 ${\ displaystyle {\ vec {\ chi}} ({\ vec {X}}, t) = {\ begin {pmatrix} x \\ y \ end {pmatrix}} = {\ begin {pmatrix} X + \ gamma Y \\ Y \ end {pmatrix}} \ quad \ rightarrow \ quad {\ dot {\ vec {\ chi}}} ({\ vec {X}}, t) = {\ begin {pmatrix} {\ dot {x }} \\ {\ dot {y}} \ end {pmatrix}} = {\ begin {pmatrix} {\ dot {\ gamma}} Y \\ 0 \ end {pmatrix}} = {\ begin {pmatrix} { \ dot {\ gamma}} y \\ 0 \ end {pmatrix}} = {\ vec {v}} ({\ vec {x}}, t)}$
from which the deformation and (spatial) speed gradient are calculated:
 ${\ displaystyle {\ begin {aligned} \ mathbf {F} = & {\ begin {pmatrix} 1 & \ gamma \\ 0 & 1 \ end {pmatrix}} \; \ rightarrow \ quad {\ dot {\ mathbf {F}} } = {\ begin {pmatrix} 0 & {\ dot {\ gamma}} \\ 0 & 0 \ end {pmatrix}} = \ operatorname {GRAD} {\ dot {\ vec {\ chi}}} \ ,, \ quad \ mathbf {F} ^ { 1} = {\ begin {pmatrix} 1 &  \ gamma \\ 0 & 1 \ end {pmatrix}} \\\ rightarrow \ mathbf {l} = & \ operatorname {grad} {\ vec {v }} = {\ begin {pmatrix} 0 & {\ dot {\ gamma}} \\ 0 & 0 \ end {pmatrix}} = {\ dot {\ mathbf {F}}} \ cdot \ mathbf {F} ^ { 1 } = {\ begin {pmatrix} 0 & {\ dot {\ gamma}} \\ 0 & 0 \ end {pmatrix}} \ cdot {\ begin {pmatrix} 1 &  \ gamma \\ 0 & 1 \ end {pmatrix}} \\\ rightarrow \ mathbf {d} = & {\ dfrac {\ dot {\ gamma}} {2}} {\ begin {pmatrix} 0 & 1 \\ 1 & 0 \ end {pmatrix}} \;, \ quad \ mathbf {w} = {\ dfrac {\ dot {\ gamma}} {2}} {\ begin {pmatrix} 0 & 1 \\  1 & 0 \ end {pmatrix}} \,. \ end {aligned}}}$
A generalization of the material law for a viscoelastic fluid (Maxwell body) with material parameters on three dimensions could look like this:
${\ displaystyle \ lambda {\ dot {\ sigma}} + \ sigma = \ eta \, {\ dot {\ varepsilon}}}$${\ displaystyle \ lambda, \ eta}$
 ${\ displaystyle \ lambda {\ stackrel {\ circ} {\ varvec {\ sigma}}} + {\ varvec {\ sigma}} = \ eta \, \ mathbf {d} \ ,.}$
Cauchy's stress tensor is deviatoric here and therefore has the form
${\ displaystyle {\ boldsymbol {\ sigma}}}$
 ${\ displaystyle {\ boldsymbol {\ sigma}} = {\ begin {pmatrix} \ sigma & \ tau \\\ tau &  \ sigma \ end {pmatrix}} \ ,.}$
The ZarembaJaumann derivation is calculated as follows:
 ${\ displaystyle {\ stackrel {\ circ} {\ boldsymbol {\ sigma}}} = {\ begin {pmatrix} {\ dot {\ sigma}} & {\ dot {\ tau}} \\ {\ dot {\ tau}} &  {\ dot {\ sigma}} \ end {pmatrix}} + {\ frac {\ dot {\ gamma}} {2}} {\ begin {pmatrix} \ sigma & \ tau \\\ tau &  \ sigma \ end {pmatrix}} \ cdot {\ begin {pmatrix} 0 & 1 \\  1 & 0 \ end {pmatrix}}  {\ frac {\ dot {\ gamma}} {2}} {\ begin {pmatrix } 0 & 1 \\  1 & 0 \ end {pmatrix}} \ cdot {\ begin {pmatrix} \ sigma & \ tau \\\ tau &  \ sigma \ end {pmatrix}} = {\ begin {pmatrix} {\ dot { \ sigma}}  {\ dot {\ gamma}} \ tau & {\ dot {\ tau}} + {\ dot {\ gamma}} \ sigma \\ {\ dot {\ tau}} + {\ dot { \ gamma}} \ sigma &  {\ dot {\ sigma}} + {\ dot {\ gamma}} \ tau \ end {pmatrix}}}$
Calculated stresses in the viscoelastic fluid at uniform shear
which leads to two differential equations for the stress components via the material law:
 ${\ displaystyle \ lambda ({\ dot {\ sigma}}  {\ dot {\ gamma}} \ tau) + \ sigma = 0 \ quad {\ textsf {and}} \ quad \ lambda ({\ dot {\ tau}} + {\ dot {\ gamma}} \ sigma) + \ tau = {\ frac {\ eta} {2}} {\ dot {\ gamma}} \ ,.}$
At constant shear rate, after eliminating the normal stress, the differential equation comes
 ${\ displaystyle {\ ddot {\ tau}} + {\ frac {2} {\ lambda}} {\ dot {\ tau}} + (\ lambda ^ { 2} + {\ dot {\ gamma}} ^ {2}) \ tau = {\ frac {\ eta {\ dot {\ gamma}}} {2 \ lambda ^ {2}}}}$
for the shear stress, which has a damped oscillation as a solution. This is a known unphysical phenomenon when using the ZarembaJaumann rate, see figure on the right.
Using the contravariant Oldroyd derivative yields a nondeviatoric stress tensor:
 ${\ displaystyle {\ stackrel {\ nabla} {\ boldsymbol {\ sigma}}} = {\ begin {pmatrix} {\ dot {\ sigma}} _ {xx} & {\ dot {\ tau}} \\ { \ dot {\ tau}} & {\ dot {\ sigma}} _ {yy} \ end {pmatrix}}  {\ begin {pmatrix} 0 & {\ dot {\ gamma}} \\ 0 & 0 \ end {pmatrix} } \ cdot {\ begin {pmatrix} \ sigma _ {xx} & \ tau \\\ tau & \ sigma _ {yy} \ end {pmatrix}}  {\ begin {pmatrix} \ sigma _ {xx} & \ tau \\\ tau & \ sigma _ {yy} \ end {pmatrix}} \ cdot {\ begin {pmatrix} 0 & 0 \\ {\ dot {\ gamma}} & 0 \ end {pmatrix}} = {\ begin {pmatrix } {\ dot {\ sigma}} _ {xx} 2 {\ dot {\ gamma}} \ tau & {\ dot {\ tau}}  {\ dot {\ gamma}} \ sigma _ {yy} \ \ {\ dot {\ tau}}  {\ dot {\ gamma}} \ sigma _ {yy} & {\ dot {\ sigma}} _ {yy} \ end {pmatrix}} \ ,.}$
The material equation gives:
${\ displaystyle \ lambda {\ stackrel {\ nabla} {\ boldsymbol {\ sigma}}} + {\ boldsymbol {\ sigma}} = \ eta \, \ mathbf {d}}$
 ${\ displaystyle {\ begin {pmatrix} \ lambda ({\ dot {\ sigma}} _ {xx} 2 {\ dot {\ gamma}} \ tau) + \ sigma _ {xx} & \ lambda ({\ dot {\ tau}}  {\ dot {\ gamma}} \ sigma _ {yy}) + \ tau \\\ lambda ({\ dot {\ tau}}  {\ dot {\ gamma}} \ sigma _ {yy}) + \ tau & \ lambda {\ dot {\ sigma}} _ {yy} + \ sigma _ {yy} \ end {pmatrix}} = {\ frac {\ eta {\ dot {\ gamma}} } {2}} {\ begin {pmatrix} 0 & 1 \\ 1 & 0 \ end {pmatrix}}}$
which can be integrated in a closed manner with initially vanishing stresses and constant shear rate:
 ${\ displaystyle {\ begin {aligned} \ sigma _ {yy} = & 0 \\\ tau = & {\ frac {\ eta {\ dot {\ gamma}}} {2}} (1e ^ { { \ frac {t} {\ lambda}}}) \\\ sigma _ {xx} = & \ eta {\ dot {\ gamma}} ^ {2} (\ lambda  \ lambda e ^ { {\ frac { t} {\ lambda}}}  te ^ { {\ frac {t} {\ lambda}}}). \ end {aligned}}}$
No vibrations occur here. The figure on the right shows the stresses calculated at a shear rate of 10 / s with the ZarembaJaumann and the contravariant Oldroyd derivative and the material parameters given in the table.
${\ displaystyle {\ dot {\ gamma}} =}$
parameter 
Relaxation time 
dynamic viscosity

Formula symbol

${\ displaystyle \ lambda}$ 
${\ displaystyle \ eta}$

unit

s 
MPa s

ZarembaJaumann derivation

1.5 
45.2

Contravariant Oldroyd derivative

1.5 
0.2

Remarks

↑ ^{a } ^{b } ^{c} The Fréchet derivative of a function according to
is the restricted linear operator which  if it exists  corresponds
to the Gâteaux differential in all directions, i.e.${\ displaystyle f}$${\ displaystyle x}$${\ displaystyle {\ mathcal {A}}}$${\ displaystyle h}$
 ${\ displaystyle {\ mathcal {A}} (h) = \ left. {\ frac {\ mathrm {d}} {\ mathrm {d} s}} f (x + sh) \ right  _ {s = 0 } = \ lim _ {s \ rightarrow 0} {\ frac {f (x + sh) f (x)} {s}} \ quad {\ text {for all}} \ quad h}$
applies. In it is scalar, vector or tensor valued but and similar. Then will too
${\ displaystyle s \ in \ mathbb {R} \ ,, f, x \, {\ textsf {and}} \, h}$${\ displaystyle x}$${\ displaystyle h}$ ${\ displaystyle {\ mathcal {A}} = {\ frac {\ partial f} {\ partial x}}}$
written.

↑ ^{a } ^{b} Because with the vortex vector results
${\ displaystyle {\ hat {e}} \ cdot \ mathbf {w} \ cdot {\ hat {e}} = {\ hat {e}} \ cdot ({\ vec {\ omega}} \ times {\ hat {e}}) = 0}$

↑ Because it
follows from :
${\ displaystyle \ mathbf {l} \ cdot {\ hat {e}} = \ lambda {\ hat {e}} \ quad {\ text {and}} \ quad {\ hat {e}} \ cdot {\ hat {e}} = 1}$
${\ displaystyle {\ begin {aligned} \ mathbf {l} \ cdot {\ hat {e}}  ({\ hat {e}} \ cdot \ mathbf {l} \ cdot {\ hat {e}}) { \ hat {e}} = \\ = \ lambda {\ hat {e}}  ({\ hat {e}} \ cdot \ lambda {\ hat {e}}) {\ hat {e}} = 0 \ end {aligned}}}$

↑ This paradox only occurs with nonmaterial objects like the axis of rotation here or the momentary pole .

↑ The symbols for the objective rates vary from source to source. The ones given here follow P. Haupt, p. 48ff. In H. Altenbach is used
for and for .${\ displaystyle \ mathbf {T} ^ {\ nabla}}$${\ displaystyle {\ stackrel {\ circ} {\ mathbf {T}}}}$${\ displaystyle \ mathbf {T} ^ {O}}$${\ displaystyle {\ stackrel {\ triangle} {\ mathbf {T}}}}$

↑ This derivation occurs in the Cauchy elasticity and is also named after C. Truesdell. He himself named the derivation after Cauchy and wrote in 1963 that this rate was named after him without an inventive reason ("came to be named, for no good reason, after [...] me") see C. Truesdell: Remarks on Hypo Elasticity , Journal of Research of the National Bureau of Standards  B. Mathematics and Mathematical Physics, Vol. 67B, No. 3, JulySeptember 1963, p. 141.
See also
literature
Individual evidence

↑ Altenbach (2012), pp. 109 and 32.

↑ after James G. Oldroyd (1921  1982), James G. Oldroyd in engl. Wikipedia (engl.)

↑ P. Haupt (2000), pp. 302ff