a. 运动变形前,参考构型中某代表性物质点 A 邻域内的线元:
d X ⃗ = d X A G ⃗ A = d x i c ⃗ i d\vec{X}=dX^A\vec{G}_A=dx^i\vec{c}_i dX=dXAGA=dxici
b. 运动变形后,线元 d X ⃗ d\vec{X} dX 映射为当前构型中的线元 d x ⃗ d\vec{x} dx:
d x ⃗ = d x i g ⃗ i = d X A C ⃗ A d\vec{x}=dx^i\vec{g}_i=dX^A\vec{C}_A dx=dxigi=dXACA
如下图所示:
根据映射关系:
x ⃗ = x ⃗ ( X 1 , X 2 , X 3 , t ) \vec{x}=\vec{x}(X^1,X^2,X^3,t) x=x(X1,X2,X3,t)
有:
d x ⃗ = ∂ x ⃗ ∂ X A d X A = ( ∂ x ⃗ ∂ X A ⊗ G ⃗ A ) ⋅ d X ⃗ ≜ F ⋅ d X ⃗ d\vec{x}=\dfrac{\partial \vec{x}}{\partial X^A}dX^A=\left(\dfrac{\partial \vec{x}}{\partial X^A}\otimes\vec{G}^A\right)\cdot d\vec{X}\triangleq \bold F\cdot d\vec{X} dx=∂XA∂xdXA=(∂XA∂x⊗GA)⋅dX≜F⋅dX
将 F \bold F F 称作 变形梯度 。可见,变形梯度(仿射量)实现了A点邻域内变形前线元到变形后线元的线性映射。
根据变形梯度的定义与不同坐标系间基的关系,有
F ≜ ∂ x ⃗ ∂ X A ⊗ G ⃗ A ≜ x ⃗ ▽ 0 = C ⃗ A ⊗ G ⃗ A = F ∙ A B G ⃗ B ⊗ G ⃗ A = x , A i g ⃗ i ⊗ G ⃗ A = g ⃗ i ⊗ c ⃗ i = F ∙ i j c ⃗ j ⊗ c ⃗ i \begin{aligned} &\bold{F}\triangleq\dfrac{\partial \vec{x}}{\partial X^A}\otimes\vec{G}^A\triangleq\vec{x}\triangledown_0\\\\ &\ \ \ =\vec{C}_A\otimes\vec{G}^A\\\\ &\ \ \ =F^B_{\bullet A}\vec{G}_B\otimes\vec{G}^A\\\\ &\ \ \ =x^i_{,A}\vec{g}_i\otimes\vec{G}^A=\vec{g}_i\otimes\vec{c}\ ^i\\\\ &\ \ \ =F^j_{\bullet i}\vec{c}_j\otimes\vec{c}\ ^i \end{aligned} F≜∂XA∂x⊗GA≜x▽0 =CA⊗GA =F∙ABGB⊗GA =x,Aigi⊗GA=gi⊗c i =F∙ijcj⊗c i
由上面的关系可知:
另外,变形梯度张量也可由位移在物质坐标系下的右梯度进行表示,由于:
X ⃗ + u ⃗ = x ⃗ + b ⃗ \vec{X}+\vec{u}=\vec{x}+\vec{b} X+u=x+b
式中, b ⃗ \vec{b} b 为参考坐标系与空间坐标系原点的位矢差,是常矢。则
F = ∂ x ⃗ ∂ X A ⊗ G ⃗ A = ∂ ∂ X A ( X ⃗ + u ⃗ ) ⊗ G ⃗ A = I + ∂ u ⃗ ∂ X A ⊗ G ⃗ A = I + u ⃗ ▽ 0 \bold F =\dfrac{\partial \vec{x}}{\partial X^A}\otimes\vec{G}^A =\dfrac{\partial }{\partial X^A}(\vec{X}+\vec{u})\otimes\vec{G}^A =\bold I+\dfrac{\partial \vec{u}}{\partial X^A}\otimes\vec{G}^A=\bold I +\vec{u}\triangledown_0 F=∂XA∂x⊗GA=∂XA∂(X+u)⊗GA=I+∂XA∂u⊗GA=I+u▽0
变形梯度的行列式:
J ≜ d e t ( F ) = d e t ( [ F ∙ A B ] ) = d e t ( [ g i B ] [ x , A i ] ) = d e t ( [ x , A i ] ) ∣ G ⃗ 1 ⋅ g ⃗ 1 G ⃗ 1 ⋅ g ⃗ 2 G ⃗ 1 ⋅ g ⃗ 3 G ⃗ 2 ⋅ g ⃗ 1 G ⃗ 2 ⋅ g ⃗ 2 G ⃗ 2 ⋅ g ⃗ 3 G ⃗ 3 ⋅ g ⃗ 1 G ⃗ 3 ⋅ g ⃗ 2 G ⃗ 3 ⋅ g ⃗ 3 ∣ = d e t ( [ x , A i ] ) [ G ⃗ 1 ⋅ ( G ⃗ 2 × G ⃗ 3 ) ] [ g ⃗ 1 ⋅ ( g ⃗ 2 × g ⃗ 3 ) ] = d e t ( [ x , A i ] ) g G ≠ 0 \mathscr{J}\triangleq det(\bold F)=det([F^B_{\bullet A}])=det([g^B_{i}][x^i_{,A}])=det([x^i_{,A}]) \begin{vmatrix} \vec{G}^1\cdot\vec{g}_1 & \vec{G}^1\cdot\vec{g}_2 & \vec{G}^1\cdot\vec{g}_3\\\\ \vec{G}^2\cdot\vec{g}_1 & \vec{G}^2\cdot\vec{g}_2 & \vec{G}^2\cdot\vec{g}_3\\\\ \vec{G}^3\cdot\vec{g}_1 & \vec{G}^3\cdot\vec{g}_2 & \vec{G}^3\cdot\vec{g}_3 \end{vmatrix}\\\ \\ =det([x^i_{,A}])[\vec{G}^1\cdot(\vec{G}^2\times\vec{G}^3)][\vec{g}_1\cdot(\vec{g}_2\times\vec{g}_3)]=det([x^i_{,A}])\sqrt{\dfrac{g}{G}}\ne0 J≜det(F)=det([F∙AB])=det([giB][x,Ai])=det([x,Ai]) G1⋅g1G2⋅g1G3⋅g1G1⋅g2G2⋅g2G3⋅g2G1⋅g3G2⋅g3G3⋅g3 =det([x,Ai])[G1⋅(G2×G3)][g1⋅(g2×g3)]=det([x,Ai])Gg=0
其中,
G = G ⃗ 1 ⋅ ( G ⃗ 2 × G ⃗ 3 ) ; g = g ⃗ 1 ⋅ ( g ⃗ 2 × g ⃗ 3 ) G=\vec{G}^1\cdot(\vec{G}^2\times\vec{G}^3);g=\vec{g}_1\cdot(\vec{g}_2\times\vec{g}_3) G=G1⋅(G2×G3);g=g1⋅(g2×g3)
或者
C = d e t ( [ C A B ] ) = d e t ( [ F ∙ A M ] T [ G M N ] [ F ∙ B N ] ) = d e t 2 ( F ) G C=det([{C}_{AB}])=det([F^M_{\bullet A}]^T[G_{MN}][F^N_{\bullet B}])=det^2(\bold F)G C=det([CAB])=det([F∙AM]T[GMN][F∙BN])=det2(F)G
则
d e t 2 ( F ) = C G ≠ 0 det^2(\bold F)=\dfrac{C}{G}\ne0 det2(F)=GC=0
变形梯度的行列式不为零,说明变形梯度是正则仿射量。
根据映射关系:
X ⃗ = X ⃗ ( x 1 , x 2 , x 3 , t ) \vec{X}=\vec{X}(x^1,x^2,x^3,t) X=X(x1,x2,x3,t)
得:
d X ⃗ = ∂ X ⃗ ∂ x i d x i = ( ∂ X ⃗ ∂ x i ⊗ g ⃗ i ) ⋅ d x ⃗ ≜ F − 1 ⋅ d x ⃗ d\vec{X}=\dfrac{\partial \vec{X}}{\partial x^i}d{x}^i=\left(\dfrac{\partial \vec{X}}{\partial x^i}\otimes\vec{g}\ ^i\right)\cdot d\vec{x}\triangleq \overset{-1}{\bold F}\cdot d\vec{x} dX=∂xi∂Xdxi=(∂xi∂X⊗g i)⋅dx≜F−1⋅dx
根据定义:
F − 1 ≜ ∂ X ⃗ ∂ x i ⊗ g ⃗ i ≜ X ⃗ ▽ = c ⃗ i ⊗ g ⃗ i = F − 1 , i j g ⃗ j ⊗ g ⃗ i = X , i A G ⃗ A ⊗ g ⃗ i = G ⃗ A ⊗ C ⃗ A = F − 1 , A B C ⃗ B ⊗ C ⃗ A = ∂ ∂ x i ( x ⃗ − u ⃗ ) ⊗ g ⃗ i = I − ∂ u ⃗ ∂ x i ⊗ g ⃗ i = I − u ⃗ ▽ \begin{aligned} &\overset{-1}{\bold F}\triangleq \dfrac{\partial \vec{X}}{\partial x^i}\otimes\vec{g}\ ^i\triangleq\vec{X}\triangledown\\\\ &\ \ \ =\vec c_i\otimes\vec{g}^i\\\\ &\ \ \ =\overset{-1}{F}\ ^{j}_{,i}\vec{g}_j\otimes\vec{g}^i\\\\ &\ \ \ =X^A_{,i}\vec{G}_A\otimes\vec{g}^i=\vec{G}_A\otimes\vec{C}^A\\\\ &\ \ \ =\overset{-1}{F}\ ^{B}_{,A}\vec{C}_B\otimes\vec{C}^A \\\\ &\ \ \ =\dfrac{\partial }{\partial x^i}(\vec{x}-\vec{u})\otimes\vec{g}^i \\\\ &\ \ \ =\bold I-\dfrac{\partial \vec{u}}{\partial x^i}\otimes\vec{g}^i =\bold I-\vec{u}\triangledown \end{aligned} F−1≜∂xi∂X⊗g i≜X▽ =ci⊗gi =F−1 ,ijgj⊗gi =X,iAGA⊗gi=GA⊗CA =F−1 ,ABCB⊗CA =∂xi∂(x−u)⊗gi =I−∂xi∂u⊗gi=I−u▽
又因为:
F − 1 ⋅ F = ( c ⃗ i ⊗ g ⃗ i ) ⋅ ( g ⃗ j ⊗ c ⃗ j ) = δ j i c ⃗ i ⊗ c ⃗ j = I \overset{-1}{\bold F}\cdot\bold{F}=(\vec c_i\otimes\vec{g}^i)\cdot(\vec{g}_j\otimes\vec{c}\ ^j)=\delta^i_j\vec c_i\otimes\vec{c}\ ^j=\bold I F−1⋅F=(ci⊗gi)⋅(gj⊗c j)=δjici⊗c j=I
因此, F − 1 \overset{-1}{\bold F} F−1 为变形梯度仿射量的逆。
定义:若某张量的分量或张量基涉及两个不互相独立的坐标系,便称之为 两点张量。
比如,变形梯度或其逆为两点张量的实例:
F = F ∙ A B G ⃗ B ⊗ G ⃗ A \bold F=F^B_{\bullet A}\vec{G}_B\otimes\vec{G}^A F=F∙ABGB⊗GA
上述形式上似乎只与物质坐标系相关,但注意到:
F ∙ A B = X , i A g B i = ∂ X A ( x ⃗ , t ) ∂ x i ∂ x i ( X ⃗ , t ) ∂ X B F^B_{\bullet A} =X^A_{,i}g^i_B =\dfrac{\partial X^A(\vec{x},t)}{\partial x^i}\dfrac{\partial x^i(\vec{X},t)}{\partial X^B} F∙AB=X,iAgBi=∂xi∂XA(x,t)∂XB∂xi(X,t)
说明其分量涉及物质坐标系与空间坐标系。
最后尤其指出:两点张量关于坐标的导数应当是全导数。具体来说,若张量 Ψ \bold \Psi Ψ 是建立在坐标系 { X ⃗ } \{\vec{X}\} {X} 与 { x ⃗ } \{\vec{x}\} {x} 上的两点张量,则
d Ψ d X A = ∂ Ψ ∂ X A + ∂ Ψ ∂ x i ∂ x i ∂ X A \dfrac{d\bold\Psi}{d X^A} =\dfrac{\partial\bold\Psi}{\partial X^A}+\dfrac{\partial\bold\Psi}{\partial x^i}\dfrac{\partial x^i}{\partial X^A} dXAdΨ=∂XA∂Ψ+∂xi∂Ψ∂XA∂xi